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# 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 # events, which allows them to find out when they have been resized by # the user. bind $w.c { after cancel $animationCallbacks(pendulum) unset animationCallbacks(pendulum) } bind $w.c { after cancel $animationCallbacks(pendulum) showPendulum %W at %x %y } bind $w.c { showPendulum %W at %x %y } bind $w.c { showPendulum %W at %x %y set animationCallbacks(pendulum) [after 15 repeat [winfo toplevel %W]] } bind $w.c { %W coords plate 0 25 %w 25 set home [expr {%w/2}] %W coords pivot [expr {$home - 5}] 20 [expr {$home + 5}] 30 } bind $w.k { 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]]