Dealing with Stationary Sinusoidal Responses of Seven Types of Multi-Fractional Vibrators Using Multi-Fractional Phasor

Abstract

The novelty and main contributions of this paper are reflected in four aspects. First, we introduce multi-fractional phasor in Theorem 1. Second, we propose the motion phasor equations of seven types of multi-fractional vibrators in Theorems 2, 12, 22, 32, 43, 54, and 65, respectively. Third, we present the analytical expressions of response phasors of seven types of multi-fractional vibrators in Theorems 10, 20, 30, 41, 52, 63, and 74, respectively. Fourth, we bring forward the analytical expressions of stationary sinusoidal responses of seven types of multi-fractional vibrators in Theorems 11, 21, 31, 42, 53, 64, and 75, respectively. In addition, by using multi-fractional phasor, we put forward the analytical expressions of vibration parameters (equivalent mass, equivalent damping, equivalent stiffness, equivalent damping ratio, equivalent damping free natural angular frequency, equivalent damped natural angular frequency, equivalent frequency ratio) and frequency transfer functions of seven types of multi-fractional vibrators. Demonstrations exhibit that the effects of multi-fractional orders on stationary sinusoidal responses of those multi-fractional vibrators are considerable.

1. Introduction

The topic of fractional vibrations remains hot, see, e.g., Uchaikin ([1], Chap. 7), Duan [2], Duan et al. [3,4], Pskhu and Rekhviashvili [5], Zelenev et al. [6], Freundlich [7,8], Al-rabtah et al. [9], Zurigat [10], Blaszczyk [11], Blaszczyk and Ciesielski [12], Blaszczyk et al. [13], Achar et al. [14,15,16]), Rossikhin and Shitikova [17,18,19,20], Rossikhin [21], Shitikova [21,22,23], Shitikova et al. [24], El-Nabulsi and Anukool [25], Liu et al. [26], Sofi [27], Li [28,29,30], simply mentioning a few.

There are three research shortcomings in the field of fractional vibrations. The first is that research works, such as those on the responses of fractional vibrations, are usually in numerical computations, see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61], to mention a few. The analytical expressions are rarely reported, with the author’s previous work being among the few to report on these expressions [28,29,30]. The second is that reports regarding the multi-fractional vibrators in the forms of (1)–(7) below

$$m{\displaystyle \frac{d^{\alpha (\omega )}{x}_1(t)}{d{t}^{\alpha (\omega )}}}+k{x}_1(t)=f(t),1<\alpha \left(\omega \right)<3,$$

(1)

$$m{\displaystyle \frac{d^2{x}_2(t)}{d{t}^2}}+c{\displaystyle \frac{d^{\beta (\omega )}{x}_2(t)}{d{t}^{\beta (\omega )}}}+k{x}_2(t)=f(t),0<\beta \left(\omega \right)<2,$$

(2)

$$m{\displaystyle \frac{d^{\alpha (\omega )}{x}_3(t)}{d{t}^{\alpha (\omega )}}}+c{\displaystyle \frac{d^{\beta (\omega )}{x}_3(t)}{d{t}^{\beta (\omega )}}}+k{x}_3(t)=f(t),1<\alpha \left(\omega \right)<3,0<\beta \left(\omega \right)<2,$$

(3)

$$m{\displaystyle \frac{d^{\alpha (\omega )}{x}_4(t)}{d{t}^{\alpha (\omega )}}}+k{\displaystyle \frac{d^{\lambda (\omega )}{x}_4(t)}{d{t}^{\lambda (\omega )}}}=f(t),1<\alpha \left(\omega \right)<3,0\le \lambda \left(\omega \right)<1,$$

(4)

$$m{\displaystyle \frac{d^2{x}_5(t)}{d{t}^2}}+k{\displaystyle \frac{d^{\lambda (\omega )}{x}_5(t)}{d{t}^{\lambda (\omega )}}}=f(t),0\le \lambda \left(\omega \right)<1,$$

(5)

$$m{\displaystyle \frac{d^{\alpha (\omega )}{x}_6(t)}{d{t}^{\alpha (\omega )}}}+c{\displaystyle \frac{d^{\beta (\omega )}{x}_6(t)}{d{t}^{\beta (\omega )}}}+k{\displaystyle \frac{d^{\lambda (\omega )}{x}_6(t)}{d{t}^{\lambda (\omega )}}}=f(t),1<\alpha \left(\omega \right)<3,0<\beta \left(\omega \right)<2,0\le \lambda \left(\omega \right)<1,$$

(6)

$$m{\displaystyle \frac{d^2{x}_7(t)}{d{t}^2}}+c{\displaystyle \frac{d^{\beta (\omega )}{x}_7(t)}{d{t}^{\beta (\omega )}}}+k{\displaystyle \frac{d^{\lambda (\omega )}{x}_7(t)}{d{t}^{\lambda (\omega )}}}=f(t),0<\beta \left(\omega \right)<2,0\le \lambda \left(\omega \right)<1,$$

(7)

are seldom seen, where α(ω), β(ω), and λ(ω) are varying orders in terms of vibration frequency ω, either deterministically or randomly. In fact, reports regarding (3)–(7) are rarely seen even in the case of α(ω), or β(ω), or λ(ω) being constant. The third is the lack of studies that investigate stationary responses to sinusoidal excitations using fractional phasor. Note that the author’s previous work [30] did not consider the seventh type of multi-fractional vibrators (7). In addition, it did not report the stationary sinusoidal responses of (1)–(7) either. This paper aims at filling up the mentioned research gaps by bringing forward the analytical expressions of stationary sinusoidal responses of seven types of multi-fractional vibration systems (1)–(7) methodologically using multi-fractional phasor, as newly introduced in this research.

The rest of the paper is organized as follows. In Section 2, we describe the fractional phasor and introduce the multi-fractional phasor. We present the analytic expressions of the results (equivalent motion phasor equations, equivalent masses, equivalent dampings, equivalent stiffnesses, equivalent damping ratios, equivalent damping free natural angular frequencies, equivalent damped natural angular frequencies, equivalent frequency ratios, frequency transfer functions, response phasors, and stationary sinusoidal responses) of seven types of multi-fractional vibrators in Section 3, Section 4, Section 5, Section 6, Section 7, Section 8 and Section 9, respectively. Discussions are provided in Section 10, followed by concluding remarks and a summary.

2. Multi-Fractional Phasor

Steinmetz introduced the theory and method of phasor in electrical engineering [62,63,64]. Den Hartog’s work [65] may be an early literature applying the phasor method to deal with stationary responses to sinusoidal excitations in vibrations. It is now a widely used method in vibrations (Jin and Xia [66], Harris [67], Nakagawa and Ringo [68], Palley et al. [69], Grote and Antonsson [70], Allemang and Avitabile [71], Soong and Grigoriu [72], Rothbart and Brown [73]).

When f(t) = F₀sin(ωt + θ), with the Euler formula, we write f(t) = Im{F₀exp[i(ωt + θ)]}. Denote the phasor of f(t) by F. It is given by

F = F₀exp(iθ).

(8)

Therefore, one can write

f(t) = Im[Fexp(iωt)].

(9)

Let n be a positive integer. Then, by using phasor,

f⁽ⁿ⁾(t) = (iω)ⁿIm[Fexp(iωt)].

(10)

When a vibration system is driven by a sinusoidal excitation below

Mx″(t) + cx′(t) + kx(t) = F₀sin(ωt + θ),

(11)

we can change the above differential equation to an algebraic one in the form

(iω)2mX + (iω)cX + kX = F,

(12)

where X is the phasor of response x(t) in the form

$$X={\displaystyle \frac{F}{\left(k-m{\omega }^2\right)+i\omega c}}.$$

(13)

Denote by H(ω) the frequency transfer function. It is in the form

$$H(\omega )={\displaystyle \frac{1}{\left(k-m{\omega }^2\right)+i\omega c}}.$$

(14)

Using the damping ratio $\varsigma ={\displaystyle \frac{c}{2\sqrt{mk}}}$ and frequency ratio $\gamma ={\displaystyle \frac{\omega }{{\omega }_n}},$ where ${\omega }_n=\sqrt{{\displaystyle \frac{k}{m}}},$ H(ω) can be written by

$$H(\omega )={\displaystyle \frac{1}{k\left(1-{\gamma }^2+i2\varsigma \gamma \right)}}.$$

(15)

Using the polar system, we have

$$H(\omega )={\displaystyle \frac{1}{k\sqrt{{\left(1-{\gamma }^2\right)}^2+{\left(2\varsigma \gamma \right)}^2}}}{e}^{-\phi (\omega )},$$

(16)

where

$$\phi (\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }^2}{\sqrt{{\left(1-{\gamma }^2\right)}^2+{\left(2\varsigma \gamma \right)}^2}}}.$$

(17)

Therefore, the response x(t) is given by

x(t) = Im[Xexp(iωt)] = |H(ω)|F₀sin(ωt + θ − φ).

(18)

Equation (18) expresses the stationary sinusoidal solution to the second-order differential Equation (11). It is algebraically derived using the phasor method. In order to deal with stationary sinusoidal solution to a fractional vibrator using the tool of phasor, fractional phasor is necessary. Further, when dealing with stationary sinusoidal solution to a multi-fractional vibrator using the method of phasor, multi-fractional phasor is essential. We express fractional phasor and multi-fractional one in Lemma 1 and Theorem 1 below, respectively.

Lemma 1 (Fractional phasor).

Suppose that y(t) = Y₀sin(ωt + θ). Let Y be the phasor of y(t). It is in the form of Y = Y₀exp(iθ). The phasor of y^(v)(t) (v ≥ 0) is given by (iω)^vY.

Proof. 

Since y^(v)(t) = Im{(iω)^vY₀exp[i(ωt + θ)]} = Im[(iω)^vYexp(iωt)], the phasor of y^(v)(t) is (iω)^vY. Lemma 1 holds. The proof is finished. □

Theorem 1 (Multi-fractional phasor).

Suppose that y(t) = Y₀sin(ωt + θ). Denote by Y the phasor of y(t). It is in the form of Y = Y₀exp(iθ). The phasor of${\displaystyle \frac{d^{v(\omega )}y(t)}{d{t}^{v(\omega )}}}$ (v ≥ 0) is given by${(i\omega )}^{v(\omega )}Y.$

Proof. 

Using the Euler formula to express y(t) produces y(t) = Im{Y₀exp[i(ωt + θ)]} = Im[Yexp(iωt)]. Thus, ${\displaystyle \frac{d^{v(\omega )}y(t)}{d{t}^{v(\omega )}}}=\mathrm{Im}\left[{(i\omega )}^{v(\omega )}Y\exp (i\omega t)\right]=\mathrm{Im}\left\{\left[{(i\omega )}^{v(\omega )}Y\right]\exp (i\omega t)\right\}.$ Hence, Theorem 1 holds. The proof is finished. □

Note that the concept of fractional phasor and its expression can be seen in the field of electronic and electrical engineering, see, e.g., Zhang and Shu [74], Sarafraz and Tavazoei [75,76], Jia et al. [77], Shamim et al. [78], Malek et al. [79], Zhao et al. [80], Adhikary et al. [81], to mention a few. However, the reports regarding the concept and expression of multi-fractional phasor are rarely seen. Therefore, we use the term lemma to describe the fractional phasor (Lemma 1) since it is a known result and the terminology theorem to express the multi-fractional phasor (Theorem 1) because it is newly introduced in this paper.

3. Results Regarding Multi-Fractional Vibrators of Type 1

In this Section, with respect to multi-fractional vibrators of type 1, we address the motion phasor equation in Section 3.1, the expressions of equivalent vibration parameters in Section 3.2, the expression of frequency transfer function in Section 3.3, and the expressions of stationary sinusoidal responses, respectively, in phasor and time to a type 1 multi-fractional vibrator in Section 3.4.

3.1. Motion Phasor Equation of Type 1 Multi-Fractional Vibrator

Theorem 2 (Motion phasor equation 1).

The motion phasor equation of a type 1 multi-fractional vibrator is expressed by

$$m{(i\omega )}^{\alpha (\omega )}{X}_1+k{X}_1=F,1<\alpha \left(\omega \right)<3,$$

(19)

where X₁ is the phasor of response x₁(t) and F is the phasor of excitation f(t) = F₀sin(ωt + θ).

Proof. 

Expressing both sides of $m{\displaystyle \frac{d^{\alpha (\omega )}{x}_1(t)}{d{t}^{\alpha (\omega )}}}+k{x}_1(t)=f(t)$ using phasor results in (19). This finishes the proof. □

Consider the principal branch of $i^{\alpha (\omega )}.$ The left side of (19) can be expressed by

$$m{(i\omega )}^{\alpha (\omega )}{X}_1+k{X}_1=\left[m{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+im{\omega }^{\alpha }\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k\right]X_1.$$

(20)

Writing the right side of the above by

$$\begin{array}{l}\left[m{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+im{\omega }^{\alpha (\omega )}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k\right]X_1\\ =\left[{(i\omega )}^2(-1)m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+(i\omega )m{\omega }^{\alpha -1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k\right]X_1,\end{array}$$

(21)

we express the motion phasor equation of a type 1 multi-fractional vibrator by

$$m{(i\omega )}^{\alpha (\omega )}{X}_1+k{X}_1=\left[{(i\omega )}^2(-1)m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+(i\omega )m{\omega }^{\alpha -1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k\right]X_1.$$

(22)

Note that $m{\displaystyle \frac{d^{\alpha (\omega )}{x}_1(t)}{d{t}^{\alpha (\omega )}}}$ implies fractional inertia force. Hence, 1 < α(ω) < 3.

3.2. Equivalent Parameters of Type 1 Multi-Fractional Vibrator

The equivalent vibration parameters for a type 1 multi-fractional vibrator include equivalent mass, equivalent damping, equivalent damping ratio, equivalent damping free natural angular frequency, equivalent damped natural angular frequency, and equivalent frequency ratio.

3.2.1. Equivalent Mass of Type 1 Multi-Fractional Vibrator

Theorem 3 (Equivalent mass 1).

Let m_eq1 be the equivalent mass of a multi-fractional vibrator of type 1. Then,

$$m_{\mathrm{eq}1}=-m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}.$$

(23)

Proof. 

From the right side of (22), we have

$$\begin{array}{l}\left[{(i\omega )}^2(-1)m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+(i\omega )m{\omega }^{\alpha -1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k\right]X_1\\ =\left[{(i\omega )}^2{m}_{\mathrm{eq}1}+(i\omega )m{\omega }^{\alpha -1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k\right]X_1.\end{array}$$

(24)

In the above, m_eq1 is the equivalent mass for a multi-fractional vibrator of type 1. This finishes the proof. □

Figure 1 indicates some plots of m_eq1.

3.2.2. Equivalent Damping of Type 1 Multi-Fractional Vibrator

Theorem 4 (Equivalent damping 1).

Let c_eq1 be the equivalent damping of a multi-fractional vibrator of type 1. Then,

$$c_{\mathrm{eq}1}=m{\omega }^{\alpha (\omega )-1}sin{\displaystyle \frac{\alpha (\omega )\pi }{2}},1<\alpha (\omega )<3.$$

(25)

Proof. 

From the second item on the right side of (24), we see that (25) is true. The proof is finished. □

Some plots of c_eq1 are indicated in Figure 2.

3.2.3. Equivalent Damping Ratio of Type 1 Multi-Fractional Vibrator

Denote by ζ_eq1 the equivalent damping ratio of a multi-fractional vibrator of type 1. Define it by

$${\varsigma }_{\mathrm{eq}1}={\displaystyle \frac{c_{\mathrm{eq}1}}{2\sqrt{m_{\mathrm{eq}1}k}}}.$$

(26)

Theorem 5 (Equivalent damping ratio 1).

The expression of ζ_eq1 is given by

$${\varsigma }_{\mathrm{eq}1}={\displaystyle \frac{{\omega }^{\frac{\alpha (\omega )}{2}}\sin \frac{\alpha (\omega )\pi }{2}}{2{\omega }_n\sqrt{-\cos \frac{\alpha (\omega )\pi }{2}}}}.$$

(27)

Proof. 

Substituting m_eq1 and c_eq1 into (26) yields (27). The proof is finished. □

Figure 3 illustrates some plots of ζ_eq1.

3.2.4. Equivalent Damping Free Natural Angular Frequency of Type 1 Multi-Fractional Vibrator

Introduce a term of equivalent damping free natural angular frequency. Let it be ω_eqn1. Define it by

$${\omega }_{\mathrm{eqn}1}=\sqrt{{\displaystyle \frac{k}{m_{\mathrm{eq}1}}}}.$$

(28)

Theorem 6 (Equivalent damping free natural angular frequency 1).

The quantity ω_eqn1 is given by

$${\omega }_{\mathrm{eqn}1}={\displaystyle \frac{{\omega }_n}{\sqrt{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}}}}.$$

(29)

Proof. 

Substituting m_eq1 into (28) produces (29). The proof is finished. □

Some plots of ω_eqn1 are shown in Figure 4.

3.2.5. Equivalent Damped Natural Angular Frequency of Type 1 Multi-Fractional Vibrator

Without losing the generality in vibration engineering, we only consider |ζ_eq1| ≤ 1 in what follows. Let ω_eqd1 be the equivalent damped natural frequency of a type 1 multi-fractional vibrator. Define it by

$${\omega }_{\mathrm{eqd}1}={\omega }_{\mathrm{eqn}1}\sqrt{1-{\varsigma }_{\mathrm{eq}1}^2}.$$

(30)

Theorem 7 (Equivalent damped natural angular frequency 1).

The quantity ω_eqd1 is expressed by

$${\omega }_{\mathrm{eqd}1}={\displaystyle \frac{{\omega }_n}{\sqrt{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha \pi }{2}}}}\sqrt{1-{\displaystyle \frac{{\omega }^{\alpha (\omega )}{\sin }^2\frac{\alpha (\omega )\pi }{2}}{4{\omega }_n^2\left|\cos \frac{\alpha (\omega )\pi }{2}\right|}}}.$$

(31)

Proof. 

Substituting ζ_eq1 and ω_eqn1 into (30) yields (31). This finishes the proof. □

Some plots of ω_eqd1 are indicated in Figure 5.

3.2.6. Equivalent Frequency Ratio of Type 1 Multi-Fractional Vibrator

Let γ_eq1 be the equivalent frequency ratio of a multi-fractional vibrator of type 1. Define it by

$${\gamma }_{\mathrm{eq}1}={\displaystyle \frac{\omega }{{\omega }_{\mathrm{eqn}1}}}.$$

(32)

Theorem 8 (Equivalent frequency ratio 1).

The quantity γ_eq1 is in the form

$${\gamma }_{\mathrm{eq}1}={\displaystyle \frac{\omega \sqrt{{\omega }^{\alpha (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|}}{{\omega }_n}}.$$

(33)

Proof. 

Substituting ω_eqn1 into (32) produces (33). The proof is finished. □

Some plots of γ_eq1 are shown in Figure 6.

3.3. Frequency Transfer Function of Type 1 Multi-Fractional Vibrator

Theorem 9 (Frequency transfer function 1).

Denote by H₁(ω) the frequency transfer function of a type 1 multi-fractional vibrator in phasor. Then,

$$H_1(\omega )={\displaystyle \frac{X_1}{F}}={\displaystyle \frac{1}{k\left(1-\frac{{\omega }^{\alpha (\omega )}}{{\omega }_n^2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|+i\frac{{\omega }^{\alpha (\omega )}}{{\omega }_n^2}\sin \frac{\alpha (\omega )\pi }{2}\right)}}.$$

(34)

Proof. 

Considering the motion phasor equation in the form

$${(i\omega )}^{\alpha (\omega )}{X}_1+{\omega }_n^2{X}_1=\left[{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+i\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+{\omega }_n^2\right]X_1={\displaystyle \frac{F}{m}},$$

(35)

we have (34). The proof is finished. □

Write the motion phasor equation by

$${(i\omega )}^2{X}_1+i2{\zeta }_{\mathrm{eqn}1}{\omega }_{\mathrm{eqn}1}{X}_1+{\omega }_{\mathrm{eqn}1}^2{X}_1={\displaystyle \frac{F}{m_{\mathrm{eq}1}}}.$$

(36)

Using γ_eq1, we can write H₁(ω) by

$$H_1(\omega )={\displaystyle \frac{1}{k\left(1-{\gamma }_{\mathrm{eq}1}^2+i2{\varsigma }_{\mathrm{eq}1}{\gamma }_{\mathrm{eq}1}\right)}}.$$

(37)

The amplitude of H₁(ω) is given by

$$\left|H_1(\omega )\right|={\displaystyle \frac{1/k}{\sqrt{{\left(1-\frac{{\omega }^{\alpha (\omega )}}{{\omega }_n^2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|\right)}^2+{\left(\frac{{\omega }^{\alpha (\omega )}}{{\omega }_n^2}\sin \frac{\alpha (\omega )\pi }{2}\right)}^2}}}$$

(38)

Or

$$\left|H_1(\omega )\right|={\displaystyle \frac{1/k}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}1}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}1}{\gamma }_{\mathrm{eq}1}\right)}^2}}}.$$

(39)

The phase of H₁(ω) is in the form

$${\phi }_1(\omega )={\tan }^{-1}{\displaystyle \frac{{\omega }^{\alpha (\omega )}\sin \frac{\alpha (\omega )\pi }{2}}{{\omega }_n^2-{\omega }^{\alpha (\omega )}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|}}$$

(40)

Or

$${\phi }_1(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }_{\mathrm{eq}1}^2}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}1}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}1}{\gamma }_{\mathrm{eq}1}\right)}^2}}},$$

(41)

Or

$${\phi }_1(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}1}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}1}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}1}\frac{\omega }{{\omega }_{\mathrm{eqn}1}}\right)}^2}}}.$$

(42)

Some illustrations of |H₁(ω)| and φ₁(ω) are displayed in Figure 7 and Figure 8, respectively.

3.4. Stationary Response to Sinusoidal Vibration of Type 1 Multi-Fractional Vibrator

There are two expressions of stationary sinusoidal response to a type 1 multi-fractional vibrator. One is the expression in the phasor domain and the other in the time domain. We explain them in Theorems 10 and 11, respectively.

Theorem 10 (Response phasor 1).

The response phasor of a type 1 multi-fractional vibrator is expressed by

$$X_1={\displaystyle \frac{F}{k\left[1-\frac{{\omega }^{\alpha (\omega )}}{{\omega }_n^2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|+i\frac{{\omega }^{\alpha (\omega )}}{{\omega }_n^2}\sin \frac{\alpha (\omega )\pi }{2}\right]}}.$$

(43)

Proof. 

Writing X₁ = H₁(ω)F yields the above. The proof is finished. □

Equivalently, X₁ can be written by

$$X_1={\displaystyle \frac{F}{k\left(1-{\gamma }_{\mathrm{eq}1}^2+i2{\varsigma }_{\mathrm{eq}1}{\gamma }_{\mathrm{eq}1}\right)}}.$$

(44)

Theorem 11 (Stationary sinusoidal response 1).

The stationary sinusoidal response to a type 1 multi-fractional vibrator is expressed by

$$\begin{array}{l}{x}_1(t)={\displaystyle \frac{F_0/k}{\sqrt{{\left[1-\frac{{\omega }^{\alpha (\omega )}}{{\omega }_n^2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|\right]}^2+{\left[\frac{{\omega }^{\alpha (\omega )}}{{\omega }_n^2}\sin \frac{\alpha (\omega )\pi }{2}\right]}^2}}}\\ \sin \left\{\omega t+\theta -{\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}1}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}1}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}1}\frac{\omega }{{\omega }_{\mathrm{eqn}1}}\right)}^2}}}\right\}.\end{array}$$

(45)

Proof. 

Note that = Im[X₁exp(iωt)]. As X₁ = H₁(ω)F = H₁(ω)F₀exp(iθ), we have the above. The proof is finished. □

Figure 9 and Figure 10 indicate some illustrations of x₁(t), exhibiting that there is a significant effect of fractional order α(ω) on the sinusoidal response x₁(t).

4. Results in Multi-Fractional Vibrators of Type 2

We explain the results of type 2 multi-fractional vibrators in this Section as follows. The motion phasor equation is described in Section 4.1, the expressions of equivalent vibration parameters are discussed in Section 4.2, the expression of frequency transfer function is represented in Section 4.3, and the expressions of stationary sinusoidal response to a type 2 multi-fractional vibrator are expressed in Section 4.4.

4.1. Motion Phasor Equation of Type 2 Multi-Fractional Vibrator

Theorem 12 (Motion phasor equation 2).

Let X₂ be the phasor of response x₂(t) of a type 2 multi-fractional vibrator. Let F be the phasor of excitation f(t) = F₀sin(ωt + θ). Then, the motion phasor equation of a type 2 multi-fractional vibrator is expressed by

$$m{(i\omega )}^2{X}_2+c{(i\omega )}^{\beta (\omega )}{X}_2+k{X}_2=F,0<\beta \left(\omega \right)<2.$$

(46)

Proof. 

Using phasor to express both sides of $m{\displaystyle \frac{d^2{x}_2(t)}{d{t}^2}}+c{\displaystyle \frac{d^{\beta (\omega )}{x}_2(t)}{d{t}^{\beta (\omega )}}}+k{x}_2(t)=f(t)$ yields (46). This completes the proof. □

Taking into account the principal branch of $i^{\beta (\omega )},$ we write the left side of (46) by

$$\left[-m{\omega }^2+c{(i\omega )}^{\beta (\omega )}+k\right]X_2=\left(-m{\omega }^2+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}+ic{\omega }^{(\omega )\beta }\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k\right)X_2.$$

(47)

Writing the right side of the above by

$$\begin{array}{l}\left(-m{\omega }^2+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}+ic{\omega }^{\beta }\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k\right)X_2\\ =\left[-m{\omega }^2+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}+(i\omega )c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k\right]X_2\end{array}$$

(48)

produces the motion phasor equation of a type 2 multi-fractional vibrator in the form

$$\begin{array}{l}{m(i\omega )}^2{X}_2+{c(i\omega )}^{\beta (\omega )}{X}_2+k{X}_2\\ =\left\{\left[m-{c\omega }^{\beta (\omega )-2}{\omega }^2\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]{(i\omega )}^2+(i\omega )c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k\right\}{X}_2.\end{array}$$

(49)

Note that $c{\displaystyle \frac{d^{\beta (\omega )}{x}_2(t)}{d{t}^{\beta (\omega )}}}$ designates a fractional damping force. Consequently, 0 < β(ω) < 2.

4.2. Equivalent Parameters of Type 2 Multi-Fractional Vibrator

The equivalent vibration parameters described in this Subsection for a type 2 multi-fractional vibrator contain equivalent mass, equivalent damping, equivalent damping ratio, equivalent damping free natural angular frequency, equivalent damped natural angular frequency, and equivalent frequency ratio.

4.2.1. Equivalent Mass of Type 2 Multi-Fractional Vibrator

Theorem 13 (Equivalent mass 2).

Let m_eq2 be the equivalent mass of a multi-fractional vibrator of type 2. Then,

$$m_{\mathrm{eq}2}=m-c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}.$$

(50)

Proof. 

Considering the first item on the right side of (49), we see that the above is true. This finishes the proof. □

Figure 11 shows some of the plots of m_eq2.

4.2.2. Equivalent Damping of Type 2 Multi-Fractional Vibrator

Theorem 14 (Equivalent damping 2).

Let c_eq2 be the equivalent damping of a multi-fractional vibrator of type 2. Then,

$$c_{\mathrm{eq}2}=c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}.$$

(51)

Proof. 

From the second item on the right side of (49), we see that the above holds. The proof is finished. □

Figure 12 illustrates some plots of c_eq2.

4.2.3. Equivalent Damping Ratio of Type 2 Multi-Fractional Vibrator

Theorem 15 (Equivalent damping ratio 2).

Let ζ_eq2 be the equivalent damping ratio of a multi-fractional vibrator of type 2. Define it by

$${\varsigma }_{\mathrm{eq}2}={\displaystyle \frac{c_{\mathrm{eq}2}}{2\sqrt{m_{\mathrm{eq}2}k}}}.$$

(52)

Then,

$${\varsigma }_{\mathrm{eq}2}={\displaystyle \frac{\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}},$$

(53)

where $\varsigma ={\displaystyle \frac{c}{2\sqrt{mk}}}$ is the primary damping ratio.

Proof. 

Substituting m_eq2 and c_eq2 into (52) results in (53). The proof is finished. □

Figure 13 shows some plots of ζ_eq2.

4.2.4. Equivalent Damping Free Natural Angular Frequency of Type 2 Multi-Fractional Vibrator

Theorem 16 (Equivalent damping free natural angular frequency 2).

Let ω_eqn2 be the equivalent damping free natural angular frequency of a type 2 multi-fractional vibrator. Define it by

$${\omega }_{\mathrm{eqn}2}=\sqrt{{\displaystyle \frac{k}{m_{\mathrm{eq}2}}}}.$$

(54)

Then,

$${\omega }_{\mathrm{eqn}2}={\displaystyle \frac{{\omega }_n}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}}.$$

(55)

Proof. 

Substituting m_eq2 into (54) produces (55). This finishes the proof. □

Some plots of ω_eqn2 are indicated in Figure 14.

4.2.5. Equivalent Damped Natural Angular Frequency of Type 2 Multi-Fractional Vibrator

Considering practical vibrations, we adopt |ζ_eq2| ≤ 1 in what follows.

Theorem 17 (Equivalent damped natural angular frequency 2).

Let ω_eqd2 be the equivalent damped natural angular frequency of a multi-fractional vibrator of type 2. Define it by

$${\omega }_{\mathrm{eqd}2}={\omega }_{\mathrm{eqn}2}\sqrt{1-{\varsigma }_{\mathrm{eq}2}^2}.$$

(56)

Then,

$${\omega }_{\mathrm{eqd}2}={\displaystyle \frac{{\omega }_n}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}}\sqrt{1-{\displaystyle \frac{{\varsigma }^2{\omega }^{2(\beta -1)}{\sin }^2\frac{\beta \pi }{2}}{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}}.$$

(57)

Proof. 

Substituting ζ_eq2 and ω_eqn2 into (56) produces (57). The proof is finished. □

Some illustrations of ω_eqd2 are given in Figure 15.

4.2.6. Equivalent Frequency Ratio of Type 2 Multi-Fractional Vibrator

Denote by γ_eq2 the equivalent frequency ratio of a type 2 multi-fractional vibrator. Define it by

$${\gamma }_{\mathrm{eq}2}={\displaystyle \frac{\omega }{{\omega }_{\mathrm{eqn}2}}}.$$

(58)

Theorem 18 (Equivalent frequency ratio 2).

Let γ be the primary frequency ratio. The quantity γ_eq2 is expressed by

$${\gamma }_{\mathrm{eq}2}=\gamma \sqrt{1-{\displaystyle \frac{c}{m}}{\omega }^{\beta -2}\cos {\displaystyle \frac{\beta \pi }{2}}}.$$

(59)

Proof. 

When substituting ω_eqn2 into (58), we have (59). This finishes the proof. □

We show some plots of γ_eq2 in Figure 16.

4.3. Frequency Transfer Function of Type 2 Multi-Fractional Vibrator

Based on the notations of m_eq2, c_eq2, ζ_eq2, and ω_eqn2, we write the motion phasor equation of type 2 multi-fractional vibrators in the form

$${(i\omega )}^2{X}_2+i2{\zeta }_{\mathrm{eqn}2}{\omega }_{\mathrm{eqn}2}{X}_2+{\omega }_{\mathrm{eqn}2}^2{X}_2={\displaystyle \frac{F}{m_{\mathrm{eq}2}}}.$$

(60)

Theorem 19 (Frequency transfer function 2).

Let H₂(ω) be the frequency transfer function in phasor for a type 2 multi-fractional vibrator. Then,

$$H_2(\omega )={\displaystyle \frac{1/k}{1-{\gamma }^2\left[1-\frac{c}{m}{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]+i\frac{2\varsigma {\omega }^{\beta (\omega )}\sin \frac{\beta (\omega )\pi }{2}}{{\omega }_n}}}.$$

(61)

Proof. 

From (60), we have

$$H_2(\omega )={\displaystyle \frac{X_2}{F}}={\displaystyle \frac{1}{m_{\mathrm{eq}2}\left[{(i\omega )}^2+i2{\zeta }_{\mathrm{eqn}2}{\omega }_{\mathrm{eqn}2}+{\omega }_{\mathrm{eqn}2}^2\right]}}.$$

(62)

Substituting m_eq2, ζ_eq2, and ω_eqn2 into the right side of the above equation yields (61). The proof is finished. □

When using γ_eq2 and ζ_eq2, we can rewrite the above by

$$H_2(\omega )={\displaystyle \frac{1}{k\left(1-{\gamma }_{\mathrm{eq}2}^2+i2{\varsigma }_{\mathrm{eq}2}{\gamma }_{\mathrm{eq}2}\right)}}.$$

(63)

The amplitude of H₂(ω) is in the form

$$\left|H_2(\omega )\right|={\displaystyle \frac{1/k}{\sqrt{{\left\{1-{\gamma }^2\left[1-\frac{c}{m}{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]\right\}}^2+{\left[\frac{1}{{\omega }_n}2\varsigma {\omega }^{\beta (\omega )}\sin \frac{\beta (\omega )\pi }{2}\right]}^2}}},$$

(64)

which equals to

$$\left|H_2(\omega )\right|={\displaystyle \frac{1/k}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}2}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}2}{\gamma }_{\mathrm{eq}2}\right)}^2}}}.$$

(65)

The phase of H₂(ω) is given by

$${\phi }_2(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }_{\mathrm{eq}2}^2}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}2}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}2}{\gamma }_{\mathrm{eq}2}\right)}^2}}}.$$

(66)

Thus,

$${\phi }_2(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}2}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}2}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}2}\frac{\omega }{{\omega }_{\mathrm{eqn}2}}\right)}^2}}}.$$

(67)

We use Figure 17 and Figure 18 to illustrate some plots of |H₂(ω)| and φ₂(ω).

4.4. Stationary Response to Sinusoidal Vibration of Type 2 Multi-Fractional Vibrator

Regarding a type 2 multi-fractional vibrator, there are two expressions of stationary sinusoidal response. One is the response in the phasor domain, see Theorem 20, and the other in the time domain, see Theorem 21.

Theorem 20 (Response phasor 2).

The response phasor of a type 2 multi-fractional vibrator is given by

$$X_2={\displaystyle \frac{F/k}{1-{\gamma }^2\left[1-\frac{c}{m}{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]+i\frac{2\varsigma {\omega }^{\beta (\omega )}\sin \frac{\beta (\omega )\pi }{2}}{{\omega }_n}}}.$$

(68)

Proof. 

Since X₂ = H₂(ω)F, according to Theorem 19, we have the above. The proof is finished. □

Equation (68) can be expressed by

$$X_2={\displaystyle \frac{F}{k\left(1-{\gamma }_{\mathrm{eq}2}^2+i2{\varsigma }_{\mathrm{eq}2}{\gamma }_{\mathrm{eq}2}\right)}}.$$

(69)

Theorem 21 (Stationary sinusoidal response 2).

The stationary sinusoidal response to a type 2 multi-fractional vibrator is given by

$$\begin{array}{l}{x}_2(t)={\displaystyle \frac{F_0/k}{\sqrt{{\left\{1-{\gamma }^2\left[1-\frac{c}{m}{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]\right\}}^2+{\left[\frac{1}{{\omega }_n}2\varsigma {\omega }^{\beta (\omega )}\sin \frac{\beta (\omega )\pi }{2}\right]}^2}}}\\ \sin \left\{\omega t+\theta -{\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}2}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}2}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}2}\frac{\omega }{{\omega }_{\mathrm{eqn}2}}\right)}^2}}}\right\}.\end{array}$$

(70)

Proof. 

Because x₂(t) = Im[X₂exp(iωt)], X₂ = H₂(ω)F, and F = F₀exp(iθ), we have x₂(t) = Im[H₂(ω)F₀exp(iθ)exp(iωt)] = Im{|H₂(ω)|F₀exp(iωt)exp(iθ)exp[−iφ(ω)]}. Thus, the above holds. The proof is finished. □

Some plots of x₂(t) are shown in Figure 19 and Figure 20, where we see that the effect of fractional order β(ω) on the sinusoidal response x₂(t) is considerable.

5. Results in Type 3 Multi-Fractional Vibrators

We present the results with respect to type 3 multi-fractional vibrators in four folds. We show the motion phasor equation of a type 3 multi-fractional vibrator in Section 5.1. The expressions of equivalent vibration parameters about a type 3 multi-fractional vibrator are discussed in Section 5.2. The expression of frequency transfer function of a type 3 multi-fractional vibrator is explained in Section 5.3. The expressions of stationary sinusoidal response to a type 3 multi-fractional vibrator in the phasor domain and the time one are put forward in Section 5.4.

5.1. Motion Phasor Equation of Type 3 Multi-Fractional Vibrator

A system that follows

$$m{\displaystyle \frac{d^{\alpha (\omega )}{x}_3(t)}{d{t}^{\alpha (\omega )}}}+c{\displaystyle \frac{d^{\beta (\omega )}{x}_3(t)}{d{t}^{\beta (\omega )}}}+k{x}_3(t)=f(t),1<\alpha \left(\omega \right)<3,0<\beta \left(\omega \right)<2$$

(71)

is called a type 3 multi-fractional vibrator. The theorem below gives its motion phasor equation.

Theorem 22 (Motion phasor equation 3).

Denote by X₃ the phasor of response x₃(t) of a type 3 multi-fractional vibrator. Let F be the phasor of excitation f(t) = F₀sin(ωt + θ). Then, the motion phasor equation of a type 3 multi-fractional vibrator is expressed by

$$m{(i\omega )}^{\alpha (\omega )}{X}_3+c{(i\omega )}^{\beta (\omega )}{X}_3+k{X}_3=F.$$

(72)

Proof. 

Using phasor to express both sides of (71) yields (72). This completes the proof. □

Considering the principal branches of i^α and i^β, we express the left side of (72) by

$$m{(i\omega )}^{\alpha (\omega )}{X}_3+c{(i\omega )}^{\beta (\omega )}{X}_3+k{X}_3=\left\{\begin{array}{l}m{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\\ +i\left[m{\omega }^{\alpha (\omega )}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]+k\end{array}\right\}{X}_3.$$

(73)

The right side of the above can be written by

$$\begin{array}{l}\left\{\begin{array}{l}m{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\\ +i\left[m{\omega }^{\alpha (\omega )}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]+k\end{array}\right\}{X}_3\\ =\left\{\begin{array}{l}-\left[m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]{(i\omega )}^2\\ +\left[m{\omega }^{(\omega )\alpha -1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}\right](i\omega )+k\end{array}\right\}{X}_3.\end{array}$$

(74)

Therefore,

$$\begin{array}{l}\left[m{(i\omega )}^{\alpha (\omega )}+c{(i\omega )}^{\beta (\omega )}+k\right]X_3\\ =\left\{\begin{array}{l}-\left[m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]{(i\omega )}^2\\ +\left[m{\omega }^{(\omega )\alpha -1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}\right](i\omega )+k\end{array}\right\}{X}_3.\end{array}$$

(75)

5.2. Equivalent Parameters of Type 3 Multi-Fractional Vibrator

In this Subsection, we present the expressions of equivalent vibration parameters of a type 3 multi-fractional vibrator. Those parameters are equivalent mass, equivalent damping, equivalent damping ratio, equivalent damping free natural angular frequency, equivalent damped natural angular frequency, and equivalent frequency ratio.

5.2.1. Equivalent Mass of Type 3 Multi-Fractional Vibrator

Theorem 23 (Equivalent mass 3).

Let m_eq3 be the equivalent mass of a multi-fractional vibrator of type 3. Then,

$$m_{\mathrm{eq}3}=-\left[m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right].$$

(76)

Proof. 

The first item on the right side of (75) stands for equivalent mass. This finishes the proof. □

Figure 21 illustrates some plots of m_eq3.

5.2.2. Equivalent Damping of Type 3 Multi-Fractional Vibrator

Theorem 24 (Equivalent damping 3).

Let c_eq3 be the equivalent damping of a multi-fractional vibrator of type 3. Then,

$$c_{\mathrm{eq}3}=m{\omega }^{\alpha (\omega )-1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}.$$

(77)

Proof. 

From the second item on the right side of (75), we see that the above is true. The proof is finished. □

Some plots of c_eq3 are indicated in Figure 22.

5.2.3. Equivalent Damping Ratio of Type 3 Multi-Fractional Vibrator

Theorem 25 (Equivalent damping ratio 3).

Let ζ_eq3 be the equivalent damping ratio of a type 3 multi-fractional vibrator. Define it by

$${\varsigma }_{\mathrm{eq}3}={\displaystyle \frac{c_{\mathrm{eq}3}}{2\sqrt{m_{\mathrm{eq}3}k}}}.$$

(78)

Then,

$${\varsigma }_{\mathrm{eq}3}={\displaystyle \frac{{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta (\omega )-1}\sin \frac{\beta (\omega )\pi }{2}}{2{\omega }_n\sqrt{-\left[{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]}}}.$$

(79)

Proof. 

Substituting m_eq3 and c_eq3 into (78) results in (79). The proof is finished. □

Some plots of ζ_eq3 are shown in Figure 23.

5.2.4. Equivalent Damping Free Natural Angular Frequency of Type 3 Multi-Fractional Vibrator

Theorem 26 (Equivalent damping free natural angular frequency 3).

Denote by ω_eqn3 the equivalent damping free natural angular frequency of a type 3 multi-fractional vibrator. Define it by

$${\omega }_{\mathrm{eqn}3}=\sqrt{{\displaystyle \frac{k}{m_{\mathrm{eq}3}}}}.$$

(80)

Then

$${\omega }_{\mathrm{eqn}3}={\displaystyle \frac{{\omega }_n}{\sqrt{-\left[{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}+\frac{c}{m}{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]}}}.$$

(81)

Proof. 

Substituting m_eq3 into (80) yields (81). The proof is finished. □

Some plots of ω_eqn3 are illustrated in Figure 24.

5.2.5. Equivalent Damped Natural Angular Frequency of Type 3 Multi-Fractional Vibrator

From a practical view of vibration engineering, one is interested in small damping. Therefore, we adopt |ζ_eq3| ≤ 1 in what follows.

Theorem 27 (Equivalent damped natural angular frequency 3).

Denote by ω_eqd3 the equivalent damped natural angular frequency of a multi-fractional vibrator of type 3. Then,

$${\omega }_{\mathrm{eqd}3}={\omega }_{\mathrm{eqn}3}\sqrt{1-{\varsigma }_{\mathrm{eq}3}^2}.$$

(82)

Thus,

$${\omega }_{\mathrm{eqd}3}={\displaystyle \frac{{\omega }_n\sqrt{1-{\left[\frac{{\left[{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta (\omega )-1}\sin \frac{\beta (\omega )\pi }{2}\right]}^2}{4{\omega }_n^2\left\{-\left[{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]\right\}}\right]}^2}}{\sqrt{-\left[{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}+\frac{c}{m}{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]}}}.$$

(83)

Proof. 

Substituting ζ_eq3 and ω_eqn3 into (82) produces (83). This finishes the proof. □

Figure 25 indicates some plots of ω_eqd3.

5.2.6. Equivalent Frequency Ratio of Type 3 Multi-Fractional Vibrator

Theorem 28 (Equivalent frequency ratio 3).

Let γ_eq3 be the equivalent frequency ratio of a multi-fractional vibrator of type 3. Define it by

$${\gamma }_{\mathrm{eq}3}={\displaystyle \frac{\omega }{{\omega }_{\mathrm{eqn}3}}}.$$

(84)

Then,

$${\gamma }_{\mathrm{eq}3}=\gamma \sqrt{-\left[{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+{\displaystyle \frac{c}{m}}{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]}.$$

(85)

Proof. 

Substituting ω_eqn3 into (84) results in (85). The proof is finished. □

Some plots of γ_eq3 are shown in Figure 26.

5.3. Frequency Transfer Function of Type 3 Multi-Fractional Vibrator

Theorem 29 (Frequency transfer function 3).

Let H₃(ω) be the frequency transfer function in phasor for a type 3 multi-fractional vibrator. Then,

$$H_3(\omega )={\displaystyle \frac{1/k}{\begin{array}{l}1-{\gamma }^2\left[{\omega }^{\alpha (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]\\ +i\frac{\gamma \left[{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta (\omega )-1}\sin \frac{\beta (\omega )\pi }{2}\right]}{{\omega }_n\left[{\omega }^{\alpha (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]}\end{array}}}.$$

(86)

Proof. 

The motion phasor equation can be written by

$$\left[m{(i\omega )}^{\alpha (\omega )}+c{(i\omega )}^{\beta (\omega )}+k\right]X_3=\left[{(i\omega )}^2{m}_{\mathrm{eq}3}+i\omega c_{\mathrm{eq}3}+k\right]X_3=F.$$

(87)

Thus,

$$H_3(\omega )={\displaystyle \frac{X_3}{F}}={\displaystyle \frac{1}{{(i\omega )}^2{m}_{\mathrm{eq}3}+i\omega c_{\mathrm{eq}3}+k}}.$$

(88)

Substituting m_eq3 and c_eq3 into the right side of the above yields (86). The proof is finished. □

Using γ_eq3 and ζ_eq3, H₃(ω) can be written by

$$H_3(\omega )={\displaystyle \frac{1}{k\left(1-{\gamma }_{\mathrm{eq}3}^2+i2{\varsigma }_{\mathrm{eq}3}{\gamma }_{\mathrm{eq}3}\right)}}.$$

(89)

The amplitude of H₃(ω) is in the form

$$\left|H_3(\omega )\right|={\displaystyle \frac{1/k}{\sqrt{\left\{\begin{array}{l}{\left[1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\left[\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\right]}^2\end{array}\right\}}}}.$$

(90)

which is equivalent to

$$\left|H_3(\omega )\right|={\displaystyle \frac{1/k}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}3}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}3}{\gamma }_{\mathrm{eq}3}\right)}^2}}}.$$

(91)

The phase of H₃(ω) is given by

$${\phi }_3(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }_{\mathrm{eq}3}^2}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}3}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}3}{\gamma }_{\mathrm{eq}3}\right)}^2}}}.$$

(92)

Thus,

$${\phi }_3(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}3}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}3}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}3}\frac{\omega }{{\omega }_{\mathrm{eqn}3}}\right)}^2}}}.$$

(93)

Figure 27 and Figure 28 show some plots of |H₃(ω)| and φ₃(ω), respectively.

5.4. Stationary Response to Sinusoidal Vibration of Type 3 Multi-Fractional Vibrator

We bring forward two expressions of stationary sinusoidal response to a type 3 multi-fractional vibrator. One is the response in phasor and the other the response in time. They are expressed by Theorems 30 and 31, respectively.

Theorem 30 (Response phasor 3).

The response phasor of a type 3 multi-fractional vibrator is expressed by

$$X_3={\displaystyle \frac{F/k}{\begin{array}{l}1-{\gamma }^2\left[{\omega }^{\alpha (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]\\ +i\frac{\gamma \left[{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta (\omega )-1}\sin \frac{\beta (\omega )\pi }{2}\right]}{{\omega }_n\left[{\omega }^{\alpha (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]}\end{array}}}.$$

(94)

Proof. 

As X₃ = H₃(ω)F, we have the above. The proof is finished. □

Equation (94) can be in the form

$$X_3={\displaystyle \frac{F}{k\left(1-{\gamma }_{\mathrm{eq}3}^2+i2{\varsigma }_{\mathrm{eq}3}{\gamma }_{\mathrm{eq}3}\right)}}.$$

(95)

Theorem 31 (Stationary sinusoidal response 3).

The stationary sinusoidal response of a type 3 multi-fractional vibrator is expressed by

$$\begin{array}{l}{x}_3(t)={\displaystyle \frac{F_0/k}{\sqrt{\left\{\begin{array}{l}{\left[1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\left[\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\right]}^2\end{array}\right\}}}}\\ \sin \left\{\omega t+\theta -{\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}3}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}3}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}3}\frac{\omega }{{\omega }_{\mathrm{eqn}3}}\right)}^2}}}\right\}.\end{array}$$

(96)

Proof. 

Since x₃(t) = Im[X₃exp(iωt)], X₃ = H₃(ω)F, and F = F₀exp(iθ), we have x₃(t) = Im{|H₃(ω)|F₀exp(iωt)exp(iθ)exp[−iφ(ω)]}. Therefore, the above is valid. The proof is finished. □

Figure 29 indicates some plots of x₃(t). It shows that there is a significant effect of fractional orders α(ω) and β(ω) on the sinusoidal response x₃(t).

6. Results about Type 4 Multi-Fractional Vibrators

Now, we propose the results regarding type 4 multi-fractional vibrators in four aspects. The motion phasor equation of a type 4 multi-fractional vibrator is proposed in Section 6.1. The expressions of equivalent vibration parameters about a type 4 multi-fractional vibrator are given in Section 6.2. The expression of frequency transfer function of a type 4 multi-fractional vibrator is in Section 6.3. The expressions of stationary sinusoidal response to a type 4 multi-fractional vibrator in the phasor domain and the time one are brought forward in Section 6.4.

6.1. Motion Phasor Equation of Type 4 Multi-Fractional Vibrator

Theorem 32 (Motion phasor equation 4).

Let X₄ be the phasor of response x₄(t) of a type 4 multi-fractional vibrator. Denote by F the phasor of excitation f(t) = F₀sin(ωt + θ). Then, the motion phasor equation of a type 4 multi-fractional vibrator is in the form

$$m{(i\omega )}^{\alpha (\omega )}{X}_4+k{(i\omega )}^{\lambda (\omega )}{X}_4=F,1<\alpha \left(\omega \right)<3,0\le \lambda \left(\omega \right)<1.$$

(97)

Proof. 

Using phasor to express both sides of $m{\displaystyle \frac{d^{\alpha (\omega )}{x}_4(t)}{d{t}^{\alpha }}}+k{\displaystyle \frac{d^{\lambda (\omega )}{x}_4(t)}{d{t}^{\lambda }}}=f(t)$ yields (97). The proof is finished. □

Taking into account the principal branches of i^α and i^λ, we write the left side of the above equation by

$$\begin{array}{l}{m(i\omega )}^{\alpha (\omega )}{X}_4+{k(i\omega )}^{\lambda (\omega )}{X}_4\\ =\left\{m{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+i\left[m{\omega }^{\alpha (\omega )}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right\}{X}_4.\end{array}$$

(98)

The right side of the above can be written by

$$\begin{array}{l}\left\{m{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+i\left[m{\omega }^{\alpha (\omega )}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right\}{X}_4\\ =\left\{\begin{array}{l}-\left[m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}\right]{(i\omega )}^2+\left[m{\omega }^{\alpha (\omega )-1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k{\omega }^{(\omega )\lambda -1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right](i\omega )\\ +k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{(\omega )\lambda \pi }{2}}\end{array}\right\}{X}_4.\end{array}$$

(99)

Therefore,

$$\begin{array}{l}{m(i\omega )}^{\alpha (\omega )}{X}_4+{k(i\omega )}^{\lambda (\omega )}{X}_4\\ =\left\{\begin{array}{l}-\left[m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}\right]{(i\omega )}^2+\left[m{\omega }^{\alpha (\omega )-1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+k{\omega }^{(\omega )\lambda -1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right](i\omega )\\ +k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{(\omega )\lambda \pi }{2}}\end{array}\right\}{X}_4.\end{array}$$

(100)

Note that $k{\displaystyle \frac{d^{\lambda (\omega )}{x}_4(t)}{d{t}^{\lambda }}}$ stands for a fractional restoration force. Hence, 0 ≤ λ(ω) < 1.

6.2. Equivalent Parameters of Type 4 Multi-Fractional Vibrator

The present results below are the expressions of equivalent vibration parameters of a type 4 multi-fractional vibrator. They are equivalent mass in Section 6.2.1, equivalent damping in Section 6.2.2, equivalent stiffness in Section 6.2.3, equivalent damping ratio in Section 6.2.4, equivalent damping free natural angular frequency in Section 6.2.5, equivalent damped natural angular frequency in Section 6.2.6, and equivalent frequency ratio in Section 6.2.7.

6.2.1. Equivalent Mass of Type 4 Multi-Fractional Vibrator

Theorem 33 (Equivalent mass 4).

Denote by m_eq4 the equivalent mass for a type 4 multi-fractional vibrator. Then,

$$m_{\mathrm{eq}4}=m_{\mathrm{eq}1}=-m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}.$$

(101)

Proof. 

From the first item on the right side of (100), we see that the above is true. The proof is finished. □

6.2.2. Equivalent Damping of Type 4 Multi-Fractional Vibrator

Theorem 34 (Equivalent damping 4).

Let c_eq4 be the equivalent damping of a multi-fractional vibrator of type 4. Then,

$$c_{\mathrm{eq}4}=m{\omega }^{\alpha -1}\sin {\displaystyle \frac{\alpha \pi }{2}}+k{\omega }^{\lambda -1}\sin {\displaystyle \frac{\lambda \pi }{2}}.$$

(102)

Proof. 

The second item on the right side of (100) implies that the above holds. The proof is finished. □

Refer Figure 30 for some plots of c_eq4.

6.2.3. Equivalent Stiffness of Type 4 Multi-Fractional Vibrator

Theorem 35 (Equivalent stiffness 4).

Denote by k_eq4 the equivalent stiffness of a type 4multi-fractional vibrator. Then,

$$k_{\mathrm{eq}4}=k{\omega }^{\lambda }\cos {\displaystyle \frac{\lambda \pi }{2}}.$$

(103)

Proof. 

From the third item on the right side of (100), we see that the above is valid. The proof is finished. □

Refer to Figure 31 for some plots of k_eq4.

6.2.4. Equivalent Damping Ratio of Type 4 Multi-Fractional Vibrator

Theorem 36 (Equivalent damping ratio 4).

Denote by ζ_eq4 the equivalent damping ratio of a type 4 multi-fractional vibrator. Define it by

$${\varsigma }_{\mathrm{eq}4}={\displaystyle \frac{c_{\mathrm{eq}4}}{2\sqrt{m_{\mathrm{eq}4}{k}_{\mathrm{eq}4}}}}.$$

(104)

Then,

$${\varsigma }_{\mathrm{eq}4}={\displaystyle \frac{m{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{mk{\omega }^{\alpha (\omega )+\lambda (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|\cos \frac{\lambda (\omega )\pi }{2}}}}.$$

(105)

Proof. 

Substituting m_eq4, c_eq4, and k_eq4 into (104) yields (105). This finishes the proof. □

Some plots of ζ_eq4 are shown in Figure 32.

6.2.5. Equivalent Damping Free Natural Angular Frequency of Type 4 Multi-Fractional Vibrator

Theorem 37 (Equivalent damping free natural angular frequency 4).

Let ω_eqn4 be the equivalent damping free natural angular frequency of a type 4 multi-fractional vibrator. Define it by

$${\omega }_{\mathrm{eqn}4}=\sqrt{{\displaystyle \frac{k_{\mathrm{eq}4}}{m_{\mathrm{eq}4}}}}.$$

(106)

Then,

$${\omega }_{\mathrm{eqn}4}={\omega }_n\sqrt{{\displaystyle \frac{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}}}}.$$

(107)

Proof. 

Substituting m_eq4 and k_eq4 into (106) produces (107). This finishes the proof. □

Figure 33 illustrates some plots of ω_eqn4.

6.2.6. Equivalent Damped Natural Angular Frequency of Type 4 Multi-Fractional Vibrator

From the point of view of practical vibrations, we adopt |ζ_eq4| ≤ 1 in what follows.

Theorem 38 (Equivalent damped natural angular frequency 4).

Let ω_eqd4 be the equivalent damped natural angular frequency of a multi-fractional vibrator of type 4. Define it by

$${\omega }_{\mathrm{eqd}4}={\omega }_{\mathrm{eqn}4}\sqrt{1-{\varsigma }_{\mathrm{eq}4}^2}.$$

(108)

Then,

$${\omega }_{\mathrm{eqd}4}={\omega }_n\sqrt{{\displaystyle \frac{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}}}}\sqrt{1-{\left[{\displaystyle \frac{m{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{mk{\omega }^{\alpha (\omega )+\lambda (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|\cos \frac{\lambda (\omega )\pi }{2}}}}\right]}^2}.$$

(109)

Proof. 

Substituting ζ_eq4 and ω_eqn4 into (108) results in (109). The proof is finished. □

Figure 34 shows some plots of ω_eqd4.

6.2.7. Equivalent Frequency Ratio of Type 4 Multi-Fractional Vibrator

Theorem 39 (Frequency ratio 4).

Denote by γ_eq4 the equivalent frequency ratio of a multi-fractional vibrator of type 4. It is defined by

$${\gamma }_{\mathrm{eq}4}={\displaystyle \frac{\omega }{{\omega }_{\mathrm{eqn}4}}}.$$

(110)

Then,

$${\gamma }_{\mathrm{eq}4}=\gamma \sqrt{{\displaystyle \frac{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}}.$$

(111)

Proof. 

Substituting ω_eqn4 into (110) produces (111). This finishes the proof. □

Some plots of γ_eq4 are illustrated in Figure 35.

6.3. Frequency Transfer Function of Type 4 Multi-Fractional Vibrator

Theorem 40 (Frequency transfer function 4).

Denote by H₄(ω) the frequency transfer function in phasor for a type 4 multi-fractional vibrator. Then,

$$H_4(\omega )={\displaystyle \frac{1/k_{\mathrm{eq}4}}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}\left[\begin{array}{l}1-{\gamma }^2\frac{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}\\ +i2\gamma \frac{m{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{mk{\omega }^{\alpha (\omega )+\lambda (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|\cos \frac{\lambda (\omega )\pi }{2}}}\\ \sqrt{\frac{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}\end{array}\right]}}.$$

(112)

Proof. 

When writing the motion phasor equation by

$$\left[m{(i\omega )}^{\alpha }+k{(i\omega )}^{\lambda }\right]X_4=\left[{(i\omega )}^2{m}_{\mathrm{eq}4}+i\omega c_{\mathrm{eq}4}+k_{\mathrm{eq}4}\right]X_4=F,$$

(113)

we have

$$H_4(\omega )={\displaystyle \frac{X_4}{F}}={\displaystyle \frac{1}{{(i\omega )}^2{m}_{\mathrm{eq}4}+i\omega c_{\mathrm{eq}4}+k_{\mathrm{eq}4}}}.$$

(114)

Substituting m_eq4, c_eq4 and k_eq4 into the above produces (112). The proof is finished. □

With γ_eq4 and ζ_eq4, we can write the above by

$$H_4(\omega )={\displaystyle \frac{1}{k_{\mathrm{eq}4}\left(1-{\gamma }_{\mathrm{eq}4}^2+i2{\varsigma }_{\mathrm{eq}4}{\gamma }_{\mathrm{eq}4}\right)}}.$$

(115)

The amplitude of H₄(ω) is

$$\left|H_4(\omega )\right|={\displaystyle \frac{1}{k_{\mathrm{eq}4}}}{\displaystyle \frac{1}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}4}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}4}{\gamma }_{\mathrm{eq}4}\right)}^2}}}.$$

(116)

The phase of H₄(ω) is

$${\phi }_4(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }_{\mathrm{eq}4}^2}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}4}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}4}{\gamma }_{\mathrm{eq}4}\right)}^2}}}.$$

(117)

We use Figure 36 and Figure 37 to show some plots of |H₄(ω)| and φ₄(ω), respectively.

6.4. Stationary Response to Sinusoidal Vibration of Type 4 Multi-Fractional Vibrator

We propose two expressions of the stationary sinusoidal response to a type 4 multi-fractional vibrator. One is in the phasor domain and the other in the time domain. They are described by Theorems 41 and 42, respectively.

Theorem 41 (Response phasor 4).

The response phasor of a type 4 multi-fractional vibrator is expressed by

$$X_4={\displaystyle \frac{F}{k{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}\left(\begin{array}{l}1-{\gamma }^2\frac{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}\\ +i2\gamma \frac{m{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{mk{\omega }^{\alpha (\omega )+\lambda (\omega )-2}\left|\cos \frac{\alpha (\omega )\pi }{2}\right|\cos \frac{\lambda (\omega )\pi }{2}}}\\ \sqrt{\frac{-{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}\end{array}\right)}}.$$

(118)

Proof. 

Since X₄ = H₄(ω)F, we see that the above is true. The proof is finished. □

The above equation can be written by

$$X_4={\displaystyle \frac{F}{k_{\mathrm{eq}4}\left(1-{\gamma }_{\mathrm{eq}4}^2+i2{\varsigma }_{\mathrm{eq}4}{\gamma }_{\mathrm{eq}4}\right)}}.$$

(119)

Theorem 42 (Stationary sinusoidal response 4).

The stationary sinusoidal response to a type 4 multi-fractional vibrator is given by

$$\begin{array}{l}{x}_4(t)={\displaystyle \frac{F_0/k_{\mathrm{eq}4}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left|\left(\begin{array}{l}1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\\ +i2\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\end{array}\right)\right|}}\\ \sin \left\{\omega t+\theta -{\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}4}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}4}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}4}\frac{\omega }{{\omega }_{\mathrm{eqn}4}}\right)}^2}}}\right\}.\end{array}$$

(120)

Proof. 

Since x₄(t) = Im[X₄exp(iωt)], X₄ = H₄(ω)F, F = F₀exp(iθ), and

$${\phi }_4(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }_{\mathrm{eq}4}^2}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}4}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}4}{\gamma }_{\mathrm{eq}4}\right)}^2}}}={\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}4}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}4}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}4}\frac{\omega }{{\omega }_{\mathrm{eqn}4}}\right)}^2}}},$$

we have x₄(t) = Im{|H₄(ω)|F₀exp(iωt)exp(iθ)exp[−iφ(ω)]}, which equals to (120). The proof is finished. □

In Figure 38, we show some plots of x₄(t) for different values of α and λ. Figure 38 exhibits that there is significant effect of fractional orders α(ω) and λ(ω) on the sinusoidal response x₄(t).

7. Results for Type 5 Multi-Fractional Vibrators

We now put forward the results with respect to type 5 multi-fractional vibrators. The motion phasor equation of a type 5 multi-fractional vibrator is presented in Section 7.1. The expressions of equivalent vibration parameters about a type 5 multi-fractional vibrator are discussed in Section 7.2. The expression of frequency transfer function of a type 5 multi-fractional vibrator is given in Section 7.3. The expressions of stationary sinusoidal response to a type 5 multi-fractional vibrator in the phasor domain and the time one are brought forward in Section 7.4.

7.1. Motion Phasor Equation of Type 5 Multi-Fractional Vibrator

Theorem 43 (Motion phasor equation 5).

Denote by X₅ the phasor of response x₅(t) of a type 5 multi-fractional vibrator. Let F be the phasor of excitation f(t) = F₀sin(ωt + θ). The motion phasor equation of a type 5 multi-fractional vibrator is expressed by

$$m{(i\omega )}^2{X}_5+k{(i\omega )}^{\lambda (\omega )}{X}_5=F,0\le \lambda \left(\omega \right)<1.$$

(121)

Proof. 

Using phasor to express both sides of $m{\displaystyle \frac{d^2{x}_5(t)}{d{t}^2}}+k{\displaystyle \frac{d^{\lambda }{x}_5(t)}{d{t}^{\lambda }}}=f(t)$ produces the above. The proof is finished. □

Considering the principal branch of i^λ, we express the left side of the above equation by

$$m{(i\omega )}^2{X}_5+k{(i\omega )}^{\lambda (\omega )}{X}_5=\left[m{(i\omega )}^2+ik{\omega }^{\lambda (\omega )}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]X_5.$$

(122)

Rewriting the right side of the above by

$$\begin{array}{l}\left[m{(i\omega )}^2+ik{\omega }^{\lambda (\omega )}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]X_5\\ =\left[m{(i\omega )}^2+k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}(i\omega )+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]X_5,\end{array}$$

(123)

we have

$$m{(i\omega )}^2{X}_5+k{(i\omega )}^{\lambda (\omega )}{X}_5=\left[m{(i\omega )}^2+k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}(i\omega )+k{\omega }^{\lambda }\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]X_5.$$

(124)

7.2. Equivalent Parameters of Type 5 Multi-Fractional Vibrator

We now present the expressions of equivalent vibration parameters of a type 5 multi-fractional vibrator, containing equivalent mass (Section 7.2.1), equivalent damping (Section 7.2.2), equivalent stiffness (Section 7.2.3), equivalent damping ratio (Section 7.2.4), equivalent damping free natural angular frequency (Section 7.2.5), equivalent damped natural angular frequency (Section 7.2.6), and equivalent frequency ratio (Section 7.2.7).

7.2.1. Equivalent Mass of Type 5 Multi-Fractional Vibrator

Theorem 44 (Equivalent mass 5).

Let m_eq5 be the equivalent mass of a type 5 multi-fractional vibrator. Then,

m_eq5 = m.

(125)

Proof. 

From the first item on the right side of (124), we see that the above is true. This finishes the proof. □

7.2.2. Equivalent Damping of Type 5 Multi-Fractional Vibrator

Theorem 45 (Equivalent damping 5).

Denote by c_eq5 the equivalent damping of a type 5 multi-fractional vibrator. Then,

$$c_{\mathrm{eq}5}=k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}.$$

(126)

Proof. 

The second item on the right side of (124) exhibits that the above is true. The proof is finished. □

Some plots of c_eq5 are illustrated in Figure 39.

7.2.3. Equivalent Stiffness of Type 5 Multi-Fractional Vibrator

Theorem 46 (Equivalent stiffness 5).

Let k_eq5 be the equivalent stiffness of a type 5 multi-fractional vibrator. Then,

$$k_{\mathrm{eq}5}=k_{\mathrm{eq}4}=k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}.$$

(127)

Proof. 

From (124) and (103), we see that the above is valid. This finishes the proof. □

7.2.4. Equivalent Damping Ratio of Type 5 Multi-Fractional Vibrator

Theorem 47 (Equivalent damping ratio 5).

Let ζ_eq5 be the equivalent damping ratio of a type 5 multi-fractional vibrator. Define it by

$${\varsigma }_{\mathrm{eq}5}={\displaystyle \frac{c_{\mathrm{eq}5}}{2\sqrt{m{k}_{\mathrm{eq}5}}}}.$$

(128)

Then,

$${\varsigma }_{\mathrm{eq}5}={\displaystyle \frac{{\omega }_n{\omega }^{\frac{\lambda (\omega )}{2}-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{\cos \frac{\lambda (\omega )\pi }{2}}}}.$$

(129)

Proof. 

Substituting c_eq5 and k_eq5 into (128) produces (129). This finishes the proof. □

Some plots of ζ_eq5 are indicated in Figure 40.

7.2.5. Equivalent Damping Free Natural Angular Frequency of Type 5 Multi-Fractional Vibrator

Theorem 48 (Equivalent damping free natural angular frequency 5).

Let ω_eqn5 be the equivalent damping free natural angular frequency of a type 5 multi-fractional vibrator. Define it by

$${\omega }_{\mathrm{eqn}5}=\sqrt{{\displaystyle \frac{k_{\mathrm{eq}5}}{m}}}.$$

(130)

Then,

$${\omega }_{\mathrm{eqn}5}={\omega }_n\sqrt{{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}}.$$

(131)

Proof. 

Substituting k_eq5 into (130) produces (131). The proof is finished. □

Figure 41 shows some plots of ω_eqn5.

7.2.6. Equivalent Damped Natural Angular Frequency of Type 5 Multi-Fractional Vibrator

From the perspective of vibrations, we use |ζ_eq5| ≤ 1 in what follows. Denote by ω_eqd5 the equivalent damped natural angular frequency of a type 5 multi-fractional vibrator. Then,

$${\omega }_{\mathrm{eqd}5}={\omega }_{\mathrm{eqn}5}\sqrt{1-{\varsigma }_{\mathrm{eq}5}^2}.$$

(132)

Theorem 49 (Equivalent damped natural angular frequency 5).

The quantity ω_eqd5 is expressed by

$${\omega }_{\mathrm{eqd}5}={\omega }_n\sqrt{{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}}\sqrt{1-{\left[{\displaystyle \frac{k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{mk{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}}\right]}^2}.$$

(133)

Proof. 

Substituting ζ_eq5 and ω_eqn5 into (132) yields (133). The proof is finished. □

Figure 42 illustrates some plots of ω_eqd5.

7.2.7. Equivalent Frequency Ratio of Type 5 Multi-Fractional Vibrator

Theorem 50 (Equivalent frequency ratio 5).

Let γ_eq5 be the equivalent frequency ratio of a type 5 multi-fractional vibrator. Define it by

$${\gamma }_{\mathrm{eq}5}={\displaystyle \frac{\omega }{{\omega }_{\mathrm{eqn}5}}}.$$

(134)

Then,

$${\gamma }_{\mathrm{eq}5}=\gamma \sqrt{{\displaystyle \frac{1}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}}.$$

(135)

Proof. 

Substituting ω_eqn5 into (134) results in (135). This finishes the proof. □

Some plots of γ_eq5 are shown in Figure 43.

7.3. Frequency Transfer Function of Type 5 Multi-Fractional Vibrator

Theorem 51 (Frequency transfer function 5).

Let H₅(ω) be the frequency transfer function in phasor for a type 5 multi-fractional vibrator. It is expressed by

$$H_5(\omega )={\displaystyle \frac{1/k_{\mathrm{eq}5}}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}\left[\begin{array}{l}1-\frac{{\gamma }^2}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}\\ +i2\gamma \frac{k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{mk{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}\sqrt{\frac{1}{{\omega }^{(\omega )\lambda }\cos \frac{\lambda (\omega )\pi }{2}}}\end{array}\right]}}.$$

(136)

Proof. 

Using the notations of c_eq5, ζ_eq5, ω_eqn5, and ω_eqd5, we write the motion phasor equation of a type 5 multi-fractional vibrator by

$${(i\omega )}^2{X}_5+i2{\zeta }_{\mathrm{eqn}5}{\omega }_{\mathrm{eqn}5}{X}_5+{\omega }_{\mathrm{eqn}5}^2{X}_5={\displaystyle \frac{F}{m}}.$$

(137)

Thus,

$$H_5(\omega )={\displaystyle \frac{X_5}{F}}={\displaystyle \frac{1/m}{{(i\omega )}^2+i2{\zeta }_{\mathrm{eqn}5}{\omega }_{\mathrm{eqn}5}+{\omega }_{\mathrm{eqn}5}^2}}.$$

(138)

Substituting ζ_eq5 and ω_eqn5 into the above yields (136). The proof is finished. □

In fact, we can write the motion phasor equation of a type 5 multi-fractional vibrator by

$$\left[m{(i\omega )}^2+k{(i\omega )}^{\lambda (\omega )}\right]X_5=\left[{(i\omega )}^{2}m+i\omega c_{\mathrm{eq}5}+k_{\mathrm{eq}5}\right]X_5=F.$$

(139)

Substituting c_eq5 and k_eq5 into the above also produces (136).

Using γ_eq5 and ζ_eq5, we write the above by

$$H_5(\omega )={\displaystyle \frac{1}{k_{\mathrm{eq}5}\left(1-{\gamma }_{\mathrm{eq}5}^2+i2{\varsigma }_{\mathrm{eq}5}{\gamma }_{\mathrm{eq}5}\right)}}.$$

(140)

The amplitude of H₅(ω) is expressed by

$$\left|H_5(\omega )\right|={\displaystyle \frac{1}{k_{\mathrm{eq}5}}}{\displaystyle \frac{1}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}5}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}5}{\gamma }_{\mathrm{eq}5}\right)}^2}}}.$$

(141)

The phase of H₅(ω) is

$${\phi }_5(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }_{\mathrm{eq}5}^2}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}5}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}5}{\gamma }_{\mathrm{eq}5}\right)}^2}}}.$$

(142)

Some plots of the amplitude and phase of H₅(ω) are shown in Figure 44 and Figure 45, respectively.

7.4. Stationary Response to Sinusoidal Vibration of Type 5 Multi-Fractional Vibrator

Below, we give two expressions of the stationary sinusoidal response to a type 5 multi-fractional vibrator. The stationary sinusoidal response in the phasor domain is expressed by Theorem 52. The one in the time domain is represented by Theorem 53.

Theorem 52 (Response phasor 5).

The response phasor of a type 5 multi-fractional vibrator is given by

$$X_5={\displaystyle \frac{F/k}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}\left[\begin{array}{l}1-\frac{{\gamma }^2}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}\\ +i2\gamma \frac{k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{mk{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}\sqrt{\frac{1}{{\omega }^{(\omega )\lambda }\cos \frac{\lambda (\omega )\pi }{2}}}\end{array}\right]}}.$$

(143)

Proof. 

Because X₅ = H₅(ω)F, we see that the above is true. The proof is finished. □

The above equation can be written by

$$X_5={\displaystyle \frac{F}{k_{\mathrm{eq}5}\left(1-{\gamma }_{\mathrm{eq}5}^2+i2{\varsigma }_{\mathrm{eq}5}{\gamma }_{\mathrm{eq}5}\right)}}.$$

(144)

Theorem 53 (Stationary sinusoidal response 5).

The stationary sinusoidal response of a type 5 multi-fractional vibrator is given by

$$\begin{array}{l}{x}_5(t)={\displaystyle \frac{F_0/k_{\mathrm{eq}5}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left|\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}+i2\gamma \frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)\right|}}\\ \sin \left\{\omega t+\theta -{\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}5}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}5}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}5}\frac{\omega }{{\omega }_{\mathrm{eqn}5}}\right)}^2}}}\right\}.\end{array}$$

(145)

Proof. 

As x₅(t) = Im[X₅exp(iωt)], X₅ = H₅(ω)F, F = F₀exp(iθ), and

$${\phi }_5(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}5}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}5}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}5}\frac{\omega }{{\omega }_{\mathrm{eqn}5}}\right)}^2}}},$$

we have x₅(t) = Im{|H₅(ω)|F₀exp(iωt)exp(iθ)exp[−iφ(ω)]}, which equals to (145). The proof is finished. □

Figure 46 shows some plots of x₅(t), demonstrating that there is a noticeable effect of fractional order λ(ω) on the response x₅(t).

8. Results Regarding Type 6 Multi-Fractional Vibrators

We bring forward the results regarding type 6 multi-fractional vibrators. The motion phasor equation of a type 6 multi-fractional vibrator is presented in Section 8.1. The expressions of equivalent vibration parameters about a type 6 multi-fractional vibrator are described in Section 8.2. The expression of frequency transfer function of a type 6 multi-fractional vibrator is expressed in Section 8.3. The expressions of stationary sinusoidal response to a type 6 multi-fractional vibrator in the phasor domain and the time one are brought forward in Section 8.4.

8.1. Motion Phasor Equation of Type 6 Multi-Fractional Vibrator

Theorem 54 (Motion phasor equation 6).

Let X₆ be the phasor of response x₆(t) to a type 6 multi-fractional vibrator. Let F be the phasor of excitation f(t) = F₀sin(ωt + θ). The motion phasor equation of a type 6 multi-fractional vibrator is given by

$$m{(i\omega )}^{\alpha (\omega )}{X}_6+c{(i\omega )}^{\beta (\omega )}{X}_6+k{(i\omega )}^{\lambda (\omega )}{X}_6=F,1<\alpha \left(\omega \right)<3,0<\beta \left(\omega \right)<2,0\le \lambda \left(\omega \right)<1.$$

(146)

Proof. 

Using phasor to express both sides of

$$m{\displaystyle \frac{d^{\alpha }{x}_6(t)}{d{t}^{\alpha }}}+c{\displaystyle \frac{d^{\beta }{x}_6(t)}{d{t}^{\beta }}}+k{\displaystyle \frac{d^{\lambda }{x}_6(t)}{d{t}^{\lambda }}}=f(t)$$

(147)

yields (146). The proof is finished. □

Now, we write the left side of (146) by

$$\begin{array}{l}{m(i\omega )}^{\alpha (\omega )}{X}_6+{c(i\omega )}^{\beta (\omega )}{X}_6+{k(i\omega )}^{\lambda (\omega )}{X}_6=\left[m{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]X_6\\ +i\left[m{\omega }^{\alpha (\omega )}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]X_6\\ +k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}{X}_6.\end{array}$$

(148)

When writing the right side of the above by

$$\begin{array}{l}\left\{\begin{array}{l}\left[m{\omega }^{\alpha (\omega )}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]\\ +i\left[m{\omega }^{\alpha (\omega )}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]\\ +k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\end{array}\right\}{X}_6\\ =\left\{\begin{array}{l}-\left[m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]{(i\omega )}^2\\ +i\omega \left[m{\omega }^{\alpha (\omega )-1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]\\ +k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\end{array}\right\}{X}_6,\end{array}$$

(149)

we have

$$\begin{array}{l}m{(i\omega )}^{\alpha }{X}_6+c{(i\omega )}^{\beta }{X}_6+k{(i\omega )}^{\lambda }{X}_6\\ =\left\{\begin{array}{l}-\left[m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]{(i\omega )}^2\\ +i\omega \left[m{\omega }^{\alpha (\omega )-1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]\\ +k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\end{array}\right\}{X}_6.\end{array}$$

(150)

8.2. Equivalent Parameters of Type 6 Multi-Fractional Vibrator

In Section 8.2, we bring forward the expressions of equivalent vibration parameters of a type 6 multi-fractional vibrator, including equivalent mass in Section 8.2.1, equivalent damping in Section 8.2.2, equivalent stiffness in Section 8.2.3, equivalent damping ratio in Section 8.2.4, equivalent damping free natural angular frequency in Section 8.2.5, equivalent damped natural angular frequency in Section 8.2.6, and equivalent frequency ratio in Section 8.2.7.

8.2.1. Equivalent Mass of Type 6 Multi-Fractional Vibrator

Theorem 55 (Equivalent mass 6).

Denote by m_eq6 the equivalent mass of a multi-fractional vibrator of type 6. Then,

$$m_{\mathrm{eq}6}=-\left[m{\omega }^{\alpha (\omega )-2}\cos {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right].$$

(151)

Proof. 

The first item on the right side of (150) stands for an inertia force in the phasor domain. The acceleration in the phasor domain is (iω)²X₆. Its coefficient is the equivalent mass. Thus, (151) holds. This finishes the proof. □

Note that

m_eq6 = m_eq3.

(152)

8.2.2. Equivalent Damping of Type 6 Multi-Fractional Vibrator

Theorem 56 (Equivalent damping 6).

Let c_eq6 be the equivalent damping of a type 6 multi-fractional vibrator. Then,

$$c_{\mathrm{eq}6}=m{\omega }^{\alpha (\omega )-1}\sin {\displaystyle \frac{\alpha (\omega )\pi }{2}}+c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}.$$

(153)

Proof. 

The second item on the right side of (150) implies a friction force in the phasor domain. The velocity in the phasor domain is (iω)X₆. Its coefficient is the equivalent damping c_eq6. Consequently, (153) is valid. The proof is finished. □

Figure 47 illustrates some plots of c_eq6.

8.2.3. Equivalent Stiffness of Type 6 Multi-Fractional Vibrator

Theorem 57 (Equivalent stiffness 6).

Let k_eq6 be the equivalent stiffness of a type 6 multi-fractional vibrator. Then,

$$k_{\mathrm{eq}6}=k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}.$$

(154)

Proof. 

The third item on the right side of (150) is a restoration force in the phasor domain. The displacement in the phasor domain is X₆. Its coefficient is the equivalent stiffness k_eq6. Therefore, (154) is true. The proof is finished. □

Note that

k_eq6 = k_eq5 = k_eq4.

(155)

8.2.4. Equivalent Damping Ratio of Type 6 Multi-Fractional Vibrator

Theorem 58 (Equivalent damping ratio 6).

Denote by ζ_eq6 the equivalent damping ratio of a type 6 multi-fractional vibrator. Define it by

$${\varsigma }_{\mathrm{eq}6}={\displaystyle \frac{c_{\mathrm{eq}6}}{2\sqrt{m_{\mathrm{eq}6}{k}_{\mathrm{eq}6}}}}.$$

(156)

Then,

$${\varsigma }_{\mathrm{eq}6}={\displaystyle \frac{m{\omega }^{\alpha (\omega )-1}\sin \frac{\alpha (\omega )\pi }{2}+c{\omega }^{\beta (\omega )-1}\sin \frac{\beta (\omega )\pi }{2}+k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{-\left[m{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}+c{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]k{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}}.$$

(157)

Proof. 

Substituting m_eq6, c_eq6, and k_eq6 into (156) yields (157). This finishes the proof. □

Some plots of ζ_eq6 are indicated in Figure 48.

8.2.5. Equivalent Damping Free Natural Angular Frequency of Type 6 Multi-Fractional Vibrator

Theorem 59 (Equivalent damping free natural angular frequency 6).

Denote by ω_eqn6 the equivalent damping free natural angular frequency of a type 6 multi-fractional vibrator. It is defined by

$${\omega }_{\mathrm{eqn}6}=\sqrt{{\displaystyle \frac{k_{\mathrm{eq}6}}{m_{\mathrm{eq}6}}}}.$$

(158)

Then,

$${\omega }_{\mathrm{eqn}6}=\sqrt{{\displaystyle \frac{k{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}{-\left[m{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}+c{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]}}}.$$

(159)

Proof. 

Substituting m_eq6 and k_eq6 into (158) results in (159). The proof is finished. □

Figure 49 indicates some plots of ω_eqn6.

8.2.6. Equivalent Damped Natural Angular Frequency of Type 6 Multi-Fractional Vibrator

From a practical perspective of vibrations, |ζ_eq6| ≤ 1 is used in what follows.

Theorem 60 (Equivalent damped natural angular frequency 6).

Let ω_eqd6 be the equivalent damped natural angular frequency of a type 6 multi-fractional vibrator. Define it by

$${\omega }_{\mathrm{eqd}6}={\omega }_{\mathrm{eqn}6}\sqrt{1-{\varsigma }_{\mathrm{eq}6}^2}.$$

(160)

Then,

$$\begin{array}{l}{\omega }_{\mathrm{eqd}6}=\sqrt{{\displaystyle \frac{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}}\\ \sqrt{1-{\left({\displaystyle \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}}\right)}^2}.\end{array}$$

(161)

Proof. 

Substituting ζ_eq6 and ω_eqn6 into (160) produces (161). The proof is finished. □

Some plots of ω_eqd6 are provided in Figure 50.

8.2.7. Equivalent Frequency Ratio of Type 6 Multi-Fractional Vibrator

Theorem 61 (Equivalent frequency ratio 6).

Denote by γ_eq6 the equivalent frequency ratio of a type 6 multi-fractional vibrator. Define it by

$${\gamma }_{\mathrm{eq}6}={\displaystyle \frac{\omega }{{\omega }_{\mathrm{eqn}6}}}.$$

(162)

Then,

$${\gamma }_{\mathrm{eq}6}=\gamma \sqrt{{\displaystyle \frac{-\left[{\omega }^{\alpha (\omega )-2}\cos \frac{\alpha (\omega )\pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]}{{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}}.$$

(163)

Proof. 

Substituting ω_eqn6 into (162) results in (163). This finishes the proof. □

Figure 51 shows some plots of γ_eq6.

8.3. Frequency Transfer Function of Type 6 Multi-Fractional Vibrator

Theorem 62 (Frequency transfer function 6).

Denote by H₆(ω) the frequency transfer function in phasor for a type 6 multi-fractional vibrator. Then,

$$H_6(\omega )={\displaystyle \frac{X_6}{F}}={\displaystyle \frac{1}{k_{\mathrm{eq}6}\left(1-{\gamma }_{\mathrm{eq}6}^2+i2{\varsigma }_{\mathrm{eq}6}{\gamma }_{\mathrm{eq}6}\right)}}.$$

(164)

Proof. 

Taking into account the notations of m_eq6, c_eq6, ζ_eq6, ω_eqn6, and ω_eqd6, we have the motion phasor equation of a type 6 multi-fractional vibrator in the form

$${(i\omega )}^2{X}_6+i2{\zeta }_{\mathrm{eqn}6}{\omega }_{\mathrm{eqn}6}{X}_6+{\omega }_{\mathrm{eqn}6}^2{X}_6={\displaystyle \frac{F}{m_{\mathrm{eq}6}}}.$$

(165)

Since H₆(ω) = X₆/F, we have (164). The proof is finished. □

The amplitude of H₆(ω) is

$$\left|H_6(\omega )\right|={\displaystyle \frac{1}{k_{\mathrm{eq}6}}}{\displaystyle \frac{1}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}6}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}6}{\gamma }_{\mathrm{eq}6}\right)}^2}}}.$$

(166)

The phase of H₆(ω) is

$${\phi }_6(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }_{\mathrm{eq}6}^2}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}6}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}6}{\gamma }_{\mathrm{eq}6}\right)}^2}}}.$$

(167)

Some plots of |H₆(ω)| and φ₆(ω) are indicated in Figure 52 and Figure 53, respectively.

8.4. Stationary Response to Sinusoidal Vibration of Type 6 Multi-Fractional Vibrator

In Section 8.4, we give two expressions of the stationary sinusoidal response to a type 6 multi-fractional vibrator. The one in Theorem 63 is the response in the phasor domain and the other in Theorem 64 is one in the time domain.

Theorem 63 (Response phasor 6).

The response phasor of a type 6 multi-fractional vibrator is in the form

$$X_6={\displaystyle \frac{F}{k_{\mathrm{eq}6}\left(1-{\gamma }_{\mathrm{eq}6}^2+i2{\varsigma }_{\mathrm{eq}6}{\gamma }_{\mathrm{eq}6}\right)}}.$$

(168)

Proof. 

Because X₆ = H₆(ω)F, the above is true. The proof is finished. □

Theorem 64 (Stationary sinusoidal response 6).

The stationary sinusoidal response of a type 6 multi-fractional vibrator is in the form

$$x_6(t)={\displaystyle \frac{1}{k_{\mathrm{eq}6}}}{\displaystyle \frac{F}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}6}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}6}{\gamma }_{\mathrm{eq}6}\right)}^2}}}\sin \left\{\omega t+\theta -{\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}6}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}6}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}6}\frac{\omega }{{\omega }_{\mathrm{eqn}6}}\right)}^2}}}\right\}.$$

(169)

Proof. 

Due to x₆(t) = Im[X₆exp(iωt)], X₆ = H₆(ω)F, F = F₀exp(iθ), and

$${\phi }_6(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}6}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}6}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}6}\frac{\omega }{{\omega }_{\mathrm{eqn}6}}\right)}^2}}},$$

we have x₆(t) = Im{|H₆(ω)|F₀exp(iωt)exp(iθ)exp[−iφ(ω)]}, which equals to (169). The proof is finished. □

Figure 54 indicates some plots of x₆(t) for different values of α, β, and λ. It demonstrates that there is a considerable effect of fractional orders α(ω), β(ω), and λ(ω) on x₆(t).

9. Results Regarding Type 7 Multi-Fractional Vibrators

We propose the results with respect to type 7 multi-fractional vibrators as follows. The motion phasor equation of a type 7 multi-fractional vibrator is presented in Section 9.1. The expressions of equivalent vibration parameters about a type 7 multi-fractional vibrator are given in Section 9.2. The expression of frequency transfer function of a type 7 multi-fractional vibrator is in Section 9.3. The expressions of stationary sinusoidal response to a type 7 multi-fractional vibrator in the phasor domain and the time one are presented in Section 9.4.

9.1. Motion Phasor Equation of Type 7 Multi-Fractional Vibrator

Theorem 65 (Motion phasor equation 7).

Let X₇ be the phasor of response x₇(t) to a type 7 multi-fractional vibrator. Let F be the phasor of excitation f(t) = F₀sin(ωt + θ). The motion phasor equation of a type 7 multi-fractional vibrator is given by

$$m{(i\omega )}^2{X}_7+c{(i\omega )}^{\beta (\omega )}{X}_7+k{(i\omega )}^{\lambda (\omega )}{X}_7=F,0<\beta \left(\omega \right)<2,0\le \lambda \left(\omega \right)<1.$$

(170)

Proof. 

Using phasor to express both sides of

$$m{\displaystyle \frac{d^2{x}_7(t)}{d{t}^2}}+c{\displaystyle \frac{d^{\beta }{x}_7(t)}{d{t}^{\beta }}}+k{\displaystyle \frac{d^{\lambda }{x}_7(t)}{d{t}^{\lambda }}}=f(t)$$

(171)

results in (170). This finishes the proof. □

With the principal branches of i^β and i^λ, the left side of the above equation is given by

$$\begin{array}{l}m{(i\omega )}^2{X}_7+c{(i\omega )}^{\beta (\omega )}{X}_7+k{(i\omega )}^{\lambda (\omega )}{X}_7=\left[-m{\omega }^2+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]X_7\\ +i\left[c{\omega }^{\beta (\omega )}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]X_7+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}{X}_7.\end{array}$$

(172)

Writing the right side of the above by

$$\begin{array}{l}\left\{\begin{array}{l}\left[-m{\omega }^2+c{\omega }^{\beta (\omega )}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]\\ +i\left[c{\omega }^{\beta (\omega )}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\end{array}\right\}{X}_7\\ =\left\{\begin{array}{l}\left[m-c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]{(i\omega )}^2\\ +i\omega \left[c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\end{array}\right\}{X}_7\end{array}$$

(173)

yields

$$\begin{array}{l}\left[m{(i\omega )}^2+c{(i\omega )}^{\beta }+k{(i\omega )}^{\lambda }\right]X_7\\ =\left\{\begin{array}{l}\left[m-c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}\right]{(i\omega )}^2\\ +i\omega \left[c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}\right]+k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}\end{array}\right\}{X}_7.\end{array}$$

(174)

9.2. Equivalent Parameters of Type 7 Multi-Fractional Vibrator

We put forward the expressions of equivalent vibration parameters of a type 7 multi-fractional vibrator in Section 9.2 below. They are equivalent mass in Section 9.2.1, equivalent damping in Section 9.2.2, equivalent stiffness in Section 9.2.3, equivalent damping ratio in Section 9.2.4, equivalent damping free natural angular frequency in Section 9.2.5, equivalent damped natural angular frequency in Section 9.2.6, and equivalent frequency ratio in Section 9.2.7.

9.2.1. Equivalent Mass of Type 7 Multi-Fractional Vibrator

Theorem 66 (Equivalent mass 7).

Denote by m_eq7 the equivalent mass of a multi-fractional vibrator of type 7. Then,

$$m_{\mathrm{eq}7}=m-c{\omega }^{\beta (\omega )-2}\cos {\displaystyle \frac{\beta (\omega )\pi }{2}}.$$

(175)

Proof. 

The first item on the right side of (174) stands for an inertia force in the phasor domain. The acceleration in the phasor domain is (iω)²X₇. Its coefficient is the equivalent mass. Thus, (175) is true. This finishes the proof. □

Note that

m_eq7 = m_eq2.

(176)

9.2.2. Equivalent Damping of Type 7 Multi-Fractional Vibrator

Theorem 67 (Equivalent damping 7).

Let c_eq7 be the equivalent damping of a type 7 multi-fractional vibrator. Then,

$$c_{\mathrm{eq}7}=c{\omega }^{\beta (\omega )-1}\sin {\displaystyle \frac{\beta (\omega )\pi }{2}}+k{\omega }^{\lambda (\omega )-1}\sin {\displaystyle \frac{\lambda (\omega )\pi }{2}}.$$

(177)

Proof. 

The second item on the right side of (174) implies a friction force in the phasor domain. The velocity in the phasor domain is (iω)X₇. Its coefficient is the equivalent damping c_eq7. Consequently, (177) holds. The proof is finished. □

Figure 55 illustrates some plots of c_eq7.

9.2.3. Equivalent Stiffness of Type 7 Multi-Fractional Vibrator

Theorem 68 (Equivalent stiffness 7).

Let k_eq7 be the equivalent stiffness of a type 7 multi-fractional vibrator. Then,

$$k_{\mathrm{eq}7}=k{\omega }^{\lambda (\omega )}\cos {\displaystyle \frac{\lambda (\omega )\pi }{2}}.$$

(178)

Proof. 

The third item on the right side of (174) is a restoration force in the phasor domain. The displacement in the phasor domain is X₆. Its coefficient is the equivalent stiffness k_eq7. Therefore, (178) is true. This finishes the proof. □

Note that

k_eq7 = k_eq6 = k_eq5 = k_eq4.

(179)

9.2.4. Equivalent Damping Ratio of Type 7 Multi-Fractional Vibrator

Theorem 69 (Equivalent damping ratio 7).

Denote by ζ_eq7 the equivalent damping ratio of a type 7 multi-fractional vibrator. Define it by

$${\varsigma }_{\mathrm{eq}7}={\displaystyle \frac{c_{\mathrm{eq}7}}{2\sqrt{m_{\mathrm{eq}7}{k}_{\mathrm{eq}7}}}}.$$

(180)

Then,

$${\varsigma }_{\mathrm{eq}7}={\displaystyle \frac{c{\omega }^{\beta (\omega )-1}\sin \frac{\beta (\omega )\pi }{2}+k{\omega }^{\lambda (\omega )-1}\sin \frac{\lambda (\omega )\pi }{2}}{2\sqrt{\left[m-c{\omega }^{\beta (\omega )-2}\cos \frac{\beta (\omega )\pi }{2}\right]k{\omega }^{\lambda (\omega )}\cos \frac{\lambda (\omega )\pi }{2}}}}.$$

(181)

Proof. 

Substituting m_eq7, c_eq7, and k_eq7 into (180) results in (181). The proof is finished. □

Some plots of ζ_eq7 are indicated in Figure 56.

9.2.5. Equivalent Damping Free Natural Angular Frequency of Type 7 Multi-Fractional Vibrator

Theorem 70 (Equivalent damping free natural angular frequency 7).

Denote by ω_eqn7 the equivalent damping free natural angular frequency of a type 7 multi-fractional vibrator. Define it by

$${\omega }_{\mathrm{eqn}7}=\sqrt{{\displaystyle \frac{k_{\mathrm{eq}7}}{m_{\mathrm{eq}7}}}}.$$

(182)

Then,

$${\omega }_{\mathrm{eqn}7}=\sqrt{{\displaystyle \frac{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}}.$$

(183)

Proof. 

Substituting m_eq7 and k_eq7 into (182) yields (183). The proof is finished. □

Figure 57 indicates some plots of ω_eqn7.

9.2.6. Equivalent Damped Natural Angular Frequency of Type 7 Multi-Fractional Vibrator

From a practical perspective of vibrations, |ζ_eq7| ≤ 1 is used in what follows.

Theorem 71 (Equivalent damped natural angular frequency 7).

Let ω_eqd7 be the equivalent damped natural angular frequency of a type 7 multi-fractional vibrator. Define it by

$${\omega }_{\mathrm{eqd}7}={\omega }_{\mathrm{eqn}7}\sqrt{1-{\varsigma }_{\mathrm{eq}7}^2}.$$

(184)

Then,

$${\omega }_{\mathrm{eqd}7}=\sqrt{{\displaystyle \frac{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}}\sqrt{1-{\left({\displaystyle \frac{c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{\left(m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}}\right)}^2}.$$

(185)

Proof. 

Substituting ζ_eq7 and ω_eqn7 into (184) yields (185). This finishes the proof. □

Some plots of ω_eqd7 are shown in Figure 58.

9.2.7. Equivalent Frequency Ratio of Type 7 Multi-Fractional Vibrator

Theorem 72 (Equivalent frequency ratio 7).

Denote by γ_eq7 the equivalent frequency ratio of a type 7 multi-fractional vibrator. Define it by

$${\gamma }_{\mathrm{eq}7}={\displaystyle \frac{\omega }{{\omega }_{\mathrm{eqn}7}}}.$$

(186)

Then,

$${\gamma }_{\mathrm{eq}7}=\gamma \sqrt{{\displaystyle \frac{1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}}.$$

(187)

Proof. 

Substituting ω_eqn7 into (186) yields (187). The proof is finished. □

Figure 59 shows some plots of γ_eq7.

9.3. Frequency Transfer Function of Type 7 Multi-Fractional Vibrator

Theorem 73 (Frequency transfer function 7).

Let H₇(ω) be the frequency transfer function in phasor for a type 7 multi-fractional vibrator. Then,

$$H_7(\omega )={\displaystyle \frac{X_7}{F}}={\displaystyle \frac{1}{k_{\mathrm{eq}7}\left(1-{\gamma }_{\mathrm{eq}7}^2+i2{\varsigma }_{\mathrm{eq}7}{\gamma }_{\mathrm{eq}7}\right)}}.$$

(188)

Proof. 

Considering m_eq7, c_eq7, ζ_eq7, ω_eqn7, and ω_eqd7, we write the motion phasor equation of a type 7 multi-fractional vibrator in the form

$${(i\omega )}^2{X}_7+i2{\zeta }_{\mathrm{eqn}7}{\omega }_{\mathrm{eqn}7}{X}_7+{\omega }_{\mathrm{eqn}7}^2{X}_7={\displaystyle \frac{F}{m_{\mathrm{eq}7}}}.$$

(189)

Owing to H₇(ω) = X₇/F, we have (188). This finishes the proof. □

|H₇(ω)| is expressed by

$$\left|H_7(\omega )\right|={\displaystyle \frac{1}{k_{\mathrm{eq}7}}}{\displaystyle \frac{1}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}7}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}7}{\gamma }_{\mathrm{eq}7}\right)}^2}}}.$$

(190)

The phase of H₇(ω) is

$${\phi }_7(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\gamma }_{\mathrm{eq}7}^2}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}7}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}7}{\gamma }_{\mathrm{eq}7}\right)}^2}}}.$$

(191)

Some plots of |H₇(ω)| and φ₇(ω) are shown in Figure 60 and Figure 61, respectively.

9.4. Stationary Response to Sinusoidal Vibration of Type 7 Multi-Fractional Vibrator

In Section 9.4, we bring forward two expressions of stationary sinusoidal response to a type 7 multi-fractional vibrator. Theorem 74 gives the expression of the response in the phasor domain while Theorem 75 the one in the time domain.

Theorem 74 (Response phasor 7).

The response phasor of a type 7 multi-fractional vibrator is in the form

$$X_7={\displaystyle \frac{F}{k_{\mathrm{eq}7}\left(1-{\gamma }_{\mathrm{eq}7}^2+i2{\varsigma }_{\mathrm{eq}7}{\gamma }_{\mathrm{eq}7}\right)}}.$$

(192)

Proof. 

Due to X₇ = H₇(ω)F, the above is valid. This finishes the proof. □

Theorem 75 (Stationary sinusoidal response 7).

The stationary sinusoidal response of a type 7 multi-fractional vibrator is in the form

$$x_7(t)={\displaystyle \frac{1}{k_{\mathrm{eq}7}}}{\displaystyle \frac{F}{\sqrt{{\left(1-{\gamma }_{\mathrm{eq}7}^2\right)}^2+{\left(2{\varsigma }_{\mathrm{eq}7}{\gamma }_{\mathrm{eq}7}\right)}^2}}}\sin \left\{\omega t+\theta -{\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}7}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}7}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}7}\frac{\omega }{{\omega }_{\mathrm{eqn}7}}\right)}^2}}}\right\}.$$

(193)

Proof. 

Since x₇(t) = Im[X₇exp(iωt)], X₇ = H₇(ω)F, F = F₀exp(iθ), and

$${\phi }_7(\omega )={\cos }^{-1}{\displaystyle \frac{1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}7}}\right)}^2}{\sqrt{{\left[1-{\left(\frac{\omega }{{\omega }_{\mathrm{eqn}7}}\right)}^2\right]}^2+{\left(2{\varsigma }_{\mathrm{eq}7}\frac{\omega }{{\omega }_{\mathrm{eqn}7}}\right)}^2}}},$$

we have x₇(t) = Im{|H₇(ω)|F₀exp(iωt)exp(iθ)exp[−iφ(ω)]}. Thus, (193) is true. The proof is finished. □

Figure 62 indicates some plots of x₇(t) for different values of β(ω) and λ(ω). It demonstrates that there is a considerable effect of fractional orders β(ω) and λ(ω) on x₇(t).

10. Discussions

With respect to seven types of multi-fractional vibrators, the expressions of equivalent parameters (mass, damping, stiffness, damping ratio, damping free natural angular frequency, damped natural angular frequency, frequency ratio) and frequency transfer functions in this paper are the same as those in the author’s previous work [28,29,30]. However, the methods deriving these expressions are different. Those in [28,29,30] are based on Fourier transform but this research uses multi-fractional phasor. Note that the concept and expression of the multi-fractional phasor are new in the field. Moreover, the present analytic expressions of stationary sinusoidal responses to seven types of multi-fractional vibrators are novel.

There are research issues with respect to seven types of multi-fractional vibrators in addition to the stationary sinusoidal responses addressed previously. There are two topics we are interested in examining in the future. One is to study what analytical expressions of −3dB frequency bandwidth are with respect to seven types of multi-fractional vibrators. The other is to explore identification methods of structures with seven types of multi-fractional vibrators.

11. Conclusions and Summary

We have introduced the multi-fractional phasor in Theorem 1. The motion phasor equations of seven types of multi-fractional vibrators are put forward in Theorems 2, 12, 22, 32, 43, 54, and 65, respectively. The analytical expressions of response phasors of seven types of multi-fractional vibrators are presented in Theorems 10, 20, 30, 41, 52, 63, and 74, respectively. The expressions of the stationary sinusoidal responses of seven types of multi-fractional vibration systems are analytically proposed in Theorems 11, 21, 31, 42, 53, 64, and 75, respectively. In addition, based on multi-fractional phasor, we have obtained the following results.

• The analytical expressions of equivalent masses of seven types of multi-fractional vibrators, which are given by Theorems 3, 13, 23, 33, 44, 55, and 66, respectively.
• The analytical expressions of equivalent dampings of seven types of multi-fractional vibrators, which are, respectively, represented by Theorems 4, 14, 24, 34, 45, 56, and 67.
• The analytical expressions of equivalent damping ratios of seven types of multi-fractional vibrators, which are expressed in Theorems 5, 15, 25, 36, 47, 58, and 69, respectively.
• The analytical expressions of equivalent stiffnesses of types 4–7 multi-fractional vibrators, which are, respectively, presented by Theorems 35, 46, 57, and 68.
• The analytical expressions of equivalent damping free natural angular frequencies of seven types of multi-fractional vibrators, which are, respectively, given by Theorems 6, 16, 26, 37, 48, 59, and 70.
• Equivalent damped free natural angular frequencies of seven types of multi-fractional vibrators, which are, respectively, described in Theorems 7, 17, 27, 38, 49, 60, and 71.
• The analytical expressions of equivalent frequency ratios of seven types of multi-fractional vibrators, which are explained in Theorems 8, 18, 28, 39, 50, 61, and 72, respectively.
• The analytical expressions of frequency transfer functions of seven types of multi-fractional vibrators, which are addressed in Theorems 9, 19, 29, 40, 51, 62, and 73, respectively.

The effects of multi-fractional orders on stationary sinusoidal responses of those multi-fractional vibrators are noticeable.