Single Degree of Freedom Oscillator

TL;DR: You can find the source code HERE

This post will describe a numerical solving procedure in time-domain of a forced vibration from an harmonic excitation. I will keep the theory as short as possible, since there are plenty of other resources covering that. For example see HERE or HERE.

Figure 1: Single Degree of Freedom Oscillator

The equation of motion shown in figure 1 is given as follows:

$m\ddot{x} + d\dot{x} + kx + F_\mu - F(t) = 0$

This equation is an ordinary differential equation (ODE) of second order. To solve this equation numerically, we will write it as a system of first order ODEs.

First, we introduce a new variable:
$y1 = x1'$
$y1$ is the velocity of the mass shown in figure 1.

Taking the new variable into account with the equation of motion results to:
$y1' = (-d \cdot y1 -k \cdot x1 - F_\mu(y1) + \rho \cdot sin(\omega t)) / m$
$y1'$ is the acceleration of the mass shown in figure 1.

These equations can be written in matrix and vector shape:

$\left[ \begin{array}{c} y1 \\ y1' \\ \end{array}\right] = \left[ \begin{array}{cc} 0 & 1 \\ -k/m & -d/m \\ \end{array}\right] \cdot \left[ \begin{array}{c} x1 \\ y1 \\ \end{array}\right] + \left[ \begin{array}{c} 0 \\ 1/m \\ \end{array}\right] \cdot \left[ \begin{array}{c} \rho \cdot sin(\omega t) \\ \end{array}\right] + \left[ \begin{array}{c} 0 \\ 1/m \\ \end{array}\right] \cdot \left[ \begin{array}{c} F_\mu(y1) \\ \end{array}\right]$

Now we can start to solve this system with the Python library Scipy!
We excite our system with its natural frequency…
… without friction
… with friction
and get: