*5.2. Equivalent Damping Ratio*

**Definition 2.** *Let ςeqj be the equivalent damping ratio of the equivalent system of a fractional oscillator in Class j. It is defined by*

$$\zeta\_{cqj} = \frac{c\_{cqj}}{2\sqrt{m\_{cqj}k}}, j = 1, 2, 3. \tag{140}$$

**Corollary 4** (Equivalent damping ratio I)**.** *The equivalent damping ratio of a fractional oscillator in Class I is expressed by*

$$\zeta\_{cq1} = \zeta\_{cq1}(\omega, a) = \frac{\omega^{\frac{q}{2}} \sin \frac{a\pi}{2}}{2\omega\_{\pi} \sqrt{-\cos \frac{a\pi}{2}}}, 1 < a \le 2. \tag{141}$$

**Proof.** Replacing *meq*1 and *ceq*1 in the expression below with the equivalent mass I and the equivalent damping I described in Section 4

$$\varepsilon\_{eq1} = \frac{c\_{eq1}}{2\sqrt{m\_{eq1}k}} \tag{142}$$

yields

$$\xi\_{eq1} = \frac{m\omega^{\kappa - 1}\sin\frac{a\pi}{2}}{2\sqrt{(-m\omega^{\kappa - 2}\cos\frac{a\pi}{2})}k} = \frac{\omega^{\frac{\kappa}{2}}\sin\frac{a\pi}{2}}{2\sqrt{-\cos\frac{a\pi}{2}}}\sqrt{\frac{m}{k}} = \frac{\omega^{\frac{\kappa}{2}}\sin\frac{a\pi}{2}}{2\omega\_n\sqrt{-\cos\frac{a\pi}{2}}}, 1 < n \le 2. \tag{143}$$

The proof finishes. 

**Remark 20.** *The damping ratio ςeq*<sup>1</sup> *follows the power law in terms of ω.*

**Remark 21.** *The damping ratio of fractional oscillators in Class I relates to the oscillation frequency ω and the fractional order α. It is increasing with respect to ω.*

$$
\zeta\_{eq1}(0,\mathfrak{a}) \;=\; 0 \; \text{and} \; \mathfrak{g}\_{eq1}(\mathfrak{so},\mathfrak{a}) \;=\; \infty. \tag{144}
$$

Figure 13 shows the curves of *<sup>ς</sup>eq*<sup>1</sup>(*<sup>ω</sup>*, *<sup>α</sup>*).

**Figure 13.** Illustrations of *<sup>ς</sup>eq*<sup>1</sup>(*<sup>ω</sup>*, *<sup>α</sup>*). Solid line: *α* = 1.3. Dot line: *α* = 1.6. Dash line: *α* = 1.9. (**a**) For *ωn* = 1. (**b**) For *ωn* = 3.

**Note 5.8:** Figure 13 indicates that the smaller the *α* the greater the *ςeq*1.

**Corollary 5** (Equivalent damping ratio II)**.** *The damping ratio of a fractional oscillator in Class II is given by*

$$\zeta\_{eq2} = \zeta\_{eq2}(\omega, \beta) = \frac{\zeta \omega^{\beta - 1} \sin \frac{\beta \pi}{2}}{\sqrt{1 - \frac{c}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}}, 0 < \beta \le 1,\tag{145}$$

*where ς* = *c* 2√*mk* . **Proof.** When replacing the *meq*2 and *ceq*2 in the following expression by the equivalent mass II and the equivalent damping II proposed in Section 4, we attain

$$\begin{split} \xi\_{cq2} &= \frac{c\_{rq2}}{2\sqrt{m\_{cq2}}k} = \frac{c\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{2\sqrt{\left(m-c\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}k} = \frac{c\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{2\sqrt{\left(1-\frac{c}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}mk} \\ &= \frac{c\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{2\sqrt{mk}\sqrt{1-\frac{c}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}} = \frac{c\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{\sqrt{1-\frac{c}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}}, 0 < \beta \le 1. \end{split}$$

This finishes the proof. 

**Remark 22.** *The damping ratio ςeq*<sup>2</sup> *obeys the power law in terms of ω.*

**Remark 23.** *The damping ratio ςeq*<sup>2</sup> *is associated with ω and the fractional order β. It is decreasing in terms of ω.*

**Note 5.9:** *ςeq*<sup>2</sup> takes *ζ* as a special case for *β* = 1. In fact,

$$\left. \zeta\_{eq2}(\omega, \beta) \right|\_{\beta=1} = \left. \frac{\zeta \omega^{\beta - 1} \sin \frac{\beta \pi}{2}}{\sqrt{1 - \frac{\zeta}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}} \right|\_{\beta=1} = \emptyset. \tag{146}$$

Figure 14 indicates the plots of *<sup>ς</sup>eq*<sup>2</sup>(*<sup>ω</sup>*, *β*) in the case of *m* = 1, *c* = 1, and *k* = 1.

**Figure 14.** Plots of *<sup>ς</sup>eq*<sup>2</sup>(*<sup>ω</sup>*, *β*) for *m* = *c* = *k* = 1. Solid line: *β* = 0.9. Dot line: *β* = 0.6. Dash line: *β* = 0.3.

**Corollary 6** (Equivalent damping ratio III)**.** *Let ςeq*<sup>3</sup> *be the damping ratio of a fractional oscillator in Class III. Then, for 1 < α* ≤ *2, 0 < β* ≤ *1,*

$$\zeta\_{\neq \neq 3} = \zeta\_{\neq 3}(\omega, a, \beta) = \frac{\omega^{a-1} \sin \frac{a\pi}{2} + 2\zeta\omega\_n \omega^{\beta - 1} \sin \frac{\beta\pi}{2}}{2\omega\_n \sqrt{-\left(\omega^{a-2} \cos \frac{a\pi}{2} + 2\zeta\omega\_n \omega^{\beta - 2} \cos \frac{\beta\pi}{2}\right)}}. \tag{147}$$

**Proof.** If replacing the *meq*3 and *ceq*3 below with the equivalent mass III and the equivalent damping III presented in Section 4, we obtain

$$\begin{split} \xi\_{\mp q}\xi\_{\mp q} &= \frac{\varepsilon\_{\mp q}\lambda}{2\sqrt{m\_{\text{eq}}\mu}} = \frac{m\omega^{\kappa-1}\sin\frac{q\pi}{2} + c\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{2\sqrt{-\left(m\omega^{\kappa-2}\cos\frac{q\pi}{2} + c\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}k} \\ &= \frac{m\left(\omega^{\kappa-1}\sin\frac{q\pi}{2} + \frac{c}{m}\omega^{\beta-1}\sin\frac{\beta\pi}{2}\right)}{2\sqrt{-\left(\omega^{\kappa-2}\cos\frac{q\pi}{2} + \frac{c}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}nk} = \frac{m\left(\omega^{\kappa-1}\sin\frac{q\pi}{2} + 2\zeta\omega\_{\kappa}\omega^{\beta-1}\sin\frac{\beta\pi}{2}\right)}{2\sqrt{mk}\sqrt{-\left(\omega^{\kappa-2}\cos\frac{q\pi}{2} + 2\zeta\omega\_{\kappa}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}}{2\omega\sqrt{-\left(\omega^{\kappa-2}\cos\frac{q\pi}{2} + 2\zeta\omega\_{\kappa}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}}, \\ &= \frac{\omega^{\kappa-1}\sin\frac{q\pi}{2} + 2\zeta\omega\_{\kappa}\omega^{\beta-1}\sin\frac{\beta\pi}{2}}{2\omega\sqrt{-\left(\omega^{\kappa-2}\cos\frac{q\pi}{2} + 2\zeta\omega\_{\kappa}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}}. \end{split}$$

Thus, we finish the proof. 

**Remark 24.** *The damping ratio ςeq*<sup>3</sup> *follows the power law in terms of ω.*

**Remark 25.** *ςeq*<sup>3</sup> *relates to ω and a pair of fractional orders (<sup>α</sup>, β).*

**Note 5.10:** *ςeq*<sup>3</sup> regards *ζ* as a special case for *α* = 2 and *β* = 1. As a matter of fact,

$$\varphi\_{\alpha\beta}(\omega, \mathbf{2}, 1) = \left. \frac{\omega^{a-1} \sin \frac{a\pi}{2} + 2\zeta\omega\_n \omega^{\beta - 1} \sin \frac{\beta\pi}{2}}{2\omega\_n \sqrt{-\left(\omega^{a-2} \cos \frac{a\pi}{2} + 2\zeta\omega\_n \omega^{\beta - 2} \cos \frac{\beta\pi}{2}\right)}} \right|\_{a=2, \beta=1} = \varsigma. \tag{148}$$

Figure 15 demonstrates the figures of *<sup>ς</sup>eq*<sup>3</sup>(*<sup>ω</sup>*, *α*, *β*) in the case of *m* = 1, *c* = 1, and *k* = 1.

**Figure 15.** Demonstrations of *<sup>ς</sup>eq*<sup>3</sup>(*<sup>ω</sup>*, *α*, *β*) in *m* = *c* = *k* = 1. Solid line: *α* = 1.9. Dot line: *α* = 1.8. Dash line: *α* = 1.7. (**a**) For *β* = 0.8. (**b**) For *β* = 0.5. (**c**) For *β* = 0.2. (**d**) For *β* = 1.
