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I have a question about the damped pendulum. I am working on an exercise where I have already numerically approximated the solution for a simple pendulum without dampening. Now, the excercise says that I can simply change the code of this simple situation to describe a pendulum with dampening.

For the simple pendulum without dampening I have used Eulers method of approximating the differential equation. Here is the code for the pendulum without damping.

Code:

```
import numpy as np
from math import *
import matplotlib.pyplot as plt
def simplePendulumSimulation(theta0,omega0,tau,m,g,l,numSteps,plotFlag):
# This function simulates a simple pendulum using the Euler-Cromer method.
# Inputs: theta0 (the initial angle, in rad)
# omega0 (the initial angular velocity, in rad/s)
# tau (the time step)
# m (mass of the pendulum)
# g (gravitational constant)
# l (length of the pendulum)
# numSteps (number of time steps to take, in s)
# plotFlag (set to 1 if you want plots, 0 otherwise)
# Outputs: t_vec (the time vector)
# theta_vec (the angle vector)
# initialize vectors
time_vec = [0]*numSteps
theta_vec = [0]*numSteps
omega_vec = [0]*numSteps
KE_vec = [0]*numSteps
PE_vec = [0]*numSteps
# set initial conditions
theta = theta0
omega = omega0
time = 0
# begin time stepping
for i in range(0,numSteps):
omega_old = omega
theta_old = theta
# update the values
omega = omega_old - (g/l)*np.sin(theta_old)*tau
theta = theta_old + omega*tau
# record the values
time_vec[i] = tau*i
theta_vec[i] = theta
omega_vec[i] = omega
KE_vec[i] = (1/2)*m*l**2*omega**2
PE_vec[i] = m*g*l*(1-np.cos(theta))
TE_vec = np.add(KE_vec,PE_vec)
# make graphs
if plotFlag == 1:
plt.figure(0)
plt.plot(time_vec,theta_vec)
plt.xlabel('Time (s)')
plt.ylabel('Displacement (rad)')
plt.savefig('plot1.png', bbox_inches='tight')
plt.figure(1)
plt.plot(time_vec,KE_vec,label='Kinetic Energy')
plt.plot(time_vec,PE_vec,label='Potential Energy')
plt.plot(time_vec,TE_vec,label='Total Energy')
plt.legend(loc='upper left')
plt.xlabel('Time (s)')
plt.ylabel('Energy (J)')
plt.savefig('plot2.png', bbox_inches='tight')
plt.figure(2)
plt.plot(theta_vec,omega_vec)
plt.xlabel('Displacement (rad)')
plt.ylabel('Velocity (rad/s)')
plt.savefig('plot3.png', bbox_inches='tight')
plt.show()
# return the vectors
simplePendulumSimulation(0.2,0,0.03,1,1,1,3000,1)
```

I was wondering if anyone could tell me if there is a simple way to change this code to introduce damping instead of beginning al over again. Thanks in advance.