Answer to Question #170163 in Python for Jordan

Question

Explore the dependence of the period of a pendulum's motion as a function of its initial amplitude. Complete tasks inside the TODO #

import numpy as np

from scipy import interpolate

from scipy import integrate

import matplotlib.pyplot as plt

class pendulum:

  ''' defines a simple pendulum '''

  def __init__(self, pendulum_length, mass):

    # TODO: Assignment Task 1: write method body

    # End of Task 1; proceed to task 2.

  def set_g(self, g):

    ''' small g to be used in calculations '''

    self._g = g

  def dydt(self, y, t):

    """ Calculate the derivatives for a pendulum

    Parameters

    ----------

    y: array-like

      vector of unknowns for the ode equation at time t

    t: float

      time t

    Returns

    -------

    ret: numpy array of float

      the derivatives of y

    """

    # TODO: Assignment Task 2: write method body

    # End of Task 2; proceed to task 3.

  def period(self, maximum_amplitude):

    ''' Calculate the period of the periodic motion.

    For amplitude=0 the small amplitude analytical solution is returned.

    Otherwise the period is calculated by integrating the ODE using

    scipy.integrate.odeint() and determining the time between release at

    maximum amplitude and the first angle=0 crossing. The exact zero

    crossing is determined using scipy.interpolate.interp1d().

    In the calculation it is assumed that the period has a value between

    90% and 150% of the small amplitude period.

    Parameters

    ----------

    maximum_amplitude: float

      maximum amplitude of the periodic motion

    Returns

    -------

    p: float

      the period

    '''

    # TODO: Assignment Task 3: write method body

    # End of Task 3; no further tasks.

if __name__ == '__main__':

  #some test code

  pen = pendulum(1, 1)

  pen.set_g(9.81)

  #generate data for figure

  angles = np.linspace(0, np.pi/2, 31)

  period_ode = [pen.period(a) for a in angles]

  # generate a plot

  fig = plt.figure()

  ax = fig.add_subplot(1, 1, 1)

  ax.plot(angles, period_ode, 'b', label='generated using ode')

  ax.set_title('Period of a 1m Pendulum as a Function of Amplitude')

  ax.set_xlabel('amplitude/rad')

  ax.set_ylabel('period/s')

  ax.legend(loc='upper left')

  plt.show()

Answer 1

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





class pendulum:

  ''' defines a simple pendulum '''



  def __init__(self, pendulum_length, mass):




  def set_g(self, g):
      self._g = g





  def dydt(self, y, t):

    """ Calculate the derivatives for a pendulum



    Parameters

    ----------

    y: array-like

      vector of unknowns for the ode equation at time t



    t: float

      time t



    Returns

    -------

    ret: numpy array of float

      the derivatives of y

    """





  def period(self, maximum_amplitude):



    Parameters

    ----------

    maximum_amplitude: float

     



    Returns

    -------

    p: float

      the period

    '''

    # TODO: Assignment Task 3: write method body

    # End of Task 3; no further tasks.


if __name__ == '__main__':

  #some test code

  pen = pendulum(1, 1)

  pen.set_g(9.81)



  #generate data for figure

  angles = np.linspace(0, np.pi/2, 31)
  period_ode = [pen.period(a) for a in angles]
  fig = plt.figure()
  ax = fig.add_subplot(1, 1, 1)
  ax.plot(angles, period_ode, 'b', label='generated using ode')
  ax.set_title('Period of a 1m Pendulum as a Function of Amplitude')
  ax.set_xlabel('amplitude/rad')
  ax.set_ylabel('period/s')
  ax.legend(loc='upper left')
  plt.show()