*5.3. Equivalent Natural Frequency II*

Now, with two parameters *<sup>ω</sup>eqn*,*j* and *ςeqj* presented above, we rewrite the equivalent oscillator (130) by

$$\frac{d^2\mathbf{x}\_j(t)}{dt^2} + 2\zeta\_{\text{eq}j}\omega\_{\text{eqn},j}\frac{d\mathbf{x}\_j(t)}{dt} + \omega\_{\text{eqn},j}^2\mathbf{x}\_j(t) = \frac{f(t)}{m\_{\text{eq}j}}, j = 1, 2, 3. \tag{149}$$

The characteristic equation of (149) is given by

$$
\omega\_j^2 + 2\varsigma\_{eqj}\omega\_{eqn,j}s\_j + \omega\_{eqn,j}^2 = 0, j = 1,2,3.\tag{150}
$$

The characteristic roots are in the form

$$s\_{j,1,2} = -\varsigma\_{\text{eq}}\omega\_{\text{eq},j} \pm \sqrt{\varsigma\_{\text{eq}}^2 \omega\_{\text{eq},j}^2 - \omega\_{\text{eq},j}^2} = -\varsigma\_{\text{eq}j}\omega\_{\text{eq},j} \pm i\omega\_{\text{eq},j}\sqrt{1 - \varsigma\_{\text{eq},j}^2}, j = 1,2,3. \tag{151}$$

Functionally, we utilize the symbol *<sup>ω</sup>eqd*,*j* for

$$
\omega\_{\text{eqd},j} = \omega\_{\text{eqn},j}\sqrt{1 - \zeta\_{\text{eqj}}^2}, j = 1, 2, 3. \tag{152}
$$

Thus, the characteristic roots are

$$s\_{j,1,2} = -\varsigma\_{\text{eq}j}\omega\_{\text{eq}n,j} \pm i\omega\_{\text{eq}d,j}, j = 1,2,3. \tag{153}$$

Note that, in practice, 0 ≤ *ςeqj* < 1 because 1 ≤ *ςeqj* means no oscillation at all.

We write those above for the sake of applying the theory of linear oscillations to fractional ones. Now, we discuss *<sup>ω</sup>eqd*,*j*.

**Corollary 7** (Equivalent natural frequency II1)**.** *Let <sup>ω</sup>eqd*,<sup>1</sup> *be the functional damped natural frequency of a fractional oscillator in Class I. It may be termed the equivalent natural frequency II1. Then,*

$$
\omega\_{cqd,1} = \omega\_{cqd,1}(\omega, a) = \frac{\omega\_n}{\sqrt{-\omega^{n-2}\cos\frac{a\pi}{2}}} \sqrt{1 - \frac{\omega^n \sin^2\frac{a\pi}{2}}{4\omega\_n^2 |\cos\frac{a\pi}{2}|}}, 1 < a \le 2. \tag{154}
$$

**Proof.** Note that

$$
\omega\_{cqd,1} = \omega\_{cqn,1}\sqrt{1 - \varsigma\_{cd1}^2}.\tag{155}
$$

Using the above *ςed*1, we have

$$
\omega\_{eqd,1} = \omega\_{eqn,1}\sqrt{1 - \zeta\_{cd1}^2} = \frac{\omega\_n}{\sqrt{-\omega^{a-2}\cos\frac{a\pi}{2}}} \sqrt{1 - \left(\frac{\omega^{\frac{a}{2}}\sin\frac{a\pi}{2}}{2\omega\_n\sqrt{|\cos\frac{a\pi}{2}|}}\right)^2}.
$$

This finishes the proof. 

The parameter *<sup>ω</sup>eqd*,<sup>1</sup> functionally takes the form of damped natural frequency as in the conventional linear oscillation theory. In this research, we do not distinguish the natural frequencies with damped or damping free. At most, we just say that it is a functional damped one. It relates to the oscillation frequency *ω* and the fractional order *α*.

**Remark 26.** *<sup>ω</sup>eqd*,<sup>1</sup> *is not a monotonic function of ω.* **Note 5.11:** *ωn* is a special case of *<sup>ω</sup>eqd*,<sup>1</sup> when *α* = 2:

$$
\omega\_{\text{eqd},1}(\omega,2) = \frac{\omega\_n}{\sqrt{-\omega^{n-2}\cos\frac{a\pi}{2}}} \sqrt{1 - \frac{\omega^a \sin^2 \frac{a\pi}{2}}{4\omega\_n^2|\cos\frac{a\pi}{2}|}}\bigg|\_{a=2} = \omega\_n. \tag{156}
$$

As a matter of fact, fractional oscillators of Class I are damping free for *α* = 2. Figure 16 illustrates the plots of *<sup>ω</sup>eqd*,<sup>1</sup>(*<sup>ω</sup>*, *<sup>α</sup>*).

**Figure 16.** Plots of *<sup>ω</sup>eqd*,<sup>1</sup>(*<sup>ω</sup>*, *<sup>α</sup>*). Solid line: *α* = 1.8. Dot line: *α* = 1.5. Dash line: *α* = 1.2. (**a**) For *ωn* = 1. (**b**) For *ωn* = 2.

**Corollary 8** (Equivalent natural frequency II2)**.** *Let <sup>ω</sup>eqd*,<sup>2</sup> *be the functional damped natural frequency of a fractional oscillator in Class II. Term it with the equivalent natural frequency II2. Then, for 0 < β* ≤ *1,*

$$
\omega\_{eqd,2} = \omega\_{eqd,2}(\omega,\beta) = \frac{\omega\_n}{\sqrt{1 - \frac{\varepsilon}{m}\omega^{\beta - 2}\cos\frac{\beta\pi}{2}}} \sqrt{1 - \frac{\xi^2 \omega^{2(\beta - 1)}\sin^2\frac{\beta\pi}{2}}{1 - \frac{\varepsilon}{m}\omega^{\beta - 2}\cos\frac{\beta\pi}{2}}}.\tag{157}
$$

**Proof.** Consider

$$
\omega\_{cqd,2} = \omega\_{cqn,2}\sqrt{1-\varsigma\_{cd2}^2}.\tag{158}
$$

Replacing *<sup>ω</sup>eqn*,<sup>2</sup> and *ςed*2 in the above yields

$$\begin{split} \omega\_{eqd,2} &= \omega\_{eqn,2} \sqrt{1 - \xi\_{cd2}^2} = \frac{\omega\_n}{\sqrt{1 - \frac{\xi\_n}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}} \sqrt{1 - \left(\frac{\xi\_n \omega^{\beta - 1} \sin \frac{\beta \pi}{2}}{2 \sqrt{1 - \frac{\xi\_n}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}}\right)^2} \\ &= \frac{\omega\_n}{\sqrt{1 - \frac{\xi\_n}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}} \sqrt{1 - \frac{\xi^2 \omega^{2(\beta - 1)} \sin^2 \frac{\beta \pi}{2}}{1 - \frac{\xi\_n}{m} \omega^{\beta - 2} \cos \frac{\beta \pi}{2}}}. \end{split}$$

Thus, Corollary 8 holds. 

**Remark 27.** *<sup>ω</sup>eqd*,<sup>2</sup> *is related to ω and the fractional order β.*

**Note 5.12:** The conventional damped natural frequency, say,

$$
\omega\_d = \omega\_n \sqrt{1 - \varsigma^2} \tag{159}
$$

is a special case of *<sup>ω</sup>eqd*,<sup>2</sup>(*<sup>ω</sup>*, *β*) for *β* = 1. Figure 17 gives the plots of *<sup>ω</sup>eqd*,<sup>2</sup>(*<sup>ω</sup>*, *β*).

**Figure 17.** Illustrating *<sup>ω</sup>eqd*,<sup>2</sup>(*<sup>ω</sup>*, *β*) for *m* = *c* = 1. Solid line: *β* = 0.8. Dot line: *β* = 0.5. Dash line: *β* = 0.2. (**a**) For *ωn* = 1. (**b**) For *ωn* = 0.3. (**c**) For *ωn* = 10.

**Corollary 9** (Equivalent natural frequency II3)**.** *Let <sup>ω</sup>eqd*,<sup>3</sup> = *<sup>ω</sup>eqd*,<sup>3</sup>(*<sup>ω</sup>*, *α*, *β*) *be the functional damped natural frequency of a fractional oscillator in Class III. Call it the equivalent natural frequency II3. Then, for 1 < α* ≤ *2 and 0 < β* ≤ *1, we have*

$$\omega\_{eqd,3} = \frac{\omega\_n \sqrt{1 - \left[\frac{\left(\omega^{a-1} \sin\frac{a\pi}{2} + 2\zeta\omega\_n \omega^{g-1} \sin\frac{\beta\pi}{2}\right)^2}{4\omega\_n^2 \left[-\left(\omega^{a-2} \cos\frac{a\pi}{2} + 2\zeta\omega\_n \omega^{g-2} \cos\frac{\beta\pi}{2}\right)\right]}\right]^2}{\sqrt{-\left(\omega^{a-2} \cos\frac{a\pi}{2} + \frac{\zeta}{m} \omega^{g-2} \cos\frac{\beta\pi}{2}\right)}}}. \tag{160}$$

**Proof.** In the expression below

$$
\omega\_{\text{eq}d,3} = \omega\_{\text{eq}v,3} \sqrt{1 - \varsigma\_{\text{cd3'}}^2} \tag{161}
$$

we replace *<sup>ω</sup>eqd*,<sup>3</sup> and *ςeq*<sup>3</sup> by those expressed above. Then, we have

$$\begin{split} \omega\_{\text{eqd},3} &= \omega\_{\text{eqn},3} \sqrt{1 - \zeta\_{\text{cd}3}^2} = \sqrt{\frac{k}{m\_{\text{eq}3}}} \sqrt{1 - \zeta\_{\text{cd}3}^2} \\ &= \frac{\omega\_{\text{eq}} \sqrt{1 - \left[ \frac{\omega^{\omega - 1} \sin \frac{\omega \pi}{2} + 2\zeta \omega \pi \omega \delta^{-1} \sin \frac{\beta \pi}{2}}{2\omega\_{\text{eq}} \sqrt{-\left(\omega^{\omega - 2} \cos \frac{\omega \pi}{2} + 2\zeta \omega \pi \omega \delta^{-1} \cos \frac{\beta \pi}{2}\right)} \right]^2}{\sqrt{-\left(\omega^{\omega - 2} \cos \frac{\omega \pi}{2} + \frac{\zeta}{\pi} \omega \delta^{-2} \cos \frac{\beta \pi}{2}\right)}} \\ &= \frac{\omega\_{\text{eq}} \sqrt{1 - \left[ \frac{\omega^{\omega - 1} \sin \frac{\omega \pi}{2} + 2\zeta \omega \pi \omega \delta^{-1} \sin \frac{\beta \pi}{2}}{4\omega\_{\text{eq}}^2 \left[ - \left(\omega^{\omega - 2} \cos \frac{\omega \pi}{2} + 2\zeta \omega \pi \omega \delta^{-2} \cos \frac{\beta \pi}{2} \right) \right]} \right]^2}{\sqrt{-\left(\omega^{\omega - 2} \cos \frac{\omega \pi}{2} + \frac{\zeta}{\pi} \omega \delta^{-2} \cos \frac{\beta \pi}{2}\right)}}. \end{split}$$

Therefore, the corollary holds. 

**Note 5.13:** The conventional damped natural frequency *ωd* is a special case of *<sup>ω</sup>eqd*,<sup>3</sup> for (*<sup>α</sup>*, *β*) = (2, 1). Indeed,

$$\omega\_{\text{eq1},3}(\omega,2,1) = \left. \frac{\omega\_n \sqrt{1 - \left[\frac{\left(\omega^{a-1}\sin\frac{\mathfrak{a}\pi}{2} + 2\xi\omega\_n\omega^{g-1}\sin\frac{\mathfrak{f}\pi}{2}\right)^2}{4\omega\_n^2\left[-\left(\omega^{a-2}\cos\frac{\mathfrak{a}\pi}{2} + 2\xi\omega\_n\omega^{g-2}\cos\frac{\mathfrak{f}\pi}{2}\right)\right]}\right]^2}}{\sqrt{-\left(\omega^{a-2}\cos\frac{\mathfrak{a}\pi}{2} + \frac{\xi}{m}\omega^{g-2}\cos\frac{\mathfrak{f}\pi}{2}\right)}} \right|\_{\substack{\omega = 1 \ \mathfrak{f} = 1}} = \omega\_n\sqrt{1 - \xi^2}.\tag{162}$$

**Remark 28.** *The natural frequency <sup>ω</sup>eqd*,<sup>3</sup> *is associated with ω and a pair of fractional orders (<sup>α</sup>, β).*

Figures 18 and 19 indicate its plots.

**Figure 18.** Demonstrations of *<sup>ω</sup>eqd*,<sup>3</sup>(*<sup>ω</sup>*, *α*, *β*) for *m* = *c* = *k* = 1. Solid line: *α* = 1.9. Dot line: *α* = 1.6. Dash line: *α* = 1.3. (**a**) For *β* = 0.9. (**b**) For *β* = 0.8. (**c**) For *β* = 0.3. (**d**) For *β* = 0.2.

**Figure 19.** *<sup>ω</sup>eqd*,<sup>3</sup>(*<sup>ω</sup>*, *α*, *β*) for *m* = *c* = 1, *β* = 0.9. Solid line: *α* = 1.9. Dot line: *α* = 1.6. Dash line: *α* = 1.3. (**a**) For *ωn* = 3. (**b**) For *ωn* = 5.

### *5.4. There Exists Infinity of Natural Frequencies of a Fractional Oscillator*

The previous discussions imply that there exists infinity of natural frequencies, for either *<sup>ω</sup>eqn*,*j* or *<sup>ω</sup>eqd*,*j*, because each is dependent on *ω* ∈ (0, ∞). We functionally derived the two characteristic roots of the frequency equation (151), namely, *sj*,1,2, actually stand for infinity of roots owing to *ω* ∈ (0, ∞).

Taking a fractional oscillator in Class I into account, its frequency equation is given by

$$s^a + \omega\_n^2 = 0, 1 < a \le 2. \tag{163}$$

Then, it is easy to see that there exists infinitely many characteristic roots in the above, also see Li et al. [18].

A contribution in this work in representing characteristic roots of three classes of fractional oscillators is that they are expressed analytically. Moreover, functionally, they take the form as that in the theory of conventional linear oscillations, making it possible to represent solutions to three classes of fractional oscillators by using elementary functions, which are easier for use in both engineering applications and theoretic analysis of fractional oscillators.

### **6. Free Responses to Three Classes of Fractional Oscillators**

We put forward the free responses in this section to three classes of fractional oscillators based on their equivalent oscillators presented in Section 4. Since the equivalent oscillators are expressed by using second-order differential equations in form, in methodology, therefore, it is easy for us to find the responses we concern with. Note that the equivalence explained in Section 4 says that

$$X\_j(\omega) = X\_j(\omega), \; j = 1, \; 2, \; 3,\tag{164}$$

where the subscript *j* stands for the Class I to III. Consequently,

$$y\_j(t) = x\_j(t), \; j = 1, \; 2, \; 3. \tag{165}$$

Therefore, our research implies three advances.

