From f50ad9196aa0fc608d58f7fef1faa5cbe26c7f26 Mon Sep 17 00:00:00 2001 From: Stefan Pfeifer Date: Mon, 22 Aug 2022 13:33:37 +0000 Subject: [PATCH] Update theory manual --- documents/theory-manual/document.tex | 4 +- documents/theory-manual/latex/equations.tex | 215 +----------------- .../theory-manual/latex/introduction.tex | 4 +- documents/theory-manual/latex/model.tex | 2 +- 4 files changed, 7 insertions(+), 218 deletions(-) diff --git a/documents/theory-manual/document.tex b/documents/theory-manual/document.tex index 4ed153d4..5b70dc31 100644 --- a/documents/theory-manual/document.tex +++ b/documents/theory-manual/document.tex @@ -25,9 +25,9 @@ % \allowdisplaybreaks % Break aligned equations over pages % Parameters -\newcommand{\version}{0.8} +\newcommand{\version}{0.9.0} \newcommand{\website}{\href{http://www.virtualbow.org/}{\texttt{\textcolor{blue}{http://www.virtualbow.org/}}}} -\newcommand{\copyrights}{Copyright (C) 2016-2021 Stefan Pfeifer} +\newcommand{\copyrights}{Copyright (C) 2016-2022 Stefan Pfeifer} \begin{document} diff --git a/documents/theory-manual/latex/equations.tex b/documents/theory-manual/latex/equations.tex index e75ecd12..9650b46f 100644 --- a/documents/theory-manual/latex/equations.tex +++ b/documents/theory-manual/latex/equations.tex @@ -738,219 +738,6 @@ \subsection{Floating Frame of Reference} \end{equation} where~$\alpha \in\ ]0,\,\nicefrac{1}{50}]$ is a tunable parameter. Tests with the bow simulation suggest that the choice of~$\alpha$ barely has any influence on the results. Therefore the maximum value~$\alpha = \nicefrac{1}{50}$ has been chosen for numerical reasons. - -\newpage -\subsection{Stiffness Matrix of a Beam Segment} - -In this section the analytical stiffness matrix for a curved beam segment as shown in figure~\ref{fig:beam-linear-1} is derived following the approach presented in \cite{bib:curved-beam-stiffness-matrix}. - -\begin{figure}[h] -\centering -\includegraphics[width=0.5\textwidth]{figures/elements/beam-linear-1} -\caption{Curved beam segment with two nodes} -\label{fig:beam-linear-1} -\end{figure} - -The segment has an initial length of~$l$. Its undeformed shape is described by the curve~$x(s)$, $y(s)$, $\varphi(s)$ with~$\varphi$ being the orientation angle measured against the $x$-axis and~$s \in [0,\,l]$ the arc length along the curve. - -In order for the segment to be in static equilibrium, the nodal forces~$p_0$,\,\ldots\,$p_5$ have to fulfill the equilibrium conditions -% -\begin{align} -&p_0 + p_3 = 0 \\ -&p_1 + p_4 = 0 \\ -&p_2 + p_5 + p_0\,\Delta y - p1\,\Delta x = 0 -\end{align} - -where~$\Delta x = x(l) - x(0)$ and~$\Delta y = y(l) - y(0)$. This can be rearranged into the following matrix equation that relates the forces on nodes A and B to each other, - -\begin{equation} -\underbrace{ -\begin{bmatrix} -p_3 \\ p_4 \\ p_5 -\end{bmatrix} -}_{\boldsymbol{p}_B} -= -\underbrace{ -\begin{bmatrix} --1 & 0 & 0 \\ - 0 & -1 & 0 \\ --\Delta y & \Delta x & -1 -\end{bmatrix} -}_{\boldsymbol{S}_{BA}} -\underbrace{ -\begin{bmatrix} -p_0 \\ p_1 \\ p_2 -\end{bmatrix} -}_{\boldsymbol{p}_A} -\label{eq:linear-beam-static-relation} -\end{equation} - -\begin{figure}[h] -\centering -\includegraphics[width=0.5\textwidth]{figures/elements/beam-linear-2} -\caption{Cross section forces of the beam segment} -\label{fig:beam-linear-2} -\end{figure} - -Forces on the cross section: - -\begin{align*} --&N(s)\cos(\varphi(s)) + Q(s)\sin(\varphi(s)) + p_3 = 0 \\ --&N(s)\sin(\varphi(s)) - Q(s)\cos(\varphi(s)) + p_4 = 0 \\ --&M(s) - p_3\,(y(l)-y(s)) + p_4\,(x(l) - x(s)) + p_5 = 0 -\end{align*} - -\begin{equation} -\begin{bmatrix} -N \\ M -\end{bmatrix} -= -\underbrace{ -\begin{bmatrix} -\cos(\varphi(s)) & \sin(\varphi(s)) & 0 \\ -y(s) - y(l) & x(l) - x(s) & 1 -\end{bmatrix} -}_{\boldsymbol{H}_B(s)} -\underbrace{ -\begin{bmatrix} -p_3 \\ p_4 \\ p_5 -\end{bmatrix} -}_{\boldsymbol{p}_B} -\end{equation} - - -\subsubsection*{Case 1: Flexibility matrix of point B when A is fixed} - -\begin{align} -V_B &= \frac{1}{2}\int_{0}^{l} -\begin{bmatrix} -N \\ M -\end{bmatrix}^\mathrm{T} -\begin{bmatrix} -\varepsilon \\ \kappa -\end{bmatrix} -ds \\ -&= \frac{1}{2}\int_{0}^{l} -\begin{bmatrix} -N \\ M -\end{bmatrix}^\mathrm{T} -\begin{bmatrix} -C_{\varepsilon\varepsilon} & C_{\varepsilon\kappa}\\ -C_{\varepsilon\kappa} & C_{\kappa\kappa} -\end{bmatrix}^{-1} -\begin{bmatrix} -N \\ M -\end{bmatrix} -ds \\ -&= \frac{1}{2}\,\boldsymbol{p}_B^\mathrm{T} -\underbrace{ -\left(\int_{0}^{l} \boldsymbol{H}_B^\mathrm{T}\, -\boldsymbol{C}^{-1}\boldsymbol{H}_B\,ds\right) -}_{\boldsymbol{K}_{BB}^{-1}} -\boldsymbol{p}_B -\end{align} - -According to Castigliano's second theorem the displacement of node B under the external loads~$\boldsymbol{p}_B$ is - -\begin{equation} -\boldsymbol{u}_{B} = \frac{\partial V_B}{\partial \boldsymbol{p}_B} = \boldsymbol{K}_{BB}^{-1}\,\boldsymbol{p}_B -\end{equation} - -with the flexibility matrix~$\boldsymbol{K}_{BB}^{-1}$ which is the inverse of the stiffness matrix. - -\subsubsection*{Case 2: Flexibility matrix of point A when B is fixed} - -\begin{equation} -\begin{bmatrix} -N \\ M -\end{bmatrix} -= \boldsymbol{H}_B\,\boldsymbol{p}_B = \boldsymbol{H}_B\,\boldsymbol{S}_{BA}\,\boldsymbol{p}_A -\end{equation} - -\begin{align} -V_A &= \frac{1}{2}\,\boldsymbol{p}_A^\mathrm{T} -\left(\int_{0}^{l} \boldsymbol{H}_A^\mathrm{T}\, -\boldsymbol{C}^{-1}\boldsymbol{H}_A\,ds\right) -\boldsymbol{p}_A \\ -&= \frac{1}{2}\,\boldsymbol{p}_A^\mathrm{T} -\boldsymbol{S}_{BA}^\mathrm{T}\left(\int_{0}^{l} \boldsymbol{H}_B^\mathrm{T}\, -\boldsymbol{C}^{-1}\boldsymbol{H}_B\,ds\right)\boldsymbol{S}_{BA}\,\boldsymbol{p}_A \\ -&= \frac{1}{2}\,\boldsymbol{p}_A^\mathrm{T} -\underbrace{ -\left(\boldsymbol{S}_{BA}^\mathrm{T}\boldsymbol{K}_{BB}^{-1}\boldsymbol{S}_{BA}\right) -}_{\boldsymbol{K}_{AA}^{-1}} -\boldsymbol{p}_A -\end{align} - -Displacement of node A under the external loads~$\boldsymbol{p}_A$, - -\begin{equation} -\boldsymbol{u}_{A} = \frac{\partial V_A}{\partial \boldsymbol{p}_A} = \boldsymbol{K}_{AA}^{-1}\,\boldsymbol{p}_A -\end{equation} - -\subsubsection*{Non block-diagonal stiffness} - -\begin{equation} -\boldsymbol{p}_B = \boldsymbol{S}_{BA}\,\boldsymbol{p}_A = -\underbrace{ -\boldsymbol{S}_{BA}\,\boldsymbol{K}_{AA} -}_{\boldsymbol{K}_{BA}} -\,\boldsymbol{u}_A -\end{equation} - -Therefore the complete stiffness relation for the beam segment is - -\begin{equation} -\begin{bmatrix} -\boldsymbol{p}_A \\ \boldsymbol{p}_B -\end{bmatrix} -= -\begin{bmatrix} -\boldsymbol{K}_{AA} & \boldsymbol{K}_{BA}^\mathrm{T} \\ -\boldsymbol{K}_{BA} & \boldsymbol{K}_{BB} -\end{bmatrix} -\begin{bmatrix} -\boldsymbol{u}_A \\ \boldsymbol{u}_B -\end{bmatrix} -\end{equation} - -Note that this is an exact result. No simplifications have been made within the linear Euler-Bernoulli beam theory. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - \newpage \section{Contact element} @@ -1121,3 +908,5 @@ \subsection{Contact Forces} \end{equation*} \subsection{Broadphase Algorithm} + +\textcolor{red}{TODO} \ No newline at end of file diff --git a/documents/theory-manual/latex/introduction.tex b/documents/theory-manual/latex/introduction.tex index aa73e2c9..04c6a056 100644 --- a/documents/theory-manual/latex/introduction.tex +++ b/documents/theory-manual/latex/introduction.tex @@ -5,5 +5,5 @@ \chapter{Introduction} \website. -This documentation, as opposed to the user manual, is about the theoretical foundations and technical details of the software. -It is meant as a reference for developers and interested users. \ No newline at end of file +This documentation is about the theoretical foundations and technical details of the software. +It is still a work in progress and is meant as a reference for developers and interested users. \ No newline at end of file diff --git a/documents/theory-manual/latex/model.tex b/documents/theory-manual/latex/model.tex index 081a5823..c45cf6b9 100644 --- a/documents/theory-manual/latex/model.tex +++ b/documents/theory-manual/latex/model.tex @@ -2,7 +2,7 @@ \chapter{The Bow Model} A scientific model is a simplification and abstraction of reality, often formulated in a mathematical way. A good model reduces the complexity of a real system by including only its most significant aspects and disregarding less significant ones. -The results of analyzing this simplified model can then be used to draw conclusions about the real system. +Analyzing this simplified model can then lead to conclusions about the real system that wouldn't have been possible otherwise. What aspects of reality a model has to reflect depends on the kinds of questions it seeks to answer. The first step for developing a bow model is therefore to clarify its scope and intended application.