Animate a pendulum
You are encouraged to solve this task according to the task description, using any language you may know.
One good way of making an animation is by simulating a physical system and illustrating the variables in that system using a dynamically changing graphical display. The classic such physical system is a simple gravity pendulum.
For this task, create a simple physical model of a pendulum and animate it.
C /C++
<lang c>#include <stdio.h>
- include <stdlib.h>
- include <stdbool.h>
- include <math.h>
- include <cairo.h>
- include <SDL.h>
- define WIDTH 320.0
- define HEIGHT 240.0
- ifndef M_PI
- define M_PI 3.14159265358979323846
- endif
const double L = HEIGHT*150.0/200.0; // length double theta = M_PI/4.0; double dtheta = 0.0; int homeX = WIDTH / 2; int homeY = HEIGHT*25.0/200.0;
static uint32_t update_anim(uint32_t iv, void *p) {
double scaling = 30.0 / (L*L); // this seems better for this impl. double firstDDtheta = -sin(theta) * scaling; double midDtheta = dtheta + firstDDtheta; double midtheta = theta + (dtheta + midDtheta)/2.0;
double midDDtheta = -sin(midtheta) * scaling; midDtheta = dtheta + (firstDDtheta + midDDtheta)/2.0; midtheta = theta + (dtheta + midDtheta)/2.0;
midDDtheta = -sin(midtheta) * scaling; double lastDtheta = midDtheta + midDDtheta; double lasttheta = midtheta + (midDtheta + lastDtheta)/2.0;
double lastDDtheta = -sin(lasttheta) * scaling; lastDtheta = midDtheta + (midDDtheta + lastDDtheta)/2.0; lasttheta = midtheta + (midDtheta + lastDtheta)/2.0;
dtheta = lastDtheta; theta = lasttheta;
return iv;
}
void show_pendulum(SDL_Surface *s) {
cairo_surface_t *surf = cairo_image_surface_create_for_data( s->pixels,
CAIRO_FORMAT_RGB24, s->w, s->h, s->pitch);
cairo_t *ct = cairo_create(surf);
double x = homeX + L*sin(theta); double y = homeY + L*cos(theta);
cairo_set_source_rgba(ct, 0, 0, 0, 1); cairo_new_path(ct); cairo_rectangle(ct, 0, 0, WIDTH, HEIGHT); cairo_fill(ct);
cairo_set_source_rgba(ct, 1, 1, 1, 1); // rod cairo_new_path(ct); cairo_move_to(ct, homeX, homeY); cairo_line_to(ct, x, y); cairo_stroke(ct);
cairo_set_source_rgba(ct, 1, 1, 0, 1); // bob cairo_new_path(ct); cairo_move_to(ct, x, y); cairo_arc(ct, x, y, 20.0, 0, 2*M_PI); cairo_fill(ct);
cairo_set_source_rgba(ct, 0.7, 0.7, 0.7, 1);
// plate cairo_new_path(ct); cairo_move_to(ct, 0, homeY); cairo_line_to(ct, WIDTH, homeY); cairo_stroke(ct);
// pivot cairo_new_path(ct); cairo_move_to(ct, homeX, homeY); cairo_arc(ct, homeX, homeY, 5.0, 0, 2*M_PI); cairo_fill(ct);
cairo_surface_destroy(surf); cairo_destroy(ct);
}
int main() {
SDL_Surface *scr, *t; SDL_Event event[1]; bool quit = false;
if ( SDL_Init(SDL_INIT_VIDEO | SDL_INIT_TIMER) >= 0 ) {
atexit(SDL_Quit);
if ( SDL_SetVideoMode(WIDTH, HEIGHT, 32, SDL_DOUBLEBUF) != NULL ) {
scr = SDL_GetVideoSurface();
SDL_AddTimer(30, update_anim, NULL);
event->type = SDL_VIDEOEXPOSE;
SDL_PushEvent(event);
while(SDL_WaitEvent(event) && !quit) {
switch(event->type) { case SDL_VIDEOEXPOSE: while(SDL_LockSurface(scr) != 0) SDL_Delay(1);
show_pendulum(scr);
SDL_UnlockSurface(scr); SDL_Flip(scr); event->type = SDL_VIDEOEXPOSE; SDL_PushEvent(event); break; case SDL_KEYDOWN: if (event->key.keysym.sym == SDLK_q) quit = true; break; }
}
SDL_FreeSurface(scr); } }
return 0;
}</lang>
C#
<lang csharp> using System; using System.Drawing; using System.Windows.Forms;
class CSharpPendulum {
Form _form;
Timer _timer;
double _angle = Math.PI / 2,
_angleAccel,
_angleVelocity = 0,
_dt = 0.1;
int _length = 50;
[STAThread]
static void Main()
{
var p = new CSharpPendulum();
}
public CSharpPendulum()
{
_form = new Form() { Text = "Pendulum", Width = 200, Height = 200 };
_timer = new Timer() { Interval = 30 };
_timer.Tick += delegate(object sender, EventArgs e)
{
int anchorX = (_form.Width / 2) - 12,
anchorY = _form.Height / 4,
ballX = anchorX + (int)(Math.Sin(_angle) * _length),
ballY = anchorY + (int)(Math.Cos(_angle) * _length);
_angleAccel = -9.81 / _length * Math.Sin(_angle);
_angleVelocity += _angleAccel * _dt;
_angle += _angleVelocity * _dt;
Bitmap dblBuffer = new Bitmap(_form.Width, _form.Height);
Graphics g = Graphics.FromImage(dblBuffer);
Graphics f = Graphics.FromHwnd(_form.Handle);
g.DrawLine(Pens.Black, new Point(anchorX, anchorY), new Point(ballX, ballY));
g.FillEllipse(Brushes.Black, anchorX - 3, anchorY - 4, 7, 7);
g.FillEllipse(Brushes.DarkGoldenrod, ballX - 7, ballY - 7, 14, 14);
f.Clear(Color.White);
f.DrawImage(dblBuffer, new Point(0, 0));
};
_timer.Start();
Application.Run(_form);
}
} </lang>
Clojure
Clojure solution using an atom and a separate rendering thread
<lang clojure> (ns pendulum
(:import (javax.swing JFrame) (java.awt Canvas Graphics Color)))
(def length 200) (def width (* 2 (+ 50 length))) (def height (* 3 (/ length 2))) (def dt 0.1) (def g 9.812) (def k (- (/ g length))) (def anchor-x (/ width 2)) (def anchor-y (/ height 8)) (def angle (atom (/ (Math/PI) 2)))
(defn draw [#^Canvas canvas angle]
(let [buffer (.getBufferStrategy canvas)
g (.getDrawGraphics buffer)
ball-x (+ anchor-x (* (Math/sin angle) length))
ball-y (+ anchor-y (* (Math/cos angle) length))]
(try
(doto g
(.setColor Color/BLACK)
(.fillRect 0 0 width height)
(.setColor Color/RED)
(.drawLine anchor-x anchor-y ball-x ball-y)
(.setColor Color/YELLOW)
(.fillOval (- anchor-x 3) (- anchor-y 4) 7 7)
(.fillOval (- ball-x 7) (- ball-y 7) 14 14))
(finally (.dispose g)))
(if-not (.contentsLost buffer)
(.show buffer)) ))
(defn start-renderer [canvas]
(->> (fn [] (draw canvas @angle) (recur)) (new Thread) (.start)))
(defn -main [& args]
(let [frame (JFrame. "Pendulum")
canvas (Canvas.)]
(doto frame
(.setSize width height)
(.setDefaultCloseOperation JFrame/EXIT_ON_CLOSE)
(.setResizable false)
(.add canvas)
(.setVisible true))
(doto canvas
(.createBufferStrategy 2)
(.setVisible true)
(.requestFocus))
(start-renderer canvas)
(loop [v 0]
(swap! angle #(+ % (* v dt)))
(Thread/sleep 15)
(recur (+ v (* k (Math/sin @angle) dt)))) ))
(-main) </lang>
E
(Uses Java Swing for GUI. The animation logic is independent, however.)
The angle of a pendulum with length and acceleration due to gravity with all its mass at the end and no friction/air resistance has an acceleration at any given moment of
This simulation uses this formula directly, updating the velocity from the acceleration and the position from the velocity; inaccuracy results from the finite timestep.
The event flow works like this: The clock object created by the simulation steps the simulation on the specified in the interval. The simulation writes its output to angle, which is a Lamport slot which can notify of updates. The whenever set up by makeDisplayComponent listens for updates and triggers redrawing as long as interest has been expressed, which is done whenever the component actually redraws, which happens only if the component's window is still on screen. When the window is closed, additionally, the simulation itself is stopped and the application allowed to exit. (This logic is more general than necessary; it is designed to be suitable for a larger application as well.)
<lang e>#!/usr/bin/env rune pragma.syntax("0.9")
def pi := (-1.0).acos() def makeEPainter := <unsafe:com.zooko.tray.makeEPainter> def makeLamportSlot := <import:org.erights.e.elib.slot.makeLamportSlot> def whenever := <import:org.erights.e.elib.slot.whenever> def colors := <import:java.awt.makeColor>
- --------------------------------------------------------------
- --- Definitions
def makePendulumSim(length_m :float64,
gravity_mps2 :float64,
initialAngle_rad :float64,
timestep_ms :int) {
var velocity := 0
def &angle := makeLamportSlot(initialAngle_rad)
def k := -gravity_mps2/length_m
def timestep_s := timestep_ms / 1000
def clock := timer.every(timestep_ms, fn _ {
def acceleration := k * angle.sin()
velocity += acceleration * timestep_s
angle += velocity * timestep_s
})
return [clock, &angle]
}
def makeDisplayComponent(&angle) {
def c
def updater := whenever([&angle], fn { c.repaint() })
bind c := makeEPainter(def paintCallback {
to paintComponent(g) {
try {
def originX := c.getWidth() // 2
def originY := c.getHeight() // 2
def pendRadius := (originX.min(originY) * 0.95).round()
def ballRadius := (originX.min(originY) * 0.04).round()
def ballX := (originX + angle.sin() * pendRadius).round()
def ballY := (originY + angle.cos() * pendRadius).round()
g.setColor(colors.getWhite())
g.fillRect(0, 0, c.getWidth(), c.getHeight())
g.setColor(colors.getBlack())
g.fillOval(originX - 2, originY - 2, 4, 4)
g.drawLine(originX, originY, ballX, ballY)
g.fillOval(ballX - ballRadius, ballY - ballRadius, ballRadius * 2, ballRadius * 2)
updater[] # provoke interest provided that we did get drawn (window not closed)
} catch p {
stderr.println(`In paint callback: $p${p.eStack()}`)
}
}
})
c.setPreferredSize(<awt:makeDimension>(300, 300))
return c
}
- --------------------------------------------------------------
- --- Application setup
def [clock, &angle] := makePendulumSim(1, 9.80665, pi*99/100, 10)
- Initialize AWT, move to AWT event thread
when (currentVat.morphInto("awt")) -> {
# Create the window
def frame := <unsafe:javax.swing.makeJFrame>("Pendulum")
frame.setContentPane(def display := makeDisplayComponent(&angle))
frame.addWindowListener(def mainWindowListener {
to windowClosing(_) {
clock.stop()
interp.continueAtTop()
}
match _ {}
})
frame.setLocation(50, 50)
frame.pack()
# Start and become visible frame.show() clock.start()
}
interp.blockAtTop()</lang>
Factor
Approximation of the pendulum for small swings : theta = theta0 * cos(omega0 * t) <lang factor>USING: accessors alarms arrays calendar colors.constants kernel locals math math.constants math.functions math.rectangles math.vectors opengl sequences system ui ui.gadgets ui.render ; IN: pendulum
CONSTANT: g 9.81 CONSTANT: l 20 CONSTANT: theta0 0.5
- current-time ( -- time ) nano-count -9 10^ * ;
- T0 ( -- T0 ) 2 pi l g / sqrt * * ;
- omega0 ( -- omega0 ) 2 pi * T0 / ;
- theta ( -- theta ) current-time omega0 * cos theta0 * ;
- relative-xy ( theta l -- xy )
[ [ sin ] [ cos ] bi ] [ [ * ] curry ] bi* bi@ 2array ;
- theta-to-xy ( origin theta l -- xy ) relative-xy v+ ;
TUPLE: pendulum-gadget < gadget alarm ;
- O ( gadget -- origin ) rect-bounds [ drop ] [ first 2 / ] bi* 0 2array ;
- window-l ( gadget -- l ) rect-bounds [ drop ] [ second ] bi* ;
- gadget-xy ( gadget -- xy ) [ O ] [ drop theta ] [ window-l ] tri theta-to-xy ;
M: pendulum-gadget draw-gadget*
COLOR: black gl-color [ O ] [ gadget-xy ] bi gl-line ;
M:: pendulum-gadget graft* ( gadget -- )
[ gadget relayout-1 ] 20 milliseconds every gadget (>>alarm) ;
M: pendulum-gadget ungraft* alarm>> cancel-alarm ;
- <pendulum-gadget> ( -- gadget )
pendulum-gadget new
{ 500 500 } >>pref-dim ;
- pendulum-main ( -- )
[ <pendulum-gadget> "pendulum" open-window ] with-ui ;
MAIN: pendulum-main </lang>
F#
A nice application of F#'s support for units of measure. <lang fsharp>open System open System.Drawing open System.Windows.Forms
// define units of measurement [<Measure>] type m; // metres [<Measure>] type s; // seconds
// a pendulum is represented as a record of physical quantities type Pendulum =
{ length : float<m>
gravity : float<m/s^2>
velocity : float<m/s>
angle : float
}
// calculate the next state of a pendulum let next pendulum deltaT : Pendulum =
let k = -pendulum.gravity / pendulum.length
let acceleration = k * Math.Sin pendulum.angle * 1.0<m>
let newVelocity = pendulum.velocity + acceleration * deltaT
let newAngle = pendulum.angle + newVelocity * deltaT / 1.0<m>
{ pendulum with velocity = newVelocity; angle = newAngle }
// paint a pendulum (using hard-coded screen coordinates) let paint pendulum (gr: System.Drawing.Graphics) =
let homeX = 160 let homeY = 50 let length = 140.0 // draw plate gr.DrawLine( new Pen(Brushes.Gray, width=2.0f), 0, homeY, 320, homeY ) // draw pivot gr.FillEllipse( Brushes.Gray, homeX-5, homeY-5, 10, 10 ) gr.DrawEllipse( new Pen(Brushes.Black), homeX-5, homeY-5, 10, 10 ) // draw the pendulum itself let x = homeX + int( length * Math.Sin pendulum.angle ) let y = homeY + int( length * Math.Cos pendulum.angle ) // draw rod gr.DrawLine( new Pen(Brushes.Black, width=3.0f), homeX, homeY, x, y ) // draw bob gr.FillEllipse( Brushes.Yellow, x-15, y-15, 30, 30 ) gr.DrawEllipse( new Pen(Brushes.Black), x-15, y-15, 30, 30 )
// defines an operator "-?" that calculates the time from t2 to t1
// where t2 is optional
let (-?) (t1: DateTime) (t2: DateTime option) : float =
match t2 with | None -> 0.0// only one timepoint given -> difference is 0 | Some t -> (t1 - t).TotalSeconds * 1.0
// our main window is double-buffered form that reacts to paint events type PendulumForm() as self =
inherit Form(Width=325, Height=240, Text="Pendulum")
let mutable pendulum = { length = 1.0<m>;
gravity = 9.81<m/s^2>
velocity = 0.0<m/s>
angle = Math.PI / 2.0
}
let mutable lastPaintedAt = None
let updateFreq = 0.01
do self.DoubleBuffered <- true
self.Paint.Add( fun args ->
let now = DateTime.Now
let deltaT = now -? lastPaintedAt |> min 0.01
lastPaintedAt <- Some now
pendulum <- next pendulum deltaT
let gr = args.Graphics
gr.Clear( Color.LightGray )
paint pendulum gr
// initiate a new paint event after a while (non-blocking)
async { do! Async.Sleep( int( 1000.0 * updateFreq / 1.0 ) )
self.Invalidate()
}
|> Async.Start
)
[<STAThread>] Application.Run( new PendulumForm( Visible=true ) )</lang>
Haskell
Using
from HackageDB
<lang haskell>import Graphics.HGL.Draw.Monad (Graphic, ) import Graphics.HGL.Draw.Picture import Graphics.HGL.Utils import Graphics.HGL.Window import Graphics.HGL.Run
import Control.Exception (bracket, ) import Control.Arrow
toInt = fromIntegral.round
pendulum = runGraphics $
bracket (openWindowEx "Pendulum animation task" Nothing (600,400) DoubleBuffered (Just 30)) closeWindow (\w -> mapM_ ((\ g -> setGraphic w g >> getWindowTick w).
(\ (x, y) -> overGraphic (line (300, 0) (x, y)) (ellipse (x - 12, y + 12) (x + 12, y - 12)) )) pts)
where dt = 1/30 t = - pi/4 l = 1 g = 9.812 nextAVT (a,v,t) = (a', v', t + v' * dt) where
a' = - (g / l) * sin t v' = v + a' * dt
pts = map (\(_,t,_) -> (toInt.(300+).(300*).cos &&& toInt. (300*).sin) (pi/2+0.6*t) )
$ iterate nextAVT (- (g / l) * sin t, t, 0)</lang> Use (interpreter ghci):
*Main> pendulum
HicEst
DIFFEQ and the callback procedure pendulum numerically integrate the pendulum equation. The display window can be resized during the run, but for window width not equal to 2*height the pendulum rod becomes a rubber band instead: <lang HicEst>REAL :: msec=10, Lrod=1, dBob=0.03, g=9.81, Theta(2), dTheta(2) BobMargins = ALIAS(ls, rs, ts, bs) ! box margins to draw the bob
Theta = (1, 0) ! initial angle and velocity
start_t = TIME()
DO i = 1, 1E100 ! "forever"
end_t = TIME() ! to integrate in real-time sections: DIFFEQ(Callback="pendulum", T=end_t, Y=Theta, DY=dTheta, T0=start_t) xBob = (SIN(Theta(1)) + 1) / 2 yBob = COS(Theta(1)) - dBob
! create or clear window and draw pendulum bob at (xBob, yBob): WINDOW(WIN=wh, LeftSpace=0, RightSpace=0, TopSpace=0, BottomSpace=0, Up=999) BobMargins = (xBob-dBob, 1-xBob-dBob, yBob-dBob, 1-yBob-dBob) WINDOW(WIN=wh, LeftSpace=ls, RightSpace=rs, TopSpace=ts, BottomSpace=bs) WRITE(WIN=wh, DeCoRation='EL=4, BC=4') ! flooded red ellipse as bob
! draw the rod hanging from the center of the window: WINDOW(WIN=wh, LeftSpace=0.5, TopSpace=0, RightSpace=rs+dBob) WRITE(WIN=wh, DeCoRation='LI=0 0; 1 1, FC=4.02') ! red pendulum rod
SYSTEM(WAIT=msec) start_t = end_t
ENDDO
END
SUBROUTINE pendulum ! Theta" = - (g/Lrod) * SIN(Theta)
dTheta(1) = Theta(2) ! Theta' = Theta(2) substitution dTheta(2) = -g/Lrod*SIN(Theta(1)) ! Theta" = Theta(2)' = -g/Lrod*SIN(Theta(1))
END</lang>
J
<lang j>require 'gl2 trig' coinsert 'jgl2'
DT =: %30 NB. seconds ANGLE=: 0.25p1 NB. radians L =: 1 NB. metres G =: 9.80665 NB. ms_2 VEL =: 0 NB. ms_1
PEND=: noun define pc pend;pn "Pendulum"; xywh 0 0 320 200;cc isi isigraph rightmove bottommove; pas 0 0;pcenter; rem form end; )
pend_run =: verb def ' wd PEND,;pshow;timer ,":DT * 1000 ' pend_close =: verb def ' wd timer 0; pclose ' pend_isi_paint=: verb def ' drawPendulum ANGLE '
sys_timer_z_=: verb define
recalcAngle wd 'psel pend; setinvalid isi'
)
recalcAngle=: verb define
accel=. - (G % L) * sin ANGLE VEL =: VEL + accel * DT ANGLE=: ANGLE + VEL * DT
)
drawPendulum=: verb define
width=. {. glqwh
ps=. (-: width) , 40
pe=. ps + 280 <.@* (cos , sin) 0.5p1 + y NB. adjust orientation
glrgb 91 91 91
glbrush
gllines ps , pe
glellipse (,~ ps - -:) 40 15
glellipse (,~ pe - -:) 20 20
glrect 0 0 ,width, 40
)
pend_run NB. run animation</lang>
Java
<lang java>import java.awt.*; import javax.swing.*;
public class Pendulum extends JPanel implements Runnable {
private double angle = Math.PI / 2; private int length;
public Pendulum(int length) {
this.length = length;
setDoubleBuffered(true);
}
@Override
public void paint(Graphics g) {
g.setColor(Color.WHITE);
g.fillRect(0, 0, getWidth(), getHeight());
g.setColor(Color.BLACK);
int anchorX = getWidth() / 2, anchorY = getHeight() / 4;
int ballX = anchorX + (int) (Math.sin(angle) * length);
int ballY = anchorY + (int) (Math.cos(angle) * length);
g.drawLine(anchorX, anchorY, ballX, ballY);
g.fillOval(anchorX - 3, anchorY - 4, 7, 7);
g.fillOval(ballX - 7, ballY - 7, 14, 14);
}
public void run() {
double angleAccel, angleVelocity = 0, dt = 0.1;
while (true) {
angleAccel = -9.81 / length * Math.sin(angle);
angleVelocity += angleAccel * dt;
angle += angleVelocity * dt;
repaint();
try { Thread.sleep(15); } catch (InterruptedException ex) {}
}
}
@Override
public Dimension getPreferredSize() {
return new Dimension(2 * length + 50, length / 2 * 3);
}
public static void main(String[] args) {
JFrame f = new JFrame("Pendulum");
Pendulum p = new Pendulum(200);
f.add(p);
f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
f.pack();
f.setVisible(true);
new Thread(p).start();
}
}</lang>
JavaScript + <canvas>
(plus gratuitous motion blur)
<lang javascript><html><head>
<title>Pendulum</title>
</head><body style="background: gray;">
<canvas id="canvas" width="600" height="600">
Sorry, your browser does not support the <canvas> used to display the pendulum animation.
</canvas> <script>
function PendulumSim(length_m, gravity_mps2, initialAngle_rad, timestep_ms, callback) {
var velocity = 0;
var angle = initialAngle_rad;
var k = -gravity_mps2/length_m;
var timestep_s = timestep_ms / 1000;
return setInterval(function () {
var acceleration = k * Math.sin(angle);
velocity += acceleration * timestep_s;
angle += velocity * timestep_s;
callback(angle);
}, timestep_ms);
}
var canvas = document.getElementById('canvas');
var context = canvas.getContext('2d');
var prev=0;
var sim = PendulumSim(1, 9.80665, Math.PI*99/100, 10, function (angle) {
var rPend = Math.min(canvas.width, canvas.height) * 0.47;
var rBall = Math.min(canvas.width, canvas.height) * 0.02;
var rBar = Math.min(canvas.width, canvas.height) * 0.005;
var ballX = Math.sin(angle) * rPend;
var ballY = Math.cos(angle) * rPend;
context.fillStyle = "rgba(255,255,255,0.51)"; context.globalCompositeOperation = "destination-out"; context.fillRect(0, 0, canvas.width, canvas.height); context.fillStyle = "yellow"; context.strokeStyle = "rgba(0,0,0,"+Math.max(0,1-Math.abs(prev-angle)*10)+")"; context.globalCompositeOperation = "source-over";
context.save();
context.translate(canvas.width/2, canvas.height/2);
context.rotate(angle);
context.beginPath();
context.rect(-rBar, -rBar, rBar*2, rPend+rBar*2);
context.fill();
context.stroke();
context.beginPath();
context.arc(0, rPend, rBall, 0, Math.PI*2, false);
context.fill();
context.stroke();
context.restore();
prev=angle;
});
</script>
</body></html></lang>
Logo
<lang logo>make "angle 45 make "L 1 make "bob 10
to draw.pendulum
clearscreen seth :angle+180 ; down on screen is 180 forward :L*100-:bob penup forward :bob pendown arc 360 :bob
end
make "G 9.80665 make "dt 1/30 make "acc 0 make "vel 0
to step.pendulum
make "acc -:G / :L * sin :angle make "vel :vel + :acc * :dt make "angle :angle + :vel * :dt wait :dt*60 draw.pendulum
end
hideturtle until [key?] [step.pendulum]</lang>
MATLAB
pendulum.m <lang MATLAB>%This is a numerical simulation of a pendulum with a massless pivot arm.
%% User Defined Parameters %Define external parameters g = -9.8; deltaTime = 1/50; %Decreasing this will increase simulation accuracy endTime = 16;
%Define pendulum rodPivotPoint = [2 2]; %rectangular coordinates rodLength = 1; mass = 1; %of the bob radius = .2; %of the bob theta = 45; %degrees, defines initial position of the bob velocity = [0 0]; %cylindrical coordinates; first entry is radial velocity,
%second entry is angular velocity
%% Simulation assert(radius < rodLength,'Pendulum bob radius must be less than the length of the rod.');
position = rodPivotPoint - (rodLength*[-sind(theta) cosd(theta)]); %in rectangular coordinates
%Generate graphics, render pendulum figure; axesHandle = gca; xlim(axesHandle, [(rodPivotPoint(1) - rodLength - radius) (rodPivotPoint(1) + rodLength + radius)] ); ylim(axesHandle, [(rodPivotPoint(2) - rodLength - radius) (rodPivotPoint(2) + rodLength + radius)] );
rectHandle = rectangle('Position',[(position - radius/2) radius radius],...
'Curvature',[1,1],'FaceColor','g'); %Pendulum bob
hold on plot(rodPivotPoint(1),rodPivotPoint(2),'^'); %pendulum pivot lineHandle = line([rodPivotPoint(1) position(1)],...
[rodPivotPoint(2) position(2)]); %pendulum rod
hold off
%Run simulation, all calculations are performed in cylindrical coordinates for time = (deltaTime:deltaTime:endTime)
drawnow; %Forces MATLAB to render the pendulum
%Find total force
gravitationalForceCylindrical = [mass*g*cosd(theta) mass*g*sind(theta)];
%This code is just incase you want to add more forces,e.g friction
totalForce = gravitationalForceCylindrical;
%If the rod isn't massless or is a spring, etc., modify this line
%accordingly
rodForce = [-totalForce(1) 0]; %cylindrical coordinates
totalForce = totalForce + rodForce;
acceleration = totalForce / mass; %F = ma
velocity = velocity + acceleration * deltaTime;
rodLength = rodLength + velocity(1) * deltaTime;
theta = theta + velocity(2) * deltaTime;
position = rodPivotPoint - (rodLength*[-sind(theta) cosd(theta)]);
%Update figure with new position info
set(rectHandle,'Position',[(position - radius/2) radius radius]);
set(lineHandle,'XData',[rodPivotPoint(1) position(1)],'YData',...
[rodPivotPoint(2) position(2)]);
end</lang>
Oz
Inspired by the E and Ruby versions.
<lang oz>declare
[QTk] = {Link ['x-oz://system/wp/QTk.ozf']}
Pi = 3.14159265
class PendulumModel
feat
K
attr
angle velocity
meth init(length:L <= 1.0 %% meters
gravity:G <= 9.81 %% m/s² initialAngle:A <= Pi/2.) %% radians self.K = ~G / L angle := A velocity := 0.0
end
meth nextAngle(deltaT:DeltaTMS %% milliseconds
?Angle) %% radians
DeltaT = {Int.toFloat DeltaTMS} / 1000.0 %% seconds
Acceleration = self.K * {Sin @angle}
in
velocity := @velocity + Acceleration * DeltaT
angle := @angle + @velocity * DeltaT
Angle = @angle
end
end
%% Animates a pendulum on a given canvas.
class PendulumAnimation from Time.repeat
feat
Pend
Rod
Bob
home:pos(x:160 y:50)
length:140.0
delay
meth init(Pendulum Canvas delay:Delay <= 25) %% milliseconds
self.Pend = Pendulum self.delay = Delay %% plate and pivot
{Canvas create(line 0 self.home.y 320 self.home.y width:2 fill:grey50)}
{Canvas create(oval 155 self.home.y-5 165 self.home.y+5 fill:grey50 outline:black)}
%% the pendulum itself self.Rod = {Canvas create(line 1 1 1 1 width:3 fill:black handle:$)}
self.Bob = {Canvas create(oval 1 1 2 2 fill:yellow outline:black handle:$)}
%%
{self setRepAll(action:Animate delay:Delay)}
end
meth Animate
Theta = {self.Pend nextAngle(deltaT:self.delay $)} %% calculate x and y from angle X = self.home.x + {Float.toInt self.length * {Sin Theta}} Y = self.home.y + {Float.toInt self.length * {Cos Theta}}
in
%% update canvas try {self.Rod setCoords(self.home.x self.home.y X Y)} {self.Bob setCoords(X-15 Y-15 X+15 Y+15)} catch system(tk(alreadyClosed ...) ...) then skip end
end
end
Pendulum = {New PendulumModel init}
Canvas
GUI = td(title:"Pendulum"
canvas(width:320 height:210 handle:?Canvas)
action:proc {$} {Animation stop} {Window close} end
)
Window = {QTk.build GUI}
Animation = {New PendulumAnimation init(Pendulum Canvas)}
in
{Window show}
{Animation go}</lang>
PureBasic
If the code was part of a larger application it could be improved by specifying constants for the locations of image elements. <lang PureBasic>Procedure handleError(x, msg.s)
If Not x
MessageRequester("Error", msg)
End
EndIf
EndProcedure
- ScreenW = 320
- ScreenH = 210
handleError(OpenWindow(0, 0, 0, #ScreenW, #ScreenH, "Animated Pendulum", #PB_Window_SystemMenu), "Can't open window.") handleError(InitSprite(), "Can't setup sprite display.") handleError(OpenWindowedScreen(WindowID(0), 0, 0, #ScreenW, #ScreenH, 0, 0, 0), "Can't open screen.")
Enumeration ;sprites
#bob_spr #ceiling_spr #pivot_spr
EndEnumeration
TransparentSpriteColor(#PB_Default, RGB(255, 0, 255)) CreateSprite(#bob_spr, 32, 32) StartDrawing(SpriteOutput(#bob_spr))
Box(0, 0, 32, 32, RGB(255, 0, 255)) Circle(16, 16, 15, RGB(253, 252, 3)) DrawingMode(#PB_2DDrawing_Outlined) Circle(16, 16, 15, RGB(0, 0, 0))
StopDrawing()
CreateSprite(#pivot_spr, 10, 10) StartDrawing(SpriteOutput(#pivot_spr))
Box(0, 0, 10, 10, RGB(255, 0, 255)) Circle(5, 5, 4, RGB(125, 125, 125)) DrawingMode(#PB_2DDrawing_Outlined) Circle(5, 5, 4, RGB(0,0 , 0))
StopDrawing()
CreateSprite(#ceiling_spr,#ScreenW,2) StartDrawing(SpriteOutput(#ceiling_spr))
Box(0,0,SpriteWidth(#ceiling_spr), SpriteHeight(#ceiling_spr), RGB(126, 126, 126))
StopDrawing()
Structure pendulum
length.d ; meters constant.d ; -g/l gravity.d ; m/s² angle.d ; radians velocity.d ; m/s
EndStructure
Procedure initPendulum(*pendulum.pendulum, length.d = 1.0, gravity.d = 9.81, initialAngle.d = #PI / 2)
With *pendulum \length = length \gravity = gravity \angle = initialAngle \constant = -gravity / length \velocity = 0.0 EndWith
EndProcedure
Procedure updatePendulum(*pendulum.pendulum, deltaTime.d)
deltaTime = deltaTime / 1000.0 ;ms Protected acceleration.d = *pendulum\constant * Sin(*pendulum\angle) *pendulum\velocity + acceleration * deltaTime *pendulum\angle + *pendulum\velocity * deltaTime
EndProcedure
Procedure drawBackground()
ClearScreen(RGB(190,190,190)) ;draw ceiling DisplaySprite(#ceiling_spr, 0, 47) ;draw pivot DisplayTransparentSprite(#pivot_spr, 154,43) ;origin in upper-left
EndProcedure
Procedure drawPendulum(*pendulum.pendulum)
;draw rod Protected x = *pendulum\length * 140 * Sin(*pendulum\angle) ;scale = 1 m/140 pixels Protected y = *pendulum\length * 140 * Cos(*pendulum\angle) StartDrawing(ScreenOutput()) LineXY(154 + 5,43 + 5, 154 + 5 + x, 43 + 5 + y) ;draw from pivot-center to bob-center, adjusting for origins StopDrawing() ;draw bob DisplayTransparentSprite(#bob_spr, 154 + 5 - 16 + x, 43 + 5 - 16 + y) ;adj for origin in upper-left
EndProcedure
Define pendulum.pendulum, event initPendulum(pendulum) drawPendulum(pendulum)
AddWindowTimer(0, 1, 50) Repeat
event = WindowEvent()
Select event
Case #pb_event_timer
drawBackground()
Select EventTimer()
Case 1
updatePendulum(pendulum, 50)
drawPendulum(pendulum)
EndSelect
FlipBuffers()
Case #PB_Event_CloseWindow
Break
EndSelect
ForEver</lang>
Python
<lang python>import pygame, sys from pygame.locals import * from math import sin, cos, radians
pygame.init()
WINDOWSIZE = 250 TIMETICK = 100 BOBSIZE = 15
window = pygame.display.set_mode((WINDOWSIZE, WINDOWSIZE)) pygame.display.set_caption("Pendulum")
screen = pygame.display.get_surface() screen.fill((255,255,255))
PIVOT = (WINDOWSIZE/2, WINDOWSIZE/10) SWINGLENGTH = PIVOT[1]*4
class BobMass(pygame.sprite.Sprite):
def __init__(self):
pygame.sprite.Sprite.__init__(self)
self.theta = 45
self.dtheta = 0
self.rect = pygame.Rect(PIVOT[0]-SWINGLENGTH*cos(radians(self.theta)),
PIVOT[1]+SWINGLENGTH*sin(radians(self.theta)),
1,1)
self.draw()
def recomputeAngle(self):
scaling = 3000.0/(SWINGLENGTH**2)
firstDDtheta = -sin(radians(self.theta))*scaling
midDtheta = self.dtheta + firstDDtheta
midtheta = self.theta + (self.dtheta + midDtheta)/2.0
midDDtheta = -sin(radians(midtheta))*scaling
midDtheta = self.dtheta + (firstDDtheta + midDDtheta)/2
midtheta = self.theta + (self.dtheta + midDtheta)/2
midDDtheta = -sin(radians(midtheta)) * scaling
lastDtheta = midDtheta + midDDtheta
lasttheta = midtheta + (midDtheta + lastDtheta)/2.0
lastDDtheta = -sin(radians(lasttheta)) * scaling
lastDtheta = midDtheta + (midDDtheta + lastDDtheta)/2.0
lasttheta = midtheta + (midDtheta + lastDtheta)/2.0
self.dtheta = lastDtheta
self.theta = lasttheta
self.rect = pygame.Rect(PIVOT[0]-
SWINGLENGTH*sin(radians(self.theta)),
PIVOT[1]+
SWINGLENGTH*cos(radians(self.theta)),1,1)
def draw(self):
pygame.draw.circle(screen, (0,0,0), PIVOT, 5, 0)
pygame.draw.circle(screen, (0,0,0), self.rect.center, BOBSIZE, 0)
pygame.draw.aaline(screen, (0,0,0), PIVOT, self.rect.center)
pygame.draw.line(screen, (0,0,0), (0, PIVOT[1]), (WINDOWSIZE, PIVOT[1]))
def update(self):
self.recomputeAngle()
screen.fill((255,255,255))
self.draw()
bob = BobMass()
TICK = USEREVENT + 2 pygame.time.set_timer(TICK, TIMETICK)
def input(events):
for event in events:
if event.type == QUIT:
sys.exit(0)
elif event.type == TICK:
bob.update()
while True:
input(pygame.event.get()) pygame.display.flip()</lang>
RLaB
The pendulum motion is an interesting and easy problem in which the facilities of RLaB for numerical computation and simulation are easily accessible. The parameters of the problem are , the length of the arm, and the magnitude of the gravity.
We start with the mathematical transliteration of the problem. We solve it in terms of describing the angle between the -axis and the arm of the pendulum, where the downwards direction is taken as positive. The Newton equation of motian, which is a second-order non-linear ordinary differential equation (ODE) reads
In our example, we will solve the problem as, so called, initial value problem (IVP). That is, we will specify that at the time t=0 the pendulum was at rest , extended at an angle radians (equivalent to 30 degrees).
RLaB has the facilities to solve ODE IVP which are accessible through odeiv solver. This solver requires that the ODE be written as the first order differential equation,
Here, we introduced , for which the original ODE reads
- .
Ruby
This does not have the window resizing handling that Tcl does -- I did not spend enough time in the docs to figure out how to get the new window size out of the configuration event. Of interest when running this pendulum side-by-side with the Tcl one: the Tcl pendulum swings noticibly faster.
<lang ruby>require 'tk'
$root = TkRoot.new("title" => "Pendulum Animation") $canvas = TkCanvas.new($root) do
width 320 height 200 create TkcLine, 0,25,320,25, 'tags' => 'plate', 'width' => 2, 'fill' => 'grey50' create TkcOval, 155,20,165,30, 'tags' => 'pivot', 'outline' => "", 'fill' => 'grey50' create TkcLine, 1,1,1,1, 'tags' => 'rod', 'width' => 3, 'fill' => 'black' create TkcOval, 1,1,2,2, 'tags' => 'bob', 'outline' => 'black', 'fill' => 'yellow'
end $canvas.raise('pivot') $canvas.pack('fill' => 'both', 'expand' => true)
$Theta = 45.0 $dTheta = 0.0 $length = 150 $homeX = 160 $homeY = 25
def show_pendulum
angle = $Theta * Math::PI / 180
x = $homeX + $length * Math.sin(angle)
y = $homeY + $length * Math.cos(angle)
$canvas.coords('rod', $homeX, $homeY, x, y)
$canvas.coords('bob', x-15, y-15, x+15, y+15)
end
def recompute_angle
scaling = 3000.0 / ($length ** 2) # first estimate firstDDTheta = -Math.sin($Theta * Math::PI / 180) * scaling midDTheta = $dTheta + firstDDTheta midTheta = $Theta + ($dTheta + midDTheta)/2 # second estimate midDDTheta = -Math.sin(midTheta * Math::PI / 180) * scaling midDTheta = $dTheta + (firstDDTheta + midDDTheta)/2 midTheta = $Theta + ($dTheta + midDTheta)/2 # again, first midDDTheta = -Math.sin(midTheta * Math::PI / 180) * scaling lastDTheta = midDTheta + midDDTheta lastTheta = midTheta + (midDTheta + lastDTheta)/2 # again, second lastDDTheta = -Math.sin(lastTheta * Math::PI/180) * scaling lastDTheta = midDTheta + (midDDTheta + lastDDTheta)/2 lastTheta = midTheta + (midDTheta + lastDTheta)/2 # Now put the values back in our globals $dTheta = lastDTheta $Theta = lastTheta
end
def animate
recompute_angle
show_pendulum
$after_id = $root.after(15) {animate}
end
show_pendulum $after_id = $root.after(500) {animate}
$canvas.bind('<Destroy>') {$root.after_cancel($after_id)}
Tk.mainloop</lang>
Scheme
This is a direct translation of the Ruby/Tk example into Scheme + PS/Tk.
<lang scheme>#!r6rs
- R6RS implementation of Pendulum Animation
(import (rnrs)
(lib pstk main) ; change this for your pstk installation
)
(define PI 3.14159) (define *conv-radians* (/ PI 180)) (define *theta* 45.0) (define *d-theta* 0.0) (define *length* 150) (define *home-x* 160) (define *home-y* 25)
- estimates new angle of pendulum
(define (recompute-angle)
(define (avg a b) (/ (+ a b) 2))
(let* ((scaling (/ 3000.0 (* *length* *length*)))
; first estimate
(first-dd-theta (- (* (sin (* *theta* *conv-radians*)) scaling)))
(mid-d-theta (+ *d-theta* first-dd-theta))
(mid-theta (+ *theta* (avg *d-theta* mid-d-theta)))
; second estimate
(mid-dd-theta (- (* (sin (* mid-theta *conv-radians*)) scaling)))
(mid-d-theta-2 (+ *d-theta* (avg first-dd-theta mid-dd-theta)))
(mid-theta-2 (+ *theta* (avg *d-theta* mid-d-theta-2)))
; again first
(mid-dd-theta-2 (- (* (sin (* mid-theta-2 *conv-radians*)) scaling)))
(last-d-theta (+ mid-d-theta-2 mid-dd-theta-2))
(last-theta (+ mid-theta-2 (avg mid-d-theta-2 last-d-theta)))
; again second
(last-dd-theta (- (* (sin (* last-theta *conv-radians*)) scaling)))
(last-d-theta-2 (+ mid-d-theta-2 (avg mid-dd-theta-2 last-dd-theta)))
(last-theta-2 (+ mid-theta-2 (avg mid-d-theta-2 last-d-theta-2))))
; put values back in globals
(set! *d-theta* last-d-theta-2)
(set! *theta* last-theta-2)))
- The main event loop and graphics context
(let ((tk (tk-start)))
(tk/wm 'title tk "Pendulum Animation") (let ((canvas (tk 'create-widget 'canvas)))
;;; redraw the pendulum on canvas
;;; - uses angle and length to compute new (x,y) position of bob
(define (show-pendulum canvas)
(let* ((pendulum-angle (* *conv-radians* *theta*))
(x (+ *home-x* (* *length* (sin pendulum-angle))))
(y (+ *home-y* (* *length* (cos pendulum-angle)))))
(canvas 'coords 'rod *home-x* *home-y* x y)
(canvas 'coords 'bob (- x 15) (- y 15) (+ x 15) (+ y 15))))
;;; move the pendulum and repeat after 20ms
(define (animate)
(recompute-angle)
(show-pendulum canvas)
(tk/after 20 animate))
;; layout the canvas (tk/grid canvas 'column: 0 'row: 0) (canvas 'create 'line 0 25 320 25 'tags: 'plate 'width: 2 'fill: 'grey50) (canvas 'create 'oval 155 20 165 30 'tags: 'pivot 'outline: "" 'fill: 'grey50) (canvas 'create 'line 1 1 1 1 'tags: 'rod 'width: 3 'fill: 'black) (canvas 'create 'oval 1 1 2 2 'tags: 'bob 'outline: 'black 'fill: 'yellow)
;; get everything started (show-pendulum canvas) (tk/after 500 animate) (tk-event-loop tk)))
</lang>
Tcl
and
<lang tcl>package require Tcl 8.5 package require Tk
- Make the graphical entities
pack [canvas .c -width 320 -height 200] -fill both -expand 1 .c create line 0 25 320 25 -width 2 -fill grey50 -tags plate .c create line 1 1 1 1 -tags rod -width 3 -fill black .c create oval 1 1 2 2 -tags bob -fill yellow -outline black .c create oval 155 20 165 30 -fill grey50 -outline {} -tags pivot
- Set some vars
set points {} set Theta 45.0 set dTheta 0.0 set pi 3.1415926535897933 set length 150 set homeX 160
- How to respond to a changing in size of the window
proc resized {width} {
global homeX
.c coords plate 0 25 $width 25
set homeX [expr {$width / 2}]
.c coords pivot [expr {$homeX-5}] 20 [expr {$homeX+5}] 30
showPendulum
}
- How to actually arrange the pendulum, mapping the model to the display
proc showPendulum {} {
global Theta dTheta pi length homeX
set angle [expr {$Theta * $pi/180}]
set x [expr {$homeX + $length*sin($angle)}]
set y [expr {25 + $length*cos($angle)}]
.c coords rod $homeX 25 $x $y
.c coords bob [expr {$x-15}] [expr {$y-15}] [expr {$x+15}] [expr {$y+15}]
}
- The dynamic part of the display
proc recomputeAngle {} {
global Theta dTheta pi length
set scaling [expr {3000.0/$length**2}]
# first estimate
set firstDDTheta [expr {-sin($Theta * $pi/180)*$scaling}]
set midDTheta [expr {$dTheta + $firstDDTheta}]
set midTheta [expr {$Theta + ($dTheta + $midDTheta)/2}]
# second estimate
set midDDTheta [expr {-sin($midTheta * $pi/180)*$scaling}]
set midDTheta [expr {$dTheta + ($firstDDTheta + $midDDTheta)/2}]
set midTheta [expr {$Theta + ($dTheta + $midDTheta)/2}]
# Now we do a double-estimate approach for getting the final value
# first estimate
set midDDTheta [expr {-sin($midTheta * $pi/180)*$scaling}]
set lastDTheta [expr {$midDTheta + $midDDTheta}]
set lastTheta [expr {$midTheta + ($midDTheta + $lastDTheta)/2}]
# second estimate
set lastDDTheta [expr {-sin($lastTheta * $pi/180)*$scaling}]
set lastDTheta [expr {$midDTheta + ($midDDTheta + $lastDDTheta)/2}]
set lastTheta [expr {$midTheta + ($midDTheta + $lastDTheta)/2}]
# Now put the values back in our globals
set dTheta $lastDTheta
set Theta $lastTheta
}
- Run the animation by updating the physical model then the display
proc animate {} {
global animation
recomputeAngle showPendulum
# Reschedule set animation [after 15 animate]
} set animation [after 500 animate]; # Extra initial delay is visually pleasing
- Callback to handle resizing of the canvas
bind .c <Configure> {resized %w}
- Callback to stop the animation cleanly when the GUI goes away
bind .c <Destroy> {after cancel $animation}</lang>