python - Solving a system of mass, spring, damper and Coulomb friction

Question

Consider the system below:

Fig.1 - Mass, spring, damper and Coulomb frction (image courtesy of Wikimedia).

with a dynamic equation of:

where Ff is the Amontons-Columb friction defined as:

and consequently, the no-slip condition is defined as

Following this example, I have a vague code in mind which I don't know how to complete:

from scipy.integrate import odeint
import numpy as np

m = 1.0
k = 2.0
c = 0.1
mus = 0.3
muk = 0.2
g = 9.8
vf = 0.01

def eq(X, t, Xi):
  Ff = k * (Xi[0] - X[0]) + c * (Xi[1] - X[1]) # - m * dydt

  if np.abs(X[1]) < vf and np.abs(Ff) < mus * m * g :
    Ff = k * (Xi[0] - X[0]) + c * (Xi[1] - X[1]) # - m * dydt
  else:
    Ff = -np.sign(X[1]) * muk * m * g
    pass

  dxdt = X[1]
  dydt = (k * (Xi[0] - X[0]) + c * (Xi[1] - X[1]) - Ff) / m
  return [dxdt, dydt]

t = np.linspace(0, 10, 1000)
Xi0 = np.piecewise(t, [t < 1, t >= 1], [0, 1])
X0 = [0, 0]
sol = odeint(eq, X0, t)

where Xi0 is a step function. My main issue is that when I want to define Ff it depends on dydt which is to be defined later in that scope!

I would appreciate if you could help me know what is the most canonical way to numerically solve this system. Thanks in advance.

Answer 1

I'm gonna put a simplified/temporary solution here till someone comes with a better one:

from scipy.integrate import odeint
import numpy as np
import matplotlib.pyplot as plt

m = 0.2
k = 2.0
c = 0.1
mus = 0.3
muk = 0.2
g = 9.8
vf = 0.01
v0 = 0.0
t1 = 10
sign = lambda x: np.tanh(100*x)

def Xi(t):
  if t < 1 :
    return 0
  else:
    return 1

vXi = np.vectorize(Xi)

def eq(X, t, Xi):
  Ff = k * (Xi(t) - X[0])

  if np.abs(X[1]) < vf and np.abs(Ff) < mus * m * g :
    Ff = k * (Xi(t) - X[0])
  else:
    Ff = sign(X[1]) * muk * m * g

  d2x = (k * (Xi(t) - X[0]) - Ff) / m
  return [X[1], d2x]

t = np.linspace(0, t1, 1000)
X0 = [v0, 0]
sol = odeint(func = eq, y0 = X0, t = t, args = (Xi, ), mxstep = 50000, atol = 1e-5)

plt.plot(t, sol[:, 0], 'r-', label = 'Output (x(t))')
plt.plot(t, vXi(t), 'b-', label = 'Input (xi(t))')
plt.ylabel('values')
plt.xlabel('time')
plt.legend(loc='best')
plt.show()

and the result is:

I used this, this and this posts to write the code. I ignored damping and inertia to simplify the problem.