*6.1. General Form of Free Responses*

Consider the free response to the functional equivalent oscillator in Class *j* in the form

$$\begin{cases} m\_{eqj} \frac{d^2 x\_j(t)}{dt^2} + c\_{eqj} \frac{dx\_j(t)}{dt} + kx\_j(t) = 0\\ x\_j(0) = x\_{j0}, \frac{dx\_j(t)}{dt} \Big|\_{t=0} = v\_{j0} \end{cases}, j = 1, 2, 3. \tag{166}$$

Following the representation style in engineering, we rewrite it by

$$\begin{cases} \frac{d^2x\_j(t)}{dt^2} + 2\xi\_{cqj}\omega\_{cqn,j}\frac{dx\_j(t)}{dt} + \omega\_{cqn,j}^2x\_j(t) = 0\\ x\_j(0) = x\_{j0}, \frac{dx\_j(t)}{dt}\Big|\_{t=0} = v\_{j0} \end{cases}, j = 1, 2, 3. \tag{167}$$

Therefore (Timoshenko ([1], p. 34), Jin and Xia ([79], p. 11)), we have, for *t* ≥ 0,

$$\mathbf{x}\_{j}(t) = e^{-\xi\_{\rm eqj}\omega\_{\rm eqn,j}t} \left( \mathbf{x}\_{j0}\cos\omega\_{\rm eqd,j}t + \frac{\upsilon\_{j0} + \zeta\_{\rm eqj}\omega\_{\rm eqn,j}\mathbf{x}\_{j0}}{\omega\_{\rm eqd,j}}\sin\omega\_{\rm eqd,j}t \right). \tag{168}$$

The above may be rewritten in the form

$$\mathbf{x}\_{j}(t) = A\_{\mathbf{c}qj} e^{-\xi\_{\mathbf{c}qj}\omega\_{\mathbf{c}q;j}t} \cos \left(\omega\_{\mathbf{c}qd,j}t - \theta\_{\mathbf{c}qj} \right), t \ge 0,\tag{169}$$

where the equivalent amplitude *Aeqj* is given by

$$A\_{eqj} = \sqrt{\mathbf{x}\_{j0}^2 + \left[\frac{\upsilon\_{j0} + \varsigma\_{eqj}\omega\_{eqp,j}\mathbf{x}\_{j0}}{\omega\_{eqd,j}}\right]^2},\tag{170}$$

and the equivalent phase *<sup>θ</sup>eqj* is

$$\theta\_{eqj} = \tan^{-1} \frac{\upsilon\_{j0} + \varsigma\_{eqj}\omega\_{eqn,j}\mathbf{x}\_{j0}}{\omega\_{eqd,j}\mathbf{x}\_{j0}}.\tag{171}$$

Note that, for *<sup>ω</sup>eqn*,*j*, *ςeqj*, *Aeqj*, and *<sup>θ</sup>eqj*, each is not constant for fractional oscillators. Instead, each is generally a function of oscillation frequency *ω* and fractional order.
