*3.2. Research Thoughts*

Let us qualitatively consider possible performances of equivalent mass and damping. In engineering, people may purposely connect an auxiliary mass *ma* to the primary mass *m* so that the equivalent mass of the total system is related to the oscillation frequency *ω* (Harris ([4], p. 6.4)). In the field of ship hull

vibrations, added mass has to be taken into account in the equivalent mass (i.e., total mass) of a ship hull (Korotkin [78]) so that the equivalent mass is ω-varying. In fact, the three dimensional fluid coefficient with respect to the added mass to a ship hull relates to the oscillation frequency, see, e.g., Jin and Xia ([79], pp. 135–136), Nakagawa et al. [80].

In addition, damping may be also ω-varying. A well-known case of ω-varying damping is the Coulomb damping (Timoshenko ([2], Chapter 1), Harris ([4], Equation (30.4))). Frequency varying damping is a technique used in damping treatments, see, e.g., Harris ([4], Equation (37.8)). Besides, commonly used damping assumptions in ship hull vibrations, such as the Copoknh's, the Voigt's, the Rayleigh's, are all ω-varying (Jin and Xia ([79], pp. 157–158)). Therefore, with the concept of ω-varying mass and damping, I purposely generalize the simple oscillation model expressed by (1) in the form

$$\begin{cases} m\_{c\eta}(\omega) \frac{d^2 \eta(t)}{dt^2} + c\_{c\eta}(\omega) \frac{dq(t)}{dt} + kq(t) = \varepsilon(t) \\\ q(0) = q\_0, q'(0) = v\_0. \end{cases} \tag{51}$$

The above second-order equation may not be equivalent to a fractional oscillator unless *meq* and or *ceq* are appropriately expressed and properly related to the fractional order *α* for Class I oscillators, or *β* for Class II oscillators, or (*<sup>α</sup>*, *β*) for oscillators in Class III. For those reasons, we further generalize (51) by

$$\begin{cases} m\_{\text{eq1}}(\omega, a) \frac{d^2 x\_1(t)}{dt^2} + c\_{\text{eq1}}(\omega, a) \frac{dx\_1(t)}{dt} + kx\_1(t) = e(t) \\\ x\_1(0) = x\_{10}, \dot{x}\_1(0) = v\_{10} \end{cases} \tag{52}$$

for Class I oscillators. As for Class II oscillators, (51) should be generalized by

$$\begin{cases} m\_{\varepsilon q2}(\omega,\theta) \frac{d^2 x\_2(t)}{dt^2} + c\_{\varepsilon q2}(\omega,\theta) \frac{dx\_2(t)}{dt} + kx\_2(t) = e(t) \\\ x\_2(0) = x\_{10}, \dot{x}\_2(0) = v\_{20}. \end{cases} \tag{53}$$

Similarly, for Class III oscillators, we generalize (51) to be the form

$$\begin{cases} m\_{eq3}(\omega, \mathfrak{a}, \mathfrak{z}) \frac{d^2 x\_3(t)}{dt^2} + c\_{eq3}(\omega, \mathfrak{a}, \mathfrak{z}) \frac{d x\_2(t)}{dt} + k x\_3(t) = \mathfrak{e}(t) \\\ x\_3(0) = x\_{30}, \dot{x}\_3(0) = v\_{30}. \end{cases} \tag{54}$$

Three generalized oscillation Equations (52)–(54), can be unified in the form

$$\begin{cases} m\_{eq} \frac{d^2 \mathbf{x}\_j(t)}{dt^2} + c\_{eq} \frac{d \mathbf{x}\_j(t)}{dt} + k \mathbf{x}\_j(t) = \mathbf{e}(t) \\\ \mathbf{x}\_j(0) = \mathbf{x}\_{j0}, \dot{\mathbf{x}}\_j(0) = vy\_0 \end{cases}, j = 1, 2, 3. \tag{55}$$

By introducing the symbols *<sup>ω</sup>eqn*,*j* = *k meqj* and *ςeqj* = *ceqj* <sup>2</sup>√*meqj<sup>k</sup>* for *j* = 1, 2, 3, we rewrite the above by

$$\begin{cases} \frac{d^2x\_j(t)}{dt^2} + 2\xi\_{\rm eqj}\omega\_{\rm eqn,j}\frac{dx\_j(t)}{dt} + \omega\_{\rm eqn,j}^2x\_j(t) = \frac{\mathbf{c}(t)}{m\_{\rm eqj}} \\\ x\_j(0) = x\_{j0}, \dot{x}\_j(0) = v\_{30} \end{cases}, j = 1, 2, 3. \tag{56}$$

Let *Yj*(*ω*) be the Fourier transform of *yj*(*t*), where *yj*(*t*)(*j* = 1, 2, 3) respectively corresponds to the one in (31), (42), and (43). Denote by *Xj*(*ω*) the Fourier transform of *xj*(*t*). Then, if we find proper *meqj* and *ceqj* such that

$$Y\_j(\omega) = X\_j(\omega), \; j \;=\; 1, \; 2, \; 3,\tag{57}$$

the second-order equation (52), or (53), or (54) is equal to the fractional oscillation Equation (31), or (42), or (43), respectively.

Obviously, once we discover the equivalent equations of the fractional oscillation Equations (52), or (53), and (54), all problems stated previously can be readily solved.

### **4. Equivalent Systems of Three Classes of Fractional Oscillators**

In this section, we first present an equivalent system and then its equivalent mass and damping in Sections 4.1–4.3, respectively for each class of fractional oscillators.

### *4.1. Equivalent System for Fractional Oscillators in Class I*

4.1.1. Equivalent Oscillation Equation of Fractional Oscillators in Class I

Theorem 1 gives the equivalent oscillator with the integer order for the fractional oscillators in Class I.

**Theorem 1** (Equivalent oscillator I)**.** *Denote a fractional oscillator in Class I by*

$$m\frac{d^\alpha y\_1(t)}{dt^\alpha} + ky\_1(t) = 0,\\ 1 < \alpha \le 2. \tag{58}$$

*Then, its equivalent oscillator with the equation of order 2 is in the form*

$$-m\omega^{a-2}\cos\frac{a\pi}{2}\frac{d^2\mathbf{x}\_1(t)}{dt^2} + m\omega^{a-1}\sin\frac{a\pi}{2}\frac{d\mathbf{x}\_1(t)}{dt} + k\mathbf{x}\_1(t) = 0, 1 < a \le 2. \tag{59}$$

**Proof.** Consider the frequency response of (58) with the excitation of the Dirac-delta function *δ*(*t*). In doing so, we study

$$m\frac{d^a h\_{y1}(t)}{dt^a} + kh\_{y1}(t) = \delta(t), 1 < a \le 2. \tag{60}$$

Doing the Fourier transform on the both sides of (60) produces

$$\left[m(i\omega)^a + k\right]H\_{\mathfrak{Y}^1}(\omega) = 1, 1 < a \le 2,\tag{61}$$

where *Hy*1(*ω*) is the Fourier transform of *hy*1(*t*). Using the principal value of *i*, we have

$$\dot{\mu}^{\alpha} = \cos \frac{\alpha \pi}{2} + i \sin \frac{\alpha \pi}{2}. \tag{62}$$

Thus, (61) implies

$$\begin{cases} \left[m(i\omega)^{a} + k\right]H\_{y1}(\omega) = \left\{m\left(\cos\frac{a\pi}{2} + i\sin\frac{a\pi}{2}\right)\omega^{a} + k\right\}H\_{y1}(\omega) \\ = \left(m\omega^{a}\cos\frac{a\pi}{2} + im\omega^{a}\sin\frac{a\pi}{2} + k\right)H\_{y1}(\omega) = 1. \end{cases} \tag{63}$$

Therefore, we have the frequency response of (60) in the form

$$H\_{y1}(\omega) = \frac{1}{m\omega^a \cos\frac{a\pi}{2} + im\omega^a \sin\frac{a\pi}{2} + k}.\tag{64}$$

On the other hand, for 1 < *α* ≤ 2, we consider (59) by

$$-m\omega^{a-2}\cos\frac{a\pi}{2}\frac{d^2h\_{x1}(t)}{dt^2} + m\omega^{a-1}\sin\frac{a\pi}{2}\frac{dh\_{x1}(t)}{dt} + kl\_{x1}(t) = \delta(t).\tag{65}$$

Performing the Fourier transform on the both sides of (65) yields

$$\begin{cases} \left[ -m\omega^{a-2}\cos\frac{a\pi}{2}\left(-\omega^2\right) + m\omega^{a-1}\sin\frac{a\pi}{2}(i\omega) + k \right] H\_{x1}(\omega) \\ = \left( m\omega^a \cos\frac{a\pi}{2} + im\omega^a \sin\frac{a\pi}{2} + k \right) H\_{x1}(\omega) = 1, \end{cases} \tag{66}$$

where *Hx*1(*ω*) is the Fourier transform of *hx*1(*t*). Therefore, we have

$$H\_{\mathbf{x}1}(\omega) = \frac{1}{m\omega^a \cos\frac{a\pi}{2} + im\omega^a \sin\frac{a\pi}{2} + k}.\tag{67}$$

By comparing (64) with (67), we see that

$$H\_{\mathfrak{Y}1}(\omega) = H\_{\mathfrak{x}1}(\omega). \tag{68}$$

Thus, (59) is the equivalent equation of (58). The proof completes. 

4.1.2. Equivalent Mass of Fractional Oscillators in Class I

From the first item on the left side of (59), we obtain the equivalent mass for the fractional oscillators of Class I type.

**Theorem 2** (Equivalent mass I)**.** *The equivalent mass of the fractional generators in Class I, denoted by meq*1, *is expressed by*

$$m\_{cq1} = m\_{cq1}(\omega, \mathfrak{a}) = -\left(\omega^{\mathfrak{a}-2} \cos \frac{\mathfrak{a}\pi}{2}\right) m\_\uparrow 1 < \mathfrak{a} \le 2. \tag{69}$$

**Proof.** According to the Newton's second law, the inertia force in the system of the fractional oscillator (58) corresponds to the first item on the left side of its equivalent system (59). That is, −*mωα*−<sup>2</sup> cos *απ*2 *<sup>d</sup>*2*x*1(*t*) *dt*<sup>2</sup> . Thus, the coefficient of *<sup>d</sup>*2*x*1(*t*) *dt*<sup>2</sup> is an equivalent mass expressed by (69). Hence, the proof finishes. 

From Theorem 2, we reveal a power law phenomenon with respect to *meq*1 in terms of *ω*.

**Remark 1.** *The equivalent mass I, meq*1, *follows the power law in terms of oscillation frequency ω in the form*

$$m\_{eq1}(\omega, \mathfrak{a}) \sim \omega^{a-2} m, 1 < \mathfrak{a} \le 2. \tag{70}$$

The equivalent mass *meq*1 relates to the oscillation frequency *ω*, the fractional order *α*, and the primary mass *m*. Denote by

$$R\_{\mathfrak{m}1}(\omega, a) = -\omega^{a-2} \cos \frac{a\pi}{2}, 1 < a \le 2. \tag{71}$$

Then, we have

$$m\_{cq1} = m\_{cq1}(\omega, \mathfrak{a}) = R\_{m1}(\omega, \mathfrak{a}) \mathfrak{m}, 1 < \mathfrak{a} \le 2. \tag{72}$$

**Note 4.1**: Since

$$R\_{m1}(\omega, \mathbf{2}) = \mathbf{1},\tag{73}$$

*meq*1(*<sup>ω</sup>*, *α*) reduces to the primary mass *m* when *α* = 2. That is,

$$m\_{eq1}(\omega, \mathbf{2}) = m.\tag{74}$$

In the case of *α* = 2, therefore, both (58) and (59) reach the conventional harmonic oscillation with damping free in the form

$$m\frac{d^2\mathbf{x}\_1(t)}{dt^2} + k\mathbf{x}\_1(t) = 0.$$

**Note 4.2:** If *α* → 1, we have

$$\lim\_{\omega \to 1} m\_{\epsilon q1}(\omega, \mathfrak{a}) = 0 \text{ for } \omega \neq 0. \tag{75}$$

The above implies that *meq*1 vanishes if *α* → 1. Consequently, any oscillation disappears in that case.

**Note 4.3:** When 1 < *α* ≤ 2, we attain

$$0 < R\_{m1}(\omega, \mathfrak{a}) \le 1 \text{ for } \omega > 1. \tag{76}$$

Thus, we reveal an interesting phenomenon expressed by

$$m\_{eq1}(\omega, \mathfrak{a}) \le m \text{ for } 1 < \mathfrak{a} \le 2, \omega > 1. \tag{77}$$

The coefficient *Rm*1(*<sup>ω</sup>*, *α*) is plotted in Figure 1.

**Figure 1.** Plots of *Rm*1(*<sup>ω</sup>*, *<sup>α</sup>*). Solid line: *α* = 1.2. Dot line: *α* = 1.5. Dash line: *α* = 1.8.

**Remark 2.** *For α* ∈ *(0, 2), we have*

$$\lim\_{\omega \to \infty} m\_{\text{eq1}}(\omega, \mathfrak{a}) = 0. \tag{78}$$

The interesting and novel behavior, described above, implies that a fractional oscillator in Class I does not oscillate for *ω* → ∞ because it is equivalently massless in that case.

**Remark 3.** *For α* ∈ *(0, 2), we have*

$$\lim\_{\omega \to 0} m\_{\text{eq}1}(\omega, \mathfrak{a}) = \infty. \tag{79}$$

The interesting behavior, revealed above, says that a fractional oscillator of Class I type does not oscillate at *ω* = 0 because its mass is equivalently infinity in addition to the explanation of static status conventionally described by *ω* = 0.

4.1.3. Equivalent Damping of Fractional Oscillators of Class I

We now propose the equivalent damping.

**Theorem 3** (Equivalent damping I)**.** *The equivalent damping of a fractional oscillator in Class I, denoted by ceq*1, *is expressed by*

$$\mathcal{L}\_{eq1} = \mathcal{L}\_{eq1}(\omega, \mathfrak{a}) = \left(\omega^{\mathfrak{a}-1} \sin \frac{\mathfrak{a}\pi}{2}\right) m, 1 < \mathfrak{a} \le 2. \tag{80}$$

**Proof.** The second term on the left side of (59) is the friction with the linear viscous damping coefficient denoted by (80). The proof completes. 

Denote

$$R\_{c1}(\omega, a) = \omega^{a-1} \sin \frac{a\pi}{2}, 1 < a \le 2. \tag{81}$$

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

Then, we have

$$c\_{eq1}(\omega, \mathfrak{a}) = \mathcal{R}\_{\mathfrak{cl}}(\omega, \mathfrak{a})\mathfrak{m}.\tag{82}$$

The coefficient *Rc*1(*<sup>ω</sup>*, *α*) is indicated in Figure 2.

**Figure 2.** *Rc*1(*<sup>ω</sup>*, *<sup>α</sup>*). Solid line: *α* = 1.2. Dot line: *α* = 1.5. Dash line: *α* = 1.8.

**Remark 4.** *The equivalent damping I relies on ω, m, and α. It obeys the power law in terms of ω in the form*

$$
\omega\_{eq1}(\omega, \mathfrak{a}) \sim \omega^{a-1} m, 1 < a \le 2. \tag{83}
$$

**Note 4.4:** Because

$$\left.c\_{eq1}(\omega,\mathfrak{a})\right|\_{\mathfrak{a}=\mathfrak{2}} = 0,\tag{84}$$

we see again that a fractional oscillator of Class I type reduces to the conventional harmonic one when *α* = 2.

**Remark 5.** *An interesting behavior of ceq*1, *we found, is expressed by*

$$\lim\_{\omega \to \infty} c\_{eq1}(\omega, \mathfrak{a}) = \infty, 1 < \mathfrak{a} < 2. \tag{85}$$

The above says that the equivalent oscillator (59), as well as the fractional oscillator (58), never oscillates at *ω* → ∞ for 1 < *α* < 2 because its damping is infinitely large in that case. Due to

$$\lim\_{\omega \to 0} c\_{eq1}(\omega, a) = 0, 1 < a < 2,\tag{86}$$

we reveal a new damping behavior of a fractional oscillator in Class I in that it is equivalently dampingless for 1 < *α* < 2 at *ω* = 0.

*4.2. Equivalent Oscillation System for Fractional Oscillators of Class II Type*

4.2.1. Equivalent Oscillation Equation of Fractional Oscillators in Class II

Theorem 4 below describes the equivalent oscillator for the fractional oscillators of Class II type.

**Theorem 4** (Equivalent oscillator II)**.** *Denote a fractional oscillator in Class II by*

$$m\frac{d^2y\_2(t)}{dt^2} + c\frac{d^\beta y\_2(t)}{dt^\beta} + ky\_2(t) = 0,\\ 0 < \beta \le 1. \tag{87}$$

*Then, its equivalent 2-order oscillation equation is given by*

$$\int \left(m - c\omega^{\beta - 2} \cos \frac{\beta \pi}{2}\right) \frac{d^2 \mathbf{x}\_2(t)}{dt^2} + \left(c\omega^{\beta - 1} \sin \frac{\beta \pi}{2}\right) \frac{d \mathbf{x}\_2(t)}{dt} + k \mathbf{x}\_2(t) = 0,\\ 0 < \beta \le 1. \tag{88}$$

**Proof.** Consider the following equation:

$$m\frac{d^2h\_{y2}(t)}{dt^2} + c\frac{d^\beta h\_{y2}(t)}{dt^\beta} + kh\_{y2}(t) = \delta(t), 0 < \beta \le 1. \tag{89}$$

Denote by *Hy*2(*ω*) the Fourier transform of *hy*2(*t*). Then, it is its frequency transfer function. Taking the Fourier transform on the both sides of (89) yields

$$\left[-m\omega^2 + c(i\omega)^\beta + k\right]H\_{y2}(\omega) = 1, 0 < \beta \le 1. \tag{90}$$

With the principal value of *iβ*, (90) becomes

$$\begin{cases} \left[ -m\omega^2 + c(i\omega)^\beta + k \right] H\_\beta(\omega) = \left\{ -m\omega^2 + c\left( \cos\frac{\beta\pi}{2} + i\sin\frac{\beta\pi}{2} \right) \omega^\beta + k \right\} H\_\beta(\omega) \\ = \left( -m\omega^2 + c\omega^\beta \cos\frac{\beta\pi}{2} + k + ic\omega^\beta \sin\frac{\beta\pi}{2} \right) H\_{\beta2}(\omega) = 1. \end{cases} \tag{91}$$

The above means

$$H\_{y2}(\omega) = \frac{1}{-m\omega^2 + c\omega^6 \cos\frac{\beta\pi}{2} + k + ic\omega^6 \sin\frac{\beta\pi}{2}}.\tag{92}$$

On the other hand, we consider the equivalent oscillation equation II with the Dirac-δ excitation by

$$\left(m - c\omega^{\beta - 2}\cos\frac{\beta\pi}{2}\right)\frac{d^2h\_{\lambda 2}(t)}{dt^2} + \left(c\omega^{\beta - 1}\sin\frac{\beta\pi}{2}\right)\frac{dh\_{\lambda 2}(t)}{dt} + k\frac{d^2h\_{\lambda 2}(t)}{dt^2} = \delta(t), 0 < \beta \le 1. \tag{93}$$

Performing the Fourier transform on the both sides of the above produces

$$\begin{cases} -m\omega^2 + c\omega^6 \cos\frac{\beta\pi}{2} + ic\omega^6 \sin\frac{\beta\pi}{2}(i\omega) + k \Big] H\_{x2}(\omega) \\ = \left( -m\omega^2 + c\omega^6 \cos\frac{\beta\pi}{2} + k + ic\omega^6 \sin\frac{\beta\pi}{2} \right) H\_{x2}(\omega) = 1, \end{cases} \tag{94}$$

where *Hx*2(*ω*) the Fourier transform of *hx*2(*t*). Thus, from the above, we have

$$H\_{\rm x2}(\omega) = \frac{1}{-m\omega^2 + c\omega\vartheta\cos\frac{\beta\pi}{2} + k + ic\omega\vartheta\sin\frac{\beta\pi}{2}}.\tag{95}$$

Equations (92) and (95) imply

$$H\_{\mathfrak{Y}2}(\omega) = H\_{\mathfrak{x}2}(\omega). \tag{96}$$

Hence, (88) is the equivalent oscillation equation of the fractional oscillators of Class II. This completes the proof.

4.2.2. Equivalent Mass of Fractional Oscillators of Class II

The equivalent mass of the fractional oscillators of Class II type is presented in Theorem 5.

**Theorem 5** (Equivalent mass II)**.** *Let meq*2 *be the equivalent mass of the fractional oscillators of Class II type. Then,*

$$m\_{c\eta2} = m\_{c\eta2}(\omega, \beta) = m - c\omega\ell^{\beta - 2}\cos\frac{\beta\pi}{2}, 0 < \beta \le 1. \tag{97}$$

**Proof.** Consider the Newton's second law. Then, we see that the inertia force in the equivalent oscillator II is *m* − *cωβ*−<sup>2</sup> cos *βπ*2 *d*2*x*2 *dt*<sup>2</sup> . Therefore, (97) holds. The proof completes. 

From Theorem 5, we reveal a power law phenomenon with respect to the equivalent mass II.

**Remark 6.** *The equivalent mass meq*2 *obeys the power law in terms of ω in the form*

$$
\omega m\_{eq2} \sim -c\omega^{\beta - 2}, 0 < \beta \le 1. \tag{98}
$$

**Note 4.5:** Equation (97) exhibits that *meq*2 is related to the oscillation frequency *ω*, the fractional order *β*, the primary mass *m*, and the primary damping *c*.

**Remark 7.** *For 0 < β* ≤ *1, we have*

$$\lim\_{\omega \to \infty} m\_{\text{eq2}}(\omega, \beta) = m. \tag{99}$$

Figure 3 shows its plots for *m* = *c* = 1 with the part of *meq*2(*<sup>ω</sup>*, *β*)> 0.

**Figure 3.** Plots of *meq*2(*<sup>ω</sup>*, *β*) > 0 for *m* = *c* = 1.

**Remark 8.** *For 0 < β < 1, we have*

$$\lim\_{\omega \to 0} m\_{eq2}(\omega, \beta) = -\infty. \tag{100}$$

**Note 4.6:** The equivalent mass II is negative if *ω* is small enough. Figure 4 exhibits the negative part of *meq*2(*<sup>ω</sup>*, *β*).

**Figure 4.** Illustrating negative part of *meq*2(*<sup>ω</sup>*, *β*) for *m* = *c* = 1. (**a**) *β* = 0.9. (**b**) *β* = 0.7. (**c**) *β* = 0.5. (**d**) *β* = 0.3.

**Remark 9.** *We restrict our research for meq*2(*<sup>ω</sup>*, *β*) *> 0.*

**Note 4.7:** The equivalent mass II reduces to the primary mass *m* for *β* = 1 as indicated below.

$$\left.m\_{eq2}(\omega,\beta)\right|\_{\beta=1} = m.\tag{101}$$

In fact, a fractional oscillator in Class II reduces to the conventional oscillator below if *β* = 1

$$m\frac{d^2x\_2}{dt^2} + c\frac{dx\_2}{ct} + kx\_2 = 0.$$

### 4.2.3. Equivalent Damping of Fractional Oscillators in Class II

Let *ceq*2 be the equivalent damping of a fractional oscillator in Class II. Then, we put forward the expression of *ceq*2 with Theorem 6.

**Theorem 6** (Equivalent damping II)**.** *The equivalent damping of the fractional oscillators in Class II is in the form*

$$\mathfrak{c}\_{\text{eq2}} = \mathfrak{c}\_{\text{eq2}}(\omega, \beta) = \mathfrak{c}\omega^{\beta - 1} \sin \frac{\beta \pi}{2}, 0 < \beta \le 1. \tag{102}$$

**Proof.** The second term on the left side of (88) is the friction force with the linear viscous damping coefficient denoted by (102). The proof completes. 

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

> Denote by

$$R\_{c2}(\omega, \beta) = \omega^{\beta - 1} \sin \frac{\beta \pi}{2}, 0 < \beta \le 1. \tag{103}$$

Then, we have

$$c\_{eq2}(\omega,\beta) = \mathcal{R}\_{c2}(\omega,\beta)\varepsilon.\tag{104}$$

Figure 5 indicates *Rc*2(*<sup>ω</sup>*, *β*).

**Figure 5.** Indication of *Rc*2(*<sup>ω</sup>*, *β*) Solid line: *β* = 0.9. Dot line: *β* = 0.6. Dash line: *β* = 0.3.

**Remark 10.** *The equivalent damping ceq*2 *is associated with the oscillation frequency ω, the primary damping c, and the fractional order β. It follows the power law in terms of ω in the form*

$$
\omega\_{eq2}(\omega,\beta) \sim \omega^{\beta - 1}c, 0 < \beta \le 1. \tag{105}
$$

**Note 4.8:** The following says that *ceq*2 reduces to the primary damping *c* if *β* = 1.

$$\left.c\_{\text{eq2}}(\omega,\beta)\right|\_{\beta=1} = \text{c.}\tag{106}$$

**Remark 11.** *The equivalent damping ceq*2 *has, for β* ∈ *(0, 1), the property given by*

$$\lim\_{\omega \to \infty} c\_{eq2}(\omega, \beta) = 0.\tag{107}$$

**Note 4.9:** The equivalent oscillation equation of Class II fractional oscillators reduces to *md*2*x*2(*t*) *dt*<sup>2</sup> + *kx*2(*t*) = 0 in the two cases. One is *ω* → <sup>∞</sup>, see Remark 7 and Remark 12. The other is *c* = 0.

**Note 4.10:** Remark 5 for lim*ω*→ ∞*ceq*1(*<sup>ω</sup>*, *β*) = ∞ and Remark 11 just above sugges<sup>t</sup> a substantial difference between two types of fractional oscillators from the point of view of the damping at *ω* → ∞.

**Remark 12.** *The equivalent damping ceq*2 *has, for β*∈ *(0, 1), the asymptotic property for ω* → *0 in the form*

$$\lim\_{\omega \to 0} c\_{\text{eq2}}(\omega, \beta) = \infty. \tag{108}$$

The above property implies that a fractional oscillator in Class II does not oscillate at *ω* → 0 because not only it is in static status but also its equivalent damping is infinitely large.

### *4.3. Equivalent Oscillation System for Fractional Oscillators of Class III*

### 4.3.1. Equivalent Oscillation Equation of Fractional Oscillators in Class III

We present Theorem 7 below to explain the equivalent oscillation equation for the fractional oscillators of Class III.

**Theorem 7** (Equivalent oscillator III)**.** *Denote a fractional oscillation equation in Class III by*

$$m\frac{d^a y\_3(t)}{dt^a} + c\frac{d^\beta y\_3(t)}{dt^\beta} + ky\_3(t) = 0,\\ 1 < a \le 2, \ 0 < \beta \le 1. \tag{109}$$

*Then, its equivalent oscillator of order 2 for 1 < α* ≤ *2 and 0 < β* ≤ *1 is in the form*

$$\begin{cases} -\left(m\omega^{\alpha-2}\cos\frac{a\pi}{2} + c\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right)\frac{d^2x\_3(t)}{dt^2} \\ + \left(m\omega^{\alpha-1}\sin\frac{a\pi}{2} + c\omega^{\beta-1}\sin\frac{\beta\pi}{2}\right)\frac{d\mathbf{x}\_3(t)}{dt} + k\mathbf{x}\_3(t) = 0. \end{cases} \tag{110}$$

**Proof.** Let us consider the equation

$$m\frac{d^a h\_{\mathcal{Y}\mathcal{S}}(t)}{dt^a} + c\frac{d^\beta h\_{\mathcal{Y}\mathcal{S}}(t)}{dt^\beta} + kh\_{\mathcal{Y}\mathcal{S}}(t) = \delta(t), 1 < a \le 2, \ 0 < \beta \le 1. \tag{111}$$

Let *Hy*3(*ω*) be the Fourier transform of *hy*3(*t*). Doing the Fourier transform on the both sides of the above results in

$$\left[m\left(i\omega\right)^{a} + \varepsilon\left(i\omega\right)^{\beta} + k\right]H\_{\mathcal{Y}}\left(\omega\right) = 1, 1 < a \le 2, \ 0 < \beta \le 1. \tag{112}$$

Taking into account the principal values of *iα* and *iβ*, (112) becomes

$$\begin{aligned} &\left[m(i\omega)^a + c(i\omega)^\beta + k\right]H\_{y3}(\omega) \\ &= \left[m\left(\cos\frac{a\pi}{2} + i\sin\frac{a\pi}{2}\right)\omega^a + c\left(\cos\frac{\beta\pi}{2} + i\sin\frac{\beta\pi}{2}\right)\omega^\beta + k\right]H\_{y3}(\omega) \\ &= \left[m\omega^a \cos\frac{a\pi}{2} + c\omega^\beta \cos\frac{\beta\pi}{2} + k + i\left(m\omega^a \sin\frac{a\pi}{2} + c\omega^\beta \sin\frac{\beta\pi}{2}\right)\right]H\_{y3}(\omega) = 1. \end{aligned} \tag{113}$$

Consequently, we have

$$H\_{y\Im}(\omega) = \frac{1}{m\omega^a \cos\frac{a\pi}{2} + c\omega^\beta \cos\frac{\beta\pi}{2} + k + i\left(m\omega^a \sin\frac{a\pi}{2} + c\omega^\beta \sin\frac{\beta\pi}{2}\right)}}.\tag{114}$$

On the other hand, considering the equivalent oscillator III driven by the Dirac-δ function, we have

$$-\left(m\omega^{a-2}\cos\frac{\eta\pi}{2} + c\omega^{6-2}\cos\frac{\theta\pi}{2}\right)\frac{d^2h\_{\pi3}(t)}{dt^2} + \left(m\omega^{a-1}\sin\frac{\theta\pi}{2} + c\omega^{6-1}\sin\frac{\theta\pi}{2}\right)\frac{dh\_{\pi3}(t)}{dt} + kh\_{\pi3}(t) = \delta(t).\tag{115}$$

When doing the Fourier transform on the both sides of the above, we obtain

$$\left(m\omega^{\mu}\cos\frac{a\pi}{2} + c\omega^{\beta}\cos\frac{\beta\pi}{2}\right)H\_{13}(\omega) + i\left(m\omega^{\mu}\sin\frac{a\pi}{2} + c\omega^{\beta}\sin\frac{\beta\pi}{2}\right)H\_{13}(\omega) + kH\_{13}(\omega) = 1,\tag{116}$$

where *Hx*3(*ω*) is the Fourier transform of *hx*3(*t*). Therefore, from the above, we ge<sup>t</sup>

$$H\_{\mathbf{x}\Im}(\omega) = \frac{1}{m\omega^{\mathfrak{a}}\cos\frac{a\pi}{2} + c\omega^{\mathfrak{b}}\cos\frac{\beta\pi}{2} + k + i\left(m\omega^{\mathfrak{a}}\sin\frac{a\pi}{2} + c\omega^{\mathfrak{b}}\sin\frac{\beta\pi}{2}\right)}.\tag{117}$$

Two expressions, (114) and (117), imply that

$$H\_{\mathbf{x}\mathfrak{J}}(\omega) = H\_{\mathfrak{Y}\mathfrak{J}}(\omega). \tag{118}$$

Thus, Theorem 7 holds. 

4.3.2. Equivalent Mass of Fractional Oscillators in Class III

From Section 4.3.1, we propose the equivalent mass of the fractional oscillators in Class III type by Theorem 8.

**Theorem 8** (Equivalent mass III)**.** *Let meq*3 *be the equivalent mass of the fractional oscillators in Class III. Then, for 1 < α* ≤ *2 and 0 < β* ≤ *1,*

$$m\_{\text{eq3}} = m\_{\text{eq3}}(\omega, a, \beta) = -\left(m\omega^{a-2}\cos\frac{a\pi}{2} + c\omega^{\beta-2}\cos\frac{\beta\pi}{2}\right). \tag{119}$$

**Proof.** When considering the Newton's second law in the equivalent oscillator III (110), we immediately see that Theorem 8 holds. 

**Remark 13.** *The equivalent mass meq*3 *obeys the power law in terms of ω.*

**Note 4.11:** The equivalent mass *meq*3 is related to *ω*, *m*, and *c*, as well as a pair of fractional orders (*<sup>α</sup>*, *β*).

**Note 4.12:** If *α* = 2 and *β* = 1, *meq*3 reduces to the primary *m*, i.e.,

$$m\_{eq3}(\omega,\alpha,\beta)|\_{\alpha=2,\beta=1} = m. \tag{120}$$

As a matter of fact, a fractional oscillator of Class III reduces to the ordinary oscillator when *α* = 2 and *β* = 1.

**Remark 14.** *In the case of ω* → <sup>∞</sup>*, we obtain*

$$\lim\_{\omega \to \infty} m\_{\text{eq}3}(\omega, \mathfrak{a}, \mathfrak{z} \mathfrak{z}) = 0, 1 < \mathfrak{a} < 2, \ 0 < \mathfrak{z} < 1. \tag{121}$$

Therefore, we sugges<sup>t</sup> that a fractional oscillator in Class III does not oscillate for *ω* → ∞ because its equivalent mass disappears in that case. Figure 6 shows its positive part for *α* = 1.5, *β* = 0.9, *m* = *c* = 1.

**Figure 6.** Indicating the positive part of *meq*3(*<sup>ω</sup>*, *α*, *β*) for *α* = 1.5, *β* = 0.9, *m* = *c* = 1.

**Remark 15.** *In the case of ω* → *0, we obtain*

$$\lim\_{\omega \to 0} m\_{\text{eq}\beta}(\omega, \mathfrak{a}, \mathfrak{z}, \beta) = -\infty, 1 < \mathfrak{a} < 2, \ 0 < \beta < 1. \tag{122}$$

In fact, if *ω* is small enough, *meq*3(*<sup>ω</sup>*, *α*, *β*) will be negative, see Figure 7.

**Figure 7.** Negative part of *meq*3(*<sup>ω</sup>*, *α*, *β*) for *m* = *c* = 1 and *β* = 0.9. Solid line: *α* = 1.9. Dot line: *α* = 1.6. Dash line: *α* = 1.3.

**Remark 16.** *This research restricts meq*3(*<sup>ω</sup>*, *α*, *β*) ∈ *(0,* ∞*).*

4.3.3. Equivalent Damping of Fractional Oscillators in Class III

Let *ceq*3 be the equivalent damping of a fractional oscillator of Class III type. Then, we propose its expression with Theorem 9.

**Theorem 9** (Equivalent damping III)**.** *The equivalent damping of the fractional oscillators in Class III is given by, for 1 < α* ≤ *2 and 0 < β* ≤ *1,*

$$\mathcal{L}\_{\text{eq3}} = \mathcal{L}\_{\text{eq3}}(\omega, \mathfrak{a}, \beta) = m\omega^{a-1} \sin \frac{a\pi}{2} + c\omega^{\beta - 1} \sin \frac{\beta\pi}{2}. \tag{123}$$

**Proof.** The second term on the left side of the equivalent oscillator III is the friction force with the linear viscous damping coefficient denoted by (123). Thus, the proof completes. 

**Remark 17.** *The equivalent damping ceq*3 *relates to ω, m, c, and a pair of fractional orders (<sup>α</sup>, β). It obeys the power law in terms of ω. It contains two terms. The first term is hyperbolically increasing in ωα*−<sup>1</sup> *as α > 1 and the second hyperbolically decayed with ωβ*−<sup>1</sup> *since β < 1.*

**Note 4.13:** From (123), we see that *ceq*3 reduces to the primary damping *c* for *α* = 2 and *β* = 1. That is,

$$\left.c\_{eq3}(\omega, a, \beta)\right|\_{a=2, \beta=1} = c.\tag{124}$$

**Remark 18.** *One asymptotic property of ceq*3 *for ω* → <sup>∞</sup>*, due to* lim*ω*→∞*ωα*−<sup>1</sup> = ∞ *for 1 < α* ≤ *2, is given by*

$$\lim\_{\omega \to \infty} c\_{eq3}(\omega, \mathfrak{a}, \beta) = \infty. \tag{125}$$

The above says that a fractional oscillator of Class III does not vibrate for *ω* → ∞.

**Remark 19.** *Another asymptotic property of ceq*3 *in terms of ω for ω* → *0, owing to* lim*ω*→0*<sup>ω</sup>β*−<sup>1</sup> = ∞ *for 0 < β < 1, is expressed by*

$$\lim\_{\omega \to 0} c\_{\text{eq}3}(\omega, \alpha, \beta) = \infty. \tag{126}$$

A system does not vibrate obviously in the case of *ω* → 0 but Remark 19 suggests a new view about that. Precisely, its equivalent damping is infinitely large at *ω* → 0. Figures 8 and 9 illustrate *ceq*3(*<sup>ω</sup>*, *α*, *β*) for *m* = *c* = 1.

**Figure 8.** Plots of equivalent damping III for *m* = *c* = 1. Solid line: *α* = 1.9. Dot line: *α* = 1.6. Dash line: *α* = 1.3. (**a**) For *β* = 0.9. (**b**) For *β* = 0.7. (**c**) For *β* = 0.5. (**d**) For *β* = 0.3.

**Figure 9.** Plots of *ceq*3(*<sup>ω</sup>*, *α*, *β*) for *m* = *c* = 1. Solid line: *β* = 0.8. Dot line: *β* = 0.5. Dash line: *β* = 0.3. (**a**) For *α* = 1.8. (**b**) For *α* = 1.3.

**Note 4.14:** The equivalent damping *ceq*3= 0 if both *α* = 2 and *c* = 0:

$$\left.c\_{\text{eq3}}(\omega, a, \beta)\right|\_{\mathfrak{a}=2, \mathfrak{c}=\mathfrak{0}} = 0.\tag{127}$$
