pendulum_sympy.ipynb

Kernel: Python 3

Modeling and Simulation in Python

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International

import numpy as np

from sympy import *

init_printing()

This notebook uses SymPy to derive the equations of motion for a springy pendulum and a rigid pendulum.

Here are the symbols we need:

x, y, vx, vy, ax, ay = symbols('x, y, vx, vy, ax, ay')
length0, k, m, g, R, t = symbols('length0, k, m, g, R, t')

We'll use vectors P, V, and A to represent position, velocity and acceleration. The Vector class in modsim.py doesn't play nicely with SymPy, so I'll use NumPy arrays instead:

P = np.array([x, y])
P

array([x, y], dtype=object)

V = np.array([vx, vy])
V

array([vx, vy], dtype=object)

A = np.array([ax, ay])
A

array([ax, ay], dtype=object)

The only vector operations we need are mag and hat:

def mag(P):
    return sqrt(P[0]**2 + P[1]**2)

mag(P)

$$\sqrt{x^{2} + y^{2}}$$

def hat(P):
    return P / mag(P)

hat(P)

array([x/sqrt(x**2 + y**2), y/sqrt(x**2 + y**2)], dtype=object)

For convenience, I'll define intermediate variables like length:

length = mag(P)
length

$$\sqrt{x^{2} + y^{2}}$$

f_spring is the force on the particle due to the spring

f_spring = -k * (length - length0) * hat(P)
f_spring

array([-k*x*(-length0 + sqrt(x**2 + y**2))/sqrt(x**2 + y**2),
       -k*y*(-length0 + sqrt(x**2 + y**2))/sqrt(x**2 + y**2)], dtype=object)

xhat and yhat are unit vectors along the x and y axes:

xhat = np.array([1, 0])
yhat = np.array([0, 1])

Now I can write force due to gravity as a Vector

f_grav = -m * g * yhat
f_grav

array([0, -g*m], dtype=object)

To write $\sum F = ma$, I'll define the left-hand side and right-hand sides of the equations separately.

lhs = f_spring + f_grav

rhs = m * A

Now I can make two equations, one for each component of F and A:

eq1 = Eq(lhs[0], rhs[0])
eq1

$$- \frac{k x}{\sqrt{x^{2} + y^{2}}} \left(- length_{0} + \sqrt{x^{2} + y^{2}}\right) = ax m$$

eq2 = Eq(lhs[1], rhs[1])
eq2

$$- g m - \frac{k y}{\sqrt{x^{2} + y^{2}}} \left(- length_{0} + \sqrt{x^{2} + y^{2}}\right) = ay m$$

Now I want equations that are explicit in ax and ay. In this case I can get them easily by dividing through by m. But for the rigid pendulum we will need to use solve, so I want to demonstrate that here.

soln = solve([eq1, eq2], [ax, ay])

Now we can extract the expressions for ax and ay

soln[ax]

$$\frac{k length_{0} x}{m \sqrt{x^{2} + y^{2}}} - \frac{k x}{m}$$

soln[ay]

$$- g + \frac{k length_{0} y}{m \sqrt{x^{2} + y^{2}}} - \frac{k y}{m}$$

And we can get SymPy to format the result for LaTeX:

print(latex(soln[ax]))

\frac{k length_{0} x}{m \sqrt{x^{2} + y^{2}}} - \frac{k x}{m}

print(latex(soln[ay]))

- g + \frac{k length_{0} y}{m \sqrt{x^{2} + y^{2}}} - \frac{k y}{m}

Or generate Python code we can paste into a slope function:

print(python(soln[ax]))

k = Symbol('k')
length0 = Symbol('length0')
x = Symbol('x')
m = Symbol('m')
y = Symbol('y')
e = k*length0*x/(m*sqrt(x**2 + y**2)) - k*x/m

print(python(soln[ay]))

g = Symbol('g')
k = Symbol('k')
length0 = Symbol('length0')
y = Symbol('y')
m = Symbol('m')
x = Symbol('x')
e = -g + k*length0*y/(m*sqrt(x**2 + y**2)) - k*y/m

To see these equations run, see pendulum.ipynb

Rigid pendulum

Solving the rigid pendulum is almost the same, except we need a third equation to represent the geometric constraint. The simplest form of the constraint is:

$x^2 + y ^2 = length$

But this equation doesn't involve vx, vy, ax, and ay, so it's not much help. However, if we take the time derivative of both sides, we get

$2 x vx + 2 y vy = 0$

And if we take the time derivative one more time (and divide through by 2), we get

$x ax + y ay + vx^2 + vy^2 = 0$

And that's just what we need.

eq3 = Eq(x*ax + y*ay + vx**2 + vy**2, 0)
eq3

$$ax x + ay y + vx^{2} + vy^{2} = 0$$

Now we can represent the force due to tension as a vector with unknown magnitude, R, and direction opposite P.

f_tension = -R * hat(P)

Again, we can write $\sum F = ma$

lhs = f_grav + f_tension

rhs = m * A

And make one equation for each dimension:

eq4 = Eq(lhs[0], rhs[0])
eq4

$$- \frac{R x}{\sqrt{x^{2} + y^{2}}} = ax m$$

eq5 = Eq(lhs[1], rhs[1])
eq5

$$- \frac{R y}{\sqrt{x^{2} + y^{2}}} - g m = ay m$$

Now we have three equations in three unknowns:

soln = solve([eq3, eq4, eq5], [ax, ay, R])

And we can get explicit expressions for ax and ay

soln[ax]

$$\frac{x}{x^{2} + y^{2}} \left(g y - vx^{2} - vy^{2}\right)$$

soln[ay]

$$- \frac{1}{x^{2} + y^{2}} \left(g x^{2} + y \left(vx^{2} + vy^{2}\right)\right)$$

Again, we can get the results in LaTeX or Python.

print(latex(soln[ax]))

\frac{x}{x^{2} + y^{2}} \left(g y - vx^{2} - vy^{2}\right)

print(latex(soln[ay]))

- \frac{1}{x^{2} + y^{2}} \left(g x^{2} + y \left(vx^{2} + vy^{2}\right)\right)

To see these equations run, see pendulum2.ipynb

print(python(soln[ax]))

x = Symbol('x')
g = Symbol('g')
y = Symbol('y')
vx = Symbol('vx')
vy = Symbol('vy')
e = x*(g*y - vx**2 - vy**2)/(x**2 + y**2)

print(python(soln[ay]))

g = Symbol('g')
x = Symbol('x')
y = Symbol('y')
vx = Symbol('vx')
vy = Symbol('vy')
e = -(g*x**2 + y*(vx**2 + vy**2))/(x**2 + y**2)