From 951878985aaba4e6911a17624f4de0e8d696b5b9 Mon Sep 17 00:00:00 2001 From: Stefan Pfeifer Date: Sat, 10 Aug 2019 19:07:36 +0200 Subject: [PATCH] Move documentation into the main repository --- Changelog.md | 12 +- {examples => documents/examples}/flatbow.bow | 0 .../examples}/holmegaard.bow | 0 {examples => documents/examples}/longbow.bow | 0 .../examples}/mollegabet.bow | 0 .../examples}/python/example.py | 0 .../examples}/versions/0.4/default.bow | 0 .../examples}/versions/0.5/default.bow | 0 .../examples}/versions/0.6/default.bow | 0 documents/theory-manual/.gitignore | 1 + documents/theory-manual/document.tex | 51 + .../benchmark_examples/bar_truss_1.png | Bin 0 -> 459357 bytes .../benchmark_examples/circular_beam.jpg | Bin 0 -> 434470 bytes .../benchmark_examples/mises_frame.jpg | Bin 0 -> 443433 bytes .../figures/elements/bar-element-1.pdf | Bin 0 -> 4862 bytes .../figures/elements/bar-element-1.svg | 565 ++++++++ .../figures/elements/bar-element-2.pdf | Bin 0 -> 12874 bytes 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documents/theory-manual/figures/logo.png | Bin 0 -> 7486 bytes .../figures/model/bow-simulator.pdf | Bin 0 -> 9671 bytes .../figures/model/bow-simulator.svg | 665 +++++++++ .../theory-manual/figures/model/hickman.png | Bin 0 -> 54764 bytes .../theory-manual/figures/model/kooi.png | Bin 0 -> 72956 bytes .../figures/setup/line-distance.jpg | Bin 0 -> 29833 bytes .../figures/solution/dynamic_progress.pdf | Bin 0 -> 6431 bytes .../figures/solution/dynamic_progress.svg | 397 +++++ .../figures/validation/draw_curve.png | Bin 0 -> 30969 bytes .../figures/validation/setup.png | Bin 0 -> 352140 bytes .../figures/validation/states_blended.png | Bin 0 -> 374619 bytes documents/theory-manual/latex/appendix.tex | 0 .../theory-manual/latex/bibliography.tex | 52 + documents/theory-manual/latex/equations.tex | 1133 +++++++++++++++ .../theory-manual/latex/introduction.tex | 9 + documents/theory-manual/latex/model.tex | 117 ++ documents/theory-manual/latex/solution.tex | 384 +++++ documents/theory-manual/latex/titlepage.tex | 11 + documents/theory-manual/latex/validation.tex | 46 + .../maxima/contact_element_a.wxmx | Bin 0 -> 49360 bytes .../maxima/contact_element_b.wxmx | Bin 0 -> 17471 bytes documents/user-manual/.gitignore | 1 + documents/user-manual/document.tex | 663 +++++++++ .../figures/bending-test/sections-tapered.pdf | Bin 0 -> 4101 bytes .../figures/bending-test/sections-tapered.svg | 421 ++++++ .../figures/bending-test/sections-uniform.pdf | Bin 0 -> 2676 bytes .../figures/bending-test/sections-uniform.svg | 375 +++++ .../figures/bending-test/setup.pdf | Bin 0 -> 3212 bytes .../figures/bending-test/setup.svg | 359 +++++ documents/user-manual/figures/dimensions.pdf | Bin 0 -> 8319 bytes documents/user-manual/figures/dimensions.svg | 316 ++++ .../figures/experiments/draw_curve.png | Bin 0 -> 30969 bytes .../user-manual/figures/experiments/setup.png | Bin 0 -> 352140 bytes .../figures/experiments/states_blended.png | Bin 0 -> 374619 bytes documents/user-manual/figures/logo.png | Bin 0 -> 7486 bytes .../figures/screenshots/example.bow | 111 ++ .../figures/screenshots/input/bow-editor.png | Bin 0 -> 30214 bytes .../figures/screenshots/input/comments.png | Bin 0 -> 3671 bytes .../figures/screenshots/input/dimensions.png | Bin 0 -> 5731 bytes .../figures/screenshots/input/layers.png | Bin 0 -> 13933 bytes .../figures/screenshots/input/masses.png | Bin 0 -> 4538 bytes .../figures/screenshots/input/profile.png | Bin 0 -> 11696 bytes .../figures/screenshots/input/settings.png | Bin 0 -> 7895 bytes .../figures/screenshots/input/string.png | Bin 0 -> 5166 bytes .../figures/screenshots/input/width.png | Bin 0 -> 14706 bytes .../screenshots/output/result-window.png | Bin 0 -> 39250 bytes .../user-manual/maxima/cantilever_beam.wxm | 39 + 80 files changed, 11015 insertions(+), 6 deletions(-) rename {examples => documents/examples}/flatbow.bow (100%) rename {examples => documents/examples}/holmegaard.bow (100%) rename {examples => documents/examples}/longbow.bow (100%) rename {examples => documents/examples}/mollegabet.bow (100%) rename {examples => documents/examples}/python/example.py (100%) rename {examples => documents/examples}/versions/0.4/default.bow (100%) rename {examples => documents/examples}/versions/0.5/default.bow (100%) rename {examples => documents/examples}/versions/0.6/default.bow (100%) create mode 100644 documents/theory-manual/.gitignore create mode 100644 documents/theory-manual/document.tex create mode 100644 documents/theory-manual/figures/benchmark_examples/bar_truss_1.png create mode 100644 documents/theory-manual/figures/benchmark_examples/circular_beam.jpg create mode 100644 documents/theory-manual/figures/benchmark_examples/mises_frame.jpg create mode 100644 documents/theory-manual/figures/elements/bar-element-1.pdf create mode 100644 documents/theory-manual/figures/elements/bar-element-1.svg create mode 100644 documents/theory-manual/figures/elements/bar-element-2.pdf create mode 100644 documents/theory-manual/figures/elements/bar-element-2.svg create mode 100644 documents/theory-manual/figures/elements/beam-element-1.pdf create mode 100644 documents/theory-manual/figures/elements/beam-element-1.svg create mode 100644 documents/theory-manual/figures/elements/beam-element-2.pdf create mode 100644 documents/theory-manual/figures/elements/beam-element-2.svg create mode 100644 documents/theory-manual/figures/elements/beam-linear-1.jpg create mode 100644 documents/theory-manual/figures/elements/beam-linear-2.jpg create mode 100644 documents/theory-manual/figures/elements/beam-linear.pdf create mode 100644 documents/theory-manual/figures/elements/beam-linear.svg create mode 100644 documents/theory-manual/figures/elements/composite-sections-1.pdf create mode 100644 documents/theory-manual/figures/elements/composite-sections-1.svg create mode 100644 documents/theory-manual/figures/elements/contact_element_0.jpg create mode 100644 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documents/user-manual/figures/bending-test/sections-tapered.svg create mode 100755 documents/user-manual/figures/bending-test/sections-uniform.pdf create mode 100755 documents/user-manual/figures/bending-test/sections-uniform.svg create mode 100755 documents/user-manual/figures/bending-test/setup.pdf create mode 100755 documents/user-manual/figures/bending-test/setup.svg create mode 100644 documents/user-manual/figures/dimensions.pdf create mode 100644 documents/user-manual/figures/dimensions.svg create mode 100755 documents/user-manual/figures/experiments/draw_curve.png create mode 100755 documents/user-manual/figures/experiments/setup.png create mode 100755 documents/user-manual/figures/experiments/states_blended.png create mode 100755 documents/user-manual/figures/logo.png create mode 100644 documents/user-manual/figures/screenshots/example.bow create mode 100644 documents/user-manual/figures/screenshots/input/bow-editor.png create mode 100644 documents/user-manual/figures/screenshots/input/comments.png create mode 100644 documents/user-manual/figures/screenshots/input/dimensions.png create mode 100644 documents/user-manual/figures/screenshots/input/layers.png create mode 100644 documents/user-manual/figures/screenshots/input/masses.png create mode 100644 documents/user-manual/figures/screenshots/input/profile.png create mode 100644 documents/user-manual/figures/screenshots/input/settings.png create mode 100644 documents/user-manual/figures/screenshots/input/string.png create mode 100644 documents/user-manual/figures/screenshots/input/width.png create mode 100644 documents/user-manual/figures/screenshots/output/result-window.png create mode 100755 documents/user-manual/maxima/cantilever_beam.wxm diff --git a/Changelog.md b/Changelog.md index e1aef370..68853608 100644 --- a/Changelog.md +++ b/Changelog.md @@ -1,6 +1,6 @@ # Changelog -## 0.6.0 (10.08.2018) +## v0.6 (11.08.2018) ### Added * Support for MacOS @@ -15,7 +15,7 @@ * Crash when loading an invalid `.bow` file from the command line * Problem with face orientation of the limb geometry in the 3D view -## 0.5.0 (18.03.2018) +## v0.5 (18.03.2018) ### Added * Possibility to define laminated bows @@ -32,7 +32,7 @@ * Initial state `t = 0` missing from the dynamic output data * Program terminates when GUI is started with invalid file -## 0.4.0 (04.11.2017) +## v0.4 (04.11.2017) ### Added * More robust static simulation by using line searching @@ -43,7 +43,7 @@ ### Changed * Don't start the simulation when draw length is smaller than brace height -## 0.3.0 (04.09.2017) +## v0.3 (04.09.2017) ### Added * Command line interface for starting simulations without the GUI @@ -53,7 +53,7 @@ ### Changed * Replace sampling time with sampling rate/frequency in the dynamics settings -## 0.2.0 (06.08.2017) +## v0.2 (06.08.2017) ### Added * Tree based bow editor @@ -68,7 +68,7 @@ * Crash when section height is zero * Crash when cancel is clicked right after starting the simulation -## 0.1.0 (25.12.2016) +## v0.1 (25.12.2016) ### Added * Platform support for Windows, Linux and MacOS diff --git a/examples/flatbow.bow b/documents/examples/flatbow.bow similarity index 100% rename from examples/flatbow.bow rename to documents/examples/flatbow.bow diff --git a/examples/holmegaard.bow b/documents/examples/holmegaard.bow similarity index 100% rename from examples/holmegaard.bow rename to documents/examples/holmegaard.bow diff --git a/examples/longbow.bow b/documents/examples/longbow.bow similarity index 100% rename from examples/longbow.bow rename to documents/examples/longbow.bow diff --git a/examples/mollegabet.bow b/documents/examples/mollegabet.bow similarity index 100% rename from examples/mollegabet.bow rename to documents/examples/mollegabet.bow diff --git a/examples/python/example.py b/documents/examples/python/example.py similarity index 100% rename from examples/python/example.py rename to documents/examples/python/example.py diff --git a/examples/versions/0.4/default.bow b/documents/examples/versions/0.4/default.bow similarity index 100% rename from examples/versions/0.4/default.bow rename to documents/examples/versions/0.4/default.bow diff --git a/examples/versions/0.5/default.bow b/documents/examples/versions/0.5/default.bow similarity index 100% rename from examples/versions/0.5/default.bow rename to documents/examples/versions/0.5/default.bow diff --git a/examples/versions/0.6/default.bow b/documents/examples/versions/0.6/default.bow similarity index 100% rename from examples/versions/0.6/default.bow rename to documents/examples/versions/0.6/default.bow diff --git a/documents/theory-manual/.gitignore b/documents/theory-manual/.gitignore new file mode 100644 index 00000000..c795b054 --- /dev/null +++ b/documents/theory-manual/.gitignore @@ -0,0 +1 @@ +build \ No newline at end of file diff --git a/documents/theory-manual/document.tex b/documents/theory-manual/document.tex new file mode 100644 index 00000000..5c65ea63 --- /dev/null +++ b/documents/theory-manual/document.tex @@ -0,0 +1,51 @@ +\documentclass[12pt]{report} + +% Style +\usepackage{parskip} % Blank line between paragraphs +\usepackage[top=1in, bottom=1.25in, left=1.25in, right=1.25in]{geometry} % Page margins +\usepackage{float} % Placement of figures with H +\renewcommand{\familydefault}{\sfdefault} % Use sans-serif font (https://tex.stackexchange.com/a/2097) + + +% Maths +\usepackage{graphicx} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{units} +\usepackage{relsize} % \mathlarger + +\usepackage[hidelinks, pdfusetitle]{hyperref} +\usepackage{color} + +% Settings +% \allowdisplaybreaks % Break aligned equations over pages + +% Parameters +\newcommand{\swtitle}{VirtualBow} +\newcommand{\version}{0.6} +\newcommand{\website}{\href{http://www.virtualbow.org/}{\texttt{\textcolor{blue}{http://www.virtualbow.org/}}}} + +\begin{document} + +\input{latex/titlepage.tex} + +\tableofcontents + +\input{latex/introduction.tex} +\input{latex/model.tex} +\input{latex/equations.tex} +\input{latex/solution.tex} +\input{latex/validation.tex} + +\appendix + +\input{latex/appendix.tex} +\input{latex/bibliography.tex} + +\end{document} + +% Todo: +% - Set version number and website. +% Perhaps define them somewhere central. +% - Add metadata? +% - Unify Bib entries. 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z^z@|m_n+5GFTdhjj*Bw*EaU&n%ADCa4n7pqP-(fAOCV?;MqSbq$n-DHjct&HY@EB< zxIP_#KUUQDBQgQg@S{S~d+V(?>m^O13xFR1oZD`Gn|E{%(RI3{=Qw!9sR@BW(RB)N zzS37+MI!b$-24FKk`a=Iq@%`_kX7D-G6xwp3_HteD#1M~R&9 zZJ7d1S;QuB>MweyAlux`HezB<}|xjwLYR+ndowf}p6w=UoNGr!zp zrul=Lj?u2~_RZOIVqw&ePBhFr{{;nEH~=YSVw2IX6G)kc8!0&n&cD1k*Fj(u?pi*4 z^vDVK_U&7_(zTP#%8OS@Uky@oi*-qHE*&JH-CdTGV=$IV^kHwhfDE>S!4eC%M{i2( zyi(`NcY9ABn^*{KVg=*@u3uUOU>XpT6 0,\\ +\Leftrightarrow\quad &(x_B-x_A)(y_C-y_A)-(y_B-y_A)(x_C-x_A) > 0. +\end{align*} + +\textbf{Contact detection 3: Penetration} + +\begin{align*} +\sin(\alpha) &= \frac{e}{|P_2 - Q_0|} = \frac{(Q_1 - Q_0)\times(P_2 - Q_0)}{|Q_1 - Q_0||P_2 - Q_0|}\\ +\\ +\Leftrightarrow\:e &= \frac{(Q_1 - Q_0)\times(P_2 - Q_0)}{|Q_1 - Q_0|}\\ +&= \frac{(x_{\rm{Q1}}-x_{\rm{Q0}})(y_{\rm{P2}}-y_{\rm{Q0}})-(y_{\rm{Q1}}-y_{\rm{Q0}})(x_{\rm{P2}}-x_{\rm{Q0}})}{\sqrt{(x_{\rm{Q1}}-x_{\rm{Q0}})^2+(y_{\rm{Q1}}-y_{\rm{Q0}})^2}}\\ +&= \frac{a_1 a_4 - a_2 a_3}{\sqrt{a_1^2 + a_2^2}} +\intertext{With the abbreviations} +a_1 &= x_1 - x_0 - h_0 \sin(\varphi_0) + h_1 \sin(\varphi_1)\\ +a_2 &= y_1 - y_0 + h_0 \cos(\varphi_0) - h_1 \cos(\varphi_1)\\ +a_3 &= x_2 - x_0 - h_0 \sin(\varphi_0)\\ +a_4 &= y_2 - y_0 + h_0 \cos(\varphi_0) +\end{align*} + +\subsection{Contact Forces} + +\textbf{Elastic forces} + +The potential energy of the system can be determined as + +\begin{align*} +V &= \int_{0}^{e}F(s)\,ds +\end{align*} + +which yields the vector of elastic forces + +\begin{align} +\boldsymbol{q}(\boldsymbol{u}) &= \frac{\partial V}{\partial \boldsymbol{u}} = \frac{\partial V}{\partial e}\frac{\partial e}{\partial \boldsymbol{u}} = F(e)\,\frac{\partial e}{\partial \boldsymbol{u}}. +\end{align} + +Taking the derivatives: + +\begin{align*} +\frac{\partial e}{\partial \boldsymbol{u}} &= \frac{\partial}{\partial \boldsymbol{u}}\left(\frac{a_1 a_4 - a_2 a_3}{\sqrt{a_1^2 + a_2^2}}\right)\\ +&= \frac{\partial}{\partial \boldsymbol{u}}\left(\frac{1}{\sqrt{a_1^2 + a_2^2}}\right)\left(a_1 a_4 - a_2 a_3\right) + \frac{1}{\sqrt{a_1^2 + a_2^2}}\,\frac{\partial}{\partial \boldsymbol{u}}\left(a_1 a_4 - a_2 a_3\right)\\ +&= +\underbrace{\frac{1}{\sqrt{a_1^2 + a_2^2}}}_{b_1} \underbrace{\left(a_4\frac{\partial a_1}{\partial \boldsymbol{u}} - a_3\frac{\partial a_2}{\partial \boldsymbol{u}} - a_2\frac{\partial a_3}{\partial \boldsymbol{u}} + a_1\frac{\partial a_4}{\partial \boldsymbol{u}}\right)}_{\boldsymbol{v}_1} +\underbrace{+\frac{a_2 a_3 - a_1 a_4}{(a_1^2 + a_2^2)^\frac{3}{2}}}_{b_2} +\underbrace{\left(a_1\frac{\partial a_1}{\partial \boldsymbol{u}} + a_2\frac{\partial a_2}{\partial \boldsymbol{u}}\right)}_{\boldsymbol{v}_2}\\ +&= b_1\boldsymbol{v}_1 + b_2\boldsymbol{v}_2\\ +\\ +\frac{\partial a_1}{\partial \boldsymbol{u}} &= +\begin{bmatrix} +-1\\ 0\\ -h_0 \cos(\varphi_0)\\ 1\\ 0\\ h_1 \cos(\varphi_1)\\ 0\\ 0 +\end{bmatrix}, +\frac{\partial a_2}{\partial \boldsymbol{u}} = +\begin{bmatrix} +0\\ -1\\ -h_0 \sin(\varphi_0)\\ 0\\ 1\\ h_1 \sin(\varphi_1)\\ 0\\ 0 +\end{bmatrix}, +\frac{\partial a_3}{\partial \boldsymbol{u}} = +\begin{bmatrix} +-1\\ 0\\ -h_0 \cos(\varphi_0)\\ 0\\ 0\\ 0\\ 1\\ 0 +\end{bmatrix}, +\frac{\partial a_4}{\partial \boldsymbol{u}} = +\begin{bmatrix} +0\\ -1\\ -h_0 \sin(\varphi_0)\\ 0\\ 0\\ 0\\ 0\\ 1 +\end{bmatrix} +\end{align*} + +\textbf{Tangent stiffness matrix} + +\begin{align*} +\boldsymbol{K}(\boldsymbol{u}) &= \frac{\partial \boldsymbol{q}}{\partial \boldsymbol{u}} = \frac{\partial F}{\partial \boldsymbol{u}}\,\left(\frac{\partial e}{\partial \boldsymbol{u}}\right)^{\rm{T}} + F(e)\,\frac{\partial^2 e}{\partial \boldsymbol{u}^2} = \frac{\partial F}{\partial e}\,\frac{\partial e}{\partial \boldsymbol{u}}\left(\frac{\partial e}{\partial \boldsymbol{u}}\right)^{\rm{T}} + F(e)\,\frac{\partial^2 e}{\partial \boldsymbol{u}^2}\\ +\\ +\frac{\partial^2 e}{\partial \boldsymbol{u}^2} &= \frac{\partial}{\partial \boldsymbol{u}}\left(b_1\boldsymbol{v}_1 + b_2\boldsymbol{v}_2\right) = \frac{\partial b_1}{\partial \boldsymbol{u}}\,\boldsymbol{v}_1^{\rm{T}} + \frac{\partial b_2}{\partial \boldsymbol{u}}\,\boldsymbol{v}_2^{\rm{T}} + b_1\,\frac{\partial \boldsymbol{v}_1}{\partial \boldsymbol{u}} + b_2\,\frac{\partial \boldsymbol{v}_2}{\partial \boldsymbol{u}}\\ +\frac{\partial b_1}{\partial \boldsymbol{u}} &= -\frac{1}{\left(a_1^2 + a_2^2\right)^{\frac{3}{2}}}\left(a_1\,\frac{\partial a_1}{\partial \boldsymbol{u}} + a_2\,\frac{\partial a_2}{\partial \boldsymbol{u}}\right)\\ +\frac{\partial b_2}{\partial \boldsymbol{u}} &= \frac{\sqrt{a_1^2 + a_2^2}}{a_1^6 + 3\,a_1^4\,a_2^2 + 3\,a_1^2\,a_2^4 + a_2^6}\bigg(\left(2\,a_1^2\,a_4 - a_2^2\,a_4 - 3\,a_1\,a_2\,a_3\right)\frac{\partial a_1}{\partial \boldsymbol{u}}\\ +&+ \left(a_1^2\,a_3 - 2\,a_2^2\,a_3 + 3\,a_1\,a_2\,a_4\right)\frac{\partial a_2}{\partial \boldsymbol{u}} + \left(a_2^3 + a_1^2\,a_2\right)\frac{\partial a_3}{\partial \boldsymbol{u}} + \left(-a_1^3 - a_1\,a_2^2\right)\frac{\partial a_4}{\partial \boldsymbol{u}}\bigg)\\ +\\ +\frac{\partial \boldsymbol{v}_1}{\partial \boldsymbol{u}} &= \frac{\partial a_4}{\partial \boldsymbol{u}}\left(\frac{\partial a_1}{\partial \boldsymbol{u}}\right)^{\rm{T}} - \frac{\partial a_3}{\partial \boldsymbol{u}}\left(\frac{\partial a_2}{\partial \boldsymbol{u}}\right)^{\rm{T}} - \frac{\partial a_2}{\partial \boldsymbol{u}}\left(\frac{\partial a_3}{\partial \boldsymbol{u}}\right)^{\rm{T}} + \frac{\partial a_1}{\partial \boldsymbol{u}}\left(\frac{\partial a_4}{\partial \boldsymbol{u}}\right)^{\rm{T}}\\ +&+ a_4\,\frac{\partial^2 a_1}{\partial \boldsymbol{u}^2} - a_3\,\frac{\partial^2 a_2}{\partial \boldsymbol{u}^2} - a_2\,\frac{\partial^2 a_3}{\partial \boldsymbol{u}^2} + a_1\,\frac{\partial^2 a_4}{\partial \boldsymbol{u}^2}\\ +\frac{\partial \boldsymbol{v}_2}{\partial \boldsymbol{u}} &= \frac{\partial a_1}{\partial \boldsymbol{u}}\left(\frac{\partial a_1}{\partial \boldsymbol{u}}\right)^{\rm{T}} + \frac{\partial a_2}{\partial \boldsymbol{u}}\left(\frac{\partial a_2}{\partial \boldsymbol{u}}\right)^{\rm{T}} + a_1\,\frac{\partial^2 a_1}{\partial \boldsymbol{u}^2} + a_2\,\frac{\partial^2 a_2}{\partial \boldsymbol{u}^2} +\end{align*} + +Second derivatives: + +\begin{equation*} +\begin{aligned} +\left(\frac{\partial^2 a_1}{\partial \boldsymbol{u}^2}\right)_{ij} &= +\begin{cases} +\:\:\:h_0\,\sin(\varphi_0) & \text{$ij = (3,3)$}\\ +-h_1\,\sin(\varphi_1) & \text{$ij = (6,6)$}\\ +\:\:\:0 & \text{otherwise} +\end{cases}\\ +\left(\frac{\partial^2 a_3}{\partial \boldsymbol{u}^2}\right)_{ij} &= +\begin{cases} +\:\:\:h_0\,\sin(\varphi_0) & \text{$ij = (3,3)$}\\ +\:\:\:0 & \text{otherwise} +\end{cases}\\ +\end{aligned} +\qquad +\begin{aligned} +\left(\frac{\partial^2 a_2}{\partial \boldsymbol{u}^2}\right)_{ij} &= +\begin{cases} +-h_0\,\cos(\varphi_0) & \text{$ij = (3,3)$}\\ +\:\:\:h_1\,\cos(\varphi_1) & \text{$ij = (6,6)$}\\ +\:\:\:0 & \text{otherwise} +\end{cases}\\ +\left(\frac{\partial^2 a_4}{\partial \boldsymbol{u}^2}\right)_{ij} &= +\begin{cases} +-h_0\,\cos(\varphi_0) & \text{$ij = (3,3)$}\\ +\:\:\:0 & \text{otherwise} +\end{cases} +\end{aligned} +\end{equation*} + +\subsection{Broadphase Algorithm} diff --git a/documents/theory-manual/latex/introduction.tex b/documents/theory-manual/latex/introduction.tex new file mode 100644 index 00000000..711ff647 --- /dev/null +++ b/documents/theory-manual/latex/introduction.tex @@ -0,0 +1,9 @@ +\chapter{Introduction} + +This is the theory manual for \swtitle, an open-source software tool for simulating bow and arrow physics. +For the latest version of the software and this document visit + +\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 diff --git a/documents/theory-manual/latex/model.tex b/documents/theory-manual/latex/model.tex new file mode 100644 index 00000000..3f9d8b22 --- /dev/null +++ b/documents/theory-manual/latex/model.tex @@ -0,0 +1,117 @@ +\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. + +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. + +\section{Scope of the Model} +\label{section:model-scope} + +In the case of \swtitle\ the purpose of the bow model is to give users the ability to evaluate the bow designs they come up with. +So what characteristics make a bow design a good one and what are their implications for the bow model? +Many things come to mind, but we limit ourselves to the following three categories: + +\begin{enumerate} +\item \textbf{Viability}: Can the design be realised? Do the materials withstand the arising loads when the bow is being used? +To be able to judge this the model has to accurately predict the distribution of stress across the limbs as well as the tensile forces in the string that arise when drawing and shooting the bow. + +\item \textbf{Performance}: Important performance characteristics are the arrow's velocity, kinetic energy and the bow's degree of efficiency, i.e. the percentage of the stored elastic energy that is converted into kinetic energy of the arrow during the shot. +To capture these things the model has to reflect all major mechanisms of efficiency loss in a bow. +We will later see what those are when we look at scientific research. + +\item \textbf{Comfort}: Viability and raw performance aren't everything. +A bow design will also be judged by it's influence on the archer, like the draw force, the shape of the draw curve (no "stacking", i.e. increasing stiffness towards the end of draw) and the dynamic forces exerted to the shooter ("hand shock"). +\end{enumerate} + +Any physical effects in archery that don't (significantly) affect one of those categories above are outside the scope of \swtitle\ and the bow model doesn't need to include them. +Exterior ballistics is an example for this, another one is the bending motion of the arrow (Archer's Paradox). + +Now the next question is: How does a bow model have to look like in order to capture the things listed above? +That's not easy to answer, but luckily there has been done quite some scientific research about bow and arrow physics in the past, so we don't have to start completely from scratch. +The next section therefore examines some of that research before we define our own model. + +\section{Literature Review} + +Research about the mechanics of bow and arrow goes back to at least the 1930s, when C.N.~Hickman, P.E.~Klopstek and others performed systematic experiments and developed simple mathematical models. +Without modern computers available these early models were still limited in their complexity by what could be solved analytically. +The focus was therefore more on qualitative insight rather than accurate prediction. +One of those early models is the~\textit{Hickman Model}~\cite{bib:hi37}, published in~1937. +It assumes two rigid limbs that are connected to the stationary rest of the bow via elastic hinges as shown in figure~\ref{fig:model:hickman}. +Even though this is a huge simplification, this model can already be used to investigate the efficiency of a bow, i.e. how it transfers its stored elastic energy to an arrow. + +\begin{figure}[h] +\centering +\includegraphics[width=0.4\textwidth]{figures/model/hickman} +\caption{Hickman model as used by W.C. Marlow \cite{bib:ma80}} +\label{fig:model:hickman} +\end{figure} + +An important modification to the Hickman model was suggested by W.C.~Marlow in 1980~\cite{bib:ma80}. +It was already well-known that the original Hickman model with an inextensible string severely overpredicted a bow's degree of efficiency. +(Predicted efficiencies typically over 90\% while measurements suggest values between 70\% and 85\% for most bows.) +Popular explanations for this discrepancy were effects like hysteresis and air resistance. +Marlow however found another reason: With an inextensible string (as in Hickman's original model) the arrow exits the bow exactly at brace height, where the limbs have zero velocity. +Only the kinetic energy of the string remains in the bow, the rest is transferred to the arrow. +Marlow then showed that accounting for the elasticity of the string shifts the point of exit for the arrow away from the braced position, such that the limbs still have substantial kinetic energy left. In his numerical example the modification reduces the predicted efficiency to~78\% as opposed to~92\% previously. +Of the energy remaining in the bow,~11\% were the kinetic energy of the limbs,~9\% the kinetic energy of the string and~2\% the potential energy of limbs and string. + +This leads to an important realization: The efficiency of a bow is mainly determined by the kinetic energy that stays in the system due to mass and elasticity of the string and the mass distribution of the limbs. The contribution of dissipative effects like air resistance and hysteresis to the overall efficiency was estimated by Marlow to be less than~2\% and~3\% respectively. + +The widespread availability of digital computers eventually made it possible to simulate much more detailed models and therefore make more accurate predictions. +Much notable work in this area has been done by Dr.~Bob~Kooi and his co-authors. +Over the course of the~1980s and~90s they published a good amount of papers about the mechanics of bow and arrow in which they utilize numerical methods to solve increasingly complex mathematical models. + +The first one of those papers is titled \textit{On the static deformation of a bow}~\cite{bib:kosp80} and investigates the storage of deformation energy in bows with and without recurve. Figure~\ref{fig:model:kooi} shows the model used there. +The bow is assumed to be symmetric and has an inextensible string. +The limbs are represented by an Euler-Bernoulli beam, which is an infinitely thin, elastic line with a certain (varying) bending stiffness for which the Euler-Bernoulli assumptions hold. +These are assumptions about the kinematics of the deformation, namely that the cross sections of the beam always stay flat and perpendicular to its centerline. This is typically a good approximation for slender beams, that is beams whose length is much larger than their cross section dimensions. Kooi uses the general version of the Euler-Bernoulli equation that accounts for arbitrarily large deformation. Combining this with geometric constraints for the string and some boundary conditions leads to the bow's equations of equilibrium, a free-boundary value problem for a set of ordinary differential equations. The equations of equilibrium are solved numerically for specific draw lengths of the bow. + +\begin{figure}[h] +\centering +\includegraphics[width=0.8\textwidth]{figures/model/kooi} +\caption{Bow model used by Bob Kooi \cite{bib:kosp80}} +\label{fig:model:kooi} +\end{figure} + +In subsequent papers titled \textit{On the mechanics of the bow and arrow}~\cite{bib:kooi81} and \textit{On the mechanics of the modern working-recurve bow}~\cite{bib:kooi91} Kooi also considers the dynamic case. Like Marlow he uses a linear-elastic string and assumes the arrow as a concentrated point mass. The equations of motion for the system form a moving-boundary value problem for a set of partial differential equations. They are solved numerically by using a finite-difference method with complete discretization. (Both space and time are approximated by a discrete "grid" of values. The partial derivatives in the governing equations are then approximated by finite difference quotients operating on those grid values. The resulting large number of difference equations can then be solved with a computer program.) + +In the paper \textit{On the Mechanics of the Arrow: Archer's Paradox}~\cite{bib:kosp97}, Kooi refines his model even more by including an elastic arrow with a sliding contact to the bow's grip. This makes it possible to analyze the Archer's Paradox: The phenomenon of arrows going around the grip of a bow by performing a bending motion in the horizontal plane. The dynamics of the arrow are however treated partly decoupled from the bow's dynamics: Only the influence of the bow on the arrow's motion is considered, the other way around is neglected. This is in line with our assumption from section~\ref{section:model-scope} that the bending motion of the arrow doesn't contribute much to a bow's overall performance and is therefore out of scope for our model. + +\section{Model Definition} + +We have now learned enough in order to define our own bow model. First of all a few words on the general modeling approach. As we saw in the previous section, any decently complex bow model cannot be solved analytically but has to be combined with some sort of numerical method for calculating an approximate solution. Models based on continuum mechanics (e.g. Euler-Bernoulli beam theory) always have an infinite number of degrees of freedom and will eventually have to be approximated by a discrete system with finitely many degrees of freedom in order to be solvable by a computer program. This step is called discretization. Kooi for example, like a true mathematician, formulates the complete equations of motion for his continuous bow model and only then starts to employ a discretization technique (in this case a finite difference method) to solve those equations. This results in a clear separation between the mathematical model and the numerical solution methods. + +We're going to use more of an engineering approach here instead and start with a discrete system in the first place. We do this by constructing the continuous parts of the bow from a large number of so-called finite elements. A finite element is an approximation of a little part of a continuous structure, e.g. a beam or a bar, but with only finitely many degrees of freedom, namely the displacements of the nodes at which the elements are connected to each other. The more elements are used, the better the approximation gets and the closer the results will be to the exact solution. + +The finite element method leads to a (potentially large) system of ordinary differential equations as the equations of motion, which can be solved by standard numerical methods as we will see later. + +\begin{figure}[h] +\centering +\includegraphics[width=0.8\textwidth]{figures/model/bow-simulator} +\caption{Finite element model of the bow} +\label{fig:model:bow-simulator} +\end{figure} + +Figure~\ref{fig:model:bow-simulator} shows the finite element approximation of our bow model. Like Kooi's model it is symmetric with respect to the horizontal plane, assumes the limbs to satisfy the Euler-Bernoulli assumptions and neglects any dissipative effects like air resistance and hysteresis. The limbs are therefore made up of Euler-Bernoulli beam elements. A concentrated point mass is placed at the tips to account for any added weight, like overlays for example. + +The string, as opposed to the limbs, only transfers longitudinal forces and is therefore approximated by linear-elastic bar elements. We limit ourselves to linear elasticity for pragmatic reasons: Obtaining the exact nonlinear stress-strain curve of a particular string material is pretty difficult. Instead, any material nonlinearity (which can be the case especially for modern synthetic materials) will have to be accounted for by using an "average" stiffness. In the previous section we saw that the mass properties of the string are fairly important, so in addition to the mass of the string itself there are two additional point masses at the tips and center to account for things like servings, nocking point, etc. + +The arrow is another point mass, as already justified earlier. It is connected to the string by an ideal, frictionless contact without any release force. This means that the arrow will be released as soon as the string doesn't push against/accelerate it anymore. + +Not shown in figure~\ref{fig:model:bow-simulator} are the contact elements that prevent the string nodes from passing through the limb, thereby making it possible to simulate recurves. These are also ideal contacts without any friction. + +All in all, there are four different types of elements needed: + +\begin{itemize} +\item Beam Elements (Limbs) +\item Bar Elements (String) +\item Mass Elements (Arrow + additional masses) +\item Contact Elements (String to limb contact) +\end{itemize} + +All of them have to account for arbitrarily large displacements and are therefore geometrically nonlinear (even though all the materials are linear-elastic). + +In the next chapters we will first show how the equations of motion for a finite element system can be assembled from the equations of motion for the single elements. We will then derive the equations of motion for the element types listed above. Finally we're going to discuss the numerical solution of those equations for the static and dynamic case and how this is implemented in \swtitle. \ No newline at end of file diff --git a/documents/theory-manual/latex/solution.tex b/documents/theory-manual/latex/solution.tex new file mode 100644 index 00000000..7355b0a7 --- /dev/null +++ b/documents/theory-manual/latex/solution.tex @@ -0,0 +1,384 @@ +\chapter{Solving the Equations of Motion} + +\section{Static Solution} + +\textcolor{red}{Static solution in context of bow: Bracing and equilibrium path, necessity for load control and displacement control} + +A System with the equation of motion + +\begin{align} +\boldsymbol{M}\,\ddot{\boldsymbol{u}} + \boldsymbol{q}(\boldsymbol{u},\,\dot{\boldsymbol{u}}) = \boldsymbol{p} +\end{align} + +is in static equilibrium if~$\dot{\boldsymbol{u}},\,\ddot{\boldsymbol{u}} = \boldsymbol{0}$, leading to the static equilibrium condition + +\begin{align} +\boldsymbol{q}(\boldsymbol{u},\,\boldsymbol{0}) = \boldsymbol{p}\label{eq:statics:equilibrium} +\end{align} + +for the displacements $\boldsymbol{u}$ and the external loads $\boldsymbol{p}$. +This is a nonlinear system of equations and can be interpreted as the internal elastic forces~$\boldsymbol{q}$ being in balance with the external loads. + +\subsection{The Basic Newton-Raphson Method} + +Most frequently, when solving static problems, the external loads are given and the task is to calculate the corresponding equilibrium displacements. +Equation~(\ref{eq:statics:equilibrium}) is then a nonlinear system of equations in $\boldsymbol{u}$. Solving it can only be done numerically except for very simple examples. + +An often used method for this kind of problem is the Newton-Raphson method. The basic idea behind it is to linearise the equilibrium equation~(\ref{eq:statics:equilibrium}) around a known displacement vector~$\boldsymbol{u}_i$ and solve the resulting linear equation to obtain an improved approximation~$\boldsymbol{u}_{i+1}$. Starting with an initial value~$\boldsymbol{u}_0$ the procedure is repeated until the solution is accurate enough by some convergence criterion. + +Linearising of (\ref{eq:statics:equilibrium}) yields + +\begin{align} +\boldsymbol{q}(\boldsymbol{u}_i) + +\underbrace{ +\left(\frac{\partial \boldsymbol{q}}{\partial \boldsymbol{u}}\bigg \vert_{\boldsymbol{u}_i}\right) +}_{\boldsymbol{K}(\boldsymbol{u}_i)} +\underbrace{ +(\boldsymbol{u}_{i+1} - \boldsymbol{u}_i) +}_{\Delta\boldsymbol{u}_i} &= \boldsymbol{p} +\end{align} + +Solving for $\Delta\boldsymbol{u}_i$ leads to the iterative procedure + +\begin{align} +\Delta\boldsymbol{u}_i &= \boldsymbol{K}^{-1}(\boldsymbol{u}_i)\left(\boldsymbol{p} - \boldsymbol{q}(\boldsymbol{u}_i)\right),\\ +\boldsymbol{u}_{i+1} &= \boldsymbol{u}_i + \Delta\boldsymbol{u}_i.\label{eq:statics:newton_iteration} +\end{align} + +In the implementation the tangent stiffness matrix is not actually inverted, because that's expensive and potentially inaccurate. Instead,~$\Delta\boldsymbol{u}_i$ is obtained by solving the equivalent linear system of equations using a decomposition of the stiffness matrix. The symmetry of the tangent stiffness can be used here. (It isn't necessarily positive definite though, like the stiffness matrices of linear mechanical systems.) + +Convergence criteria are usually formulated in terms of change in displacement~$\Delta\boldsymbol{u}$, the residuum $\boldsymbol{p} - \boldsymbol{q}$ or their product $\Delta\boldsymbol{u}^\mathsf{T}(\boldsymbol{p}-\boldsymbol{q})$ (which has the dimension of an energy). We will use the energy criterion + +\begin{equation*} +\Delta\boldsymbol{u}_{i}^\mathsf{T}(\boldsymbol{p}-\boldsymbol{q}(\boldsymbol{u}_i)) < \varepsilon +\end{equation*} + +The basic Newton-Raphson method can be improved and extended in many ways, for example by combining it with a line search or by avoiding the calculation of the tangent stiffness matrix at every iteration (simplified Newton-Raphson method, BFGS method \textcolor{red}{[Source]}). + +\subsection{Constrained Newton-Raphson method} + +In the previous section the Newton-Raphson method was applied to the problem of calculating the equilibrium displacements of a system when all applied forces are known. This method is used in \swtitle for finding the braced equilibrium state of the bow (see section \textcolor{red}{[Reference]}). However, when calculating the draw curve we don't actually know the draw forces, just the displacement of the string center. \textcolor{red}{Also: Load decrease.} + +A solution to this is displacement control, where we prescribe a value for one of the displacement components and instead treat the corresponding external forces as a variable that is determined during solution. + +The following derivation has been adapted from \cite{fem_script_uni_bochum}. Displacement control is a special case of a wider class of methods where the equilibrium equations are modified as +% +\begin{align} +\boldsymbol{q}(\boldsymbol{u}) - \lambda\,\boldsymbol{p} &= 0,\label{eq:equilibrium_dc}\\ +c(\boldsymbol{u},\,\lambda) &= 0.\label{eq:constraint_dc} +\end{align} +% +Here the external forces are scaled by the unknown load factor $\lambda$. To compensate for this additional unknown, a scalar constraint~$c(\boldsymbol{u},\,\lambda) = 0$ on the displacements and the load parameter is introduced. Depending on the choice of constraint the method can be turned into load control, displacement control or other more sophisticated schemes like the arc-length method. We will carry out the derivation for the general case and specify the constraints later. This is also how these methods are implemented in \swtitle: A general implementation for arbitrary constraints that is then specialized into load controlled and displacement controlled methods by providing the necessary constraints. + +The system of nonlinear equations (\ref{eq:equilibrium_dc}), (\ref{eq:constraint_dc}) has to be solved for the displacements~$\boldsymbol{u}$ and the load parameter~$\lambda$. Like before, the Newton-Raphson method is used: The equations are linearised around a given approximation~$\{\boldsymbol{u}_i,\,\lambda_i\}$ and solved to obtain an improved approximiation~$\{\boldsymbol{u}_{i+1},\,\lambda_{i+1}\}$. This is repeated iteratively until displacements and load parameter satisfy the equilibrium equations~(\ref{eq:equilibrium_dc}) as well as the constraint equation~(\ref{eq:constraint_dc}) reasonably well as determined by some convergence criterion. + +Carrying out the linearisation leads to the two equations +% +\begin{align} +\boldsymbol{q}(\boldsymbol{u}) - \lambda\,\boldsymbol{p}\:&\approx\:\boldsymbol{q}(\boldsymbol{u_{i}}) - \lambda_i\,\boldsymbol{p} + +\underbrace{ +\frac{\partial \boldsymbol{q}}{\partial \boldsymbol{u}}(\boldsymbol{u_{i}}) +}_{\boldsymbol{K_{i}}} +\underbrace{ +(\boldsymbol{u_{i+1}} - \boldsymbol{u_{i}}) +}_{\Delta\boldsymbol{u_{i}}} +- \boldsymbol{p} +\underbrace{ +\,(\lambda_{i+1} - \lambda_{i}) +}_{\Delta \lambda_{i}} +\notag\\ +&\approx\ \boldsymbol{q}(\boldsymbol{u_{i}}) - \lambda_i\,\boldsymbol{p} + \boldsymbol{K_{i}}\,\Delta\boldsymbol{u_{i}} - \boldsymbol{p}\,\Delta \lambda_{i}\label{eq:equilibrium_dc_lin}\\ +\notag\\ +c(\boldsymbol{u},\,\lambda) \ &\approx\ c(\boldsymbol{u_{i}},\,\lambda_{i}) + \frac{\partial c}{\partial \boldsymbol{u}}(\boldsymbol{u_{i}},\,\lambda_{i})(\boldsymbol{u_{i+1}} - \boldsymbol{u_{i}}) + \frac{\partial c}{\partial \lambda}(\boldsymbol{u_{i}},\,\lambda_{i})(\lambda_{i+1} - \lambda_{i})\notag\\ +&\approx\ c(\boldsymbol{u_{i}},\,\lambda_{i}) + \frac{\partial c}{\partial \boldsymbol{u}}(\boldsymbol{u_{i}},\,\lambda_{i})\,\Delta\boldsymbol{u_{i}} + \frac{\partial c}{\partial \lambda}(\boldsymbol{u_{i}},\,\lambda_{i})\,\Delta \lambda_{i}\label{eq:constraints_dc_lin} +\end{align} + +Equations~(\ref{eq:equilibrium_dc_lin}) and~(\ref{eq:constraints_dc_lin}) can written as the single linear system of equations +% +\begin{align} +\begin{bmatrix} +\boldsymbol{K_{i}} & -\boldsymbol{p}\\ +\frac{\partial c}{\partial \boldsymbol{u}}(\boldsymbol{u_{i}},\,\lambda_{i}) & \frac{\partial c}{\partial \lambda}(\boldsymbol{u_{i}},\,\lambda_{i}) +\end{bmatrix} +\begin{bmatrix} +\Delta\boldsymbol{u_{i}}\\ +\Delta \lambda_{i} +\end{bmatrix} += +\begin{bmatrix} +\lambda_i\,\boldsymbol{p} - \boldsymbol{q}(\boldsymbol{u_{i}})\\ +-\boldsymbol{c}(\boldsymbol{u_{i}},\,f_{i}) +\end{bmatrix}.\label{eq:total_dc_lin} +\end{align} +% +Solving this for the increments in displacement and load factor $\{\Delta\boldsymbol{u_{i}},\,\Delta \lambda_{i}\}$ allows us to calculate the next values as +% +\begin{align} +\boldsymbol{u_{i+1}} = \boldsymbol{u_{i}} + \Delta\boldsymbol{u_{i}}\\ +\lambda_{i+1} = \lambda_{i} + \Delta \lambda_{i}. +\end{align} +% +However, directly solving~(\ref{eq:total_dc_lin}) is not numerically efficient because the augmented stiffness matrix is no longer symmetric. A much better way to solve this is block elimination. We use the first row of~(\ref{eq:total_dc_lin}) to express~$\Delta\boldsymbol{u_{i}}$ as +% +\begin{align} +\Delta\boldsymbol{u_{i}} &= \boldsymbol{\alpha} + \boldsymbol{\beta}\,\Delta \lambda_{i}.\label{eq:auxiliary_disp_dc} +\end{align} +% +With the auxiliary vectors +% +\begin{align} +\boldsymbol{\alpha} &= \boldsymbol{K_{i}^{-1}}(\lambda_i\,\boldsymbol{p} - \boldsymbol{q}(\boldsymbol{u_{i}}))\label{eq:auxiliary_1}\\ +\boldsymbol{\beta} &= \boldsymbol{K_{i}^{-1}}\,\boldsymbol{p}\label{eq:auxiliary_2} +\end{align} +% +These can be computed efficiently by exploiting the symmetry of the stiffness matrix $\boldsymbol{K_{i}}$. +% +To calculate $\Delta \lambda_i$, we substitute~(\ref{eq:auxiliary_disp_dc}) into the second row of~(\ref{eq:total_dc_lin}) which gives us +% +\begin{align} +&\frac{\partial c}{\partial \boldsymbol{u}}(\boldsymbol{u_{i}},\,\lambda_{i})\Delta\boldsymbol{u_{i}} + \frac{\partial c}{\partial \lambda}(\boldsymbol{u_{i}},\,\lambda_{i})\,\Delta \lambda_{i} = -c(\boldsymbol{u_{i}},\,\lambda_{i})\notag\\ +&\frac{\partial c}{\partial \boldsymbol{u}}(\boldsymbol{u_{i}},\,\lambda_{i})(\boldsymbol{\alpha} + \boldsymbol{\beta}\,\Delta \lambda_{i}) + \frac{\partial c}{\partial \lambda}(\boldsymbol{u_{i}},\,\lambda_{i})\,\Delta \lambda_{i} = -c(\boldsymbol{u_{i}},\,\lambda_{i})\notag\\ +&\ \ \vdots\notag\\ +&\Delta \lambda_{i} = -\frac{c(\boldsymbol{u_{i}},\,\lambda_{i}) + \frac{\partial c}{\partial \boldsymbol{u}}(\boldsymbol{u_{i}},\,\lambda_{i})\,\boldsymbol{\alpha}}{\frac{\partial c}{\partial \boldsymbol{u}}(\boldsymbol{u_{i}},\,\lambda_{i})\,\boldsymbol{\beta} + \frac{\partial c}{\partial \lambda}(\boldsymbol{u_{i}},\,\lambda_{i})}.\label{eq:res_load_factor} +\end{align} + +The Newton-Raphson iteration is now complete. To summarize: First, calculate $\boldsymbol{\alpha}$ and~$\boldsymbol{\beta}$ by solving~(\ref{eq:auxiliary_1}) and~(\ref{eq:auxiliary_2}). Then calculate the increment of the load parameter by using~(\ref{eq:res_load_factor}). The increment in displacements is then obtained by equation~(\ref{eq:auxiliary_disp_dc}).\\ + +\subsection{Constraints for load- and displacement control} + +Finally, the constraint function and the parametrisation of the load vector have to be chosen. To get a displacement controlled method as initially stated we choose the constraint function as +% +\begin{align} +\boldsymbol{c}(\boldsymbol{u},\,\lambda) = u^{(k)} - \overline{u} +\end{align} +% +where~$u^k$ is the $k$-th component of the displacement vector that we want to control and $\overline{u}$ is the prescribed value for it. The partial derivatives are +% +\begin{align*} +\frac{\partial c}{\partial \boldsymbol{u}} &= \boldsymbol{e}_k,\quad +\frac{\partial c}{\partial \lambda} = 0, +\end{align*} +% +with $\boldsymbol{e}_k$ being the $k$-th unit vector. If a load controlled method is needed instead, the constraint becomes +% +\begin{align} +\boldsymbol{c}(\boldsymbol{u},\,\lambda) = \lambda - \overline{\lambda}, +\end{align} +% +with the desired load factor $\overline{\lambda}$. Partial derivatives: +% +\begin{align*} +\frac{\partial c}{\partial \boldsymbol{u}} &= \boldsymbol{0},\quad +\frac{\partial c}{\partial \lambda} = 1, +\end{align*} + +\subsection{Line search} + +There is one last extension to the Newton-Raphson method that is needed for our purposes and that is line search. + +\textcolor{red}{Todo: Explain line search and why it helps with contact handling} + +\newpage +\subsection{Finding the Braced Equilibrium State} + +\textcolor{red}{Length of the string...} + +\section{Dynamic Solution} + +The equations of motion together with the initial displacements~$\boldsymbol{u}_0$ and velocities~$\boldsymbol{v}_0$ of the system form the second-order initial value problem + +\begin{align} +\boldsymbol{M}\ddot{\boldsymbol{u}} &+ \boldsymbol{q}(\boldsymbol{u},\,\ddot{\boldsymbol{u}}) = \boldsymbol{p},\quad \boldsymbol{u}(0) = \boldsymbol{u}_0,\ \ddot{\boldsymbol{u}}(0) = \boldsymbol{v}_0.\label{eq:dynamics:system_equation} +\end{align} + +for the displacements~$\boldsymbol{u}(t)$. + +Numerical solution methods for initial value problems can be grouped into explicit and implicit ones. +Explicit methods use the current state of the system at time~$t$ to calculate the next state at~$t + \Delta t$. +Implicit methods instead find the next state by solving an equation that involves the current and the next state. +In the context of finite element analysis, implicit methods require equilibrium iterations at every time step involving the system's tangent stiffness matrix, while explicit methods only require simple vector operations. +Implicit methods therefore have a higher computational cost per step, but can make up for it by requiring less steps overall, as their better stability allows them to use much larger step sizes than explicit methods. + +Explicit methods are easier to implement and are a good choice for small time scales, like impact and wave propagation problems. +They are most effective when the computation of the time steps is fast, so they work well with constant, diagonal (lumped) mass matrices and simple elements with low-order shape functions. + +Implicit methods in contrast are well suited for larger time scales, where explicit algorithms would need too many steps. +Examples are structural response problems or numerically stiff systems, i.e. systems whose eigenfrequencies range from very small to very large. + +\swtitle\ currently uses the central difference method \cite{bib:dynamic_solution}, which is an explicit method often used in FEM. +Based on it's properties and the advantages of explicit methods it seems like a good choice. +However, implicit methods have not yet been tested with \swtitle. The Newmark-beta method \textcolor{red}{Source} for example might be worth a try. + +\swtitle\ previously used the Runge-Kutta-Fehlberg-7-8 method provided by the \textcolor{red}{the odeint library}. +It performed worse than the central difference method, probably because of the high number of function evaluations carried out by an order~7/8 Runge-Kutta scheme. +The big advantage however was the timestep control based on error estimation, which meant that absolutely no user input regarding the timestep was necessary. + +\subsection{The Central Difference Method} + +The continuous solution $\boldsymbol{u}(t)$ is approximated by the discrete values~$\boldsymbol{u}_i = \boldsymbol{u}(t_{i})$ at the equidistant time steps~$t_i = i\cdot\Delta t$,\ \ $i \in \mathbb{N}$. +The acceleration~$\ddot{\boldsymbol{u}}_i$ can be approximated by a central difference quotient as + +\begin{align} +\ddot{\boldsymbol{u}}_i &= \frac{\boldsymbol{u}_{i+1} - 2\,\boldsymbol{u}_{i} + \boldsymbol{u}_{i-1}}{\Delta t^2}\label{eq:dynamics:difference_acceleration} +\end{align} + +Solving for~$\boldsymbol{u}_{i+1}$ and calculating the accelerations~$\ddot{\boldsymbol{u}}_{i}$ from the equation of motion~(\ref{eq:dynamics:system_equation}) leads to + +\begin{align} +\boldsymbol{u}_{i+1} &= 2\,\boldsymbol{u}_{i} - \boldsymbol{u}_{i-1} + \Delta t^2\,\ddot{\boldsymbol{u}}_{i}\notag\\ +&= 2\,\boldsymbol{u}_{i} - \boldsymbol{u}_{i-1} + \Delta t^2\,\boldsymbol{M}^{-1}(\boldsymbol{p}_i - \boldsymbol{q}_i). +\end{align} + +This expression is already the core of the method, it allows the calculation of the next displacements~$\boldsymbol{u}_{i+1}$ from the current and previous system states. +It only involves cheap vector operations as the mass matrix is diagonal and doesn't have to be "actually" inverted. + +Only the velocity is still missing. For this the central difference quotients of the displacements~$\boldsymbol{u}_{i+1}$,~$\boldsymbol{u}_{i}$ and~$\boldsymbol{u}_{i-1}$ are used to calculate the intermediate velocities + +\begin{align} +\dot{\boldsymbol{u}}_{i+\frac{1}{2}} &= \frac{\boldsymbol{u}_{i+1} - \boldsymbol{u}_{i}}{\Delta t},\\ +\dot{\boldsymbol{u}}_{i-\frac{1}{2}} &= \frac{\boldsymbol{u}_{i} - \boldsymbol{u}_{i-1}}{\Delta t}. +\end{align} + +The actual new velocity~$\dot{\boldsymbol{u}}_{i+1}$ is linearly extrapolated from those, + +\begin{align} +\dot{\boldsymbol{u}}_{i+1} &= \dot{\boldsymbol{u}}_{i+\frac{1}{2}} + \frac{1}{2}\,\left(\dot{\boldsymbol{u}}_{i+\frac{1}{2}} - \dot{\boldsymbol{u}}_{i-\frac{1}{2}}\right),\notag\\ +&= \frac{1}{\Delta t}\,\left(\frac{3}{2}\,\dot{\boldsymbol{u}}_{i+1} - 2\,\dot{\boldsymbol{u}}_{i} + \frac{1}{2}\,\dot{\boldsymbol{u}}_{i-1}\right). +\end{align} + +This last step is actually a custom modification as far as I can tell. +All descriptions of the central difference method that I've seen end with the calculation of~$\dot{\boldsymbol{u}}_{i+\frac{1}{2}}$. + +The initialisation of the procedure needs some special attention, as the method uses the previous displacements~$\boldsymbol{u}_{i-1}$ that don't exist yet at time~$t = 0$. +Instead,~$\boldsymbol{u}_{-1}$ is estimated by using finite difference approximations of the system's initial velocity and acceleration: + +\begin{align} +\dot{\boldsymbol{u}}_0 &= \frac{\boldsymbol{u}_{1} - \boldsymbol{u}_{-1}}{2\,\Delta t},\\ +\ddot{\boldsymbol{u}}_0 &= \frac{\boldsymbol{u}_{1} - 2\,\boldsymbol{u}_{0} + \boldsymbol{u}_{-1}}{\Delta t^2},\\ +&\Rightarrow \boldsymbol{u}_{-1} = \boldsymbol{u}_{0} - \Delta t\,\dot{\boldsymbol{u}}_{0} + \frac{\Delta t^2}{2}\,\ddot{\boldsymbol{u}}_{0}. +\end{align} + +A last but very important question is how to choose the timestep~$\Delta t$. +The timestep influences the computational effort, the accuracy of the solution but most importantly the stability of the algorithm. +A too large time step will lead to a solution that "explodes" numerically. +For linear, conservative mechanical systems of the form + +\begin{align} +\boldsymbol{M}\ddot{\boldsymbol{u}} + \boldsymbol{K}\boldsymbol{u} = \boldsymbol{p} +\end{align} + +the stability condition is + +\begin{align} +\Delta t \le \frac{2}{\omega_{max}}\label{eq:timestep_bound} +\end{align} + +where~$\omega_{max}$ is the maximum natural frequency \cite{bib:dynamic_solution} of the system. +Unfortunately there is no such analytical bound on~$\Delta t$ that guarantees stability for nonlinear systems. +A common thing to do is to linearise the system around the current configuration and apply (\ref{eq:timestep_bound}) anyway. +A safety factor~$0 < \beta < 1$ is introduced to account for the uncertainties and the timestep is set to + +\begin{align} +\Delta t = \beta \frac{2}{\omega_{max}}. +\end{align} + +This doesn't guarantee stability in a strict sense, but it's okay for practical purposes. A common choice is~$\beta = 0.9$ but~\swtitle\ uses a lower value for the sake of robustness. The frequency $\omega_{max}$ is sometimes estimated by using the bound + +\begin{align} +\omega_{max} \le \max_{i}\,\{\omega_{max,\,i}\} +\end{align} + +where $\omega_{max,\,i}$ is the maximum natural frequency of the respective finite element~$i$. But this leads to larger time steps than necessary and actually causes more implementation effort than just solving the generalised eigenvalue problem + +\begin{align} +\left(\boldsymbol{M} - \omega^2\boldsymbol{K}\right)\boldsymbol{\bar{u}} = 0\label{eq:natural_frequencies} +\end{align} + +which is what \swtitle\ does. + +\textcolor{red}{No damping here. For a version with damping see the LS-Dyna theoretical manual of 2006.} + +\subsection{Stopping Criterion and Progress Estimation} + +Everything has to stop at some point, even the dynamic simulation. But when exactly? +The simulated time interval has to include the separation of the arrow from the bow plus some additional time, because some interesting things like e.g. the maximum force on the string only happen after the arrow left the bow. +A related requirement is that there needs to be some estimation for the simulation progress that can be shown the user. +But how can the progress be estimated even though the ending time of the simulation isn't known in advance? + +The current approach is to simulate the time interval~$[0,\,\alpha\,T]$ where~$T$ is the (yet unknown) time for the arrow to reach brace height and~$\alpha$ is an arbitrary positive factor. +The default value is~$\alpha = 1.5$. The simulation progress at time~$\overline{t}$ can now be formally written as + +\begin{equation} +\rm{progress} = \frac{\overline{t}}{\alpha\,T}\cdot 100\%.\label{eq:solution:progress} +\end{equation} + +But only formally, because~$T$ is unknown as long as~$\overline{t} < T$, i.e. as long as the arrow has not yet actually reached brace height. For this first part of the simulation~$T$ is estimated by using the current time~$\overline{t}$ and position~$\overline{u}$ of the arrow and extrapolate to the time when the arrow will reach brace height. + +\begin{figure}[h] +\centering +\includegraphics[width=0.8\textwidth]{figures/solution/dynamic_progress} +\caption{Arrow position $u(t)$ during the shot} +\label{fig:solution:dynamic_progress} +\end{figure} + +The progression of the arrow position, denoted by~$u(t)$, typically looks like shown in figure~\ref{fig:solution:dynamic_progress}. +It starts at time~$t = 0$ at full draw~$u_0$ and reaches brace height~$u_T$ at the time~$t = T$ by definition. +If the draw curve of the bow is assumed to be approximately linear then~$u(t)$ will resemble the cosine quarter-wave + +\begin{equation} +u(t) \approx u_{0}\,\cos{\left(C\,t\right)}.\label{eq:solution:progress:ansatz} +\end{equation} + +Not all draw curves might be approximately linear, but this is only about crude progess estimation after all. +The constant~$C$ is obtained by applying equation~(\ref{eq:solution:progress:ansatz}) to the current time and position, + +\begin{equation} +u(\overline{t}) = \overline{u} \Rightarrow \ C = \frac{1}{\overline{t}}\,\arccos{\left(\frac{\overline{u}}{u_0}\right)}.\label{eq:solution:progress:constant} +\end{equation} + +Now $T$ can be extrapolated from~(\ref{eq:solution:progress:ansatz}) using the condition~$u(T) = u_T$, + +\begin{align} +T = \frac{1}{C}\,\arccos{\left(\frac{u_T}{u_0}\right)} = \overline{t}\ \frac{\arccos{\left(\nicefrac{u_T}{u_0}\right)}}{\arccos{\left(\nicefrac{\overline{u}}{u_0}\right)}}. +\end{align} + +This approximation for~$T$ is used in the progress estimation~(\ref{eq:solution:progress}) until the arrow has reached brace height. Then the actual value is known and used for the rest of the simulation. + + + + +\newpage +\section{Model setup} + +\subsection{Distributing string elements} + +\begin{figure}[h] +\centering +\includegraphics[width=0.8\textwidth]{figures/setup/line-distance} +\caption{Problem} +\label{fig:solution:dynamic_progress} +\end{figure} + +Problem: Find a point $P$ on the line between $A$ and $B$ which has the euclidean distance $d$ to point $C$. + +Parametrisation of the points $P$: + +\begin{align*} +P(\eta) &= A + \eta\,(B-A),\quad \eta \in [0,\,1] +\end{align*} + +Distance condition: + +\begin{align*} +&\lVert C - P(\eta)\rVert = d\\ +\Leftrightarrow\:&\lVert +\begin{bmatrix} +c_x - a_x - \eta\,(b_x - a_x) \\ c_y - a_y - \eta\,(b_y - a_y) +\end{bmatrix} +\rVert = d\\ +\Leftrightarrow\:&\left(c_x - a_x - \eta\,(b_x - a_x)\right)^2 + \left(c_y - a_y - \eta\,(b_y - a_y)\right)^2 = d^2\\ +\Leftrightarrow\:&\left((b_x - a_x)^2 + (b_y - a_y)^2\right)\,\eta^2-2\left((c_x - a_x)(b_x - a_x) + (c_y - a_y)(b_y - a_y)\right)\,\eta\\ +&+(c_x - a_x)^2 + (c_y - a_y)^2 - d^2 = 0 +\end{align*} + +This is a quadratic equation for $\eta$. Take smallest solution in $[0,\,1]$. diff --git a/documents/theory-manual/latex/titlepage.tex b/documents/theory-manual/latex/titlepage.tex new file mode 100644 index 00000000..986c0c87 --- /dev/null +++ b/documents/theory-manual/latex/titlepage.tex @@ -0,0 +1,11 @@ +\begin{titlepage} + +\vspace*{5em} +\centering{\Huge\textbf{\swtitle~v\version \\[1em] Theory Manual}} + +\vspace{5em} + +\centerline{\includegraphics[width=0.2\textwidth]{figures/logo.png}} + +\thispagestyle{empty} +\end{titlepage} diff --git a/documents/theory-manual/latex/validation.tex b/documents/theory-manual/latex/validation.tex new file mode 100644 index 00000000..c62a8c1d --- /dev/null +++ b/documents/theory-manual/latex/validation.tex @@ -0,0 +1,46 @@ +\chapter{Model Validation} + +A very important task in the development of this software is to make sure that the results obtained by simulation agree reasonably well with real world examples. +This section shows the validation efforts made so far. +It is still very sparse, so if you have used this program for a real world application, let me know about your results and they will be added here. +\subsubsection{Statics of a straight steel bow} + +In this experiment the draw curve and limb shapes of a small steel bow with a constant cross section have been measured and compared to version 2014.4 of Bow Simulation Tool (now \swtitle). +The bow is shown in figure~\ref{fig:validation:setup} and has been made from an old saw blade. +Steel is a good material for this kind of test because of its homogenous mechanical properties. +The elastic modulus was assumed to be $E = 210\ \rm{GPa}$ which is a good estimate for most types of steel. + +\begin{figure}[H] +\centering +\includegraphics[width=0.5\textwidth]{figures/validation/setup.png} +\caption{Steel bow. Cross section:~16.85~$\times$~0.75mm. Length:~269mm. Brace heighth:~49.8mm} +\label{fig:validation:setup} +\end{figure} + +The experiment was carried out by hanging a plastic bag at the string center. +The draw force was then gradually applied by counting steel balls with a known mass into the bag. +After every load step the draw length was measured and a photo of the bow was taken. + +Figure~\ref{fig:validation:draw_curve} shows a comparison between the measured and the simulated draw curve and figure~\ref{fig:validation:limb_shapes} compares the pictures of the limb against the simulated limb shapes. + +\begin{figure}[H] +\centering +\includegraphics[width=0.85\textwidth]{figures/validation/draw_curve.png} +\caption{Experimental and simulated draw curves} +\label{fig:validation:draw_curve} +\end{figure} + +\begin{figure}[H] +\centering +\includegraphics[width=0.65\textwidth]{figures/validation/states_blended.png} +\caption{Experimental and simulated limb shapes} +\label{fig:validation:limb_shapes} +\end{figure} + +The agreement between experiment and simulation is surprisingly good here. +It really shows the potential of such kinds of simulations, provided that the material properties are well known and a the bow can be built exactly as simulated, with low tolerances. + +But this is still a very simple example. +The next step would be to repeat this experiment with bows that have varying cross sections and non-straight profiles. +Another open question are the dynamic simulation results. +It's unclear if they can match experiments as good as the static results do, because there are much more uncertainties involved. \ No newline at end of file diff --git a/documents/theory-manual/maxima/contact_element_a.wxmx b/documents/theory-manual/maxima/contact_element_a.wxmx new file mode 100644 index 0000000000000000000000000000000000000000..42ab568fc6917a7603a858c093f8692a3f0b6e66 GIT binary patch literal 49360 zcmeHQ&5q;75uU_}VaNv?hymY%j)1^3>m6w{v+JLYRz~biU}Lk$9yUM#!yZslYs9rk 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+\documentclass[12pt]{article} + +% Style +\usepackage{parskip} % Blank line between paragraphs +\usepackage[top=1in, bottom=1.25in, left=1.25in, right=1.25in]{geometry} % Page margins +\usepackage{float} % Placement of figures with H +\renewcommand{\familydefault}{\sfdefault} % Use sans-serif font (https://tex.stackexchange.com/a/2097) + +% Maths +\usepackage{graphicx} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{units} + +\usepackage[hidelinks, pdfusetitle]{hyperref} + +% Utillities +\usepackage{framed} % Draw frames around paragraphs +\usepackage{listings} % Code listings +\usepackage{color} +\usepackage{array} + +% Settings +\allowdisplaybreaks % Break aligned equations over pages + +\usepackage{perpage} % Reset footnote counter per page (1) +\MakePerPage{footnote} % Reset footnote counter per page (2) + +% Change itemize label items +\renewcommand{\labelitemi}{$\bullet$} +\renewcommand{\labelitemii}{$\bullet$} +\renewcommand{\labelitemiii}{$\bullet$} + +% Parameters +\newcommand{\swtitle}{VirtualBow} +\newcommand{\version}{0.6} +\newcommand{\website}{\href{http://www.virtualbow.org/}{\texttt{\textcolor{blue}{http://www.virtualbow.org/}}}} +\newcommand{\copyrights}{Copyright (C) 2016-2018 Stefan Pfeifer} + +\begin{document} + +\begin{titlepage} + +\vspace*{5em} +\centering{\Huge\textbf{\swtitle~v\version \\[1em] User Manual}} + +\vspace{5em} + +\centerline{\includegraphics[width=0.2\textwidth]{figures/logo.png}} + +\null +\vfill +\centerline{\copyrights} + +\thispagestyle{empty} +\end{titlepage} + +\newpage +\tableofcontents + +\newpage +\section{Introduction} +\bigskip + +This is the user manual for \swtitle, a software tool for bow and arrow physics simulation. +It shows how to use the program, explains the various input parameters and results and also contains some additional background information. + +For the latest version of the software and this manual visit the project's website at\\ +\website.\\ + +\newpage +\section{The Bow Editor} + +The bow editor is the main window of the application. Here you can load, edit and save bow models and start static and dynamic simulations. + +\begin{figure}[H] +\centering +\includegraphics[width=\textwidth]{figures/screenshots/input/bow-editor} +\caption{Bow Editor} +\label{fig:bow-editor} +\end{figure} + +Double-click any of the items on the left to edit the respective model properties. The 3D view on the right shows the current limb geometry. +Use the mouse to rotate (left mouse button), shift (middle mouse button) and zoom the view. +More view options are available through the buttons on the bottom-right corner. + +\newpage +\subsection{Profile} + +The profile curve is the geometric centerline of the bow limbs in unbraced state. +Use the table on the left to edit the parameters. +The result is shown on the plot on the right. + +\begin{figure}[H] +\centering +\includegraphics[width=\textwidth]{figures/screenshots/input/profile} +\caption{Profile dialog} +\label{fig:profile} +\end{figure} + +The profile curve consists of a series of arc segments that each have a certain length and curvature. Each row in the table defines one such segment. + +\textbf{Note:} Mathematically, the curvature $\kappa$ of a segment is the reciprocal of its radius $r$, so you can calculate the curvature via $\kappa = \frac{1}{r}$ if you know the radius and vice versa. + +\newpage +\subsection{Width} + +The width dialog is used to define the limb's width along its profile curve. This width is the same for all layers of the bow. + +\begin{figure}[H] +\centering +\includegraphics[width=\textwidth]{figures/screenshots/input/width} +\caption{Width dialog} +\label{fig:width} +\end{figure} + +On the table on the left you can specify values for the width at certain relative positions (between 0 and 1) along the limb's profile curve. This definition of cross section properties relative to the total length of the profile curve makes it possible to change the profile without having to adjust any cross sections. + +The actual width distribution of the limb is constructed as a smooth curve (monotonic cubic spline) passing through the supplied values as shown on the plot on the right. + +\newpage +\subsection{Layers} + +With the layer dialog you can create any number of layers and specify their height/thickness and material properties. + +\begin{figure}[H] +\centering +\includegraphics[width=\textwidth]{figures/screenshots/input/layers} +\caption{Layer dialog} +\label{fig:width} +\end{figure} + +Click the plus button on the top left to add layers. You can later rearrange them by dragging the tabs. Double-click on a tab to rename a layer. + +The table on the left sets the height distribution of the layer. It works the same way as the limb's width: You specify a number of values at different relative positions and the program creates an interpolating curve. + +The layer material is specified by the following two constants, + +\begin{itemize} +\item \textbf{rho:} Density (Mass per unit volume) +\item \textbf{E:} Elastic modulus (Measure for the stiffness of a solid material) +\end{itemize} + +For manufactured materials like e.g. fiberglass or steel you might find those numbers in a datasheet provided by the manufacturer. + +Wood is a bit more difficult, because the properties can vary quite a bit even within the same type. +You can find average numbers on the internet, for example at \url{http://www.wood-database.com}. +Those are probably good enough as a first reference. +However, in order to be really sure about a specific piece of wood there is no other way than to test it. +One possibility is a bending test as shown in Appendix~\ref{sec:bending-test}. + +\newpage +\subsection{String} + +Here you can define the mechanical properties of the string by providing data for the string material and the number of strands being used. + +\begin{figure}[H] +\centering +\includegraphics[width=0.4\textwidth]{figures/screenshots/input/string} +\caption{String dialog} +\label{fig:string} +\end{figure} + +\begin{itemize} +\item \textbf{Strand density:} Linear density of the strands (mass per unit length) +\item \textbf{Strand stiffness:} Stiffness of the strands against elongation (force per unit strain) +\item \textbf{Number of strands:} Total number of strands in the string +\end{itemize} + +The linear density of a string material can be easily determined with a kitchen scale (weight divided by length), but the stiffness is much more difficult to obtain. +Table~\ref{tbl:string-materials} shows some reference values taken from the SuperTiller V6.6 Excel spreadsheet by Alan Case\footnote{\url{http://www.buildyourownbow.com/build-alongs/how-to-use-supertiller-build-along/}}. + +\textbf{Note:} The stiffness of the string material is an important parameter in dynamic analysis. +The static results however aren't affected very much by it as long as it is high enough to prevent significant elongation. + +\begin{table}[H] +\centering +\begin{tabular}{ | p{60pt} | p{50pt} | p{50pt} | p{65pt} | p{140pt} | } +\hline +\textbf{Material} & \textbf{Stiffness} [\unitfrac{N}{100\%}] & \textbf{Density} [\unitfrac{kg}{m}] & \textbf{Breaking strength} [\unit{N}] & \textbf{Source/Comment}\\ \hline +Dacron B50 & 3113.76 & 0.000333 & 217.96 & BCY assuming 7\% elongation at break (linearized) \\ \hline +Fast Flight & 14086.04 & 0.000182 & 422.58 & BCY assuming 3\% elongation at break (linearized) \\ \hline +Dyneema & 18860.46 & 0.000160 & 578.27 & Calculated assuming linear stress-strain (to break) \\ \hline +Linen 40/3 & 1668.08 & 0.0642 & 49.38 & Maurice Taylor, Archery The Technical Side, 1947 \\ \hline +Silk & 1026.51 & 0.0930 & 65.83 & Maurice Taylor, Archery The Technical Side, 1947 \\ \hline +\end{tabular} +\caption{Properties of common bowstring materials according to SuperTiller V6.6} +\label{tbl:string-materials} +\end{table} + +\newpage +\subsection{Masses} + +This dialog is usede to define the various masses of the bow. All of them except the arrow mass are optional. + +\begin{figure}[H] +\centering +\includegraphics[width=0.4\textwidth]{figures/screenshots/input/masses} +\caption{Masses dialog} +\label{fig:masses} +\end{figure} + +\begin{itemize} +\item \textbf{Arrow:} Mass of the arrow +\item \textbf{String center:} Additional mass at the string center (serving, nocking point) +\item \textbf{String tip:} Additional mass at the ends of the string (serving, silencers) +\item \textbf{Limb tip:} Additional mass at the limb tip (nocks, overlays) +\end{itemize} + +\subsection{Dimensions} + +The parameters listed in this dialog define the lengths and angles of the bow and its optional middle section/grip. See also figure~\ref{fig:dimensions-2} for a visual explanation of those parameters. + + +\begin{figure}[H] +\centering +\includegraphics[width=0.4\textwidth]{figures/screenshots/input/dimensions} +\caption{Dimensions dialog} +\label{fig:dimensions-1} +\end{figure} + +\newpage +\begin{itemize} +\item{Draw} +\begin{itemize} +\item \textbf{Brace height:} Distance of the string center to the coordinate origin in braced state +\item \textbf{Draw length:} Distance of the string center to the coordinate origin in fully drawn state +\end{itemize} +\item{Handle} +\begin{itemize} +\item \textbf{Length:} Length of the riser or stiff middle section +\item \textbf{Setback:} Depth of the riser or stiff middle section +\item \textbf{Angle:} Angle of the riser or stiff middle section +\end{itemize} +\end{itemize} + +\begin{figure}[H] +\centering +\includegraphics[width=0.4\textwidth]{figures/dimensions} +\caption{Dimensions of the bow} +\label{fig:dimensions-2} +\end{figure} + +\newpage +\subsection{Comments} + +The comments are meant for documenting the bow model. Any notes about the bow and the simulation results can be added here. + +\begin{figure}[H] +\centering +\includegraphics[width=0.7\textwidth]{figures/screenshots/input/comments} +\caption{Comments dialog} +\label{fig:comments} +\end{figure} + +\subsection{Settings} + +These are numerical settings that can be used to tweak the simulation. +Most of the time the default values should be fine though. +However, as the default values favor accuracy and realiability over performance there might be use cases where finding faster settings is worth it. +Think about running a large number of simulations with a script for example. + +\begin{figure}[H] +\centering +\includegraphics[width=0.4\textwidth]{figures/screenshots/input/settings} +\caption{Settings dialog} +\label{fig:settings} +\end{figure} + +\newpage +These are the individual settings: + +\begin{itemize} +\item \textbf{General} + \begin{itemize} + \item \textbf{Limb elements:} Number of finite elements that are used to approximate the limb. More elements increase the accuracy but also the computing time. + \item \textbf{String elements:} Same as above. + \end{itemize} +\item \textbf{Statics} + \begin{itemize} + \item \textbf{Draw steps:} Number of steps that are performed by the static simulation from brace height to full draw. This determines the resolution of the static results. You can usually decrease this value to speed up the simulation. + \end{itemize} +\item \textbf{Dynamics} + \begin{itemize} + \item \textbf{Time span factor:} This controls the time period that is simulated. A value of $1$ corresponds to the time at which the arrow reaches brace height. The default value is larger than that in order to capture some of the things that occur after the arrow left the bow (maximum dynamic loads on limb and string). + \item \textbf{Time step factor:} When carrying out the dynamic simulation the program will repeatedly use the current state of the bow at time $t$ to calculate the next state at time $t + \Delta t$ where $\Delta t$ is some small timestep. This timestep has to be chosen small enough to get an accurate and stable solution but also as large as possible to keep the number of steps low. The program can estimate this, but to be on the safe side the estimation is multiplied with a safety factor between $0$ and $1$ that you can choose here. + \item \textbf{Sampling rate:} Limits the time resolution of the output data. This is done because the dynamic simulation usually produces much finer grained data than is actually useful. Not including all of that in the final output saves time and memory. + \end{itemize} +\end{itemize} + +\newpage +\section{Simulation Results} + +You can start static or dynamic simulations by using the yellow and green toolbar buttons or the simulation menu. +A new window with the results will open once the simulation has finished. + +\begin{figure}[H] +\centering +\includegraphics[width=\textwidth]{figures/screenshots/output/result-window} +\caption{Simulation results} +\label{fig:result-window} +\end{figure} + +With the buttons on the bottom left you can switch between the static and dynamic (if available) results. +The results themselves are organized in different tabs. +At the bottom of the window there is a slider where you can change the current draw length (statics) or time (dynamics). +This value applies to all of the result tabs. + +\newpage +These are the contents of the individual result tabs: + +\begin{itemize} +\item \textbf{Shape:} Shows the shape of the limb and string as well as the position of the arrow at different stages of either the draw (statics) or the shot (dynamics) +\item \textbf{Stress:} Shows the distribution of material stress along the length of the limb for the back and belly of each layer +\item \textbf{Curvature:} Shows the curvature of the limb (difference to unbraced state) +\item \textbf{Energy:} Shows how potential and kinetic energy of the different parts of the bow develop during the simulation +\item \textbf{Other Plots:} Here you can combine arbitrary simulation results and plot them together, e.g. things like the draw curve of the bow or the velocity of the arrow. +\\ +\item \textbf{Special Values (Statics):} + \begin{itemize} + \item \textbf{String length:} Initial length of the string such that the bow meets the specified brace height + \item \textbf{Final draw force:} Draw force in fully drawn state + \item \textbf{Drawing work:} Total work done by drawing the bow. + \item \textbf{Storage ratio:} This is an indicator of the bow's capability to store energy and is defined (/made up by the author) as +$$\mathrm{storage\_ ratio} = \frac{\mathrm{drawing\_ work}}{1/2\cdot \mathrm{draw\_ force}\cdot (\mathrm{draw\_ length} - \mathrm{brace\_ height})}.$$ +It describes the amount of energy stored by the bow's draw curve in relation to a fictious linear draw curve with the same final draw force. +\\ + \end{itemize} +\item \textbf{Special Values (Dynamics):} + \begin{itemize} + \item \textbf{Arrow velocity:} Velocity of the arrow when leaving the bow + \item \textbf{Arrow energy:} Kinetic energy of the arrow when leaving the bow + \item \textbf{Efficiency:} Degree of efficiency of the bow. Useful energy output (kinetic energy of the arrow) divided by energy input (static drawing work). + \end{itemize} +\end{itemize} + +\newpage +\section{Command Line Interface} + +The command line interface can be used to start simulations in batch mode, without opening the GUI. +This way \swtitle can be called from other programs for performing more advanced computations like parameter studies and optimizations. + +The command line parameters are as follows: + +\smallskip + +\begin{framed} +\textbf{virtualbow [input] [output] [options]} +\end{framed} + +\begin{itemize} +\item \textbf{input:} Path to an input file (\texttt{.bow}) +\item \textbf{output:} Path for the output file (\texttt{.dat}) +\item \textbf{options:} Simulation options, \texttt{--static} or \texttt{--dynamic} +\end{itemize} + +\smallskip + +All of the arguments are optional. +Calling \swtitle with either no arguments or only an input file will open the GUI. +If in addition to that either an output file or simulation options are provided, the simulation is carried out silently and the results are written to disk. +If not specified, a default output file named after the input file is created. + +\textbf{Note:} To use the command line interface on Windows you have to either specify the complete path to the \texttt{virtualbow.exe} executable or add the installation directory to your \texttt{PATH} environment variable. + +\subsection*{Input and Output Files} + +\swtitle's input files use the JSON\footnote{\url{http://json.org/}} format, a human readable text format that stores different types of data in a hierarchical way. +The output files use MessagePack\footnote{\url{http://msgpack.org/}}, a more compact binary format that is otherwise very similar to JSON. +Both are very common formats with implementations available in many programming languages. +An example for using \swtitle with Python can be found in Appendix~\ref{sec:scripting-examples}. + +The exact definition and layout of the data contained in the input and output files is documented in Appendix~\ref{sec:input-structure} and~\ref{sec:output-structure}, respectively. However please note that those aren't stable yet. Future releases are very likely to introduce breaking changes. + +\newpage +\section{Background Information} + +This section is intended to give interested users an overview of the mathematical bow model behind \swtitle, i.e. how the different components of the bow are modeled and what the assumptions and limitations are. +This section will eventually be replaced by a separate technical documentation of the simulation model.\\ + +\textbf{Limb:} The limb is regarded as an Euler-Bernoulli beam. +This means that all cross-sections of the beam are assumed to stay flat and perpendicular to the beam axis during deformation. +The Euler-Bernoulli beam theory therefore only accounts for bending deformation and neglects shear deformation, which is usually a valid thing to do for long, slender beams.\\ + +The material of the limb is considered linear-elastic, so the relation between material stress $\sigma$ and strain $\epsilon$ at any point in the limb is given by the linear equation $\sigma = E\cdot \epsilon$ with the elastic modulus $E$ as a material constant. +The overall behaviour of the limb however is nonlinear due to the nonlinear kinematics/geometry of large deformations.\\ + +\textbf{String:} Contrary to the limb, the string only transfers longitudinal forces and has no flexural rigidity. +The material is considered linear-elastic as well. +The string has a constant cross section and is internally implemented as a chain of point masses connected by springs. +Additional point masses at the center and the tips represent things like servings and nocking point.\\ + +\textbf{Arrow:} The arrow is modeled as a point mass. +Deformation and vibration of the arrow (known as \textit{archers paradox}) is neglected/not captured by this model. +That's because the scope of this program is only to evaluate overall bow performance, things like final arrow velocity, degree of efficiency, etc. +For this purpose a point mass is sufficient.\\ + +\textbf{Symmetry:} The bow is assumed to be symmetric. +This is often only an approximation as most bows besides crossbow prods are actually slightly asymmetric. +The assumption of symmetry simplifies the definition of the parameters by the user (no need to define the limb twice). +It also allows the program to simulate only one half of the bow, which reduces the computing time. +(As a user you don't have to take this into account, all input and output data of the program corresponds to the complete bow.) + +\appendix + +\newpage +\section{Input File Structure} +\label{sec:input-structure} + +\newcommand{\tablespace}{\rule{0pt}{3ex}} + +\begin{table}[H] +\resizebox{35em}{!}{\begin{tabular}{ l | l | l | l } +\textbf{Field} & \textbf{Type} & \textbf{Units} & \textbf{Description} \\ +\hline +\tablespace meta & & & \\ +\quad version & \texttt{string} & -- & Internally used version string \\ +\quad comments & \texttt{string} & -- & User comments \\ +\tablespace settings & & & \\ +\quad n\_limb\_elements & \texttt{integer} & -- & Number of limb elements \\ +\quad n\_string\_elements & \texttt{integer} & -- & Number of string elements \\ +\quad n\_draw\_steps & \texttt{integer} & -- & Number of steps for the static simulation \\ +\tablespace \quad time\_span\_factor & \texttt{double} & -- & Factor for modifying total simulation time \\ +\quad time\_step\_factor & \texttt{double} & -- & Factor for modifying simulation time steps \\ +\quad sampling\_rate & \texttt{double} & \unit[]{Hz} & Time resolution for the dynamic output \\ +\tablespace profile & \texttt{double[][]} & \unit[]{m},\ \unit[]{1/m} & Table with segment lengths and curvatures \\ +\tablespace width & \texttt{double[][]} & \unit[]{--},\ \unit[]{m} & Table with positions and widths \\ +\tablespace layers & & & \\ +\quad \texttt{\{} & & & \\ +\quad\quad name & \texttt{string} & \unit[]{--} & Name of the layer \\ +\quad\quad height & \texttt{double[][]} & \unit[]{--},\ \unit[]{m} & Table with positions and heights \\ +\quad\quad rho & \texttt{double} & \unit[]{kg/m$^3$} & Density of the layer material \\ +\quad\quad E & \texttt{double} & \unit[]{Pa} & Elastic modulus of the layer material \\ +\quad \texttt{\}} & & & \\ +\quad\texttt{\{\ \ldots\ \}} & & & \\ +\tablespace string & & & \\ +\quad strand\_stiffness & \texttt{double} & \unit[]{N} & Stiffness of the string material\\ +\quad strand\_density & \texttt{double} & \unit[]{kg/m} & Density of the string material \\ +\quad n\_strands & \texttt{integer} & -- & Number of strands \\ +\tablespace masses & & & \\ +\quad arrow & \texttt{double} & \unit[]{kg} & Mass of the arrow \\ +\quad string\_center & \texttt{double} & \unit[]{kg} & Additional mass at string center \\ +\quad string\_tip & \texttt{double} & \unit[]{kg} & Additional mass at string tips \\ +\quad limb\_tip & \texttt{double} & \unit[]{kg} & Additional mass at limb tips \\ +\tablespace dimensions & & & \\ +\quad brace\_height & \texttt{double} & \unit[]{m} & Brace height \\ +\quad draw\_length & \texttt{double} & \unit[]{m} & Draw length \\ +\quad handle\_length & \texttt{double} & \unit[]{m} & Handle length \\ +\quad handle\_setback & \texttt{double} & \unit[]{m} & Handle setback \\ +\quad handle\_angle & \texttt{double} & \unit[]{m} & Handle angle +\end{tabular}} +\end{table} + +\newpage +\section{Output File Structure} +\label{sec:output-structure} + +\footnotesize{ +\texttt{P}: Number of limb nodes\\ +\texttt{Q}: Number of string nodes\\ +\texttt{R}: Number of layer nodes +} + +\begin{table}[H] +\resizebox{35em}{!}{\begin{tabular}{ l | l | l | l } +\textbf{Field} & \textbf{Type} & \textbf{Unit} & \textbf{Description} \\ +\hline +\tablespace limb\_properties & & & \\ +\quad length & \texttt{double[P]} & \unit[]{m} & Arc lengths of the limb nodes (unbraced) \\ +\tablespace\quad angle & \texttt{double[P]} & \unit[]{rad} & Orientation angles of the limb nodes (unbraced) \\ +\quad x\_pos & \texttt{double[P]} & \unit[]{m} & X coordinates of the limb nodes (unbraced) \\ +\quad y\_pos & \texttt{double[P]} & \unit[]{m} & Y coordinates of the limb nodes (unbraced) \\ +\tablespace\quad width & \texttt{double[P]} & \unit[]{m} & Cross section width \\ +\quad height & \texttt{double[P]} & \unit[]{m} & Cross section height (total) \\ +\tablespace\quad rhoA & \texttt{double[P]} & \unit[]{kg/m} & Linear density of the cross sections \\ +\quad Cee & \texttt{double[P]} & \unit[]{N} & Longitudinal stiffness of the cross sections \\ +\quad Ckk & \texttt{double[P]} & \unit[]{Nm$^2$} & Bending stiffness of the cross sections \\ +\quad Cek & \texttt{double[P]} & \unit[]{Nm} & Coupling between bending and elongation \\ +\tablespace\quad layers & & & \\ +\quad\quad \texttt{\{} & & & \\ +\quad\quad\quad length & \texttt{double} & \unit[]{m} & Arc lengths of the layer nodes \\ +\quad\quad\quad He\_back & \texttt{double[R][P]} & \unit[]{N/m$^2$} & Stress evaluation matrix\textsuperscript{1} (back) \\ +\quad\quad\quad Hk\_back & \texttt{double[R][P]} & \unit[]{N/m} & Stress evaluation matrix\textsuperscript{1} (back) \\ +\quad\quad\quad He\_belly & \texttt{double[R][P]} & \unit[]{N/m$^2$} & Stress evaluation matrix\textsuperscript{1} (belly) \\ +\quad\quad\quad Hk\_belly & \texttt{double[R][P]} & \unit[]{N/m} & Stress evaluation matrix\textsuperscript{1} (belly) \\ +\quad\quad \texttt{\}} & & & \\ +\quad\quad \texttt{\{\ \ldots\ \}} & & & \\ +\tablespace statics & & & \\ +\quad states & & & \\ +\quad\quad \texttt{\{\ \ldots\ \}} & & & Sequence of bow states (see table below) \\ +\tablespace\quad string\_length & \texttt{double} & \unit[]{m} & Initial length of the string \\ +\quad final\_draw\_force & \texttt{double} & \unit[]{N} & Final draw force \\ +\quad drawing\_work & \texttt{double} & \unit[]{J} & Drawing work \\ +\quad storage\_ratio & \texttt{double} & -- & Storage ratio \\ +\tablespace dynamics & & & \\ +\quad states & & & \\ +\quad\quad \texttt{\{\ \ldots\ \}} & & & Sequence of bow states (see table below) \\ +\tablespace\quad final\_arrow\_velocity & \texttt{double} & \unit[]{m/s} & Final velocity of the arrow \\ +\quad final\_arrow\_energy & \texttt{double} & \unit[]{J} & Final energy of the arrow \\ +\quad efficiency & \texttt{double} & -- & Degree of efficiency \smallskip\bigskip +\end{tabular}} +\end{table} + +\newpage +\footnotesize{ +\texttt{N}: Number of simulation steps\\ +\texttt{P}: Number of limb nodes\\ +\texttt{Q}: Number of string nodes +} +\begin{table}[H] +\resizebox{28em}{!}{\begin{tabular}{ l | l | l | l } +\textbf{Field} & \textbf{Type} & \textbf{Unit} & \textbf{Description} \\ +\hline +\tablespace states & & & \\ +\quad time & \texttt{double[N]} & \unit[]{s} & Time \\ +\quad draw\_length & \texttt{double[N]} & \unit[]{m} & Draw length \\ +\quad draw\_force & \texttt{double[N]} & \unit[]{N} & Draw force \\ +\quad string\_force & \texttt{double[N]} & \unit[]{N} & String force (total) \\ +\quad strand\_force & \texttt{double[N]} & \unit[]{N} & String force (strand) \\ +\quad grip\_force & \texttt{double[N]} & \unit[]{N} & Grip force \\ +\tablespace \quad pos\_arrow & \texttt{double[N]} & \unit[]{m} & Arrow position \\ +\quad vel\_arrow & \texttt{double[N]} & \unit[]{m/s} & Arrow velocity \\ +\quad acc\_arrow & \texttt{double[N]} & \unit[]{m/s$^2$} & Arrow acceleration \\ +\tablespace \quad x\_pos\_limb & \texttt{double[N][P]} & \unit[]{m} & X coordinates of the limb nodes \\ +\quad y\_pos\_limb & \texttt{double[N][P]} & \unit[]{m} & Y coordinates of the limb nodes \\ +\quad angle\_limb & \texttt{double[N][P]} & \unit[]{rad} & Rotation angles of the limb nodes \\ +\tablespace \quad epsilon & \texttt{double[N][P]} & \unit[]{--} & Longitudinal strain at the limb nodes \\ +\quad kappa & \texttt{double[N][P]} & \unit[]{m} & Bending curvature at the limb nodes \\ +\tablespace\quad x\_pos\_string & \texttt{double[N][Q]} & \unit[]{m} & X coordinates of the string nodes \\ +\quad y\_pos\_string & \texttt{double[N][Q]} & \unit[]{m} & Y coordinates of the string nodes \\ +\tablespace \quad e\_pot\_limbs & \texttt{double[N]} & \unit[]{J} & Potential energy of the limbs \\ +\quad e\_kin\_limbs & \texttt{double[N]} & \unit[]{J} & Kinetic energy of the limbs \\ +\quad e\_pot\_string & \texttt{double[N]} & \unit[]{J} & Potential energy of the string \\ +\quad e\_kin\_string & \texttt{double[N]} & \unit[]{J} & Kinetic energy of the string \\ +\quad e\_kin\_arrow & \texttt{double[N]} & \unit[]{J} & Kinetic energy of the arrow +\end{tabular}} +\end{table} + +\textsuperscript{1}\textbf{Note:} For space efficiency reasons, the stresses for each layer aren't stored directly in the output data. Instead they can be calculated as needed by multiplying the layer's stress evaluation matrices with the strain and curvature of the limb at the given bow state, +% +\begin{align*} +&\mathrm{\texttt{sigma\_back}} = \mathrm{\texttt{He\_back}} \cdot \mathrm{\texttt{epsilon}} + \mathrm{\texttt{Hk\_back}} \cdot \mathrm{\texttt{kappa}}\\ +&\mathrm{\texttt{sigma\_belly}} = \mathrm{\texttt{He\_belly}} \cdot \mathrm{\texttt{epsilon}} + \mathrm{\texttt{Hk\_belly}} \cdot \mathrm{\texttt{kappa}} +\end{align*} +% +The result is a vector of stresses corresponding to the nodes of the layer. + +%\textbf{Note:} Matrices are stored in row major order. This is also what Pythons \texttt{numpy} package uses. +%MATLAB for example uses column major order, which means that matrices loaded from \swtitle output will probably have to be transposed first. + +\newpage +\section{Python Scripting Example} +\label{sec:scripting-examples} + +The code example below shows how simulations can be automated with the Python programming language. The script loads, modifies and saves an input file, runs a static simulation with it, loads the output results and calculates the maximum stress of the first layer. + +Two external packages are needed: \texttt{msgpack} for reading the output files and \texttt{numpy} for evaluating stresses (matrix multiplication). They can be installed via + +\smallskip + +\begin{framed} +\texttt{pip install msgpack}\\ +\texttt{pip install numpy} +\end{framed} + +Reading and writing the input file is possible out of the box by using Python's \texttt{json} standard library module. \swtitle is called like any other command line tool either with \texttt{os.system} or \texttt{subprocess.call}. + +\definecolor{myblue}{rgb}{0,0,0.5} +\definecolor{myred}{rgb}{0.4,0.4,0.4} +\definecolor{mygreen}{rgb}{0,0.5,0} + +\lstdefinestyle{mystyle}{ + showstringspaces=false, + columns=flexible, + keywordstyle=\bfseries\color{myblue}, + commentstyle=\itshape\color{myred}, + stringstyle=\color{mygreen}, +} + +\lstdefinelanguage{mypython}{ + morekeywords={import, open, with, as}, + morecomment=[l]\#, + morestring=[s]{"}{"}, +}[keywords,comments,strings] + +\bigskip + +\begin{framed} +\begin{lstlisting}[language=mypython, style=mystyle] +import json # Loading and saving input files +import msgpack # Loading output files +import numpy # Evaluating stresses +import os # Runnig the simulation command + +# Load input file +with open("input.bow", "r") as file: + input = json.load(file) + +# Modify input +input["string"]["n_strands"] += 1 + +# Save modified input +with open("input.bow", "w") as file: + json.dump(input, file) + +# Run a static simulation +os.system("virtualbow input.bow output.dat --static") + +# Load the output file +with open("output.dat", "rb") as file: + output = msgpack.unpackb(file.read()) + +# Calculate the maximum stress at the back of layer 0 +He_back = numpy.array(output[b"limb_properties"][b"layers"][0][b"He_back"]) +Hk_back = numpy.array(output[b"limb_properties"][b"layers"][0][b"Hk_back"]) + +epsilon = numpy.array(output[b"statics"][b"states"][b"epsilon"][-1]) +kappa = numpy.array(output[b"statics"][b"states"][b"kappa"][-1]) + +sigma_back = He_back.dot(epsilon) + Hk_back.dot(kappa) +print(sigma_back.max()) +\end{lstlisting} +\end{framed} + +\newpage +\section{A Simple Bending Test} +\label{sec:bending-test} + +A bending test is an easy way to determine the elastic modulus of a material. It can be done without any special equipment. +Figure~\ref{fig:bending-test} shows the setup. A test piece with length~$l$ is clamped on one side and subjected to a vertical force~$F$ at its free end. +The deflection~$s$ due to this load is measured. + +\begin{figure}[H] +\centering +\includegraphics[width=0.5\textwidth]{figures/bending-test/setup.pdf} +\caption{Experimental setup} +\label{fig:bending-test} +\end{figure} + +The elastic modulus can then be calculated depending on the cross section geometry using the equations in Table~\ref{tbl:bending-test}. + +\begin{table}[H] +\centering +\begin{tabular}{| c | l |} +\hline +\textbf{Test geometry} & \textbf{Elastic Modulus} \\ \hline +\raisebox{-.5\height}{\includegraphics[width=0.4\textwidth]{figures/bending-test/sections-uniform.pdf}} & $\displaystyle E = \frac{4}{wh^3}\,\frac{Fl^3}{s}$ \\ \hline +\raisebox{-.5\height}{\includegraphics[width=0.4\textwidth]{figures/bending-test/sections-tapered.pdf}} & $\displaystyle E = \frac{12\,\ln(h_{l}\,l)+6}{w\,(h_{l}-h_{0})^3}\,\frac{Fl^3}{s}$ \\ \hline +\end{tabular} +\caption{Elastic modulus for different test geometries} +\label{tbl:bending-test} +\end{table} + +Here are a few practical considerations: + +\begin{itemize} +\item The precision of the cross sections is very important, especially the height. +\item The equations above hold for slender beams and small deflections. 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input start ] */ +WL: ratsimp(ev(W, x=L)); +/* [wxMaxima: input end ] */ + +/* [wxMaxima: input start ] */ +e: ratsimp(solve([WL = s], [E])); +/* [wxMaxima: input end ] */ + +/* [wxMaxima: input start ] */ +logcontract(%); +/* [wxMaxima: input end ] */ + +/* [wxMaxima: input start ] */ +factor(%); +/* [wxMaxima: input end ] */ + +/* Maxima can't load/batch files which end with a comment! */ +"Created with wxMaxima"$

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