*7.1. General Form of Impulse Responses*

Given a following functional form of equivalent oscillators for finding their impulse responses, we denote by *hj*(*t*) the impulse response to the equivalent oscillator in Class *j* in the form

$$c\_{cqj}\frac{d^2h\_j(t)}{dt^2} + c\_{cqj}\frac{dh\_j(t)}{dt} + kh\_j(t) = \delta(t), j = 1, 2, 3. \tag{193}$$

Rewrite the above in the form

$$\frac{d^2h\_j(t)}{dt^2} + \frac{c\_{eqj}}{m\_{eqj}}\frac{dh\_j(t)}{dt} + \frac{k}{m\_{eqj}}h\_j(t) = \frac{\delta(t)}{m\_{eqj}}, j = 1, 2, 3. \tag{194}$$

According to the results in the previous sections, we have

$$\frac{d^2h\_j(t)}{dt^2} + 2\xi\_{\text{eqj}}\omega\_{\text{equ},j}\frac{dh\_j(t)}{dt} + \omega\_{\text{eqn},j}^2h\_j(t) = \frac{\delta(t)}{m\_{\text{eqj}}}, j = 1, 2, 3. \tag{195}$$

Therefore, functionally, we have

$$h\_{\dot{\jmath}}(t) = \frac{e^{-\zeta\_{eq\dot{\jmath}}\omega\_{eq\dot{\jmath}}t}}{m\_{eq\dot{\jmath}}\omega\_{eqd\_{\dot{\jmath}}\dot{\jmath}}}\sin\omega\_{eqd\_{\dot{\jmath}}\dot{\jmath}}t \ge 0. \tag{196}$$

Equation (196) is a general form of the impulse response to fractional oscillators for Class *j* (*j* = 1, 2, 3). Its specific form for each Class is discussed as follows.

### *7.2. Impulse Response to Fractional Oscillators in Class I*

**Theorem 13** (Impulse response I)**.** *Let h*1(*t*) *be the impulse response to a fractional oscillator in Class I. Then, for t* ≥ *0 and 1 < α* ≤ *2, we have*

$$h\_{1}(t) = \frac{e^{-\frac{\omega r \sin \frac{\alpha \pi}{2}}{2|\cos \frac{\alpha \pi}{2}|}t} \sin \left(\frac{\omega\_{n}}{\sqrt{\omega^{a-2}|\cos \frac{\alpha \pi}{2}|}} \sqrt{1 - \frac{\omega^{2a} \sin^{2} \frac{\alpha \pi}{2}}{4\omega\_{n}^{2} |\cos \frac{\alpha \pi}{2}|}t}\right)}{m\omega\_{n} \sqrt{\omega^{a-2} |\cos \frac{\alpha \pi}{2}|} \sqrt{1 - \frac{\omega^{2a} \sin^{2} \frac{\alpha \pi}{2}}{4\omega\_{n}^{2} |\cos \frac{\alpha \pi}{2}|}}}. \tag{197}$$

**Proof.** From (196), we have

$$h\_1(t) = \frac{e^{-\zeta\_{eq1}\omega\_{eqn,1}t}}{m\_{eq1}\omega\_{eqd,1}}\sin\omega\_{eqd,1}t, t \ge 0. \tag{198}$$

When replacing *meq*1 by that in Section 4, *ςeq*<sup>1</sup> and *<sup>ω</sup>eqd*,<sup>1</sup> as well as *<sup>ω</sup>eqn*,<sup>1</sup> with those in Section 5, respectively, we obtain

*h*1(*t*) = *e* <sup>−</sup>*ςeq*1*<sup>ω</sup>eqn*,1*<sup>t</sup> meq*1*ωeqd*,<sup>1</sup> sin *<sup>ω</sup>eqd*,1*<sup>t</sup>* = *e* − *ω α* 2 sin *απ*2 <sup>2</sup>*ωn*√|cos *απ*2 | √ *ωn ωα*−<sup>2</sup>|cos *απ*2 | *t* sin *ωn* "##\$1− *ω*2*<sup>α</sup>* sin<sup>2</sup> *απ*2 <sup>4</sup>*ω*2*n*|cos *απ*2 | *t ωα*−<sup>2</sup>|cos *απ*2 | *ωα*−<sup>2</sup>|cos *απ*2 |*m ωn ωα*−<sup>2</sup>|cos *απ*2 | !1− *ω*2*<sup>α</sup>* sin<sup>2</sup> *απ*2 <sup>4</sup>*ω*2*n*|cos *απ*2 | = *e* − *ω* sin *απ*2 <sup>2</sup>|cos *απ*2 | *t* sin *ωn* "##\$1− *ω*2*<sup>α</sup>* sin<sup>2</sup> *απ*2 <sup>4</sup>*ω*2*n*|cos *απ*2 | *t* √*ωα*−<sup>2</sup>|cos *απ*2 | *<sup>m</sup>ωn <sup>ω</sup>α*−<sup>2</sup>|cos *απ*2 |!1− *ω*2*<sup>α</sup>* sin<sup>2</sup> *απ*2 <sup>4</sup>*ω*2*n*|cos *απ*2 | .

This finishes the proof. 

Figure 33 shows the plots of *h*1(*t*), where the oscillation frequency *ω* is fixed. Note that *ω* is an argumen<sup>t</sup> of *h*1(*t*). Therefore, its pictures in time domain are indicated in Figure 34. Figure 35 indicates its figures in t-ω plane.

**Figure 33.** Plots of impulse response *h*1(*t*) with *ωn* = 1. (**a**) *α* = 1.9, solid line: *ω* =1(*ςeq*<sup>1</sup> = 0.08); dot line: *ω* = 0.7 (*ςeq*<sup>1</sup> = 0.04). (**b**) *α* = 1.6, solid line: *ω* =1(*ςeq*<sup>1</sup> = 0.33); dot line: *ω* = 0.7 (*ςeq*<sup>1</sup> = 0.19). (**c**) *α* = 1.3, solid line: *ω* =1(*ςeq*<sup>1</sup> = 0.66); dot line: *ω* = 0.7 (*ςeq*<sup>1</sup> = 0.42). (**d**) *α* = 2, solid line: *ω* =1(*ςeq*<sup>1</sup> = 0); dot line: *ω* = 0.7 (*ςeq*<sup>1</sup> = 0).

**Figure 34.** Plots of impulse response *h*1(*t*) for *ωn* = 5, *ω* = 0, 1, ..., 5. (**a**) For *α* = 1.8 (0 ≤ *ςeq*<sup>1</sup> ≤ 0.57). (**b**) For *α* = 1.5 (0 ≤ *ςeq*<sup>1</sup> ≤ 0.94). (**c**) For *α* = 1.3 (0 ≤ *ςeq*<sup>1</sup> ≤ 1.07). (**d**) For *α* =2(*ςeq*<sup>1</sup> = 0).

**Figure 35.** Impulse response *h*1(*t*) in t-ω plane with *m* = 1, *ωn* = 0.3 for *t* = 0, 1, ..., 30; *ω* = 1, 2, ..., 5. (**a**) *α* = 1.9 (0.26 ≤ *ςeq*<sup>1</sup> ≤ 5.58). (**b**) *α* =2(*ςeq*<sup>1</sup> = 0).

**Note 7.1:** The impulse response *h*1(*t*) reduces to the conventional one if *α* = 2. In fact,

$$|h\_{1}(t)|\_{a=2} = \left(\frac{e^{-\frac{\omega\sin\frac{4\pi t}{2}}{2}}t}{m\omega\_{n}\sqrt{\omega^{a-2}|\cos\frac{4\pi t}{2}|}}\sqrt{1-\frac{\omega^{2a}\sin^{2}\frac{\pi t}{2}}{4\omega\_{n}^{2}|\cos\frac{\pi t}{2}|}}t\right)\_{a=2} = \frac{1}{m\omega\_{n}}\sin\omega\_{n}t.\tag{199}$$
