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# Configure Jupyter so figures appear in the notebook
%matplotlib inline
# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'
# import functions from the modsim.py module
from modsim import *
Tables in Chinese restaurants often have a rotating tray or turntable that makes it easy for customers to share dishes. These turntables are supported by low-friction bearings that allow them to turn easily and glide. However, they can be heavy, especially when they are loaded with food, so they have a high moment of inertia.
Suppose I am sitting at a table with a pot of tea on the turntable directly in front of me, and the person sitting directly opposite asks me to pass the tea. I push on the edge of the turntable with 1 Newton of force until it has turned 0.5 radians, then let go. The turntable glides until it comes to a stop 1.5 radians from the starting position. How much force should I apply for a second push so the teapot glides to a stop directly opposite me?
The following figure shows the scenario, where F is the force I apply to the turntable at the perimeter, perpendicular to the moment arm, r, and tau is the resulting torque. The blue circle near the bottom is the teapot.
We'll answer this question in these steps:
We'll use the results from the first push to estimate the coefficient of friction for the turntable.
We'll use that coefficient of friction to estimate the force needed to rotate the turntable through the remaining angle.
Our simulation will use the following parameters:
The radius of the turntable is 0.5 meters, and its weight is 7 kg.
The teapot weights 0.3 kg, and it sits 0.4 meters from the center of the turntable.
As usual, I'll get units from Pint.
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radian = UNITS.radian
m = UNITS.meter
s = UNITS.second
kg = UNITS.kilogram
N = UNITS.newton
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And store the parameters in a Params object.
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params = Params(radius_disk=0.5*m,
mass_disk=7*kg,
radius_pot=0.4*m,
mass_pot=0.3*kg,
force=1*N,
torque_friction=0.2*N*m,
theta_end=0.5*radian,
t_end=20*s)
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make_system creates the initial state, init, and computes the total moment of inertia for the turntable and the teapot.
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def make_system(params):
"""Make a system object.
params: Params object
returns: System object
"""
mass_disk, mass_pot = params.mass_disk, params.mass_pot
radius_disk, radius_pot = params.radius_disk, params.radius_pot
init = State(theta=0*radian, omega=0*radian/s)
I_disk = mass_disk * radius_disk**2 / 2
I_pot = mass_pot * radius_pot**2
return System(params, init=init, I=I_disk+I_pot)
Here's the System object we'll use for the first phase of the simulation, while I am pushing the turntable.
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system1 = make_system(params)
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When I stop pushing on the turntable, the angular acceleration changes abruptly. We could implement the slope function with an if statement that checks the value of theta and sets force accordingly. And for a coarse model like this one, that might be fine. But we will get more accurate results if we simulate the system in two phases:
During the first phase, force is constant, and we run until theta is 0.5 radians.
During the second phase, force is 0, and we run until omega is 0.
Then we can combine the results of the two phases into a single TimeFrame.
Here's the slope function we'll use:
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def slope_func(state, t, system):
"""Computes the derivatives of the state variables.
state: State object
t: time
system: System object
returns: sequence of derivatives
"""
theta, omega = state
radius_disk, force = system.radius_disk, system.force
torque_friction, I = system.torque_friction, system.I
torque = radius_disk * force - torque_friction
alpha = torque / I
return omega, alpha
As always, we'll test the slope function before running the simulation.
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slope_func(system1.init, 0, system1)
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Here's an event function that stops the simulation when theta reaches theta_end.
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def event_func1(state, t, system):
"""Stops when theta reaches theta_end.
state: State object
t: time
system: System object
returns: difference from target
"""
theta, omega = state
return theta - system.theta_end
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event_func1(system1.init, 0, system1)
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Now we can run the first phase.
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results1, details1 = run_ode_solver(system1, slope_func, events=event_func1)
details1
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And look at the results.
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results1.tail()
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t_0 = results1.last_label() * s
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And make an initial State object for Phase 2.
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init2 = results1.last_row()
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And a new System object with zero force.
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system2 = System(system1, t_0=t_0, init=init2, force=0*N)
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Here's an event function that stops when angular velocity is 0.
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def event_func2(state, t, system):
"""Stops when omega is 0.
state: State object
t: time
system: System object
returns: omega
"""
theta, omega = state
return omega
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event_func2(system2.init, 0, system2)
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Now we can run the second phase.
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slope_func(system2.init, system2.t_0, system2)
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results2, details2 = run_ode_solver(system2, slope_func, events=event_func2)
details2
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And check the results.
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results2.tail()
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Pandas provides combine_first, which combines results1 and results2.
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results = results1.combine_first(results2)
results.tail()
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Now we can plot theta for both phases.
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def plot_theta(results):
plot(results.theta, label='theta')
decorate(xlabel='Time (s)',
ylabel='Angle (rad)')
plot_theta(results)
And omega.
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def plot_omega(results):
plot(results.omega, label='omega', color='C1')
decorate(xlabel='Time (s)',
ylabel='Angular velocity (rad/s)')
plot_omega(results)
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subplot(2, 1, 1)
plot_theta(results)
subplot(2, 1, 2)
plot_omega(results)
savefig('figs/chap25-fig01.pdf')
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def run_two_phases(force, torque_friction, params):
"""Run both phases.
force: force applied to the turntable
torque_friction: friction due to torque
params: Params object
returns: TimeFrame of simulation results
"""
# put the specified parameters into the Params object
params = Params(params, force=force, torque_friction=torque_friction)
# run phase 1
system1 = make_system(params)
results1, details1 = run_ode_solver(system1, slope_func,
events=event_func1)
# get the final state from phase 1
t_0 = results1.last_label() * s
init2 = results1.last_row()
# run phase 2
system2 = System(system1, t_0=t_0, init=init2, force=0*N)
results2, details2 = run_ode_solver(system2, slope_func,
events=event_func2)
# combine and return the results
results = results1.combine_first(results2)
return TimeFrame(results)
Let's test it with the same parameters.
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force = 1*N
torque_friction = 0.2*N*m
results = run_two_phases(force, torque_friction, params)
results.tail()
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And check the results.
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theta_final = results.last_row().theta
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Here's the error function we'll use with root_bisect.
It takes a hypothetical value for torque_friction and returns the difference between theta_final and the observed duration of the first push, 1.5 radian.
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def error_func1(torque_friction, params):
"""Error function for root_scalar.
torque_friction: hypothetical value
params: Params object
returns: offset from target value
"""
force = 1 * N
results = run_two_phases(force, torque_friction, params)
theta_final = results.last_row().theta
print(torque_friction, theta_final)
return theta_final - 1.5 * radian
Testing the error function.
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guess1 = 0.1*N*m
error_func1(guess1, params)
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guess2 = 0.3*N*m
error_func1(guess2, params)
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And running root_scalar.
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res = root_bisect(error_func1, [guess1, guess2], params)
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The result is the coefficient of friction that yields a total rotation of 1.5 radian.
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torque_friction = res.root
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Here's a test run with the estimated value.
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force = 1 * N
results = run_two_phases(force, torque_friction, params)
theta_final = get_last_value(results.theta)
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Looks good.
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from matplotlib.patches import Circle
from matplotlib.patches import Arrow
def draw_func(state, t):
theta, omega = state
# draw a circle for the table
radius_disk = magnitude(params.radius_disk)
circle1 = Circle([0, 0], radius_disk)
plt.gca().add_patch(circle1)
# draw a circle for the teapot
radius_pot = magnitude(params.radius_pot)
center = pol2cart(theta, radius_pot)
circle2 = Circle(center, 0.05, color='C1')
plt.gca().add_patch(circle2)
# make the aspect ratio 1
plt.axis('equal')
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state = results.first_row()
draw_func(state, 0)
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animate(results, draw_func)
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# Solution
def error_func2(force, params):
"""Error function for root_scalar.
force: hypothetical value
params: Params object
returns: offset from target value
"""
# notice that this function uses the global value of torque_friction
results = run_two_phases(force, torque_friction, params)
theta_final = get_last_value(results.theta)
print(force, theta_final)
remaining_angle = np.pi - 1.5
return theta_final - remaining_angle * radian
Test the error function with force=1
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# Solution
guess1 = 0.5 * N
error_func2(guess1, params)
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# Solution
guess2 = 2 * N
error_func2(guess2, params)
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And run root_bisect to find the desired force.
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# Solution
res = root_bisect(error_func2, [guess1, guess2], params)
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force = res.root
results = run_two_phases(force, torque_friction, params)
theta_final = get_last_value(results.theta)
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remaining_angle = np.pi - 1.5
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Exercise: Now suppose my friend pours 0.1 kg of tea and puts the teapot back on the turntable at distance 0.3 meters from the center. If I ask for the tea back, how much force should they apply, over an arc of 0.5 radians, to make the teapot glide to a stop back in front of me? You can assume that torque due to friction is proportional to the total mass of the teapot and the turntable.
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# Solution
mass_before = params.mass_pot + params.mass_disk
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# Solution
params2 = Params(params, mass_pot=0.2*kg, radius_pot=0.3*m)
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# Solution
mass_after = params2.mass_pot + params2.mass_disk
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# Solution
torque_friction
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# Solution
torque_friction2 = torque_friction * mass_after / mass_before
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# Solution
guess = 2 * N
results = run_two_phases(guess, torque_friction2, params2)
results.tail()
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# Solution
subplot(2, 1, 1)
plot_theta(results)
subplot(2, 1, 2)
plot_omega(results)
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# Solution
# Note: this is so similar to error_func2, it would be better
# to generalize it, but for expediency, I will make a modified
# verison.
def error_func3(force, params):
"""Error function for root_scalar.
force: hypothetical value
params: Params object
returns: offset from target value
"""
# notice that this function uses the global value of torque_friction2
results = run_two_phases(force, torque_friction2, params)
theta_final = get_last_value(results.theta)
print(force, theta_final)
remaining_angle = np.pi * radian
return theta_final - remaining_angle
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# Solution
guess1 = 1 * N
error_func3(guess1, params)
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# Solution
guess2 = 3 * N
error_func3(guess2, params)
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# Solution
res = root_bisect(error_func3, [guess1, guess2], params2)
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# Solution
error_func3(res.root, params)
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