*8.5. Application to Represetenting Mittag-Leffler Function (3)*

The step response to fractional oscillators in Class I by using the generalized Mittag-Leffler function is in the form (Uchaikin ([38], Chapter 7))

$$\log\_1(t) = t^a E\_{a,a+1} \left[ - (\omega\_n t)^a \right], 1 < a \le 2, t \ge 0. \tag{229}$$

In the following corollary, we propose the representation of (229) by elementary functions.

**Corollary 17.** *The generalized Mittag-Leffler function expressed by (229) can be represented by using the elementary functions described in Theorem 16. Precisely, for t* ≥ *0 and 1 < α* ≤ *2, we have*

$$t^{a}E\_{a,a+1}\left[-\left(\omega\_{n}t\right)^{a}\right]=\frac{1}{k}\left[1-\frac{e^{-\frac{\omega\sin\frac{a\pi}{2}}{2\left[\cos\frac{a\pi}{2}\right]}t}{\omega}\cos\left(\frac{\omega\_{n}\sqrt{1-\frac{\omega^{a}\sin\frac{a\pi}{2}}{4\omega\_{n}^{2}\left[\cos\frac{a\pi}{2}\right]}}t}{\sqrt{1-\left(\frac{\omega^{a}\sin\frac{a\pi}{2}}{2\omega\_{n}\sqrt{1-\cos\frac{a\pi}{2}}}\right)^{2}}}\right)\right],\tag{230}$$

*where φ*1 *is given by (212).* **Proof.** The left side of (230) equals to *g*1(*t*) following Theorem 16. According to (229), therefore, (230) holds. This completes the proof. 

### **9. Frequency Responses to Three Classes of Fractional Oscillators**

We put forward frequency responses to three classes of fractional oscillators in this section. They are expressed by elementary functions based on the theory of three equivalent oscillators addressed in Section 4.

### *9.1. General Form of Frequency Responses to Three Classes of Fractional Oscillators*

Denote by *Hj*(*ω*) the Fourier transform of the impulse response *hj*(*t*) to a fractional oscillator in Class *j* (*j* = 1, 2, 3), where *hj*(*t*) is given by (196). Then, it is the frequency response function to a fractional oscillator in Class *j* (*j* = 1, 2, 3).

In fact, doing the Fourier transform on the both sides of (195) produces

$$\left(-\omega^2 + i2\varsigma\_{eqj}\omega\_{eqn,j}\omega + \omega\_{eqn,j}^2\right)H\_j(\omega) = \frac{1}{m\_{eqj}}.\tag{231}$$

Thus, we have

$$H\_{\vec{l}}(\omega) = \frac{1}{m\_{\text{cyl}} \left(\omega\_{\text{eq},j}^2 - \omega^2 + i2\zeta\_{\text{eq}\vec{l}}\omega\_{\text{eq},\text{\%}}\omega\right)} = \frac{1}{m\_{\text{cyl}}\omega\_{\text{eq},j}^2 \left(1 - \frac{\omega^2}{\omega\_{\text{eq},j}^2} + i2\zeta\_{\text{eq}\vec{l}}\frac{\omega}{\omega\_{\text{eq},j}}\right)}.\tag{232}$$

Note that

$$m\_{eqj}\omega\_{eq\eta,j}^2 = m\_{eqj}\frac{k}{m\_{eqj}} = k.\tag{233}$$

Therefore, by letting *γeqj* be the equivalent frequency ratio of a fractional oscillator in Class *j*, *Hj*(*ω*) may be expressed by

$$H\_{\vec{j}}(\omega) = \frac{1}{k\left(1 - \gamma\_{eqj}^2 + i2\varsigma\_{eqj}\gamma\_{eqj}\right)}, j = 1, 2, 3. \tag{234}$$

The amplitude of *Hj*(*ω*) is

$$\left|H\_j(\omega)\right| = \frac{1}{k} \frac{1}{\sqrt{\left(1 - \gamma\_{eqj}^2\right)^2 + \left(2\zeta\_{eqj}\gamma\_{eqj}\right)^2}}, j = 1, 2, 3. \tag{235}$$

Its phase frequency response is given by

$$\langle \varphi\_{\hat{\jmath}}(\omega) = \tan^{-1} \frac{2\zeta\_{c\eta j}\gamma\_{c\eta j}}{1 - \gamma\_{c\eta j}^2}, j = 1, 2, 3. \tag{236}$$

### *9.2. Frequency Response to a Fractional Oscillator in Class I*

**Theorem 19** (Frequency response I)**.** *Let <sup>H</sup>*1(*ω*) *be the frequency response to a fractional oscillator in Class I. Then, for 1 < α* ≤ *2, it is in the form*

$$H\_1(\omega) = \frac{1}{k\left(1 - \frac{\omega^n \left|\cos\frac{\omega\pi}{2}\right|}{\omega\_n^2} + i\frac{\omega^n \sin\frac{\omega\pi}{2}}{\omega\_n^2}\right)}}\tag{237}$$

*Symmetry* **2018**, *10*, 40

**Proof.** In the equation below,

$$H\_1(\omega) = \frac{1}{k\left(1 - \gamma\_{eq1}^2 + i2\zeta\_{eq1}\gamma\_{eq1}\right)},\tag{238}$$

when replacing *γeq*<sup>1</sup> by

$$\gamma\_{\mathbf{cq}1} = \gamma\_{\mathbf{cq}1}(\omega, \mathbf{a}) = \frac{\omega}{\omega\_{\mathbf{cq}n, 1}} = \frac{\omega \sqrt{\omega^{n-2} |\cos \frac{\mathbf{a} \pi}{2}|}}{\omega\_n},\tag{239}$$

and <sup>2</sup>*ςeq*1*γeq*<sup>1</sup> by

$$2\zeta\_{eq1}\gamma\_{eq1} = 2\frac{\omega^{\frac{\mu}{2}}\sin\frac{a\pi}{2}}{2\omega\_n\sqrt{|\cos\frac{a\pi}{2}|}}\frac{\omega\sqrt{\omega^{a-2}|\cos\frac{a\pi}{2}|}}{\omega\_n} = \frac{\omega^a\sin\frac{a\pi}{2}}{\omega\_n^2},\tag{240}$$

we have (237). This completes the proof. 

> From Theorem 19, we have the amplitude of *<sup>H</sup>*1(*ω*) given by

$$|H\_1(\omega)| = \frac{1/k}{\sqrt{\left(1 - \frac{\omega^n |\cos\frac{\omega\pi}{2}|}{\omega\_n^2}\right)^2 + \left(\frac{\omega^n \sin\frac{\omega\pi}{2}}{\omega\_n^2}\right)^2}}\tag{241}$$

and the phase in the form

$$\varphi\_1(\omega) = \tan^{-1} \frac{\frac{\omega^a \sin \frac{a\pi}{2}}{\omega\_n^2}}{1 - \frac{\omega^a \left| \cos \frac{a\pi}{2} \right|}{\omega\_n^2}} = \tan^{-1} \frac{\omega^a \sin \frac{a\pi}{2}}{\omega\_n^2 - \omega^a \left| \cos \frac{a\pi}{2} \right|}. \tag{242}$$

**Note 9.1 (Equivalent frequency ratio I):** The equivalent frequency ratio *γeq*<sup>1</sup> is a function of oscillation frequency *ω* and the fractional order *α*. It may be denoted by *<sup>γ</sup>eq*<sup>1</sup>(*<sup>ω</sup>*, *<sup>α</sup>*).

Figure 51 shows the plot of *γeq*1. Figure 52 indicates the illustrations of *<sup>H</sup>*1(*ω*).

**Figure 51.** Equivalent frequency ratio *<sup>γ</sup>eq*<sup>1</sup>(*<sup>ω</sup>*, *α*) for fractional oscillators in Class I with *m* = *k* = 1. Solid line: *α* = 1.8. Dot line: *α* = 1.5. Dash line: *α* = 1.2.

**Figure 52.** Frequency response *<sup>H</sup>*1(*ω*) to fractional oscillators in Class I with *m* = *k* = 1. Solid line: *α* = 1.8 (0.04 ≤ *ςeq*<sup>1</sup> ≤ 0.06). Dot line: *α* = 1.5 (0.13 ≤ *ςeq*<sup>1</sup> ≤ 0.19). Dash line: *α* = 1.2 (0.33 ≤ *ςeq*<sup>1</sup> ≤ 0.46). (**a**) Amplitude |*<sup>H</sup>*1(*ω*)| in ordinary coordinate. (**b**) |*<sup>H</sup>*1(*ω*)| in log-log. (**c**) Phase *ϕ*1(*ω*) in ordinary coordinate. (**d**) *ϕ*1(*ω*) in log-log.

**Note 9.2:** If *α* = 2, *<sup>H</sup>*1(*ω*) → ∞ at *ω* = *ωn*. In that case, *<sup>H</sup>*1(*ω*) turns to be the ordinary frequency response with damping free in the form

$$\left. \right| H\_1(\omega)|\_{a=2} = \left. \frac{1/k}{1 - \frac{\omega^a \left| \cos \frac{a\pi}{2} \right|}{\omega\_n^2} + i \frac{\omega^a \sin \frac{a\pi}{2}}{\omega\_n^2}} \right|\_{a=2} = \frac{1/k}{1 - \frac{\omega^2}{\omega\_n^2}} = \frac{1/k}{1 - \gamma^2}. \tag{243}$$

### *9.3. Frequency Response to a Fractional Oscillator in Class II*

**Theorem 20** (Frequency response II)**.** *Denote by <sup>H</sup>*2(*ω*) *the frequency response to a fractional oscillator in Class II. Then, for 0 < β* ≤ *1, it is given by*

$$H\_2(\omega) = \frac{1/k}{1 - \gamma^2 \left(1 - \frac{\varepsilon}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}\right) + i \frac{2 \zeta \omega^{\beta} \sin \frac{\beta \pi}{2}}{\omega\_n}}\tag{244}$$

*where γ* = *ωωnis the ordinary frequency ratio.*

**Proof.** Consider

$$H\_2(\omega) = \frac{1}{k\left(1 - \gamma\_{eq2}^2 + i2\zeta\_{eq2}\gamma\_{eq2}\right)}.\tag{245}$$

Note that

$$\gamma\_{eq2} = \gamma\_{eq2}(\omega, \beta) = \frac{\omega}{\omega\_{eqn,2}} = \frac{\omega}{\omega\_n} \sqrt{1 - \frac{c}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}} = \gamma \sqrt{1 - \frac{c}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}.\tag{246}$$

Besides,

$$2\zeta\_{\rm eq}\gamma\_{\rm eq2} = \frac{2\zeta\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{\sqrt{1-\frac{c}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}}\left(\frac{\omega}{\omega\_n}\sqrt{1-\frac{c}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}\right) = \frac{2\zeta\omega^{\beta}\sin\frac{\beta\pi}{2}}{\omega\_n}.\tag{247}$$

Therefore, (245) becomes

$$H\_{2}(\omega) = \frac{1}{k\left(1 - \gamma\_{eq2}^{2} + i2\zeta\_{eq2}\gamma\_{eq2}\right)}$$

$$=\frac{1/k}{1 - \left(\frac{\omega}{\omega\_{\text{Tr}}}\sqrt{1 - \frac{\xi}{m}\omega^{\beta - 2}\cos\frac{\beta\pi}{2}}\right)^{2} + i\frac{2\varsigma\omega^{\beta}\sin\frac{\beta\pi}{2}}{\omega\gamma\_{\text{Tr}}}}$$

$$=\frac{1/k}{1 - \gamma^{2}\left(1 - \frac{\xi}{m}\omega^{\beta - 2}\cos\frac{\beta\pi}{2}\right) + i\frac{2\varsigma\omega^{\beta}\sin\frac{\beta\pi}{2}}{\omega\gamma\_{\text{Tr}}}}\cdot$$

This finishes the proof. 

> From Theorem 20, we have the amplitude of *<sup>H</sup>*2(*ω*) in the form

$$|H\_2(\omega)| = \frac{1/k}{\sqrt{\left[1 - \gamma^2 \left(1 - \frac{\xi}{m}\omega^{\beta - 2}\cos\frac{\beta\pi}{2}\right)\right]^2 + \left(\frac{2\xi\omega^{\beta}\sin\frac{\beta\pi}{2}}{\omega\_n}\right)^2}},\tag{248}$$

and its phase given by

$$\varphi\_2(\omega) = \tan^{-1} \frac{\frac{2\xi\omega^{\beta}\sin\frac{\beta\pi}{2}}{\omega\_n}}{1 - \gamma^2 \left(1 - \frac{c}{m}\omega^{\beta - 2}\cos\frac{\beta\pi}{2}\right)}}.\tag{249}$$

**Note 9.3 (Equivalent frequency ratio II):** The equivalent frequency ratio *γeq*<sup>2</sup> is dependent on oscillation frequency *ω* and the fractional order *β* as can be seen from (9.16). We denote it by *<sup>γ</sup>eq*<sup>2</sup>(*<sup>ω</sup>*, *β*). Figure 53 indicates the plot of *<sup>γ</sup>eq*<sup>2</sup>(*<sup>ω</sup>*, *β*). *<sup>H</sup>*2(*ω*) is shown in Figure 54.

**Figure 53.** Equivalent frequency ratio *<sup>γ</sup>eq*<sup>2</sup>(*<sup>ω</sup>*, *β*) of fractional oscillators in Class II with *m* = *c* = *k* = 1. Solid line: *β* = 0.8. Dot line: *β* = 0.5. Dash line: *β* = 0.2.

**Figure 54.** *Cont.*

**Figure 54.** Frequency response *<sup>H</sup>*2(*ω*) to fractional oscillators of Class II type with *m* = *c* = 1 and *k* = 4. Solid line: *β* = 0.8 (0.15 ≤ *ςeq*<sup>2</sup> ≤ 0.29). Dot line: *β* = 0.5 (0.06 ≤ *ςeq*<sup>2</sup> ≤ 0.33). Dash line: *β* = 0.2 (0.01 ≤ *ςeq*<sup>2</sup> ≤ 0.35). (**a**) Amplitude |*<sup>H</sup>*2(*ω*)| in ordinary coordinate. (**b**) |*<sup>H</sup>*2(*ω*)| in log-log. (**c**) Phase *ϕ*2(*ω*) in ordinary coordinate. (**d**) *ϕ*2(*ω*) in log-log.

**Note 9.4:** When *β* = 1, *<sup>H</sup>*2(*ω*) reduces to that of an ordinary oscillator's in the form (also see Figure 55).

$$\left.H\_{2}(\omega)\right|\_{\beta=1} = \left.\frac{1/k}{1 - \gamma^{2}\left(1 - \frac{\varepsilon}{\pi}\omega^{\beta - 2}\cos\frac{\beta\pi}{2}\right) + i\frac{2\varepsilon\rho^{\beta}\sin\frac{\beta\pi}{2}}{\omega\_{0}}}\right|\_{\beta=1} \tag{250}$$
  $\omega = \frac{1/k}{1 - \gamma^{2} + i2\varepsilon\frac{\omega}{\omega\_{0}}} = \frac{1/k}{1 - \gamma^{2} + i2\varepsilon\gamma}$ 

**Figure 55.** *<sup>H</sup>*2(*ω*) for *β* = 1 with *m* = *c* = 1 and *k* =4(*ζeq*<sup>2</sup> = 0.25). (**a**) |*<sup>H</sup>*2(*ω*)| in ordinary coordinate. (**b**) |*<sup>H</sup>*2(*ω*)| in log-log. (**c**) Phase *ϕ*2(*ω*) in log-log.

### *9.4. Frequency Response to a Fractional Oscillator in Class III*

**Theorem 21** (Frequency response III)**.** *Let <sup>H</sup>*3(*ω*) *be the frequency response to a fractional oscillator of Class III type. Then, for 1 < α* ≤ *2 and 0 < β* ≤ *1, <sup>H</sup>*3(*ω*) *is in the form*

$$H3(\omega) = \frac{1/k}{1 - \gamma^2 \left(\omega^{a-2} \left|\cos\frac{a\pi}{2}\right| - \frac{c\omega^{\beta-2}\cos\frac{\beta\pi}{2}}{m}\right) + i\frac{\gamma\left(\omega^{a-1}\sin\frac{a\pi}{2} + 2\varphi\omega\_n\omega^{\beta-1}\sin\frac{\beta\pi}{2}\right)}{\omega\_n\left(\omega^{a-2} \left|\cos\frac{a\pi}{2}\right| - 2\varphi\omega\_n\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}}.\tag{251}$$

**Proof.** In the equation below

$$H\_{\mathfrak{P}}(\omega) = \frac{1}{k\left(1 - \gamma\_{c\eta3}^2 + i2\zeta\_{c\eta3}\gamma\_{c\eta3}\right)},\tag{252}$$

 1 *Symmetry* **2018**, *10*, 40

we notice

$$\begin{split} \gamma\_{\text{eq3}} &= \gamma\_{\text{eq3}}(\omega, \mathfrak{a}, \beta) = \frac{\omega}{\omega\_{\text{eq3}}} = \frac{\omega}{\omega\_{\text{n}}} \sqrt{-\left(\omega^{\mathfrak{a}-2}\cos\frac{\mathfrak{a}\pi}{2} + \frac{\mathfrak{c}}{m}\omega^{\mathfrak{b}-2}\cos\frac{\beta\pi}{2}\right)} \\ &= \gamma \sqrt{-\left(\omega^{\mathfrak{a}-2}\cos\frac{\mathfrak{a}\pi}{2} + \frac{\mathfrak{c}}{m}\omega^{\mathfrak{b}-2}\cos\frac{\beta\pi}{2}\right)} . \end{split} \tag{253}$$

In addition,

$$\begin{split} 2\zeta\_{\alpha\beta}\gamma\epsilon\_{\ell\ell} &= 2\frac{\left[\begin{array}{c} \left(\omega^{a-1}\sin\frac{a\pi}{2} + 2\zeta\omega\gamma\_{n}\omega^{\beta-1}\sin\frac{\beta\pi}{2} \\\\ \gamma\sqrt{-\left(\omega^{a-2}\cos\frac{a\pi}{2} + \frac{c}{m}\omega\beta^{\beta-2}\cos\frac{\beta\pi}{2}\right)} \end{array}\right] \\ &\frac{\gamma\sqrt{-\left(\omega^{a-2}\cos\frac{a\pi}{2} + 2\zeta\omega\gamma\_{n}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}}{2\omega\_{n}\sqrt{-\left(\omega^{a-2}\cos\frac{a\pi}{2} + 2\zeta\omega\gamma\_{n}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}} \\ &= \frac{\gamma\left(\omega^{a-1}\sin\frac{a\pi}{2} + 2\zeta\omega\gamma\_{n}\omega^{\beta-1}\sin\frac{\beta\pi}{2}\right)}{\omega\gamma\_{n}\left(\omega^{a-2}\left|\cos\frac{a\pi}{2}\right| - 2\zeta\omega\omega\_{n}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}. \end{split} \tag{254}$$

Thus, (252) becomes

$$H\_3(\omega) = \frac{1/k}{1 - \gamma^2 \left(\omega^{a-2} \left|\cos\frac{a\pi}{2}\right| - \frac{c\omega^{\beta-2}\cos\frac{\beta\pi}{2}}{m}\right) + i\frac{\gamma\left(\omega^{a-1}\sin\frac{a\pi}{2} + 2\zeta\omega\_n\omega^{\beta-1}\sin\frac{\beta\pi}{2}\right)}{\omega\_n\left(\omega^{a-2} \left|\cos\frac{a\pi}{2}\right| - 2\zeta\omega\_n\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}}.$$

Therefore, the proof completes. 

> From Theorem 21, we obtain |*<sup>H</sup>*3(*ω*)| in the form

$$|H\_3(\omega)| = \frac{1/k}{\sqrt{\left\{\begin{aligned} \left[1-\gamma^2\left(\omega^{\mu-2}|\cos\frac{\theta\pi}{2}| - \frac{\xi}{m}\omega^{\beta-2}\cos\frac{\theta\pi}{2}\right)\right]^2\\ &+ \left[\frac{\gamma\left(\omega^{\mu-1}\sin\frac{\theta\pi}{2} + 2\varepsilon\omega\_{\mu\theta}\omega^{\beta-1}\sin\frac{\theta\pi}{2}\right)}{\omega\_{\mu}\left(\omega^{\mu-2}|\cos\frac{\theta\pi}{2}| - 2\varepsilon\omega\_{\mu}\omega^{\beta-2}\cos\frac{\theta\pi}{2}\right)}\right]^2\end{aligned}}\tag{255}$$

The phase *ϕ*3(*ω*) is given by

$$\varphi\_3(\omega) = \tan^{-1} \frac{\frac{\gamma \left(\omega^{a-1} \sin \frac{a\pi}{2} + 2\zeta\omega\omega\_{\text{n}}\omega^{\beta - 1} \sin \frac{\beta\pi}{2}\right)}{\omega \sqrt{\left(\omega^{a-2} \left|\cos \frac{a\pi}{2}\right| - 2\zeta\omega\omega\_{\text{n}}\omega^{\beta - 2} \cos \frac{\beta\pi}{2}\right)}}{1 - \gamma^2 \left(\omega^{a-2} \left|\cos \frac{a\pi}{2}\right| - \frac{c}{m}\omega^{\beta - 2}\cos \frac{\beta\pi}{2}\right)}. \tag{256}$$

**Note 9.5 (Equivalent frequency ratio III):** *γeq*<sup>3</sup> relates to *ω* and a pair of fractional orders (*<sup>α</sup>*, *β*). Figure 56 indicates its plots. Figure 57 demonstrates *<sup>H</sup>*3(*ω*).

**Figure 56.** Plots of equivalent frequency ratio *<sup>γ</sup>eq*<sup>3</sup>(*<sup>ω</sup>*, *α*, *β*) for *m* = *c* = *k* = 1. (**a**) Solid line: (*<sup>α</sup>*, *β*) = (1.8, 0.8). Dot line: (*<sup>α</sup>*, *β*) = (1.5, 0.8). (**b**) Solid line: (*<sup>α</sup>*, *β*) = (1.5, 0.8). Dot line: (*<sup>α</sup>*, *β*) = (1.5, 0.6).

**Figure 57.** *Cont.*

**Figure 57.** Illustrations of frequency response *<sup>H</sup>*3(*ω*) to fractional oscillators of Class III with *m* = *c* = 1, *k* = 25. (**a**) |*<sup>H</sup>*3(*ω*)| and *ϕ*3(*ω*). Solid line: (*<sup>α</sup>*, *β*) = (1.8, 0.9) (0.23 ≤ *ςeq*<sup>3</sup> ≤ 0.54). Dot line: (*<sup>α</sup>*, *β*) = (1.5, 0.9) (0.36 ≤ *ςeq*<sup>3</sup> ≤ 1.04). (**b**) |*<sup>H</sup>*3(*ω*)| and *ϕ*3(*ω*). Solid line: (*<sup>α</sup>*, *β*) = (1.8, 0.7) (0.27 ≤ *ςeq*<sup>3</sup> ≤ 0.50). Dot line: (*<sup>α</sup>*, *β*) = (1.5, 0.7) (0.50 ≤ *ςeq*<sup>3</sup> ≤ 0.95). (**c**) |*<sup>H</sup>*3(*ω*)| and *ϕ*3(*ω*). Solid line: (*<sup>α</sup>*, *β*) = (1.8, 0.55) (0.31 ≤ *ςeq*<sup>3</sup> ≤ 0.46). Dot line: (*<sup>α</sup>*, *β*) = (1.5, 0.55) (0.86 ≤ *ςeq*<sup>3</sup> ≤ 0.97).

**Note 9.6:** If (*<sup>α</sup>*, *β*) = (2, 1), *<sup>H</sup>*3(*ω*) reduces to the ordinary one given by

$$H\_3(\omega)|\_{(a,\emptyset)=(2,1)} = \frac{1/k}{1-\gamma^2+i2\xi\gamma}.\tag{257}$$
