*5.1. Equivalent Natural Frequency I*

**Definition 1.** *Denote by <sup>ω</sup>eqn*,*j a natural frequency of a fractional oscillator in the jth class (j = 1, 2, 3). It takes the form*

$$
\omega\_{c\eta u,j} = \sqrt{\frac{k}{m\_{c\eta j}}}, j = 1, 2, 3,\tag{129}
$$

*where meqj is the equivalent mass of the fractional oscillator in the jth class.*

With the above definition, we write (128) by

$$\frac{d^2 \mathbf{x}\_j(t)}{dt^2} + \frac{c\_{eqj}}{m\_{eqj}} \frac{d \mathbf{x}\_j(t)}{dt} + \frac{\mathbf{k}}{m\_{eqj}} \mathbf{x}\_j(t) = \frac{d^2 \mathbf{x}\_j(t)}{dt^2} + \frac{c\_{eqj}}{m\_{eqj}} \frac{d \mathbf{x}\_j(t)}{dt} + \omega\_{eqnj}^2 \mathbf{x}\_j(t) = \frac{f(t)}{m\_{eqj}}, j = 1, 2, 3. \tag{130}$$

**Note 5.1:** *<sup>ω</sup>eqn*,*j* may take the conventional natural frequency, denoted by

$$
\omega\_n = \sqrt{\frac{k}{m'}}\tag{131}
$$

as a special case.

**Corollary 1** (Equivalent natural frequency I1)**.** *The equivalent natural frequency I1, which we denote it by <sup>ω</sup>eqn*,1, *of a fractional oscillator in Class I is given by*

$$
\omega\_{c\eta u, 1} = \frac{\omega\_n}{\sqrt{-\omega^{a-2} \cos \frac{a\pi}{2}}}, 1 < a \le 2. \tag{132}
$$

**Proof.** According to (129), we have, for 1 < *α* ≤ 2,

$$
\omega\_{c\eta u,1} = \sqrt{\frac{k}{m\_{c\eta 1}}} = \sqrt{\frac{k}{-m\omega^{a-2}\cos\frac{a\pi}{2}}} = \sqrt{\frac{1}{-\omega^{a-2}\cos\frac{a\pi}{2}}}\sqrt{\frac{k}{m}} = \frac{\omega\_n}{\sqrt{-\omega^{a-2}\cos\frac{a\pi}{2}}}.\tag{133}
$$

The proof finishes. 

> Figure 10 shows the plots of *<sup>ω</sup>eqn*,1.

**Figure 10.** Natural frequency *<sup>ω</sup>eqn*,1. Solid line: *α* = 1.8. Dot line: *α* = 1.5. Dash line: *α* = 1.2. (**a**) *ωn* = 1. (**b**) *ωn* = 2.

**Note 5.2:** From Figure 10, we see that *<sup>ω</sup>eqn*,<sup>1</sup> is an increasing function with *ω*. Besides, the greater the value of *α* the smaller the *<sup>ω</sup>eqn*,1.

**Note 5.3:** *<sup>ω</sup>eqn*,<sup>1</sup> becomes *ωn* if *α* = 2. In fact,

$$\left.\omega\_{\text{eqn},1}\right|\_{a=2} = \sqrt{\frac{k}{m\_{\text{eq}1}}}\bigg|\_{a=2} = \left.\frac{\omega\_n}{\sqrt{-\omega^{a-2}\cos\frac{a\pi}{2}}}\right|\_{a=2} = \omega\_n. \tag{134}$$

**Corollary 2** (Equivalent natural frequency I2)**.** *The natural frequency I2, <sup>ω</sup>eqn*,2, *of a fractional oscillator in Class II is given by*

$$
\omega\_{eqn,2} = \frac{\omega\_n}{\sqrt{1 - \frac{\varepsilon}{m}\omega^{\beta - 2}\cos\frac{\beta \pi}{2}}} \cdot \tag{135}
$$

**Proof.** Following (129), we have

$$
\omega\_{\text{eqn},2} = \sqrt{\frac{k}{m\_{\text{eq}2}}} = \sqrt{\frac{k}{m - c\omega^{\theta - 2}\cos\frac{\theta \pi}{2}}} = \sqrt{\frac{k}{m\left(1 - \frac{\xi}{m}\omega^{\theta - 2}\cos\frac{\theta \pi}{2}\right)}} = \frac{\omega\_n}{\sqrt{1 - \frac{\xi}{m}\omega^{\theta - 2}\cos\frac{\theta \pi}{2}}}.
$$

Hence, the proof completes. 

> Figure 11 indicates the curves of *<sup>ω</sup>eqn*,2.

**Figure 11.** Curves of *<sup>ω</sup>eqn*,<sup>2</sup> for *m* = *c* = 1. Solid line: *β* = 0.8. Dot line: *β* = 0.5. Dash line: *β* = 0.3. (**a**) *ωn* = 1. (**b**) *ωn* = 2.

**Note 5.4:** Figure 11 shows that *<sup>ω</sup>eqn*,<sup>2</sup> is a decreasing function with *ω*. The greater the value of *β* the smaller the *<sup>ω</sup>eqn*,2.

**Note 5.5:** *<sup>ω</sup>eqn*,<sup>2</sup> takes *ωn* as a special case for *β* = 1. As a matter of fact,

$$\left.\omega\_{c\eta n,2}\right|\_{\beta=1} = \left.\frac{\omega\_n}{\sqrt{1-\frac{c}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}}}\right|\_{\beta=1} = \omega\_n.\tag{136}$$

**Corollary 3** (Equivalent natural frequency I3)**.** *The natural frequency I3, denoted by <sup>ω</sup>eqn*,3, *of a fractional oscillator in Class III is given by*

$$
\omega\_{eqn,3} = \frac{\omega\_n}{\sqrt{-\left(\omega^{a-2}\cos\frac{a\pi}{2} + \frac{c}{m}\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)}}\tag{137}
$$

**Proof.** With (129), we write

$$
\omega\_{\varepsilon qu,3} = \sqrt{\frac{k}{m\_{q3}}} = \sqrt{\frac{k}{-\left(\frac{m\omega^{a-2}\cos\frac{\omega\pi}{2} + \varepsilon\omega\theta^{-2}\cos\frac{\xi\pi}{2}}{2}\right)}} = \frac{\omega\_{u}}{\sqrt{-\left(\omega^{a-2}\cos\frac{\omega\pi}{2} + \frac{\varepsilon\omega\theta^{-2}\cos\frac{\xi\pi}{2}}{2}\right)}}.\tag{138}
$$

The above completes the proof. 

> Figure 12 gives the illustrations of *<sup>ω</sup>eqn*,3.

**Figure 12.** Illustrations of *<sup>ω</sup>eqn*,<sup>3</sup> for *m* = *c* = 1. Solid line: *α* = 1.8, *β* = 0.9. Dot line: *α* = 1.5, *β* = 0.9. Dash line: *α* = 1.2, *β* = 0.9. (**a**) *ωn* = 1. (**b**) *ωn* = 2.

**Note 5.6:** Figure 12 exhibits that *<sup>ω</sup>eqn*,<sup>3</sup> is an increasing function in terms of *ω*. **Note 5.7:** *<sup>ω</sup>eqn*,<sup>3</sup> takes *ωn* as a special case for *α* = 2 and *β* = 1. Indeed,

$$\left.\omega\_{\text{eqn},3}\right|\_{a=2,\delta=0} = \sqrt{\frac{k}{-\left(m\omega^{a-2}\cos\frac{a\pi}{2} + c\omega^{\delta-2}\cos\frac{\delta\pi}{2}\right)}}\bigg|\_{a=2,\delta=1} = \sqrt{\frac{k}{m}} = \omega\_n. \tag{139}$$
