HEV-Example 2 - Determine the period of beating for spring-mass system

A spring-mass system is subjected to a harmonic force whose frequency is close to the natural frequency of the system. If the forcing frequency is 39.8 [Hz] and the natural frequency is 40 [Hz], determine the period of beating.
The forcing frequency is 39.8 [Hz] which will be defined as: $$\omega = 2\pi39.8$$ The natural frequency is 40 [Hz] which will be defined as: $$\omega_n = 2\pi40$$ The period of beating will be calcualted using formula: $$\tau_d = \frac{2\pi}{(\omega_n - \omega)}$$ $$\tau_d = \frac{2\pi}{2\pi(40 - 39.8)} = 5 [\mathrm{s}]$$

Solution in Python

To solve this example in Python first we will need to import the numpy library.

import numpy as np

This library is required for $\pi$ number. The math library can also be used. The first step is to define the forcing frequency and the natural frequency.

omega = 39.8
omega_n = 40

The period of beating can now be easily calculated:

tau_d = (2*np.pi)/(2*np.pi*(omega_n - omega))
print("tau_d = {}".format(tau_d))

The output of the previous code is:

taud=4.9999 

If we want the result to be rounded we will use the Python built-in round function in taud formula.

tau_d = round((2*np.pi)/(2*np.pi*(omega_n - omega)),1)
print("tau_d = {}".format(tau_d))

The output is given below.

tau_d = 5