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.
Ada
This does not use a GUI, it simply animates the pendulum and prints out the positions. If you want, you can replace the output method with graphical update methods.
X and Y are relative positions of the pendulum to the anchor.
pendulums.ads: <lang Ada>generic
type Float_Type is digits <>; Gravitation : Float_Type;
package Pendulums is
type Pendulum is private; function New_Pendulum (Length : Float_Type; Theta0 : Float_Type) return Pendulum; function Get_X (From : Pendulum) return Float_Type; function Get_Y (From : Pendulum) return Float_Type; procedure Update_Pendulum (Item : in out Pendulum; Time : in Duration);
private
type Pendulum is record Length : Float_Type; Theta : Float_Type; X : Float_Type; Y : Float_Type; Velocity : Float_Type; end record;
end Pendulums;</lang>
pendulums.adb: <lang Ada>with Ada.Numerics.Generic_Elementary_Functions; package body Pendulums is
package Math is new Ada.Numerics.Generic_Elementary_Functions (Float_Type);
function New_Pendulum (Length : Float_Type; Theta0 : Float_Type) return Pendulum is Result : Pendulum; begin Result.Length := Length; Result.Theta := Theta0 / 180.0 * Ada.Numerics.Pi; Result.X := Math.Sin (Theta0) * Length; Result.Y := Math.Cos (Theta0) * Length; Result.Velocity := 0.0; return Result; end New_Pendulum;
function Get_X (From : Pendulum) return Float_Type is begin return From.X; end Get_X;
function Get_Y (From : Pendulum) return Float_Type is begin return From.Y; end Get_Y;
procedure Update_Pendulum (Item : in out Pendulum; Time : in Duration) is Acceleration : constant Float_Type := Gravitation / Item.Length * Math.Sin (Item.Theta); begin Item.X := Math.Sin (Item.Theta) * Item.Length; Item.Y := Math.Cos (Item.Theta) * Item.Length; Item.Velocity := Item.Velocity + Acceleration * Float_Type (Time); Item.Theta := Item.Theta + Item.Velocity * Float_Type (Time); end Update_Pendulum;
end Pendulums;</lang>
example main.adb: <lang Ada>with Ada.Text_IO; with Ada.Calendar; with Pendulums;
procedure Main is
package Float_Pendulum is new Pendulums (Float, -9.81); use Float_Pendulum; use type Ada.Calendar.Time;
My_Pendulum : Pendulum := New_Pendulum (10.0, 30.0); Now, Before : Ada.Calendar.Time;
begin
Before := Ada.Calendar.Clock; loop Delay 0.1; Now := Ada.Calendar.Clock; Update_Pendulum (My_Pendulum, Now - Before); Before := Now; -- output positions relative to origin -- replace with graphical output if wanted Ada.Text_IO.Put_Line (" X: " & Float'Image (Get_X (My_Pendulum)) & " Y: " & Float'Image (Get_Y (My_Pendulum))); end loop;
end Main;</lang>
Output:
X: 5.00000E+00 Y: 8.66025E+00 X: 4.95729E+00 Y: 8.68477E+00 X: 4.87194E+00 Y: 8.73294E+00 X: 4.74396E+00 Y: 8.80312E+00 X: 4.57352E+00 Y: 8.89286E+00 X: 4.36058E+00 Y: 8.99919E+00 X: 4.10657E+00 Y: 9.11790E+00 X: 3.81188E+00 Y: 9.24498E+00 X: 3.47819E+00 Y: 9.37562E+00 X: 3.10714E+00 Y: 9.50504E+00 X: 2.70211E+00 Y: 9.62801E+00 X: 2.26635E+00 Y: 9.73980E+00 X: 1.80411E+00 Y: 9.83591E+00 X: 1.32020E+00 Y: 9.91247E+00 X: 8.20224E-01 Y: 9.96630E+00 X: 3.10107E-01 Y: 9.99519E+00 X: -2.03865E-01 Y: 9.99792E+00 X: -7.15348E-01 Y: 9.97438E+00 X: -1.21816E+00 Y: 9.92553E+00 X: -1.70581E+00 Y: 9.85344E+00 X: -2.17295E+00 Y: 9.76106E+00 X: -2.61452E+00 Y: 9.65216E+00 X: -3.02618E+00 Y: 9.53112E+00 X: -3.40427E+00 Y: 9.40271E+00 X: -3.74591E+00 Y: 9.27190E+00 X: -4.04873E+00 Y: 9.14373E+00 X: -4.31141E+00 Y: 9.02285E+00 X: -4.53271E+00 Y: 8.91373E+00 X: -4.71186E+00 Y: 8.82034E+00 X: -4.84868E+00 Y: 8.74587E+00 X: -4.94297E+00 Y: 8.69293E+00 X: -4.99459E+00 Y: 8.66337E+00 X: -5.00352E+00 Y: 8.65822E+00 ...
BBC BASIC
<lang bbcbasic> MODE 8
*FLOAT 64 VDU 23,23,4;0;0;0; : REM Set line thickness theta = RAD(40) : REM initial displacement g = 9.81 : REM acceleration due to gravity l = 0.50 : REM length of pendulum in metres REPEAT PROCpendulum(theta, l) WAIT 1 PROCpendulum(theta, l) accel = - g * SIN(theta) / l / 100 speed += accel / 100 theta += speed UNTIL FALSE END DEF PROCpendulum(a, l) LOCAL pivotX, pivotY, bobX, bobY pivotX = 640 pivotY = 800 bobX = pivotX + l * 1000 * SIN(a) bobY = pivotY - l * 1000 * COS(a) GCOL 3,6 LINE pivotX, pivotY, bobX, bobY GCOL 3,11 CIRCLE FILL bobX + 24 * SIN(a), bobY - 24 * COS(a), 24 ENDPROC</lang>
C
<lang c>#include <stdlib.h>
- include <math.h>
- include <GL/glut.h>
- include <GL/gl.h>
- include <sys/time.h>
- define length 5
- define g 9.8
double alpha, accl, omega = 0, E; struct timeval tv;
double elappsed() { struct timeval now; gettimeofday(&now, 0); int ret = (now.tv_sec - tv.tv_sec) * 1000000 + now.tv_usec - tv.tv_usec; tv = now; return ret / 1.e6; }
void resize(int w, int h) { glViewport(0, 0, w, h); glMatrixMode(GL_PROJECTION); glLoadIdentity();
glMatrixMode(GL_MODELVIEW); glLoadIdentity(); glOrtho(0, w, h, 0, -1, 1); }
void render() { double x = 320 + 300 * sin(alpha), y = 300 * cos(alpha); resize(640, 320);
glClear(GL_COLOR_BUFFER_BIT);
glBegin(GL_LINES); glVertex2d(320, 0); glVertex2d(x, y); glEnd(); glFlush();
double us = elappsed(); alpha += (omega + us * accl / 2) * us; omega += accl * us;
/* don't let precision error go out of hand */ if (length * g * (1 - cos(alpha)) >= E) { alpha = (alpha < 0 ? -1 : 1) * acos(1 - E / length / g); omega = 0; } accl = -g / length * sin(alpha); }
void init_gfx(int *c, char **v) { glutInit(c, v); glutInitDisplayMode(GLUT_RGB); glutInitWindowSize(640, 320); glutIdleFunc(render); glutCreateWindow("Pendulum"); }
int main(int c, char **v) { alpha = 4 * atan2(1, 1) / 2.1; E = length * g * (1 - cos(alpha));
accl = -g / length * sin(alpha); omega = 0;
gettimeofday(&tv, 0); init_gfx(&c, v); glutMainLoop(); 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>
Euphoria
DOS32 version
<lang euphoria>include graphics.e include misc.e
constant dt = 1E-3 constant g = 50
sequence vc sequence suspension atom len
procedure draw_pendulum(atom color, atom len, atom alfa)
sequence point point = (len*{sin(alfa),cos(alfa)} + suspension) draw_line(color, {suspension, point}) ellipse(color,0,point-{10,10},point+{10,10})
end procedure
function wait()
atom t0 t0 = time() while time() = t0 do if get_key() != -1 then return 1 end if end while return 0
end function
procedure animation()
atom alfa, omega, epsilon if graphics_mode(18) then end if vc = video_config() suspension = {vc[VC_XPIXELS]/2,vc[VC_YPIXELS]/2} len = vc[VC_YPIXELS]/2-20 alfa = PI/2 omega = 0
while 1 do draw_pendulum(BRIGHT_WHITE,len,alfa) if wait() then exit end if draw_pendulum(BLACK,len,alfa) epsilon = -len*sin(alfa)*g omega += dt*epsilon alfa += dt*omega end while
if graphics_mode(-1) then end if
end procedure
animation()</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 0.01lastPaintedAt <- 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>
Icon and Unicon
The following code uses features exclusive to Unicon, specifically the object-oriented gui library.
<lang Unicon> import gui $include "guih.icn"
- some constants to define the display and pendulum
$define HEIGHT 400 $define WIDTH 500 $define STRING_LENGTH 200 $define HOME_X 250 $define HOME_Y 21 $define SIZE 30 $define START_ANGLE 80
class WindowApp : Dialog ()
# draw the pendulum on given context_window, at position (x,y) method draw_pendulum (x, y) # reference to current screen area to draw on cw := Clone(self.cwin)
# clear screen WAttrib (cw, "bg=grey") EraseRectangle (cw, 0, 0, WIDTH, HEIGHT)
# draw the display WAttrib (cw, "fg=dark gray") DrawLine (cw, 10, 20, WIDTH-20, 20) WAttrib (cw, "fg=black") DrawLine (cw, HOME_X, HOME_Y, x, y) FillCircle (cw, x, y, SIZE+2) WAttrib (cw, "fg=yellow") FillCircle (cw, x, y, SIZE)
# free reference to screen area Uncouple (cw) end
# find the average of given two arguments method avg (a, b) return (a + b) / 2 end
# this method gets called by the ticker # it computes the next position of the pendulum and # requests a redraw method tick () static x, y static theta := START_ANGLE static d_theta := 0 # update x,y of pendulum scaling := 3000.0 / (STRING_LENGTH * STRING_LENGTH) # -- first estimate first_dd_theta := -(sin (dtor (theta)) * scaling) mid_d_theta := d_theta + first_dd_theta mid_theta := theta + avg (d_theta, mid_d_theta) # -- second estimate mid_dd_theta := - (sin (dtor (mid_theta)) * 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 (dtor (mid_theta_2)) * 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 (dtor (last_theta)) * 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) # -- update stored angles d_theta := last_d_theta_2 theta := last_theta_2 # -- update x, y pendulum_angle := dtor (theta) x := HOME_X + STRING_LENGTH * sin (pendulum_angle) y := HOME_Y + STRING_LENGTH * cos (pendulum_angle)
# draw pendulum draw_pendulum (x, y) end
# set up the window method component_setup () # some cosmetic settings for the window attrib("size="||WIDTH||","||HEIGHT, "bg=light gray", "label=Pendulum") # make sure we respond to window close event connect (self, "dispose", CLOSE_BUTTON_EVENT) # start the ticker, to update the display periodically self.set_ticker (20) end
end
procedure main ()
w := WindowApp () w.show_modal ()
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>
Liberty BASIC
<lang lb>nomainwin
WindowWidth = 400 WindowHeight = 300
open "Pendulum" for graphics_nsb_nf as #main #main, "down;fill white; flush" #main, "color black" #main, "trapclose [quit.main]"
Angle = asn(1) DeltaT = 0.1 PendLength = 150 FixX = int(WindowWidth / 2) FixY = 40
timer 30, [swing]
wait
[swing]
#main, "cls" #main, "discard"
PlumbobX = FixX + int(sin(Angle) * PendLength) PlumbobY = FixY + int(cos(Angle) * PendLength) AngAccel = -9.81 / PendLength * sin(Angle) AngVelocity = AngVelocity + AngAccel * DeltaT Angle = Angle + AngVelocity * DeltaT
#main, "backcolor black" #main, "place ";FixX;" ";FixY #main, "circlefilled 3" #main, "line ";FixX;" ";FixY;" ";PlumbobX;" ";PlumbobY #main, "backcolor red" #main, "circlefilled 10"
wait
[quit.main]
close #main end</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>
Mathematica
<lang Mathematica>freq = 8; length = freq^(-1/2); Animate[Graphics[
List[{Line[{{0, 0}, length {Sin[T], -Cos[T]}} /. {T -> (Pi/6) Cos[2 Pi freq t]}], PointSize[Large], Point[{length {Sin[T], -Cos[T]}} /. {T -> (Pi/6) Cos[2 Pi freq t]}]}], PlotRange -> {{-0.3, 0.3}, {-0.5, 0}}], {t, 0, 1}, AnimationRate -> 0.07]</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>
ooRexx
ooRexx does not have a portable GUI, but this version is similar to the Ada version and just prints out the coordinates of the end of the pendulum. <lang ooRexx> pendulum = .pendulum~new(10, 30)
before = .datetime~new do 100 -- somewhat arbitrary loop count
call syssleep .2 now = .datetime~new pendulum~update(now - before) before = now say " X:" pendulum~x " Y:" pendulum~y
end
- class pendulum
- method init
expose length theta x y velocity use arg length, theta x = rxcalcsin(theta) * length y = rxcalccos(theta) * length velocity = 0
- attribute x GET
- attribute y GET
- constant g -9.81 -- acceleration due to gravity
- method update
expose length theta x y velocity use arg duration acceleration = self~g / length * rxcalcsin(theta) durationSeconds = duration~microseconds / 1000000 x = rxcalcsin(theta, length) y = rxcalccos(theta, length) velocity = velocity + acceleration * durationSeconds theta = theta + velocity * durationSeconds
- requires rxmath library
</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>
PicoLisp
A minimalist solution. The pendulum consists of the center point '+', and the swinging xterm cursor. <lang PicoLisp>(load "@lib/math.l")
(de pendulum (X Y Len)
(let (Angle pi/2 V 0) (call 'clear) (call 'tput "cup" Y X) (prin '+) (call 'tput "cup" 1 (+ X Len)) (until (key 25) # 25 ms (let A (*/ (sin Angle) -9.81 1.0) (inc 'V (*/ A 40)) # DT = 25 ms = 1/40 sec (inc 'Angle (*/ V 40)) ) (call 'tput "cup" (+ Y (*/ Len (cos Angle) 2.2)) # Compensate for aspect ratio (+ X (*/ Len (sin Angle) 1.0)) ) ) ) )</lang>
Test (hit any key to stop): <lang PicoLisp>(pendulum 40 1 36)</lang>
Prolog
SWI-Prolog has a graphic interface XPCE. <lang Prolog>:- use_module(library(pce)).
pendulum :- new(D, window('Pendulum')), send(D, size, size(560, 300)), new(Line, line(80, 50, 480, 50)), send(D, display, Line), new(Circle, circle(20)), send(Circle, fill_pattern, colour(@default, 0, 0, 0)), new(Boule, circle(60)), send(Boule, fill_pattern, colour(@default, 0, 0, 0)), send(D, display, Circle, point(270,40)), send(Circle, handle, handle(h/2, w/2, in)), send(Boule, handle, handle(h/2, w/2, out)), send(Circle, connect, Boule, link(in, out, line(0,0,0,0,none))), new(Anim, animation(D, 0.0, Boule, 200.0)), send(D, done_message, and(message(Anim, free), message(Boule, free), message(Circle, free), message(@receiver,destroy))), send(Anim?mytimer, start), send(D, open).
- - pce_begin_class(animation(window, angle, boule, len_pendulum), object).
variable(window, object, both, "Display window"). variable(boule, object, both, "bowl of the pendulum"). variable(len_pendulum, object, both, "len of the pendulum"). variable(angle, object, both, "angle with the horizontal"). variable(delta, object, both, "increment of the angle"). variable(mytimer, timer, both, "timer of the animation").
initialise(P, W:object, A:object, B : object, L:object) :->
"Creation of the object":: send(P, window, W), send(P, angle, A), send(P, boule, B), send(P, len_pendulum, L), send(P, delta, 0.01),
send(P, mytimer, new(_, timer(0.01,message(P, anim_message)))).
% method called when the object is destroyed % first the timer is stopped % then all the resources are freed unlink(P) :-> send(P?mytimer, stop), send(P, send_super, unlink).
% message processed by the timer
anim_message(P) :->
get(P, angle, A),
get(P, len_pendulum, L),
calc(A, L, X, Y),
get(P, window, W),
get(P, boule, B),
send(W, display, B, point(X,Y)),
% computation of the next position
get(P, delta, D),
next_Angle(A, D, NA, ND),
send(P, angle, NA),
send(P, delta, ND).
- - pce_end_class.
% computation of the position of the bowl. calc(Ang, Len, X, Y) :- X is Len * cos(Ang)+ 250, Y is Len * sin(Ang) + 20.
% computation of the next angle
% if we reach 0 or pi, delta change.
next_Angle(A, D, NA, ND) :-
NA is D + A,
(((D > 0, abs(pi-NA) < 0.01); (D < 0, abs(NA) < 0.01))->
ND = - D;
ND = D).
</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>
Racket
<lang scheme>#lang racket
(require 2htdp/image
2htdp/universe)
(define (pendulum)
(define (accel θ) (- (sin θ))) (define θ (/ pi 2.5)) (define θ′ 0) (define θ′′ (accel (/ pi 2.5))) (define (x θ) (+ 200 (* 150 (sin θ)))) (define (y θ) (* 150 (cos θ))) (λ (n) (define p-image (underlay/xy (add-line (empty-scene 400 200) 200 0 (x θ) (y θ) "black") (- (x θ) 5) (- (y θ) 5) (circle 5 "solid" "blue"))) (set! θ (+ θ (* θ′ 0.04))) (set! θ′ (+ θ′ (* (accel θ) 0.04))) p-image))
(animate (pendulum))</lang>
RLaB
The plane 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 plane (2-D) 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 a vector , for which the original ODE reads
- .
The RLaB script that solves the problem is
<lang RLaB> // // example: solve ODE for pendulum //
// we first define the first derivative function for the solver dudt = function(t, u, p) {
// t-> time // u->[theta, dtheta/dt ] // p-> g/L, parameter rval = zeros(2,1); rval[1] = u[2]; rval[2] = -p[1] * sin(u[1]); return rval;
};
// now we solve the problem // physical parameters L = 5; // (m), the length of the arm of the pendulum p = mks.g / L; // RLaB has a built-in list 'mks' which contains large number of physical constants and conversion factors T0 = 2*const.pi*sqrt(L/mks.g); // approximate period of the pendulum
// initial conditions theta0 = 30; // degrees, initial angle of deflection of pendulum u0 = [theta0*const.pi/180, 0]; // RLaB has a built-in list 'const' of mathematical constants.
// times at which we want solution t = [0:4:1/64] * T0; // solve for 4 approximate periods with at time points spaced at T0/64
// prepare ODEIV solver optsode = <<>>; optsode.eabs = 1e-6; // relative error for step size optsode.erel = 1e-6; // absolute error for step size optsode.delta_t = 1e-6; // maximum dt that code is allowed optsode.stdout = stderr(); // open the text console and in it print the results of each step of calculation optsode.imethod = 5; // use method No. 5 from the odeiv toolkit, Runge-Kutta 8th order Prince-Dormand method //optsode.phase_space = 0; // the solver returns [t, u1(t), u2(t)] which is default behavior optsode.phase_space = 1; // the solver returns [t, u1(t), u2(t), d(u1)/dt(t), d(u2)/dt]
// solver do my bidding y = odeiv(dudt, p, t, u0, optsode);
// Make an animation. We choose to use 'pgplot' rather then 'gnuplot' interface because the former is // faster and thus less cache-demanding, while the latter can be very cache-demanding (it may slow your // linux system quite down if one sends lots of plots for gnuplot to plot). plwins (1); // we will use one pgplot-window
plwin(1); // plot to pgplot-window No. 1; necessary if using more than one pgplot window plimits (-L,L, -1.25*L, 0.25*L); xlabel ("x-coordinate"); ylabel ("z-coordinate"); plegend ("Arm"); for (i in 1:y.nr) {
// plot a line between the pivot point at (0,0) and the current position of the pendulum arm_line = [0,0; L*sin(y[i;2]), -L*cos(y[i;2])]; // this is because theta is between the arm and the z-coordinate plot (arm_line); sleep (0.1); // sleep 0.1 seconds between plots
}
</lang>
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>
<lang ruby>Shoes.app(:width => 320, :height => 200) do
@centerX = 160 @centerY = 25 @length = 150 @diameter = 15
@Theta = 45.0 @dTheta = 0.0
stroke gray strokewidth 3 line 0,25,320,25 oval 155,20,10
stroke black @rod = line(@centerX, @centerY, @centerX, @centerY + @length) @bob = oval(@centerX - @diameter, @centerY + @length - @diameter, 2*@diameter)
animate(24) do |i| recompute_angle show_pendulum end
def show_pendulum angle = (90 + @Theta) * Math::PI / 180 x = @centerX + (Math.cos(angle) * @length).to_i y = @centerY + (Math.sin(angle) * @length).to_i
@rod.remove strokewidth 3 @rod = line(@centerX, @centerY, x, y) @bob.move(x-@diameter, y-@diameter) 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
end</lang>
Scala
Inspired by Java <lang Scala>import scala.swing._ import scala.swing.Swing._ import scala.actors._ import scala.actors.Actor._
import java.awt.{Color, Graphics}
object Pendulum extends SimpleSwingApplication {
val length = 100 val prefSizeX = 2*length+50 val prefSizeY = length/2*3 lazy val ui = new Panel { import math._ background = Color.white preferredSize = (prefSizeX, prefSizeY) peer.setDoubleBuffered(true) var angle: Double = Pi/2; def pendular = new Actor { var angleAccel, angleVelocity = 0.0; var dt = 0.1 def act() { while (true) { angleAccel = -9.81 / length * sin(angle) angleVelocity += angleAccel * dt angle += angleVelocity * dt repaint() Thread.sleep(15) } } }
override def paintComponent(g: Graphics2D) = { super.paintComponent(g) g.setColor(Color.white); g.fillRect(0, 0, size.width, size.height); val anchorX = size.width / 2 val anchorY = size.height / 4 val ballX = anchorX + (sin(angle) * length).toInt val ballY = anchorY + (cos(angle) * length).toInt g.setColor(Color.lightGray) g.drawLine(anchorX-2*length, anchorY, anchorX+2*length, anchorY) g.setColor(Color.black) g.drawLine(anchorX, anchorY, ballX, ballY) g.fillOval(anchorX - 3, anchorY - 4, 7, 7) g.drawOval(ballX - 7, ballY - 7, 14, 14) g.setColor(Color.yellow) g.fillOval(ballX - 7, ballY - 7, 14, 14) } }
def top = new MainFrame { title = "Rosetta Code >>> Task: Animate a pendulum | Language: Scala" contents = ui ui.pendular.start }
}</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
<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>
XPL0
<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations
proc Ball(X0, Y0, R, C); \Draw a filled circle int X0, Y0, R, C; \center coordinates, radius, color int X, Y; for Y:= -R to R do
for X:= -R to R do if X*X + Y*Y <= R*R then Point(X+X0, Y+Y0, C);
def L = 2.0, \pendulum arm length (meters)
G = 9.81, \acceleration due to gravity (meters/second^2) Pi = 3.14, DT = 1.0/72.0; \delta time = screen refresh rate (seconds)
def X0=640/2, Y0=480/2; \anchor point = center coordinate real S, V, A, T; \arc length, velocity, acceleration, theta angle int X, Y; \ball coordinates
[SetVid($101); \set 640x480x8 graphic display mode T:= Pi*0.75; V:= 0.0; \starting angle and velocity S:= T*L; repeat A:= -G*Sin(T);
V:= V + A*DT; S:= S + V*DT; T:= S/L; X:= X0 + fix(L*100.0*Sin(T)); \100 scales to fit screen Y:= Y0 + fix(L*100.0*Cos(T)); Move(X0, Y0); Line(X, Y, 7); \draw pendulum Ball(X, Y, 10, $E\yellow\); while port($3DA) & $08 do []; \wait for vertical retrace to go away repeat until port($3DA) & $08; \wait for vertical retrace signal Move(X0, Y0); Line(X, Y, 0); \erase pendulum Ball(X, Y, 10, 0\black\);
until KeyHit; \keystroke terminates program SetVid(3); \restore normal text screen ]</lang>
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