2.1.1. Simple Oscillation Model

The simplest model of an oscillator of order 2 is with single degree of freedom (SDOF). It consists of a constant mass *m* and a massless damper with a linear viscous damping constant *c*. The stiffness of spring is denoted by spring constant *k*. That SDOF mass-spring system is described by

$$\begin{cases} \begin{array}{c} m \frac{d^2 q(t)}{dt^2} + c \frac{dq(t)}{dt} + kq(t) = e(t) \\\\ q(0) = q\_0, q'(0) = v\_0 \end{array} \end{cases} \tag{1}$$

where *e*(*t*) is the forcing function. The solution *q*(*t*) may be the displacement in mechanical engineering [1–7] or current in electronics engineering [8].

In physics and engineering, for facilitating the analysis, one usually rewrites (1) by

$$\begin{cases} \frac{d^2q(t)}{dt^2} + \frac{c}{m} \frac{dq(t)}{dt} + \frac{k}{m}q(t) = \frac{c(t)}{m} \\\\ q(0) = q\_{0\prime}q'(0) = v\_{0\prime} \end{cases} \tag{2}$$

and further rewrites it by

$$\begin{cases} \frac{d^2q(t)}{dt^2} + 2\xi\omega\_n \frac{dq(t)}{dt} + \omega\_n^2 q(t) = \frac{\varepsilon(t)}{m} \\\\ q(0) = q\_{0\prime}q'(0) = v\_{0\prime} \end{cases} \tag{3}$$

where *ωn* is called the natural angular frequency (natural frequency for short) with damping free given by

$$
\omega\_{\text{fl}} = \sqrt{\frac{k}{m'}} \tag{4}
$$

and the parameter *ς* is the damping ratio expressed by

$$
\zeta = \frac{c}{2\sqrt{mk}}.\tag{5}
$$

The characteristic equation of (3) is in the form

$$p^2 + 2\zeta\omega\_n p + \omega\_n^2 = 0,\tag{6}$$

which is usually called the frequency equation in engineering [1–7]. The solution to the above is given by

$$p\_{1,2} = -\varsigma \omega\_{\mathfrak{n}} \pm i\omega\_{\mathfrak{n}} \sqrt{1 - \varsigma^2} \,\,\, \,\, \tag{7}$$

where *i* = √−1. Taking into account damping, one uses the term damped natural frequency denoted by *ωd*. It is given by

$$
\omega\_d = \omega\_n \sqrt{1 - \zeta^2}.\tag{8}
$$

**Note 2.1:** All parameters above, namely, *m*, *c*, *k*, *ζ*, *ωn*, and *ωd*, are constants.
