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질량 스프링 댐퍼 TikZ 애니메이션

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질량 스프링 댐퍼 TikZ 애니메이션

다음 ODE로 설명되는 질량-스프링-댐퍼 시스템의 TikZ를 만들었습니다.

dx1/dt = x2
dx2/dt = -w0^2*x1 -2*zeta*w0*x2 + u
y = K*w0*x1 + 2*zeta*w0*x2

여기서 u는 입력이고 y는 출력입니다. w0=K=1및 zeta=0.1초기 조건을 가정해 보겠습니다 x1(0)=x2(0)=0.

TikZ는

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows, shapes, patterns, calc, decorations.pathmorphing, decorations.markings, positioning, animations}

\begin{document}
    \begin{tikzpicture}
        \tikzstyle{wheel} = [draw, circle, minimum size = 0.75cm]
        \tikzstyle{mass} = [draw, rectangle, minimum height = 1cm, minimum width = 2cm]
        \tikzstyle{spring} = [decorate, decoration = {zigzag, pre length = 0.3cm, post length = 0.3cm, segment length = 6}]
        \tikzstyle{damper} = [decoration = {markings, mark connection node = dmp, mark = at position 0.5 with 
        {
            \node (dmp) [inner sep = 0pt, transform shape, rotate = -90, minimum width = 5pt, minimum height = 10pt, draw=none] {};
            \draw ($(dmp.north east)+(1.5pt,0)$) -- (dmp.south east) -- (dmp.south west) -- ($(dmp.north west)+(1.5pt,0)$);
            \draw ($(dmp.north)+(0,-1.5pt)$) -- ($(dmp.north)+(0,1.5pt)$);
        }
        }, decorate]
        \tikzstyle{groundflat} = [fill, pattern = north east lines, draw = none, minimum width = 0.75cm, minimum height = 0.3cm]
        \draw[fill, pattern = north east lines] (-.5,-.25) cos (0,0) sin (.5,.25) coordinate (top) cos (1,0) sin (1.5,-.25) cos (2,0) sin (2.5,.25) cos (3,0) sin (3.5,-.25) |- (-.5,-1)--cycle;
        \node[wheel, above = 0pt of top] (u) {$u$};
        \node[mass, above = 2cm of u] (m) {$y$};
        \draw [-] (u.north) |- ++(0.5,0.25cm)coordinate (uright);
        \draw [-] (u.north) |- ++(-0.5,0.25cm)coordinate (uleft);
        \draw [spring] (uleft) -- ($(m.south) +(-0.5,0)$);
        \draw [damper] (uright) -- ($(m.south) +(0.5,0)$);
    \end{tikzpicture}
\end{document}

이제 특정 시간에 따라 이 st를 어떻게 u=sin(a*t)다르게 애니메이션화할 수 있습니까 ? 물론 ODE의 해를 따라야 합니다. 그리고 바퀴는 지면 정현파를 따라 "타야" 합니다.atyu

답변1

업데이트:OP 및 의견에서 요청한 대로 도로에서 바퀴를 "타고" 이동하는 관찰자입니다.

(이미지를 클릭하면 대화형 SVG가 실행됩니다. [4.2MiB] 브라우저에서.)

\pstODEsolve지배적인 ODE 시스템은 from 패키지 로 해결되었습니다.pst-ode.

바퀴가 도로 프로필을 "타는" 경우유(엑스) 일정한 속도에서 세 번째 미분 방정식 d엑스/디티, 해결해야합니다. 의 움직임을 기술하고 있다.엑스바퀴와 도로의 접촉점 좌표. 미분 방정식 시스템은 이제

디엑스/디티= Ω 바퀴 아르 자형휠 / sqrt(1+(유'(엑스)) 2 )

디와이질량 /d티=V대량의

디V질량 /d티= −Ω 0 2 (와이질량 -와이오프셋 -와이휠 ) − 2ζΩ 0 (V질량 -V바퀴 )

통합 매개변수티는 시간을 나타내고, Ω 휠은 휠의 각속도를 나타내며,아르 자형그 반경을 휠 . (그러므로 Ω 휠 아르 자형바퀴 티는 곡선 도로 프로파일을 따라 이동한 거리입니다.) Ω 0 은 감쇠되지 않은 고유 주파수이고 ζ는 감쇠 비율입니다. d를 기준으로엑스/디티,엑스(티), 휠 반경아르 자형바퀴 , 지방도 표고유(엑스), 경사유'(엑스) 및 2차 미분유''(엑스) 휠 축 좌표엑스바퀴 ,와이바퀴 와 그 수직. 속도V바퀴 가 계산됩니다.와이바퀴 와V휠은 이제 질량-스프링-댐퍼 시스템에 대한 입력 역할을 합니다.

도로 프로필유(엑스)는 파형 버스트로 모델링됩니다.

유(엑스) = 왜냐하면(엑스-엑스오프셋 )이자형-λ(엑스-엑스오프셋 ) 2

바퀴가 "구멍"에 부딪히는 그다지 조화롭지 않은 궤적과 그에 따른 응답 곡선을 확인하세요.

애니메이션 플롯의 이동 창은 환경의 xmin및 xmax옵션을 동적으로 설정하여 구현됩니다 axis.

ODE 솔버(RKF45메소드)는 pst-odePostScript로 코딩되었습니다. 최근 Marcel Krüger는 다음을 구현했습니다.포스트스크립트 인터프리터Lua에서는 ps2pdf더 이상 필요하지 않습니다.

lualatex3번 실행하여 PDF로 조판합니다 . GIF 및 SVG 출력의 경우 소스 시작 부분에 있는 줄 중 하나의 주석 처리를 해제하고 해당 지침을 따르세요.

%\PassOptionsToPackage{export}{animate} % lualatex <file> ; convert -density 300 -alpha remove -delay 4 <file>.pdf <file>.gif
%\PassOptionsToPackage{dvisvgm,autoplay}{animate} % dvilualatex <file> ; dvisvgm --zoom=-1 --font-format=woff2 --bbox=papersize <file>.dvi
\documentclass[margin=3pt]{standalone}

\usepackage{pst-ode}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{arrows.meta, calc, decorations}
\usepackage{listofitems} % read space separated items
\usepackage{animate}
\usepackage{xsavebox} %\xsbox

% adjustable parameters & definitions
\def\tEnd{80} % max. time (integration parameter)
\def\animationframes{201} % number of output points for animated objects
\def\outputpoints{1001} % number of output points for curves
\def\omegaWheel{1.0} % wheel angular velocity
\def\rWheel{1.0} % wheel radius
\def\rCenterOfMass{0.16em} % center-of-mass symbol size
\def\springlength{6} % length of spring in neutral state (and of damper cylinder)
\def\springturns{10} % (integer!) number of spring windings
\def\windowsizeright{5} % moving window x-size around moving object
\def\windowsizeleft{25}
\def\ymin{-1} % y axis limits
\def\ymax{17}
\pstVerb{
  tx@Dict begin
  % mass-spring-damper properties
  /w0 1.0 def
  /zeta 0.1 def
  % initial conditions
  /x0     0.0 def % x(t0)=0 starting x coordinate
  /yMass0 0.0 def % y coord of mass centre (/wo vert. offset)
  /vMass0 0.0 def % y velocity of mass centre
  % road profile U(x) with wave-form burst, and its first and second derivatives U'(x), U''(x)
  /A 1 def        % amplitude
  /a 1 def        % mode
  /lambda (1/40) AlgParser cvx exec def % burst growth/decay rate
  /xOff 20.0 def  % burst offset
  /U (A*Euler^(-lambda*(x[0]-xOff)^2)*cos(a*(x[0]-xOff))) AlgParser cvx bind def
  /dUdx (-A*Euler^(-lambda*(x[0]-xOff)^2)*a*sin(a*(x[0]-xOff))-2*lambda*(x[0]-xOff)*U) AlgParser cvx bind def
  /d2Udx2 (-(a^2+2*lambda+4*lambda^2*(x[0]-xOff)^2)*U-4*lambda*(x[0]-xOff)*dUdx) AlgParser cvx bind def
  % do not change anything below
  /rWheel \rWheel\space def
  /omegaWheel \omegaWheel\space def
  /massWheelOff (rWheel+1.5*\springlength+2)  AlgParser cvx exec def
  /dxdt (rWheel*omegaWheel/sqrt(1+dUdx^2)) AlgParser cvx bind def
  % local wheel hub coordinates
  /xWheel (x[0]-rWheel*dUdx/sqrt(1+dUdx^2)) AlgParser cvx bind def
  /yWheel (U+rWheel/sqrt(1+dUdx^2)) AlgParser cvx bind def
  /dyWheeldx (dUdx*(1-d2Udx2/(1+dUdx^2)*(yWheel-U))) AlgParser cvx bind def
  % angular position of wheel in terms of t
  /thetaWheel {omegaWheel t mul 2 Pi mul div dup cvi sub neg 360 mul} bind def
  end
}

% solve equations of motion with many output points for smooth curves
\pstODEsolve[algebraicAll,saveData]{curves}{% PS variable and file that take result table
                           % table format to be saved in `curves' and `curves.dat':
  x[0] |                   %    x coordinate of contact point
  U |                      %    y coordinate U(x) of contact point
  xWheel | yWheel |        %    wheel hub x (same as that of mass) and y coordinates
  x[1]+massWheelOff+rWheel %    y coordinate of mass (response with const. vertical offset)
}{0}{\tEnd}{\outputpoints}{% t_0, t_end, number of t steps plus 1
  x0 | yMass0 | vMass0     % initial conditions
}{                         % ODE system's RHS:
  dxdt |                   %   dx/dt, movement of contact point (x coordinate)
  x[2] |                   %   centre of mass vert. velocity
  -w0^2*(x[1]-yWheel+rWheel)%  centre of mass vert. acceleration
    -2*zeta*w0*(x[2]-dyWheeldx*dxdt)
}

% solve equations with fewer output points for mass-spring-damper animation; save solution
% table (wheel and mass centre coordinates, valve angular position) in `objects' and `objects.dat'
\pstODEsolve[algebraicAll,saveData]{objects}{
  xWheel | yWheel | x[1]+massWheelOff+rWheel | thetaWheel-90
}{0}{\tEnd}{\animationframes}{x0 | yMass0 | vMass0}{
  dxdt | x[2] | -w0^2*(x[1]-yWheel+rWheel)-2*zeta*w0*(x[2]-dyWheeldx*dxdt)
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \fileopenr{<file stream>}{<file name>}, opens file for reading
\newcommand\fileopenr[2]{%
  \newread#1%
  \immediate\openin#1=#2%
}
% \readtolist[<sep char>]{<file stream>}{\list}
% reads a line from file stream and splits at <sep char> into \list[1], \list[2], ...
\newcommand\readtolist[3][,]{{%
  \setsepchar{#1}%
  \immediate\read#2 to \inputline%
    \ifeof#2
      \immediate\closein#2%
      \ifdefined\multiframebreak\multiframebreak\fi%
    \else%
      \greadlist*#3\inputline%
    \fi%
}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\tikzset{spring/.style 2 args= {decorate, decoration = {zigzag, segment length = \dimexpr(#1-1pt)/\springturns\relax, amplitude=#2}}}%

\begin{document}
\IfFileExists{curves.dat}{}{dummy text\end{document}}%
\xsbox{centreofmass}{%
  \tikz[radius=\rCenterOfMass] {% from https://tex.stackexchange.com/questions/23453
    \fill (0,0) -- ++(\rCenterOfMass,0) arc [start angle=0,end angle=90] -- ++(0,{-2*\rCenterOfMass}) arc [start angle=270, end angle=180];%
    \draw [thin] (0,0) circle;%
  }%
}%
\begin{animateinline}[controls]{25}
  \fileopenr{\data}{objects.dat}
  \readtolist[ ]{\data}{\table}
  \multiframe{10000}{}{
    \begin{tikzpicture}[line cap=rect, line join=bevel]
      \begin{axis}[
        unit vector ratio = 1, axis on top,
        xmin={\table[1]-\windowsizeleft}, xmax={\table[1]+\windowsizeright},
        ymin=\ymin, ymax=\ymax,
        xtick distance=5,
        ytick distance=2,
        xlabel={$x$}, ylabel={$y$}, ylabel style={rotate=-90},
        extra x tick labels={\phantom{\strut00}},extra x ticks={\pgfkeysvalueof{/pgfplots/xmax}}, % prevents wobbling of axis
      ]
        % road profile; off-window coordinates filtered away for smaller file size
        \addplot [name path=road,no markers,unbounded coords=jump,x filter/.expression={x<\pgfkeysvalueof{/pgfplots/xmin}-0.1 || x>\pgfkeysvalueof{/pgfplots/xmax}+0.1 ? nan : x}]
          table [x index=0, y index=1] {curves.dat} -- (\pgfkeysvalueof{/pgfplots/xmax},0);
        \path [name path=bottom] (0,\pgfkeysvalueof{/pgfplots/ymin}) -- (\pgfkeysvalueof{/pgfplots/xmax},\pgfkeysvalueof{/pgfplots/ymin});
        \addplot [fill=black!20!white] fill between[of=road and bottom];
        % wheel hub trajectory
        \addplot [no markers,blue,unbounded coords=jump,x filter/.expression={x<\pgfkeysvalueof{/pgfplots/xmin}-0.1 || x>\table[1] ? nan : x}]
          table [x index=2, y index=3] {curves.dat};
        % centre of mass trajectory
        \addplot [no markers,red,unbounded coords=jump,x filter/.expression={x<\pgfkeysvalueof{/pgfplots/xmin}-0.1 || x>\table[1] ? nan : x}]
          table [x index=2, y index=4] {curves.dat};
        \coordinate (w) at (\table[1],\table[2]);  % wheel hub
        \coordinate (m) at (\table[1],\table[3]);  % centre of mass
        % wheel with valve
        \fill[even odd rule] (w) circle [radius={0.7*\rWheel}] (w) circle [radius=\rWheel];
        \draw[thick] ($(w)+(axis direction cs:{0.6*\rWheel*cos(\table[4])},{0.6*\rWheel*sin(\table[4])})$)
          -- ++(axis direction cs:{0.1*\rWheel*cos(\table[4])},{0.1*\rWheel*sin(\table[4])});
        % mass
        \node at (m) {\thecentreofmass};
        \draw (m) ++(axis direction cs:-1,-1) rectangle ++(axis direction cs:2,2);
        % spring/damper support
        \draw (m) ++(axis direction cs:-1,-1) -- ++(axis direction cs:0,{-0.25-0.25*\springlength}) coordinate (sprtop);
        \path (m) ++(axis direction cs:1,-1) coordinate (dmptop);
        \draw (w) ++(axis direction cs:0,{\rWheel+0.5}) -| ++(axis direction cs:-1,{0.25+0.25*\springlength}) coordinate (sprbot);
        \draw let \p1=(axis direction cs:0.35,0) in [{Circle[open, length=\x1]}-, shorten <=-0.5*\x1] (w) -- ++(axis direction cs:0,{\rWheel+0.5}) -| ++(axis direction cs:1,0.5) coordinate (dmpbot);
        % spring
        \path let \p1=($(sprtop)-(sprbot)$), \p2=(axis direction cs:0.6,0) % length and amplitude (half width)
              in [draw,spring={\y1}{\x2}] (sprbot) -- (sprtop);
        % damper lower part
        \draw (dmpbot) -| +(axis direction cs:0.4,\springlength) (dmpbot) -| +(axis direction cs:-0.4,\springlength);
        % damper upper part
        \draw (dmptop) -- ++(axis direction cs:0,-\springlength) ++(axis direction cs:-0.3,0) -- ++(axis direction cs:0.6,0);
      \end{axis}
    \end{tikzpicture}%
    \readtolist[ ]{\data}{\table}
  }
\end{animateinline}
\end{document}

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