Driven Harmonic Oscillator

Driven Harmonic Oscillator#

Author: Ludovic Charleux (ludovic.charleux@univ-smb.fr)

In this example, you will simulate an harmonic oscillator and compare the numerical solution to the closed form one.

Read about the theory of harmonic oscillators on Wikipedia

The case of the one dimensional mechanical oscillator leads to the following equation:

m \ddot{x} + μ \dot{x} + k x = m {\ddot{x}}_{d}

Where:

$x$ is the position,
$\dot{x}$ and $\ddot{x}$ are respectively the speed and acceleration,
$m$ is the mass,
$μ$ the
$k$ the stiffness,
and ${\ddot{x}}_{d}$ the driving acceleration which is null if the oscillator is free.

Most 1D oscilators follow the same canonical equation:

\ddot{x} + 2 ζ ω_{0} \dot{x} + ω_{0}^{2} x = {\ddot{x}}_{d}

Where:

In the case of the mechanical oscillator:

ω_{0} = \sqrt{\frac{k}{m}}

ζ = \frac{μ}{2 \sqrt{m k}}

omega0 = 2.0 * np.pi * 1.0
zeta = 0.1
ad = 1.0
omegad = 2.0 * np.pi * 1.2

First, you need to code: the harmonic oscillator oscillator using the standarde ODE formulation:

\dot{X} = f (X, t)

def f(X, t):
    """
    Driven Harmonic oscillator.
    """
    return

Solve the ODE using odeint and plot it.

Determine the steady state amplitude $a (ω_{d})$ of the oscillator.

Plot the evolution of the amplitude of the driven oscillator and compare it with the theory.