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z2ku#{l|?x08FCBQZ<_a8_6uej4FWzu&C#8C_-gUTmjdf-ka|Wm>C~1(pa}m1sD}_J zKM2$j9H@5?s2~VbBLwOX94LGo3?w87+CT*XK@#>11L#;B!)rq81&fXN;~&P63fr7SOVaiK0)wF!p#T5? literal 0 HcmV?d00001 diff --git a/main.tex b/main.tex index 497be80..63cba0e 100755 --- a/main.tex +++ b/main.tex @@ -10,7 +10,7 @@ % Silbentrennung % ----------------------------------------- \hyphenation{ -ANSYS DOWEC WindPACT +ANSYS CATIA DOWEC WindPACT } % wird nach \selectlanguage{ngerman} zurückgesetzt diff --git a/sections/Modellentwicklung.tex b/sections/Modellentwicklung.tex index c94d540..f81f889 100755 --- a/sections/Modellentwicklung.tex +++ b/sections/Modellentwicklung.tex @@ -60,7 +60,7 @@ Strukturdämpfung (über alle Moden) & \unit{0,\!477465}{\%} \\ \end{tabular} \end{table}\vspace{-1em} \ac{NREL} richtet sich bei den Rotorblätter an das DOWEC"=Konzeptmodell. -Somit versteht sich der in der Tabelle~\ref{tab:NRELRotor} angegebene Skalierungsfaktor darauf, die aus dem DOWEC"=Konzeptmodell übernommene Massenverteilung auf die Gesamtmasse der Rotorblätter von der REpower\,5M Muster"=\ac{WEA} zu skalieren. +Somit versteht sich der, in der Tabelle~\ref{tab:NRELRotor} angegebene, Skalierungsfaktor darauf, die aus dem DOWEC"=Konzeptmodell übernommene Massenverteilung auf die Gesamtmasse der Rotorblätter von der REpower\,5M Muster"=\ac{WEA} zu skalieren. \ac{NREL} listet ebenfalls in Abhängigkeit von dem Nabenradius in 49 Stellen Informationen zu Steifigkeiten und Massenschwerpunkte. Das Masterprojekt \emph{Entwicklung eines Abschnittes des NREL Rotorblattes mit anisotropen Material anhand gegebener Steifigkeitsdaten}~\cite{MP15} setzte sich bereits mit diesen Informationen auseinander und werden hier übernommen. @@ -142,11 +142,11 @@ Strukturdämpfung (über alle Moden) & \unit{1}{\%} \\ \subsubsection{Frequenzen und stationäres Verhalten der Forschungsanlage} -Das \ac{NREL} berechnete mit zwei verschiedenen Modellen Eigenfrequenzen einer linearisierten Eigenwertanalyse ihrer küstenzugewandten Ausgangs"=\ac{WEA}. Dabei lauten die beiden Modelle FAST und ADAMS und beschreiben eine balkenähnliche Simulation. +\ac{NREL} berechnete mit zwei verschiedenen Modellen Eigenfrequenzen einer linearisierten Eigenwertanalyse ihrer küstenzugewandten Ausgangs"=\ac{WEA}. Benannt sind die beiden Modelle als FAST"=Modell und ADAMS"=Modell und beschreiben eine balkenähnliche Simulation. %Beide Modelle berechnen das linearisierte Eigenwertproblem. Das Ergebnis dieser Eigenfrequenzanalyse listet die Tabelle \ref{tab:NRELFreq} auf. -Die Eigenfrequenz"=Abweichung zwischen den Modellen FAST und ADAMS, besonders zur zweiten Rotorblatt"=Eigenfrequenz mit der asymmetrischen flatternden Eigenform, ist dahingehend begründet, dass das ADAMS"=Modell gegenüber dem FAST"=Modell Positionen zum Massenschwerpunk, Moden höherer Ordnung und Torsionsfreiheitsgrade im Turm berücksichtigt. +Die, in der Tabelle~\ref{tab:NRELFreq}, zu erkennende Eigenfrequenz"=Abweichung zwischen den Modellen FAST und ADAMS, besonders zur zweiten Rotorblatt"=Eigenfrequenz mit der asymmetrischen flatternden Eigenform, ist dahingehend begründet, dass das ADAMS"=Modell gegenüber dem FAST"=Modell Positionen zum Massenschwerpunk, Moden höherer Ordnung und Torsionsfreiheitsgrade im Turm berücksichtigt. \begin{table}[H] \caption[Frequenzen der NREL 5-MW Ausgangs"=Windenergieanlage]{Turmeigenschaften der NREL 5-MW Ausgangs"=Windenergieanlage~\cite{NREL09}}\label{tab:NRELFreq}\centering \begin{tabular}{clccllll} @@ -175,7 +175,7 @@ Freq. & Beschreibung der Eigenform & FAST & ADAMS \\ \midrule \subsection{Computermodell} Die Forschungsanlage wird als Flächenmodell angenähert. -Ein grundlegendes Computermodell ist durch die Arbeit \cite{Prange14} von \citeauthor{Prange14} entstanden und validiert. +Ein grundlegendes Computermodell ist durch die Arbeit \emph{Entwicklung eines 3D"=Geometriemodells und numerische Strömungssimulation einer Windenergieanlage} \cite{Prange14} von \citeauthor{Prange14} entstanden und validiert. Das dabei verwendete \ac{CAD}-Programm ist \emph{CATIA\,V5} von der Firma \emph{Dassault Systèmes}, mit dem die dreidimensionale Geometrieerzeugung erfolgte. Alle erforderliche Modifikationen zur Simulation werden ebenfalls mit dem \ac{CAD}-Programm \emph{CATIA\,V5} durchgeführt. Unter den Modifikationen fallen diverse Schnitte, die in das 3D"=Flächenmodell hinzugefügt werden und damit die nötige Möglichkeit bieten Materialparameter vorgeben zu können. @@ -186,13 +186,13 @@ Die Abbildung \ref{fig:ComputermodellCATIA} links zeigt das übernommene und rec \caption{Computermodell der Windenergieanlage} \label{fig:ComputermodellCATIA} \end{figure} \vspace{-1.5em} -Für ein hinreichendes mechanisches Modell beziehungsweise \ac{FEM}"=Modell sind +Für ein hinreichendes mechanisches Modell beziehungsweise FE"=Modell sind Materialeigenschaften wie %sind für gewünschte \ac{FEM}"=Simulationen Steifigkeiten und Massen zuzuweisen. % weitere Elemente hinzugefügt werden, damit es zu einem hinreichenden mechanisches Modell beziehungsweise \ac{FEM}"=Modell wird. Gemeint sind insbesondere Steifigkeiten -Dabei zeigten anfängliche Versuche im Forschungsprojekt \ac{WindNumSim}, dass das Nachbilden der Steifigkeiten für die Rotorblätter nicht trivial ist. +Dabei zeigten anfängliche Versuche im Forschungsprojekt \ac{WindNumSim}, dass das Nachbilden der Steifigkeiten für die Rotorblätter mit den zur Verfügung stehenden Daten nicht trivial sind. Somit ist es beispielsweise unmöglich das Verhalten der \ac{WEA} mit ausschließlich isotrope Werkstoffe nachzuempfinden. %% TODO: Referenz zu einer Arbeit -Erste Überlegungen die Steifigkeit für das Rotorblatt abzubilden wurden in einem vorangegangenen Masterprojekt~\citep{MP15} durchgeführt. Als Resultat wird eine Modellierung nach \ac{SNL} gewählt, welches ein in \acs{MATLAB} geschriebenes Programm speziell zur Berechnung von Rotorblättern entwickelte. +Erste Überlegungen die Steifigkeit für das Rotorblatt abzubilden wurden in einem vorangegangenen Masterprojekt~\citep{MP15} durchgeführt. Als Resultat wird eine Modellierung nach \ac{SNL} \cite{SANDIA13} gewählt, welches ein in \acs{MATLAB} geschriebenes Programm speziell zur Berechnung von Rotorblättern entwickelte. Dieses Programm -- genannt \acs{NuMAD} -- erstellt ein in \emph{\ac{APDL}} geschriebene Datei. Das Ergebnis der Rotorblattmodellierung welche die vorgegebenen Steifigkeiten nachbilden zeigt die Abbildung \ref{fig:LageraufbautenAPDL}. \begin{figure}[H]\centering \includegraphics[width=0.95\textwidth]{Lageraufbauten_iso_ou.png} @@ -200,8 +200,8 @@ Dieses Programm -- genannt \acs{NuMAD} -- erstellt ein in \emph{\ac{APDL}} gesch \label{fig:LageraufbautenAPDL} \end{figure} \vspace{-1.5em} In Abbildung~\ref{fig:LageraufbautenAPDL} ist zusehen, dass das Modell aus vielen kleinen Einzelflächen besteht. -Auf diesen Einzelflächen sind jeweils eigene Materialparameter in Form von unterschiedliche Lagenaufbauten verbunden. -Die hauptsächliche Modifikation besteht darin, das CATIA"=Modell um die gezeigten Einzelflächen zu erweitern. +Auf diesen Einzelflächen sind jeweils eigene Materialparameter in Form von unterschiedliche Lagenaufbauten mit unterschiedlichen Materialien verbunden. +Die hauptsächliche Modifikation besteht darin, das CATIA"=Modell um die dargestellten Einzelflächen zu erweitern, sodass diese in das vom Forschungsprojekt verwendete ANSYS"=Workbench"=Modell importiert und mit den Lageraufbauten erweitert werden kann. \subsubsection{Rotorblatt} @@ -218,7 +218,7 @@ Die Abbildung~\ref{fig:SektionenBatch} zeigt die benötigten Sektionen. \caption{Sektionen im Rotorblatt} \label{fig:SektionenBatch} \end{figure} \vspace{-1.5em} -Anschließend werden Punkte und Linien in Längsrichtung erzeugt. Die Linien werden in radialer Richtung extrudiert und wiederum mit dem CATIA-Modell geschnitten. +Im Anschluss werden Punkte und Linien in Längsrichtung erzeugt. Die Linien werden in radialer Richtung extrudiert und wiederum mit dem CATIA-Modell geschnitten. Sämtliche Informationen zum Lagenaufbau werden aus der NuMAD-generierten ANSYS-Einlesedatei \texttt{shell7.src} des Masterprojekts \citep{MP15} rausgeschrieben, in MATLAB als CATIA-Makro aufbereitet und anschließend in CATIA eingelesen und ausgeführt. \paragraph{Sektionen und Punkte}~\\ @@ -273,7 +273,7 @@ Der Hauptteil zur Erzeugung von Ebenen in CATIA zeigt folgender Programmausdruck % 'part1.Update\r\n'], planeDistances(i), i); %end %\end{lstlisting} -\begin{lstlisting}[language=VBScript,showstringspaces=false, caption={VBScript Makro\_open\_file\_and\_create\_elements.catvbs}, label=lst:CATIA-Ebenen] +\begin{lstlisting}[language=VBScript,firstnumber=22,showstringspaces=false, caption={VBScript Makro\_open\_file\_and\_create\_elements.catvbs}, label=lst:CATIA-Ebenen] Set hybridShapePlaneOffset1 = hybridShapeFactory1.AddNewPlaneOffset(reference1, 7000, False) hybridShapePlaneOffset1.Name = "Ebene.1001" hybridBody1.AppendHybridShape hybridShapePlaneOffset1 @@ -285,6 +285,79 @@ und dabei, der in der ersten Zeile angegebene Abstand von 7000, mit den Werten a Die zweite Zeile benennt die Ebenen mit den Bezeichnungen von \glqq \texttt{Ebene.1001}\grqq\ bis \glqq \texttt{Ebene.1018}\grqq. Mit den abschließenden Zeilen werden die Ebenen mit dem Modell verknüpft. Zuvor werden noch einige benötigte Variable zur Verfügung gestellt die hier nicht weiter aufgezeigt werden. Für den interessierten Leser und zur Vollständigkeit beziehungsweise zur Reproduzierbarkeit ist das gesamte MATLAB-Skript, welches das CATIA-Makro erzeugt, im Anhang als Programmausdruck \ref{lst:MATLAB-CATIA-Makro} angefügt. +Mit den nun zur Verfügung stehenden Ebenen kann das Modell geschnitten aufgetrennt werden. +Der Programmausdruck~\ref{lst:CATIA-Trennen} zeigt den Teil aus dem mit MATLAB generierten CATIA"=Makro welche die Trennung beschreibt. +\begin{lstlisting}[language=VBScript,firstnumber=159,showstringspaces=false, caption={VBScript Makro\_open\_file\_and\_create\_elements.catvbs}, label=lst:CATIA-Trennen] +Set hybridShapeAssemble2 = hybridShapes2.Item("Trennen.1001r") +Set reference1 = part1.CreateReferenceFromObject(hybridShapeAssemble2) +Set hybridShapePlaneOffset1 = hybridShapes1.Item("Ebene.1002") +Set reference2 = part1.CreateReferenceFromObject(hybridShapePlaneOffset1) + +Set hybridShapeSplit1 = hybridShapeFactory1.AddNewHybridSplit(reference1, reference2, -1) +hybridShapeFactory1.GSMVisibility reference1, 0 +hybridShapeSplit1.Name = "Trennen.1002" +hybridBody3.AppendHybridShape hybridShapeSplit1 +part1.InWorkObject = hybridShapeSplit1 +part1.Update +part1.Update +Set hybridShapeSplit1 = hybridShapeFactory1.AddNewHybridSplit(reference1, reference2, 1) +hybridShapeFactory1.GSMVisibility reference1, 0 +hybridShapeSplit1.Name = "Trennen.1002r" +hybridBody3.AppendHybridShape hybridShapeSplit1 +part1.InWorkObject = hybridShapeSplit1 +part1.Update +part1.Update +\end{lstlisting} +% +Anschließend werden die geschnittenen Abschnitte wieder zu einer Gesamtoberfläche zusammengesetzt, siehe dazu Programmausdruck~\ref{lst:CATIA-VerbindenEbenen}. +\begin{lstlisting}[language=VBScript,firstnumber=499,showstringspaces=false, caption={VBScript Makro\_open\_file\_and\_create\_elements.catvbs}, label=lst:CATIA-VerbindenEbenen] +Set hybridBody4 = hybridBodies1.Add() +hybridBody4.Name = "Gesamtoberfläche" +part1.Update +Set hybridShapeSplit1 = hybridShapes2.Item("Trennen.1001") +Set reference1 = part1.CreateReferenceFromObject(hybridShapeSplit1) + +Set hybridShapeSplit1 = hybridShapes2.Item("Trennen.1002") +Set reference2 = part1.CreateReferenceFromObject(hybridShapeSplit1) +Set hybridShapeAssemble1 = hybridShapeFactory1.AddNewJoin(reference1, reference2) + +Set hybridShapeSplit1 = hybridShapes2.Item("Trennen.1003") +Set reference3 = part1.CreateReferenceFromObject(hybridShapeSplit1) +hybridShapeAssemble1.AddElement reference3 +\end{lstlisting} +% +% +Da mit der Einlesedatei \texttt{shell7.src} die kleinen Einzelflächen mit den zuvor definierten \emph{Keypoints} erstellt werden, wird vorerst nachvollzogen welche \emph{Keypoints} zur Flächenerzeugung verwendet wurden. +Die Zeilen 18.977 bis 21.484 der Einlesedatei \texttt{shell7.src} beschreiben die Flächenerzeugung. +Der Programmausdruck~\ref{lst:APDL-Flächen} stellt den Beginn mit der ersten Fläche dar. +\begin{lstlisting}[language=Fortran,firstnumber=18977, caption={APDL shell7.src: Informationen zu Einzelflächen},label=lst:APDL-Flächen] +! GENERATE SKIN AREAS ================================== + +*if,hide_warndlg_areas,eq,1,then + /UIS, MSGPOP, 3 +*endif + + asel,none + lsel,all + csys,0 + a,1,9,1009,1001 + aatt,,,32,1001,2 + asel,none + lsel,all +\end{lstlisting} +Notiert werden Zeilen mit dem \texttt{a}"=Befehl, wie den in Zeile 18986; \texttt{a,1,9,1009,1001}, und in eine Textdatei Namens \texttt{sections.txt} zwischengespeichert. +Die dem \texttt{a}"=Befehl folgenden vier Ziffern, sind die eindeutigen Bezeichner der \emph{Keypoints}. +Mit dieser Methode reduziert sich die Anzahl der Keypoint von insgesamt 37.192 auf 498, wenn auf duplizierte sowie \emph{Keypoints} achtet die sich wirklich auf den Ebenen befinden. Die Umsetzung mit MATLAB zeigt Programmausdruck~\ref{lst:MATLAB-Keypoints}. +\begin{lstlisting}[language=Matlab, firstnumber=45, caption={MATLAB Create\_CATIA\_Makro.m: Keypoint-Aufbereitung},label=lst:MATLAB-Keypoints] +uniqueSectionKeypoints = unique(sectionKeypoints(:)); +% +% KEYPOINTS AND SECTIONS +iskeypointInSection = (ismember(keypoints(:,1) , uniqueSectionKeypoints)); +\end{lstlisting} + +Linien ... + + \paragraph{Längsschnitte}~\\\nopagebreak ~ % @@ -609,7 +682,11 @@ In Abbildung~\ref{fig:konvergenz:modal} ist die Konvergenz zu den Eigenfrequenze \subsubsection{Randbedingungen} -~ +\begin{figure}[H]\centering +\includegraphics[width=0.48\textwidth]{Randbedingungen_Beschleunigung.png} +\caption{Statische Randbedingungen} +\label{fig:Kontakte} +\end{figure} \vspace{-1.5em} \subsubsection{Analyseparameter} ~ diff --git a/sections/References.bib b/sections/References.bib index 0045016..d45fe26 100755 --- a/sections/References.bib +++ b/sections/References.bib @@ -31,6 +31,17 @@ isbn = {3-540-43511-5}, gender={sm}, } +@ARTICLE{newmark59, + author = {Newmark, Nathan M.}, + journaltitle = {Journal of Engineering Mechanics Division}, + year = {1959}, + volume = {85}, + number = {3}, + title= {A method of computation for structural dynamics}, + publisher = {ASCE}, + location = {Urbana, IL}, + pages = {67--94}, +} @BOOK{tm409, author = {Gross, Dietmar and Hauger, Werner and Wriggers, Peter}, title = {Technische Mechanik}, @@ -77,3 +88,11 @@ address = {Golden, Colorado}, year = {2009} } +@TECHREPORT{SANDIA13, + type = {Forschungsbericht}, + title={Definition of a 5MW/61.5m Wind Turbine Blade Reference Model}, + author={Resor, Brian R.}, + publisher = {Sandia National Laboratories}, + address = {Albuquerque, New Mexico 87185 and Livermore, California 94550}, + year = {2013} +} diff --git a/sections/Theorie.tex b/sections/Theorie.tex index 5378724..2586080 100755 --- a/sections/Theorie.tex +++ b/sections/Theorie.tex @@ -6,8 +6,8 @@ \subsection{Rechnerunterstütztes Konstruieren} \ac{CAD} -\subsubsection{Flächendarstellung} -\subsubsection{Datenaustausch} +%\subsubsection{Flächendarstellung} +%\subsubsection{Datenaustausch} \subsection{Faserverbundwerkstoffe} @@ -23,23 +23,24 @@ Die \ac{FEM} umfasst eine Vielzahl von Methodiken physikalische Fragestellungen zu beantworten. So sind für strukturmechanische Probleme einer \ac{WEA} beispielsweise die statische Durchbiegung der Rotorblätter aufgrund Eigengewicht und Windlasten als auch Eigenformen und die Anlagenbelastung bei drehenden Rotorblätter von Interesse. -In den nachfolgenden Abschnitten wird auf die Grundlagen der statischen und dynamischen Analyse eingegangen. -Zur weiteren Vertiefung der Themengebiete beziehungsweise bei Interesse zur Lösung von anderen Problemstellungen sei unter anderem auf die Literatur \citep{bathe86}, \cite{klein05} und \cite{becker02} sowie das \acs{ANSYS}"=Programmhandbuch %\cite{ansys} -verwiesen. +Dazu wird in den nachfolgenden Abschnitten auf die Grundlagen der statischen und dynamischen Analyse eingegangen. +%Zur weiteren Vertiefung der Themengebiete beziehungsweise bei Interesse zur Lösung von anderen Problemstellungen sei unter anderem auf die Literatur \citep{bathe86}, \cite{klein05} und \cite{becker02} sowie das \acs{ANSYS}"=Programmhandbuch %\cite{ansys} +%verwiesen. +Die Grundlage dieser Darstellung wiederum sind die Literatur \citep{bathe86}, \cite{klein05} und \cite{becker02} sowie das \acs{ANSYS}"=Programmhandbuch. Im Allgemeinen werden mit der \ac{FEM} Differentialgleichungen beziehungsweise Systeme von Differentialgleichungen gelöst. Am Beispiel eines Biegebalkens ist es die Biegedifferentialgleichung -und bei dynamischen Problemen eine Bewegungsdifferentialgleichung. +und bei allgemeinen dynamischen Problemen eine Bewegungsdifferentialgleichung. Gewonnen werden diese Differentialgleichung durch Gleichgewichtsbetrachtungen eines beliebigen Festkörpers, der entweder statisch bestimmt oder statisch überbestimmt gelagert und mit äußeren Lasten beaufschlagt ist. Die \ac{FEM} umfasst zur Lösung der Feldgröße dabei drei grundlegende Schritte mit der die Differentialgleichung als FE"=Gleichungssystem beschrieben wird; die Partitionierung wobei das Berechnungsgebiet in Elementen mit Knoten eingeteilt wird und das Gebiet als Summe der Elemente anzusehen ist, die Approximation bei der die gesuchte Verformung durch Polynome angenähert wird und die Elemente fortan in Abhängigkeit der Knotenverformung beschrieben sind und schließlich -die Assemblierung welche das globale \glqq diskrete\grqq\ Gleichgewicht (FE"=Gleichungssystem) aufstellt mit der das Gebiet als Summe der beschriebenen Elemente anzusehen ist, wobei kinematische Bedingungen gleicher Verformungen zu berücksichtigen sind. +die Assemblierung die das globale \glqq diskrete\grqq\ Gleichgewicht (FE"=Gleichungssystem) aufstellt mit der das Gebiet als Summe der beschriebenen Elemente anzusehen ist, wobei kinematische Bedingungen gleicher Verformungen zu berücksichtigen sind. \subsubsection{Statische Analysen} Mit dem Kräftegleichgewicht am infinitesimalen Element ergibt sich für ein beliebigen Körper~\(V\) im allgemeinen dreidimensionalen Fall die klassische elliptische Differentialgleichung mit den Randbedingungen auf der Körperoberfläche~\(\partial V\) -- in differentielle Tensor"=Formulierung -- \[ - \begin{Array}{rcl} + \begin{Array}{rll} \nabla\cdot\tensorII{\sigma} + \tensorI{f} &= \tensorI{0} & \quad\text{in } V, \\ \tensorI{u} &= \tensorI{u}_0 & \quad\text{auf } \partial V_1, \\ \tensorII{\sigma}\cdot\tensorI{n} &= \tensorI{t} & \quad\text{auf } \partial V_2, \\ @@ -52,7 +53,7 @@ Die Summe der gleichgerichteten, auf das Volumen bezogenen, differentiellen Spannungen% ~\(\sigma_{ji,j}(\tensorI{x})\) stehen mit der ortsabhängigen Volumenkraft~\(f_i(\tensorI{x})\) der jeweiligen Richtung im Gleichgewicht, siehe dazu auch Abbildung~\ref{pgfplots:Kraeftegleichgewicht} rechts mit Darstellung der in \(x\) gerichteten Kräften. Die Randbedingungen beschreiben dabei zum einen Verschiebungen~\(\tensorI{u}_0(\tensorI{x})\) auf einem Teil der Körperoberfläche~\(\partial V_1\) und zum anderen Belastungen in Form eines Spannungsvektors~\(\tensorI{t}(\tensorI{x})\) auf den restlichen Teil der Körperoberfläche~\(\partial V_2\). -Zur näherungsweisen Verarbeitung der Differentialgleichung wird hier das Variationsprinzip mit dem sogenannte \emph{Prinzip der virtuellen Verrückung} herangezogen, mit der ein Ersatzgleichgewichtsgleichung formuliert wird. +Zur näherungsweisen Verarbeitung der Differentialgleichung wird hier vorerst das Variationsprinzip mit dem sogenannte \emph{Prinzip der virtuellen Verrückung} herangezogen, mit dem ein Ersatzgleichgewichtsgleichung formuliert wird. %\begin{tikzpicture} @@ -184,7 +185,7 @@ Zur näherungsweisen Verarbeitung der Differentialgleichung wird hier das Variat \end{figure} \vspace{-1.5em} \paragraph{Globales Gleichgewicht über schwache Formulierung des Randwertproblems}~\\ -Auf Grundlage der Differentialgleichung (starke Formulierung) erfolgt die schwache Formulierung durch Multiplikation einer Testfunktion beziehungsweise hier mit einer virtuellen Verrückung \(\delta\tensorI{u}\), welche die kinematischen beziehungsweise wesentlichen Randbedingungen erfüllen muss (\(\forall \delta\tensorI{u}\in C^1(V)\cap C(\overline{V}) \text{ mit } \overline{V}=V \cup \partial V,~ \delta\tensorI{u} = \tensorI{u}_0 \text{ auf } \partial V_1\)), mit anschließender Integration über das Berechnungsgebiet \(V\) +Auf Grundlage der Differentialgleichung (starke Formulierung) erfolgt die schwache Formulierung durch Multiplikation einer Testfunktion beziehungsweise hier mit einer virtuellen Verrückung \(\delta\tensorI{u}\), welche die kinematischen beziehungsweise wesentlichen Randbedingungen erfüllen muss (\(\forall \delta\tensorI{u}\in C^1(V)\cap C(\overline{V}) \text{ mit } \overline{V}=V \cup \partial V,~ \delta\tensorI{u} = \tensorI{u}_0 \text{ auf } \partial V_1\)) und anschließender Integration über das Berechnungsgebiet \(V\) \[ \underbrace{\int\limits_{V\vphantom{V_1}}\!\!\!(\tensorII{\sigma} : \nabla\delta\tensorI{u})\dif V }_{\delta W\ti{i}} = \underbrace{\!\!\int\limits_V\!\!\!(\tensorI{f}\cdot\delta\tensorI{u})\dif V + \!\!\int\limits_{\partial V_2}\!\!\!(\tensorI{t}\cdot\delta\tensorI{u})\dif A % + \sum\tensorI{F}_i\cdot\delta\tensor{u}_i @@ -198,7 +199,7 @@ wobei \( \int_V \nabla\cdot\tensorII{\sigma}\cdot\delta\tensorI{u}\dif V = \int_A(\tensorII{\sigma}\cdot\tensorI{n})\cdot\delta\tensorI{u}\dif A - \int_V\tensorII{\sigma}:(\nabla\delta\tensorI{u})\dif V \) (Green'sche Integralsatz) sowie -\( \nabla\delta\tensorI{u} = \tensorII{\varepsilon}(\delta\tensorI{u}) = \delta\tensorII{\varepsilon} \) im linearen Fall. Die Integration erfolgt für kleine Verformungen über das unverformte und bei dem nichtlinearen Fall über das verformte Volumen. +\( \nabla\delta\tensorI{u} = \tensorII{\varepsilon}(\delta\tensorI{u}) = \delta\tensorII{\varepsilon} \) im linearen Fall. Es sei darauf hingewiesen, das die Integration für kleine Verformungen über das unverformte und bei dem nichtlinearen Fall über das verformte Volumen erfolgt. \paragraph{Materialgesetz}~\\ Im linear elastischen Fall besteht zwischen der Spannung und der Dehnung das folgende Materialgesetz @@ -208,7 +209,7 @@ Im linear elastischen Fall besteht zwischen der Spannung und der Dehnung das fol \text{bzw. in Ingeniersnotation}~~~~~ & \tensor{\sigma}_{6\times1} &=& \tensor{C}_{6\times6} \tensor{\varepsilon}_{6\times1} & \quad\text{da } \tensorII{\sigma}^\T = \tensorII{\sigma} ~\wedge~ \tensorII{\varepsilon}^\T = \tensorII{\varepsilon} \end{Array} \] -Im allgemeinem Fall entsprechen die enthaltene Richtungen der Dehnung nicht die enthaltene Richtung der Kraftkomponente. Es kann somit jede Spannungskomponente von jeder Dehnungskomponente abhängen, weshalb entgegen dem bekannteren eindimensionalen Fall \(\sigma(x) = E\varepsilon(x)\) das Verhalten nicht mit einer einzigen Größe beschrieben wird. +Im allgemeinem Fall entsprechen die enthaltene Richtungen der Dehnung nicht die enthaltene Richtung der Kraftkomponente. Es kann somit jede Spannungskomponente von jeder Dehnungskomponente abhängen, weshalb entgegen dem bekannteren eindimensionalen Fall \(\sigma(x) = E\varepsilon(x)\) das Verhalten nicht mit einer einzigen Zahl oder Größe beschrieben wird. Damit ist die Elastizitätsbeziehung beziehungsweise der Elastizitätstensor \(\tensorIV{C}\), der das Materialverhalten beschreibt, kein Skalar sondern ein Tensor vierter Stufe. Für den physikalisch nichtlinearen Fall sind die Spannungen allgemein von dem Verlauf der Verformung abhängig \(\tensorII{\sigma} = \tensorII{\sigma}(\tensorI{u})\). @@ -249,7 +250,7 @@ Das Aufstellen des FE"=Gleichungssystems beginnt mit der \emph{Vernetzung} bezie \] Hiermit ist das Gesamtvolumen die Vereinigung der diskreten Einzelvolumen oder Elemente. -Die \emph{Approximation} mittels Ansatzfunktionen als Linearkombination von Formfunktionen mit Knotenverformungen als Koeffizienten lautet +Die \emph{Approximation}, mittels Ansatzfunktionen als Linearkombination von Formfunktionen mit Knotenverformungen als Koeffizienten, lautet \[ \tensor{u}\big|_{V^{(e)}} \approx \tensor{u}\ti{fe}\big|_{V^{(e)}} = \tensor{N}\big|_{V^{(e)}} \tensor{\hat{u}} \quad \text{mit } \forall \tensor{u}\ti{fe} \in C^0(V) \] @@ -260,20 +261,23 @@ Werden für die Volumenvernetzung Hexaeder-Elemente (Würfel) mit trilinearen Fo \] Mit der \emph{Assemblierung} wird das Gebiet als Summe aller Elementintegrale abgebildet. \[ + \begin{Array}{lll} \displaystyle \underbrace{\int\limits_{V\vphantom{V_1}}\!\!\!\tensorII{\sigma} : \nabla\delta\tensorI{u} \dif V }_{\delta W\ti{i}} = \underbrace{\sum\limits_{(e)} \int\limits_{~V^{(e)}}\!\!\!\tensorII{\sigma} : \nabla\delta\tensorI{u} \dif V }_{\delta W\ti{i}\big|_{V^{(e)}}} - \quad , \quad - \underbrace{\int\limits_{V\vphantom{V_1}}\!\!\!\tensorI{f}\cdot\delta\tensorI{u}\dif V}_{\delta W\ti{a}} = - \underbrace{\sum\limits_{(e)} \int\limits_{~V^{(e)}}\!\!\!\tensorI{f}\cdot\delta\tensorI{u}\dif V}_{\delta W\ti{a}\big|_{V^{(e)}}} + \quad , \\[4.5em] + \displaystyle + \underbrace{\int\limits_{V\vphantom{V_1^(e)}}\!\!\!\tensorI{f}\cdot\delta\tensorI{u}\dif V + \!\!\int\limits_{\partial V_2\vphantom{V_1^(e)}}\!\!\!(\tensorI{t}\cdot\delta\tensorI{u})\dif A}_{\delta W\ti{a}} = + \underbrace{\sum\limits_{(e)} \int\limits_{~V^{(e)}\vphantom{V_1^(e)}}\!\!\!\tensorI{f}\cdot\delta\tensorI{u}\dif V + \sum\limits_{(e)} \int\limits_{~\partial V_2^{(e)}}\!\!\!\tensorI{t}\cdot\delta\tensorI{u}\dif A}_{\delta W\ti{a}\big|_{V^{(e)}}} + \end{Array} \] Für kleine Verformungen ergibt sich nun, mit den Materialgesetz, der kinematischen Beziehung und der Assemblierung, die virtuelle innere Arbeit zu \[ \begin{Array}{rll} \displaystyle \sum\limits_{(e)} \int\limits_{~V^{(e)}}\!\!\!\tensorII{\sigma} : \nabla\delta\tensorI{u} \dif V & \displaystyle = \sum\limits_{(e)} \int\limits_{~V^{(e)}}\!\!\!\tensor{C} \tensor{\varepsilon} \cdot \delta\tensor{\varepsilon} \dif V = - \sum\limits_{(e)} \delta\tensor{\hat{u}}^\T \!\!\!\!\!\int\limits_{~V^{(e)}}\!\!\! \underbrace{(\tensor{D}\tensor{N})^\T}_{\tensor{B}^\T} \tensor{C} \underbrace{(\tensor{D}\tensor{N})}_{\tensor{B}\vphantom{B^\T}} \dif V \tensor{\hat{u}} \\ + \sum\limits_{(e)} \delta\tensor{\hat{u}}^\T \!\!\!\!\!\int\limits_{~V^{(e)}}\!\!\! \underbrace{(\tensor{D}\tensor{N})^\T}_{\tensor{B}^\T} \tensor{C} \underbrace{(\tensor{D}\tensor{N})}_{\tensor{B}\vphantom{B^\T}} \dif V \tensor{\hat{u}} \\[2em] & \displaystyle = - \sum\limits_{(e)} \delta\tensor{\hat{u}}^\T \tensor{K}^{(e)} \tensor{\hat{u}} \quad\text{mit } \tensor{K}^{(e)} = \!\!\int\limits_{~V^{(e)}}\!\!\! \tensor{B}^\T \tensor{C} \tensor{B} \dif V \\ + \sum\limits_{(e)} \delta\tensor{\hat{u}}^\T \tensor{K}^{(e)} \tensor{\hat{u}} \quad\text{mit } \tensor{K}^{(e)} = \!\!\int\limits_{~V^{(e)}}\!\!\! \tensor{B}^\T \tensor{C} \tensor{B} \dif V \\[1.5em] & \displaystyle = \delta\tensor{\hat{u}}^\T \tensor{K} \tensor{\hat{u}} \quad \text{mit } \tensor{K} = \sum\limits_{(e)} \tensor{K}^{(e)} \end{Array} @@ -284,10 +288,10 @@ Weiter sind \(\tensor{K}^{(e)}\) die Elementsteifigkeitmatrix und \(\tensor{K}\) Und die äußere virtuelle Arbeit wird entsprechend umgeformt \[ \begin{Array}{rll} \displaystyle - \sum\limits_{(e)} \int\limits_{~V^{(e)}}\!\!\!\tensorI{f}\cdot\delta\tensorI{u}\dif V &\displaystyle = - \sum\limits_{(e)} \delta\tensor{\hat{u}}^\T \!\!\!\!\!\int\limits_{~V^{(e)}}\!\!\!\tensor{N}^\T\tensorI{f}\dif V \\ + \sum\limits_{(e)} \int\limits_{~V^{(e)}}\!\!\!\tensorI{f}\cdot\delta\tensorI{u}\dif V + \sum\limits_{(e)} \int\limits_{~\partial V_2^{(e)}}\!\!\!\tensorI{t}\cdot\delta\tensorI{u}\dif A &\displaystyle = + \sum\limits_{(e)} \delta\tensor{\hat{u}}^\T \!\!\!\!\!\int\limits_{~V^{(e)}}\!\!\!\tensor{N}^\T\tensorI{f}\dif V + \sum\limits_{(e)} \delta\tensorI{\hat{u}}^\T\!\!\!\!\!\!\!\int\limits_{~\partial V_2^{(e)}}\!\!\!\!\!\!\tensor{N}^\T\tensorI{t}\dif A \\[2em] & \displaystyle = - \sum\limits_{(e)} \delta\tensor{\hat{u}}^\T\tensor{\hat{r}}^{(e)} \quad \text{mit } \tensor{\hat{r}}^{(e)} = \!\!\int\limits_{~V^{(e)}}\!\!\!\tensor{N}^\T\tensorI{f}\dif V\\ + \sum\limits_{(e)} \delta\tensor{\hat{u}}^\T\tensor{\hat{r}}^{(e)} \quad \text{mit } \tensor{\hat{r}}^{(e)} = \!\!\int\limits_{~V^{(e)}}\!\!\!\tensor{N}^\T\tensorI{f}\dif V + \!\!\!\!\!\!\int\limits_{~\partial V_2^{(e)}}\!\!\!\!\!\!\tensor{N}^\T\tensorI{t}\dif A\\[1.5em] & \displaystyle = \delta\tensor{\hat{u}}^\T\tensor{\hat{r}} \quad \text{mit } \tensor{\hat{r}} = \sum\limits_{(e)} \tensor{\hat{r}}^{(e)} \end{Array} @@ -318,12 +322,76 @@ mit den Lagerbedingung \(\tensor{\hat{u}}_0 = 0\) sich zu einem eindeutig lösba \subsubsection{Dynamische Analysen} - +Im Gegensatz zu dem statischen Fall~\(\tensorI{u}(\tensorI{x})\) sind bei dynamischen zeitabhängigen Problemen~\(\tensorI{u}(\tensorI{x},t)\) -- die Bewegungen beschreiben -- Trägheitskräfte \(\rho\tensorI{\ddt{u}}\) zu berücksichtigen. +Diese Trägheitskräfte wirken entgegengesetzt zur angenommenen Bewegungsrichtung. +Damit lautet die starke Form des Gleichgewicht oder der allgemeine dynamische Impulsbilanz für ein Materialpunkt \[ - \begin{Array}{rcl} + \begin{Array}{rlll} \nabla\cdot\tensorII{\sigma} + \rho\tensorI{b} &= \rho\tensorI{\ddt{u}} & \quad \forall \tensorI{x}\in V ~\wedge~ \forall t \in [0,T] , \\ - \tensorI{u} &= \tensorI{u}_0 & \quad\text{auf } \partial V_1, \\ - \tensorII{\sigma}\cdot\tensorI{n} &= \tensorI{t} & \quad\text{auf } \partial V_2, \\ + \tensorI{u} &= \tensorI{u}_0, & \quad\forall \tensorI{x} \in \partial V_1 & \text{(Dirichlet'sche Randbedingungen)} \\ + \tensorII{\sigma}\cdot\tensorI{n} &= \tensorI{t} & \quad\forall \tensorI{x} \in \partial V_2 ~\wedge~ \forall t \in [0,T], & \text{(Neumann'sche Randbedingungen)} \\ &&\quad\text{wobei }\partial V_1 \cup \partial V_2 = \partial V. \end{Array} -\] \ No newline at end of file +\] +Hierin ist das Intervall \([0,T]\) das betrachtete Zeitintervall. + +Sind geschwindigkeitsabhängige Dämpfungskräfte zu simulieren, ist in der partiellen Differentialgleichung ein weiterer Dämpfungsterm \(d\tensorI{\dt{u}}\) zu berücksichtigen +\[ + \begin{Array}{rlll} + \nabla\cdot\tensorII{\sigma} + \rho\tensorI{b} &= \rho\tensorI{\ddt{u}} + d\tensorI{\dt{u}} & \quad \forall \tensorI{x}\in V ~\wedge~ \forall t \in [0,T] + \end{Array} +\] + +Zu den Anfangsbedingungen gehören +\[ + \begin{Array}{rlll} + \tensorI{u}(\tensorI{x},0) = \tensorI{u}_0 \\ + \tensorI{\dt{u}}(\tensorI{x},0) = \tensorI{\dt{u}}_0 \\ + \tensorI{\ddt{u}}(\tensorI{x},0) = \tensorI{\ddt{u}}_0 \\ + \end{Array} +\] +Da diese möglichen Anfangsbedingungen nicht unabhängig voneinander sind, werden nur zwei der drei Anfangsbedingungen je Materialpunkt vorgeschrieben. +Die verbleibende Größe ergibt sich aus der Auswertung der partiellen Differentialgleichung zum Zeitpunkt \(t=0\). + +\paragraph{Globales Gleichgewicht über schwache Formulierung des Randwertproblems}~\\ +Die Umformung in die schwache Formulierung erfolgt analog zum statischen Fall +\[ + \int\limits_{V}\!\!\!\tensorII{\sigma} : \nabla\delta\tensorI{u} \dif V + \!\!\int\limits_V\!\!\!\rho\tensorI{\ddt{u}}\cdot\delta\tensorI{u}\dif V + = \!\!\int\limits_V\!\!\!\rho\tensorI{b}\cdot\delta\tensorI{u}\dif V + \!\!\int\limits_{\partial V_2}\!\!\!\tensorI{t}\cdot\delta\tensorI{u}\dif A +% + \sum\tensorI{F}_i\cdot\delta\tensor{u}_i + \quad \forall t \in [0,T] +\] + + +\paragraph{Das FE"=Gleichungssystem}~\\ +... +\[ + \tensor{K}\tensor{\hat{u}} + \tensor{M}\tensor{\hat{\ddt{u}}} = \tensor{\hat{r}} +\] +... + + +\paragraph{Eigenfrequenzanalyse}~\\ +Die Differentialgleichung für ungedämpfte freie Schwingung +\[ + \tensor{K}\tensor{\hat{u}} + \tensor{M}\tensor{\hat{\ddt{u}}} = \tensor{0} +\] +Einsetzen des Lösungsansatzes +\[ + \tensor{\hat{u}} = \tensor{\Phi}\euler^{i\,\omega\,t} \quad \text{mit } i = \sqrt{-1} +\] +in die Differentialgleichung liefert das (reelle) Eigenwertproblem +\[ + (-\lambda \tensor{M} + \tensor{K}) \tensor{\Phi} = \tensor{0} \quad \text{mit } \lambda=\omega^2 +\] +wobei allgemein \(\euler^{i\,\omega\,t} \neq 0 \) gilt. + +... + +\paragraph{Transiente Analyse}~\\ +Neben der Volumendiskretisierung wird für die Zeitintegration von transienten Feldgleichungen das zu untersuchende Zeitintervall in diskrete Zeitabschnitte \(\Delta t\) unterteilt +\[ + [0,T] = \bigcup \Delta t \quad \text{mit } \Delta t = t_{n+1} - t_n +\] + +Zeitintegration nach Newmark \cite{newmark59}... \ No newline at end of file diff --git a/sections/Untersuchung.tex b/sections/Untersuchung.tex index 7ffcfe4..539c462 100755 --- a/sections/Untersuchung.tex +++ b/sections/Untersuchung.tex @@ -161,12 +161,8 @@ coordinates \toprule Art der Rechnung & Verformung in mm \\ \midrule -\emph{Eigengewicht} \\ -Lineare Ansatzfunktionen & 268,69 \\ -[.5em] -\emph{mit Einzellast \(F=\unit{80}{kN}\) an Rotorblattspitze} \\ -Lineare Ansatzfunktionen & 2346,1 \\ -Quadratische Ansatzfunktionen & 2250,0 \\ +Eigengewicht & 268,69 \\ +mit Einzellast \(F=\unit{80}{kN}\) an Rotorblattspitze & 2346,1 \\ \bottomrule \end{tabular} \end{table}\vspace{-1em}