from vpython import *
# Double pendulum
# The analysis is in terms of Lagrangian mechanics.
# The Lagrangian variables are angle of upper bar, angle of lower bar,
# measured from the vertical.
# Bruce Sherwood
# Corrections to the Lagrangian calculations by Owen Long, UC. Riverside
scene.width = scene.height = 600
scene.range = 1.8
scene.title = "A double pendulum"
def display_instructions():
s = "In VPython programs:\n"
s += " Rotate the camera by dragging with the right mouse button,\n or hold down the Ctrl key and drag.\n"
s += " To zoom, drag with the left+right mouse buttons,\n or hold down the Alt/Option key and drag,\n or use the mouse wheel.\n"
s += "Touch screen: pinch/extend to zoom, swipe or two-finger rotate."
scene.caption = s
# Display text below the 3D graphics:
display_instructions()
g = 9.8
M1 = 2.0
M2 = 1.0
d = 0.05 # thickness of each bar
gap = 2*d # distance between two parts of upper, U-shaped assembly
L1 = 0.5 # physical length of upper assembly; distance between axles
L1display = L1+d # show upper assembly a bit longer than physical, to overlap axle
L2 = 1 # physical length of lower bar
L2display = L2+d/2 # show lower bar a bit longer than physical, to overlap axle
# Coefficients used in Lagrangian calculation
A = (1/4)*M1*L1**2+(1/12)*M1*L1**2+M2*L1**2
B = (1/2)*M2*L1*L2
C = g*L1*(M1/2+M2)
D = M2*L1*L2/2
E = (1/12)*M2*L2**2+(1/4)*M2*L2**2
F = g*L2*M2/2
hpedestal = 1.3*(L1+L2) # height of pedestal
wpedestal = 0.1 # width of pedestal
tbase = 0.05 # thickness of base
wbase = 8*gap # width of base
offset = 2*gap # from center of pedestal to center of U-shaped upper assembly
pedestal_top = vec(0,hpedestal/2,0) # top of inner bar of U-shaped upper assembly
theta1 = 1.3*pi/2 # initial upper angle (from vertical)
theta1dot = 0 # initial rate of change of theta1
theta2 = 0 # initial lower angle (from vertical)
theta2dot = 0 # initial rate of change of theta2
pedestal = box( pos=pedestal_top-vec(0,hpedestal/2,offset),
size=vec(wpedestal,1.1*hpedestal,wpedestal),
color=vec(0.4,0.4,0.5) )
base = box( pos=pedestal_top-vec(0,hpedestal+tbase/2,offset),
size=vec(wbase,tbase,wbase),
color=pedestal.color )
axle1 = cylinder( pos=pedestal_top-vec(0,0,gap/2-d/4), axis=vec(0,0,-1),
size=vec(offset,d/4,d/4), color=color.yellow )
bar1 = box( pos=pedestal_top+vec(L1display/2-d/2,0,-(gap+d)/2),
size=vec(L1display,d,d), color=color.red )
bar1.rotate( angle=-pi/2, axis=vec(0,0,1), origin=vec(axle1.pos.x, axle1.pos.y, bar1.pos.z) )
bar1.rotate( angle=theta1, axis=vec(0,0,1), origin=vec(axle1.pos.x, axle1.pos.y, bar1.pos.z) )
bar1b = box( pos=pedestal_top+vec(L1display/2-d/2,0,(gap+d)/2),
size=vec(L1display,d,d), color=bar1.color )
bar1b.rotate( angle=-pi/2, axis=vec(0,0,1), origin=vec(axle1.pos.x, axle1.pos.y, bar1b.pos.z) )
bar1b.rotate( angle=theta1, axis=vec(0,0,1), origin=vec(axle1.pos.x, axle1.pos.y, bar1b.pos.z) )
pivot1 = vec(axle1.pos.x, axle1.pos.y, 0)
axle2 = cylinder( pos=pedestal_top+vec(L1,0,-(gap+d)/2), axis=vec(0,0,1),
size=vec(gap+d,axle1.size.y/2,axle1.size.y/2), color=axle1.color )
axle2.rotate( angle=-pi/2, axis=vec(0,0,1), origin=vec(axle1.pos.x, axle1.pos.y, axle2.pos.z) )
axle2.rotate( angle=theta1, axis=vec(0,0,1), origin=vec(axle1.pos.x, axle1.pos.y, axle2.pos.z) )
bar2 = box( pos=axle2.pos+vec(L2display/2-d/2,0,(gap+d)/2),
size=vec(L2display,d,d), color=color.green )
bar2.rotate( angle=-pi/2, axis=vec(0,0,1), origin=vec(axle2.pos.x, axle2.pos.y, bar2.pos.z) )
bar2.rotate( angle=theta2, axis=vec(0,0,1), origin=vec(axle2.pos.x, axle2.pos.y, bar2.pos.z) )
dt = 0.001
t = 0
while True:
rate(1/dt)
# Calculate accelerations of the Lagrangian coordinates=
atheta1 = ((E*C/B)*sin(theta1)-F*sin(theta2))/(D-E*A/B)
atheta2 = -(A*atheta1+C*sin(theta1))/B
# Update velocities of the Lagrangian coordinates=
theta1dot = theta1dot+atheta1*dt
theta2dot = theta2dot+atheta2*dt
# Update Lagrangian coordinates=
dtheta1 = theta1dot*dt
dtheta2 = theta2dot*dt
theta1 = theta1+dtheta1
theta2 = theta2+dtheta2
bar1.rotate( angle=dtheta1, axis=vec(0,0,1), origin=pivot1 )
bar1b.rotate( angle=dtheta1, axis=vec(0,0,1), origin=pivot1 )
pivot2 = vec(axle2.pos.x, axle2.pos.y, pivot1.z)
axle2.rotate( angle=dtheta1, axis=vec(0,0,1), origin=pivot1 )
bar2.rotate( angle=dtheta2, axis=vec(0,0,1), origin=pivot2 )
pivot2 = vec(axle2.pos.x, axle2.pos.y, pivot1.z)
bar2.pos = pivot2 + bar2.axis/2
t = t+dt