From 4e356b38bfc24fc25a2fcf0ddbf0dee3a72d5f41 Mon Sep 17 00:00:00 2001 From: plaa Date: Sat, 17 Jul 2010 22:52:23 +0000 Subject: [PATCH] Major refactoring for 1.1.1 git-svn-id: https://openrocket.svn.sourceforge.net/svnroot/openrocket/trunk@56 180e2498-e6e9-4542-8430-84ac67f01cd8 --- .classpath | 13 +- .settings/org.eclipse.core.resources.prefs | 4 + .settings/org.eclipse.jdt.core.prefs | 64 + TODO | 42 +- build.xml | 8 +- doc/design/simulation-sequence.uml | 2 + doc/design/simulation-sequence.umlseq | 5 + doc/design/simulation-sequence.uxf | 120 + doc/properties.txt | 44 + doc/simulation-2.0.uml | 91 - doc/simulation-2.0.umlclass | 479 - .../chapter-aerodynamic-properties.tex | 2235 ++ doc/techdoc/chapter-appendices.tex | 769 + .../chapter-basics-of-model-rockets.tex | 360 + doc/techdoc/chapter-conclusion.tex | 59 + doc/techdoc/chapter-experimental.tex | 353 + doc/techdoc/chapter-flight-simulation.tex | 781 + doc/techdoc/chapter-introduction.tex | 168 + 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mode 100644 doc/techdoc/skin-friction-laminar.tex create mode 100644 doc/techdoc/techdoc.pdf create mode 100644 doc/techdoc/techdoc.tex create mode 100644 src/net/sf/openrocket/aerodynamics/AbstractAerodynamicCalculator.java delete mode 100644 src/net/sf/openrocket/aerodynamics/AtmosphericModel.java delete mode 100644 src/net/sf/openrocket/aerodynamics/ConeDragTest.java delete mode 100644 src/net/sf/openrocket/aerodynamics/ExactAtmosphericConditions.java delete mode 100644 src/net/sf/openrocket/aerodynamics/WindSimulator.java create mode 100644 src/net/sf/openrocket/database/MotorSetDatabase.java create mode 100644 src/net/sf/openrocket/database/ThrustCurveMotorSet.java create mode 100644 src/net/sf/openrocket/file/DirectoryIterator.java create mode 100644 src/net/sf/openrocket/file/RegularDirectoryIterator.java create mode 100644 src/net/sf/openrocket/file/ZipDirectoryIterator.java create mode 100644 src/net/sf/openrocket/masscalc/AbstractMassCalculator.java create mode 100644 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test/net/sf/openrocket/database/MotorSetDatabaseTest.java create mode 100644 test/net/sf/openrocket/database/ThrustCurveMotorSetTest.java create mode 100644 test/net/sf/openrocket/motor/ThrustCurveMotorTest.java create mode 100644 test/net/sf/openrocket/util/MutableTest.java create mode 100644 web/html/cc-by-nc-nd-80x15.png create mode 100644 web/html/cc-by-sa-80x15.png create mode 100644 web/html/layout-backup.css delete mode 100644 web/html/layout-original.css delete mode 120000 web/html/layout.css diff --git a/.classpath b/.classpath index b97a4aa3..9df09db5 100644 --- a/.classpath +++ b/.classpath @@ -3,11 +3,20 @@ - + + + + + + - + + + + + diff --git a/.settings/org.eclipse.core.resources.prefs b/.settings/org.eclipse.core.resources.prefs new file mode 100644 index 00000000..311a8658 --- /dev/null +++ b/.settings/org.eclipse.core.resources.prefs @@ -0,0 +1,4 @@ +#Sun Jun 13 15:09:13 EEST 2010 +eclipse.preferences.version=1 +encoding//doc/design/simulation-sequence.uml=UTF-8 +encoding//doc/design/simulation-sequence.umlseq=UTF-8 diff --git a/.settings/org.eclipse.jdt.core.prefs b/.settings/org.eclipse.jdt.core.prefs new file mode 100644 index 00000000..0a8ce2f5 --- /dev/null +++ b/.settings/org.eclipse.jdt.core.prefs @@ -0,0 +1,64 @@ +#Wed Jun 09 17:40:23 EEST 2010 +eclipse.preferences.version=1 +org.eclipse.jdt.core.compiler.problem.annotationSuperInterface=warning +org.eclipse.jdt.core.compiler.problem.autoboxing=ignore +org.eclipse.jdt.core.compiler.problem.comparingIdentical=warning +org.eclipse.jdt.core.compiler.problem.deadCode=warning +org.eclipse.jdt.core.compiler.problem.deprecation=warning +org.eclipse.jdt.core.compiler.problem.deprecationInDeprecatedCode=enabled +org.eclipse.jdt.core.compiler.problem.deprecationWhenOverridingDeprecatedMethod=enabled +org.eclipse.jdt.core.compiler.problem.discouragedReference=warning +org.eclipse.jdt.core.compiler.problem.emptyStatement=ignore +org.eclipse.jdt.core.compiler.problem.fallthroughCase=ignore +org.eclipse.jdt.core.compiler.problem.fatalOptionalError=enabled +org.eclipse.jdt.core.compiler.problem.fieldHiding=warning +org.eclipse.jdt.core.compiler.problem.finalParameterBound=warning +org.eclipse.jdt.core.compiler.problem.finallyBlockNotCompletingNormally=warning +org.eclipse.jdt.core.compiler.problem.forbiddenReference=error +org.eclipse.jdt.core.compiler.problem.hiddenCatchBlock=warning +org.eclipse.jdt.core.compiler.problem.incompatibleNonInheritedInterfaceMethod=warning +org.eclipse.jdt.core.compiler.problem.incompleteEnumSwitch=ignore +org.eclipse.jdt.core.compiler.problem.indirectStaticAccess=ignore +org.eclipse.jdt.core.compiler.problem.localVariableHiding=warning +org.eclipse.jdt.core.compiler.problem.methodWithConstructorName=warning +org.eclipse.jdt.core.compiler.problem.missingDeprecatedAnnotation=warning +org.eclipse.jdt.core.compiler.problem.missingHashCodeMethod=warning +org.eclipse.jdt.core.compiler.problem.missingOverrideAnnotation=warning +org.eclipse.jdt.core.compiler.problem.missingSerialVersion=ignore +org.eclipse.jdt.core.compiler.problem.missingSynchronizedOnInheritedMethod=ignore +org.eclipse.jdt.core.compiler.problem.noEffectAssignment=warning +org.eclipse.jdt.core.compiler.problem.noImplicitStringConversion=warning +org.eclipse.jdt.core.compiler.problem.nonExternalizedStringLiteral=ignore +org.eclipse.jdt.core.compiler.problem.nullReference=warning +org.eclipse.jdt.core.compiler.problem.overridingPackageDefaultMethod=warning +org.eclipse.jdt.core.compiler.problem.parameterAssignment=ignore +org.eclipse.jdt.core.compiler.problem.possibleAccidentalBooleanAssignment=warning +org.eclipse.jdt.core.compiler.problem.potentialNullReference=warning +org.eclipse.jdt.core.compiler.problem.rawTypeReference=warning +org.eclipse.jdt.core.compiler.problem.redundantNullCheck=warning +org.eclipse.jdt.core.compiler.problem.redundantSuperinterface=ignore +org.eclipse.jdt.core.compiler.problem.specialParameterHidingField=disabled +org.eclipse.jdt.core.compiler.problem.staticAccessReceiver=warning +org.eclipse.jdt.core.compiler.problem.suppressWarnings=enabled +org.eclipse.jdt.core.compiler.problem.syntheticAccessEmulation=ignore +org.eclipse.jdt.core.compiler.problem.typeParameterHiding=warning +org.eclipse.jdt.core.compiler.problem.uncheckedTypeOperation=warning +org.eclipse.jdt.core.compiler.problem.undocumentedEmptyBlock=ignore +org.eclipse.jdt.core.compiler.problem.unhandledWarningToken=warning +org.eclipse.jdt.core.compiler.problem.unnecessaryElse=ignore +org.eclipse.jdt.core.compiler.problem.unnecessaryTypeCheck=ignore +org.eclipse.jdt.core.compiler.problem.unqualifiedFieldAccess=ignore +org.eclipse.jdt.core.compiler.problem.unusedDeclaredThrownException=ignore +org.eclipse.jdt.core.compiler.problem.unusedDeclaredThrownExceptionExemptExceptionAndThrowable=enabled +org.eclipse.jdt.core.compiler.problem.unusedDeclaredThrownExceptionIncludeDocCommentReference=enabled +org.eclipse.jdt.core.compiler.problem.unusedDeclaredThrownExceptionWhenOverriding=disabled +org.eclipse.jdt.core.compiler.problem.unusedImport=warning +org.eclipse.jdt.core.compiler.problem.unusedLabel=warning +org.eclipse.jdt.core.compiler.problem.unusedLocal=warning +org.eclipse.jdt.core.compiler.problem.unusedParameter=ignore +org.eclipse.jdt.core.compiler.problem.unusedParameterIncludeDocCommentReference=enabled +org.eclipse.jdt.core.compiler.problem.unusedParameterWhenImplementingAbstract=disabled +org.eclipse.jdt.core.compiler.problem.unusedParameterWhenOverridingConcrete=disabled +org.eclipse.jdt.core.compiler.problem.unusedPrivateMember=warning +org.eclipse.jdt.core.compiler.problem.unusedWarningToken=warning +org.eclipse.jdt.core.compiler.problem.varargsArgumentNeedCast=warning diff --git a/TODO b/TODO index 1645b66a..af114709 100644 --- a/TODO +++ b/TODO @@ -16,6 +16,13 @@ Maybe: - Water rocket modeling +Done: + +- Reading .RKT +- Central logging system +- Simulation refactoring + + More ideas: ----------- @@ -30,20 +37,6 @@ Memory usage: - Warn user when 90% of memory is used, offer to send info to server -Logging: - -- Implement a central logging system - - takes messages with different priorities - - converts them into LogLine's that are passed to log handlers - - printing log handler and storing log handler - - bug reports to contain last ~50 log lines -- LogLine - - contains TraceException for log position information - - contains sequential number -- log storing handler stores separately debug and info/warn levels -- remove all System.out/err writing, replace with logging - - Motor support: - Combine duplicates into one display in the selection dialog @@ -56,7 +49,7 @@ Motor support: - Integration with thrustcurve.org (syncing?) - Reading thrust curves from external directory - Plot motor thrust curve -- Water rocket modelling +- Water rocket modeling Optimization: @@ -70,8 +63,8 @@ Optimization: Running: -- Windows executable wrapper (launch4j) -- Allow only one instance of OpenRocket running (RMI communication) +- Windows executable wrapper (launch4j?) +- Allow only one instance of OpenRocket running (RMI communication?) - Running/simulating from command line @@ -88,7 +81,6 @@ UI issues: - Drag and drop of components / other table items - Ctrl+scroll wheel to zoom in/out - Move components by dragging on the figure -- Progress bar during startup Simulation: @@ -121,27 +113,17 @@ Component support: File support: -- Reading (and writing) .RKT format +- Reading (and writing) SpaceCad format +- Writing .RKT format ? Refactoring tasks: - Change databases and icon loading to bean instances, implement stubs for unit tests -- Move startup class to src14 directory, remove reflection - Remove database etc. initialization from class initialization, create separate set of test motors - Extract event rules and data saving from Simulator into listeners -- How to handle different simulation situations elegantly - (launch rod, free flight, recovery) - Change SimulationStatus to include methods for obtaining basic position (maybe even change to an interface, implements Cloneable) -- Modularize calculators, simulators, atmospheric conditions, etc. -- Change Motor (immutable) to be a factory of MotorInstance (stateful) - - - -Done: ------ -- Reading .RKT format \ No newline at end of file diff --git a/build.xml b/build.xml index 58155ea3..6262bc0d 100644 --- a/build.xml +++ b/build.xml @@ -23,7 +23,6 @@ - @@ -53,10 +52,9 @@ - Compiling main classes - - Compiling startup classes - + Compiling main classes + + Copying build.properties diff --git a/doc/design/simulation-sequence.uml b/doc/design/simulation-sequence.uml new file mode 100644 index 00000000..d77e438c --- /dev/null +++ b/doc/design/simulation-sequence.uml @@ -0,0 +1,2 @@ + + diff --git a/doc/design/simulation-sequence.umlseq b/doc/design/simulation-sequence.umlseq new file mode 100644 index 00000000..57fe0720 --- /dev/null +++ b/doc/design/simulation-sequence.umlseq @@ -0,0 +1,5 @@ + + + + + diff --git a/doc/design/simulation-sequence.uxf b/doc/design/simulation-sequence.uxf new file mode 100644 index 00000000..07574a27 --- /dev/null +++ b/doc/design/simulation-sequence.uxf @@ -0,0 +1,120 @@ +// Uncomment the following line to change the fontsize: +// fontsize=14 + + +////////////////////////////////////////////////////////////////////////////////////////////// +// Welcome to UMLet! +// +// Double-click on UML elements to add them to the diagram, or to copy them +// Edit elements by modifying the text in this panel +// Hold Ctrl 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framecom.umlet.element.base.Relation250185075050lt=<<- +30;30;730;30com.umlet.element.base.Relation250182075070lt=<<- + postThrustCalculation() +730;50;30;50com.umlet.element.base.Relation250173075050lt=<<- +30;30;730;30com.umlet.element.base.Relation250170075070lt=<<- + preThrustCalculation() +730;50;30;50com.umlet.element.custom.Systemborder2401710780190Thrust computation +bg=#ffdddd +fg=graycom.umlet.element.custom.SeqObjectActive98017402030com.umlet.element.custom.SeqObjectActive98018602030com.umlet.element.base.Relation220176022680lt=<<- + thrustComputation() +70;60;120;60;120;30;60;30com.umlet.element.custom.SeqObjectActive27018102040 \ No newline at end of file diff --git a/doc/properties.txt b/doc/properties.txt new file mode 100644 index 00000000..202bc87e --- /dev/null +++ b/doc/properties.txt @@ -0,0 +1,44 @@ + +System properties used by OpenRocket: +===================================== + + +Logging options +--------------- + +Possible values are ERROR, WARN, USER, INFO, DEBUG and ALL. + + +openrocket.log.stderr + Defines from which log level upwards log events are written to STDERR. Defaults to no output. + +openrocket.log.stdout + Defines from which log level upwards log events are written to STDOUT. Defaults to no output. + +openrocket.log.tracelevel + Defines from which log level upwards the logging position is stored along with the log line. + Defaults to INFO. + + + + +Debugging options +----------------- + +openrocket.debug.menu + If defined the "Debug" menu will be displayed in the main application window. + +openrocket.debug.bugurl + URL used for sending bug reports. + +openrocket.debug.updateurl + URL used for retrieving update notifications. + +openrocket.debug.coordinatecount + If defined, the number of instantiations of the Coordinate class are counted and reported + every 1M instantiations, or as often as defined by this parameter. + +openrocket.debug.quaternioncount + If defined, the number of instantiations of the Quaternion class are counted and reported + every 1M instantiations, or as often as defined by this parameter. + diff --git a/doc/simulation-2.0.uml b/doc/simulation-2.0.uml deleted file mode 100644 index 64b54822..00000000 --- a/doc/simulation-2.0.uml +++ /dev/null @@ -1,91 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/doc/simulation-2.0.umlclass b/doc/simulation-2.0.umlclass deleted file mode 100644 index eab546de..00000000 --- a/doc/simulation-2.0.umlclass +++ /dev/null @@ -1,479 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/doc/techdoc/chapter-aerodynamic-properties.tex b/doc/techdoc/chapter-aerodynamic-properties.tex new file mode 100644 index 00000000..a875bb94 --- /dev/null +++ b/doc/techdoc/chapter-aerodynamic-properties.tex @@ -0,0 +1,2235 @@ + + +\chapter{Aerodynamic properties of model rockets~~} +\label{chap-aerodynamics} + +A model rocket encounters three basic forces during its flight: +thrust from the motors, gravity, and aerodynamical forces. Thrust is +generated by the motors by exhausting high-velocity gases in the +opposite direction. The thrust of a motor is directly proportional to +the velocity of the escaping gas and the mass per time unit that is +exhausted. The thrust of commercial model rocket motors as a function +of time have been measured in static motor tests and are readily +available online~\cite{thrust-curve-database}. Normally the thrust of +a rocket motor is aligned on the center axis of the rocket, so that it +produces no angular moment to the rocket. + +Every component of the rocket is also affected by gravitational +force. When the forces and moments generated are summed up, the +gravitational force can be seen as a single force originating from the +{\it center of gravity} (CG). A homogeneous gravitational field does +not generate any angular moment on a body relative to the CG. +Calculating the gravitational force is therefore a simple matter of +determining the total mass and CG of the rocket. + +Aerodynamic forces, on the other hand, produce both net forces and +angular moments. To determine the effect of the aerodynamic +forces on the rocket, the total force and moment must be calculated +relative to some reference point. In this chapter, a method for +determining these forces and moments will be presented. + + + +\section{General aerodynamical properties} +\label{sec-general-aerodynamics} + +The aerodynamic forces acting on a rocket are usually split into +components for further examination. The two most important +aerodynamic force components of interest in a typical model rocket are +the {\it normal force} and {\it drag}. The aerodynamical normal +force is the force component that generates the corrective moment +around the CG and provides stabilization of the rocket. The +drag of a rocket is defined as the force component parallel to the +velocity of the rocket. This is the aerodynamical force that opposes +the movement of the rocket through air. + +Figure~\ref{fig-aerodynamic-forces}(a) shows the thrust, gravity, +normal force and drag of a rocket in free flight. It should be noted +that if the rocket is flying at an angle of attack $\alpha>0$, then +the normal force and drag are not perpendicular. In order to have +independent force components, it is necessary to define component +pairs that are always perpendicular to one another. Two such pairs +are the normal force and axial drag, or side force and drag, shown in +Figure~\ref{fig-aerodynamic-forces}(b). The two pairs coincide if the +angle of attack is zero. The component pair that will be used as a +basis for the flight simulations is the normal force and axial drag. + + +\begin{figure} +\centering +\parbox{35mm}{\centering +\epsfig{file=figures/aerodynamics/free-flight-forces,width=35mm} \\ (a)} +\hfill +\parbox{35mm}{\centering +\epsfig{file=figures/aerodynamics/aero-force-components,width=35mm} \\ (b)} +\hfill +\parbox{35mm}{\centering +\epsfig{file=figures/aerodynamics/pitch-yaw-roll,width=35mm} \\ (c)} +\caption{(a) Forces acting on a rocket in free flight: gravity $G$, + motor thrust $T$, drag $D$ and normal force $N$. (b) Perpendicular + component pairs of the total aerodynamical force: normal force $N$ + and axial drag $D_A$; side force $S$ and drag $D$. (c) The pitch, + yaw and roll directions of a model rocket.} +\label{fig-aerodynamic-forces} +\end{figure} + + +The three moments around the different axis are called the +{\it pitch}, {\it yaw} and {\it roll moments}, as depicted in +Figure~\ref{fig-aerodynamic-forces}(c). Since a typical rocket has no +``natural'' roll angle of flight as an aircraft does, we may choose +the pitch angle to be in the same plane as the angle of attack, \ie +the plane defined by the velocity vector and the centerline of the +rocket. Thus, the normal force generates the pitching moment and no +other moments. + + + + + +\subsection{Aerodynamic force coefficients} + +When studying rocket configurations, the absolute force values are +often difficult to interpret, since many factors affect them. In +order to get a value better suited for comparison, the forces are +normalized by the current dynamic pressure $q=\frac{1}{2}\rho v_0^2$ +and some characteristic area \Aref\ to get a non-dimensional force +coefficient. Similarly, the moments are normalized by the dynamic +pressure, characteristic area and characteristic length $d$. Thus, +the normal force coefficient corresponding to the normal force $N$ is +defined as +% +\begin{equation} +C_N = \frac{N}{\frac{1}{2}\rho v_0^2 \, \Aref} +\label{eq-CN-def} +\end{equation} +% +and the pitch moment coefficient for a pitch moment $m$ as +% +\begin{equation} +C_m = \frac{m}{\frac{1}{2}\rho v_0^2 \, \Aref\, d}. +\label{eq-Cm-def} +\end{equation} +% +A typical choice of reference area is the base of the rocket's nose +cone and the reference length is its diameter. + +The pitch moment is always calculated around some reference point, +while the normal force stays constant regardless of the point of +origin. If the moment coefficient $C_m$ is known for some reference +point, the moment coefficient at another point $C_m'$ can be +calculated from +% +\begin{equation} +C_m'd = C_md - C_N\Delta x +\label{eq-moment-reference} +\end{equation} +% +where $\Delta x$ is the distance along the rocket centerline. +Therefore it is sufficient to calculate the moment coefficient only at +some constant point along the rocket body. In this thesis the +reference point is chosen to be the tip of the nose cone. + +The {\it center of pressure} (CP) is defined as the position from +which the total normal force alone produces the current pitching +moment. Therefore the total normal force produces no moment around +the CP itself, and an equation for the location of the CP +can be obtained from (\ref{eq-moment-reference}) by selecting setting +$C_m'=0$: +% +\begin{equation} +X = \frac{C_m}{C_N}\,d +\end{equation} +% +Here $X$ is the position of the CP along the rocket centerline from +the nose cone tip. This equation is valid when $\alpha>0$. As +$\alpha$ approaches zero, both $C_m$ and $C_N$ approach zero. The CP +is then obtained as a continuous extension using l'Hôpital's rule +% +\begin{equation} +X = \left.\frac{\;\frac{\partial C_m}{\partial\alpha}\;} + {\;\frac{\partial C_N}{\partial\alpha}\;}\,d\right|_{\alpha=0} + = \frac{\Cma}{\CNa}\,d +\label{eq-CP-position} +\end{equation} +% +where the normal force coefficient and pitch moment coefficient +derivatives have been defined as +% +\begin{equation} +\CNa = \left.\frac{\partial C_N}{\partial\alpha}\right|_{\alpha=0} +\hspace{5mm}\mbox{and}\hspace{5mm} +\Cma = \left.\frac{\partial C_m}{\partial\alpha}\right|_{\alpha=0}. +\label{eq-CNa-derivative} +\end{equation} + +At very small angles of attack we may approximate $C_N$ and $C_m$ to +be linear with $\alpha$, so to a first approximation +% +\begin{equation} +C_N \approx \CNa\,\alpha +\hspace{5mm}\mbox{and}\hspace{5mm} +C_m \approx \Cma\,\alpha. +\label{eq-CNa-approx} +\end{equation} +% +The Barrowman method uses the coefficient derivatives to determine the +CP position using equation~(\ref{eq-CP-position}). However, there are +some significant nonlinearities in the variation of $C_N$ as a +function of $\alpha$. These will be accounted for by holding the +approximation of equation~(\ref{eq-CNa-approx}) exact and letting +\CNa\ and \Cma\ be a function of $\alpha$. Therefore, for the +purposes of this thesis we define +% +\begin{equation} +\CNa = \frac{C_N}{\alpha} +\hspace{5mm}\mbox{and}\hspace{5mm} +\Cma = \frac{C_m}{\alpha} +\label{eq-CNa-definition} +\end{equation} +% +for $\alpha>0$ and by equation~(\ref{eq-CNa-derivative}) for +$\alpha=0$. These definitions are compatible, since +equation~(\ref{eq-CNa-definition}) simplifies to the partial +derivative~(\ref{eq-CNa-derivative}) at the limit +$\alpha\rightarrow0$. This definition also allows us to stay true to +Barrowman's original method which is familiar to many rocketeers. + + +Similar to the normal force coefficient, the drag coefficient is +defined as +% +\begin{equation} +C_D = \frac{D}{\frac{1}{2}\rho v_0^2 \, \Aref}. +\label{eq-CD-def} +\end{equation} +% +Since the size of the rocket has been factored out, the drag +coefficient at zero angle of attack $C_{D0}$ allows a straightforward +method of comparing the effect of different rocket shapes on drag. +However, this coefficient is not constant and will vary with \eg the +speed of the rocket and its angle of attack. + + +If each of the fins of a rocket are canted at some angle $\delta>0$ +with respect to the rocket centerline, the fins will produce a roll +moment on the rocket. Contrary to the normal force and pitching +moment, canting the fins will produce a non-zero rolling moment but no +corresponding net force. Therefore the only quantity computed is the +roll moment coefficient, defined by +% +\begin{equation} +C_l = \frac{l}{\frac{1}{2}\rho v_0^2 \, \Aref\, d} +\label{eq-Cl-def} +\end{equation} +% +where $l$ is the roll moment. + +It shall be shown later that rockets with axially-symmetrical fin +configurations experience no forces that would produce net yawing +moments. However, a single fin may produce all six types of forces and +moments. The equations for the forces and moments of a single fin will +not be explicitly written out, and they can be computed from the +geometry in question. + + + +\subsection{Velocity regions} + +Most of the aerodynamic properties of rockets vary with the velocity +of the rocket. The important parameter is the {\it Mach number}, +which is the free-stream velocity of the rocket divided by the local +speed of sound +% +\begin{equation} +M = \frac{v_0}{c}. +\end{equation} +% +The velocity range encountered by rockets is divided into regions with +different impacts on the aerodynamical properties, listed in +Table~\ref{tab-sonics}. + +In {\it subsonic flight} all of the airflow around the rocket occurs +below the speed of sound. This is the case for approximately $M<0.8$. +At very low Mach numbers air can be effectively treated as an +incompressible fluid, but already above $M\approx 0.3$ some +compressibility issues may have to be considered. + +In {\it transonic flight} some of the air flowing around the rocket +accelerates above the speed of sound, while at other places it remains +subsonic. Some local shock waves are generated and hard-to-predict +interference effects may occur. The drag of a rocket has a sharp +increase in the transonic region, making it hard to pass into the +supersonic region. Transonic flight occurs at Mach numbers of +approximately 0.8--1.2. + +In {\it supersonic flight} all of the airflow is faster than the +speed of sound (with the exception of \eg the nose cone tip). A shock +wave is generated by the nose cone and fins. In supersonic flight +the drag reduces from that of transonic flight, but is generally +greater than that of subsonic flight. Above approximately Mach 5 new +phenomena begin to emerge that are not encountered at lower supersonic +speeds. This region is called {\it hypersonic flight}. + +\begin{table} +\caption{Velocity regions of rocket flight} +\label{tab-sonics} +\begin{center} +\begin{tabular}{cr@{ -- }l} +Region & \multicolumn{2}{c}{Mach number ($M$)} \\ +\hline +Subsonic & \hspace{10mm} 0 & 0.8 \\ +Transonic & 0.8 & 1.2 \\ +Supersonic & 1.2 & $\sim5$ \\ +Hypersonic & $\sim5$ & \\ +\hline +\end{tabular} +\end{center} +\end{table} + + +Methods for predicting the aerodynamic properties of subsonic flight +and some extensions to supersonic flight will be presented. Since the +analytical prediction of aerodynamic properties in the transonic +region is quite difficult, this region will be accounted for by using +some suitable interpolation function that corresponds reasonably to +actual measurements. Hypersonic flight will not be considered, since +practically no model or high power rockets ever achieve such speeds. + + +\subsection{Flow and geometry parameters} + +There exist many different parameters that characterize aspects of +flow or a rocket's geometry. One of the most important flow +parameters is the {\it Reynolds number} $R$. It is a dimensionless +quantity that characterizes the ratio of inertial forces and viscous +forces of flow. Many aerodynamic properties depend on the Reynolds +number, defined as +% +\begin{equation} +R = \frac{v_0\; L}{\nu}. +\end{equation} +% +Here $v_0$ is the free-stream velocity of the rocket, $L$ is a +characteristic length and $\nu$ is the kinematic viscosity of air. It +is notable that the Reynolds number is dependent on a characteristic +length of the object in question. In most cases, the length used is +the length of the rocket. A typical 30~cm sport model flying at +50~m/s has a corresponding Reynolds number of approximately +1\s000\s000. + +Another term that is frequently encountered in aerodynamical equations +has been defined its own parameter $\beta$, which characterizes the +flow speed both in subsonic and supersonic flow: +% +\begin{equation} +\beta = \sqrt{\envert{M^2-1}} = +\left\{ +\begin{array}{ll} +\sqrt{1-M^2}, & {\rm if\ } M<1 \\ +\sqrt{M^2-1}, & {\rm if\ } M>1 +\end{array} +\right. +\end{equation} +% +As the flow speed approaches the transonic region $\beta$ approaches +zero. This term appears for example in the {\it Prandtl factor} $P$ +which corrects subsonic force coefficients for compressible flow: +% +\begin{equation} +P = \frac{1}{\beta} = \frac{1}{\sqrt{1-M^2}} +\label{eq-prandtl-factor} +\end{equation} + +It is also often useful to define parameters characterizing general +properties of a rocket. One such parameter is the {\it caliber}, +defined as the maximum body diameter. The caliber is often used to +indicate relative distances on the body of a rocket, such as the +stability margin. Another common parameter characterizes the +``slenderness'' of a rocket. It is the {\it fineness ratio} of a +rocket $f_B$, defined as the length of the rocket body divided by the +maximum body diameter. Typical model rockets have a fineness ratio in +the range of 10--20, but extreme models may have a fineness ratio as +low as 5 or as large as 50. + + + + +\subsection{Coordinate systems} + +During calculation of the aerodynamic properties a coordinate system +fixed to the rocket will be used. The origin of the coordinates is at +the nose cone tip with the positive $x$-axis directed along the rocket +centerline. This convention is also followed internally in the +produced software. In the following sections the position of the $y$- +and $z$-axes are arbitrary; the parameter $y$ is used as a general +spanwise coordinate when discussing the fins. During simulation, +however, the $y$- and $z$-axes are fixed in relation to the rocket, +and do not necessarily align with the plane of the pitching moments. + + + +\clearpage +\section{Normal forces and pitching moments} + +Barrowman's method~\cite{barrowman-thesis} for determining the total +normal force coefficient derivative \CNa, the pitch moment +coefficient derivative \Cma\ and the CP location at subsonic speeds +first splits the rocket into simple separate components, then +calculates the CP location and \CNa\ for each component separately and +then combines these to get the desired coefficients and CP +location. The general assumptions made by the derivation are: +% +\begin{enumerate} +\item The angle of attack is very close to zero. +\item The flow around the body is steady and non-rotational. +\item The rocket is a rigid body. +\item The nose tip is a sharp point. +\item The fins are flat plates. +\item The rocket body is axially symmetric. +\end{enumerate} + +The components that will be discussed are nose cones, cylindrical body +tube sections, shoulders, boattails and fins, in an arbitrary +order. The interference effect between the body and fins will be +taken into account by a separate correction term. Extensions to +account for body lift and arbitrary fin shapes will also be derived. + + +\subsection{Axially symmetric body components} + +The body of the rocket is assumed to be an axially symmetric body of +rotation. The entire body could be considered to be a single +component, but in practice it is divided into nose cones, shoulders, +boattails and cylindrical body tube sections. The geometry of typical +nose cones, shoulders and boattails are described in +Appendix~\ref{app-nosecone-geometry}. + +The method presented by Barrowman for calculating the normal force and +pitch moment coefficients at supersonic speeds is based on a +second-order shock expansion method. However, this assumes that the +body of the rocket is very streamlined, and it cannot handle areas +with a slope larger than than $\sim30^\circ$. Since the software +allows basically any body shape, applying this method would be +difficult. + +Since the emphasis is on subsonic flow, for the purposes of this +thesis the normal force and pitching moments produced by the body are +assumed to be equal at subsonic and supersonic speeds. The assumption +is that the CP location is primarily affected by the fins. The effect +of supersonic flight on the drag of the body will be accounted for in +Section~\ref{sec-drag}. + + + +\subsubsection{\CNa\ of body components at subsonic speeds} + +The normal force for an axially symmetric body at position $x$ in +subsonic flow is given by +% +\begin{equation} +N(x) = \rho v_0 \; \frac{\partial}{\partial x}[A(x)w(x)] +\label{eq-normal-force} +\end{equation} +% +where $A(x)$ is the cross-sectional area of the body, and the $w(x)$ +is the local downwash, given as a function of the angle of attack as +% +\begin{equation} +w(x) = v_0 \sin\alpha. +\end{equation} +% +For angles of attack very close to zero $\sin\alpha\approx\alpha$, but +contrary to the original derivation, we shall not make this +simplification. From the definition of the normal force +coefficient~(\ref{eq-CN-def}) and equation~(\ref{eq-normal-force}) we +obtain +% +\begin{equation} +C_N(x) = \frac{N(x)}{\frac{1}{2}\rho v_0^2\;\Aref} + = \frac{2\; \sin\alpha}{\Aref}\; \frac{\dif A(x)}{\dif x}. +\label{eq-CNx} +\end{equation} +% +Assuming that the derivative $\frac{\dif A(x)}{\dif x}$ is +well-defined, we can integrate over the component length $l$ to obtain +% +\begin{equation} +C_N = \frac{2\; \sin\alpha}{\Aref}\; + \int_0^l \frac{\dif A(x)}{\dif x}\dif x + = \frac{2\; \sin\alpha}{\Aref}\; [A(l)-A(0)]. +\end{equation} +% +We then have +% +\begin{equation} +\CNa = \frac{C_N}{\alpha} + = \frac{2}{\Aref}\; [A(l)-A(0)]\; + \underbrace{\frac{\sin\alpha}{\alpha}}_ + {\parbox{10mm}{\scriptsize\centering + $\rightarrow 1$ as \\ $\alpha\rightarrow0$}}. +\label{eq-body-CNa} +\end{equation} +% +This is the same equation as derived by Barrowman with the exception +of the correction term $\sin\alpha/\alpha$. + +Equation~(\ref{eq-body-CNa}) shows that as long as the cross-sectional +area of the component changes smoothly, the normal force coefficient +derivative does not depend on the component shape, only the difference +of the cross-sectional area at the beginning and end. As a +consequence, according to Barrowman's theory, a cylindrical body tube +has no effect on the normal force coefficient or CP location. +However, the lift due to cylindrical body tube sections has been noted +to be significant for long, slender rockets even at angles of attack +of only a few degrees~\cite{galejs}. An extension +for the effect of body lift will be given shortly. + + + + +\subsubsection{\Cma\ of body components at subsonic speeds} + +A normal force $N(x)$ at position $x$ produces a pitching moment +% +\begin{equation} +m(x) = xN(x). +\end{equation} +% +at the nose cone tip. Therefore the pitching moment coefficient is +% +\begin{equation} +C_m(x) = \frac{m(x)}{\frac{1}{2}\rho v_0^2\;\Aref\, d} + = \frac{xN(x)}{\frac{1}{2}\rho v_0^2\;\Aref\, d}. +\end{equation} +% +Substituting equation~(\ref{eq-CNx}) we obtain +% +\begin{equation} +C_m(x) = \frac{x\;C_N(x)}{d} + = \frac{2\; \sin\alpha\; x}{\Aref\, d}\; \frac{\dif A(x)}{\dif x}. +\end{equation} +% +This can be integrated over the length of the body to obtain +% +\begin{equation} +C_m = \frac{2\;\sin\alpha}{\Aref\,d} + \int_0^l x \left(\od{A(x)}{x}\right) \dif x + = \frac{2\;\sin\alpha}{\Aref\,d} + \sbr{ lA(l)-\int_0^l A(x) \dif x }. +\end{equation} +% +The resulting integral is simply the volume of the body $V$. +Therefore we have +% +\begin{equation} +C_m = \frac{2\;\sin\alpha}{\Aref\,d} \sbr{ lA(l)-V } +\end{equation} +% +and +% +\begin{equation} +\Cma = \frac{2}{\Aref\,d}\sbr{ lA(l)-V }\; \frac{\sin\alpha}{\alpha}. +\label{eq-body-Cma} +\end{equation} +% +This is, again, the result derived by Barrowman with the additional +correction term $\sin\alpha/\alpha$. + + +\subsubsection{Effect of body lift} +\label{sec-body-lift} + +The analysis thus far has neglected the effect of body lift as +negligible at small angles of attack. However, in the flight of long, +slender rockets the lift may be quite significant at angles of attack +of only a few degrees, which may occur at moderate wind +speeds~\cite{galejs}. + +Robert Galejs suggested adding a correction term to the body +component \CNa\ to account for body lift~\cite{galejs}. The normal +force exerted on a cylindrical body at an angle of attack $\alpha$ +is~\cite[p.~3-11]{hoerner} +% +\begin{equation} +C_N = K\; \frac{A_{\rm plan}}{\Aref}\; \sin^2\alpha +\end{equation} +% +where $A_{\rm plan} = d\cdot l$ is the planform area of the cylinder +and K is a constant $K\approx 1.1$. Galejs had simplified the +equation with $\sin^2\alpha\approx\alpha^2$, but this shall not be +performed here. At small angles of attack, when the approximation is +valid, this yields a linear correction to the value of \CNa. + +It is assumed that the lift on non-cylindrical components can be +approximated reasonably well with the same equation. The CP location +is assumed to be the center of the planform area, that is +% +\begin{equation} +X_{\rm lift} = \frac{\int_0^l x\; 2r(x)\dif x}{A_{\rm plan}}. +\end{equation} +% +This is reminiscent of the CP of a rocket flying at an angle of attack +of $90^\circ$. For a cylinder the CP location is at the center of the +body, which is also the CP location obtained at the limit with +equation~(\ref{eq-body-CP-position}). However, for nose cones, +shoulders and boattails it yields a slightly different position than +equation~(\ref{eq-body-CP-position}). + +%The value of $K$ has been experimentally fitted to experimental data +%from wind tunnels. + + + + + + +\subsubsection{Center of pressure of body components} + +The CP location of the body components can be calculated by +inserting equations~(\ref{eq-body-CNa}) and (\ref{eq-body-Cma}) into +equation~(\ref{eq-CP-position}): +% +\begin{equation} +X_B = \frac{(\Cma)_B}{(\CNa)_B}\;d + = \frac{lA(l)-V}{A(l)-A(0)} +\label{eq-body-CP-position} +\end{equation} +% +It is worth noting that the correction term $\sin\alpha/\alpha$ +cancels out in the division, however, it is still present in the +value of \CNa\ and is therefore significant at large angles of attack. + +The whole rocket body could be numerically integrated and the +properties of the whole body computed. However, it is often more +descriptive to split the body into components and calculate the +parameters separately. The total CP location can be calculated from +the separate CP locations $X_i$ and normal force coefficient +derivatives $(\CNa)_i$ by the moment sum +% +\begin{equation} +X = \frac{\sum_{i=1}^n X_i(\CNa)_i}{\sum_{i=1}^n (\CNa)_i}. +\label{eq-moment-sum} +\end{equation} +% +In this manner the effect of the separate components can be more +easily analyzed. + + + +\subsection{Planar fins} +\label{sec-planar-fins} + +The fins of the rocket are considered separately from the body. Their +CP location and normal force coefficient are determined and added to +the total moment sum~(\ref{eq-moment-sum}). The interference between +the fins and the body is taken into account by a separate correction +term. + +In addition to the corrective normal force, the fins can induce a roll +rate if each of the fins are canted at an angle $\delta$. The roll +moment coefficient will be derived separately in +Section~\ref{sec-roll-dynamics}. + +Barrowman's original report and thesis derived the equations for +trapezoidal fins, where the tip chord is parallel to the body +(Figure~\ref{fig-fin-geometry}(a)). The equations can be extended to +\eg elliptical fins~\cite{barrowman-elliptical-fins} +(Figure~\ref{fig-fin-geometry}(b)), but many model rocket fin +designs depart from these basic shapes. Therefore an +extension is presented that approximates the aerodynamical +properties for a free-form fin defined by a list of $(x,y)$ +coordinates (Figure~\ref{fig-fin-geometry}(c)). + + +\begin{figure} +\centering +\parbox{35mm}{\centering +\epsfig{file=figures/fin-geometry/fin-trapezoidal,scale=0.5} \\ (a)} +\hfill +\parbox{35mm}{\centering +\epsfig{file=figures/fin-geometry/fin-elliptical,scale=0.5} \\ (b)} +\hfill +\parbox{35mm}{\centering +\epsfig{file=figures/fin-geometry/fin-free,scale=0.5} \\ (c)} +\caption{Fin geometry of (a) a trapezoidal fin, (b) an elliptical fin + and (c) a free-form fin.} +\label{fig-fin-geometry} +\end{figure} + + + +Additionally, Barrowman considered only cases with three or four +fins. This shall be extended to allow for any reasonable number of +fins, even single fins. + + +\subsubsection{Center of pressure of fins at subsonic and supersonic + speeds} + +Barrowman argued that since the CP of a fin is located along its mean +aerodynamic chord (MAC) and on the other hand at low subsonic speeds +on its quarter chord, then the CP must be located at the intersection +of these two (depicted in Figure~\ref{fig-fin-geometry}(a)). He +proceeded to calculate this intersection point analytically from the +fin geometry of a trapezoidal fin. + +Instead of following the derivation Barrowman used, an alternative +method will be presented that allows simpler extension to free-form +fins. The two methods yield identical results for trapezoidal fins. +The length of the MAC $\bar c$, its spanwise position $y_{\rm MAC}$, +and the effective leading edge location $x_{\rm MAC,LE}$ are given +by~\cite{appl-comp-aero-fins} +% +\begin{align} +\bar c &= \frac{1}{\Afin} \int_0^s c^2(y) \dif y + \label{eq-MAC-length} \\ +y_{\rm MAC} &= \frac{1}{\Afin} \int_0^s yc(y) \dif y + \label{eq-MAC-ypos} \\ +x_{\rm MAC,LE} &= \frac{1}{\Afin} \int_0^s x_{\rm LE}(y)c(y) \dif y + \label{eq-MAC-xpos} +\end{align} +% +where $\Afin$ is the one-sided area of a single fin, $s$ is the span of +one fin, and $c(y)$ is the length of the fin chord and $x_{\rm LE}(y)$ +the leading edge position at spanwise position $y$. + +When these equations are applied to trapezoidal fins and the +lengthwise position of the CP is selected at the quarter chord, +$X_f=x_{\rm MAC,LE}+0.25\,\bar c$, +one recovers exactly the results derived by Barrowman: +% +\begin{align} +y_{\rm MAC} &= \frac{s}{3}\,\frac{C_r+2C_t}{C_r+C_t} \\ +X_f &= \frac{X_t}{3}\,\frac{C_r+2C_t}{C_r+C_t} + + \frac{1}{6}\,\frac{C_r^2+C_t^2+C_rC_t}{C_r+C_t} +\end{align} +% +However, equations~(\ref{eq-MAC-length})--(\ref{eq-MAC-xpos}) may also +be directly applied to elliptical or free-form fins. + +Barrowman's method assumes that the lengthwise position of the CP +stays at a constant 25\% of the MAC at subsonic speeds. However, the +position starts moving rearward above approximately Mach 0.5. For +$M>2$ the relative lengthwise position of the CP is given by an empirical +formula~\cite[p.~33]{fleeman} +% +\begin{equation} +\frac{X_f}{\bar c} = \frac{\AR\beta - 0.67}{2\AR\beta-1} +\label{eq-fin-CP-mach2} +\end{equation} +% +where $\beta=\sqrt{M^2-1}$ for $M>1$ and \AR\ is the aspect ratio of +the fin defined using the span $s$ as $\AR=2s^2/\Afin$. +% +Between Mach 0.5 and 2 the lengthwise position of the CP is +interpolated. A suitable function that gives a curve similar to that +of Figure~2.18 of reference~\cite[p.~33]{fleeman} was found to be a +fifth order polynomial $p(M)$ with the constraints +% +\begin{equation} + \begin{split} +p(0.5) & = 0.25 \\ +p'(0.5) & = 0 \\ +p(2) & = f(2) \\ +p'(2) & = f'(2) \\ +p''(2) & = 0 \\ +p'''(2) & = 0 + \end{split} +\end{equation} +% +where $f(M)$ is the function of equation~(\ref{eq-fin-CP-mach2}). + + + +The method presented here can be used to estimate the CP location of +an arbitrary thin fin. However, problems arise with the method if the +fin shape has a jagged edge as shown in +Figure~\ref{fig-fin-jagged}(a). If $c(y)$ would include only the sum +of the two separate chords in the area containing the gap, then the +equations would yield the same result as for a fin shown in +Figure~\ref{fig-fin-jagged}(b). This clearly would be incorrect, +since the position of the latter fin portion would be neglected. To +overcome this problem, $c(y)$ is chosen as the length from the leading +edge to the trailing edge of the fin, effectively adding the portion +marked by the dotted line to the fin. This corrects the CP position +slightly rearwards. The fin area used in +equations~(\ref{eq-MAC-length})--(\ref{eq-MAC-xpos}) must in this case +also be calculated including this extra fin area, but the extra area +must not be included when calculating the normal force coefficient. + +This correction is also approximate, since in reality such a jagged +edge would cause some unknown interference factor between the two fin +portions. Simulating such jagged edges using these methods should +therefore be avoided. + + +\begin{figure} +\centering +\parbox{35mm}{\centering +\epsfig{file=figures/fin-geometry/fin-jagged,scale=0.5} \\ (a)} +\hspace{5mm} +\parbox{35mm}{\centering +\epsfig{file=figures/fin-geometry/fin-jagged-equivalent,scale=0.5} \\ (b)} +\caption{(a) A jagged fin edge, and (b) an equivalent fin if $c(y)$ is + chosen to include only the actual fin area.} +\label{fig-fin-jagged} +\end{figure} + + + + + + + +\subsubsection{Single fin \CNa\ at subsonic speeds} +\label{sec-average-angle} + +Barrowman derived the normal force coefficient derivative value based +on Diederich's semi-empirical method~\cite{diederich}, which states that +for one fin +% +\begin{equation} +\del{\CNa}_1 = \frac{\CNa_0\; F_D \left(\frac{\Afin}{\Aref}\right) + \cos\Gamma_c} + {2+F_D\sqrt{1+\frac{4}{F_D^2}}}, +\label{eq-fin-CNa-base} +\end{equation} +% +where +% +\begin{itemize} +\item[$\CNa_0$] = normal force coefficient derivative of a 2D airfoil +\item[$F_D$] = Diederich's planform correlation parameter +\item[$\Afin$] = area of one fin +\item[$\Gamma_c$] = midchord sweep angle (depicted in + Figure~\ref{fig-fin-geometry}(a)). +\end{itemize} +% +Based on thin airfoil theory of potential flow corrected for +compressible flow +% +\begin{equation} +\CNa_0 = \frac{2\pi}{\beta} +\label{eq-fin-CNa0} +\end{equation} +% +where $\beta=\sqrt{1-M^2}$ for $M<1$. $F_D$ is a parameter that +corrects the normal force coefficient for the sweep of the fin. +According to Diederich, $F_D$ is given by +% +\begin{equation} +F_D=\frac{\AR}{\frac{1}{2\pi}\CNa_0\cos\Gamma_c}. +\label{eq-fin-FD} +\end{equation} +% +Substituting equations~(\ref{eq-fin-CNa0}), (\ref{eq-fin-FD}) and +$\AR=2s^2/\Afin$ into (\ref{eq-fin-CNa-base}) and simplifying one +obtains +% +\begin{equation} +%\del{\CNa}_1 = \frac{2\pi\; \AR \del{\frac{\Afin}{\Aref}}} +% {2+\sqrt{4 + \del{\frac{\beta \AR}{\cos\Gamma_c}}^2}}. +\del{\CNa}_1 = \frac{2\pi\; \frac{s^2}{\Aref}} + {1+\sqrt{1 + \del{\frac{\beta s^2}{\Afin\cos\Gamma_c}}^2}}. +\label{eq-CNa1} +\end{equation} +% +This is the normal force coefficient derivative for one fin, where the +angle of attack is between the airflow and fin surface. + +The value of equation~(\ref{eq-CNa1}) can be calculated directly for +trapezoidal and elliptical fins. However, in the case of free-form +fins, the question arises of how to define the mid-chord angle +$\Gamma_c$. If the angle $\Gamma_c$ is taken as the angle from the +middle of the root chord to the tip of the fin, the result may not be +representative of the actual shape, as shown by angle $\Gamma_{c1}$ in +Figure~\ref{fig-midchord-angle}. + +\begin{figure} +\centering +\epsfig{file=figures/fin-geometry/fin-midchord-angle,scale=0.7} +\caption{A free-form fin shape and two possibilities for the midchord + angle $\Gamma_c$.} +\label{fig-midchord-angle} +\end{figure} + +Instead the fin planform is divided into a large number of chords, and +the angle between the midpoints of each two consecutive chords is +calculated. The midchord angle used in equation~(\ref{eq-CNa1}) is +then the average of all these angles. This produces an angle better +representing the actual shape of the fin, as angle $\Gamma_{c2}$ in +Figure~\ref{fig-midchord-angle}. The angle calculated by this method +is also equal to the natural midchord angles for trapezoidal and +elliptical fins. + + + + +\subsubsection{Single fin \CNa\ at supersonic speeds} +\label{sec-single-fin-CNa-supersonic} + +The method for calculating the normal force coefficient of fins at +supersonic speed presented by Barrowman is based on a third-order +expansion according to Busemann theory~\cite{barrowman-fin}. The +method divides the fin into narrow streamwise strips, the normal force +of which are calculated separately. In this presentation the method +is further simplified by assuming the fins to be flat plates and by +ignoring a third-order term that corrects for fin-tip Mach cone +effects. + +% +% Angle of Inclination = between ray and surface +% Angle of Incidence = between ray and normal of surface +% + + +The local pressure coefficient of strip $i$ is calculated by +% +\begin{equation} +C_{P_i} = K_1 \,\eta_i + K_2 \,\eta_i^2 + K_3 \,\eta_i^3 +\label{eq-local-pressure-coefficient} +\end{equation} +% +where $\eta_i$ is the inclination of the flow at the surface and the +coefficients are +% +\begin{align} +K_1 &= \frac{2}{\beta} \\ +K_2 &= \frac{(\gamma+1)M^4 - 4\,\beta^2}{4\,\beta^4} \\ +K_3 &= \frac{(\gamma+1)M^8 + (2\gamma^2-7\gamma-5)M^6 + + 10(\gamma+1)M^4 + 8}{6\,\beta^7} +\end{align} +% +It is noteworthy that the coefficients $K_1$, $K_2$ and $K_3$ can be +pre-calculated for various Mach numbers, which makes the pressure +coefficient of a single strip very fast to compute. At small +angles of inclination the pressure coefficient is nearly linear, as +presented in Figure~\ref{fig-fin-strip-pressure-coefficient}. + + +\begin{figure} +\centering +\epsfig{file=figures/fin-geometry/Cp-supersonic,scale=0.6} +\caption{The local pressure coefficient as a function of the + strip inclination angle at various Mach numbers. The dotted line + depicts the linear component of + equation~(\ref{eq-local-pressure-coefficient}).} +\label{fig-fin-strip-pressure-coefficient} +\end{figure} + + +%If the rocket is not rolling, the inclinations $\eta_i$ are the +%same for all strips of a fin and only one pressure coefficient needs +%to be computed. However, presence of a roll velocity generates +%varying inclinations for all strips. Therefore the current +%examination is performed with separate inclinations and later +%simplified for non-rolling conditions. The effects of roll are +%further discussed in Section~\ref{sec-roll-dynamics}. + +The lift force of strip $i$ is equal to +% +\begin{equation} +F_i = C_{P_i} \cdot \frac{1}{2} \rho v_0^2 \cdot + \underbrace{c_i \Delta y}_{\rm area}. +\label{eq-supersonic-strip-lift-force} +\end{equation} +% +The total lift force of the fin is obtained by summing up the +contributions of all fin strips. The normal force coefficient is then +calculated in the usual manner as +% +\begin{align} +C_N &= \frac{\sum_i F_i}{\frac{1}{2}\rho v_0^2\; \Aref} \\ + &= \frac{1}{\Aref}\sum_i C_{P_i} \cdot c_i\Delta y. +\end{align} + +When computing the corrective normal force coefficient of the fins the +effect of roll is not taken into account. In this case, and +assuming that the fins are flat plates, the inclination angles +$\eta_i$ of all strips are the same, and the pressure coefficient is +constant over the entire fin. Therefore the normal force coefficient +is simply +% +\begin{equation} +(C_N)_1 = \frac{\Afin}{\Aref} \;C_P. +\end{equation} +% +Since the pressure coefficient is not linear with the angle of attack, +the normal force coefficient slope is defined using +equation~(\ref{eq-CNa-definition}) as +% +\begin{equation} +(\CNa)_1 = \frac{(C_N)_1}{\alpha} = + \frac{\Afin}{\Aref} \; \del{K_1 + K_2\,\alpha + K_3\,\alpha^2}. +\end{equation} + + + + +\subsubsection{Multiple fin \CNa} +\label{update-roll-angle} + +In his thesis, Barrowman considered only configurations with three and +four fins, one of which was parallel to the lateral airflow. For +simulation purposes, it is necessary to lift these restrictions to +allow for any direction of lateral airflow and for any number of +fins. + +The lift force of a fin is perpendicular to the fin and originates +from its CP. Therefore a single fin may cause a rolling and yawing +moment in addition to a pitching moment. In this case all of the +forces and moments must be computed from the geometry. If there are +two or more fins placed symmetrically around the body then the yawing +moments cancel, and if additionally there is no fin cant then the +total rolling moment is also zero, and these moments need not be +computed. + +The geometry of an uncanted fin configuration is depicted in +Figure~\ref{fig-dihedral-angle}. The dihedral angle between each of +the fins and the airflow direction is denoted $\Lambda_i$. The fin +$i$ encounters a local angle of attack of +% +\begin{equation} +\alpha_i = \alpha \sin\Lambda_i +\end{equation} +% +for which the normal force component (the component parallel to the +lateral airflow) is then +% +\begin{equation} +\del{\CNa}_{\Lambda_i} = \del{\CNa}_1 \sin^2 \Lambda_i. +\end{equation} + + +\begin{figure} +\centering +\epsfig{file=figures/fin-geometry/dihedral-angle,scale=1} +\caption{The geometry of an uncanted three-fin configuration (viewed + from rear).} +\label{fig-dihedral-angle} +\end{figure} + + +The sum of the coefficients for $N$ fins then yields +% +\begin{equation} +\sum_{k=1}^N \del{\CNa}_{\Lambda_k} = + \del{\CNa}_1 \sum_{k=1}^N \sin^2\Lambda_k. +\label{N-fin-equation} +\end{equation} +% +However, when $N\geq 3$ and the fins are spaced equally around the +body of the rocket, the sum simplifies to a constant +% +\begin{equation} +\sum_{k=1}^N \sin^2 (2\pi k/N + \theta) = \frac{N}{2}. +\label{N-fin-simplification} +\end{equation} +% +This equation predicts that the normal force produced by three or more +fins is independent of the roll angle $\theta$ of the vehicle. +Investigation by Pettis~\cite{pettis} showed that the normal force +coefficient derivative of a four-finned rocket at Mach~1.48 decreased +by approximately 6\% at a roll angle of $45^\circ$, and the roll angle +had negligible effect on an eight-finned rocket. Experimental data of +a four-finned sounding rocket at Mach speeds from 0.60 to 1.20 +supports the 6\% estimate~\cite{experimental-transonic}. + +The only experimental data available to the author of three-fin +configurations was of a rocket with a rounded triangular body cross +section~\cite{triform-fin-data}. This data suggests an effect of +approximately 15\% on the normal force coefficient derivative +depending on the roll angle. However, it is unknown how much of this +effect is due to the triangular body shape and how much from the fin +positioning. + +It is also hard to predict such an effect when examining singular +fins. If three identical or very similar singular fins are placed on +a rocket body, the effect should be the same as when the fins belong +to the same three-fin configuration. Due to these facts the effect of +the roll angle on the normal force coefficient derivative is ignored +when a fin configuration has three or more fins. +% +\footnote{In OpenRocket versions prior to 0.9.6 a sinusoidal reduction +of 15\% and 6\% was applied to three- and four-fin configurations, +respectively. However, this sometimes caused a significantly +different predicted CP location compared to the pure Barrowman method, +and also caused a discrepancy when such a fin configuration was +decomposed into singular fins. It was deemed better to follow the +tested and tried Barrowman method instead of introducing additional +terms to the equation.} + +However, in configurations with many fins the fin--fin +interference may cause the normal force to be less than that estimated +directly by equation~(\ref{N-fin-equation}). According to +reference~\cite[p.~5-24]{MIL-HDBK}, the normal force coefficients +for six and eight-fin configurations are 1.37 and 1.62 times that of +the corresponding four-fin configuration, respectively. The values +for five and seven-fin configurations are interpolated between these +values. + +\pagebreak[4] +Altogether, the normal force coefficient derivative $(\CNa)_N$ is +calculated by: +% +\begin{equation} +(\CNa)_N = \del{\sum_{k=1}^N \sin^2\Lambda_k} \del{\CNa}_1 \cdot + \left\{ +\begin{array}{ll} + 1.000\; & N_{\rm tot}=1,2,3,4 \vspace{1mm}\\ + 0.948\; & N_{\rm tot}=5 \vspace{1mm}\\ + 0.913\; & N_{\rm tot}=6 \vspace{1mm}\\ + 0.854\; & N_{\rm tot}=7 \vspace{1mm}\\ + 0.810\; & N_{\rm tot}=8 \vspace{1mm}\\ + 0.750\; & N_{\rm tot}>8 +\end{array} + \right. +\end{equation} +% +%\begin{equation} +%(\CNa)_N = +% \left\{ +%\begin{array}{ll} +% \del{\sum_{k=1}^N \sin^2\Lambda_k} \del{\CNa}_1 & N=1,2 \vspace{1mm}\\ +% 1.50\; \del{\CNa}_1 & N=3 \vspace{1mm}\\ +% 2.00\; \del{\CNa}_1 & N=4 \vspace{1mm}\\ +% 2.37\; \del{\CNa}_1 & N=5 \vspace{1mm}\\ +% 2.74\; \del{\CNa}_1 & N=6 \vspace{1mm}\\ +% 2.99\; \del{\CNa}_1 & N=7 \vspace{1mm}\\ +% 3.24\; \del{\CNa}_1 & N=8 +%\end{array} +% \right. +%\end{equation} +% +Here $N$ is the number of fins in this fin set, while $N_{\rm tot}$ is +the total number of parallel fins that have an interference effect. +The sum term simplifies to $N/2$ for $N\geq3$ according to +equation~(\ref{N-fin-simplification}). The interference effect for +$N_{\rm tot}>8$ is assumed at 25\%, as data for such configurations is +not available and such configurations are rare and eccentric in any +case. + +\subsubsection{Fin--body interference} + +The normal force coefficient must still be corrected for fin--body +interference, which increases the overall produced normal force. Here +two distinct effects can be identified: the normal force on the fins +due to the presence of the body and the normal force on the body due +to the presence of fins. Of these the former is significantly larger; +the latter is therefore ignored. The effect of the extra fin lift is +taken into account using a correction term +% +\begin{equation} +\del{\CNa}_{T(B)} = K_{T(B)}\;\del{\CNa}_N +\end{equation} +% +where $\del{\CNa}_{T(B)}$ is the normal force coefficient derivative +of the tail in the presence of the body. The term $K_{T(B)}$ can be +approximated by~\cite{barrowman-rd} +% +\begin{equation} +K_{T(B)} = 1 + \frac{r_t}{s + r_t}, +\end{equation} +% +where $s$ is the fin span from root to tip and $r_t$ is the body +radius at the fin position. The value $\del{\CNa}_{T(B)}$ is then +used as the final normal force coefficient derivative of the fins. + + +%The normal force coefficient must still be corrected for fin--body +%interference, which increases the produced normal force. The effect +%of the interference can be split into two components, the normal force +%of the fins in the presence of the body $\del{\CNa}_{T(B)}$ and of the +%body in the presence of the fins $\del{\CNa}_{B(T)}$. (The subscript +%$T$ refers to {\it tail}.) The interference is taken into account +%using the correction factors +%% +%\begin{align} +%\del{\CNa}_{T(B)} &= K_{T(B)}\;\del{\CNa}_N \\ +%\del{\CNa}_{B(T)} &= K_{B(T)}\;\del{\CNa}_N. +%\end{align} +%% +%In his original report, Barrowman simplified the factor $K_{T(B)}$ to +%% +%\begin{equation} +%K_{T(B)} = 1 + \frac{r_t}{s + r_t}, +%\end{equation} +%% +%where $s$ is the fin span from root to tip and $r_t$ is the body +%radius at the fin position, and ignored the effect of $K_{B(T)}$ as +%small compared to $K_{T(B)}$ for typical fin dimensions. However, +%$K_{T(B)}$ may be significant for fins with a span short compared to +%the body radius. In his thesis, a more accurate equation for +%$K_{T(B)}$ is provided, and $K_{B(T)}$ is given +%as~\cite[p.~31]{barrowman-thesis} +%% +%\begin{equation} +%K_{B(T)} = \del{1 + \frac{r_t}{s + r_t}}^2 - K_{T(B)}. +%\end{equation} +%% +%Therefore the total interference effect can be accounted for by a +%factor +%% +%\begin{equation} +%K_T = K_{T(B)} + K_{B(T)} = \del{1 + \frac{r_t}{s + r_t}}^2 +%\end{equation} +%% +%and +%% +%\begin{equation} +%\del{\CNa}_{\rm fins} = K_T\; (\CNa)_N. +%\end{equation} + +%This equation takes the increase on body lift and adds it as an +%additional force on the fins. The equation holds at subsonic speeds +%and also at supersonic speeds until the fin tip Mach cone intersects +%the body. In the latter case more complex interference factors would +%be required, which have been ignored in the current software. + +% TODO: FUTURE: supersonic interference effects, MIL-HDBK page 5-25 or B'man + + + +\pagebreak[4] +\subsection{Pitch damping moment} + +So far the effect of the current pitch angular velocity has been +ignored as marginal. This is the case during the upward flight of a +stable rocket. However, if a rocket is launched nearly vertically in +still air, the rocket flips over rather rapidly at apogee. In +some cases it was observed that the rocket was left wildly oscillating +during descent. The pitch damping moment opposes the fast rotation of +the rocket thus damping the oscillation. + +Since the pitch damping moment is notable only at apogee, and +therefore does not contribute to the overall flight characteristics, +only a rough estimate of its magnitude is required. A cylinder in +perpendicular flow has a drag coefficient of approximately $C_D=1.1$, +with the reference area being the planform area of the +cylinder~\cite[p.~3-11]{hoerner}. Therefore a short piece of cylinder +$\dif\xi$ at a distance $\xi$ from a rotation axis, as shown in +Figure~\ref{fig-pitch-velocity}, produces a force +% +\begin{equation} +\dif F = 1.1 \cdot \frac{1}{2}\rho(\omega\xi)^2 \cdot +\underbrace{2r_t\,\dif\xi}_{\rm ref.area} +\end{equation} +% +when the cylinder is rotating at an angular velocity $\omega$. The +produced moment is correspondingly $\dif m = \xi\dif F$. Integrating +this over $0\ldots l$ yields the total pitch moment +% +\begin{equation} +m = 0.275 \cdot \rho\, r_t\, l^4 \omega^2 +\end{equation} +% +and thus the moment damping coefficient is +% +\begin{equation} +C_{\rm damp} = + 0.55 \cdot \frac{l^4\; r_t}{\Aref\, d}\cdot\frac{\omega^2}{v_0^2}. +\end{equation} +% +This value is computed separately for the portions of the rocket body +fore and aft of the CG using an average body radius as $r_t$. + +\begin{figure} +\centering +\epsfig{file=figures/components/body-pitch-rate,scale=0.8} +\caption{Pitch damping moment due to a pitching body component.} +\label{fig-pitch-velocity} +\end{figure} + + + +Similarly, a fin with area $\Afin$ at a distance $\xi$ from the CG +produces a moment of approximately +% +\begin{equation} +C_{\rm damp} = + 0.6\cdot \frac{N\,\Afin\;\xi^3}{\Aref\,d}\cdot\frac{\omega^2}{v_0^2} +\end{equation} +% +where the effective area of the fins is assumed to be +$\Afin\cdot N/2$. For $N>4$ the value $N=4$ is used, since the other +fins are not exposed to any direct airflow. + +The damping moments are applied to the total pitch moment in the +opposite direction of the current pitch rate. It is noteworthy +that the damping moment coefficients are proportional to $\omega^2/v_0^2$, +confirming that the damping moments are insignificant during most of +the rocket flight, where the angles of deflection are small and the +velocity of the rocket large. Through roll coupling the yaw rate may also +momentarily become significant, and therefore the same correction is +also applied to the yaw moment. + + + + + +\clearpage +\section{Roll dynamics} +\label{sec-roll-dynamics} + +When the fins of a rocket are canted at some angle $\delta>0$, the +fins induce a rolling moment on the rocket. On the other hand, when a +rocket has a specific roll velocity, the portions of the fin far from +the rocket centerline encounter notable tangential velocities +which oppose the roll. Therefore a steady-state roll velocity, +dependent on the current velocity of the rocket, will result. + +The effect of roll on a fin can be examined by dividing the fin into +narrow streamwise strips and later integrating over the strips. A +strip $i$ at distance $\xi_i$ from the rocket centerline encounters a +radial velocity +% +\begin{equation} +u_i = \omega \xi_i +\end{equation} +% +where $\omega$ is the angular roll velocity, as shown in +Figure~\ref{fig-roll-velocity}. The radial velocity induces an angle +of attack +% +\begin{equation} +\eta_i = \tan^{-1} \del{\frac{u_i}{v_0}} = +\tan^{-1}\del{\frac{\omega\xi_i}{v_0}} +\approx \frac{\omega\xi_i}{v_0} +\label{eq-tan-approx} +\end{equation} +% +to the strip. The approximation $\tan^{-1} \eta \approx \eta$ is +valid for $u_i\ll v_0$, that is, when the velocity of the rocket is +large compared to the radial velocity. The approximation is +reasonable up to angles of $\eta \approx 20^\circ$, above which angle +most fins stall, which limits the validity of the equation in any +case. + +When a fin is canted at an angle $\delta$, the total +inclination of the strip to the airflow is +% +\begin{equation} +\alpha_i = \delta - \eta_i. +\label{eq-roll-aoa-variation} +\end{equation} + +Assuming that the force produced by a strip is directly proportional +to the local angle of attack, the force on strip $i$ is +% +\begin{equation} +F_i = k_i \alpha_i = k_i (\delta - \eta_i) +\end{equation} +% +for some $k_i$. The total moment produced by the fin is then +% +\begin{equation} +%l = \int_0^s (r+y) k (\delta-\eta(y))\dif y +% = \int_0^s (r+y) k \delta \dif y - \int_0^s (r+y) k \eta(y) \dif y +l = \sum_i \xi_i F_i = \sum_i \xi_i k_i (\delta - \eta_i) + = \sum_i \xi_i k_i \delta - \sum_i \xi_i k_i \eta_i. +\end{equation} +% +This shows that the effect of roll can be split into two components: +the first term $\sum_i \xi_i k_i \delta$ is the roll moment induced by +a fin canted at the angle $\delta$ when flying at zero roll rate +($\omega=0$), while the second term $\sum_i \xi_i k_i \eta_i$ is the +opposing moment generated by an uncanted fin ($\delta=0$) when flying +at a roll rate $\omega$. These two moments are called the roll +forcing moment and roll damping moment, respectively. These +components will be analyzed separately. + + +\begin{figure} +\centering +\epsfig{file=figures/fin-geometry/roll-velocity,scale=0.8} +\caption{Radial velocity at different positions of a fin. Viewed from + the rear of the rocket.} +\label{fig-roll-velocity} +\end{figure} + + + +\subsection{Roll forcing coefficient} + +As shown previously, the roll forcing coefficient can be computed by +examining a rocket with fins canted at an angle $\delta$ flying at +zero roll rate ($\omega=0$). In this case, the cant angle $\delta$ acts +simply as an angle of attack for each of the fins. Therefore, the +methods computed in the previous section can be directly applied. +Because the lift force of a fin originates from the mean aerodynamic +chord, the roll forcing coefficient of $N$ fins is equal to +% +\begin{equation} +C_{lf} = \frac{N (y_{\rm MAC}+r_t) \del{\CNa}_1 \delta}{d} +\label{eq-roll-forcing-moment} +\end{equation} +% +where $y_{\rm MAC}$ and $\del{\CNa}_1$ are computed using the methods +described in Section~\ref{sec-planar-fins} and $r_t$ is the radius of +the body tube at the fin position. This result is applicable +for both subsonic and supersonic speeds. + + + +\subsection{Roll damping coefficient} + +The roll damping coefficient is computed by examining a rocket with +uncanted fins ($\delta=0$) flying at a roll rate $\omega$. Since +different portions of the fin encounter different local angles of +attack, the damping moment must be computed from the separate +streamwise airfoil strips. + +At subsonic speeds the force generated by strip $i$ is equal to +% +\begin{equation} +F_i = \CNa_0 \; \frac{1}{2}\rho v_0^2 \; +\underbrace{c_i \Delta\xi_i}_{\rm area} \; \eta_i. +\end{equation} +% +Here $\CNa_0$ is calculated by equation~(\ref{eq-fin-CNa0}) and +$c_i \Delta\xi_i$ is the area of the strip. The roll damping moment +generated by the strip is then +% +\begin{equation} +\del{C_{ld}}_i + = \frac{F_i\,\xi_i}{\frac{1}{2}\rho v_0^2\,\Aref\, d} + = \frac{\CNa_0}{\Aref\,d} \; \xi_i c_i\Delta\xi_i \; \eta_i. +\end{equation} +% +By applying the approximation~(\ref{eq-tan-approx}) and summing +(integrating) the airfoil strips the total roll damping moment for $N$ +fins is obtained as: +% +\begin{align} +C_{ld} & = N \sum_i (C_{ld})_i \nonumber \\ +& = \frac{N \;\CNa_0\;\omega}{\Aref\,d\,v_0} \sum_i c_i\xi_i^2\Delta\xi_i. +\label{eq-roll-damping-moment} +\end{align} +% +The sum term is a constant for a specific fin shape. It can be +computed numerically from the strips or analytically for specific +shapes. For trapezoidal fins the term can be integrated as +% +\begin{equation} +\sum_i c_i\xi_i^2\Delta\xi_i = +\frac{C_r+C_t}{2}\;r_t^2s + \frac{C_r+2C_t}{3}\;r_ts^2 + +\frac{C_r+3C_t}{12}\;s^3 +\end{equation} +% +and for elliptical fins +% +\begin{equation} +\sum_i c_i\xi_i^2\Delta\xi_i = +C_r \del{ \frac{\pi}{4}\;r_t^2s + \frac{2}{3}\;r_ts^2 + + \frac{\pi}{16}\;s^3 }. +\end{equation} + + +The roll damping moment at supersonic speeds is calculated +analogously, starting from the supersonic strip lift force, +equation~(\ref{eq-supersonic-strip-lift-force}), where the angle of +inclination of each strip is calculated using +equation~(\ref{eq-tan-approx}). The roll moment at supersonic speeds +is thus +% +\begin{equation} +C_{ld} = \frac{N}{\Aref\,d} \sum_i C_{P_i}\, c_i \xi_i \Delta \xi_i. +\end{equation} +% +The dependence on the incidence angle $\eta_i$ is embedded within the +local pressure coefficient $C_{P_i}$, +equation~(\ref{eq-local-pressure-coefficient}). Since the dependence +is non-linear, the sum term is a function of the Mach number as well +as the fin shape. + + + + +\subsection{Equilibrium roll frequency} + +One quantity of interest when examining rockets with canted fins +is the steady-state roll frequency that the fins induce on a rocket +flying at a specific velocity. This is obtained by equating the roll +forcing moment~(\ref{eq-roll-forcing-moment}) and roll damping +moment~(\ref{eq-roll-damping-moment}) and solving for the roll rate +$\omega$. The equilibrium roll frequency at subsonic speeds is +therefore +% +\begin{equation} +f_{\rm eq} = \frac{\omega_{\rm eq}}{2\pi} = +\frac{\Aref\; \beta v_0 \; y_{\rm MAC} \; (\CNa)_1 \; \delta} +{4\pi^2\; \sum_i c_i\xi_i^2\Delta\xi_i} +\label{eq-subsonic-roll-rate} +\end{equation} +% +It is worth noting that the arbitrary reference area \Aref\ is +cancelled out by the reference area appearing within $(\CNa)_1$, +as is to be expected. + +At supersonic speeds the dependence on the incidence angle is +non-linear and therefore the equilibrium roll frequency must be solved +numerically. Alternatively, the second and third-order terms of the +local pressure coefficient of +equation~(\ref{eq-local-pressure-coefficient}) may be ignored, in +which case an approximation for the equilibrium roll frequency nearly +identical to the subsonic case is obtained: +% +\begin{equation} +f_{\rm eq} = \frac{\omega_{\rm eq}}{2\pi} = +\frac{\Aref\; \beta v_0 \; y_{\rm MAC} \; (\CNa)_1 \; \delta} +{4\pi\; \sum_i c_i\xi_i^2\Delta\xi_i} +\label{eq-supersonic-roll-rate} +\end{equation} +% +The value of $(\CNa)_1$ must, of course, be computed using different +methods in the subsonic and supersonic cases. + + + + +%\subsection{Roll sensitivity} +% +%The vast majority of model rockets have uncanted fins and in +%principle no roll is induced to these rockets. However, in practice +%imprecision in the attachment of the fins and other protrusions always +%cause some roll to the rocket during flight. In some applications, +%such as launching rockets with onboard video cameras, it is desirable +%to design the rockets so as to minimize the roll rate. To assist this +%design, a quantity called the {\it roll sensitivity} of the rocket is +%defined as +%% +%\begin{equation} +%f_{\rm sens} = \frac{1}{N}\eval{\pd{f_{\rm eq}}{\delta}}_{\delta=0}. +%\end{equation} +%% +%This is the slope of the equilibrium roll frequency at $\delta=0$, +%divided by the number of fins. If measured in units of +%$\rm Hz/^\circ$, the quantity indicates the number of rotations per +%second induced by one fin being attached at an angle of $1^\circ$. By +%minimizing the roll sensitivity of a rocket, the effect of +%construction imperfections on the roll rate can be minimized. From +%equation~(\ref{eq-roll-rate}) the subsonic roll sensitivity is +%obtained as +%% +%\begin{equation} +%f_{\rm sens} = +%\frac{\Aref\; \beta v_0 \; y_{\rm MAC} \; (\CNa)_1} +%{N\; 4\pi^2\; \sum_i c_i\xi_i^2\Delta\xi_i} +%\end{equation} +%% +%or conversely, +%% +%\begin{equation} +%f_{\rm eq} = N\; f_{\rm sens}\,\delta. +%\end{equation} +%% +%Similarly, in supersonic flight the roll sensitivity may either be +%solved numerically, or computed using the linear approximation for +%$C_{P_i}$ yielding +%% +%\begin{equation} +%f_{\rm sens} = +%\frac{\Aref\; \beta v_0 \; y_{\rm MAC} \; (\CNa)_1} +%{N\; 4\pi\; \sum_i c_i\xi_i^2\Delta\xi_i}. +%\end{equation} +% +% +%When the fins are canted by design, the roll sensitivity loses its +%significance. Therefore if all the fins on a rocket are uncanted, the +%quantity of intrest is the roll sensitivity, while for a rocket with +%canted fins it is the equilibrium roll frequency. + + + + + + +\clearpage +\section{Drag forces} +\label{sec-drag} + +Air flowing around a solid body causes drag, which resists the +movement of the object relative to the air. Drag forces arise from +two basic mechanisms, the air pressure distribution around the rocket +and skin friction. The pressure distribution is further divided into +body pressure drag (including shock waves generated as supersonic +speeds), parasitic pressure drag due to protrusions such as launch +lugs and base drag. Additional sources of drag include interference +between the fins and body and vortices generated at fin tips when +flying at an angle of attack. The different drag sources are depicted +in Figure~\ref{fig-drag-components}. Each drag source will be analyzed +separately; the interference drag and fin-tip vortices will be +ignored as small compared to the other sources. + +As described in Section~\ref{sec-general-aerodynamics}, two different +drag coefficients can be defined: the (total) drag coefficient $C_D$ +and the axial drag coefficient $C_A$. At zero angle of attack these +two coincide, $C_{D_0} = C_{A_0}$, but at other angles a distinction +between the two must be made. The value of significance in the +simulation is the axial drag coefficient $C_A$ based on the choice of +force components. However, the drag coefficient $C_D$ describes the +deceleration force on the rocket, and is a more commonly known value +in the rocketry community, so it is informational to calculate its +value as well. + +In this section the zero angle-of-attack drag coefficient +$C_{D_0} = C_{A_0}$ will be computed first. Then, in +Section~\ref{sec-axial-drag} this will be extended for angles of +attack and $C_A$ and $C_D$ will be computed. Since the drag force of +each component is proportional to its particular size, the subscript +$\bullet$ will be used for coefficients that are computed using the +reference area of the specific component. This reference area is the +frontal area of the component unless otherwise noted. Conversion to +the global reference area is performed by +% +\begin{equation} +C_{D_0} = \frac{A_{\rm component}}{\Aref} \cdot C_{D\bullet}. +\end{equation} + + +\begin{figure} +\centering +\epsfig{file=figures/aerodynamics/drag-components,width=13.5cm} +\caption{Types of model rocket drag at subsonic speeds.} +\label{fig-drag-components} +\end{figure} + + +\subsection{Laminar and turbulent boundary layers} + +At the front of a streamlined body, air flows smoothly around the +body in layers, each of which has a different velocity. The layer +closest to the surface ``sticks'' to the object having zero velocity. +Each layer gradually increases the speed until the free-stream +velocity is reached. This type of flow is said to be {\it laminar} +and to have a {\it laminar boundary layer}. The thickness of the +boundary layer increases with the distance the air has flowed along +the surface. At some point a transition occurs and the layers of air +begin to mix. The boundary layer becomes {\it turbulent} and thickens +rapidly. This transition is depicted in +Figure~\ref{fig-drag-components}. + +A turbulent boundary layer induces a notably larger skin friction drag +than a laminar boundary layer. It is therefore necessary to consider +how large a portion of a rocket is in laminar flow and at what point +the flow becomes turbulent. The point at which the flow becomes +turbulent is the point that has a {\it local critical Reynolds number} +% +\begin{equation} +R_{\rm crit} = \frac{v_0 \; x}{\nu}, +\label{eq-transition-Re} +\end{equation} +% +where $v_0$ is the free-stream air velocity, $x$ is the distance along +the body from the nose cone tip and +$\nu\approx 1.5\cdot10^{-5}\;\rm m^2/s$ is the kinematic viscosity of +air. The critical Reynolds number is approximately +$R_{\rm crit} = 5\cdot10^5$~\cite[p.~43]{barrowman-thesis}. Therefore, +at a velocity of 100~m/s the transition therefore occurs approximately +7~cm from the nose cone tip. + +% Air viscosity: +% http://www.engineeringtoolbox.com/air-absolute-kinematic-viscosity-d_601.html + + +%Since the drag force is approximately proportional to the square of +%the free-stream velocity, the value of $C_D$ is most critical at high +%velocities. Equation~(\ref{eq-transition-Re}) shows that at a velocity +%of 100~m/s the transition to turbulent flow occurs about 7~cm +%from the nose cone tip. Therefore at these speeds most of the wetted +%area of a typical model rocket is in turbulent flow. + +Surface roughness or even slight protrusions may also trigger the +transition to occur prematurely. At a velocity of 60~m/s the critical +height for a cylindrical protrusion all around the body is of the +order of 0.05~mm~\cite[p.~348]{advanced-model-rocketry}. The +body-to-nosecone joint, a severed paintbrush hair or some other +imperfection on the surface may easily exceed this limit and cause +premature transition to occur. + +Barrowman presents methods for computing the drag of both fully +turbulent boundary layers as well as partially-laminar layers. Both +methods were implemented and tested, but the difference in apogee +altitude was less than 5\% in with all tested designs. Therefore, +the boundary layer is assumed to be fully turbulent in all cases. + +%A typical model rocket may therefore be assumed to have a fully +%turbulent boundary layer. Only sport models which have been +%finished to fine precision may benefit from a partial laminar +%flow around the rocket. These different types of rockets will be +%taken into account by having two modes of calculation, one for typical +%model rockets that assumes a fully turbulent boundary layer, and +%another one which assumes very fine precision finish. + + + + + + +\subsection{Skin friction drag} + +Skin friction is one of the most notable sources of model rocket +drag. It is caused by the friction of the viscous flow of air +around the rocket. In his thesis Barrowman presented formulae for +estimating the skin friction coefficient for both laminar and +turbulent boundary layers as well as the transition between the +two~\cite[pp.~43--47]{barrowman-thesis}. As discussed above, a fully +turbulent boundary layer will be assumed in this thesis. + +The skin friction coefficient $C_f$ is defined as the drag coefficient +due to friction with the reference area being the total wetted area +of the rocket, that is, the body and fin area in contact with the +airflow: +% +\begin{equation} +C_f = \frac{D_{\rm friction}}{\frac{1}{2} \rho v_0^2\;A_{\rm wet}} +\end{equation} +% +The coefficient is a function of the rocket's Reynolds number $R$ and +the surface roughness. The aim is to first calculate the skin +friction coefficient, then apply corrections due to compressibility +and geometry effects, and finally to convert the coefficient to the +proper reference area. + + +\subsubsection{Skin friction coefficients} +\label{sec-skin-friction-coefficient} + +The values for $C_f$ are given by different formulae depending on the +Reynolds number. For fully turbulent flow the coefficient is given by +% +\begin{equation} +C_f = \frac{1}{(1.50\; \ln R - 5.6)^2}. +\label{eq-turbulent-friction} +\end{equation} + +The above formula assumes that the surface is ``smooth'' and the +surface roughness is completely submerged in a thin, laminar sublayer. +At sufficient speeds even slight roughness may have an effect on the +skin friction. The critical Reynolds number corresponding to the +roughness is given by +% +\begin{equation} +R_{\rm crit} = 51\left(\frac{R_s}{L}\right)^{-1.039}, +\end{equation} +% +where $R_s$ is an approximate roughness height of the surface. A few +typical roughness heights are presented in Table~\ref{tab-roughnesses}. +For Reynolds numbers above the critical value, the skin friction +coefficient can be considered independent of Reynolds number, and has +a value of +% +\begin{equation} +C_f = 0.032\left(\frac{R_s}{L}\right)^{0.2}. +\label{eq-critical-friction} +\end{equation} +% + + +\begin{table} +\caption{Approximate roughness heights of different + surfaces~\cite[p.~5-3]{hoerner}} +\label{tab-roughnesses} +\begin{center} +\begin{tabular}{lc} +Type of surface & Height / \um \\ +\hline +Average glass & 0.1 \\ +Finished and polished surface & 0.5 \\ +Optimum paint-sprayed surface & 5 \\ +Planed wooden boards & 15 \\ +% planed = höylätty ??? +Paint in aircraft mass production & 20 \\ +Smooth cement surface & 50 \\ +Dip-galvanized metal surface & 150 \\ +Incorrectly sprayed aircraft paint & 200 \\ +Raw wooden boards & 500 \\ +Average concrete surface & 1000 \\ +\hline +\end{tabular} +\end{center} +\end{table} + + +Finally, a correction must be made for very low Reynolds numbers. The +experimental formulae are applicable above approximately +$R\approx10^4$. This corresponds to velocities typically below 1~m/s, +which therefore have negligible effect on simulations. Below this +Reynolds number, the skin friction coefficient is assumed to be equal +as for $R=10^4$. + +Altogether, the skin friction coefficient for turbulent flow is +calculated by +% +\begin{equation} +C_f = \left\{ +\begin{array}{ll} +1.48\cdot10^{-2}, & \mbox{if $R<10^4$} \\ +\mbox{Eq.~(\ref{eq-turbulent-friction})}, & \mbox{if $10^4R_{\rm crit}$} +\end{array} +\right. . +\end{equation} +% +These formulae are plotted with a few different surface roughnesses in +Figure~\ref{fig-skinfriction-plot}. Included also is the laminar and +transitional skin friction values for comparison. + +\begin{figure} +\centering +\epsfig{file=figures/drag/skin-friction-coefficient,width=11cm} +\caption{Skin friction coefficient of turbulent, laminar and + roughness-limited boundary layers.} +\label{fig-skinfriction-plot} +\end{figure} + + +\subsubsection{Compressibility corrections} + +A subsonic speeds the skin friction coefficient turbulent and +roughness-limited boundary layers need to be corrected for +compressibility with the factor +% +\begin{equation} +{C_f}_c = C_f\; (1-0.1\, M^2). +\end{equation} +% +In supersonic flow, the turbulent skin friction coefficient must be +corrected with +% +\begin{equation} +{C_f}_c = \frac{C_f}{(1+0.15\, M^2)^{0.58}} +\end{equation} +% +and the roughness-limited value with +% +\begin{equation} +{C_f}_c = \frac{C_f}{1 + 0.18\, M^2}. +\end{equation} +% +However, the corrected roughness-limited value should not be used if +it would yield a value smaller than the corresponding turbulent +value. + + +\subsubsection{Skin friction drag coefficient} +\label{sec-skin-friction-drag} + +After correcting the skin friction coefficient for compressibility +effects, the coefficient can be converted into the actual drag +coefficient. This is performed by scaling it to the correct reference +area. The body wetted area is corrected for its cylindrical geometry, +and the fins for their finite thickness. +%effect of finite fin thickness which Barrowman handled +%separately is also included~\cite[p.~55]{barrowman-thesis}. +The total friction drag coefficient is then +% +\begin{equation} +(C_D)_{\rm friction} = {C_f}_c \; \frac{ + \del{1 + \frac{1}{2f_B}} \cdot A_{\rm wet,body} + + \del{1 + \frac{2t}{\bar c}} \cdot A_{\rm wet,fins}} + {\Aref} +\label{eq-friction-drag-scale} +\end{equation} +% +where $f_B$ is the fineness ratio of the rocket, and $t$ the thickness +and $\bar c$ the mean aerodynamic chord length of the fins. The +wetted area of the fins $A_{\rm wet,fins}$ includes both sides of the +fins. + + + + + +\subsection{Body pressure drag} + +Pressure drag is caused by the air being forced around the rocket. A +special case of pressure drag are shock waves generated at supersonic +speeds. In this section methods for estimating the pressure drag of +nose cones will be presented and reasonable estimates also for +shoulders and boattails. + + +\subsubsection{Nose cone pressure drag} + +At subsonic speeds the pressure drag of streamlined nose cones is +significantly smaller than the skin friction drag. In fact, suitable +shapes may even yield negative pressure drag coefficients, producing a +slight reduction in drag. Figure~\ref{fig-nosecone-cd} presents +various nose cone shapes and their respective measured pressure drag +coefficients.~\cite[p.~3-12]{hoerner} + +It is notable that even a slight rounding at the joint between the nose +cone and body reduces the drag coefficient dramatically. Rounding the +edges of an otherwise flat head reduces the drag coefficient from 0.8 +to 0.2, while a spherical nose cone has a coefficient of only 0.01. +The only cases where an appreciable pressure drag is present is when +the joint between the nose cone and body is not smooth, which may +cause slight flow separation. + +The nose pressure drag is approximately +proportional to the square of the sine of the joint angle $\phi$ +(shown in +Figure~\ref{fig-nosecone-cd})~\cite[p.~237]{handbook-supersonic-aerodynamics}: +% +\begin{equation} +(C_{D\bullet,M=0})_p = 0.8 \cdot \sin^2\phi. +\label{eq-nosecone-pressure-drag} +\end{equation} +% +This yields a zero pressure drag for all nose cone shapes that have a +smooth transition to the body. The equation does not take into +account the effect of extremely blunt nose cones (length less than +half of the diameter). Since the main drag cause is slight flow +separation, the coefficient cannot be corrected for compressibility +effects using the Prandtl coefficient, and the value is applicable +only at low subsonic velocities. + +\begin{figure} +\centering +\epsfig{file=figures/nose-geometry/nosecone-cd-top,width=11cm} +\caption{Pressure drag of various nose cone + shapes~\cite[p.~3-12]{hoerner}.} +\label{fig-nosecone-cd} +\end{figure} + + +At supersonic velocities shock waves increase the pressure drag +dramatically. In his report Barrowman uses a second-order +shock-expansion method that allows determining the pressure +distribution along an arbitrary slender rotationally symmetrical +body~\cite{second-order-shock-expansion-method}. However, +the method has some problematic limitations. The method cannot handle +body areas that have a slope larger than approximately $30^\circ$, +present in several typical nose cone shapes. The local airflow in +such areas may decrease below the speed of sound, and the method +cannot handle transonic effects. Drag in the transonic +region is of special interest for rocketeers wishing to build rockets +capable of penetrating the sound barrier. + +Instead of a general piecewise computation of the air pressure around +the nose cone, a simpler semi-empirical method for estimating the +transonic and supersonic pressure drag of nose cones is used. The +method, described in detail in +Appendix~\ref{app-nosecone-drag-method}, combines theoretical and +empirical data of different nose cone shapes to allow estimating the +pressure drag of all the nose cone shapes described in +Appendix~\ref{app-nosecone-geometry}. + +The semi-empirical method is used at Mach numbers above 0.8. +At high subsonic velocities the pressure drag is interpolated between +that predicted by equation~(\ref{eq-nosecone-pressure-drag}) and the +transonic method. The pressure drag is assumed to be non-decreasing +in the subsonic region and to have zero derivative at $M=0$. A +suitable interpolation function that resembles the shape of the +Prandtl factor is +% +\begin{equation} +(C_{D\bullet})_{\rm pressure} = a\cdot M^b + (C_{D\bullet,M=0})_p +\label{eq-nosecone-pressure-interpolator} +\end{equation} +% +where $a$ and $b$ are computed to fit the drag coefficient and its +derivative at the lower bound of the transonic method. + + + +\subsubsection{Shoulder pressure drag} + +Neither Barrowman nor Hoerner present theoretical or experimental +data on the pressure drag of transitions at subsonic velocities. In +the case of shoulders, the pressure drag coefficient is assumed to be +the same as that of a nose cone, except that the reference area is the +difference between the aft and fore ends of the transition. The +effect of a non-smooth transition at the beginning of the shoulder is +ignored, since this causes an increase in pressure and thus cannot +cause flow separation. + +While this assumption is reasonable at subsonic velocities, it is +somewhat dubious at supersonic velocities. However, no comprehensive +data set of shoulder pressure drag at supersonic velocities was +found. Therefore the same assumption is made for supersonic +velocities and a warning is generated during such simulations (see +Section~\ref{sec-warnings}). The refinement of the supersonic +shoulder pressure drag estimation is left as a future enhancement. + + + +\subsubsection{Boattail pressure drag} + +The estimate for boattail pressure drag is based on the body base +drag estimate, which will be presented in Section~\ref{sec-base-drag}. +At one extreme, the transition length is zero, in which case the +boattail pressure drag will be equal to the total base drag. On the +other hand, a gentle slope will allow a gradual pressure change +causing approximately zero pressure drag. Hoerner has presented +pressure drag data for wedges, which suggests that at a +length-to-height ratio below 1 has a constant pressure drag +corresponding to the base drag and above a ratio of 3 the pressure +drag is negligible. Based on this and the base drag +equation~(\ref{eq-base-drag}), an approximation for the pressure drag +of a boattail is given as +% +\begin{equation} +(C_{D\bullet})_{\rm pressure} = +\frac{A_{\rm base}}{A_{\rm boattail}} \cdot (C_{D\bullet})_{\rm base} +\cdot +\left\{ +\begin{array}{cl} +1 & \mbox{if\ } \gamma < 1 \\ +\frac{3-\gamma}{2} & \mbox{if\ } 1 < \gamma < 3 \\ +0 & \mbox{if\ } \gamma > 3 +\end{array} +\right. +\end{equation} +% +where the length-to-height ratio $\gamma = l/(d_1-d_2)$ is calculated +from the length and fore and aft diameters of the boattail. The +ratios 1 and 3 correspond to reduction angles of $27^\circ$ and +$9^\circ$, respectively, for a conical boattail. The base drag +$(C_{D\bullet})_{\rm base}$ is calculated using +equation~(\ref{eq-base-drag}). + +Again, this approximation is made primarily based on subsonic data. +At supersonic velocities expansion fans exist, the counterpart of +shock waves in expanding flow. However, the same equation is used for +subsonic and supersonic flow and a warning is generated during +transonic simulation of boattails. + + + +\subsection{Fin pressure drag} + +The fin pressure drag is highly dependent on the fin profile shape. +Three typical shapes are considered, a rectangular profile, rounded +leading and trailing edges, and an airfoil shape with rounded leading +edge and tapering trailing edge. Barrowman estimates the fin pressure +drag by dividing the drag further into components of a finite +thickness leading edge, thick trailing edge and overall fin +thickness~\cite[p.~48--57]{barrowman-thesis}. In this report the fin +thickness was already taken into account as a correction to the skin +friction drag in Section~\ref{sec-skin-friction-drag}. The division +to leading and trailing edges also allows simple extension to the +different profile shapes. + +The drag of a rounded leading edge can be considered as a circular +cylinder in cross flow with no base drag. Barrowman derived +an empirical formula for the leading edge pressure drag as +% +\begin{equation} +(C_{D\bullet})_{LE\perp} = +\left\{ +\begin{array}{ll} + (1-M^2)^{-0.417} - 1 & \mbox{for $M<0.9$} \\ + 1-1.785(M-0.9) & \mbox{for $0.9 < M < 1$} \\ + 1.214 - \frac{0.502}{M^2} + \frac{0.1095}{M^4} & \mbox{for $M>1$} +\end{array} + \right. . +\end{equation} +% +The subscript $\perp$ signifies the the flow is perpendicular to the +leading edge. + +In the case of a rectangular fin profile the leading edge pressure +drag is equal to the stagnation pressure drag as derived in +equation~\ref{eq-blunt-cylinder-drag} of +Appendix~\ref{app-blunt-cylinder-drag}: +\begin{equation} +(C_{D\bullet})_{LE\perp} = (C_{D\bullet})_{\rm stag} +\end{equation} + +The leading edge pressure drag of a slanted fin is obtained from the +cross-flow principle~\cite[p.~3-11]{hoerner} as +% +\begin{equation} +(C_{D\bullet})_{LE} = (C_{D\bullet})_{LE\perp} \cdot \cos^2\Gamma_L +\end{equation} +% +where $\Gamma_L$ is the leading edge angle. Note that in the equation +both coefficients are relative to the frontal area of the cylinder, so +the ratio of their reference areas is also $\cos\Gamma_L$. In the +case of a free-form fin the angle $\Gamma_L$ is the average leading +edge angle, as described in Section~\ref{sec-average-angle}. + +The fin base drag coefficient of a square profile fin is the same as +the body base drag coefficient in equation~\ref{eq-base-drag}: +% +\begin{equation} +(C_{D\bullet})_{TE} = (C_{D\bullet})_{\rm base} +\end{equation} +% +For fins with rounded edges the value is taken as half of the total +base drag, and for fins with tapering trailing edges the base +drag is assumed to be zero. + +The total fin pressure drag is the sum of the leading and trailing +edge drags +% +\begin{equation} +(C_{D\bullet})_{\rm pressure} = +(C_{D\bullet})_{LE} + (C_{D\bullet})_{TE}. +\end{equation} +% +The reference area is the fin frontal area $N\cdot ts$. + +% TODO: FUTURE: supersonic shock wave drag??? + + + + + +\subsection{Base drag} +\label{sec-base-drag} + +Base drag is caused by a low-pressure area created at the base of the +rocket or in any place where the body radius diminishes rapidly +enough. The magnitude of the base drag can be estimated using the +empirical formula~\cite[p.~23]{fleeman} +% +\begin{equation} +(C_{D\bullet})_{\rm base} = + \left\{ +\begin{array}{ll} + 0.12+0.13M^2, & \mbox{if $M<1$} \\ + 0.25/M, & \mbox{if $M>1$} +\end{array} + \right. . +\label{eq-base-drag} +\end{equation} +% +The base drag is disrupted when a motor exhausts into the area. A +full examination of the process would need much more detailed +information about the motor and would be unnecessarily complicated. A +reasonable approximation is achieved by subtracting the area of the +thrusting motors from the base reference area~\cite[p.~23]{fleeman}. +Thus, if the base is the same size as the motor itself, no base drag +is generated. On the other hand, if the base is large with only a +small motor in the center, the base drag is approximately the same as +when coasting. + +The equation presented above ignores the effect that the rear body +slope angle has on the base pressure. A boattail at the end of the +rocket both diminishes the reference area of base drag, thus reducing +drag, but the slope also directs air better into the low pressure +area. This effect has been neglected as small compared to the effect +of reduced base area. + + + + + + + + + +\subsection{Parasitic drag} + +Parasitic drag refers to drag caused by imperfections and protrusions +on the rocket body. The most significant source of parasitic drag in +model rockets are the launch guides that protrude from the rocket +body. The most common type of launch guide is one or two launch lugs, +which are pieces of tube that hold the rocket on the launch rod during +takeoff. Alternatives to launch lugs include replacing the tube with +metal wire loops or attaching rail pins that hold the rocket on a +launch rail. These three guide types are depicted in +Figure~\ref{fig-launch-guides}. The effect of launch lugs on the +total drag of a model rocket is small, typically in the range of +0--10\%, due to their comparatively small size. However, studying +this effect may be of notable interest for model rocket designers. + + +\begin{figure} +\centering +\epsfig{file=figures/components/launch-guides,width=12cm} +\caption{Three types of common launch guides.} +\label{fig-launch-guides} +\end{figure} + + +A launch lug that is long enough that no appreciable airflow occurs +through the lug may be considered a solid cylinder next to the main +rocket body. A rectangular protrusion that has a length at least +twice its height has a drag coefficient of 0.74, with reference area +being its frontal area~\cite[p.~5-8]{hoerner}. The drag coefficient +varies proportional to the stagnation pressure as in the case of a +blunt cylinder in free airflow, presented in +Appendix~\ref{app-blunt-cylinder-drag}. + +A wire held perpendicular to airflow has instead a drag coefficient of +1.1, where the reference area is the planform area of the +wire~\cite[p.~3-11]{hoerner}. A wire loop may be thought of as a +launch lug with length and wall thickness equal to the thickness of +the wire. However, in this view of a launch lug the reference area +must not include the inside of the tube, since air is free to flow +within the loop. + +These two cases may be unified by changing the used reference area as +a function of the length of the tube $l$. At the limit $l=0$ the +reference area is the simple planform area of the loop, and when the +length is greater than the diameter $l>d$ the reference area includes +the inside of the tube as well. The slightly larger drag coefficient +of the wire may be taken into account as a multiplier to the blunt +cylinder drag coefficient. + +Therefore the drag coefficient of a launch guide can be approximately +calculated by +% +\begin{equation} +(C_{D\bullet})_{\rm parasitic} = +\max\{1.3-0.3\;l/d, 1\} \cdot (C_{D\bullet})_{\rm stag} +\end{equation} +% +where $(C_{D\bullet})_{\rm stag}$ is the stagnation pressure +coefficient calculated in equation~(\ref{eq-blunt-cylinder-drag}), and +the reference area is +% +\begin{equation} +A_{\rm parasitic} = \pi r_{ext}^2 - \pi r_{int}^2 \cdot +\max\{1-l/d,0\}. +\end{equation} + +This approximation may also be used to estimate the drag of rail +pins. A circular pin protruding from a wall has a drag coefficient of +0.80~\cite[p.~5-8]{hoerner}. Therefore the drag of the pin is +approximately equal to that of a lug with the same frontal area. The +rail pins can be approximated in a natural manner as launch lugs with +the same frontal area as the pin and a length equal to their +diameter. + + + + +\subsection{Axial drag coefficient} +\label{sec-axial-drag} + +The total drag coefficient may be calculated by simply scaling the +coefficients to a common reference area and adding them together: +% +\begin{equation} +C_{D_0} = \sum_T \frac{A_T}{\Aref}(C_{D\bullet})_T + + (C_D)_{\rm friction} +\end{equation} +% +where the sum includes the pressure, base and parasitic drags. The +friction drag was scaled to the reference area \Aref\ already in +equation~(\ref{eq-friction-drag-scale}). + +This yields the total drag coefficient at zero angle of attack. At an +angle of attack the several phenomena begin to affect the drag. +More frontal area is visible to the airflow, the pressure gradients +along the body change and fin-tip vortices emerge. On the other hand, +the drag force is no longer axial, so the axial drag force is less +than the total drag force. + +Based on experimental data an empirical formula was produced for +calculating the axial drag coefficient at an angle of attach $\alpha$ +from the zero-angle drag coefficient. The scaling function is a +two-part polynomial function that starts from 1 at $\alpha=0^\circ$, +increases to 1.3 at $\alpha=17^\circ$ and then decreases to zero at +$\alpha=90^\circ$; the derivative is also zero at these points. Since +the majority of the simulated flight is at very small angles of +attack, this approximation provides a sufficiently accurate estimate +for the purposes of this thesis. + + diff --git a/doc/techdoc/chapter-appendices.tex b/doc/techdoc/chapter-appendices.tex new file mode 100644 index 00000000..c8abf7df --- /dev/null +++ b/doc/techdoc/chapter-appendices.tex @@ -0,0 +1,769 @@ + + +\chapter{Nose cone and transition geometries} +\label{app-nosecone-geometry} + +Model rocket nose cones are available in a wide variety of shapes and +sizes. In this appendix the most common shapes and their defining +parameters are presented. + + +\section{Conical} + +The most simple nose cone shape is a right circular cone. They are +easy to make from a round piece of cardboard. A conical nose cone is +defined simply by its length and base diameter. An additional +parameter is the opening angle $\phi$, shown in +Figure~\ref{fig-nosecone-shapes}(a). The defining equation of a +conical nose cone is +% +\begin{equation} +r(x) = \frac{x}{L}\cdot R. +\end{equation} + + +\section{Ogival} + +Ogive nose cones have a profile which is an arc of a circle, as shown +in Figure~\ref{fig-nosecone-shapes}(b). The most common ogive shape +is the {\it tangent ogive} nose cone, which is formed when radius of +curvature of the circle $\rho_t$ is selected such that the joint +between the nose cone and body tube is smooth, +% +\begin{equation} +\rho_t = \frac{R^2+L^2}{2R}. +\end{equation} +% +If the radius of curvature $\rho$ is greater than this, then the +resulting nose cone has an angle at the joint between the nose cone +and body tube, and is called a {\it secant ogive}. The secant ogives +can also be viewed as a larger tangent ogive with its base cropped. +At the limit $\rho\rightarrow\infty$ the secant ogive becomes a +conical nose cone. + +The parameter value $\kappa$ used for ogive nose cones is the ratio of +the radius of curvature of a corresponding tangent ogive $\rho_t$ to +the radius of curvature of the nose cone $\rho$: +% +\begin{equation} +\kappa = \frac{\rho_t}{\rho} +\end{equation} +% +$\kappa$ takes values from zero to one, where $\kappa=1$ produces a +tangent ogive and $\kappa=0$ produces a conical nose cone (infinite +radius of curvature). + +With a given length $L$, radius $R$ and parameter $\kappa$ the radius +of curvature is computed by +% +\begin{equation} +\rho^2 = \frac{ \del{L^2+R^2}\cdot + \del{ \del{\del{2-\kappa}L}^2 + \del{\kappa R}^2 } +}{ 4\del{\kappa R}^2 }. +\end{equation} +% +Using this the radius at position $x$ can be computed as +% +\begin{equation} +r(x) = \sqrt{\rho^2 - \del{L/\kappa - x}^2} - \sqrt{\rho^2-\del{L/\kappa}^2} +\end{equation} + + + +\section{Elliptical} + +Elliptical nose cones have the shape of an ellipsoid with one major +radius is $L$ and the other two $R$. The profile has a shape of a +half-ellipse with major axis $L$ and $R$, +Figure~\ref{fig-nosecone-shapes}(c). It is a simple geometric shape +common in model rocketry. The special case $R=L$ corresponds to a +half-sphere. + +The equation for an elliptical nose cone is obtained by stretching the +equation of a unit circle: +\begin{equation} +r(x) = R \cdot \sqrt{1-\del{1-\frac{x}{L}}^2} +\end{equation} + + +\section{Parabolic series} + +A parabolic nose cone is the shape generated by rotating a section of +a parabola around a line perpendicular to its symmetry axis, +Figure~\ref{fig-nosecone-shapes}(d). This is +distinct from a paraboloid, which is rotated around this symmetry +axis (see Appendix~\ref{app-power-series}). + +Similar to secant ogives, the base of a ``full'' parabolic nose cone +can be cropped to produce nose cones which are not tangent with the +body tube. The parameter $\kappa$ describes the portion of the larger +nose cone to include, with values ranging from zero to one. The most +common values are $\kappa=0$ for a conical nose cone, $\kappa=0.5$ for +a 1/2~parabola, $\kappa=0.75$ for a 3/4~parabola and $\kappa=1$ for a +full parabola. The equation of the shape is +% +\begin{equation} +r(x) = R\cdot\frac{x}{L} \del{ \frac{2 - \kappa\frac{x}{L}} + {2-\kappa}}. +\end{equation} + + + +\section{Power series} +\label{app-power-series} + +The power series nose cones are generated by rotating the segment +% +\begin{equation} +r(x) = R\del{\frac{x}{L}}^\kappa +\end{equation} +% +around the x-axis, Figure~\ref{fig-nosecone-shapes}(e). The parameter +value $\kappa$ can range from zero to one. Special cases are +$\kappa=1$ for a conical nose cone, $\kappa=0.75$ for a 3/4~power nose +cone and $\kappa=0.5$ for a 1/2~power nose cone or an ellipsoid. +The limit $\kappa\rightarrow0$ forms a blunt cylinder. + + +\section{Haack series} + +In contrast to the other shapes which are formed from rotating +geometric shapes or simple formulae around an axis, the Haack series +nose cones are mathematically derived to minimize the theoretical +pressure drag. Even though they are defined as a series, two specific +shapes are primarily used, the {\it LV-Haack}\ shape and the +{\it LD-Haack}\ or {\it Von Kárman}\ shape. The letters LV and LD +refer to length-volume and length-diameter, and they minimize the +theoretical pressure drag of the nose cone for a specific length and +volume or length and diameter, respectively. Since the parameters +defining the dimensions of the nose cone are its length and radius, +the Von Kárman nose cone (Figure~\ref{fig-nosecone-shapes}(f)) should, +in principle, be the optimal nose cone shape. + +The equation for the series is +% +\begin{equation} +r(x) = \frac{R}{\sqrt{\pi}} \; + \sqrt{\theta - \frac{1}{2}\sin(2\theta) + \kappa \sin^3\theta} +\end{equation} +% +where +% +\begin{equation} +\theta = \cos^{-1} \del{1-\frac{2x}{L}}. +\end{equation} +% +The parameter value $\kappa=0$ produces the Von Kárman of LD-Haack +shape and $\kappa=1/3$ produces the LV-Haack shape. In principle, +values of $\kappa$ up to $2/3$ produce monotonic nose cone shapes. +However, since there is no experimental data available for the +estimation of nose cone pressure drag for $\kappa > 1/3$ (see +Appendix~\ref{app-haack-series-pressure-drag}), the selection of the +parameter value is limited in the software to the range +$0 \ldots 1/3$. + + +\section{Transitions} + +The vast majority of all model rocket transitions are conical. +However, all of the nose cone shapes may be adapted as transition +shapes as well. The transitions are parametrized with the fore and +aft radii $R_1$ and $R_2$, length $L$ and the optional shape parameter +$\kappa$. + +Two choices exist when adapting the nose cones as transition shapes. +One is to take a nose cone with base radius $R_2$ and crop the tip of +the nose at the radius position $R_1$. The length of the nose cone +must be selected suitably that the length of the transition is $L$. +Another choice is to have the profile of the transition resemble two +nose cones with base radius $R_2-R_1$ and length $L$. These two +adaptations are called {\it clipped} and {\it non-clipped} transitions, +respectively. A clipped and non-clipped elliptical transition is +depicted in Figure~\ref{fig-transition-clip}. + +For some transition shapes the clipped and non-clipped adaptations are +the same. For example, the two possible ogive transitions have equal +radii of curvature and are therefore the same. Specifically, the +conical and ogival transitions are equal whether clipped or not, and +the parabolic series are extremely close to each other. + + + + +\begin{figure}[p] +\centering +\begin{tabular}{ccc} + +\epsfig{file=figures/nose-geometry/geometry-conical,scale=0.7} && +\epsfig{file=figures/nose-geometry/geometry-ogive,scale=0.7} \\ +(a) & \hspace{1cm} & (b) \\ +&& \\ +&& \\ +&& \\ +%&& \\ + +\epsfig{file=figures/nose-geometry/geometry-elliptical,scale=0.7} && +\epsfig{file=figures/nose-geometry/geometry-parabolic,scale=1.0} \\ +(c) && (d) \\ +&& \\ +&& \\ +&& \\ +%&& \\ + +\epsfig{file=figures/nose-geometry/geometry-power,scale=0.6} && +\epsfig{file=figures/nose-geometry/geometry-haack,scale=0.6} \\ +(e) && (f) \\ +&& \\ +%&& \\ +\end{tabular} +\caption{Various nose cone geometries: (a)~conical, (b)~secant ogive, + (c)~elliptical, (d)~parabolic, (e)~1/2~power (ellipsoid) and + (f)~Haack series (Von Kárman).} +\label{fig-nosecone-shapes} +\end{figure} + +\begin{figure}[p] +\vspace{5mm} +\begin{center} +\epsfig{file=figures/nose-geometry/geometry-transition,scale=0.7} +\end{center} +\caption{A clipped and non-clipped elliptical transition.} +\label{fig-transition-clip} +\end{figure} + + + + + +\chapter{Transonic wave drag of nose cones} +\label{app-nosecone-drag-method} + +The wave drag of different types of nose cones vary largely in the +transonic velocity region. Each cone shape has its distinct +properties. In this appendix methods for calculating and +interpolating the drag of various nose cone shapes at transonic and +supersonic speeds are presented. A summary of the methods is +presented in Appendix~\ref{app-transonic-nosecone-summary}. + + + +\section{Blunt cylinder} +\label{app-blunt-cylinder-drag} + +A blunt cylinder is the limiting case for every nose cone shape at the +limit $f_N\rightarrow 0$. Therefore it is useful to have a formula +for the front pressure drag of a circular cylinder in longitudinal +flow. As the object is not streamline, its drag coefficient does not +vary according to the Prandtl factor~(\ref{eq-prandtl-factor}). +Instead, the coefficient is approximately proportional to the +{\it stagnation pressure}, or the pressure at areas perpendicular to +the airflow. The stagnation pressure can be approximated by the +function~\cite[pp.~15-2,~16-3]{hoerner} +% +\begin{equation} +\frac{q_{\rm stag}}{q} = +\left\{ +\begin{array}{ll} +1 + \frac{M^2}{4} + \frac{M^4}{40}, & \mbox{for\ } M < 1 \\ +1.84 - \frac{0.76}{M^2} + \frac{0.166}{M^4} + \frac{0.035}{M^6}, & + \mbox{for\ } M > 1 +\end{array} +\right. . +\end{equation} +% +The pressure drag coefficient of a blunt circular cylinder as a +function of the Mach number can then be written as +% +\begin{equation} +(C_{D\bullet})_{\rm pressure} = +(C_{D\bullet})_{\rm stag} = 0.85 \cdot \frac{q_{\rm stag}}{q}. +\label{eq-blunt-cylinder-drag} +\end{equation} + + + + + +\section{Conical nose cone} + +A conical nose cone is simple to construct and closely resembles many +slender nose cones. The conical shape is also the limit of several +parametrized nose cone shapes, in particular the secant ogive with +parameter value 0.0 (infinite circle radius), the power series nose +cone with parameter value 1.0 and the parabolic series with parameter +value 0.0. + +Much experimental data is available on the wave drag of conical nose +cones. Hoerner presents formulae for the value of $C_{D\bullet}$ at +supersonic speeds, the derivative $\dif C_{D\bullet}/\dif M$ at +$M=1$, and a figure of $C_{D\bullet}$ at +$M=1$~\cite[pp.~16-18\ldots16-20]{hoerner}. Based on these and +the low subsonic drag coefficient~(\ref{eq-nosecone-pressure-drag}), a +good interpolation of the transonic region is possible. + +The equations presented by Hoerner are given as a function of the +half-apex angle $\varepsilon$, that is, the angle between the conical +body and the body centerline. The half-apex angle is related to the +nose cone fineness ratio by +% +\begin{equation} +\tan\varepsilon = \frac{d/2}{l} = \frac{1}{2f_N}. +\end{equation} + +The pressure drag coefficient at supersonic speeds ($M\gtrsim1.3$) is +given by +% +\begin{align} +(C_{D\bullet})_{\rm pressure} +& = 2.1\;\sin^2\varepsilon + 0.5\; + \frac{\sin\varepsilon}{\sqrt{M^2-1}} \nonumber\\ +& = \frac{2.1}{1+4f_N^2} + \frac{0.5}{\sqrt{(1+4f_N^2)\; (M^2-1)}} . +\label{eq-conical-supersonic-drag} +\end{align} +% +It is worth noting that as the Mach number increases, the drag +coefficient tends to the constant value $2.1\sin^2\epsilon$. At $M=1$ +the slope of the pressure drag coefficient is equal to +% +\begin{equation} +\eval{\frac{\partial (C_{D\bullet})_{\rm pressure}}{\partial M}}_{M=1} = + \frac{4}{\gamma+1} \cdot (1-0.5\;C_{D\bullet,M=1}) +\label{eq-conical-sonic-drag-derivative} +\end{equation} +% +where $\gamma=1.4$ is the specific heat ratio of air and the drag +coefficient at $M=1$ is approximately +% +\begin{equation} +C_{D\bullet,M=1} = 1.0\; \sin\varepsilon. +\label{eq-conical-sonic-drag} +\end{equation} + +The pressure drag coefficient between Mach~0 and Mach~1 is +interpolated using equation~(\ref{eq-nosecone-pressure-drag}). +Between Mach~1 and Mach~1.3 the coefficient is calculated using +polynomial interpolation with the boundary conditions from +equations~(\ref{eq-conical-supersonic-drag}), +(\ref{eq-conical-sonic-drag-derivative}) and +(\ref{eq-conical-sonic-drag}). + + + + + + +\section{Ellipsoidal, power, parabolic and Haack series nose cones} +\label{app-haack-series-pressure-drag} + +A comprehensive data set of the pressure drag coefficient for all nose +cone shapes at all fineness ratios at all Mach numbers is not +available. However, Stoney has collected a compendium of nose cone +drag data including data on the effect of the fineness ratio $f_N$ on +the drag coefficient and an extensive study of drag coefficients of +different nose cone shapes at fineness ratio +3~\cite{nosecone-cd-data}. The same report suggests that the effects +of fineness ratio and Mach number may be separated. + +The curves of the pressure drag coefficient as a function of the nose +fineness ratio $f_N$ can be closely fitted with a function of the form +% +\begin{equation} +(C_{D\bullet})_{\rm pressure} = \frac{a}{(f_N + 1)^b}. +\label{eq-fineness-ratio-drag-interpolator} +\end{equation} +% +The parameters $a$ and $b$ can be calculated from two data points +corresponding to fineness ratios 0 (blunt cylinder, +Appendix~\ref{app-blunt-cylinder-drag}) and ratio 3. Stoney includes +experimental data of the pressure drag coefficient as a function of +Mach number at fineness ratio 3 for power series $x^{1/4}$, $x^{1/2}$, +$x^{3/4}$ shapes, $1/2$, $3/4$ and full parabolic shapes, ellipsoidal, +L-V~Haack and Von Kárman nose cones. These curves are written into +the software as data curve points. For parametrized nose cone shapes +the necessary curve is interpolated if +necessary. Typical nose cones of model rockets have fineness ratios +in the region of 2--5, so the extrapolation from data of fineness +ratio 3 is within reasonable bounds. + + + + + +\section{Ogive nose cones} + +One notable shape missing from the data in Stoney's report are secant +and tangent ogives. These are common shapes for model rocket nose +cones. However, no similar experimental data of the pressure drag as +a function of Mach number was found for ogive nose cones. + +At supersonic velocities the drag of a tangent ogive is approximately +the same as the drag of a conical nose cone with the same length and +diameter, while secant ogives have a somewhat smaller +drag~\cite[p.~239]{handbook-supersonic-aerodynamics}. The minimum +drag is achieved when the secant ogive radius is approximately twice +that of a corresponding tangent ogive, corresponding to the parameter +value 0.5. The minimum drag is consistently 18\% less than that of a +conical nose at Mach numbers in the range of 1.6--2.5 and for fineness +ratios of 2.0--3.5. Since no better transonic data is available, it +is assumed that ogives follow the conical drag profile through +the transonic and supersonic region. The drag of the corresponding +conical nose is diminished in a parabolic fashion with the ogive +parameter, with a minimum of -18\% at a parameter value of 0.5. + + + + + +\section{Summary of nose cone drag calculation} +\label{app-transonic-nosecone-summary} + +The low subsonic pressure drag of nose cones is calculated using +equation~(\ref{eq-nosecone-pressure-drag}): +% +\begin{equation*} +(C_{D\bullet,M=0})_p = 0.8 \cdot \sin^2\phi. +\end{equation*} +% +The high subsonic region is interpolated using a function of the form +presented in equation~(\ref{eq-nosecone-pressure-interpolator}): +% +\begin{equation*} +(C_{D\bullet})_{\rm pressure} = a\cdot M^b + (C_{D\bullet,M=0})_p +\end{equation*} +% +where $a$ and $b$ are selected according to the lower boundary of the +transonic pressure drag and its derivative. + +The transonic and supersonic pressure drag is calculated depending on +the nose cone shape as follows: +% +\begin{itemize} + +\item[\bf Conical:] At supersonic velocities ($M > 1.3$) the + pressure drag is calculated using + equation~(\ref{eq-conical-supersonic-drag}). Between Mach 1 and 1.3 + the drag is interpolated using a polynomial with boundary conditions + given by equations~(\ref{eq-conical-supersonic-drag}), + (\ref{eq-conical-sonic-drag-derivative}) and + (\ref{eq-conical-sonic-drag}). +\\ + + + +\item[\bf Ogival:] The pressure drag at transonic and supersonic + velocities is equal to the pressure drag of a conical nose cone with + the same diameter and length corrected with a shape factor: +%, multiplied by the shape factor +% +\begin{equation} +(C_{D\bullet})_{\rm pressure} = +\del{0.72 \cdot (\kappa - 0.5)^2 + 0.82} \cdot +(C_{D\bullet})_{\rm cone}. +\end{equation} +% +The shape factor is one at $\kappa = 0, 1$ and 0.82 at $\kappa=0.5$. +\\ + + + +\item[\bf Other shapes:] The pressure drag calculation is based on + experimental data curves: +% +\begin{enumerate} +\item Determine the pressure drag $C_3$ of a similar nose cone + with fineness ratio $f_N=3$ from experimental data. If data for a + particular shape parameter is not available, interpolate the data + between parameter values. + +\item Calculate the pressure drag of a blunt cylinder $C_0$ + using equation~(\ref{eq-blunt-cylinder-drag}). + +\item Interpolate the pressure drag of the nose cone using + equation~(\ref{eq-fineness-ratio-drag-interpolator}). + After parameter substitution the equation takes the form +% +\begin{equation} +(C_{D\bullet})_{\rm pressure} \;=\; +\frac{C_0}{(f_N+1)^{\log_4 C_0/C_3}} \;=\; +C_0 \cdot \del{\frac{C_3}{C_0}}^{\log_4(f_N+1)} +\end{equation} +% + The last form is computationally more efficient since the exponent + $\log_4(f_N+1)$ is constant during a simulation. + +\end{enumerate} + +\end{itemize} + + + + + + + + + + + +\chapter{Streamer drag coefficient estimation} +\label{app-streamers} + + +A streamer is a typically rectangular strip of plastic or other +material that is used as a recovery device especially in small model +rockets. The deceleration is based on the material flapping in the +passing air, thus causing air resistance. Streamer optimization has +been a subject of much interest in the rocketry +community~\cite{streamer-optimization}, and contests on streamer +landing duration are organized regularly. In order to estimate the +drag force of a streamer a series of experiments were performed and an +empirical formula for the drag coefficient was developed. + +One aspect that is not taken into account in the present investigation +is the fluctuation between the streamer and rocket. At one extreme a +rocket with a very small streamer drops head first to the ground with +almost no deceleration at all. At the other extreme there is a very +large streamer creating significant drag, and the rocket falls below +it tail-first. Between these two extremes is a point where the +orientation is labile, and the rocket as a whole twirls +around during descent. This kind of interaction between the rocket +and streamer cannot be investigated in a wind tunnel and would require +an extensive set of flight tests to measure. Therefore it is not +taken into account, instead, the rocket is considered effectively a +point mass at the end of the streamer, the second extreme mentioned +above. + + +\subsubsection*{Experimental methods} + +A series of experiments to measure the drag coefficients of streamers +was performed using the $40\times40\times120$~cm wind tunnel of +Pollux~\cite{pollux-wind-tunnel}. The experiments were performed +using various materials, widths and lengths of streamers and at +different wind speeds. The effect of the streamer size and shape was +tested separately from the effect of the streamer material. + +A tube with a rounded $90^\circ$ angle at one end was installed in the +center of the wind tunnel test section. A line was drawn through the +tube so that one end of the line was attached to a the streamer and the +other end to a weight which was placed on a digital scale. When the +wind tunnel was active the force produced by the streamer was read +from the scale. A metal wire was taped to the edge of the streamer to +keep it rigid and the line attached to the midpoint of the wire. + +A few different positions within the test section and free line +lengths were tried. All positions seemed to produce approximately +equal results, but the variability was significantly lower when the +streamer fit totally into the test section and had a only 10~cm length +of free line between the tube and streamer. This configuration was +used for the rest of the experiments. + +Each streamer was measured at three different velocities, 6~m/s, 9~m/s +and 12~m/s. The results indicated that the force produced is +approximately proportional to the square of the airspeed, signifying +that the definition of a drag coefficient is valid also for streamers. + +The natural reference area for a streamer is the area of the strip. +However, since in the simulation we are interested in the total drag +produced by a streamer, it is better to first produce an equation for +the drag coefficient normalized to unit area, $C_D \cdot \Aref$. +These coefficient values were calculated separately for the different +velocities and then averaged to obtain the final normalized drag +coefficient of the streamer. + + +\subsubsection*{Effect of streamer shape} + +\begin{figure}[p] +\centering +\hspace*{-7mm} +\epsfig{file=figures/experimental/streamerCDvsWL,width=155mm} +\caption{The normalized drag coefficient of a streamer as a function + of the width and length of the streamer. The points are the + measured values and the mesh is cubically interpolated between the + points.} +\label{fig-streamer-CD-vs-shape} +\end{figure} + + +\begin{figure}[p] +\centering +\hspace*{-7mm} +\epsfig{file=figures/experimental/streamerCDvsWLestimate,width=155mm} +\caption{Estimated and measured normalized drag coefficients of a + streamer as a function of the width and length of the streamer. The + lines from the points lead to their respective estimate values.} +\label{fig-streamer-shape-estimate} +\end{figure} + + + +Figure~\ref{fig-streamer-CD-vs-shape} presents the normalized drag +coefficient as a function of the streamer width and length for a fixed +material of $\rm80~g/m^2$ polyethylene plastic. It was noticed that +for a specific streamer length, the normalized drag coefficient was +approximately linear with the width, +% +\begin{equation} +C_D \cdot \Aref = k\cdot w, +\label{eq-streamer-first-approx} +\end{equation} +% +where $w$ is the width and $k$ is dependent on the streamer length. +The slope $k$ was found to be approximately linear with +the length of the streamer, with a linear regression of +% +\begin{equation} +k = 0.034 \cdot (l+\rm 1~m). +\label{eq-streamer-second-approx} +\end{equation} +% +Substituting equation (\ref{eq-streamer-second-approx}) into +(\ref{eq-streamer-first-approx}) yields +% +\begin{equation} +C_D \cdot \Aref = 0.034 \cdot (l+{\rm 1~m})\cdot w +\label{eq-streamer-estimate} +\end{equation} +% +or using $\Aref = wl$ +% +\begin{equation} +C_D = 0.034 \cdot \frac{l+\rm 1~m}{l}. +\label{eq-streamer-shape-estimate} +\end{equation} + + +The estimate as a function of the width and length is presented in +Figure~\ref{fig-streamer-shape-estimate} along with the measured data +points. The lines originating from the points lead to their +respective estimate values. The average relative error produced by +the estimate was 9.7\%. + + +\subsubsection*{Effect of streamer material} + + + +The effect of the streamer material was studied by creating +$4\times40$~cm and $8\times48$~cm streamers from various household +materials commonly used in streamers. The tested materials were +polyethylene plastic of various thicknesses, cellophane and crêpe +paper. The properties of the materials are listed in +Table~\ref{table-streamer-materials}. + + +Figure~\ref{fig-streamer-material} presents the normalized drag +coefficient as a function of the material thickness and surface +density. It is evident that the thickness is not a good +parameter to characterize the drag of a streamer. On the other hand, +the drag coefficient as a function of surface density is nearly +linear, even including the crêpe paper. While it is not as +definitive, both lines seem to intersect with the $x$-axis at +approximately $\rm-25~g/m^2$. Therefore the coefficient of the +$\rm80~g/m^2$ polyethylene estimated by +equation~(\ref{eq-streamer-shape-estimate}) is corrected for a +material surface density $\rho_m$ with +% +\begin{equation} +C_{D_m} = \left(\frac{\rho_m + \rm 25~g/m^2}{\rm 105~g/m^2}\right) + \cdot C_D. +\end{equation} +% +Combining these two equations, one obtains the final empirical +equation +% +\begin{equation} +C_{D_m} = 0.034 \cdot + \left(\frac{\rho_m + \rm 25~g/m^2}{\rm 105~g/m^2}\right) \cdot + \left(\frac{l + 1~{\rm m}}{l}\right). +\label{eq-streamer-CD-estimate} +\end{equation} + +This equation is also reasonable since it produces positive and finite +normalized drag coefficients for all values of $w$, $l$ and $\rho_m$. +However, this equation does not obey the rule-of-thumb of rocketeers +that the optimum width-to-length ratio for a streamer would be 1:10. +According to equation~(\ref{eq-streamer-estimate}), the maximum drag +for a fixed surface area is obtained at the limit $l\rightarrow0$, +$w\rightarrow\infty$. In practice the rocket dimensions limit the +practical dimensions of a streamer, from which the 1:10 rule-of-thumb +may arise. + + +\subsubsection*{Equation validation} + +To test the validity of the equation, several additional streamers +were measured for their drag coefficients. These were of various +materials and of dimensions that were not used in the fitting of the +empirical formulae. These can therefore be used as an independent +test set for validating equation~(\ref{eq-streamer-CD-estimate}). + +Table~\ref{table-streamer-validation} presents the tested streamers +and their measured and estimated normalized drag coefficients. The +results show relative errors in the range of 12--27\%. While rather +high, they are considered a good result for estimating such a random +and dynamic process as a streamer. Furthermore, due to the +proportionality to the square of the velocity, a 25\% error in the +normalized force coefficient translates to a 10--15\% error in the +rocket's descent velocity. This still allows the rocket designer to +get a good estimate on how fast a rocket will descend with a +particular streamer. + + +\begin{figure}[p] +\centering +\parbox{70mm}{\centering +\epsfig{file=figures/experimental/streamerCDvsThickness2,width=70mm} +\\ (a) +}\parbox{70mm}{\centering +\epsfig{file=figures/experimental/streamerCDvsDensity2,width=70mm} \\ (b)} +\caption{The normalized drag coefficient of a streamer as a function + of (a) the material thickness and (b) the material surface density.} +\label{fig-streamer-material} +\end{figure} + + + +\begin{table}[p] +\caption{Properties of the streamer materials experimented with.} +\label{table-streamer-materials} +\begin{center} +\begin{tabular}{ccc} +\hline +Material & Thickness / \um & Density / $\rm g/m^2$ \\ +\hline +Polyethylene & 21 & 19 \\ +Polyethylene & 22 & 10 \\ +Polyethylene & 42 & 41 \\ +Polyethylene & 86 & 80 \\ +Cellophane & 20 & 18 \\ +Crêpe paper & 110$\dagger$ & 24 \\ +\hline +\end{tabular} \\ +{\footnotesize $\dagger$ Dependent on the amount of pressure applied.} +\end{center} +\end{table} + + + +\begin{table}[p] +\caption{Streamers used in validation and their results.} +\label{table-streamer-validation} +\begin{center} +\begin{tabular}{ccccccc} +\hline +Material & Width & Length & Density & Measured & Estimate & Error \\ + & m & m & $\rm g/m^2$ & + \multicolumn{2}{c}{$10^{-3} (C_D\cdot\Aref)$} & \\ +\hline +Polyethylene & 0.07 & 0.21 & 21 & 0.99 & 1.26 & 27\% \\ +Polyethylene & 0.07 & 0.49 & 41 & 1.81 & 2.23 & 23\% \\ +Polyethylene & 0.08 & 0.24 & 10 & 0.89 & 1.12 & 26\% \\ +Cellophane & 0.06 & 0.70 & 20 & 1.78 & 1.49 & 17\% \\ +Crêpe paper & 0.06 & 0.50 & 24 & 1.27 & 1.43 & 12\% \\ +\hline +\end{tabular} +\end{center} +\end{table} + diff --git a/doc/techdoc/chapter-basics-of-model-rockets.tex b/doc/techdoc/chapter-basics-of-model-rockets.tex new file mode 100644 index 00000000..60b60acf --- /dev/null +++ b/doc/techdoc/chapter-basics-of-model-rockets.tex @@ -0,0 +1,360 @@ + + +\chapter{Basics of model rocket flight} +\label{chap-basics} + + +As rockets and rocket motors come in a huge variety of shapes and +sizes, different categories are defined for different levels of +rocketry. {\it Model rocketry} itself is governed by the NAR Model +Rocket Safety Code~\cite{nar-safety-code} in the U.S. and other +similar regulations in other countries. The safety code requires that +the model rockets be constructed only of light-weight materials +without any metal structural parts, and have a maximum lift-off weight +of 1.5~kg. They may only used pre-manufactured motors of classes A--G +(see Section~\ref{sec-motor-classes} for the classification). + +{\it High power rocketry} (HPR) is basically scaled up model +rocketry. There are no weight restrictions, and they can use +pre-manufactured solid or hybrid rocket motors in the range of H--O. +The combined total impulse of all motors may not exceed 81\s920~Ns. + +{\it Experimental} or {\it amateur rocketry} includes any rocketry +activities beyond model and high power rocketry. This may include +for example using motor combinations that exceed the limits placed by +high power rocketry, building self-made motors or utilizing liquid +fueled motors. Finally there is {\it professional rocketry} which is +conducted for profit, usually by governments or large corporations. + +Even though rockets come in many different sizes, the same principles +apply to all of them. In this thesis the emphasis will be on model +rocketry, but the results are just as valid for larger rockets as long +as the assumptions of for example the speed range remain valid. In +this chapter the basics of model rocketry and differences to high +power rocketry are explained. + + +\section{Model rocket flight} + +A typical flight of a model rocket can be characterized by the four +phases depicted in Figure~\ref{fig-model-flight}: +% +\begin{enumerate} +\item Launch: The model rocket is launched from a vertical launch + guide. +\item Powered flight: The motor accelerates the rocket during the + powered flight period. +\item Coasting flight: The rocket coasts freely until approximately + at its apogee. +\item Recovery: The recovery device opens and the rocket descends + slowly to the ground. +\end{enumerate} + +\begin{figure} +\centering +\epsfig{file=figures/model-flight,scale=0.8} +\caption{The basic phases of a typical model rocket flight: + 1.~Launch, 2.~Powered flight, 3.~Coasting and 4.~Recovery.} +\label{fig-model-flight} +\end{figure} + +Model rockets are launched from a vertical launch guide that keeps the +rocket in an upright position until it has sufficient velocity for the +fins to aerodynamically stabilize the flight. The NAR safety code +forbids launching a model rocket at an angle greater than +$30^\circ$ from vertical. A typical launch guide for small rockets is +a metal rod about 3-5~mm in diameter, and the launch lug is a short +piece of plastic tube glued to the body tube. Especially in +larger rockets this may be replaced by two extruding bolts, the ends +of which slide along a metal rail. Use of a launch lug can be avoided +by a tower launcher, which has 3--4 metal bars around the rocket +that hold it in an upright position. + +After clearing the launch guide, the rocket is in free, powered flight. +During this phase the motor accelerates the rocket while it is +aerodynamically stabilized to keep its vertical orientation. When the +propellant has been used, the rocket is typically at its maximum +velocity. It then coasts freely for a short period while the motor +produces smoke to help follow the rocket, but provides no +additional thrust. Finally, at approximately the point of apogee, a +small pyrotechnical ejection charge is fired upwards from the motor +which pressurizes the model rocket and opens the recovery device. + +High-power rocket motors usually have no ejection charges incorporated +in them. Instead, the rocket carries a small flight computer that +measures the acceleration of the rocket or the outside pressure change +to detect the point of apogee and to open the recovery device. +Frequently only a small drogue parachute is opened at apogee, and the +main parachute is opened at some pre-defined lower altitude around +100--300 meters. + +The typical recovery device of a model rocket is either a parachute or +a {\it streamer}. The parachutes are usually a simple planar circle +of plastic or fabric with 4--10 shroud lines attached. A streamer is +a strip of plastic or fabric connected to the rocket, intended to +flutter in the air and thus slow down the descent of the rocket. +Especially small rockets often use streamers as their recovery device, +since even light wind can cause a light-weight rocket with a +parachute to drift a significant distance. + + + + +\section{Rocket motor classification} +\label{sec-motors} +\label{sec-motor-classes} + +The motors used in model and high power rocketry are categorized based +on their total impulse. A class `A' motor may have a total impulse in +the range of 1.26--2.50~Ns. Every consecutive class doubles the +allowed total impulse of the motor. Thus, a B-motor can have an +impulse in the range 2.51--5.00~Ns and a C-motor in the range +5.01--10.0~Ns. There are also classes \half A and \quarter A which +have impulse ranges half and one quarter of those of an A-motor, +respectively. Commercial rocket motors are available up to +class~O with a total impulse of 30\s000~Ns~\cite{all-certified-motors}. +Table~\ref{tab-motor-classes} lists the impulse ranges for model +and high-power rocket motors. + +\begin{table} +\caption{Total impulse ranges for motor classes \quarter A--O.} +\label{tab-motor-classes} +\begin{center} +\begin{tabular}{cr@{--}l|cr@{--}l|cr@{--}l} +\hline +\quarter A & 0.0 & 0.625~Ns & E & 20.01 & 40.0~Ns & K & 1280.01 & 2560~Ns \\ +\half A & 0.626 & 1.25~Ns & F & 40.01 & 80.0~Ns & L & 2560.01 & 5120~Ns \\ +A & 1.26 & 2.50~Ns & G & 80.01 & 160~Ns & M & 5120.01 & 10240~Ns \\ +B & 2.51 & 5.00~Ns & H & 160.01 & 320~Ns & N & 10240.01 & 20480~Ns \\ +C & 5.01 & 10.0~Ns & I & 320.01 & 640~Ns & O & 20480.01 & 40960~Ns \\ +D & 10.01 & 20.0~Ns & J & 640.01 & 1280~Ns & \\ +\hline +\end{tabular} +\end{center} +\end{table} + +Another important parameter of a rocket motor is the thrust given by +the motor. This defines the mass that may be lifted by the motor and +the acceleration achieved. Small model rocket motors typically have +an average thrust of about 3--10~N, while high-power rocket motors can +have thrusts in excess of 5\s000~N. + +The third parameter used to classify a model rocket motor is the +length of the delay between the motor burnout and the ignition of the +ejection charge. Since the maximum velocity of different rockets +using the same type of motor can be vastly different, also the length +of the coasting phase varies. Therefore motors with otherwise the +same specifications are often manufactured with several different +delay lengths. These delay lengths do not apply to high-power rocket +motors, since they do not have ejections charges incorporated in them. + +Model rocket motors are given a classification code based on these +three parameters, for example ``D7-3''. The letter specifies the +total impulse range of the motor, while the first number specifies the +average thrust in Newtons and the second number the delay of the +ejection charge in seconds. The delay number can also be replaced by +`P', which stands for {\it plugged}, \ie the motor does not have an +ejection charge. Some manufacturers may also use an additional letter +at the end of the classification code specifying the +propellant type used in the motor. + +Even motors with the same classification code may have slight +variations to them. First, the classification only specifies the +impulse range of the motor, not the exact impulse. In principle, a +D-motor in the lower end of the range might have a total impulse only +1~Ns larger than a C-motor in the upper end of its range. Second, +the code only specifies the average thrust of the motor. The thrust +rarely is constant, but varies with time. +Figure~\ref{fig-thrust-curve} shows the typical thrust curve of a +small black powder rocket motor. The motors typically have a short +thrust peak at ignition that gives the rocket an initial acceleration +boost before stabilizing to a thrust level a little below the average +thrust. Statically measured thrust curves of most commercial rocket +motors are readily available on the +Internet~\cite{thrust-curve-database}. + +\begin{figure} +\centering +\epsfig{file=figures/motors/D12-thrustcurve,width=9cm} +\caption{A typical thrust curve of an Estes D12-3 rocket motor and + its average thrust.~\cite{D12-curve}} +\label{fig-thrust-curve} +\end{figure} + +Also the propellant type may affect the characteristics of the motor. +Most model rocket motors are made up of a solid, pyrotechnical +propellant---typically black powder---that is cast into a suitable +shape and ignited on launch. Since the propellant burns on its +surface, different thrust curves can be achieved by different mold +shapes. + +% vesiraketit! + +A significantly different motor type, {\it hybrid motors}, were +commercially introduced in 1995. These motors typically include the +propellant and oxidizer in different states, typically a composite +plastic as the fuel and a separate tank of liquid nitrous oxide +($\rm N_2O$) as the oxidizer. The plastic on its own does not +burn very well, but provides ample thrust when the nitrous oxide is +fed through its core. The nitrous oxide tank is +self-pressurized by its natural vapor pressure. However, since +temperature greatly affects the vapor pressure of nitrous oxide, the +thrust of a hybrid motor is also diminished if the oxidizer is cold. +On the other hand, the motor will burn longer in this case, and since +nitrous oxide is denser when cold, the motor may even yield a greater +total impulse. + +The significance of this effect was observed when analyzing the video +footage of the launch of the first Finnish hybrid rocket, +``Haisunäätä''~\cite{haisunaata-launch}. The average thrust during the +first 0.5~seconds was determined to be only about 70~N, whereas the +static tests suggest the thrust should have been over 200~N. +Instead, the motor burned for over 10~seconds, while the normal thrust +curves indicate a burning time of 5--6~seconds. This shows that the +temperature of the hybrid motor oxidizer can have a dramatic effect on +the thrust given by the motor, and the static test curve should be +assumed to be valid only in similar operating conditions as during the +test. + +One further non-pyrotechnical rocket type is {\it water rockets}. +These are especially popular first rockets, as they require no special +permits and are easy to construct. The water rocket includes a bottle +or other chamber that has water and pressurized air inside it. On +launch the pressure forces the water out of a nozzle, providing thrust +to the rocket. While simulating water rockets is beyond the scope of +this thesis, it is the aim that methods for modeling water rockets can +be added to the produced software in the future. + + + +\section{Clustering and staging} + +Two common methods used to achieve greater altitudes with model +rockets are {\it clustering} and {\it staging}. A cluster has two or +more rocket motors burning concurrently, while staging uses motors +that burn consecutively. The motor configuration of a cluster and +staged rocket is depicted in Figure~\ref{fig-cluster-stages}. + +When a cluster is launched, the total thrust is the sum of the thrust +curves of the separate motors. This allows greater acceleration and +a greater liftoff weight. Staging is usually performed by using +zero-delay motors, that ignite the ejection charge immediately at +burnout. The ejection charge fires towards the upper stage motor and +ignites the next motor. High power motors with no ejection charges +can be clustered by using an onboard accelerometer or timer that +ignites the subsequent stages. Staging provides a longer duration of +powered flight, thus increasing the altitude. + + +\begin{figure} +\centering +\parbox{65mm}{\centering +\epsfig{file=figures/motors/cluster,width=60mm} \\ (a)} +\hspace{10mm} +\parbox{40mm}{\centering +\epsfig{file=figures/motors/staged,width=30mm} \\ (b)} +\caption{The motor configuration for (a) a cluster rocket and (b) a + two-staged rocket.} +\label{fig-cluster-stages} +\end{figure} + +Clustering provides a greater acceleration at launch, but staging +typically provides greater altitude than a cluster with similar +motors. This is because a clustered rocket accelerates quickly to a +greater speed thus also increasing the aerodynamic drag. A staged +rocket has a smaller thrust for a longer period of time, which reduces +the overall effect of drag during the flight. + + +\section{Stability of a rocket} +\label{sec-stability} + +When designing a new rocket, its stability is of paramount +importance. A small gust of wind or some other disturbance may cause +the rocket to tilt slightly from its current orientation. When this +occurs, the rocket centerline is no longer parallel to the +velocity of the rocket. This condition is called flying at an +{\it angle of attack $\alpha$}, where $\alpha$ is the angle between +the rocket centerline and the velocity vector. + +When a stable rocket flies at an angle of attack, its fins produce a +moment to correct the rocket's flight. The corrective moment is +produced by the aerodynamic forces perpendicular to the axis of the +rocket. Each component of the rocket can be seen as producing a +separate normal force component originating from the component's CP, +as depicted in Figure~\ref{fig-normal-forces}. + +\begin{figure} +\centering +\epsfig{file=figures/aerodynamics/component-normal-forces,width=130mm} +\caption{Normal forces produced by the rocket components.} +\label{fig-normal-forces} +\end{figure} + +The effect of the separate normal forces can be combined into a single +force, the magnitude of which is the sum of the separate forces and +which effects the same moment as the separate forces. The point on +which the total force acts is defined as the center of pressure or the +rocket. As +can be seen from Figure~\ref{fig-normal-forces}, the moment produced +attempts to correct the rocket's flight only if the CP is located aft +of the CG. If this condition holds, the rocket is said to be +{\it statically stable}. A statically stable rocket always produces a +corrective moment when flying at a small angle of attack. + +The argument for static stability above may fail in two conditions: +First, the normal forces might cancel each other out exactly, in which +case a moment would be produced but with zero total force. Second, +the normal force at the CP might be in the wrong direction (downward +in the figure), yielding an uncorrective moment. However, we shall +see that the only component to produce a downward force is a boattail, +and the force is equivalent to the corresponding broadening of the +body. Therefore the total force acting on the rocket cannot be zero +nor in a direction to produce an uncorrective moment when aft of the +CG. + +The {\it stability margin} of a rocket is defined as the distance between +the CP and CG, measured in {\it calibers}, where one caliber is the +maximum body diameter of the rocket. A rule of thumb among model +rocketeers is that the CP should be approximately 1--2 calibers aft of +the CG. However, the CP of a rocket typically moves upwards as the +angle of attack increases. In some cases, a 1--2 caliber stability +margin may totally disappear at an angle of attack of only a few +degrees. As side wind is the primary cause of angles of attack, this +effect is called +{\it wind caused instability}~\cite{galejs}. + +Another stability issue concerning rocketeers is the +{\it dynamic stability} of a rocket. A rocket that is statically +stable may still be poor at returning the rocket to the original +orientation quickly enough. Model rockets may encounter several types of +dynamic instability depending on their shape, size and +mass~\cite[pp.~140--141]{stine}: +% +\begin{enumerate} +\item {\it Too little oscillation damping.} In short, light-weight + rockets the corrective moment may significantly over-correct the + perturbation, requiring a corrective moment in the opposite + direction. This may lead to continuous oscillation during the + flight. +\item {\it Too small corrective moment.} This is the case of over-damped + oscillation, where the corrective moment is too small compared to + the moment of inertia of the rocket. Before the rocket has been + able to correct its orientation, the thrust of the motors may have + already significantly affected the direction of flight. +\item {\it Roll-pitch coupling.} If the model has a natural roll + frequency (caused \eg by canting the fins) close to the oscillation + frequency of the rocket, roll-pitch resonance may occur and cause + the model to go unstable. +\end{enumerate} + +By definition, dynamic stability issues are such that they occur over +time during the flight of the rocket. A full flight simulation that +takes into account all corrective moments automatically also simulates +the possible dynamic stability problems. Therefore the dynamic +stability of rockets will not be further considered in this +thesis. For an analytical consideration of the problem, refer to +\cite{advanced-model-rocketry}. + + + diff --git a/doc/techdoc/chapter-conclusion.tex b/doc/techdoc/chapter-conclusion.tex new file mode 100644 index 00000000..a2e6a06f --- /dev/null +++ b/doc/techdoc/chapter-conclusion.tex @@ -0,0 +1,59 @@ + + +\chapter{Conclusion} +\label{chap-conclusion} + + +Model rocketry is an intriguing sport which combines various fields +ranging from aerodynamic design to model construction to +pyrotechnics. At its best, it works as an inspiration for youngsters +to study engineering and sciences. + +This thesis work provides one of the computer-age tools for everybody +intrested in model rocket design. Providing everybody free access to +a full-fledged rocket simulator allows many more hobbyists to +experiment with different kinds of rocket designs and become more +involved in the sport. The most enthusiastic rocketeers may dive +even deeper and get to examine not only the simulation results, but +also how those simulations are actually performed. + +The software produced contains an easy-to-use interface, which allows +new users to start experimenting with the minimum effort. The +back-end is designed to be easily extensible, in anticipation of +future enhancements. This thesis also includes a step-by-step process +for computing the aerodynamical characteristics of a rocket and for +simulating its flight. These are the current default implementations +used by the software. + +Comparison to experimental data shows that the most important +aerodynamical parameters for flight simulation---the center of +pressure location and drag coefficient---are simulated with an +accuracy of approximately 10\% at subsonic velocities. In this +velocity regime the accuracy of the simulated altitude is on par with +the commercial simulation software RockSim. While comparison with +supersonic rockets was not possible, it is expected that the +simulation is reasonably accurate to at least Mach~1.5. + +The six degree of freedom simulator also allows simulating rocket roll +in order to study the effect of roll stabilization, a feature +not available in other hobby-level rocket simulators. While the +comparison with experimental data of a rolling rocket was +inconclusive as to its accuracy, it is still expected to give valuable +insight into the effects of roll during flight. + +The external listener classes that can be attached to the simulator +allow huge potential for custom extensions. For example testing the +active roll reduction controller that will be included in the +successor project of Haisunäätä would have been exceedingly difficult +without such support. By interfacing the actual controller with a +simulated flight environment it was possible to discover various bugs +in the controller software that would otherwise have gone undetected. + +Finally, it must be emphasized that the release of the OpenRocket +software is not the end of this project. In line with the Open Source +philosophy, it is just the beginning of its development cycle, +where anybody with the know-how can contribute to making OpenRocket an +even better simulation environment. + + + diff --git a/doc/techdoc/chapter-experimental.tex b/doc/techdoc/chapter-experimental.tex new file mode 100644 index 00000000..eecb07f8 --- /dev/null +++ b/doc/techdoc/chapter-experimental.tex @@ -0,0 +1,353 @@ + +\chapter{Comparison with experimental data} +\label{chap-experimental} + +In order to validate the results produced by the software, several +test flights were made and compared to the results simulated by the +software. In addition to the software produced, the same simulations +were performed in the current {\it de facto} standard model rocket simulator +RockSim~\cite{rocksim}. The software used was the free demonstration +version of RockSim version 8.0.1f9. This is the latest demo version +of the software available at the time of writing. The RockSim site +states that the demo version is totally equivalent to the normal +version except that it can only be used a limited time and it does not +simulate the rocket's descent after apogee. + +Comparisons were performed using both a typical model rocket design, +presented in Section~\ref{sec-comparison-small}, and a large hybrid +rocket, Section~\ref{sec-comparison-large}. A small model with +canted fins was also constructed and flown to test the roll +simulation, presented in Section~\ref{sec-comparison-roll}. Finally +in Section~\ref{sec-comparison-windtunnel} some of the the aerodynamic +properties calculated by the software are compared to actual +measurements performed in a wind tunnel. + + + + +\section{Comparison with a small model rocket} +\label{sec-comparison-small} + +For purposes of gathering experimental flight data, a small model +rocket representing the size and characteristics of a typical model +rocket was constructed and flown in various configurations. The +rocket model was 56~cm long with a body diameter of 29~mm. The nose +cone was a 10~cm long tangent ogive, and the fins simple trapezoidal +fins. The entire rocket was painted using an airbrush but not +finished otherwise and the fin profiles were left rectangular, so as +to represent a typical non-competition model rocket. The velocity of +the rocket remained below 0.2~Mach during the entire flight. + +In the payload section of the rocket was included an Alt15K/WD Rev2 +altimeter from PerfectFlite~\cite{perfectflite}. The altimeter +measures the altitude of the rocket based on atmospheric pressure +changes ten times per second. The manufacturer states the accuracy of +the altimeter to be $\pm (0.25\% + \rm 0.6~m)$. The altimeter logs +the flight data, which can later be retrieved to a computer for +further analysis. + +Four holes, each 1~mm in diameter were drilled evenly around the +payload body to allow the ambient air pressure to reach the pressure +sensor, as per the manufacturer's instructions. The rocket was +launched from a 1~m high tower launcher, which removed the need for +any launch lugs. Figure~\ref{fig-rocket-picture} presents a +picture of the test rocket and the tower launcher. + + +\begin{figure} +\centering +\parbox{75mm}{\centering % width 7.4cm +\epsfig{file=figures/pix/rocket-tower,height=11cm} \\ (a)} +\hspace{10mm} +\parbox{35mm}{\centering % width 3.4cm +\epsfig{file=figures/pix/rocket-closeup,height=11cm} \\ (b)} +% +\caption{The test rocket awaiting launch on the tower launcher (a) and + a close-up of its ventilation holes (b).} +\label{fig-rocket-picture} +\end{figure} + + +A design of the same rocket was created in both OpenRocket and +RockSim. During construction of the rocket each component was +individually weighed and the weight of the corresponding component +was overridden in the software for maximum accuracy. Finally, the +mass and CG position of the entire rocket was overridden with measured +values. + +One aspect of the rocket that could not be measured was the average +surface roughness. In the OpenRocket design the ``regular paint'' +finish was selected, which corresponds to an average surface roughness +of 60~\textmu m. From the available options of ``polished'', +``gloss'', ``matt'' and ``unfinished'' in RockSim, the ``matt'' option +was estimated to best describe the rocket; the corresponding +average surface roughness is unknown. + +The rocket was flown using motors manufactured by WECO Feuerwerk +(previously Sachsen Feuerwerk)~\cite{weco-feuerwerk}, which correspond +largely to the motors produced by Estes~\cite{estes}. The only source +available for the thrust curves of Sachsen Feuerwerk motors was a +German rocketry store~\cite{sf-thrustcurves}, the original source of +the measurements are unknown. The thrust curve for the C6-3 motor is +quite similar to the corresponding Estes motor, and has a total impulse +of 7.5~Ns. However, the thrust curve for the B4-4 motor yields a +total impulse of 5.3~Ns, which would make it a C-class motor, while +the corresponding Estes motor has an impulse of only 4.3~Ns. Both +OpenRocket and RockSim simulated the flight of the rocket using the +SF B4-4 motor over 60\% higher than the apogee of the experimental +results. It is likely that the thrust curve of the SF B4-4 is wrong, +and therefore the Estes B4-4 motor was used in the simulations in its +stead. + + +\begin{table} +\caption{Apogee altitude of simulated and experimental flights with + B4-4 and C6-3 motors.} +\label{tab-flight-results} +\begin{center} +\begin{tabular}{ccccc} + & \multicolumn{2}{c}{B4-4} & \multicolumn{2}{c}{C6-3} \\ +\hline +Experimental~~~~ & 64.0 m & & 151.5 m & \\ +OpenRocket~~~~ & 74.4 m & +16\% & 161.4 m & +7\% \\ +RockSim~~~~ & 79.1 m & +24\% & 180.1 m & +19\% \\ +\hline +\end{tabular} +\end{center} +\end{table} + + +Figure~\ref{fig-flight-B4} shows the experimental and simulated +results for the flight using a B4-4 motor (simulations using an Estes +motor) and figure~\ref{fig-flight-C6} using a C6-3 motor. The RockSim +simulations are truncated at apogee due to limitations of the +demonstration version of the software. A summary of the apogee +altitudes is presented in Table~\ref{tab-flight-results}. + +Both simulations produce a bit too optimistic results. OpenRocket +yielded altitudes 16\% and 7\% too high for the B4-4 and C6-3 motors, +respectively, while RockSim had errors of 24\% and 19\%. The C6-3 +flight is considered to be more accurate due to the ambiguity of the +B4-4 thrust curve. +% +Another feature that can be seen from the graphs is that the estimated +descent speed of the rocket is quite close to the actual descent +speed. The error in the descent speeds are 7\% and 13\% respectively. + + +\begin{figure}[p] +\centering +\epsfig{file=figures/experimental/flight-B4-4,width=12cm} +\caption{Experimental and simulated flight using a B4-4 motor.} +\label{fig-flight-B4} +\end{figure} + +\begin{figure}[p] +\centering +\epsfig{file=figures/experimental/flight-C6-3,width=12cm} +\caption{Experimental and simulated flight using a C6-3 motor.} +\label{fig-flight-C6} +\end{figure} + + +% B4-4 C6-3 +%Exp 64.0 151.5 +%OR 74.4 +10.4 +16% 161.4 +9.9 +7% +%RS 79.1 +15.1 +24% 180.1 +28.6 +19% + + +The rocket was also launched with a launch lug 24~mm long and 5~mm in +diameter attached first to its mid-body and then next to its fins to +test the effect of a launch lug on the aerodynamic drag. The apogee +altitudes of the tests were 147.2~m and 149.0~m, which correspond to +an altitude reduction of 2--3\%. The OpenRocket simulation with such +a launch lug yielded results approximately 1.3\% less than without the +launch lug. + + + + +\section{Comparison with a hybrid rocket} +\label{sec-comparison-large} + +The second comparison is with the Haisunäätä hybrid +rocket~\cite{haisunaata-launch}, which was launched in September 2008. +The rocket is a HyperLOC 835 model, with a length of 198~cm and a body +diameter of 10.2~cm. The nose cone is a tangent ogive with a length +of 34~cm, and the kit includes three approximately trapezoidal fins. + +The flight computer on board was a miniAlt/WD altimeter by +PerfectFlite~\cite{perfectflite}, with a stated accuracy of +$\pm0.5\%$. The flight computer calculates the altitude 20 times per +second based on the atmospheric pressure and stores the data into +memory for later analysis. + +The rocket was modeled as accurately as possible with both OpenRocket +and RockSim, but the mass and CG of each component was computed by the +software. Finally, the mass of the entire rocket excluding the motor +was overridden by the measured mass of the rocket. The surface +roughness was estimated as the same as for the small rocket, +60~\textmu m in OpenRocket and ``matt'' for RockSim. + +Figure~\ref{fig-flight-haisunaata} presents the true flight profile +and that of the simulations. Both OpenRocket and RockSim estimate a +too low apogee altitude, with an error of 16\% and 12\%, +respectively. As in the case of the small rocket model, RockSim +produces an estimate 5--10\% higher than OpenRocket. It remains +unclear which software is more accurate in its estimates. + +% Experimental 965m +% OpenRocket 814m +% RockSim 853m + + +One error factor also affecting this comparison is the use of a hybrid +rocket motor. As noted in Section~\ref{sec-motors}, the vapor +pressure of the nitrous oxide is highly dependent on temperature, +which affects the thrust of the motor. This may cause some variation +in the thrust between true flight and motor tests. + +\begin{figure}[p] +\centering +\epsfig{file=figures/experimental/flight-haisunaata,width=12cm} +\caption{Experimental and simulated flight of a hybrid rocket.} +\label{fig-flight-haisunaata} +\end{figure} + +\begin{figure}[p] +\centering +\epsfig{file=figures/experimental/flight-roll-rate,width=12cm} +\caption{Experimental and simulated roll rate results using a C6-3 + motor.} +\label{fig-flight-roll} +\end{figure} + + + +\section{Comparison with a rolling rocket} +\label{sec-comparison-roll} + +In order to test the rolling moment computation, a second +configuration of the small model rocket, described in +Section~\ref{sec-comparison-small}, was built with canted fins. The +design was identical to the previous one, but each fin was canted by +an angle of $5^\circ$. In addition, the payload section contained a +magnetometer logger, built by Antti~J. Niskanen, that measured the +roll rate of the rocket. The logger used two Honeywell HMC1051 +magnetometer sensors to measure the Earth's magnetic field and store +the values at a rate of 100~Hz for later analysis. The rocket was +launched from the tower launcher using a Sachsen Feuerwerk C6-3 +motor. Further test flights were not possible since the lower rocket +part was destroyed by a catastrophic motor failure on the second +launch. + +After the flight, a spectrogram of the magnetometer data was generated +by dividing the data into largely overlapping segments of 0.4~seconds each, +windowed by a Hamming window, and computing the Fourier transform of +these segments. For each segment the frequency with the largest power +density was chosen as the roll frequency at the midpoint of the +segment in time. The resulting roll frequency as a function of time +is plotted in Figure~\ref{fig-flight-roll} with the corresponding +simulated roll frequency. + + +The simulated roll rate differs significantly from the experimental +roll rate. During the flight the rocket peaked at a roll rate of 16 +revolutions per second, while the simulation has only about half of +this. The reason for the discrepancy is unknown and would need more +data to analyze. However, after the test flight it was noticed that +the cardboard fins of the test rocket were slightly curved, which may +have a significant effect on the roll rate. A more precise test rocket +with more rigid and straight fins would be needed for a more +definitive comparison. Still, even at a cant angle of $7^\circ$ the +simulation produces a roll rate of only 12~r/s. + +Even so, it is believed that including roll in the simulation allows +users to realistically analyze the effect of roll stabilization for +example in windy conditions. + + +\section{Comparison with wind tunnel data} +\label{sec-comparison-windtunnel} + + +Finally, the simulated results were compared with experimental wind +tunnel data. The model that was analyzed by J.~Ferris in the +transonic region~\cite{experimental-transonic} and by C.~Babb and +D.~Fuller in the supersonic region~\cite{experimental-supersonic} is +representative of the Arcas Robin meteorological rocket that has been +used in high-altitude research activities. The model is 104.1~cm long +with a body diameter of 5.72~cm. It includes a 27~cm long tangent +ogive nose cone and a 4.6~cm long conical boattail at the rear end, +which reduces the diameter to 3.7~cm. The rocket includes four +trapezoidal fins, the profiles of which are double-wedges. For +details of the configuration, refer to~\cite{experimental-transonic}. + +The design was replicated in OpenRocket as closely as possible, +given the current limitations of the software. The most notable +difference is that an airfoil profile was selected for the fins +instead of the double-wedge that is not supported by OpenRocket. The +aerodynamical properties were computed at the same Mach and Reynolds +numbers as the experimental data. + + +\begin{figure}[t] +\centering +\epsfig{file=figures/experimental/ca-vs-mach,width=11cm} +\caption{Experimental and simulated axial drag coefficient as a + function of Mach number.} +\label{fig-experimental-CA} +\end{figure} + +The most important variables affecting the altitude reached by a +rocket are the drag coefficient and CP location. The experimental and +simulated axial drag coefficient at zero angle-of-attack is presented +in Figure~\ref{fig-experimental-CA}. The general shape of the +simulated drag coefficient follows the experimental results. However, +a few aspects of the rocket break the assumptions made in the +computation methods. First, the boattail at the end of the rocket +reduces the drag by guiding the air into the void left behind it, +while the simulation software only takes into account the reduction of +base area. Second, the airfoil shape of the fins affects the drag +characteristic especially in the transonic region, where it produces +the slight reduction peak. Finally, at higher supersonic speeds the +simulation produces less reliable results as expected, producing a too +high drag coefficient. Overall, however, the drag coefficient matches +the experimental results with reasonable accuracy, and the results of +actual test flights shown in Sections~\ref{sec-comparison-small} and +\ref{sec-comparison-large} give credence to the drag coefficient +estimation. + + +\begin{figure} +\centering +\epsfig{file=figures/experimental/cp-vs-mach,width=12cm} \\ +(a) \\ +\epsfig{file=figures/experimental/cna-vs-mach,width=12cm} \\ +(b) +\caption{Experimental and simulated center of pressure location (a) + and normal force coefficient derivative (b) as a function of Mach + number.} +\label{fig-experimental-CP-CNa} +\end{figure} + +The CP location as a function of Mach number and the normal force +coefficient derivative \CNa\ are presented in +Figure~\ref{fig-experimental-CP-CNa}. The 3\% error margins in the +transonic region were added due to difficulty in estimating the normal +force and pitch moment coefficient derivatives from the printed +graphs; in the supersonic region the CP location was provided +directly. At subsonic speeds the CP location matches the experimental +results to within a few percent. At higher supersonic speeds the +estimate is too pessimistic, and due to the interpolation this is +visible also in the transonic region. However, the CP location is +quite reasonable up to about Mach~1.5. + +The simulated normal force coefficient derivative is notably lower +than the experimental values. The reason for this is unknown, since +in his thesis Barrowman obtained results accurate to about 6\%. The +effect of the lower normal force coefficient on a flight simulation is +that the rocket corrects its orientation slightly slower than in +reality. The effect on the flight altitude is considered to be small +for typical stable rockets. + diff --git a/doc/techdoc/chapter-flight-simulation.tex b/doc/techdoc/chapter-flight-simulation.tex new file mode 100644 index 00000000..f452103b --- /dev/null +++ b/doc/techdoc/chapter-flight-simulation.tex @@ -0,0 +1,781 @@ + + +\chapter{Flight simulation} +\label{chap-simulation} + +In this chapter the actual flight simulation is analyzed. First in +Section~\ref{sec-atmospheric-properties} methods for simulating +atmospheric conditions and wind are presented. Then in +Section~\ref{sec-flight-modeling} the actual simulation procedure is +developed. + + + +\section{Atmospheric properties} +\label{sec-atmospheric-properties} + +In order to calculate the aerodynamic forces acting on the rocket it +is necessary to know the prevailing atmospheric conditions. Since the +atmosphere is not constant with altitude, a model must be developed to +account for the changes. Wind also plays an important role in the +flight of a rocket, and therefore it is important to have a realistic +wind model in use during the simulation. + + +\subsection{Atmospheric model} + +The atmospheric model is responsible to estimating the atmospheric +conditions at varying altitudes. The properties that are of most +interest are the density of air $\rho$ (which is a scaling parameter +to the aerodynamic coefficients via the dynamic pressure +$\frac{1}{2}\rho v^2$) and the speed of sound $c$ (which affects the +Mach number of the rocket, which in turn affects its aerodynamic +properties). These may in turn be calculated from the air pressure +$p$ and temperature $T$. + +Several models exist that define standard atmospheric conditions as a +function of altitude, including the Internaltional Standard +Atmosphere (ISA)~\cite{international-standard-atmosphere} and the +U.S. Standard Atmosphere~\cite{US-standard-atmosphere}. These two +models yield identical temperature and pressure profiles for altitudes +up to 32~km. + +The models are based on the assumption that air follows the ideal gas +law +% +\begin{equation} +\rho = \frac{Mp}{RT} +\end{equation} +% +where $M$ is the molecular mass of air and $R$ is the ideal gas +constant. From the equilibrium of hydrostatic forces the differential +equation for pressure as a function of altitude $z$ can be found as +% +\begin{equation} +\dif p = -g_0 \rho \dif z = -g_0 \frac{Mp}{RT} \dif z +\label{eq-pressure-altitude} +\end{equation} +% +where $g_0$ is the gravitational acceleration. If the temperature of +air were to be assumed to be constant, this would yield an exponential +diminishing of air pressure. + +The ISA and U.S. Standard Atmospheres further specity a standard +temperature and pressure at sea level and a temperature profile for +the atmosphere. The temperature profile is given as eight +temperatures for different altitudes, which are then linearly +interpolated. The temperature profile and base pressures for the ISA +model are presented in Table~\ref{table-ISA-model}. These values +along with equation~(\ref{eq-pressure-altitude}) define the +temperature/pressure profile as a function of altitude. + +\begin{table} +\caption{Layers defined in the International Standard + Atmosphere~\cite{wiki-ISA-layers}} +\label{table-ISA-model} +\begin{center} +\begin{tabular}{ccccl} +\hline +Layer & Altitude$^\dagger$ & Temperature & Lapse rate & + \multicolumn{1}{c}{Pressure} \\ + & m & $^\circ$C & $^\circ$C/km & \multicolumn{1}{c}{Pa} \\ +\hline +0 & 0 & $+15.0$ & $-6.5$ & 101\s325 \\ +1 & 11\s000 & $-56.5$ & $+0.0$ & \num22\s632 \\ +2 & 20\s000 & $-56.5$ & $+1.0$ & \num\num5\s474.9 \\ +3 & 32\s000 & $-44.5$ & $+2.8$ & \num\num\num\s868.02 \\ +4 & 47\s000 & \num$-2.5$ & $+0.0$ & \num\num\num\s110.91 \\ +5 & 51\s000 & \num$-2.5$ & $-2.8$ & \num\num\num\s\num66.939 \\ +6 & 71\s000 & $-58.5$ & $-2.0$ & \num\num\num\s\num\num3.9564 \\ +7 & 84\s852 & $-86.2$ & & \num\num\num\s\num\num0.3734 \\ +\hline +\end{tabular} +\end{center} +\vspace{-3mm} +{\footnotesize $^\dagger$ Altitude is the geopotential height which + does not account for the diminution of gravity at high altitudes.} +\vspace{3mm} +\end{table} + +These models are totally static and do not take into account any local +flight conditions. Many rocketeers may be interested in flight +differences during summer and winter and what kind of effect air pressure +has on the flight. These are also parameters that can easily be +measured on site when launching rockets. On the other hand, it is +generally hard to know a specific temperature profile for a specific +day. Therefore the atmospheric model was extended to allow the user +to specify the base conditions either at mean sea level or at the +altitude of the launch site. These values are simply assigned to the +first layer of the atmospheric model. Most model rockets do not +exceed altitudes of a few kilometers, and therefore the flight +conditions at the launch site will dominate the flight. + +One parameter that also has an effect on air density and the speed of +sound is humidity. The standard models do not include any definition +of humidity as a function of altitude. Furthermore, the effect of +humidity on air density and the speed of sound is marginal. The +difference in air density and the speed of sound between completely dry +air and saturated air at standard conditions are both less than 1\%. +Therefore the effect of humidity has been ignored. + + + + +\subsection{Wind modeling} + +Wind plays a critical role in the flight of model rockets. As has +been seen, large angles of attack may cause rockets to lose a +significant amount of stability and even go unstable. Over-stable +rockets may weathercock and turn into the wind. In a perfectly static +atmosphere a rocket would, in principle, fly its entire flight +directly upwards at zero angle of attack. Therefore, the effect of +wind must be taken into account in a full rocket simulation. + +Most model rocketeers, however, do not have access to a full wind +profile of the area they are launching in. Different layers of air +may have different wind velocities and directions. Modeling such +complex patterns is beyond the scope of this project. Therefore, the +goal is to produce a realistic wind model that can be specified with +only a few parameters understandable to the user and that covers +altitudes of most rocket flights. Extensions to allow for multiple +air layers may be added in the future. + +In addition to a constant average velocity, wind always has some +degree of turbulence in it. The effect of turbulence can be modeled +by summing the steady flow of air and a random, zero-mean turbulence +velocity. Two central aspects of the turbulence velocity are the +amplitude of the variation and the frequencies at which they occur. +Therefore a reasonable turbulence model is achieved by a random +process that produces a sequence with a similar distribution and +frequency spectrum as that of real wind. + +Several models of the spectrum of wind turbulence at specific +altitudes exist. Two commonly used such spectra are the {\it Kaimal} +and {\it von Kármán} wind turbulence +spectra~\cite[p.~23]{wind-energy-handbook}: +% +\begin{eqnarray} +\mbox{Kaimal:} & & \frac{S_u(f)}{\sigma_u^2} = + \frac{4 L_{1u} / U}{(1 + 6fL_{1u}/U)^{5/3}} \label{eq-kaimal-wind} \\ +% +\mbox{von Kármán:} & & \frac{S_u(f)}{\sigma_u^2} = + \frac{4 L_{2u} / U}{(1 + 70.8(fL_{2u}/U)^2)^{5/6}} \label{eq-karman-wind} +\end{eqnarray} + +Here $S_u(f)$ is the spectral density function of the turbulence +velocity and $f$ the turbulence frequency, $\sigma_u$ the standard +deviation of the turbulence velocity, $L_{1u}$ and $L_{2u}$ length +parameters and $U$ the average wind speed. + +Both models approach the asymptotic limit +$S_u(f)/\sigma_u^2 \sim f^{-5/3}$ quite fast. Above frequencies of +0.5~Hz the difference between equation~(\ref{eq-kaimal-wind}) and the +same equation without the term 1 in the denominator is less than 4\%. +Since the time scale of a model rocket's flight is quite short, the +effect of extremely low frequencies can be ignored. Therefore +turbulence may reasonably well be modelled by utilizing +{\it pink noise} that has a spectrum of $1/f^\alpha$ with $\alpha=5/3$. +True pink noise has the additional useful property of being +scale-invariant. This means that a stream of pink noise samples may +be generated and assumed to be at any sampling rate while maintaining +their spectral properties. + +Discerete samples of pink noise with spectrum $1/f^\alpha$ can be +generated by applying a suitable digital filter to {\it white noise}, +which is simply uncorrelated pseudorandom numbers. One such filter is +the infinite impulse response (IIR) filter presented by +Kasdin~\cite{pink-filter}: +% +\begin{equation} +x_n = w_n - a_1 x_{n-1} - a_2 x_{n-2} - a_3 x_{n-3} - \ldots +\label{eq-pink-generator} +\end{equation} +% +where $x_i$ are the generated samples, $w_n$ is a generated white +random number and the coefficients are computed using +% +\begin{equation} +\begin{array}{rl} +a_0 & = 1 \\ +a_k & = \del{k-1-\frac{\alpha}{2}} \frac{a_{k-1}}{k}. +\end{array} +\label{eq-pink-coefficients} +\end{equation} +% +The infinite sum may be truncated with a suitable number of terms. +In the context of IIR filters these terms are calles {\it poles}. +Experimentation showed that already 1--3 poles provides a reasonably +accurate frequency spectrum in the high frequency range. + +One problem in using pink noise as a turbulence velocity +model is that the power spectrum of pure pink noise goes to +infinity at very low frequencies. This means that a long sequence +of random values may deviate significantly from zero. However, when +using the truncated IIR filter of equation~(\ref{eq-pink-generator}), +the spectrum density becomes constant below a certain limiting +frequency, dependent on the number of poles used. By adjusting the +number of poles used, the limiting frequency can be adjusted to a value +suitable for model rocket flight. Specifically, the number of poles +must be selected such that the limiting frequency is suitable at the +chosen sampling rate. + +It is also desirable that the simulation resolution does not affect +the wind conditions. For example, a simulation with a time step of +10~ms should experience the same wind conditions as a simulation with +a time step of 5~ms. This is achieved by selecting a constant +turbulence generation frequency and interpolating between the +generated points when necessary. The fixed frequency was chosen at +20~Hz, which can still simulate fluctuations at a time scale of 0.1 +seconds. + +\begin{figure}[p] +\centering +\epsfig{file=figures/wind/pinktime, width=105mm} +\caption{The effect of the number of IIR filter poles on two 20 second + samples of generated turbulence, normalized so that the two-pole + sequence has standard deviation one.} +\label{fig-pink-poles} +\end{figure} + +\begin{figure}[p] +\centering +\epsfig{file=figures/wind/pinkfreq, width=95mm} +\caption{The average power spectrum of 100 turbulence + simulations using a two-pole IIR filter (solid) and the Kaimal + turbulence spectrum (dashed); vertical axis arbitrary.} +\label{fig-pink-spectrum} +\end{figure} + + + +The effect of the number of poles is depicted in +Figure~\ref{fig-pink-poles}, where two pink noise sequences were +generated from the same random number source with two-pole and +ten-pole IIR filters. A small number of poles generates values strongly +centered on zero, while a larger number of poles introduces more low +frequency variability. Since the free-flight time of a typical model +rocket is of the order of 5--30 seconds, it is desireable that the +maximum gust length during the flight is substantially shorter than +this. Therefore the pink noise generator used by the wind model was +chosen to contain only two poles, which has a limiting frequency of +approximately 0.3~Hz when sampled at 20~Hz. This means that gusts of +wind longer than 3--5 seconds will be rare in the simulted turbulence, +which is a suitable gust length for modeling typical model rocket +flight. Figure~\ref{fig-pink-spectrum} depicts the resulting pink +noise spectrum of the two-pole IIR filter and the Kaimal spectrum of +equation~(\ref{eq-kaimal-wind}) scaled to match each other. + + +%, which causes frequency +%components below approximately 0.3~Hz to be subdued. Therefore, gusts +%of wind longer than 3--5 seconds will be rare in the simulated wind, a +%suitable time scale for the flight of a model rocket. +%Figure~\ref{fig-turbulence}(a) shows a 20 second sample of the +%generated turbulence, normalized to have a standard deviation of one. +%Figure~\ref{fig-turbulence}(b) depicts the actual frequency spectrum +%of the generated turbulence and the Kaimal spectrum of +%equation~(\ref{eq-kaimal-wind}) scaled to match each other. + + + +To simplify the model, the average wind speed is assumed to be +constant with altitude and in a constant direction. This allows +specifying the model parameters using just the average wind speed and +its standard deviation. An alternative parameter for specifying the +turbulence amplitude is the {\it turbulence intensity}, which is the +percentage that the standard deviation is of the average wind +velocity, +% +\begin{equation} +I_u = \frac{\sigma_u}{U}. +\end{equation} +% +Wind farm load design standards typically specify turbulence +intensities around 10\ldots20\%~\cite[p.~22]{wind-energy-handbook}. +It is assumed that these intensities are at the top of the range of +conditions in which model rockets are typically flown. + +Overall, the process to generate the wind velocity as a function of +time from the average wind velocity $U$ and standard deviation +$\sigma_u$ can be summarized in the following steps: +% +\begin{enumerate} +%\item[Input:] Average wind velocity $U$ and standard deviation +% $\sigma_u$. +% +\item Generate a pink noise sample $x_n$ from a Gaussian white noise + sample $w_n$ using equations~(\ref{eq-pink-generator}) and + (\ref{eq-pink-coefficients}) with two memory terms included. + +\item Scale the sample to a standard deviation one. This is performed + by dividing the value by a previously calculated standard deviation + of a long, unscaled pink noise sequence (2.252 for the two-pole IIR + filter). + +\item The wind velocity at time $n\cdot\Delta t$ ($\Delta t = 0.05\rm~s$) + is $U_n = U + \sigma_u x_n$. Velocities in between are interpolated. +\end{enumerate} + + + + +\section{Modeling rocket flight} +\label{sec-flight-modeling} + + +Modeling of rocket flight is based on Newton's laws. The basic forces +acting upon a rocket are gravity, thrust from the motors and +aerodynamic forces and moments. These forces and moments are +calculated and integrated numerically to yield a simulation over a +full flight. + +Since most model rockets fly at a maximum a few kilometers high, the +curvature of the Earth is not taken into account. Assuming a flat +Earth allows us to use simple Cartesian coordinates to represent the +position and altitude of the rocket. As a consequence, the coriolis +effect when flying long distances north or south is not simulated +either. + + + +\subsection{Coordinates and orientation} + +During a rocket's flight many quantities, such as the aerodynamical +forces and thrust from the motors, are relative to the rocket itself, +while others, such as the position and gravitational force, are more +naturally described relative to the launch site. Therefore two sets +of coordinates are defined, the {\it rocket coordinates}, which are +the same as used in Chapter~\ref{chap-aerodynamics}, and +{\it world coordinates}, which is a fixed coordinate system with the +origin at the position of launch. + +The position and velocity of a rocket are most naturally maintained as +Cartesian world coordinates. Following normal convensions, the +$xy$-plane is selected to be parallel to the ground and the $z$-axis +is chosen to point upwards. In flight dynamics of aircraft the +$z$-axis often points towards the earth, but in the case of rockets it +is natural to have the rocket's altitude as the $z$-coordinate. + +Since the wind is assumed to be unidirectional and the Coriolis effect +is ignored, it may be assumed that the wind is directed along the +$x$-axis. The angle of the launch rod may then be positioned relative +to the direction of the wind without any loss of generality. + +Determining the orientation of a rocket is more complicated. A +natural choise for defining the orientation would be to use the +spherical coordinate zenith and azimuth angles $(\theta, \phi)$ and an +additional roll angle parameter. Another choise common in aviation is +to use {\it Euler angles}~\cite{wiki-euler-angles}. However, both of +these systems have notable shortcomings. Both systems have +singularity points, in which the value of some parameter is +ambiguous. With spherical coordinates, this is the direction of the +$z$-axis, in which case the azimuth angle $\phi$ has no effect on the +position. Rotations that occur near these points must often be +handled as special cases. Furthermore, rotations in spherical +coordinate systems contain complex trigonometric formulae which are +prone to programming errors. + +The solution to the singularity problem is to introduce an extra +parameter and an additional constraint to the system. For example, +the direction of a rocket could be defined by a three-dimensional unit +vector $(x,y,z)$ instead of just the zenith and azimuth angles. The +additional constraint is that the vector must be of unit length. This +kind of representation has no singularity points which would require +special consideration. + +Furthermore, Euler's rotation theorem states that a rigid body can be +rotated from any orientation to any other orientation by a single +rotation around a specific axis~\cite{wiki-euler-rotation-theorem}. +Therefore instead of defining quantities that define the orientation +of the rocket we can define a three-dimensional rotation that rotates +the rocket from a known reference orientation to the current +orientation. This has the additional advantage that the same rotation +and its inverse can be used to transform any vector between world +coordinates and rocket coordinates. + +A simple and efficient way of descibing the 3D rotation is by using +{\it unit quaternions}. Each unit quaternion corresponds to a unique +3D rotation, and they are remarkably simple to combine and use. The +following section will present a brief overview of the properties of +quaternions. + +The fixed reference orientation of the rocket defines the rocket +pointing towards the positive $z$-axis in world coordinates and an +arbitrary but fixed roll angle. The orientation of the rocket is then +stored as a unit quaternion that rotates the rocket from this +reference orientation to its current orientation. +This rotation can also be used to transform vectors from world +coordinates to rocket coordinates and its inverse from rocket +coordinates to world coordinates. (Note that the rocket's initial +orientation on the launch pad may already be different than its +reference orientation if the launch rod is not completely vertical.) + + + + +\subsection{Quaternions} + +{\it Quaternions} are an extension of complex numbers into four +dimensions. The usefulness of quaternions arises from their use in +spatial rotations. Similar to the way multiplication with a complex +number of unit length $e^{i\phi}$ corresponds to a rotation of angle +$\phi$ around the origin on the complex plane, multiplication with +unit quaternions correspond to specific 3D rotations around an axis. +A more thorough review of quaternions and their use in spatial +rotations is available in Wikipedia~\cite{wiki-quaternion-rotations}. + +The typical notation of quaternions resembles the addition of a scalar +and a vector: +% +\begin{equation} +q = w + x\vi + y\vj + z\vk = w + \vect v +\end{equation} +% +Addition of quaternions and multiplication with a scalar operate as +expected. However, the multiplication of two quaternions is +non-commutative (in general $ab \neq ba$) and follows the rules +% +\begin{equation} +\vi^2 = \vj^2 = \vk^2 = \vi\vj\vk = -1. +\end{equation} +% +As a corollary, the following equations hold: +% +\begin{equation} +\begin{array}{rl} +\vi\vj = \vk \hspace{15mm}& \vj\vi = -\vk \\ +\vj\vk = \vi \hspace{15mm}& \vk\vj = -\vi \\ +\vk\vi = \vj \hspace{15mm}& \vi\vk = -\vj +\end{array} +\end{equation} +% +The general multiplication of two quaternions becomes +% +\begin{equation} +\begin{array}{rl} +(a + b\vi + c\vj + d\vk)(w + x\vi + y\vj + z\vk)\;\; = + & (aw-bx-cy-dz) \\ + & + (ax+bw+cz-dy)\;\vi \\ + & + (ay-bz+cw+dx)\;\vj \\ + & + (az+by-cx+dw)\;\vk +\end{array} +\end{equation} +% +while the norm of a quaternion is defined in the normal manner +% +\begin{equation} +|q| = \sqrt{w^2+x^2+y^2+z^2}. +\end{equation} + +The usefulness of quaternions becomes evident when we consider a +rotation around a vector $\vect u$, $|\vect u|=1$ by an angle $\phi$. +Let +% +\begin{equation} +q = \cos\frac{\phi}{2} + \vect u \sin\frac{\phi}{2}. +\label{eq-rotation-quaternion} +\end{equation} +% +Now the previously mentioned rotation of a three-dimensional vector +$\vect v$ defined by $\vi$, $\vj$ and $\vk$ is equivalent to the +quaternion product +% +\begin{equation} +\vect v \mapsto q\vect v q^{-1}. +\end{equation} +% +Similarly, the inverse rotation is equivalent to the transformation +% +\begin{equation} +\vect v \mapsto q^{-1} \vect v q. +\end{equation} +% +The problem simplifies even further, since for unit quaternions +% +\begin{equation} +q^{-1} = (w + x\vi + y\vj + z\vk)^{-1} = w - x\vi - y\vj - z\vk. +\end{equation} +% +Vectors can therefore be considered quaternions with no scalar +component and their rotation is equivalent to the left- and right-sided +multiplication with unit quaternions, requiring a total of 24 +floating-point multiplications. Even if this does not make the +rotations more efficient, it simplifies the trigonometry considerably +and therefore helps reduce programming errors. + + +\subsection{Mass and moment of inertia calculations} +\label{sec-mass-inertia} + +Converting the forces and moments into linear and angluar acceleration +requires knowledge of the rocket's mass and moments of inertia. The +mass of a component can be easily calculated from its volume and +density. Due to the highly symmetrical nature of rockets, the rocket +centerline is commonly a principal axis for the moments of inertia. +Furthermore, the moments of inertia around the in the $y$- and +$z$-axes are very close to one another. Therefore as a simplification +only two moments of inertia are calculated, the longitudal and +rotational moment of inertia. These can be easily calculated for each +component using standard formulae~\cite{wiki-moments-of-inertia} and +combined to yield the moments of the entire rocket. + +This is a good way of calculating the mass, CG and inertia of a rocket +during the design phase. However, actual rocket components often have +a slightly different density or additional sources of mass such as +glue attached to them. These cannot be effectively modeled by the +simulator, since it would be extremely tedious to define all these +properties. Instead, some properties of the components can be +overridden to utilize measured values. + +Two properties that can very easily be measured are the mass and +CG position of a component. Measuring the moments of inertia is a +much harder task. Therefore the moments of inertia are still computed +automatically, but are scaled by the overridden measurement values. + +If the mass of a component is overridden by a measured value, the +moments of inertia are scaled linearly according to the mass. This +assumes that the extra weight is distributed evenly along the +component. If the CG position is overridden, there is no knowledge +where the extra weight is at. Therefore as a best guess the moments +of inertia are updated by shifting the moment axis according to the +parallel axis theorem. + +As the components are computed individually and then combined, the +overriding can take place either for individual components or larger +combinations. It is especially useful is to override the mass and/or CG +position of the entire rocket. This allows constructing a rocket from +components whose masses are not precisely known and afterwards scaling +the moments of inertia to closely match true values. + + + +\subsection{Flight simulation} + +The process of simulating rocket flight can be broken down into the +following steps: + +\begin{enumerate} +\setcounter{enumi}{-1} +\item Initialize the rocket in a known position and orientation at + time $t=0$. +\item Compute the local wind velocity and other atmospheric conditions. +\item Compute the current airspeed, angle of attack, lateral wind + direction and other flight parameters. +\item Compute the aerodynamic forces and moments affecting the rocket. +\item Compute the effect of motor thrust and gravity. +\item Compute the mass and moments of inertia of the rocket and from + these the linear and rotational acceleration of the rocket. +\item Numerically integrate the acceleration to the rocket's position + and orientation during a time step $\Delta t$ and update the current + time $t \mapsto t+\Delta t$. +\end{enumerate} + +Steps 1--6 are repeated until an end criteria is met, typically until +the rocket has landed. + +The computation of the atmospheric properties and instantaneous wind +velocity were discussed in Section~\ref{sec-atmospheric-properties}. +The local wind velocity is added to the rocket velocity to get the +airspeed velocity of the rocket. By inverse rotation this quantity is +obtained in rocket coordinates, from which the angle of attack and +other flight parameters can be computed. + +After the instantaneous flight parameters are known, the aerodynamic +forces can be computed as discussed in +Chapter~\ref{chap-aerodynamics}. The computed forces are in the +rocket coordinates, and can be converted to world coordinates by +applying the orientation rotation. The thrust from the motors is +similarly calculated from the thrust curves and converted to world +coordinates, while the direction of gravity is already in world +coordinates. When all of the the forces and moments acting upon the +rocket are known, the linear and rotational accelerations can be +calculated using the mass and moments of inertia discussed in +Section~\ref{sec-mass-inertia}. + +The numerical integration is performed using the Runge-Kutta~4 (RK4) +integration method. In order to simulate the differential equations +% +\begin{equation} +\begin{split} +x''(t) &= a(t) \\ +\phi''(t) &= \alpha(t) +\end{split} +\end{equation} +% +the equation is first divided into first-order equations using the +substitutions $v(t)=x'(t)$ and $\omega(t)=\phi'(t)$: +% +\begin{equation} +\begin{split} +v'(t) &= a(t) \\ +x'(t) &= v(t) \\ +\omega'(t) &= \alpha(t) \\ +\phi'(t) &= \omega(t) +\end{split} +\end{equation} +% +For brevity, this is presented in the first order representation +% +\begin{equation} +y' = f(y,\; t) +\end{equation} +% +where $y$ is a vector function containing the position and orientation +of the rocket. + +Next the right-hand side is evaluated at four positions, dependent on +the previous evaluations: +% +\begin{equation} +\begin{split} +k_1 &= f(y_0,\; t_0) \\ +k_2 &= f(y_0 + k_1\:\mbox{$\frac{\Delta t}{2}$},\; + t_0 + \mbox{$\frac{\Delta t}{2}$}) \\ +k_3 &= f(y_0 + k_2\:\mbox{$\frac{\Delta t}{2}$},\; + t_0 + \mbox{$\frac{\Delta t}{2}$}) \\ +k_4 &= f(y_0 + k_3\:\Delta t,\; t_0 + \Delta t) +\end{split} +\end{equation} +% +Finally, the result is a weighted sum of these values: +% +\begin{align} +y_1 &= y_0 + \frac{1}{6}\left(k_1+2k_2+2k_3+k_4\right)\,\Delta t \\ +t_1 &= t_0 + \Delta t +\end{align} + +Computing the values $k_1\ldots k_4$ involves performing steps~1--5 +four times per simulation iteration, but results in significantly +better simulation precision. The method is a fourth-order integration +method, meaning that the error incurred during one simulation step is +of the order $O(\Delta t^5)$ and of the total simulation +$O(\Delta t^4)$. This is a considerable improvement +over, for example, simple Euler integration, which has a total error +of the order $O(\Delta t)$. Halving the time step in an Euler +integration only halves the total error, but reduces the error of a +RK4 simulation 16-fold. + +The example above used a total rotation vector $\phi$ to contain the +orientation of the rocket. Instead, this is replaced by the rotation +quaternion, which can be utilized directly as a transformation between +world and rocket coordinates. Instead of updating the total rotation +vector, +% +\begin{equation} +\phi_1 = \phi_0 + \omega\,\Delta t, +\end{equation} +% +the orientation quaternion $o$ is updated by the same amount by +% +\begin{equation} +o_1 = \del{\cos\del{|\omega|\,\Delta t} + +\hat\omega\sin\del{|\omega|\,\Delta t}} \cdot o_0. +\end{equation} +% +The first term is simply the unit quaternion corresponding to the +3D rotation $\omega\,\Delta t$ as in +equation~(\ref{eq-rotation-quaternion}). It is applied to the +previous value $o_0$ by multiplying the quaternion from the left. +This update is performed both during the calculation of +$k_2\ldots k_4$ and when computing the final step result. Finally, in +order to improve numerical stability, the quaternion is normalized to +unit length. + +Since most of a rocket's flight occurs in a straight line, rather +large time steps can be utilized. However, the rocket may encounter +occasional oscillation, which may affect its flight notably. +Therefore the time step utilized is dynamically reduced in cases where +the angular velocity or angular acceleration exceeds a predefined +limit. This allows utilizing reasonably large time steps for most of +the flight, while maintaining the accuracy during oscillation. + + +\subsection{Recovery simulation} + +All model rockets must have some recovery system for safe landing. +This is typically done either using a parachute or a streamer. When a +parachute is deployed the rocket typically splits in half, and it is +no longer practical to compute the orientation of the rocket. +Therefore at this point the simulation changes to a simpler, three +degree of freedom simulation, where only the position of the rocket is +computed. + +The entire drag coefficient of the rocket is assumed to come from the +deployed recovery devices. For parachutes the drag coefficient is +by default 0.8~\cite[p.~13-23]{hoerner} with the reference area being the +area of the parachute. The user can also define their own drag +coefficient. + +The drag coefficient of streamers depend on the material, width and +length of the streamer. The drag coefficient and optimization of +streamers has been an item of much intrest within the rocketry +community, with competitions being held on streamer descent time +durations~\cite{streamer-optimization}. In order to estimate the drag +coefficient of streamers, a series of experiments were perfomed using +the $40\times40\times120$~cm wind tunnel of +Pollux~\cite{pollux-wind-tunnel}. The experiments were performed +using various materials, widths and lengths of streamers and at +different wind speeds. From these results an empirical formula was +devised that estimates the drag coefficient of streamers. The +experimental results and the derivation of the empirical formula are +presented in Appendix~\ref{app-streamers}. Validation performed with +an independent set of measurements indicates that the drag coefficient +is estimated with an accuracy of about 20\%, which translates to a +descent velocity accuracy within 15\% of the true value. + + + + +\subsection{Simulation events} + +Numerous different events may cause actions to be taken during a +rocket's flight. For example in high-power rockets the burnout or +ignition charge of the first stage's motor may trigger the ignition of +a second stage motor. Similarly a flight computer may deploy a small +drogue parachute when apogee is detected and the main parachute is +deployed later at a predefined lower altitude. To accomodate +different configurations a simulation event system is used, where +events may cause other events to be triggered. + +Table~\ref{tab-simulation-events} lists the available simulation +events and which of them can be used to trigger motor ignition or recovery +device deployment. Each trigger event may additionally include a +delay time. For example, one motor may be configured to ignite at +launch and a second motor to ignite using a timer at 5 seconds after +launch. Alternatively, a short delay of 0.5--1 seconds may be used to +simulate the delay of an ejection charge igniting the upper stage +motors. + +The flight events are also stored along with the simulated flight data +for later analysis. They are also available to the simulation +listeners, described in Section~\ref{sec-listeners}, to act upon +specific conditions. + +\begin{table} +\caption{Simulation events and the actions they may trigger (motor + ignition or recovery device deployment).} +\label{tab-simulation-events} +% +\begin{center} +\begin{tabular}{ll} +Event description & Triggers \\ +\hline +Rocket launch at $t=0$ & Ignition, recovery \\ +Motor ignition & None \\ +Motor burnout & Ignition \\ +Motor ejection charge & Ignition, recovery \\ +Launch rod cleared & None \\ +Apogee detected & Recovery \\ +Change in altitude & Recovery \\ +Touchdown after flight & None \\ +Deployment of a recovery device & None \\ +End of simulation & None \\ +\hline +\end{tabular} +\end{center} +\end{table} + + + + + diff --git a/doc/techdoc/chapter-introduction.tex b/doc/techdoc/chapter-introduction.tex new file mode 100644 index 00000000..d756d990 --- /dev/null +++ b/doc/techdoc/chapter-introduction.tex @@ -0,0 +1,168 @@ + +\chapter{Introduction} + +Model rocketry is a sport that involves designing, constructing and +launching self-made rockets. Model rockets vary greatly in size, +shape, weight and construction from detailed scale models of +professional rockets to lightweight and highly finished competition +models. The sport is relatively popular and is often cited as a +source of inspiration for children to become engineers and +scientists. + +The hobby started as amateur rocketry in the 1950's when hobbyists +wanted to experiment their skill with building rockets. Designing, +building and firing self-made {\it motors} was, however, extremely dangerous, +and the American Rocket Society (now the American Institute of +Aeronautics and Astronautics, AIAA) has estimated that about one in seven +amateur rocketeers during the time were injured in their hobby. This +changed in 1958 when the first commercially-built model rocket +motors became available. Having industrially-made, reasonably-priced +and safe motors available removed the most dangerous aspect of amateur +rocketry. This along with strict guidelines to the design and +launching of model rockets formed the foundation for a safe and +widespread hobby.~\cite[pp.~1--3]{stine} + +Since then model rocketry has spread around the globe and among all +age groups. Thousands of rockets ranging from 10~cm high miniatures +to large models reaching altitudes in excess of 10~km are launched +annually. Model rocket motors with thrusts from a few Newtons up to +several kilo-Newtons are readily available. Since its forming in +1957, over 90\s000 people have joined the National Association of +Rocketry (NAR) in the U.S. alone. +% Model rocketry is used as an +%educational device in numerous of schools and by many youth +%organizations. + +In designing rockets, the {\it stability} of a rocket is of central +priority. A stable rocket corrects its course if some outside +force disturbs it slightly. A disturbance of an unstable rocket +instead increases until the rocket starts spinning in the +air erratically. As shall be discussed in +Section~\ref{sec-stability}, a rocket is deemed +{\it statically stable} if its center of pressure (CP) is aft of its +center of gravity (CG)\footnote{An alternative term would be + {\it center of mass}, but in the context of model rocketry, we are + interested in the effect of gravity on the rocket. Thus, the term + center of gravity is widely used in model rocketry texts, and this + convention will be followed in this thesis.}. +The center of gravity of a rocket can be easily calculated in advance +or determined experimentally. The center of pressure, on the other +hand, has been quite hard to determine either analytically or +experimentally. In 1966 James and Judith Barrowman developed an +analytical method for determining the CP of a slender-bodied rocket at +subsonic speeds and presented their results as a research and +development project at the 8th National Association of Rocketry Annual +Meeting (NARAM-8)~\cite{barrowman-rd}, and later as a part of James +Barrowman's Master's thesis~\cite{barrowman-thesis}. This method has +become known as the {\it Barrowman method} of determining the CP of a +rocket within the model rocketry community, and has a major role in +determining the aerodynamic characteristics of model rockets. + +Another important aerodynamic quantity of interest is the +{\it aerodynamic drag} of a rocket. Drag is caused by the flow of air +around the rocket and it can easily reduce the maximum altitude of a +rocket by 50--80\% of the otherwise theoretical maximum. Estimating +the drag of a model rocket is a rather complex task, and the effects +of different design choices are not always very evident to a +hobbyist. + +Knowing the fundamental aerodynamic properties of a rocket allows one +to simulate its free flight. This involves numerically integrating +the flight forces and determining the velocity, rotation and position +of the rocket as a function of time. This is best performed by +software designed for the purpose of model rocket design. + +RockSim~\cite{rocksim} is one such piece of software. It is a +commercial, proprietary program that allows one to define the geometry +and configuration of a model rocket, estimate its aerodynamic +properties and simulate a launch with different rocket motors. It has +become the {\it de facto} standard software for model rocket +performance estimation. However, as a proprietary program, it is +essentially a ``black-box'' solution. Someone wishing to study or +validate the methods will not be able to do so. Similarly extending +or customizing the functionality or refining the calculations methods +to fit ones needs is impossible. The software is also only available +on select operating systems. Finally, the cost of the software may be +prohibitive especially for younger hobbyists, voluntary organizations, +clubs and schools. + +Open Source software, on the other hand, has become an increasingly +competitive alternative to proprietary software. Open Source allows +free access to the source code of the programs and encourages +users with the know-how to enhance the software and share their +changes~\cite{oss-principles}. Success stories such as the Linux +operating system, the OpenOffice.org office suite, the Firefox web +browser and countless others have shown that Open Source software can +often achieve and even exceed the quality of expensive proprietary +software. + + +\section{Objectives of the thesis} + +The objectives of this thesis work are to: +% +\begin{enumerate} +\item Develop and document relatively easy, yet reasonably accurate + methods for the calculation of the fundamental aerodynamic + properties of model rockets and their numerical simulation; + +\item Test the methods developed and compare the results with other + estimates and actual experimental data; and + +\item Implement a cross-platform, Open Source model rocket design and + simulation software that uses the aforementioned methods, is at the + same time easy to use and yet versatile, and which is easily + extensible and customizable for user requirements, new types of rocket + components and new estimation methods. +\end{enumerate} + +The methods presented will largely follow the methods developed by +Barrowman~\cite{barrowman-rd,barrowman-thesis}, since these are +already familiar to the rocketry community. Several extensions to the +methods will be added to allow for more accurate calculation at larger +angles of attack and for fin shapes not accounted for in the original +paper. The emphasis will be on subsonic flight, but extensions will +be made for reasonable estimation at transonic and low supersonic +velocities. + +The software developed as part of the thesis is the OpenRocket +project~\cite{openrocket}. It is an Open Source rocket development +and simulation environment written totally in Java. The program +structure has been designed to make full use of object oriented +programming, allowing one to easily extend its features. The software +also includes a framework for creating user-made +{\it listener components} (discussed in Section~\ref{sec-listeners}) +that can listen to and interact with the simulation while it is +running. This allows a powerful and easy way of interacting with the +simulation and allows simulating for example guidance systems. + +One possible future enhancement that has also specifically been +considered throughout the development is calculating the aerodynamic +properties using computational fluid dynamics (CFD). CFD calculates +the exact airflow in a discretized mesh around the rocket. This would +allow for even more accurate calculation of the aerodynamic forces for +odd-shaped rockets, for which the methods explained herein do not +fully apply. + +It is anticipated that the software will allow more hobbyists the +possibility of simulating their rocket designs prior to building them +and experimenting with different configuration, thus giving them a +deeper understanding of the aerodynamics of rocket flight. It will +also provide a more versatile educational tool since the simulation +methods are open and everyone will be able to ``look under the hood'' +and see how the software performs the calculations. + +In Chapter~\ref{chap-basics} a brief overview of model rocketry and +its different aspects will be given. Then in +Chapter~\ref{chap-aerodynamics} methods for calculating the +aerodynamic properties of a general model rocket will be presented. +In Chapter~\ref{chap-simulation} the aspects of simulating a rocket's +flight are considered. Chapter~\ref{chap-software} then explains how +the aerodynamic calculations and simulation are implemented in the +OpenRocket software and presents some of its features. In +Chapter~\ref{chap-experimental} the results of the software simulation +are compared with the performance of constructed and launched rockets. +Chapter~\ref{chap-conclusion} then presents a summary of the +achievements and identifies areas of further work. + + diff --git a/doc/techdoc/chapter-software.tex b/doc/techdoc/chapter-software.tex new file mode 100644 index 00000000..f66a98ca --- /dev/null +++ b/doc/techdoc/chapter-software.tex @@ -0,0 +1,475 @@ + + +\chapter{The OpenRocket simulation software} +\label{chap-software} + +The flight simulation described in the previous chapters was +implemented in the {\it OpenRocket} simulation +software~\cite{openrocket}. The software was written entirely in Java +for maximum portability between different operating systems. A +significant amount of effort was put into making the graphical user +interface (UI) robust, intuitive and easy to use. As of the first +public release the source code contained over 300 classes and over +47\s000 lines of code (including comments and blank lines). + +The software was released under the copyleft GNU General Public +License (GPL)~\cite{gnu-gpl}, allowing everybody access to the source code +and permitting use and modification for any purposes. The only major +restriction placed is that if a modified version is distributed, the +source code for the modifications must also be available under the GNU +GPL. This ensures that the program stays free and Open Source; a +company cannot simply take the code, enhance it and start selling it +as their own without contributing anything back to the community. + +In this section the basic architectural designs are discussed and the +main features of the UI are explained. + + + +\section{Architectural design} + +The software has been split into various components within their own +Java packages. This enables use of the components without needing the +other components. For example, all of the UI code is within the +\code{gui} package, and the simulation system can be used +independently of it. + + +The rocket structure is composed of components, each with their +own class defined within the package \code{rocketcomponent}. This +provides the base for defining a rocket. The components are described +in more detail in Section~\ref{sec-rocket-components}. The simulation +code is separated from the aerodynamic calculation code in the +\code{simulation} and \code{aerodynamics} packages, respectively; +these are discussed in Section~\ref{sec-simulator-calculator}. + +The package naming convention recommended in the Java Language +Specification is followed~\cite{java-packages}. Therefore all package +names discussed herein are relative to the package +\url{net.sf.openrocket}. For example, the rocket components are +defined in the package \url{net.sf.openrocket.rocketcomponent}. + + + + +\subsection{Rocket components} +\label{sec-rocket-components} + +The structure of the rocket is split up into {\it components}, for +example nose cones, body tubes and components within the rocket body. +Each component type is defined as a subclass of the abstract +\code{RocketComponent} class. Each component can contain zero or more +components attached to it, which creates a tree structure containing +the entire rocket design. The base component of every design is a +\code{Rocket} object, which holds one or more \code{Stage} objects. +These represent the stages of the rocket and hold all of the physical +components. + +Inheritance has been highly used in the class hierarchy of the +components in order to combine common features of various components. +There are also some abstract classes, such as \code{BodyComponent}, +whose sole purpose currently is to allow future extensions. The +complete component class hierarchy is presented as a UML diagram in +Figure~\ref{fig-rocketcomponent-uml}, and a more detailed description +of each class is presented in Table~\ref{table-rocketcomponents}. + +\begin{sidewaysfigure} +\centering +\hspace{-1cm} +\epsfig{file=figures/software/rocketcomponents,width=20cm} +\caption{A UML diagram of the rocket component classes. Abstract + classes are shown in italics.} +\label{fig-rocketcomponent-uml} +\end{sidewaysfigure} + + +\begin{table} +\caption{Descriptions of the rocket component classes and their + functionality. Abstract classes are shown in italics.} +\label{table-rocketcomponents} +\sloppy\footnotesize +\vspace{\baselineskip} +\hspace{-5mm}\begin{tabular}{lp{80mm}} +Component class & Description \\ +\hline +\code{\textit{RocketComponent}}: & + The base class of all components, + including common features such as child component handling. \\ + +\hspace{3mm}\code{Rocket}: & + The base component of a + rocket design, provides change event notifications. \\ + +\hspace{3mm}\code{\textit{ComponentAssembly}}: & + A base component for an assembly of external components. This could + in the future be extended to allow multiple rocket bodies next to + each other. \\ + +\hspace{6mm}\code{Stage}: & + A separable stage of the rocket. \\ + +\hspace{3mm}\code{\textit{ExternalComponent}}: & + An external component that has an effect on the aerodynamics of the + rocket. \\ + +\hspace{6mm}\code{\textit{BodyComponent}}: & + A portion of the main rocket body, defined in cylindrical coordinates + by $r = f(x, \theta)$. \\ + +\hspace{9mm}\code{\textit{SymmetricComponent}}: & + An axisymmetrical body component. \\ + +\hspace{12mm}\code{Transition}: & + A symmetrical transition (shoulder or boattail). \\ + +\hspace{15mm}\code{NoseCone}: & + A transition with the initial radius zero. \\ + +\hspace{12mm}\code{BodyTube}: & + A cylindrical body tube. Can be used as a motor mount. \\ + +\hspace{6mm}\code{\textit{FinSet}}: & + A set of one or more fins. \\ + +\hspace{9mm}\code{TrapezoidalFinSet}: & + A set of trapezoidal fins. \\ + +\hspace{9mm}\code{EllipticalFinSet}: & + A set of elliptical fins. \\ + +\hspace{9mm}\code{FreeformFinSet}: & + A set of free-form fins. \\ + +\hspace{6mm}\code{LaunchLug}: & + A launch lug or rail pin. \\ + +\hspace{3mm}\code{\textit{InternalComponent}}: & + An internal component that may affect the mass of the rocket but not + its aerodynamics. \\ + +\hspace{6mm}\code{\textit{StructuralComponent}}: & + A structural internal component, with specific shape and density. \\ + +\hspace{9mm}\code{\textit{RingComponent}}: & + A hollow cylindrical component. \\ + +\hspace{12mm}\code{\textit{ThicknessRingComponent}}: & + A component defined by an outer radius and shell thickness. \\ + +\hspace{15mm}\code{InnerTube}: & + An inner tube. Can be used as a motor mount and can be clustered. \\ + +\hspace{15mm}\code{TubeCoupler}: & + A coupler tube. \\ + +\hspace{15mm}\code{EngineBlock}: & + An engine block. \\ + +\hspace{12mm}\code{\textit{RadiusRingComponent}}: & + A component defined by an inner and outer radius. \\ + +\hspace{15mm}\code{CenteringRing}: & + A ring for centering components. \\ + +\hspace{15mm}\code{Bulkhead}: & + A solid bulkhead (inner radius zero). \\ + +\hspace{6mm}\code{\textit{MassObject}}: & + An internal component shaped approximately like a solid cylinder and + with a specific mass. \\ + +\hspace{9mm}\code{MassComponent}: & + A generic component with specific mass, for example payload. \\ + +\hspace{9mm}\code{\textit{RecoveryDevice}}: & + A recovery device. \\ + +\hspace{12mm}\code{Parachute}: & + A parachute. \\ + +\hspace{12mm}\code{Streamer}: & + A streamer. \\ + +\hspace{9mm}\code{ShockCord}: & + A shock cord with a specified material and length. \\ + +\hline +\end{tabular} +\end{table} + + +Additionally three interfaces are defined for the components, +\code{MotorMount}, \code{Clusterable} and \code{RadialParent}. +Components implementing the \code{Motor\-Mount} interface, currently +\code{BodyTube} and \code{InnerTube}, can function as motor mounts and +have motors loaded in them. The \code{Clusterable} interface +signifies that the component can be clustered in various +configurations. Currently only the \code{InnerTube} component can be +clustered. Components and motors that are attached to a clustered +inner tube are automatically replicated to all tubes within the +cluster. The \code{RadialParent} interface allows inner components to +automatically identify their correct inner and outer radii based on +their parent and sibling components. For example, a coupler tube can +automatically detect its radius based on the inner radius of +the parent body tube. + + +Since the software functionality is divided into different packages, +all component similarities cannot be directly be exploited through +inheritance. For example, the method of drawing a nose cone shape +belongs to the \url{gui.rocketfigure} package, however, it can share +the same code that draws a transition. For these purposes, reflective +programming is used extensively. The code for drawing both nose cones +and transitions is provided in the class +\code{gui.rocketfigure.SymmetricComponentShapes}, while the simpler +body tube is drawn by the class \code{BodyTubeShapes}. The correct +class is derived and instantiated dynamically based on the component +class. This allows easily sharing functionality common to different +components while still having loose coupling between the rocket +structure, presentation, computation and storage methods. + + + + +\subsection{Aerodynamic calculators and simulators} +\label{sec-simulator-calculator} + +One of the key aspects in the design of the simulation implementation +was extensibility. Therefore all aerodynamic calculation code is +separated in the package \code{aerodynamics} and all simulation code +is in the package \code{simulator}. + +The basis for aerodynamic calculations is the abstract class +\code{Aerodynamic\-Calculator}, while the simulators are subclasses of +the \code{Flight\-Simulator} class. This allows adding new +implementations of the aerodynamic calculators and simulators +independently. For example, a simulator using Euler integration was +written in the early stages of development, and later replaced by the +Runge-Kutta~4 simulator. Similarly, a different method of calculating +the aerodynamic forces, such as CFD, could be implemented and used by +the existing simulators. + +Similar abstraction has been performed for the atmospheric temperature +and pressure model by the abstract class \code{AtmosphericModel} and +different rocket motor types by the \code{Motor} class, among others. + + + + + +\subsection{Simulation listeners} +\label{sec-listeners} + +Simulation listeners are pieces of code that can dynamically be +configured to listen to and interact with a simulation while it is +running. The listeners are called after each simulation step, at each +simulation event and both when the flight conditions and aerodynamic +forces have been calculated. The listeners may simply gather flight +data for use outside the simulation or modify the rocket or simulation +during the flight. This allows great potential for extensibility both +internally and externally. + +Listeners are used internally for various purposes such as retrieving +flight progress information when the user is running simulations and +cancelling the simulations when necessary. Implementing such +functionality otherwise would have required a lot of special case +handling directly within the simulation code. + +Listeners can also be used to modify the simulation or the rocket +during its flight. The successor project of Haisunäätä will include +an active roll stabilization system, where a flight computer will +measure the roll rate using two magnetometers and use a PID controller +to adjust two auxiliary fins to cancel out the roll inevitably +produced by imperfections in the main fins. A simulation listener was +written that first simulated the PID controller purely in Java, which +modified the cant angle of the auxiliary fins during the simulation. +Later a similar listener interfaced the external flight computer +directly using a serial data link. The listener fed the flight data +to the controller which computed and reported the control actions back +to the simulator. This system helped identify and fix numerous bugs +in the flight computer software, which would have otherwise been +nearly impossible to fully test. It is expected that the simulation +listeners will be an invaluable tool for more ambitious model rocket +enthusiasts. + +A listener is produced by implementing the \code{SimulationListener} +interface or by extending the \code{AbstractSimulationListener} +class. The UI includes the option of defining custom simulation +listeners to be utilized during flight simulation. + + +\subsection{Warnings} +\label{sec-warnings} + +The aerodynamic calculations and simulations are based on certain +assumptions and simplifications, such as a low angle of attack and a +smooth, continuous rocket body. The rocket component architecture +makes it possible to create designs that break these assumptions. +Instead of limiting the options of the design, the aerodynamic +calculator and simulator can produce warnings about such issues. +These warnings are presented to the user during the design of the +rocket or after simulations. It is then left up to the user to judge +whether such violations are significant enough to cast doubt to the +validity of the results. + + +\subsection{File format} + +An XML-based file format was devised for storing the rocket designs +and simulations. The use of XML allows using many existing tools for +reading and writing the files, allows easy extensibility and makes the +files human-readable. The user has the option of including all +simulated data in the file, storing the data at specific time +intervals or storing only the simulation launch conditions. To reduce +the file size, the files can additionally be compressed using the +standard GZIP compression algorithm~\cite{GZIP}. The files are +compressed and uncompressed automatically by the software. The file +extension .ORK was chosen for the design files, an abbreviation of the +software name that at the same time had no previous uses as a file +extension. + + + + + +\section{User interface design} + +The user interface was designed with the intent of being robust but +yet easy to use even for inexperienced users. The main window, +depicted in Figure~\ref{fig-main-window}(a) with the design of the +original Haisunäätä rocket, consists of a schematic drawing of the +rocket, the tree structure of the rocket components and +buttons for adding new components to the structure. Components can +be selected or edited by clicking and double-clicking either the tree +view or the component in the schematic diagram. The selected +components are drawn in bold to give a visual clue to the position +of the component. + +The schematic drawing can be viewed either from the side or from the +rear, can be zoomed in and out and rotated along the centerline. The +schematic diagram view also presents basic information about the +rocket, such as the design name, length, maximum diameter, mass and +possible warnings about the design. It also calculates the CG and CP +positions continuously during design and shows them both numerically +and on the diagram. Additionally, a simulation is automatically run +in the background after each modification and the main results are +presented in the lower left corner of the diagram. Many users are +interested in the maximum altitude or velocity of the rocket, and this +allows an immediate feedback on the effect of the changes they are +making to the design. The flight information typically takes less +than a second to update. + +The upper part of the main window can also be changed to view +simulation results, Figure~\ref{fig-main-window}(b). Many simulations +can be added with different launch conditions and motor configurations +to investigate their effects. Each simulation has a row which +presents the basic information about the simulation. The first column +gives an immediate visual clue to the status of the simulation; a gray +ball indicates that the simulation has not been run yet, green +indicates an up-to-date simulation, red indicates that the design has +been changed after the simulation was run and yellow indicates that +the simulation information was loaded from a file, but that the file +states it to be up-to-date. The simulations can be run one or several +at a time. The software automatically utilizes several threads when +running multiple simulations on a multi-CPU computer to utilize the +full processing capacity. + +Figure~\ref{fig-various-dialogs} shows two dialogs that are used +to modify and analyze the designs. The components are edited using a +small dialog window that allows the user to either fill in the exact +numerical values specifying the shape of the component or use sliders +to modify them. The user can change the units by clicking on them, or +set default values from the preferences. Different tabs allow control +over the mass and CG override settings, figure color options, motor +mount and cluster options and more. The Component analysis dialog +shown in the figure can be used to analyze the effect of individual +components on the total stability, drag and roll characteristics of +the rocket. + +Similarly, the launch conditions and simulator options can be edited +in the corresponding dialog. The simulator options also allow the +user to define custom simulation listeners to use during the +simulations. The simulation edit dialog is also used for later data +analysis. The simulated data can be plotted in a variety of ways as +shown in Figure~\ref{fig-plotting}. The user can use predefined plot +settings or define their own. Up to 15 different variables out of the +47 quantities computed can be plotted at a time. The variable on the +horizontal axis can be freely selected, and the other variables can be +plotted on one of two vertical axis, on either side of the plot. The +user can either specify whether a variable should plot on the left or +right axis or let the software decide the most suitable axis. Typical +plots include the altitude, vertical velocity and acceleration of the +rocket with time or the drag coefficient as a function of the Mach +number. + + +\begin{figure} +\hspace{-7mm} +\begin{tabular}{m{1mm}m{10cm}} +\hspace{-5mm}(a) & +\epsfig{file=figures/pix/openrocket-main-haisunaata,scale=0.5} \\ +\hspace{-5mm}(b) & +\epsfig{file=figures/pix/openrocket-simulations-haisunaata,scale=0.5} + +\end{tabular} +\caption{The main design window of OpenRocket; (a) the design + view and (b) the simulation view.} +\label{fig-main-window} +\end{figure} + +\begin{figure} +\centering +\epsfig{file=figures/pix/openrocket-dialog-edit,scale=0.5} +\vspace{2cm} \\ +\epsfig{file=figures/pix/openrocket-dialog-analysis,scale=0.5} +\vspace{1cm} \\ +\caption{Dialog windows for editing the properties of a nose cone + and for analyzing the influence of individual components on + the stability, drag and roll characteristics of the rocket.} +\label{fig-various-dialogs} +\end{figure} + +\begin{figure} +\centering +\epsfig{file=figures/pix/openrocket-dialog-plot-options,scale=0.5} +\vspace{2cm} \\ +\epsfig{file=figures/pix/openrocket-dialog-plot,scale=0.5} +\vspace{1cm} \\ +\caption{Dialog window for changing the simulation plot options and a + plot of the simulated flight of the Haisunäätä hybrid rocket.} +\label{fig-plotting} +\end{figure} + + +\section{Future enhancements} + +Numerous features have been planned and taken into account during the +design of the software. Below are listed a few of the planned +features and how they have been taken into account: + +{\it Alternative aerodynamic calculators.} For example CFD could be +used to calculate the aerodynamic properties, allowing even better +simulation accuracy. The calculators have been abstracted by the +\code{AerodynamicCalculator} class so they can easily be +interchanged. + +{\it Alternative simulators.} These could take into account for +example the curvature of the Earth and include the Coriolis effect. +New simulators can be created by extending the abstract +\code{FlightSimulator} class. + +{\it Export and import of flight data.} The simulated data could be +exported for further analysis as comma separated values (CSV). +Similarly, experimental data could be imported either from files or +directly from altimeters. Support for imported data already exists in +the core functionalities. + +{\it Importing database files.} The motor database is easily +extendable to read external thrust curves. Also data of commercially +available rocket components could be imported and available in the +component edit dialog. + +{\it Further UI enhancements.} These could include for example a 3D +view of the rocket, an animation of the rocket flight, a ``wizard'' +dialog for easily creating new designs, {\it etc.} + diff --git a/doc/techdoc/figures/aerodynamics/aero-force-components.eps b/doc/techdoc/figures/aerodynamics/aero-force-components.eps new file mode 100644 index 00000000..5b7e8d87 --- /dev/null +++ b/doc/techdoc/figures/aerodynamics/aero-force-components.eps @@ -0,0 +1,338 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: aero-force-components.fig +%%Creator: fig2dev Version 3.2 Patchlevel 5 +%%CreationDate: Tue Jun 10 15:56:26 2008 +%%For: sampo@porro (Sampo Niskanen,,,) +%%BoundingBox: 0 0 139 240 +%Magnification: 1.0000 +%%EndComments +%%BeginProlog +/$F2psDict 200 dict def +$F2psDict begin +$F2psDict /mtrx matrix put +/col-1 {0 setgray} bind def +/col0 {0.000 0.000 0.000 srgb} bind def 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state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 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{MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup 3 mul string + currentfile + 3 index 0 eq {/ASCIIHexDecode filter} + {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } + ifelse exch readstring pop + dup 0 3 index getinterval /rbmap xdef + dup 2 index dup getinterval /gbmap xdef + 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + rbmap gbmap bbmap true 3 colorimage gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 33 246 577 545 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 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2407 mt 4707 2357 L +4707 2357 mt 4754 2349 L +4707 2357 mt 4754 2349 L +4754 2349 mt 4713 2308 L +4747 1993 mt 4705 1948 L +4705 1948 mt 4752 1939 L +4705 1948 mt 4752 1939 L +4752 1939 mt 4711 1896 L +4743 2457 mt 4702 2416 L +4702 2416 mt 4749 2407 L +4702 2416 mt 4749 2407 L +4749 2407 mt 4707 2357 L +4741 2046 mt 4700 2001 L +4700 2001 mt 4747 1993 L +4700 2001 mt 4747 1993 L +4747 1993 mt 4705 1948 L +4696 2465 mt 4743 2457 L +4743 2457 mt 4702 2416 L +4736 2099 mt 4694 2055 L +4694 2055 mt 4741 2046 L +4694 2055 mt 4741 2046 L +4741 2046 mt 4700 2001 L +gr + +4741 3176 mt +(Width / m) s +gs 848 268 5060 2919 rc +4730 2152 mt 4689 2108 L +4689 2108 mt 4736 2099 L +4689 2108 mt 4736 2099 L +4736 2099 mt 4694 2055 L +4728 1736 mt 4687 1692 L +4687 1692 mt 4734 1684 L +4687 1692 mt 4734 1684 L +4734 1684 mt 4692 1639 L +4724 2205 mt 4683 2161 L +4683 2161 mt 4730 2152 L +4683 2161 mt 4730 2152 L +4730 2152 mt 4689 2108 L +4722 1790 mt 4681 1745 L +4681 1745 mt 4728 1736 L +4681 1745 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mt 4634 1755 L +4666 2316 mt 4625 2273 L +4625 2273 mt 4672 2265 L +4625 2273 mt 4672 2265 L +4672 2265 mt 4630 2222 L +4664 1905 mt 4623 1860 L +4623 1860 mt 4670 1851 L +4623 1860 mt 4670 1851 L +4670 1851 mt 4628 1808 L +4660 2365 mt 4619 2324 L +4619 2324 mt 4666 2316 L +4619 2324 mt 4666 2316 L +4666 2316 mt 4625 2273 L +4659 1956 mt 4617 1913 L +4617 1913 mt 4664 1905 L +4617 1913 mt 4664 1905 L +4664 1905 mt 4623 1860 L +4655 2425 mt 4613 2373 L +4613 2373 mt 4660 2365 L +4613 2373 mt 4660 2365 L +4660 2365 mt 4619 2324 L +4653 2010 mt 4612 1965 L +4612 1965 mt 4659 1956 L +4612 1965 mt 4659 1956 L +4659 1956 mt 4617 1913 L +4649 2473 mt 4608 2434 L +4608 2434 mt 4655 2425 L +4608 2434 mt 4655 2425 L +4655 2425 mt 4613 2373 L +4647 2063 mt 4606 2018 L +4606 2018 mt 4653 2010 L +4606 2018 mt 4653 2010 L +4653 2010 mt 4612 1965 L +4645 1648 mt 4604 1604 L +4604 1604 mt 4651 1595 L +4604 1604 mt 4651 1595 L +4651 1595 mt 4610 1551 L +4643 2533 mt 4602 2481 L +4602 2481 mt 4649 2473 L +4602 2481 mt 4649 2473 L +4649 2473 mt 4608 2434 L +4642 2117 mt 4600 2072 L +4600 2072 mt 4647 2063 L +4600 2072 mt 4647 2063 L +4647 2063 mt 4606 2018 L +4640 1701 mt 4598 1657 L +4598 1657 mt 4645 1648 L +4598 1657 mt 4645 1648 L +4645 1648 mt 4604 1604 L +4597 2540 mt 4643 2533 L +4643 2533 mt 4602 2481 L +4636 2170 mt 4595 2125 L +4595 2125 mt 4642 2117 L +4595 2125 mt 4642 2117 L +4642 2117 mt 4600 2072 L +4634 1755 mt 4593 1710 L +4593 1710 mt 4640 1701 L +4593 1710 mt 4640 1701 L +4640 1701 mt 4598 1657 L +4630 2222 mt 4589 2179 L +4589 2179 mt 4636 2170 L +4589 2179 mt 4636 2170 L +4636 2170 mt 4595 2125 L +4628 1808 mt 4587 1764 L +4587 1764 mt 4634 1755 L +4587 1764 mt 4634 1755 L +4634 1755 mt 4593 1710 L +4625 2273 mt 4583 2231 L +4583 2231 mt 4630 2222 L +4583 2231 mt 4630 2222 L +4630 2222 mt 4589 2179 L +4623 1860 mt 4582 1817 L +4582 1817 mt 4628 1808 L +4582 1817 mt 4628 1808 L +4628 1808 mt 4587 1764 L +4619 2324 mt 4578 2282 L +4578 2282 mt 4625 2273 L +4578 2282 mt 4625 2273 L +4625 2273 mt 4583 2231 L +0.5 sg +4577 1515 mt 4577 2308 L +4577 1515 mt 4577 2308 L +0 sg +4617 1913 mt 4576 1869 L +4576 1869 mt 4623 1860 L +4576 1869 mt 4623 1860 L +4623 1860 mt 4582 1817 L +4613 2373 mt 4572 2332 L +4572 2332 mt 4619 2324 L +4572 2332 mt 4619 2324 L +4619 2324 mt 4578 2282 L +4612 1965 mt 4570 1922 L +4570 1922 mt 4617 1913 L +4570 1922 mt 4617 1913 L +4617 1913 mt 4576 1869 L +4608 2434 mt 4566 2381 L +4566 2381 mt 4613 2373 L +4566 2381 mt 4613 2373 L +4613 2373 mt 4572 2332 L +4606 2018 mt 4565 1973 L +4565 1973 mt 4612 1965 L +4565 1973 mt 4612 1965 L +4612 1965 mt 4570 1922 L +4604 1604 mt 4563 1559 L +4563 1559 mt 4610 1551 L +4153 1945 mt 4200 1937 L +4200 1937 mt 4159 1893 L +0.5 sg +4163 1949 mt 4163 2543 L +4163 1949 mt 4163 2543 L +0 sg +4195 1990 mt 4153 1945 L +4153 1945 mt 4200 1937 L +0.5 sg +4203 1115 mt 4203 2183 L +0 sg +4200 1937 mt 4159 1893 L +4159 1893 mt 4206 1885 L +4200 1937 mt 4247 1930 L +4247 1930 mt 4206 1885 L +4206 1885 mt 4253 1877 L +4253 1877 mt 4212 1832 L +4247 1930 mt 4206 1885 L +4206 1885 mt 4253 1877 L +4253 1877 mt 4212 1832 L +4212 1832 mt 4259 1824 L +4253 1877 mt 4300 1869 L +4300 1869 mt 4259 1824 L +4259 1824 mt 4305 1815 L +4305 1815 mt 4264 1771 L +4300 1869 mt 4259 1824 L +4259 1824 mt 4305 1815 L +4305 1815 mt 4264 1771 L +4264 1771 mt 4311 1763 L +4305 1815 mt 4352 1807 L +4352 1807 mt 4311 1763 L +4311 1763 mt 4358 1755 L +4358 1755 mt 4317 1711 L +4352 1807 mt 4311 1763 L +4311 1763 mt 4358 1755 L +4358 1755 mt 4317 1711 L +4317 1711 mt 4364 1702 L +4358 1755 mt 4405 1747 L +4405 1747 mt 4364 1702 L +4364 1702 mt 4411 1694 L +4411 1694 mt 4369 1648 L +4405 1747 mt 4364 1702 L +4364 1702 mt 4411 1694 L +4411 1694 mt 4369 1648 L +4369 1648 mt 4416 1640 L +4411 1694 mt 4458 1685 L +4458 1685 mt 4416 1640 L +4458 1685 mt 4416 1640 L +4416 1640 mt 4463 1631 L +4458 1685 mt 4505 1675 L +4505 1675 mt 4463 1631 L +4463 1631 mt 4510 1622 L +4510 1622 mt 4469 1576 L +4505 1675 mt 4463 1631 L +4463 1631 mt 4510 1622 L +4510 1622 mt 4469 1576 L +4469 1576 mt 4516 1568 L +4510 1622 mt 4557 1613 L +4557 1613 mt 4516 1568 L +4557 1613 mt 4516 1568 L +4516 1568 mt 4563 1559 L +gs 1939 557 3545 2356 rc +60 W +4577 1515 PD +gr + +gs 1939 557 3545 2356 rc +60 W +4203 1115 PD +gr + +4191 1160 mt 4238 1153 L +4238 1153 mt 4197 1107 L +gs 1939 557 3545 2356 rc +60 W +4577 1515 PD +gr + +4563 1559 mt 4610 1551 L +4610 1551 mt 4568 1506 L +4602 2481 mt 4561 2444 L +4561 2444 mt 4608 2434 L +4561 2444 mt 4608 2434 L +4608 2434 mt 4566 2381 L +4600 2072 mt 4559 2027 L +4559 2027 mt 4606 2018 L +4559 2027 mt 4606 2018 L +4606 2018 mt 4565 1973 L +4598 1657 mt 4557 1613 L +4557 1613 mt 4604 1604 L +4557 1613 mt 4604 1604 L +4604 1604 mt 4563 1559 L +4597 2540 mt 4555 2490 L +4555 2490 mt 4602 2481 L +4555 2490 mt 4602 2481 L +4602 2481 mt 4561 2444 L +4595 2125 mt 4553 2081 L +4553 2081 mt 4600 2072 L +4553 2081 mt 4600 2072 L +4600 2072 mt 4559 2027 L +4593 1710 mt 4551 1666 L +4551 1666 mt 4598 1657 L +4551 1666 mt 4598 1657 L +4598 1657 mt 4557 1613 L +4550 2548 mt 4597 2540 L +4597 2540 mt 4555 2490 L +4589 2179 mt 4548 2134 L +4548 2134 mt 4595 2125 L +4548 2134 mt 4595 2125 L +4595 2125 mt 4553 2081 L +4587 1764 mt 4546 1720 L +4546 1720 mt 4593 1710 L +4546 1720 mt 4593 1710 L +4593 1710 mt 4551 1666 L +4583 2231 mt 4542 2187 L +4542 2187 mt 4589 2179 L +4542 2187 mt 4589 2179 L +4589 2179 mt 4548 2134 L +4582 1817 mt 4540 1773 L +4540 1773 mt 4587 1764 L +4540 1773 mt 4587 1764 L +4587 1764 mt 4546 1720 L +4578 2282 mt 4536 2239 L +4536 2239 mt 4583 2231 L +4536 2239 mt 4583 2231 L +4583 2231 mt 4542 2187 L +4576 1869 mt 4535 1825 L +4535 1825 mt 4582 1817 L +4535 1825 mt 4582 1817 L +4582 1817 mt 4540 1773 L +4572 2332 mt 4531 2291 L +4531 2291 mt 4578 2282 L +4531 2291 mt 4578 2282 L +4578 2282 mt 4536 2239 L +4570 1922 mt 4529 1879 L +4529 1879 mt 4576 1869 L +4529 1879 mt 4576 1869 L +4576 1869 mt 4535 1825 L +4566 2381 mt 4525 2340 L +4525 2340 mt 4572 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+4101 2006 mt 4148 1998 L +4148 1998 mt 4106 1953 L +4142 2048 mt 4101 2006 L +4101 2006 mt 4148 1998 L +4148 1998 mt 4106 1953 L +4106 1953 mt 4153 1945 L +gs 1939 557 3545 2356 rc +60 W +4163 1949 PD +gr + +4048 2064 mt 4095 2056 L +4095 2056 mt 4054 2014 L +3714 2170 mt 3673 2123 L +3673 2123 mt 3720 2116 L +3714 2170 mt 3761 2163 L +3761 2163 mt 3720 2116 L +3849 2203 mt 3896 2195 L +3896 2195 mt 3855 2148 L +3896 2195 mt 3855 2148 L +3855 2148 mt 3902 2141 L +3896 2195 mt 3943 2188 L +3943 2188 mt 3902 2141 L +3984 2234 mt 4031 2226 L +4031 2226 mt 3990 2180 L +4031 2226 mt 3990 2180 L +3990 2180 mt 4037 2172 L +4031 2226 mt 4078 2218 L +4078 2218 mt 4037 2172 L +4073 2272 mt 4031 2226 L +4031 2226 mt 4078 2218 L +4073 2272 mt 4120 2264 L +4120 2264 mt 4078 2218 L +4120 2264 mt 4078 2218 L +4078 2218 mt 4125 2210 L +gs 1939 557 3545 2356 rc +60 W +4096 2238 PD +gr + +4054 2014 mt 4101 2006 L +4101 2006 mt 4059 1960 L +4095 2056 mt 4054 2014 L +4054 2014 mt 4101 2006 L +4101 2006 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+3697 2330 mt 3744 2324 L +3697 2330 mt 3744 2324 L +3744 2324 mt 3703 2279 L +3736 1958 mt 3695 1913 L +3695 1913 mt 3742 1906 L +3695 1913 mt 3742 1906 L +3742 1906 mt 3701 1860 L +3735 1547 mt 3693 1503 L +3693 1503 mt 3740 1495 L +3693 1503 mt 3740 1495 L +3740 1495 mt 3699 1451 L +3733 1125 mt 3691 1082 L +3691 1082 mt 3738 1075 L +3691 1082 mt 3738 1075 L +3738 1075 mt 3697 1031 L +3733 2415 mt 3691 2377 L +3691 2377 mt 3738 2371 L +3691 2377 mt 3738 2371 L +3738 2371 mt 3697 2330 L +3731 2011 mt 3689 1965 L +3689 1965 mt 3736 1958 L +3689 1965 mt 3736 1958 L +3736 1958 mt 3695 1913 L +3729 1596 mt 3688 1554 L +3688 1554 mt 3735 1547 L +3688 1554 mt 3735 1547 L +3735 1547 mt 3693 1503 L +3727 1182 mt 3686 1132 L +3686 1132 mt 3733 1125 L +3686 1132 mt 3733 1125 L +3733 1125 mt 3691 1082 L +3727 2463 mt 3686 2421 L +3686 2421 mt 3733 2415 L +3686 2421 mt 3733 2415 L +3733 2415 mt 3691 2377 L +3725 2064 mt 3684 2018 L +3684 2018 mt 3731 2011 L +3684 2018 mt 3731 2011 L +3731 2011 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+3485 2145 mt 3532 2140 L +3485 2145 mt 3532 2140 L +3532 2140 mt 3490 2094 L +3524 1780 mt 3483 1734 L +3483 1734 mt 3530 1727 L +3483 1734 mt 3530 1727 L +3530 1727 mt 3489 1680 L +3522 1375 mt 3481 1328 L +3481 1328 mt 3528 1321 L +3481 1328 mt 3528 1321 L +3528 1321 mt 3487 1274 L +3522 2666 mt 3481 2614 L +3481 2614 mt 3528 2609 L +3481 2614 mt 3528 2609 L +3528 2609 mt 3487 2556 L +3520 955 mt 3479 909 L +3479 909 mt 3526 903 L +3479 909 mt 3526 903 L +3526 903 mt 3485 855 L +3520 2244 mt 3479 2196 L +3479 2196 mt 3526 2191 L +3479 2196 mt 3526 2191 L +3526 2191 mt 3485 2145 L +3519 1832 mt 3477 1786 L +3477 1786 mt 3524 1780 L +3477 1786 mt 3524 1780 L +3524 1780 mt 3483 1734 L +3517 1427 mt 3475 1381 L +3475 1381 mt 3522 1375 L +3475 1381 mt 3522 1375 L +3522 1375 mt 3481 1328 L +3517 2718 mt 3475 2671 L +3475 2671 mt 3522 2666 L +3475 2671 mt 3522 2666 L +3522 2666 mt 3481 2614 L +3515 1007 mt 3474 962 L +3474 962 mt 3520 955 L +3474 962 mt 3520 955 L +3520 955 mt 3479 909 L +3515 2296 mt 3473 2247 L +3473 2247 mt 3520 2244 L +3473 2247 mt 3520 2244 L +3520 2244 mt 3479 2196 L +3513 1885 mt 3472 1837 L +3472 1837 mt 3519 1832 L +3472 1837 mt 3519 1832 L +3519 1832 mt 3477 1786 L +3511 1479 mt 3470 1433 L +3470 1433 mt 3517 1427 L +3470 1433 mt 3517 1427 L +3517 1427 mt 3475 1381 L +3470 2726 mt 3517 2718 L +3517 2718 mt 3475 2671 L +3509 1058 mt 3468 1014 L +3468 1014 mt 3515 1007 L +3468 1014 mt 3515 1007 L +3515 1007 mt 3474 962 L +3468 2301 mt 3515 2296 L +3515 2296 mt 3473 2247 L +3507 1938 mt 3466 1891 L +3466 1891 mt 3513 1885 L +3466 1891 mt 3513 1885 L +3513 1885 mt 3472 1837 L +3505 1531 mt 3464 1485 L +3464 1485 mt 3511 1479 L +3464 1485 mt 3511 1479 L +3511 1479 mt 3470 1433 L +3504 1108 mt 3462 1065 L +3462 1065 mt 3509 1058 L +3462 1065 mt 3509 1058 L +3509 1058 mt 3468 1014 L +3500 1582 mt 3458 1537 L +3458 1537 mt 3505 1531 L +3458 1537 mt 3505 1531 L +3505 1531 mt 3464 1485 L +3498 1162 mt 3457 1115 L +3457 1115 mt 3504 1108 L +3457 1115 mt 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+3395 1694 mt 3442 1687 L +3395 1694 mt 3442 1687 L +3442 1687 mt 3400 1645 L +3434 1335 mt 3393 1288 L +3393 1288 mt 3440 1281 L +3393 1288 mt 3440 1281 L +3440 1281 mt 3398 1234 L +3434 2619 mt 3393 2565 L +3393 2565 mt 3440 2560 L +3393 2565 mt 3440 2560 L +3440 2560 mt 3398 2508 L +3432 915 mt 3391 866 L +3391 866 mt 3438 861 L +3391 866 mt 3438 861 L +3438 861 mt 3397 810 L +3432 2201 mt 3391 2154 L +3391 2154 mt 3438 2150 L +3391 2154 mt 3438 2150 L +3438 2150 mt 3396 2103 L +3430 1791 mt 3389 1746 L +3389 1746 mt 3436 1740 L +3389 1746 mt 3436 1740 L +3436 1740 mt 3395 1694 L +3428 1388 mt 3387 1342 L +3387 1342 mt 3434 1335 L +3387 1342 mt 3434 1335 L +3434 1335 mt 3393 1288 L +3428 2677 mt 3387 2623 L +3387 2623 mt 3434 2619 L +3387 2623 mt 3434 2619 L +3434 2619 mt 3393 2565 L +3427 968 mt 3385 921 L +3385 921 mt 3432 915 L +3385 921 mt 3432 915 L +3432 915 mt 3391 866 L +3427 2251 mt 3385 2205 L +3385 2205 mt 3432 2201 L +3385 2205 mt 3432 2201 L +3432 2201 mt 3391 2154 L 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Version 7.6.0.324 (R2008a). Operating System: Linux 2.6.28-11-server #37-Ubuntu SMP Mon Mar 23 17:33:24 UTC 2009 x86_64. +%%Title: /m/fs/user3/a/38/spniskan/thesis/flight-roll-rate.eps +%%CreationDate: 05/09/2009 01:39:32 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%LanguageLevel: 2 +%%Pages: 1 +%%BoundingBox: 33 200 577 592 +%%EndComments + +%%BeginProlog +% MathWorks dictionary +/MathWorks 160 dict begin +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rm /rmoveto ldef +/rl /rlineto ldef +/s {show newpath} bdef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinewidth 2 div 0 360 arc fill + PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup 3 mul string + currentfile + 3 index 0 eq {/ASCIIHexDecode filter} + {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } + ifelse exch readstring pop + dup 0 3 index getinterval /rbmap xdef + dup 2 index dup getinterval /gbmap xdef + 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + rbmap gbmap bbmap true 3 colorimage gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 33 200 577 592 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 12 def +portraitMode 0396 7104 csm + + 0 0 6528 4703 rc +85 dict begin %Colortable dictionary +/c0 { 0.000000 0.000000 0.000000 sr} bdef +/c1 { 1.000000 1.000000 1.000000 sr} bdef +/c2 { 0.900000 0.000000 0.000000 sr} bdef +/c3 { 0.000000 0.820000 0.000000 sr} bdef +/c4 { 0.000000 0.000000 0.800000 sr} bdef +/c5 { 0.910000 0.820000 0.320000 sr} bdef +/c6 { 1.000000 0.260000 0.820000 sr} bdef +/c7 { 0.000000 0.820000 0.820000 sr} bdef +c0 +1 j +1 sg + 0 0 6529 4704 rf +6 w +0 3833 5059 0 0 -3833 849 4186 4 MP +PP +-5059 0 0 3833 5059 0 0 -3833 849 4186 5 MP stroke +4 w +DO +SO +6 w +0 sg + 849 4186 mt 5908 4186 L + 849 353 mt 5908 353 L + 849 4186 mt 849 353 L +5908 4186 mt 5908 353 L + 849 4186 mt 5908 4186 L + 849 4186 mt 849 353 L + 849 4186 mt 849 4135 L + 849 353 mt 849 403 L +%%IncludeResource: font Helvetica +/Helvetica /ISOLatin1Encoding 120 FMSR + + 816 4331 mt +(0) s +1692 4186 mt 1692 4135 L +1692 353 mt 1692 403 L +1659 4331 mt +(1) s +2535 4186 mt 2535 4135 L +2535 353 mt 2535 403 L +2502 4331 mt +(2) s +3378 4186 mt 3378 4135 L +3378 353 mt 3378 403 L +3345 4331 mt +(3) s +4221 4186 mt 4221 4135 L +4221 353 mt 4221 403 L +4188 4331 mt +(4) s +5064 4186 mt 5064 4135 L +5064 353 mt 5064 403 L +5031 4331 mt +(5) s +5908 4186 mt 5908 4135 L +5908 353 mt 5908 403 L +5875 4331 mt +(6) s + 849 4186 mt 899 4186 L +5908 4186 mt 5857 4186 L + 748 4230 mt +(0) s + 849 3760 mt 899 3760 L +5908 3760 mt 5857 3760 L + 748 3804 mt +(2) s + 849 3334 mt 899 3334 L +5908 3334 mt 5857 3334 L + 748 3378 mt +(4) s + 849 2908 mt 899 2908 L +5908 2908 mt 5857 2908 L + 748 2952 mt +(6) s + 849 2482 mt 899 2482 L +5908 2482 mt 5857 2482 L + 748 2526 mt +(8) s + 849 2056 mt 899 2056 L +5908 2056 mt 5857 2056 L + 681 2100 mt +(10) s + 849 1630 mt 899 1630 L +5908 1630 mt 5857 1630 L + 681 1674 mt +(12) s + 849 1204 mt 899 1204 L +5908 1204 mt 5857 1204 L + 681 1248 mt +(14) s + 849 778 mt 899 778 L +5908 778 mt 5857 778 L + 681 822 mt +(16) s + 849 353 mt 899 353 L +5908 353 mt 5857 353 L + 681 397 mt +(18) s + 849 4186 mt 5908 4186 L + 849 353 mt 5908 353 L + 849 4186 mt 849 353 L +5908 4186 mt 5908 353 L +gs 849 353 5060 3834 rc +4 0 7 1 7 2 7 1 7 1 6 1 7 1 7 1 +6 1 7 2 6 1 6 1 6 1 7 1 6 1 6 1 +6 2 5 1 6 1 6 1 6 1 5 1 6 1 5 2 +6 1 5 1 6 1 5 1 5 1 5 1 5 2 5 1 +5 1 5 1 5 1 5 1 4 1 5 2 5 1 4 1 +5 1 4 1 5 1 4 2 4 1 5 1 4 1 4 1 +4 1 4 1 4 2 4 1 4 1 4 1 4 1 3 1 +4 1 4 2 3 1 4 1 4 1 3 1 3 1 4 1 +3 2 4 1 3 1 3 1 3 1 4 1 3 1 3 2 +3 1 3 1 3 1 3 1 3 1 3 1 2 2 3 1 +3 1 3 1 3 1 2 1 3 1 3 2 2 1 3 1 +2 1 3 1 2 1 3 1 2 2 3 1 2 1 2 1 +3 1 2 1 2 1 3 2 2 1 2 1 2 1 2 1 +3 1 2 2 2 1 2 1 2 1 2 1 2 1 2 1 +2 2 2 1 2 1 2 1 2 1 2 1 2 2 1 1 +2 1 2 1 2 1 2 1 1 1 2 2 2 1 2 1 +1 1 2 1 2 1 2 2 1 1 2 1 1 1 2 1 +2 1 1 1 2 2 1 1 2 1 2 1 1 1 2 1 +1 2 2 1 1 1 2 1 1 1 1 1 2 2 1 1 +2 1 1 1 2 1 1 1 1 1 2 2 1 1 1 1 +2 1 1 1 1 1 2 2 1 1 1 1 2 1 1 1 +2 2 1 1 1 1 2 1 1 1 1 2 2 1 1 1 +2 1 1 1 2 1 1 2 1 1 2 1 1 1 2 1 +1 2 2 1 1 1 2 1 1 1 2 2 1 1 2 1 +1 1 2 1 2 1 1 2 2 1 1 1 2 1 1 1 +2 2 2 1 1 1 2 1 2 1 1 1 2 2 2 1 +1 1 2 1 2 1 1 2 2 1 2 1 1 1 2 1 +2 2 2 1 1 1 2 1 2 1 1 1 2 2 2 1 +2 1 1 1 2 1 2 2 2 1 1 1 2 1 2 1 +2 1 2 2 2 1 2 1 2 1 2 1 2 2 2 1 +2 1 3 1 2 2 2 1 3 1 3 1 3 1 3 2 +4 1 3 1 3 1 4 1 3 1 3 2 3 1 4 1 +3 1 3 1 4 1 3 2 3 1 3 1 4 1 3 1 +3 2 3 1 3 1 4 1 3 1 3 1 3 2 3 1 +4 1 3 1 3 1 3 1 3 2 3 1 3 1 4 1 +3 1 3 1 3 2 3 1 3 1 3 1 3 1 3 2 +3 1 3 1 3 1 3 1 3 1 3 2 3 1 3 1 +3 1 3 1 3 1 5073 3796 300 MP stroke +3 2 3 1 3 1 3 1 3 1 3 2 3 1 3 1 +3 1 2 1 3 1 3 2 3 1 3 1 3 1 3 1 +3 2 3 1 3 1 3 1 2 1 3 1 3 2 3 1 +3 1 3 1 3 1 3 2 2 1 3 1 3 1 3 1 +3 1 3 2 2 1 3 1 3 1 3 1 3 1 3 2 +2 1 3 1 3 1 3 1 3 2 2 1 3 1 3 1 +3 1 2 1 3 2 3 1 3 1 3 1 2 1 3 2 +3 1 3 1 2 1 3 1 3 1 3 2 2 1 3 1 +3 1 3 1 2 1 3 2 3 1 3 1 2 1 3 1 +3 2 2 1 3 1 3 1 2 1 3 1 3 2 2 1 +3 1 3 1 2 1 3 2 3 1 2 1 3 1 3 1 +2 1 3 2 3 1 2 1 3 1 3 1 2 2 3 1 +3 1 2 1 3 1 3 1 3 2 2 1 3 1 3 1 +2 1 3 2 3 1 2 1 3 1 3 1 2 1 3 2 +3 1 2 1 3 1 3 1 2 2 3 1 3 1 2 1 +3 1 2 1 3 2 3 1 2 1 3 1 3 1 2 2 +3 1 3 1 2 1 3 1 3 1 2 2 3 1 3 1 +2 1 3 1 3 2 2 1 3 1 3 1 2 1 3 1 +3 2 2 1 3 1 2 1 3 1 3 1 2 2 3 1 +2 1 3 1 3 1 2 2 3 1 2 1 3 1 2 1 +3 1 3 2 2 1 3 1 2 1 3 1 2 2 3 1 +2 1 3 1 3 1 2 1 3 2 2 1 3 1 2 1 +3 1 2 2 3 1 2 1 3 1 3 1 2 1 3 2 +2 1 3 1 2 1 3 1 2 1 3 2 2 1 3 1 +2 1 3 1 2 2 3 1 2 1 3 1 2 1 3 2 +2 1 3 1 2 1 3 1 2 1 3 2 3 1 2 1 +3 1 2 1 3 2 2 1 3 1 2 1 3 1 2 1 +3 2 2 1 3 1 2 1 3 1 3 2 2 1 3 1 +2 1 3 1 2 1 3 2 2 1 3 1 2 1 3 1 +2 2 3 1 2 1 3 1 2 1 3 1 3 2 2 1 +3 1 2 1 3 1 2 2 3 1 2 1 3 1 2 1 +3 1 2 2 3 1 2 1 3 1 2 1 3 2 2 1 +3 1 2 1 3 1 2 1 3 2 2 1 3 1 2 1 +2 1 3 1 2 2 3 1 2 1 3 1 2 1 2 1 +3 2 2 1 3 1 2 1 2 1 3 2 2 1 3 1 +2 1 2 1 3 2 2 1 3 1 2 1 2 1 3 1 +2 2 3 1 2 1 2 1 3 1 2 2 2 1 3 1 +2 1 3 1 2 1 4289 3443 300 MP stroke +2 2 3 1 2 1 2 1 3 1 2 1 3 2 2 1 +2 1 3 1 2 1 2 2 3 1 2 1 2 1 3 1 +2 1 2 2 3 1 2 1 2 1 3 1 2 2 2 1 +2 1 3 1 2 1 2 1 3 2 2 1 2 1 3 1 +2 1 2 2 3 1 2 1 2 1 3 1 2 1 2 2 +2 1 3 1 2 1 2 1 3 2 2 1 2 1 3 1 +2 1 2 1 2 2 3 1 2 1 2 1 3 1 2 2 +2 1 2 1 3 1 2 1 2 1 3 2 2 1 2 1 +2 1 3 1 2 2 2 1 2 1 3 1 2 1 2 1 +2 2 3 1 2 1 2 1 3 1 2 2 2 1 2 1 +3 1 2 1 2 1 2 2 3 1 2 1 2 1 3 1 +2 2 2 1 2 1 3 1 2 1 2 1 2 2 3 1 +2 1 2 1 3 1 2 2 2 1 2 1 3 1 2 1 +2 1 2 2 3 1 2 1 2 1 3 1 2 2 2 1 +2 1 3 1 2 1 2 1 2 2 3 1 2 1 2 1 +2 1 3 2 2 1 2 1 3 1 2 1 2 1 2 2 +3 1 2 1 2 1 2 1 3 2 2 1 2 1 2 1 +2 1 3 1 2 2 2 1 2 1 3 1 2 1 2 2 +2 1 3 1 2 1 2 1 2 1 2 2 3 1 2 1 +2 1 2 1 3 2 2 1 2 1 2 1 2 1 3 1 +2 2 2 1 2 1 3 1 2 1 2 2 2 1 2 1 +3 1 2 1 2 1 2 2 2 1 3 1 2 1 2 1 +2 2 2 1 3 1 2 1 2 1 2 1 2 2 2 1 +3 1 2 1 2 1 2 2 2 1 3 1 2 1 2 1 +2 1 3 2 2 1 2 1 2 1 2 1 3 2 2 1 +2 1 2 1 2 1 3 1 2 2 2 1 2 1 2 1 +3 1 2 2 2 1 2 1 2 1 3 1 2 1 2 2 +2 1 2 1 3 1 2 1 2 1 2 2 2 1 2 1 +3 1 2 1 2 2 2 1 2 1 2 1 2 1 2 1 +3 2 2 1 2 1 2 1 2 1 2 2 2 1 2 1 +2 1 2 1 3 1 2 2 2 1 2 1 2 1 2 1 +2 1 2 2 2 1 2 1 2 1 2 1 2 2 2 1 +2 1 3 1 2 1 2 1 2 2 2 1 2 1 2 1 +2 1 2 2 2 1 2 1 3 1 2 1 2 2 2 1 +2 1 2 1 2 1 3 1 2 2 2 1 2 1 2 1 +3 1 2 2 2 1 2 1 2 1 2 1 3 1 2 2 +2 1 2 1 2 1 2 1 3 1 2 2 2 1 2 1 +2 1 2 1 2 2 3625 3089 300 MP stroke +3 1 2 1 2 1 2 1 2 1 2 2 2 1 3 1 +2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 1 +3 2 2 1 2 1 2 1 2 1 2 2 2 1 2 1 +2 1 2 1 3 1 2 2 2 1 2 1 2 1 2 1 +2 2 2 1 2 1 2 1 2 1 2 1 3 2 2 1 +2 1 2 1 2 1 2 2 2 1 2 1 2 1 2 1 +2 1 2 2 2 1 2 1 2 1 2 1 2 1 2 2 +3 1 2 1 2 1 2 1 2 2 2 1 2 1 2 1 +2 1 2 2 2 1 2 1 2 1 2 1 2 1 2 2 +2 1 2 1 2 1 2 1 2 2 2 1 3 1 2 1 +2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 1 +2 2 2 1 2 1 2 1 2 1 2 2 2 1 2 1 +2 1 2 1 2 1 2 2 2 1 2 1 2 1 2 1 +2 2 2 1 2 1 2 1 2 1 2 1 2 2 2 1 +2 1 2 1 1 1 2 1 2 2 2 1 2 1 2 1 +2 1 2 2 2 1 2 1 2 1 2 1 2 1 2 2 +2 1 2 1 2 1 1 1 2 2 2 1 2 1 2 1 +2 1 2 2 2 1 2 1 2 1 2 1 2 1 2 2 +2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 +2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 1 +2 2 2 1 2 1 1 1 2 1 2 2 2 1 2 1 +2 1 2 1 2 1 2 2 2 1 2 1 2 1 1 1 +2 2 2 1 2 1 2 1 2 1 2 1 2 2 2 1 +2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 1 +2 2 2 1 2 1 2 1 2 1 2 1 2 2 2 1 +2 1 1 1 2 1 2 2 2 1 2 1 2 1 2 1 +2 1 2 2 2 1 2 1 2 1 2 1 1 2 2 1 +2 1 2 1 2 1 2 1 2 2 2 1 2 1 2 1 +2 1 2 2 2 1 1 1 2 1 2 1 2 1 2 2 +2 1 2 1 2 1 2 1 2 2 2 1 2 1 2 1 +1 1 2 1 2 2 2 1 2 1 2 1 2 1 2 2 +2 1 2 1 2 1 2 1 1 1 2 2 2 1 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1 2 1 2 1 1 1 2 1 2 2 1 1 2 1 +2 1 1 1 2 2 2 1 1 1 2 1 2 1 1 1 +2 2 2 1 2 1 1 1 2 1 2 2 1 1 2 1 +2 1 1 1 2 1 2 2 1 1 2 1 2 1 1 1 +2 2 2 1 1 1 2 1 2 1 2 1 1 2 2 1 +2 1 1 1 2 1 2 2 1 1 2 1 2 1 1 1 +2 1 2 2 1 1 2 1 2 1 1 1 2 2 2 1 +2 1 1 1 2 1 2 2 1 1 2 1 2 1 2 1 +1 1 2 2 2 1 2 1 1 1 2 1 2 2 2 1 +2 1 2 1 1 1 2 1 2 2 2 1 2 1 2 1 +2 1 2 2 2 1 2 1 2 1 2 1 2 2 2 1 +2 1 3 1 2 1 2 2 3 1 3 1 2 1 3 1 +4 2 3 1 4 1 6 2 7 0 6 0 6 0 6 -1 +4 -1 3 0 3 -1 2472 2393 300 MP stroke +2 -1 3 -1 2 -1 2 -1 2 -1 1 -1 2 -1 2 -1 +1 -2 2 -1 1 -1 2 -1 1 -1 2 -1 1 -1 1 -1 +2 -2 1 -1 1 -1 2 -1 1 -1 1 -1 1 -2 2 -1 +1 -1 1 -1 2 -1 1 -2 1 -1 2 -1 1 -1 1 -1 +1 -1 2 -2 1 -1 1 -1 2 -1 1 -1 1 -2 2 -1 +1 -1 1 -1 2 -1 1 -1 1 -2 2 -1 1 -1 1 -1 +2 -1 1 -2 1 -1 2 -1 1 -1 1 -1 2 -1 1 -2 +1 -1 1 -1 2 -1 1 -1 1 -2 2 -1 1 -1 1 -1 +2 -1 1 -2 1 -1 2 -1 1 -1 1 -1 2 -1 1 -2 +1 -1 2 -1 1 -1 1 -1 2 -2 1 -1 1 -1 2 -1 +1 -1 1 -1 2 -2 1 -1 2 -1 1 -1 1 -1 2 -2 +1 -1 1 -1 2 -1 1 -1 1 -1 2 -2 1 -1 1 -1 +2 -1 1 -1 1 -2 2 -1 1 -1 1 -1 2 -1 1 -1 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z9wNJzGCtH%PT-(bw1w;_;*R4$alo{ROMKHE_^koI0S1lW1;n7J>BfI1pCN>Ws=D$2 zs;86v$=-K7ADM_)Bh~A-139)$ZIq*}j5x4=Xjnq^PUczOGnq57;j~xd5$90zsSMT$ z+=8Cq4(YQ!Nc{wCq4&H1+__}!Dum=cVH4WUc)C~sV zi#l(Rsyw&=Uc~6@)N<_E+_7x<8=sxe=I)HdvzXfMWJb^Cv-i= +] setcolorspace } bdef mwscm +gs np 885 353 mt 0 3837 rl 5275 0 rl 0 -3837 rl cp c np +[5332 0 0 3882 856 337] cc +<< % Image dictionary + /ImageType 1 + /Width 91 /Height 114 + /BitsPerComponent 8 + /Decode [0 255] + /ImageMatrix [91.000000 0 0 114.000000 0 0] + /DataSource currentfile + /ASCII85Decode filter + /RunLengthDecode filter +>> +image +ZiC+0!W2p%!#mgqEZiC12!sf,1 +$O.+O&.oHc&aB?,!sAc-"9o#0"p4i.!<<3&q#CBqrVup!l2V=u!t,GF$k3aX&J5]l'G_MU!!!*!!$MFF +!!W]8"U55SO'c>p1!!1[O"p"c/ +"pY/2rW!-*#7h=Y'c-KC!!2or"Teo7"pP/.!!EE1#7Ln:!!NN1"UG55r;Zg"rVup#qu@!)!!!-%!rr<% +qu@!)!=&i:$OmFIrW!$*%hf`W(BXX- +"p4i+!WrE(rW!9.!!39'"TSN'!!3'!'*A4="q(hN$k!.?"qMCc)]Te9)XRM=!XT5D$OR1K&.oNf%h/aI +"UYVB"Q'1i"ptkL%LEUT'c%`.*ZZ:Bli7Y$#Rq7V'G_H!'bh8p'G:i^#Qrl_%L*4H&J,Zm)&F53*ZlLI 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Operating System: Linux 2.6.24-24-generic #1 SMP Wed Apr 15 15:11:35 UTC 2009 x86_64. +%%Title: /m/fs/user3/a/38/spniskan/thesis/streamerCDvsWL.eps +%%CreationDate: 05/14/2009 17:26:52 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%LanguageLevel: 2 +%%Pages: 1 +%%BoundingBox: -30 221 643 570 +%%EndComments + +%%BeginProlog +% MathWorks dictionary +/MathWorks 160 dict begin +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rm /rmoveto ldef +/rl /rlineto ldef +/s {show newpath} bdef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinewidth 2 div 0 360 arc fill + PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup 3 mul string + currentfile + 3 index 0 eq {/ASCIIHexDecode filter} + {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } + ifelse exch readstring pop + dup 0 3 index getinterval /rbmap xdef + dup 2 index dup getinterval /gbmap xdef + 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + rbmap gbmap bbmap true 3 colorimage gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: -30 221 643 570 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 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Operating System: Linux 2.6.24-24-generic #1 SMP Wed Apr 15 15:11:35 UTC 2009 x86_64. +%%Title: /m/fs/user3/a/38/spniskan/thesis/streamerCDvsWLestimate.eps +%%CreationDate: 05/14/2009 17:26:36 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%LanguageLevel: 2 +%%Pages: 1 +%%BoundingBox: -30 221 643 570 +%%EndComments + +%%BeginProlog +% MathWorks dictionary +/MathWorks 160 dict begin +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rm /rmoveto ldef +/rl /rlineto ldef +/s {show newpath} bdef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinewidth 2 div 0 360 arc fill + PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup 3 mul string + currentfile + 3 index 0 eq {/ASCIIHexDecode filter} + {/ASCII85Decode filter 3 index 2 eq {/RunLengthDecode filter} if } + ifelse exch readstring pop + dup 0 3 index getinterval /rbmap xdef + dup 2 index dup getinterval /gbmap xdef + 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + rbmap gbmap bbmap true 3 colorimage gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: -30 221 643 570 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 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zlcyuv@|d|j(E&-?J6Qv#arS_4G#C_2qOI(mfNar3XVBv@5@pJk_HIP9wL8!wO9v-6 z=sXJb{Z|rGEbZ-WouS&9*%KX)pcP1hQ&`lt#YhG11ia{hqfEdWw50w6g#6plTRGO2(*seu0X0%(2Y30%9-%W_FV za3GqyK!q}M0ZzHl7EW$NFpC|5{2^M9!NSJf(VFPu?qE-J2Y$k=og6KJcP}EWcWBx1 zF*uw0TgVbcYwByDb+vY*b@dFc7HE)OW!ApW!8H5ir3N|F7|K8A2{$4g#M);iaP)$p_yb5EGDO`M-jg zC^BUJ?;s`$K<57$VgiS$BvAV=5EE$BGVJ>=5EF$aLFWGoV*Yy8^d}=v6fNQiOcO2% zQcUvH2>v`F+R@1o8Xo|zT`b+)T^zw{=Eb6aWfub8{*__)C%X`=KFj&K1Vq42frjOe zhT)%a4wwN;RsX^;{Hg7mS} zB=YxLm+!j~Dt^420NVtczdg1V@ZvRW0PGlH1a|yNz!$k27q|!k?`;qf-`sHH4qv(x zB!S1_@9!0mGrk`iRw4qQ+)mbz1E2>o1HM-|=3;3D&i#N>U~p!&_z#?Z;J{&;6*wPV z{0C`)|F|3%^DvUgJE9-*L`B8niNcS0l2~xw{bL?DHUVcpKjle^fs>XW^Dv@f1aM>l z|9$UQOdL@5a~=l%3EiK{VSl9=Y)*b@4}<@uUomj4;pg#+feVd5u0Qr6Nx=U?izFVL zTK`y16z~eD*$;KaaPSS>k9inyh3uC+K=Lnj@uI(s7tjX|UVm&4965k@c|YgjC4T0! z7&zeixnD7G-1-X-z-5bH%fW-uANv4r5W&&ok9ioJ1UP#ADUTrfD^J0Z@6R-2Bqeam zcmU2qfcIY)6#Vnay0+eyP> matrix makepattern +/Pat1 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke + 0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke} +>> matrix makepattern +/Pat2 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L + 8 8 L 8 0 L 0 0 L fill} +>> matrix makepattern +/Pat3 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L + 0 12 M 12 0 L stroke} +>> matrix makepattern +/Pat4 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L + 0 -4 M 12 8 L stroke} +>> matrix makepattern +/Pat5 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L + 0 12 M 8 -4 L 4 12 M 10 0 L stroke} +>> matrix makepattern +/Pat6 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L + 0 -4 M 8 12 L 4 -4 M 10 8 L stroke} +>> matrix makepattern +/Pat7 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L + 12 0 M -4 8 L 12 4 M 0 10 L stroke} +>> matrix makepattern +/Pat8 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L + -4 0 M 12 8 L -4 4 M 8 10 L stroke} +>> matrix makepattern +/Pat9 exch def +/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def +/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def +/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def +/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def +/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def +/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def +/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def +} def +% +% +%End of PostScript Level 2 code +% +/PatternBgnd { + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse +} def +% +% Substitute for Level 2 pattern fill codes with +% grayscale if Level 2 support is not selected. +% +/Level1PatternFill { +/Pattern1 {0.250 Density} bind def +/Pattern2 {0.500 Density} bind def +/Pattern3 {0.750 Density} bind def +/Pattern4 {0.125 Density} bind def +/Pattern5 {0.375 Density} bind def +/Pattern6 {0.625 Density} bind def +/Pattern7 {0.875 Density} bind def +} def +% +% Now test for support of Level 2 code +% +Level1 {Level1PatternFill} {Level2PatternFill} ifelse +% +/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont +dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall +currentdict end definefont pop +/MFshow { + { dup 5 get 3 ge + { 5 get 3 eq {gsave} {grestore} ifelse } + {dup dup 0 get findfont exch 1 get scalefont setfont + [ currentpoint ] exch dup 2 get 0 exch R dup 5 get 2 ne {dup dup 6 + get exch 4 get {show} {stringwidth pop 0 R} ifelse }if dup 5 get 0 eq + {dup 3 get {2 get neg 0 exch R pop} {pop aload pop M} ifelse} {dup 5 + get 1 eq {dup 2 get exch dup 3 get exch 6 get stringwidth pop -2 div + dup 0 R} {dup 6 get stringwidth pop -2 div 0 R 6 get + show 2 index {aload pop M neg 3 -1 roll neg R pop pop} {pop pop pop + pop aload pop M} ifelse }ifelse }ifelse } + ifelse } + forall} bind def +/MFwidth {0 exch { dup 5 get 3 ge { 5 get 3 eq { 0 } { pop } ifelse } + {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont + 6 get stringwidth pop add} {pop} ifelse} ifelse} forall} bind def +/MLshow { currentpoint stroke M + 0 exch R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/MRshow { currentpoint stroke M + exch dup MFwidth neg 3 -1 roll R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/MCshow { currentpoint stroke M + exch dup MFwidth -2 div 3 -1 roll R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/XYsave { [( ) 1 2 true false 3 ()] } bind def +/XYrestore { [( ) 1 2 true false 4 ()] } bind def +end +%%EndProlog +gnudict begin +gsave +50 50 translate +0.050 0.050 scale +0 setgray +newpath +(Helvetica) findfont 140 scalefont setfont +gsave % colour palette begin +/maxcolors 64 def +/HSV2RGB { exch dup 0.0 eq {pop exch pop dup dup} % achromatic gray + { /HSVs exch def /HSVv exch def 6.0 mul dup floor dup 3 1 roll sub + /HSVf exch def /HSVi exch cvi def /HSVp HSVv 1.0 HSVs sub mul def + /HSVq HSVv 1.0 HSVs HSVf mul sub mul def + /HSVt HSVv 1.0 HSVs 1.0 HSVf sub mul sub mul def + /HSVi HSVi 6 mod def 0 HSVi eq {HSVv HSVt HSVp} + {1 HSVi eq {HSVq HSVv HSVp}{2 HSVi eq {HSVp HSVv HSVt} + {3 HSVi eq {HSVp HSVq HSVv}{4 HSVi eq {HSVt HSVp HSVv} + {HSVv HSVp HSVq} ifelse} ifelse} ifelse} ifelse} ifelse + } ifelse} def +/Constrain { + dup 0 lt {0 exch pop}{dup 1 gt {1 exch pop} if} ifelse} def +/YIQ2RGB { + 3 copy -1.702 mul exch -1.105 mul add add Constrain 4 1 roll + 3 copy -0.647 mul exch -0.272 mul add add Constrain 5 1 roll + 0.621 mul exch -0.956 mul add add Constrain 3 1 roll } def +/CMY2RGB { 1 exch sub exch 1 exch sub 3 2 roll 1 exch sub 3 1 roll exch } def +/XYZ2RGB { 3 copy -0.9017 mul exch -0.1187 mul add exch 0.0585 mul exch add + Constrain 4 1 roll 3 copy -0.0279 mul exch 1.999 mul add exch + -0.9844 mul add Constrain 5 1 roll -0.2891 mul exch -0.5338 mul add + exch 1.91 mul exch add Constrain 3 1 roll} def +/SelectSpace {ColorSpace (HSV) eq {HSV2RGB}{ColorSpace (XYZ) eq { + XYZ2RGB}{ColorSpace (CMY) eq {CMY2RGB}{ColorSpace (YIQ) eq {YIQ2RGB} + if} ifelse} ifelse} ifelse} def +/InterpolatedColor true def +/grayindex {/gidx 0 def + {GrayA gidx get grayv ge {exit} if /gidx gidx 1 add def} loop} def +/dgdx {grayv GrayA gidx get sub GrayA gidx 1 sub get + GrayA gidx get sub div} def +/redvalue {RedA gidx get RedA gidx 1 sub get + RedA gidx get sub dgdxval mul add} def +/greenvalue {GreenA gidx get GreenA gidx 1 sub get + GreenA gidx get sub dgdxval mul add} def +/bluevalue {BlueA gidx get BlueA gidx 1 sub get + BlueA gidx get sub dgdxval mul add} def +/interpolate { + grayindex grayv GrayA gidx get sub abs 1e-5 le + {RedA gidx get GreenA gidx get BlueA gidx get} + {/dgdxval dgdx def redvalue greenvalue bluevalue} ifelse} def +/GrayA [0 .0159 .0317 .0476 .0635 .0794 .0952 .1111 .127 .1429 .1587 .1746 + .1905 .2063 .2222 .2381 .254 .2698 .2857 .3016 .3175 .3333 .3492 .3651 + .381 .3968 .4127 .4286 .4444 .4603 .4762 .4921 .5079 .5238 .5397 .5556 + .5714 .5873 .6032 .619 .6349 .6508 .6667 .6825 .6984 .7143 .7302 .746 + .7619 .7778 .7937 .8095 .8254 .8413 .8571 .873 .8889 .9048 .9206 .9365 + .9524 .9683 .9841 1 ] def +/RedA [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .0238 .0873 .1508 + .2143 .2778 .3413 .4048 .4683 .5317 .5952 .6587 .7222 .7857 .8492 .9127 + .9762 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .9444 .881 .8175 .754 .6905 .627 + .5635 .5 ] def +/GreenA [0 0 0 0 0 0 0 0 .0079 .0714 .1349 .1984 .2619 .3254 .3889 .4524 + .5159 .5794 .6429 .7063 .7698 .8333 .8968 .9603 1 1 1 1 1 1 1 1 1 1 1 1 1 + 1 1 1 .9603 .8968 .8333 .7698 .7063 .6429 .5794 .5159 .4524 .3889 .3254 + .2619 .1984 .1349 .0714 .0079 0 0 0 0 0 0 0 0 ] def +/BlueA [.5 .5635 .627 .6905 .754 .8175 .881 .9444 1 1 1 1 1 1 1 1 1 1 1 1 1 + 1 1 1 .9762 .9127 .8492 .7857 .7222 .6587 .5952 .5317 .4683 .4048 .3413 + .2778 .2143 .1508 .0873 .0238 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 0 ] def +/pm3dround {maxcolors 0 gt {dup 1 ge + {pop 1} {maxcolors mul floor maxcolors 1 sub div} ifelse} if} def +/pm3dGamma 1.0 1.5 div def +/ColorSpace (RGB) def +Color true and { % COLOUR vs. GRAY map + InterpolatedColor { %% Interpolation vs. RGB-Formula + /g {stroke pm3dround /grayv exch def interpolate + SelectSpace setrgbcolor} bind def + }{ + /g {stroke pm3dround dup cF7 Constrain exch dup cF5 Constrain exch cF15 Constrain + SelectSpace setrgbcolor} bind def + } ifelse +}{ + /g {stroke pm3dround pm3dGamma exp setgray} bind def +} ifelse +1.000 UL +LTb +546 420 M +63 0 V +6353 0 R +-63 0 V +stroke +0.00 0.00 0.00 C 462 420 M +[ [(Helvetica) 140.0 0.0 true true 0 (0)] +] -46.7 MRshow +1.000 UL +LTb +546 1162 M +63 0 V +6353 0 R +-63 0 V +stroke +0.00 0.00 0.00 C 462 1162 M +[ [(Helvetica) 140.0 0.0 true true 0 (5)] +] -46.7 MRshow +1.000 UL +LTb +546 1904 M +63 0 V +6353 0 R +-63 0 V +stroke +0.00 0.00 0.00 C 462 1904 M +[ [(Helvetica) 140.0 0.0 true true 0 (10)] +] -46.7 MRshow +1.000 UL +LTb +546 2646 M +63 0 V 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