3 # This demonstration illustrates how Tcl/Tk can be used to construct
4 # simulations of physical systems.
6 if {![info exists widgetDemo]} {
7 error "This script should be run from the \"widget\" demo."
15 wm title $w "Pendulum Animation Demonstration"
16 wm iconname $w "pendulum"
19 label $w.msg -font $font -wraplength 4i -justify left -text "This demonstration shows how Tcl/Tk can be used to carry out animations that are linked to simulations of physical systems. In the left canvas is a graphical representation of the physical system itself, a simple pendulum, and in the right canvas is a graph of the phase space of the system, which is a plot of the angle (relative to the vertical) against the angular velocity. The pendulum bob may be repositioned by clicking and dragging anywhere on the left canvas."
22 ## See Code / Dismiss buttons
23 set btns [addSeeDismiss $w.buttons $w]
24 pack $btns -side bottom -fill x
26 # Create some structural widgets
27 pack [panedwindow $w.p] -fill both -expand 1
28 $w.p add [labelframe $w.p.l1 -text "Pendulum Simulation"]
29 $w.p add [labelframe $w.p.l2 -text "Phase Space"]
31 # Create the canvas containing the graphical representation of the
33 canvas $w.c -width 320 -height 200 -background white -bd 2 -relief sunken
34 $w.c create text 5 5 -anchor nw -text "Click to Adjust Bob Start Position"
35 # Coordinates of these items don't matter; they will be set properly below
36 $w.c create line 0 25 320 25 -tags plate -fill grey50 -width 2
37 $w.c create oval 155 20 165 30 -tags pivot -fill grey50 -outline {}
38 $w.c create line 1 1 1 1 -tags rod -fill black -width 3
39 $w.c create oval 1 1 2 2 -tags bob -fill yellow -outline black
40 pack $w.c -in $w.p.l1 -fill both -expand true
42 # Create the canvas containing the phase space graph; this consists of
43 # a line that gets gradually paler as it ages, which is an extremely
44 # effective visual trick.
45 canvas $w.k -width 320 -height 200 -background white -bd 2 -relief sunken
46 $w.k create line 160 200 160 0 -fill grey75 -arrow last -tags y_axis
47 $w.k create line 0 100 320 100 -fill grey75 -arrow last -tags x_axis
48 for {set i 90} {$i>=0} {incr i -10} {
49 # Coordinates of these items don't matter; they will be set properly below
50 $w.k create line 0 0 1 1 -smooth true -tags graph$i -fill grey$i
53 $w.k create text 0 0 -anchor ne -text "\u03b8" -tags label_theta
54 $w.k create text 0 0 -anchor ne -text "\u03b4\u03b8" -tags label_dtheta
55 pack $w.k -in $w.p.l2 -fill both -expand true
57 # Initialize some variables
61 set pi 3.1415926535897933
65 # This procedure makes the pendulum appear at the correct place on the
66 # canvas. If the additional arguments "at $x $y" are passed (the 'at'
67 # is really just syntactic sugar) instead of computing the position of
68 # the pendulum from the length of the pendulum rod and its angle, the
69 # length and angle are computed in reverse from the given location
70 # (which is taken to be the centre of the pendulum bob.)
71 proc showPendulum {canvas {at {}} {x {}} {y {}}} {
72 global Theta dTheta pi length home
73 if {$at eq "at" && ($x!=$home || $y!=25)} {
75 set x2 [expr {$x - $home}]
76 set y2 [expr {$y - 25}]
77 set length [expr {hypot($x2, $y2)}]
78 set Theta [expr {atan2($x2, $y2) * 180/$pi}]
80 set angle [expr {$Theta * $pi/180}]
81 set x [expr {$home + $length*sin($angle)}]
82 set y [expr {25 + $length*cos($angle)}]
84 $canvas coords rod $home 25 $x $y
86 [expr {$x-15}] [expr {$y-15}] [expr {$x+15}] [expr {$y+15}]
90 # Update the phase-space graph according to the current angle and the
91 # rate at which the angle is changing (the first derivative with
93 proc showPhase {canvas} {
94 global Theta dTheta points psw psh
95 lappend points [expr {$Theta+$psw}] [expr {-20*$dTheta+$psh}]
96 if {[llength $points] > 100} {
97 set points [lrange $points end-99 end]
99 for {set i 0} {$i<100} {incr i 10} {
100 set list [lrange $points end-[expr {$i-1}] end-[expr {$i-12}]]
101 if {[llength $list] >= 4} {
102 $canvas coords graph$i $list
107 # Set up some bindings on the canvases. Note that when the user
108 # clicks we stop the animation until they release the mouse
109 # button. Also note that both canvases are sensitive to <Configure>
110 # events, which allows them to find out when they have been resized by
112 bind $w.c <Destroy> {
113 after cancel $animationCallbacks(pendulum)
114 unset animationCallbacks(pendulum)
117 after cancel $animationCallbacks(pendulum)
118 showPendulum %W at %x %y
120 bind $w.c <B1-Motion> {
121 showPendulum %W at %x %y
123 bind $w.c <ButtonRelease-1> {
124 showPendulum %W at %x %y
125 set animationCallbacks(pendulum) [after 15 repeat [winfo toplevel %W]]
127 bind $w.c <Configure> {
128 %W coords plate 0 25 %w 25
130 %W coords pivot [expr $home-5] 20 [expr $home+5] 30
132 bind $w.k <Configure> {
135 %W coords x_axis 2 $psh [expr %w-2] $psh
136 %W coords y_axis $psw [expr %h-2] $psw 2
137 %W coords label_dtheta [expr $psw-4] 6
138 %W coords label_theta [expr %w-6] [expr $psh+4]
141 # This procedure is the "business" part of the simulation that does
142 # simple numerical integration of the formula for a simple rotational
144 proc recomputeAngle {} {
145 global Theta dTheta pi length
146 set scaling [expr {3000.0/$length/$length}]
148 # To estimate the integration accurately, we really need to
149 # compute the end-point of our time-step. But to do *that*, we
150 # need to estimate the integration accurately! So we try this
151 # technique, which is inaccurate, but better than doing it in a
152 # single step. What we really want is bound up in the
153 # differential equation:
155 # theta + theta = -----------
157 # But my math skills are not good enough to solve this!
160 set firstDDTheta [expr {-sin($Theta * $pi/180)*$scaling}]
161 set midDTheta [expr {$dTheta + $firstDDTheta}]
162 set midTheta [expr {$Theta + ($dTheta + $midDTheta)/2}]
164 set midDDTheta [expr {-sin($midTheta * $pi/180)*$scaling}]
165 set midDTheta [expr {$dTheta + ($firstDDTheta + $midDDTheta)/2}]
166 set midTheta [expr {$Theta + ($dTheta + $midDTheta)/2}]
167 # Now we do a double-estimate approach for getting the final value
169 set midDDTheta [expr {-sin($midTheta * $pi/180)*$scaling}]
170 set lastDTheta [expr {$midDTheta + $midDDTheta}]
171 set lastTheta [expr {$midTheta + ($midDTheta + $lastDTheta)/2}]
173 set lastDDTheta [expr {-sin($lastTheta * $pi/180)*$scaling}]
174 set lastDTheta [expr {$midDTheta + ($midDDTheta + $lastDDTheta)/2}]
175 set lastTheta [expr {$midTheta + ($midDTheta + $lastDTheta)/2}]
176 # Now put the values back in our globals
177 set dTheta $lastDTheta
181 # This method ties together the simulation engine and the graphical
182 # display code that visualizes it.
184 global animationCallbacks
193 # Reschedule ourselves
194 set animationCallbacks(pendulum) [after 15 [list repeat $w]]
196 # Start the simulation after a short pause
197 set animationCallbacks(pendulum) [after 500 [list repeat $w]]