Oscillatory systems

Oscillatory systems are of particular interest in engineering. They’re commonly modelled with a second-order ODE, though can get complicated.

Our most prevalent examples are:

• Mass-spring system
• Pendulum
• RLC circuit

The number of variables that a system can displace with respect to is the number of degrees of freedom.

Terminology

For a homogeneous ODE:

$$a{x}^{\prime \prime }+b{x}^{\prime }+cx=0$$

We have two parameters, ${\omega }_0>0$ and $\zeta \ge 0$, where:

$${\omega }_0^2=\frac{c}{a}$$ $$\zeta =\sqrt{\frac{b^2}{4ac}}$$

So we can substitute into the ODE to get a standard form:

$$x^{\prime \prime }+2\zeta {\omega }_0{x}^{\prime }+{\omega }_0^{2}x=0$$

and characteristic equation with roots:

$$(s_1,s_2)={\omega }_0(-\zeta \pm \sqrt{{\zeta }^2-1})$$

The parameter ${\omega }_0$ is called the undamped natural frequency. $\zeta$ is called the damping ratio with three distinct cases:

• Two distinct real roots, overdamped system when $\zeta >1$.
• Repeated real roots, critically damped, when $s=-{\omega }_0$ and $\zeta =1$.
• Two complex roots, underdamped when $\zeta <1$.