197 lines
7.4 KiB
Tcl
197 lines
7.4 KiB
Tcl
# pendulum.tcl --
|
|
#
|
|
# This demonstration illustrates how Tcl/Tk can be used to construct
|
|
# simulations of physical systems.
|
|
|
|
if {![info exists widgetDemo]} {
|
|
error "This script should be run from the \"widget\" demo."
|
|
}
|
|
|
|
package require Tk
|
|
|
|
set w .pendulum
|
|
catch {destroy $w}
|
|
toplevel $w
|
|
wm title $w "Pendulum Animation Demonstration"
|
|
wm iconname $w "pendulum"
|
|
positionWindow $w
|
|
|
|
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."
|
|
pack $w.msg
|
|
|
|
## See Code / Dismiss buttons
|
|
set btns [addSeeDismiss $w.buttons $w]
|
|
pack $btns -side bottom -fill x
|
|
|
|
# Create some structural widgets
|
|
pack [panedwindow $w.p] -fill both -expand 1
|
|
$w.p add [labelframe $w.p.l1 -text "Pendulum Simulation"]
|
|
$w.p add [labelframe $w.p.l2 -text "Phase Space"]
|
|
|
|
# Create the canvas containing the graphical representation of the
|
|
# simulated system.
|
|
canvas $w.c -width 320 -height 200 -background white -bd 2 -relief sunken
|
|
$w.c create text 5 5 -anchor nw -text "Click to Adjust Bob Start Position"
|
|
# Coordinates of these items don't matter; they will be set properly below
|
|
$w.c create line 0 25 320 25 -tags plate -fill grey50 -width 2
|
|
$w.c create oval 155 20 165 30 -tags pivot -fill grey50 -outline {}
|
|
$w.c create line 1 1 1 1 -tags rod -fill black -width 3
|
|
$w.c create oval 1 1 2 2 -tags bob -fill yellow -outline black
|
|
pack $w.c -in $w.p.l1 -fill both -expand true
|
|
|
|
# Create the canvas containing the phase space graph; this consists of
|
|
# a line that gets gradually paler as it ages, which is an extremely
|
|
# effective visual trick.
|
|
canvas $w.k -width 320 -height 200 -background white -bd 2 -relief sunken
|
|
$w.k create line 160 200 160 0 -fill grey75 -arrow last -tags y_axis
|
|
$w.k create line 0 100 320 100 -fill grey75 -arrow last -tags x_axis
|
|
for {set i 90} {$i>=0} {incr i -10} {
|
|
# Coordinates of these items don't matter; they will be set properly below
|
|
$w.k create line 0 0 1 1 -smooth true -tags graph$i -fill grey$i
|
|
}
|
|
|
|
$w.k create text 0 0 -anchor ne -text "\u03b8" -tags label_theta
|
|
$w.k create text 0 0 -anchor ne -text "\u03b4\u03b8" -tags label_dtheta
|
|
pack $w.k -in $w.p.l2 -fill both -expand true
|
|
|
|
# Initialize some variables
|
|
set points {}
|
|
set Theta 45.0
|
|
set dTheta 0.0
|
|
set pi 3.1415926535897933
|
|
set length 150
|
|
set home 160
|
|
|
|
# This procedure makes the pendulum appear at the correct place on the
|
|
# canvas. If the additional arguments "at $x $y" are passed (the 'at'
|
|
# is really just syntactic sugar) instead of computing the position of
|
|
# the pendulum from the length of the pendulum rod and its angle, the
|
|
# length and angle are computed in reverse from the given location
|
|
# (which is taken to be the centre of the pendulum bob.)
|
|
proc showPendulum {canvas {at {}} {x {}} {y {}}} {
|
|
global Theta dTheta pi length home
|
|
if {$at eq "at" && ($x!=$home || $y!=25)} {
|
|
set dTheta 0.0
|
|
set x2 [expr {$x - $home}]
|
|
set y2 [expr {$y - 25}]
|
|
set length [expr {hypot($x2, $y2)}]
|
|
set Theta [expr {atan2($x2, $y2) * 180/$pi}]
|
|
} else {
|
|
set angle [expr {$Theta * $pi/180}]
|
|
set x [expr {$home + $length*sin($angle)}]
|
|
set y [expr {25 + $length*cos($angle)}]
|
|
}
|
|
$canvas coords rod $home 25 $x $y
|
|
$canvas coords bob \
|
|
[expr {$x-15}] [expr {$y-15}] [expr {$x+15}] [expr {$y+15}]
|
|
}
|
|
showPendulum $w.c
|
|
|
|
# Update the phase-space graph according to the current angle and the
|
|
# rate at which the angle is changing (the first derivative with
|
|
# respect to time.)
|
|
proc showPhase {canvas} {
|
|
global Theta dTheta points psw psh
|
|
lappend points [expr {$Theta+$psw}] [expr {-20*$dTheta+$psh}]
|
|
if {[llength $points] > 100} {
|
|
set points [lrange $points end-99 end]
|
|
}
|
|
for {set i 0} {$i<100} {incr i 10} {
|
|
set list [lrange $points end-[expr {$i-1}] end-[expr {$i-12}]]
|
|
if {[llength $list] >= 4} {
|
|
$canvas coords graph$i $list
|
|
}
|
|
}
|
|
}
|
|
|
|
# Set up some bindings on the canvases. Note that when the user
|
|
# clicks we stop the animation until they release the mouse
|
|
# button. Also note that both canvases are sensitive to <Configure>
|
|
# events, which allows them to find out when they have been resized by
|
|
# the user.
|
|
bind $w.c <Destroy> {
|
|
after cancel $animationCallbacks(pendulum)
|
|
unset animationCallbacks(pendulum)
|
|
}
|
|
bind $w.c <1> {
|
|
after cancel $animationCallbacks(pendulum)
|
|
showPendulum %W at %x %y
|
|
}
|
|
bind $w.c <B1-Motion> {
|
|
showPendulum %W at %x %y
|
|
}
|
|
bind $w.c <ButtonRelease-1> {
|
|
showPendulum %W at %x %y
|
|
set animationCallbacks(pendulum) [after 15 repeat [winfo toplevel %W]]
|
|
}
|
|
bind $w.c <Configure> {
|
|
%W coords plate 0 25 %w 25
|
|
set home [expr %w/2]
|
|
%W coords pivot [expr $home-5] 20 [expr $home+5] 30
|
|
}
|
|
bind $w.k <Configure> {
|
|
set psh [expr %h/2]
|
|
set psw [expr %w/2]
|
|
%W coords x_axis 2 $psh [expr %w-2] $psh
|
|
%W coords y_axis $psw [expr %h-2] $psw 2
|
|
%W coords label_dtheta [expr $psw-4] 6
|
|
%W coords label_theta [expr %w-6] [expr $psh+4]
|
|
}
|
|
|
|
# This procedure is the "business" part of the simulation that does
|
|
# simple numerical integration of the formula for a simple rotational
|
|
# pendulum.
|
|
proc recomputeAngle {} {
|
|
global Theta dTheta pi length
|
|
set scaling [expr {3000.0/$length/$length}]
|
|
|
|
# To estimate the integration accurately, we really need to
|
|
# compute the end-point of our time-step. But to do *that*, we
|
|
# need to estimate the integration accurately! So we try this
|
|
# technique, which is inaccurate, but better than doing it in a
|
|
# single step. What we really want is bound up in the
|
|
# differential equation:
|
|
# .. - sin theta
|
|
# theta + theta = -----------
|
|
# length
|
|
# But my math skills are not good enough to solve this!
|
|
|
|
# first estimate
|
|
set firstDDTheta [expr {-sin($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
|
|
}
|
|
|
|
# This method ties together the simulation engine and the graphical
|
|
# display code that visualizes it.
|
|
proc repeat w {
|
|
global animationCallbacks
|
|
|
|
# Simulate
|
|
recomputeAngle
|
|
|
|
# Update the display
|
|
showPendulum $w.c
|
|
showPhase $w.k
|
|
|
|
# Reschedule ourselves
|
|
set animationCallbacks(pendulum) [after 15 [list repeat $w]]
|
|
}
|
|
# Start the simulation after a short pause
|
|
set animationCallbacks(pendulum) [after 500 [list repeat $w]]
|