PSD and Cross-PSD of Responses of Seven Classes of Fractional Vibrations Driven by fGn, fBm, Fractional OU Process, and von Kármán Process

Abstract

This paper gives its contributions in four stages. First, we propose the analytical expressions of power spectrum density (PSD) responses and cross-PSD responses to seven classes of fractional vibrators driven by fractional Gaussian noise (fGn). Second, we put forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by fractional Brownian motion (fBm). Third, we present the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators driven by the fractional Ornstein–Uhlenbeck (OU) process. Fourth, we bring forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by the von Kármán process. We show that the statistical dependences of the responses to seven classes of fractional vibrators follow those of the excitation of fGn, fBm, the OU process, or the von Kármán process. We also demonstrate the obvious effects of fractional orders on the responses to seven classes of fractional vibrations. In addition, we newly introduce class VII fractional vibrators, their frequency transfer function, and their impulse response in this research.

1. Introduction

We start with three topics in the field, as follows. First, the topic of fractional vibrations attracts the interests of researchers; see, e.g., Uchaikin [1], Achar et al. [2,3,4], Duan [5], Coccolo et al. [6], Sidorov et al. [7], Pirrotta et al. [8,9], Lewandowski and Wielentejczyk [10], Spanos and Malara [11], Blaszczyk [12], Blaszczyk and Ciesielski [13], Blaszczyk et al. [14,15], Al-Rabtah et al. [16], Zurigat [17], Rossikhin and Shitikova [18,19,20,21], Rossikhin [22], Shitikova et al. [23], Shitikova [24,25], El-Nabulsi et al. [26], Banerjee [27], Sofi [28], Sofi and Muscolino [29], Molina-Villegas et al. [30], Parovik [31], Li et al. [32,33], Li [34,35,36], and references therein, to mention a few.

Second, the topic of fractional processes remains in the interest of researchers; see, e.g., Mandelbrot [37,38], Beran [39], Levy-Vehel and Lutton [40], Lévy-Véhel and Rams [41], Bender et al. [42], Guével and Lévy-Véhel [43], Lebovits et al. [44], Corlay et al. [45], Ayache et al. [46], Park et al. [47], Luks and Xiao [48], Li and Xiao [49], Flandrin [50,51], Zao et al. [52], Borgnat et al. [53,54], Salcedo-Sanz et al. [55], Levin et al. [56], Marković and Gros [57], Eliazar and Shlesinger [58], Campa et al. [59], Pinchas and Avraham [60,61], Mirás-Avalos et al. [62], Kaulakys et al. [63], Lubashevsky [64], Starchenko [65], Gorev et al. [66], Sheluhin et al. [67], Sousa-Vieira and Fernández-Veiga [68], Beskardes et al. [69], Ercan and Kavvas [70], Lee et al. [71], Li [72,73,74], to cite a few.

The third topic concerns fractional equation-driven fractional random functions; see, e.g., Li and Yan [75], Gao et al. [76], Guo et al. [77], Gao and Sun [78], Pei and Zhang [79], Barth and Stüwe [80], Kim and Park [81], Noupelah et al. [82], Massing [83], Lee [84], Fa et al. [85], Freundlich and Sado [86], Burlon [87], Wang et al. [88], Hu and Zhou [89], and Liu et al. [90], to simply mention a few. However, reports regarding the analytical expressions of the responses of seven classes of fractional vibrators (see the next Section) driven by fractional processes (fractional Gaussian noise (fGn), fractional Brownian motion (fBm), the fractional Ornstein–Uhlenbeck (OU) process, and the von Kármán process) are rarely seen. This paper aims at bringing forward the analytical expressions of the power spectrum density (PSD) and cross-PSD responses to seven classes of fractional vibrators driven by those fractional processes.

The rest of the paper is organized as follows. In Section 2, we explain the motion equations of seven classes of fractional vibrators, their frequency transfer functions, and their impulse response functions. In Section 3, Section 4, Section 5 and Section 6, we put forward the analytical expressions of the response to seven classes of fractional vibrators driven by fGn, fBm, the fractional OU process, and the von Kármán process, respectively. Discussions and conclusions are given in Section 7.

2. Seven Classes of Fractional Vibrators

The author’s previous work (Li [34]) addressed the first three classes of fractional vibrations in this paper. Further, Li [35,36] discusses the first six classes of fractional vibrations in this paper. The seventh class of fractional vibrators, as shown below, as well as their frequency transfer functions and impulses, are newly introduced in this research.

2.1. Motion Equations

Let m, c, and k be the primary mass, damping, and stiffness of a vibration system, respectively. We refer to a class I fractional vibrator when its motion equation is, for 0 < α < 3, in the form

$$m\frac{d^{\alpha }{x}_1(t)}{d{t}^{\alpha }}+k{x}_1(t)=f(t).$$

(1)

A system is called a class II fractional vibrator when its motion equation is, for 0 < β < 2, given by

$$m\frac{d^2{x}_2(t)}{d{t}^2}+c\frac{d^{\beta }{x}_2(t)}{d{t}^{\beta }}+k{x}_2(t)=f(t).$$

(2)

We call a system class III fractional vibrator if its motion equation is, for 0 < α < 3 and 0 < β < 2, expressed by

$$m\frac{d^{\alpha }{x}_3(t)}{d{t}^{\alpha }}+c\frac{d^{\beta }{x}_3(t)}{d{t}^{\beta }}+k{x}_3(t)=f(t).$$

(3)

We term a system a class IV fractional vibrator if its motion equation is, for 0 < α < 3 and 0 ≤ λ < 1, represented by

$$m\frac{d^{\alpha }{x}_4(t)}{d{t}^{\alpha }}+k\frac{d^{\lambda }{x}_4(t)}{d{t}^{\lambda }}=f(t).$$

(4)

A system is called a class V fractional vibrator if its motion equation is, for 0 ≤ λ < 1, expressed by

$$m\frac{d^2{x}_5(t)}{d{t}^2}+k\frac{d^{\lambda }{x}_5(t)}{d{t}^{\lambda }}=f(t).$$

(5)

We call a system a class VI fractional vibrator if its motion equation is, for 0 < α < 3, 0 < β < 2 and 0 ≤ λ < 1, in the form

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

(6)

A system is called a class VII fractional vibrator when its motion equation is, for 0 < β < 2 and 0 ≤ λ < 1, given by

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

(7)

2.2. Frequency Transfer Functions

Theorem 1. 

Let F(ω) be the Fourier transform. Let X_j(ω) be the Fourier transform of the jth class response x_j(t) (j = 1, 2, …, 7). Let H_j(ω) be the frequency transfer function of a fractional vibrator of the jth class. Then, $H_j(\omega )=\frac{X_j(\omega )}{F(\omega )}.$ For j = 1,

$$H_1(\omega )=\frac{1}{k+{(i\omega )}^{\alpha }m}.$$

(8)

If j = 2,

$$H_2(\omega )=\frac{1}{k-{\omega }^{2}m+{(i\omega )}^{\beta }c}.$$

(9)

When j = 3,

$$H_3(\omega )=\frac{1}{k+{(i\omega )}^{\alpha }m+{(i\omega )}^{\beta }c}.$$

(10)

For j = 4,

$$H_4(\omega )=\frac{1}{{(i\omega )}^{\alpha }m+{(i\omega )}^{\lambda }k}.$$

(11)

When j = 5,

$$H_5(\omega )=\frac{1}{-{\omega }^{2}m+{(i\omega )}^{\lambda }k}.$$

(12)

If j = 6,

$$H_6(\omega )=\frac{1}{{(i\omega )}^{\alpha }m+{(i\omega )}^{\beta }c+{(i\omega )}^{\lambda }k}.$$

(13)

For j = 7, we have

$$H_7(\omega )=\frac{1}{-{\omega }^{2}m+{(i\omega )}^{\beta }c+{(i\omega )}^{\lambda }k}.$$

(14)

Proof. 

Solving the Fourier transforms on both sides of (1)–(7) produces (8)–(14), respectively. The proof is finished. □

Corollary 1. 

Let ω_n be the primary damping free natural angular frequency. Let ζ be the primary damping ratio. Denote the primary frequency ratio with γ. Then, H_j(ω) (j = 1, 2, …, 7) can be expressed in the following forms. If j = 1,

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

(15)

For j = 2,

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

(16)

When j = 3,

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

(17)

For j = 4,

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

(18)

If j = 5,

$$H_5(\omega )=\frac{1}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}+i2\gamma \frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}.$$

(19)

When j = 6,

$$H_6(\omega )=\frac{1}{k\left(\begin{array}{l}{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\end{array}\right)}.$$

(20)

For j = 7,

$$H_7(\omega )=\frac{1}{k\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\gamma \left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\right]}.$$

(21)

Proof. 

Taking into account the principal branch of i^α or i^β or i^λ, (8)–(14) can be expressed by (15)–(21), respectively. This finishes the proof. □

2.3. Impulse Response Functions

Theorem 2. 

Let h_j(t) be the impulse response function of the fractional vibrator of the jth class (j = 1, 2, …, 7). Then, when j = 1,

$$h_1(t)=\frac{e^{-\frac{\omega \sin \frac{\alpha \pi }{2}}{2\left|\cos \frac{\alpha \pi }{2}\right|}t}\sin \left(\frac{{\omega }_n}{\sqrt{{\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|}}\sqrt{1-\frac{{\omega }^{2\alpha }{\sin }^2\frac{\alpha \pi }{2}}{4{\omega }_n^2\left|\cos \frac{\alpha \pi }{2}\right|}}t\right)}{m{\omega }_n\sqrt{{\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|}\sqrt{1-\frac{{\omega }^{2\alpha }{\sin }^2\frac{\alpha \pi }{2}}{4{\omega }_n^2\left|\cos \frac{\alpha \pi }{2}\right|}}}u(t),$$

(22)

where u(t) is the unit step function. For j = 2,

$$h_2(t)=\frac{e^{-\frac{\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}t}\sin \frac{{\omega }_n\sqrt{1-\frac{{\varsigma }^2{\omega }^{2(\beta -1)}{\sin }^2\frac{\beta \pi }{2}}{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}t}{{\omega }_{n}m\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}\sqrt{1-\frac{{\varsigma }^2{\omega }^{2(\beta -1)}{\sin }^2\frac{\beta \pi }{2}}{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}}u(t).$$

(23)

If j = 3,

$$\begin{array}{l}{h}_3(t)==\frac{\frac{1}{m}{e}^{-\frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{-2\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}t}}{{\omega }_n\sqrt{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\sqrt{1-{\left[\frac{{\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}^2}{4{\omega }_n^2\left[-\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}\right]}^2}}\\ \sin \frac{{\omega }_n\sqrt{1-{\left[\frac{{\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}^2}{4{\omega }_n^2\left[-\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}\right]}^2}}{\sqrt{-\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}tu(t).\end{array}$$

(24)

When j = 4,

$$\begin{array}{l}{h}_4(t)=\frac{e^{-\frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}}{\omega }_{n}t}}{{\omega }_{n}m\sqrt{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}\sqrt{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\sqrt{1-{\left(\frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\right)}^2}}\\ \sin {\omega }_n\sqrt{\frac{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}}\sqrt{1-{\left(\frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\right)}^2}tu(t).\end{array}$$

(25)

For j = 5,

$$h_5(t)=\frac{e^{-\frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{\omega }_{n}t}\sin \left[{\omega }_n\sqrt{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\sqrt{1-{\left(\frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2}t\right]u(t)}{m{\omega }_n\sqrt{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\sqrt{1-{\left(\frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2}}.$$

(26)

If j = 6,

$$h_6(t)=e^{-{\varsigma }_{\mathrm{eq}6}{\omega }_{\mathrm{eqn}6}t}\frac{1}{m_{\mathrm{eq}6}{\omega }_{\mathrm{eqd}6}}\sin {\omega }_{\mathrm{eqd}6}tu(t),$$

(27)

where

$$\begin{gathered} m_{\mathrm{eq}6}=m_{\mathrm{eq}3}=-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right), \\ {\varsigma }_{\mathrm{eq}6}=\frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}, \\ \begin{array}{l}{\omega }_{\mathrm{eqd}6}=\sqrt{\frac{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}\\ \sqrt{1-{\left(\frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2}.\end{array} \end{gathered}$$

When j = 7, we have

$$h_7(t)=e^{-{\varsigma }_{\mathrm{eq}7}{\omega }_{\mathrm{eqn}7}t}\frac{1}{m_{\mathrm{eq}7}{\omega }_{\mathrm{eqd}7}}\sin {\omega }_{\mathrm{eqd}7}t,t\ge 0,$$

(28)

where

$$\begin{gathered} m_{\mathrm{eq}7}=m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}, {\varsigma }_{\mathrm{eq}7}=\frac{c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{\left(m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}, \\ {\omega }_{\mathrm{eqn}7}=\sqrt{\frac{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}, \begin{array}{l}{\omega }_{\mathrm{eqd}7}=\sqrt{\frac{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}{m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}\\ \sqrt{1-{\left(\frac{c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{\left(m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2}.\end{array} \end{gathered}$$

Proof. 

Solving the inverse Fourier transforms of (15)–(21) yields (22)–(28), respectively. The proof is finished. □

3. PSD and Cross-PSD Responses to Seven Classes of Fractional Vibrations Driven by fGn

This section describes the contributions in three sections. First, we present the analytic expressions of the PSD and cross-PSD responses to seven classes of fractional vibration systems driven by fGn. Second, we show that the orders of fractional vibrators have considerable effects on the responses. Lastly, we show that the statistical dependences of the responses follow those of fGn.

3.1. Background

The literature regarding systems driven by fractional Gaussian noise (fGn) is quite rich; see, e.g., Wang et al. [88], Hu and Zhou [89], and Liu et al. [90]. However, closed-form expressions of PSD and cross-PSD responses to the excitation of fGn regarding seven classes of fractional vibration systems are seldom seen. This section demonstrates the closed-form expressions of PSD and cross-PSD responses to seven classes of fractional vibration systems driven by fGn.

The rest of the section is organized as follows. In Section 3.2, Section 3.3, Section 3.4, Section 3.5, Section 3.6, Section 3.7 and Section 3.8, we present the closed-form expressions of PSD and cross-PSD responses to fractional vibration systems from classes I to VII under the excitation of fGn. A summary is given in Section 3.9.

3.2. Responses of Class I Fractional Vibrators Driven by fGn

3.2.1. Computations

Let x₁(t) be the response of a class I fractional vibrator driven by f(t) of fGn. Let S_ff(ω) be the PSD of f(t). Then,

$$S_{ff}(\omega )=V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H},$$

(29)

where H ∈ (0, 1) is the Hurst parameter and

$$V_H=\Gamma (1-2H)\frac{\cos \pi H}{\pi H}$$

(30)

is the strength of fGn (Li and Lim [91]). Figure 1 indicates some plots of S_ff(ω).

Theorem 3 (PSD response I). 

Denote the PSD of x₁(t) with S_xx1(ω). Then,

$$S_{xx1}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k^2\left[{\left(1-\frac{{\omega }^{\alpha }}{{\omega }_n^2}\left|\cos \frac{\alpha \pi }{2}\right|\right)}^2+{\left(\frac{{\omega }^{\alpha }}{{\omega }_n^2}\sin \frac{\alpha \pi }{2}\right)}^2\right]}.$$

(31)

Proof. 

Note that S_xx1(ω) = S_ff(ω)|H₁(ω)|². Thus, Theorem 3 is the result. The proof is finished. □

Theorem 4 (cross-PSD response I). 

Let S_fx1(ω) be the cross-PSD between f(t) of fGn and x₁(t). Then,

$$S_{fx1}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k\left(1-\frac{{\omega }^{\alpha }}{{\omega }_n^2}\left|\cos \frac{\alpha \pi }{2}\right|+i\frac{{\omega }^{\alpha }}{{\omega }_n^2}\sin \frac{\alpha \pi }{2}\right)}.$$

(32)

Proof. 

Solving the operation of (33) below

S_fx1(ω) = S_ff(ω)H₁(ω)

(33)

yields (32). This finishes the proof. □

In the time domain, the autocorrelation function (ACF) response, denoted by r_xx1(τ), is given by

r_xx1(τ) = r_ff(τ) * h₁(τ) * h₁(−τ).

(34)

In (34), r_ff(τ) is the inverse Fourier transform of S_ff(ω). Denote the cross-correlation response between f(t) of fGn and x₁(t) with r_fx1(τ). Then,

r_fx1(τ) = r_ff(τ) * h₁(τ).

(35)

In (35), r_fx1(τ) is the inverse Fourier transform of S_fx1(ω).

3.2.2. Effects of α on Responses

Figure 2 indicates some plots of |H₁(ω)|². Some plots of S_xx1(ω) are shown in Figure 3.

When α = 2, a class I fractional vibrator reduces to be a conventional damping-free vibrator. Note that H has considerable effects on the responses; see Figure 4.

In Figure 5, we illustrate some plots of S_fx1(ω).

According to the generation method of fractional time series (Li [70], Li and Chi [92]), we illustrate some plots of driven fGn and response ones in Figure 6. From Figure 6, we can see that there are noticeable effects of the fractional order α on the responses of class I fractional vibration systems driven by fGn.

3.3. Responses of Class II Fractional Vibrators Driven by fGn

3.3.1. Computation Methods

Theorem 5 (PSD response II). 

Denote with x₂(t) the response to a class II fractional vibrator driven by f(t) of fGn. Let S_xx2(ω) be the PSD of x₂(t). Then,

$$S_{xx2}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k^2\left\{{\left[1-{\gamma }^2\left(1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2+{\left(\frac{2\varsigma {\omega }^{\beta }}{{\omega }_n}\sin \frac{\beta \pi }{2}\right)}^2\right\}}.$$

(36)

Proof. 

Let S_ff(ω) be the PSD of f(t) of fGn. Considering the operation of (37),

S_xx2(ω) = S_ff(ω)|H₂(ω)|²

(37)

produces (36). The proof is complete. □

Theorem 6 (cross-PSD response II). 

Denote with S_fx2(ω) the cross-PSD between f(t) of fGn and x₂(t). Then,

$$S_{fx2}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k\left[1-{\gamma }^2\left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\frac{2\varsigma {\omega }^{\beta }}{{\omega }_n}\sin \frac{\beta \pi }{2}\right]}.$$

(38)

Proof. 

Solving the operation in (39)

S_fx2(ω) = S_ff(ω)H₂(ω)

(39)

yields (38). The proof ends. □

Denote the ACF response with r_xx2(τ). It is the inverse Fourier transform of S_xx2(ω). It is expressed by

r_xx2(τ) = r_ff(τ) * h₂(τ) * h₂(−τ).

(40)

In (40), r_ff(τ) is the ACF of f(t) of fGn. The cross-correlation response between f(t) of fGn and x₂(t), denoted by r_fx2(τ), is

r_fx2(τ) = r_ff(τ) * h₂(τ).

(41)

In (41), r_fx2(τ)is the inverse Fourier transform of S_fx2(ω).

3.3.2. Effects of β on Responses

Some plots of |H₂(ω)|² are indicated in Figure 7. Figure 8 shows some plots of PSD response S_xx2(ω). Figure 9 shows a resonance curve when β = 1. Figure 10 illustrates some plots of the cross-PSD response of S_fx2(ω).

There are effects of the fractional order β on the responses of class II fractional vibration systems driven by fGn. Figure 11 exhibits the effects of β on the fluctuation range of the response x₂(t).

3.4. Responses of Class III Fractional Vibrators Driven by fGn

3.4.1. Computations

Theorem 7 (PSD response III). 

Let x₃(t) be the response to a class III fractional vibrator driven by f(t) of fGn. Let S_xx3(ω) be the PSD of x₃(t). Then,

$$S_{xx3}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k^2\left\{\begin{array}{l}{\left[1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\left[\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\right]}^2\end{array}\right\}}.$$

(42)

Proof. 

Considering the operation of (43),

S_xx3(ω) = S_ff(ω)|H₃(ω)|²

(43)

results in (42). The proof is finished. □

Theorem 8 (cross-PSD response III). 

Let S_fx3(ω) be the cross-PSD between the excitation of f(t) of fGn and x₃(t). Then,

$$S_{fx3}(\omega )=\frac{k^{-1}{V}_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{\begin{array}{l}1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\end{array}}.$$

(44)

Proof. 

Performing S_fx3(ω) = S_ff(ω)H₃(ω) produces (44). The proof is complete. □

Let r_xx3(τ) be the ACF of x₃(t). It is expressed by

r_xx3(τ) = r_ff(τ) * h₃(τ) * h₃(−τ).

(45)

In (45), r_xx3(τ) is the ACF response of a class III fractional vibrator driven by fGn. Denote with r_fx3(τ) the cross-correlation between the excitation f(t) of fGn and x₃(t). Then,

r_fx3(τ) = r_ff(τ) * h₃(τ).

(46)

In (46), r_fx3(τ) is the cross-correlation response of a class III fractional vibrator driven by fGn.

3.4.2. Effects of α and β on Responses

Some plots of |H₃(ω)|² are shown in Figure 12. Figure 13 indicates some plots of S_xx3(ω). Some plots of S_fx3(ω) are indicated in Figure 14. Figure 13 and Figure 14 show that there are significant effects of (α, β) on the responses of a class III fractional vibrator driven by fGn.

3.5. Responses of Class IV Fractional Vibrators Driven by fGn

3.5.1. Computations

Theorem 9 (PSD response IV). 

Let x₄(t) be the response to a class IV fractional vibrator excited by f(t) of fGn. Let S_xx4(ω) be the PSD of x₄(t). Then,

$$S_{xx4}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k^2{\omega }^{2\lambda }{\cos }^2\frac{\lambda \pi }{2}\left[\begin{array}{l}{\left(1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\right)}^2\\ +4{\left(\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2\end{array}\right]}.$$

(47)

Proof. 

Solving the operation of (48) below

S_xx4(ω) = S_ff(ω)|H₄(ω)|²

(48)

yields (47). The proof is finished. □

Theorem 10 (cross-PSD response IV). 

Denote the cross-PSD between f(t) of fGn and x₄(t) with S_fx4(ω). Then,

$$S_{fx4}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(\begin{array}{l}1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\\ +i2\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\end{array}\right)}.$$

(49)

Proof. 

Solving the operation of (50)

S_fx4(ω) = S_ff(ω)H₄(ω)

(50)

yields (49). This finishes the proof. □

Let r_xx4(τ) be the inverse Fourier transform of S_xx4(ω). According to the convolution theory, one has

r_xx4(τ) = r_ff(τ) * h₄(τ) * h₄(−τ).

(51)

In (51), r_xx4(τ) is the ACF response of a class IV fractional vibrator driven by fGn. Let r_fx4(τ) be the cross-correlation between the excitation f(t) and the response x₄(t). Then,

r_fx4(τ) = r_ff(τ) * h₄(τ).

(52)

In (52), r_fx4(τ) is the cross-correlation response of a class IV fractional vibrator driven by fGn.

3.5.2. Effects of α and λ on Responses

We illustrate some plots of |H₄(ω)|² in Figure 15. Figure 16 indicates some plots of PSD response S_xx4(ω). Some plots of cross-PSD response S_fx4(ω) are shown in Figure 17. Figure 16 and Figure 17 exhibit that there are noticeable effects of (α, λ) on the responses of a class IV fractional vibrator excited by fGn.

3.6. Responses of Class V Fractional Vibrators Driven by fGn

3.6.1. Computation Methods

Theorem 11 (PSD response V). 

Let x₅(t) be the response to a class V fractional vibrator driven by f(t) of fGn. Denote with S_xx5(ω) the PSD of x₅(t). Then,

$$S_{xx5}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k^2{\omega }^{2\lambda }{\cos }^2\frac{\lambda \pi }{2}\left[{\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\right)}^2+4{\gamma }^2{\left(\frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2\right]}.$$

(53)

Proof. 

Because of Equation (54)

S_xx5(ω) = S_ff(ω)|H₅(ω)|²,

(54)

(53) holds. The proof ends. □

Theorem 12 (cross-PSD response V). 

Let S_fx5(ω) be the cross-PSD between f(t) of fGn and x₅(t). Then,

$$S_{fx5}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}+i2\gamma \frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}.$$

(55)

Proof. 

Due to Equation (56)

S_fx5(ω) = S_ff(ω)H₅(ω),

(56)

Equation (55) is true. This finishes the proof. □

Let r_xx5(τ) be the ACF of x₅(t). Then,

r_xx5(τ) = r_ff(τ) * h₅(τ) * h₅(−τ).

(57)

In Equation (57), r_xx5(τ) is the ACF response of a class V fractional vibrator driven by fGn. Let r_fx5(τ) be the cross-correlation between f(t) of fGn and x₅(t). Then,

r_fx5(τ) = r_ff(τ) * h₅(τ).

(58)

In Equation (58), r_fx5(τ) is the cross-correlation response of a class V fractional vibrator driven by fGn.

3.6.2. Effects of λ on Responses

Figure 18 illustrates some plots of |H₅(ω)|². Figure 19 shows some plots of the PSD response S_xx5(ω). Figure 20 indicates some plots of the cross-PSD response S_fx5(ω). Figure 19 and Figure 20 show that the effects of the fractional order λ on the responses of a class V fractional vibration system under the excitation of fGn are considerable.

3.7. Responses of Class VI Fractional Vibrators Driven by fGn

3.7.1. Computations

Theorem 13 (PSD response VI). 

Let x₆(t) be the response to a class VI fractional vibrator driven by f(t) of fGn. Let S_xx6(ω) be the PSD of x₆(t). Then,

$$S_{xx6}(\omega )=\frac{1}{k^2}\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{\begin{array}{l}{\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\gamma }^2{\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)}^2\end{array}}.$$

(59)

Proof. 

Solving the operation of Equation (60)

S_xx6(ω) = S_ff(ω)|H₆(ω)|²

(60)

yields Equation (59). This finishes the proof. □

Theorem 14 (cross-PSD response VI). 

Denote with S_fx6(ω) the cross-PSD between f(t) of fGn and x₆(t). Then,

$$S_{fx6}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k\left[\begin{array}{l}{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\end{array}\right]}.$$

(61)

Proof. 

Solving the operation of Equation (62)

S_fx6(ω) = S_ff(ω)H₆(ω)

(62)

results in Equation (61). The proof ends. □

Applying the Wiener–Khinchin relation, the Wiener–Lee relation, and the convolution theory to S_xx6(ω) = S_ff(ω)|H₆(ω)|² and S_fx6(ω) = S_ff(ω)H₆(ω) produces

r_xx6(τ) = r_ff(τ) * h₆(τ) * h₆(−τ),

(63)

where r_xx6(τ) is the ACF of x₆(t), and

r_fx6(τ) = r_ff(τ) * h₆(τ),

(64)

where r_fx6(τ) is the cross-correlation between f(t) of fGn and x₆(t). In (63), r_xx6(τ) is the ACF response of a class VI fractional vibrator driven by fGn, while in (64), r_fx6(τ) is the cross-correlation response of a class VI fractional vibrator driven by fGn.

3.7.2. Effects of α, β, λ on Responses

Figure 21 demonstrates some plots of |H₆(ω)|². Figure 22 indicates some plots of S_xx6(ω). Figure 23 illustrates some plots of S_fx6(ω). Figure 22 and Figure 23 show that the effects of (α, β, λ) on the responses of a class VI fractional vibrator driven by fGn are significant.

3.8. Responses of Class VII Fractional Vibrators Driven by fGn

3.8.1. Computations

Theorem 15 (PSD response VII). 

Let x₇(t) be the response to a class VII fractional vibrator excited by f(t) of fGn. Denote with S_xx7(ω) the PSD of x₇(t). Then,

$$S_{xx7}(\omega )=\frac{1}{k^2}\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{\begin{array}{l}{\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\gamma }^2{\left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)}^2\end{array}}.$$

(65)

Proof. 

Considering the operation of Equation (66)

S_xx7(ω) = S_ff(ω)|H₇(ω)|²,

(66)

Equation (65) is true. Thus, Theorem 15 holds. The proof ends. □

Theorem 16 (cross-PSD response VII). 

Let S_fx7(ω) be the cross-PSD between f(t) of fGn and x₇(t). Then,

$$S_{fx7}(\omega )=\frac{V_H\sin (H\pi )\Gamma (2H+1){\left|\omega \right|}^{1-2H}}{k\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\gamma \left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\right]}.$$

(67)

Proof. 

Performing the operation of Equation (68)

S_fx7(ω) = S_ff(ω)H₇(ω)

(68)

produces Equation (67). The proof is finished. □

In the time domain, one has the ACF response in the form

r_xx7(τ) = r_ff(τ) * h₇(τ) * h₇(−τ).

(69)

In (69), r_xx7(τ) is the ACF of x₇(t). The cross-correlation response is given by

r_fx7(τ) = r_ff(τ) * h₇(τ).

(70)

In Equation (70), r_fx7(τ) is the cross-correlation between f(t) of fGn and x₇(t).

3.8.2. Effects of β, λ on Responses

Figure 24 indicates some plots of |H₇(ω)|², Figure 25 shows some plots of S_xx7(ω), and Figure 26 illustrates some plots of S_fx7(ω). Figure 25 and Figure 26 indicate that the effects of (β, λ) on the responses of class VII fractional vibrators excited by fGn are significant.

3.9. Summary

We have presented the analytic expressions of the PSD and cross-PSD responses of seven classes of fractional vibrators driven by fGn. We have shown that there are noticeable effects of fractional orders, say, α, β, or λ on responses. In addition, the statistical dependences (long-range dependence or short-range dependence) of the responses rely on the excitation fGn because S_xxj(0) = ∞ when 0.5 < H < 1 or S_xxj(0) < ∞ when 0 < H < 0.5 for j = 1, …, 7.

4. PSD and Cross-PSD Responses to Seven Classes of Fractional Vibrations Driven by fBm

This section studies the issue of a fractional Brownian motion (fBm) passing through seven classes of fractional vibration systems. The contributions given in this section are twofold. One brings forward the closed-form expressions of the PSD and cross-PSD responses to seven classes of fractional vibrators driven by fBm in Theorems 17 and 18, respectively. The other demonstrates that there are considerable effects of the fractional orders of seven classes of vibration systems on the responses. In addition, we show that the responses of seven classes of fractional vibrators driven by fBm are of long-range dependence.

4.1. Background

The topic of differential equations driven by fBm has received interest from researchers; see, e.g., Li and Yan [75], Gao et al. [76], Guo et al. [77], Gao and Sun [78], Pei and Zhang [79], Sun et al. [85], He et al. [93,94], Liu [95], Tuan et al. [96,97], Sharma et al. [98], Fan et al. [99], Zhang and Yuan [100], Sun et al. [9], Shahnazi-Pour et al. [101], Araya et al. [102], Gairing et al. [103], Xu et al. [104], and Heydari et al. [105,106], to mention a few. However, reports with respect to the closed-form analytic representations of the responses to seven classes of fractional vibration systems under the excitation of fBm are seldom seen. This section gives its contributions in closed-form analytic representations of the PSD and cross-PSD responses to seven classes of fractional vibration systems driven by fBm.

The rest of the section is organized as follows. In Section 4.2, Section 4.3, Section 4.4, Section 4.5, Section 4.6, Section 4.7 and Section 4.8, we put forward the analytic expressions of the PSD and cross-PSD responses to seven classes of fractional vibration systems driven by fBm. A summary is given in Section 4.9.

4.2. Responses of Class I Fractional Vibrators Driven by fBm

4.2.1. Computations

Let f(t) be a driven signal of fBm and S_ff(t, ω) be the PSD of f(t) in this section. Then (Flandrin [51], Li [72,74]),

$$S_{ff}(t,\omega )=\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right),$$

(71)

where

0 < H < 1.

(72)

Figure 27 shows some plots of S_ff(t, ω).

Theorem 17 (PSD response I). 

Denote the PSD of the response of a class I fractional vibrator x₁(t) with S_xx1(t, ω). Then,

$$S_{xx1}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k^2\left[{\left(1-\frac{{\omega }^{\alpha }}{{\omega }_n^2}\left|\cos \frac{\alpha \pi }{2}\right|\right)}^2+{\left(\frac{{\omega }^{\alpha }}{{\omega }_n^2}\sin \frac{\alpha \pi }{2}\right)}^2\right]}.$$

(73)

Proof. 

Considering the operation of Equation (74),

S_xx1(t, ω) = S_ff(t, ω)|H₁(ω)|²

(74)

results in Equation (73). The proof is finished. □

Theorem 18 (cross-PSD response I). 

Let S_fx1(t, ω) be the cross-PSD between f(t) of fBm and x₁(t). Then,

$$S_{fx1}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k\left(1-\frac{{\omega }^{\alpha }}{{\omega }_n^2}\left|\cos \frac{\alpha \pi }{2}\right|+i\frac{{\omega }^{\alpha }}{{\omega }_n^2}\sin \frac{\alpha \pi }{2}\right)}.$$

(75)

Proof. 

Solving S_fx1(t, ω) = S_ff(t, ω)H₁(ω) yields Equation (75). The proof ends. □

Let r_xx1(t, τ) be the inverse Fourier transform of S_xx1(t, ω). Denote with r_fx1(t, τ) the inverse Fourier transform of S_fx1(t, ω). Then, applying the convolution theory to S_xx1(t, ω) = S_ff(t, ω)|H₁(ω)|² results in the ACF response given by (72), in the form:

r_xx1(t, τ) = r_ff(t, τ) * h₁(τ) * h₁(−τ).

(76)

Similarly, applying the convolution theory to

S_fx1(t, ω) = S_ff(t, ω)H₁(ω)

(77)

yields Equation (78), which is the cross-correlation response in the form:

r_fx1(t, τ) = r_ff(t, τ) * h₁(τ).

(78)

4.2.2. Effects of α on Responses

Figure 28 indicates some plots of |H₁(ω)|². Some plots of the PSD response S_xx1(t, ω) are shown in Figure 29.

When α = 2, a class I fractional vibrator reduces to be a conventional damping-free vibrator. In general, there are three parameters affecting the response S_xx1(t, ω). From a system perspective, the parameter is the fractional order α. From the view of fractional processes, the parameter H is crucial to S_xx1(t, ω). Figure 30 shows the H effects on S_xx1(t, ω). In Figure 31, we illustrate some plots of the cross-PSD response S_fx1(t, ω). In Figure 32, we show the H effects on S_fx1(t, ω). Figure 29 and Figure 31 indicate that there are significant effects of the fractional order α on the responses to class I fractional vibration systems driven by fBm.

4.3. Responses of Class II Fractional Vibrators Driven by fBm

4.3.1. Computations

Theorem 19 (PSD response II). 

Let x₂(t) be the response to a class II fractional vibrator driven by f(t) of fBm. Let S_xx2(t, ω) be the PSD of x₂(t). Then,

$$S_{xx2}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k^2\left\{{\left[1-{\gamma }^2\left(1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2+{\left(\frac{2\varsigma {\omega }^{\beta }}{{\omega }_n}\sin \frac{\beta \pi }{2}\right)}^2\right\}}.$$

(79)

Proof. 

Solving the operation of Equation (80)

S_xx2(t, ω) = S_ff(t, ω)|H₂(ω)|²,

(80)

we have Equation (79). This finishes the proof. □

Theorem 20 (cross-PSD response II). 

Denote with S_fx2(t, ω) the cross-PSD between f(t) of fBm and x₂(t). Then,

$$S_{fx2}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k\left[1-{\gamma }^2\left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\frac{2\varsigma {\omega }^{\beta }}{{\omega }_n}\sin \frac{\beta \pi }{2}\right]}.$$

(81)

Proof. 

Because of Equation (82),

S_fx2(t, ω) = S_ff(t, ω)H₂(ω),

(82)

Equation (81) holds. The proof ends. □

Let r_xx2(t, τ) be the ACF of x₂(t). Denote with r_fx2(t, τ) the cross-correlation between the excitation f(t) of fBm and the response x₂(t). According to the convolution theory with respect to S_xx2(t, ω) = S_ff(t, ω)|H₂(ω)|², one has the ACF response, expressed by

r_xx2(t, τ) = r_ff(t, τ) * h₂(τ) * h₂(−τ).

(83)

In Equation (83), r_xx2(t, τ) is the inverse Fourier transform of S_xx2(t, ω). Similarly, taking into account the convolution theory with respect to S_fx2(t, ω) = S_ff(t, ω)H₂(ω) produces the cross-correlation response in the form

r_fx2(t, τ) = r_ff(t, τ) * h₂(τ).

(84)

In Equation (84), r_fx2(t, τ) is the inverse Fourier transform of S_fx2(t, ω).

4.3.2. Effects of β on Responses

Some plots of |H₂(ω)|² are indicated in Figure 33. Figure 34 shows some plots of the PSD response S_xx2(t, ω). Figure 35 shows the H effects on S_xx2(t, ω). Figure 36 illustrates some plots of the cross-PSD response S_fx2(t, ω). We use Figure 37 to exhibit the effects of β on |S_fx2(t, ω)|. Figure 35, Figure 36 and Figure 37 show that the effects of β and H on the responses to class II fractional vibration systems driven by fBm are considerable.

4.4. Responses of Class III Fractional Vibrators Driven by fBm

4.4.1. Computations

Theorem 21 (PSD response III). 

Let x₃(t) be the response to a class III fractional vibrator driven by f(t) of fBm. Let S_xx3(t, ω) be the PSD of x₃(t). Then,

$$S_{xx3}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k^2\left\{\begin{array}{l}{\left[1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\left[\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\right]}^2\end{array}\right\}}.$$

(85)

Proof. 

With Equation (86) below,

S_xx3(t, ω) = S_ff(t, ω)|H₃(ω)|²,

(86)

we have Equation (85). The proof is finished. □

Theorem 22 (cross-PSD response III). 

Let S_fx3(t, ω) be the cross-PSD between f(t) of fBm and x₃(t). Then,

$$S_{fx3}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{\begin{array}{l}1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\end{array}}.$$

(87)

Proof. 

Solving the operation of Equation (88)

S_fx3(t, ω) = S_ff(t, ω)H₃(ω)

(88)

yields Equation (87). The proof ends. □

Denote with r_xx3(t, τ) the ACF of x₃(t). Let r_fx3(t, τ) be the cross-correlation between f(t) of fBm and x₃(t). According to the Wiener–Khinchin relation and the Wiener–Lee relation, we have the ACF response, expressed by

r_xx3(t, τ) = r_ff(t, τ) * h₃(τ) * h₃(−τ),

(89)

and the cross-correlation response, given by

r_fx3(t, τ) = r_ff(t, τ) * h₃(τ).

(90)

In Equation (89), r_xx3(t, τ) is the inverse Fourier transform of S_xx3(t, ω) while r_fx3(t, τ) in Equation (90) is the inverse Fourier transform of S_fx3(t, ω).

4.4.2. Effects of α and β on Responses

Some plots of |H₃(ω)|² are shown in Figure 38. Figure 39 indicates some plots of the PSD response S_xx3(ω). Some plots of |S_fx3(ω)| are indicated in Figure 40. Figure 39 and Figure 40 exhibit that the effects of (α, β) and H on the responses to class III fractional vibrators driven by fBm are significant.

4.5. Responses of Class IV Fractional Vibrators Driven by fBm

4.5.1. Computations

Theorem 23 (PSD response IV). 

Let x₄(t) be the response to a class IV fractional vibrator driven by f(t) of fBm. Let S_xx4(t, ω) be the PSD of x₄(t). Then,

$$S_{xx4}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k^2{\omega }^{2\lambda }{\cos }^2\frac{\lambda \pi }{2}\left[\begin{array}{l}{\left(1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\right)}^2\\ +4{\left(\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2\end{array}\right]}.$$

(91)

Proof. 

Performing the operation in Equation (92)

S_xx4(t, ω) = S_ff(t, ω)|H₄(ω)|²

(92)

yields Equation (91). This finishes the proof. □

Theorem 24 (cross-PSD response IV). 

Denote with S_fx4(t, ω) the cross-PSD between f(t) of fBm and x₄(t). Then,

$$S_{fx4}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(\begin{array}{l}1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\\ +i2\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\end{array}\right)}.$$

(93)

Proof. 

Solving the operation in Equation (94)

S_fx4(t, ω) = S_ff(t, ω)H₄(ω)

(94)

produces Equation (93). This finishes the proof. □

Solving the inverse Fourier transform on both sides of S_xx4(t, ω) = S_ff(t, ω)|H₄(ω)|² yields

r_xx4(t, τ) = r_ff(t, τ) * h₄(τ) * h₄(−τ).

(95)

In Equation (95), r_xx4(t, τ) is the ACF response of a class IV fractional vibrator. Performing the inverse Fourier transform on the both sides of S_fx4(t, ω) = S_ff(t, ω)H₄(ω) results in

r_fx4(t, τ) = r_ff(t, τ) * h₄(τ).

(96)

In Equation (96), r_fx4(t, τ) is the cross-correlation response of a class IV fractional vibrator.

4.5.2. Effects of α and λ on Responses

Figure 41 illustrates some plots of |H₄(ω)|². Figure 42 shows some plots of the PSD response S_xx4(t, ω). Some plots of |S_fx4(t, ω)| are given in Figure 43. Figure 42 and Figure 43 show that there are noticeable effects of (α, λ) on the responses to class IV fractional vibrators driven by fBm.

4.6. Responses of Class V Fractional Vibrators Driven by fBm

4.6.1. Computation Methods

Theorem 25 (PSD response V). 

Let x₅(t) be the response to a class V fractional vibrator driven by f(t) of fBm. Denote the PSD of x₅(t) with S_xx5(t, ω). Then,

$$S_{xx5}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k^2{\omega }^{2\lambda }{\cos }^2\frac{\lambda \pi }{2}\left[{\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\right)}^2+4{\gamma }^2{\left(\frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2\right]}.$$

(97)

Proof. 

Solving the operation in Equation (98)

S_xx5(t, ω) = S_ff(t, ω)|H₅(ω)|²

(98)

yields Equation (97). The proof ends. □

Theorem 26 (cross-PSD response V). 

Let S_fx5(t, ω) be the cross-PSD between f(t) of fBm and x₅(t). Then,

$$S_{fx5}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}+i2\gamma \frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}.$$

(99)

Proof. 

When solving the operation in Equation (100),

S_fx5(t, ω) = S_ff(t, ω)H₅(ω),

(100)

we have Equation (99). This finishes the proof. □

Solving the inverse Fourier transform on the both sides of S_xx5(t, ω) = S_ff(t, ω)|H₅(ω)|² produces

r_xx5(t, τ) = r_ff(t, τ) * h₅(τ) * h₅(−τ).

(101)

In Equation (101), r_xx5(t, τ) is the ACF response of a class V fractional vibrator. Considering the convolution theory with respect to S_fx5(t, ω) = S_ff(t, ω)H₅(ω) yields the cross-correlation response, given by

r_fx5(t, τ) = r_ff(t, τ) * h₅(τ).

(102)

In Equation (102), r_fx5(t, τ) is the cross-correlation between f(t) of fBm and x₅(t).

4.6.2. Effects of λ on Responses

Figure 44 illustrates some plots of |H₅(ω)|², Figure 45 shows some plots of the PSD response S_xx5(t, ω), and Figure 46 shows some plots of |S_fx5(t, ω)|. Figure 45 and Figure 46 show that the effects of λ on the responses to class V fractional vibration systems driven by fBm are considerable.

4.7. Responses of Class VI Fractional Vibrators Driven by fBm

4.7.1. Computations

Theorem 27 (PSD response VI). 

Let x₆(t) be the response to a class VI fractional vibrator excited by f(t) of fBm. Let S_xx6(t, ω) be the PSD of x₆(t). Then,

$$S_{xx6}(t,\omega )=\frac{1}{k^2}\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{\begin{array}{l}{\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\gamma }^2{\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)}^2\end{array}}.$$

(103)

Proof. 

Solving the operation in Equation (104)

S_xx6(t, ω) = S_ff(t, ω)|H₆(ω)|²

(104)

yields Equation (103). The proof ends. □

Theorem 28 (cross-PSD response VI). 

Denote with S_fx6(t, ω) the cross-PSD between f(t) of fBm and x₆(t). Then,

$$S_{fx6}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k\left[\begin{array}{l}{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\end{array}\right]}.$$

(105)

Proof. 

Performing the operation in Equation (106)

S_fx6(t, ω) = S_ff(t, ω)H₆(ω)

(106)

results in Equation (105). The proof is finished. □

Solving the inverse Fourier transform on the both sides of S_xx6(t, ω) = S_ff(t, ω)|H₆(ω)|² produces

r_xx6(t, τ) = r_ff(t, τ) * h₆(τ) * h₆(−τ).

(107)

In Equation (107), r_xx6(t, τ) is the ACF of x₆(t). It is the ACF response of a class VI fractional vibrator driven by fBm. The cross-correlation response is given by

r_fx6(t, τ) = r_ff(t, τ) * h₆(τ).

(108)

In Equation (108), r_fx6(t, τ) is the cross-correlation between f(t) of fBm and x₆(t).

4.7.2. Effects of α, β, λ on Responses

Figure 47 illustrates some plots of |H₆(ω)|². Figure 48 shows some plots of the PSD response S_xx6(t, ω). Figure 49 demonstrates some plots of |S_fx6(t, ω)|. Figure 48 and Figure 49 imply that the effects of (α, β, λ) on the responses to class VI fractional vibrators under the excitation of fBm are significant.

4.8. Responses of Class VII Fractional Vibrators Driven by fBm

4.8.1. Computations

Theorem 29 (PSD response VII). 

Let x₇(t) be the response to a class VII fractional vibrator driven by f(t) of fBm. Denote with S_xx7(t, ω) the PSD of x₇(t). Then,

$$S_{xx7}(t,\omega )=\frac{1}{k^2}\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{\begin{array}{l}{\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\gamma }^2{\left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)}^2\end{array}}.$$

(109)

Proof. 

Solving the operation in Equation (110)

S_xx7(t, ω) = S_ff(t, ω)|H₇(ω)|²

(110)

results in Equation (109). The proof is complete. □

Theorem 30 (cross-PSD response VII). 

Let S_fx7(t, ω) be the cross-PSD between f(t) of fBm and x₇(t). Then,

$$S_{fx7}(t,\omega )=\frac{\frac{V_H}{{\left|\omega \right|}^{2H+1}}\left(1-2^{1-2H}\cos 2\omega t\right)}{k\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\gamma \left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\right]}.$$

(111)

Proof. 

Solving the operation in (112)

S_fx7(ω) = S_ff(ω)H₇(ω)

(112)

yields Equation (111). The proof ends. □

Solving the inverse Fourier transform on both sides of S_xx7(t, ω) = S_ff(t, ω)|H₇(ω)|² produces

r_xx7(t, τ) = r_ff(t, τ) * h₇(τ) * h₇(−τ).

(113)

In Equation (113), r_xx7(t, τ) is the ACF of x₇(t). This is the ACF response to a class VII fractional vibrator. Similarly, performing the inverse Fourier transform on both sides of S_fx7(ω) = S_ff(ω)H₇(ω) yields

r_fx7(t, τ) = r_ff(t, τ) * h₇(τ).

(114)

In Equation (114), r_fx7(t, τ) is the cross-correlation between f(t) of fBm and x₇(t). This is the cross-correlation response of a class VII fractional vibrator.

4.8.2. Effects of β, λ on Responses

Figure 50 indicates some plots of |H₇(ω)|², Figure 51 shows some plots of the PSD response S_xx7(t, ω), and Figure 52 demonstrates some plots of |S_fx7(t, ω)|. The above figures imply that the effects of (β, λ) on the responses to class VII fractional vibration systems driven by fBm are significant.

4.9. Summary

The analytic expressions of PSD and cross-PSD responses to seven classes of fractional vibrators driven by fBm are given in Theorems 17–30, respectively. The noticeable effects of fractional orders of vibration systems on the responses are illustrated. The responses are of long-range dependence for 0 < H < 0.5 and 0.5 < H < 1 because S_xxj(t, 0) = ∞ when 0 < H < 0.5 and 0.5 < H < 1 for j = 1, …, 7.

5. PSD and Cross-PSD Responses to Seven Classes of Fractional Vibrations Driven by Fractional OU Processes

This section describes the analytic expressions of the PSD and cross-PSD responses to seven classes of fractional vibrators driven by fractional OU processes. The results exhibit that the fractional orders of vibration systems as well as the order of fractional OU processes have effects on the responses. The responses of seven classes of fractional vibrators driven by fractional OU processes are of short-range dependence.

5.1. Background

The Langevin equation and the OU process, which is the solution to the Langevin equation under the excitation of white noise, remain research interests in various fields (Eliazar and Shlesinger [58], Pavliotis [107], Coffey et al. [108]). Naturally, fractional Langevin equations attract researchers (Li et al. [32], West [109], Lim et al. [110,111]). Accordingly, fractional OU processes have paid attention to, e.g., Coffey et al. [108], Shao [112], Cheridito et al. [113], Magdziarz [114], Gehringer and Li [115], and Patel and Sharma [116], to mention a few.

In the field, analytic expressions of the responses to seven classes of fractional vibrators driven by fractional OU processes are rarely reported. In this section, we contribute the closed-form analytic expressions of the PSD and cross-PSD responses to seven classes of fractional vibrators under the excitation of fractional OU processes. The present results exhibit that there are considerable effects of the fractional order, α, β, and λ, on responses.

The rest of the section is organized as follows. In Section 5.2, Section 5.3, Section 5.4, Section 5.5, Section 5.6, Section 5.7 and Section 5.8, we address the analytic expressions of the PSD and cross-PSD responses to the fractional vibration systems from classes I to VII that are driven by the fractional OU processes. A summary is given in Section 5.9.

5.2. Responses of Class I Fractional Vibrators Driven by Fractional OU Processes

5.2.1. Computations

Let f(t) be a fractional OU process of order b and S_ff(ω) be the PSD of f(t) in this section. Then (Li et al. [32], Lim et al. [110,111]),

$$S_{ff}(\omega )=\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b},$$

(115)

where b > 0, λ > 0. Without losing the generality, in what follows, we set λ = 1. Figure 53 illustrates some plots of S_ff(ω).

Theorem 31 (PSD response I). 

Let x₁(t) be the response to a class I fractional vibrator driven by f(t) of fractional OU processes. Denote with S_xx1(ω) the PSD of x₁(t). Then,

$$S_{xx1}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k^2\left[{\left(1-\frac{{\omega }^{\alpha }}{{\omega }_n^2}\left|\cos \frac{\alpha \pi }{2}\right|\right)}^2+{\left(\frac{{\omega }^{\alpha }}{{\omega }_n^2}\sin \frac{\alpha \pi }{2}\right)}^2\right]}.$$

(116)

Proof. 

Considering the operation in Equation (117)

S_xx1(ω) = S_ff(ω)|H₁(ω)|²,

(117)

we have Equation (116). The proof is finished. □

Theorem 32 (cross-PSD response I). 

Let S_fx1(ω) be the cross-PSD between f(t) of fractional OU processes and x₁(t). Then,

$$S_{fx1}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k\left(1-\frac{{\omega }^{\alpha }}{{\omega }_n^2}\left|\cos \frac{\alpha \pi }{2}\right|+i\frac{{\omega }^{\alpha }}{{\omega }_n^2}\sin \frac{\alpha \pi }{2}\right)}.$$

(118)

Proof. 

Taking into account the operation in Equation (119),

S_fx1(ω) = S_ff(ω)H₁(ω),

(119)

we have Equation (118). The proof ends. □

Let r_xx1(τ) be the ACF of x₁(t). Let r_ff(τ) be the ACF of f(t) of fractional OU processes. Let r_fx1(τ) be the cross-correlation between the excitation f(t) and the response x₁(t). Then, the ACF response is given by

r_xx1(τ) = r_ff(τ) * h₁(τ) * h₁(−τ).

(120)

In Equation (120), r_xx1(τ) is the inverse Fourier transform of S_xx1(ω). The cross-correlation response is

r_fx1(τ) = r_ff(τ) * h₁(τ).

(121)

In Equation (121), r_fx1(τ) is the inverse Fourier transform of S_fx1(ω).

5.2.2. Effects of α on Responses

Figure 54 indicates some plots of |H₁(ω)|². Some plots of the PSD response S_xx1(ω) are shown in Figure 55.

When α = 2, a class I fractional vibrator reduces to be a conventional damping-free vibrator. Figure 56 shows the effect of b on S_xx1(ω).

Figure 57 illustrates some plots of the cross-PSD response S_fx1(ω). Figure 56 and Figure 57 exhibit that the order α of class I fractional vibration systems and the order b of the fractional OU processes have noticeable effects on the responses to class I fractional vibrators. We illustrate some plots of driven fractional OU processes and response ones in Figure 58.

5.3. Responses of Class II Fractional Vibration Systems Driven by Fractional OU Processes

5.3.1. Computation Methods

Theorem 33 (PSD response II). 

Let x₂(t) be the response to a class II fractional vibrator driven by fractional OU processes. Let S_xx2(ω) be the PSD of x₂(t). Then,

$$S_{xx2}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k^2\left\{{\left[1-{\gamma }^2\left(1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2+{\left(\frac{2\varsigma {\omega }^{\beta }}{{\omega }_n}\sin \frac{\beta \pi }{2}\right)}^2\right\}}.$$

(122)

Proof. 

Solving the operation Equation (123) below

S_xx2(ω) = S_ff(ω)|H₂(ω)|²

(123)

yields Equation (122). This finishes the proof. □

Theorem 34 (cross-PSD response II). 

Denote with S_fx2(ω) the cross-PSD between f(t) of fractional OU processes and x₂(t). Then,

$$S_{fx2}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k\left[1-{\gamma }^2\left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\frac{2\varsigma {\omega }^{\beta }}{{\omega }_n}\sin \frac{\beta \pi }{2}\right]}.$$

(124)

Proof. 

Because of Equation (125) below,

S_fx2(ω) = S_ff(ω)H₂(ω),

(125)

(124) holds. The proof ends. □

Let r_xx2(τ) be the ACF of x₂(t). Let r_fx2(τ) be the cross-correlation between the excitation f(t) of fractional OU processes and the response x₂(t). According to the convolution theory, the ACF response is expressed by

r_xx2(τ) = r_ff(τ) * h₂(τ) * h₂(−τ).

(126)

In Equation (126), r_xx2(τ) is the inverse Fourier transform of S_xx2(ω). The cross-correlation response is given by

r_fx2(τ) = r_ff(τ) * h₂(τ).

(127)

In Equation (127), r_fx2(τ) is the inverse Fourier transform of S_fx2(ω).

5.3.2. Effects of β on Responses

Some plots of |H₂(ω)|² are indicated in Figure 59. Figure 60 shows some plots of the PSD response S_xx2(ω). Figure 61 shows that the order b of the fractional OU processes does not obviously affect S_xx2(ω). Figure 62 illustrates some plots of |S_fx2(ω)|. Figure 60 and Figure 62 show that there are effects of β on the responses to class II fractional vibrators driven by the fractional OU processes. Figure 63 exhibits the effects of β on the fluctuation range of x₂(t).

5.4. Responses of Class III Fractional Vibrators Driven by Fractional OU Processes

5.4.1. Computations

Theorem 35 (PSD response III). 

Let x₃(t) be the response to a class III fractional vibrator excited by f(t) of fractional OU processes. Let S_xx3(ω) be the PSD of x₃(t). Then,

$$S_{xx3}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k^2\left\{\begin{array}{l}{\left[1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\left[\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\right]}^2\end{array}\right\}}.$$

(128)

Proof. 

With Equation (129) below,

S_xx3(ω) = S_ff(ω)|H₃(ω)|²,

(129)

we have Equation (128). The proof is finished. □

Theorem 36 (cross-PSD response III). 

Let S_fx3(ω) be the cross-PSD between f(t) of fractional OU processes and x₃(t). Then,

$$S_{fx3}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{\begin{array}{l}1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\end{array}}.$$

(130)

Proof. 

Using Equation (131) below

S_fx3(ω) = S_ff(ω)H₃(ω)

(131)

yields Equation (130). The proof ends. □

Let r_xx3(τ) be the ACF of x₃(t). Let r_fx3(τ) be the cross-correlation between the excitation f(t) of fractional OU processes and the response x₃(t). Applying the convolution theory to S_xx3(ω) = S_ff(ω)|H₃(ω)|² produces the ACF response, expressed by

r_xx3(τ) = r_ff(τ) * h₃(τ) * h₃(−τ).

(132)

Similarly, applying the convolution theory to S_fx3(ω) = S_ff(ω)H₃(ω) results in the cross-correlation response

r_fx3(τ) = r_ff(τ) * h₃(τ).

(133)

5.4.2. Effects of α and β on Responses

Some plots of |H₃(ω)|² are shown in Figure 64. Figure 65 indicates some plots of the PSD response S_xx3(ω). Some plots of |S_fx3(ω)| are given in Figure 66. Figure 65 and Figure 66 exhibit that the effects of (α, β) on the responses to class III fractional vibration systems driven by the fractional OU processes are significant.

5.5. Responses of Class IV Fractional Vibration Systems Driven by Fractional OU Processes

5.5.1. Computations

Theorem 37 (PSD response IV). 

Let x₄(t) be the response to a class IV fractional vibrator excited by f(t) of fractional OU processes. Let S_xx4(ω) be the PSD of x₄(t). Then,

$$S_{xx4}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k^2{\omega }^{2\lambda }{\cos }^2\frac{\lambda \pi }{2}\left[\begin{array}{l}{\left(1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\right)}^2\\ +4{\left(\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2\end{array}\right]}.$$

(134)

Proof. 

Performing the operation Equation (135) below

S_xx4(ω) = S_ff(ω)|H₄(ω)|²

(135)

yields Equation (134). This finishes the proof. □

Theorem 38 (cross-PSD response IV). 

Denote with S_fx4(ω) the cross-PSD between f(t) of fractional OU processes and x₄(t). Then,

$$S_{fx4}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(\begin{array}{l}1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\\ +i2\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\end{array}\right)}.$$

(136)

Proof. 

Solving the operation Equation (137) below

S_fx4(ω) = S_ff(ω)H₄(ω)

(137)

produces Equation (136). This finishes the proof. □

Solving the inverse Fourier transform on both sides of S_xx4(ω) = S_ff(ω)|H₄(ω)|² yields

r_xx4(τ) = r_ff(τ) * h₄(τ) * h₄(−τ).

(138)

In Equation (138), r_xx4(τ) is the ACF of x₄(t). It is the ACF response of a class IV fractional vibrator driven by the fractional OU process. In addition,

r_fx4(τ) = r_ff(τ) * h₄(τ).

(139)

In Equation (139), r_fx4(τ) is the cross-correlation between f(t) of the fractional OU process and x₄(t). It is the cross-correlation response of a class IV fractional vibrator driven by the fractional OU process.

5.5.2. Effects of α and λ on Responses

Figure 67 illustrates some plots of |H₄(ω)|². Figure 68 indicates some plots of the PSD response S_xx4(ω). Some plots of the cross-PSD response S_fx4(ω) are shown in Figure 69. They exhibit that the effects of (α, λ) and b on the responses to class IV fractional vibration systems driven by the fractional OU processes are noticeable.

5.6. Responses of Class V Fractional Vibrators Driven by Fractional OU Processes

5.6.1. Computation Methods

Theorem 39 (PSD response V). 

Let x₅(t) be the response to a class V fractional vibrator driven by f(t) of the fractional OU processes. Denote with S_xx5(ω) the PSD of x₅(t). Then,

$$S_{xx5}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k^2{\omega }^{2\lambda }{\cos }^2\frac{\lambda \pi }{2}\left[{\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\right)}^2+4{\gamma }^2{\left(\frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2\right]}.$$

(140)

Proof. 

Solving the operation Equation (141) below

S_xx5(ω) = S_ff(ω)|H₅(ω)|²

(141)

yields Equation (140). The proof is finished. □

Theorem 40 (cross-PSD response V). 

Let S_fx5(ω) be the cross-PSD between f(t) of fractional OU processes and x₅(t). Then,

$$S_{fx5}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}+i2\gamma \frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}.$$

(142)

Proof. 

When solving the operation Equation (143) below,

S_fx5(ω) = S_ff(ω)H₅(ω),

(143)

we have Equation (142). This finishes the proof. □

Performing the inverse Fourier transform on both sides of S_xx5(ω) = S_ff(ω)|H₅(ω)|² produces

r_xx5(τ) = r_ff(τ) * h₅(τ) * h₅(−τ).

(144)

In Equation (144), r_xx5(τ) is the ACF of x₅(t). It is the ACF response of a class V fractional vibrator driven by the fractional OU process. In addition,

r_fx5(τ) = r_ff(τ) * h₅(τ).

(145)

In Equation (145), r_fx5(τ) is the cross-correlation between f(t) of the fractional OU process and x₅(t). It is the cross-correlation response of a class V fractional vibrator driven by the fractional OU process.

5.6.2. Effects of λ on Responses

Figure 70 illustrates some plots of |H₅(ω)|². Figure 71 indicates some plots of S_xx5(ω). Figure 72 shows some plots of |S_fx5(ω)|. From these figures, we can see that the effects of λ and b on the responses to class V fractional vibrators driven by the fractional OU process are considerable.

5.7. Responses of Class VI Fractional Vibrators Driven by Fractional OU Processes

5.7.1. Computations

Theorem 41 (PSD response VI). 

Let x₆(t) be the response to a class VI fractional vibrator excited by f(t) of fractional OU processes. Let S_xx6(ω) be the PSD of x₆(t). Then,

$$S_{xx6}(\omega )=\frac{1}{k^2}\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{\begin{array}{l}{\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\gamma }^2{\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)}^2\end{array}}.$$

(146)

Proof. 

Considering Equation (147) below

S_xx6(ω) = S_ff(ω)|H₆(ω)|²

(147)

yields Equation (146). The proof ends. □

Theorem 42 (cross-PSD response VI). 

Denote with S_fx6(ω) the cross-PSD between f(t) of the fractional OU process and x₆(t). Then,

$$S_{fx6}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k\left[\begin{array}{l}{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\end{array}\right]}.$$

(148)

Proof. 

Performing the operation Equation (149) below

S_fx6(ω) = S_ff(ω)H₆(ω)

(149)

results in Equation (148). The proof is finished. □

Solving the inverse Fourier transform on both sides of S_xx6(ω) = S_ff(ω)|H₆(ω)|² produces

r_xx6(τ) = r_ff(τ) * h₆(τ) * h₆(−τ).

(150)

In Equation (150), r_xx6(τ) is the ACF of x₆(t). It is the ACF response of a class VI fractional vibrator driven by the fractional OU process. Furthermore,

r_fx6(τ) = r_ff(τ) * h₆(τ).

(151)

In Equation (151), r_fx6(τ) is the cross-correlation between f(t) of the fractional OU process and x₆(t). It is the cross-correlation response of a class VI fractional vibrator driven by the fractional OU process.

5.7.2. Effects of α, β, λ on Responses

Figure 73 illustrates some plots of |H₆(ω)|². Figure 74 shows some plots of r S_xx6(ω). Figure 75 demonstrates some plots of |S_fx6(ω)|. Figure 73 and Figure 74 show that the effects of (α, β, λ) on the responses to class VI fractional vibration systems driven by the fractional OU process are significant.

5.8. Responses of Class VII Fractional Vibrators Driven by Fractional OU Processes

5.8.1. Computations

Theorem 43 (PSD response VII). 

Let x₇(t) be the response to a class VII fractional vibrator driven by f(t) of the fractional OU process. Denote with S_xx7(ω) the PSD of x₇(t). Then,

$$S_{xx7}(\omega )=\frac{1}{k^2}\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{\begin{array}{l}{\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\gamma }^2{\left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)}^2\end{array}}.$$

(152)

Proof. 

Solving the operation in Equation (153)

S_xx7(ω) = S_ff(ω)|H₇(ω)|²

(153)

results in Equation (152). The proof is complete. □

Theorem 44 (cross-PSD response VII). 

Let S_fx7(ω) be the cross-PSD between f(t) of the fractional OU process and x₇(t). Then,

$$S_{fx7}(\omega )=\frac{\frac{1}{{\left({\lambda }^2+{\omega }^2\right)}^b}}{k\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\gamma \left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\right]}.$$

(154)

Proof. 

Solving the operation in Equation (155)

S_fx7(ω) = S_ff(ω)H₇(ω)

(155)

yields Equation (154). The proof ends. □

Solving the inverse Fourier transform on both sides of S_xx7(ω) = S_ff(ω)|H₇(ω)|² produces

r_xx7(τ) = r_ff(τ) * h₇(τ) * h₇(−τ).

(156)

In Equation (156), r_xx7(τ) is the ACF of x₇(t). It is the ACF response of a class VII fractional vibrator driven by the fractional OU process. Additionally,

r_fx7(τ) = r_ff(τ) * h₇(τ).

(157)

In Equation (157), r_fx7(τ) is the cross-correlation between f(t) of the fractional OU process and x₇(t). It is the cross-correlation response of a class VII fractional vibrator driven by the fractional OU process.

5.8.2. Effects of β, λ on Responses

Figure 76 shows some plots of |H₇(ω)|². Figure 77 indicates some plots of S_xx7(ω). Figure 78 illustrates some plots of |S_fx7(ω)|. Figure 77 and Figure 78 exhibit that the effects of (β, λ) on the responses to class VII fractional vibration systems driven by the fractional class VII vibrators are considerable.

5.9. Summary

We present the analytic expressions of the PSD and cross-PSD responses to seven classes of fractional vibrators driven by the fractional OU process in Theorems 31–44, respectively. We illustrate that there are considerable effects of orders of fractional vibrators on the responses. The responses of seven classes of fractional vibrators under the excitation of the fractional OU processes are short-range-dependent due to S_xxj(0) < ∞ for j = 1, …, 7.

6. PSD and Cross-PSD Responses to Seven Classes of Fractional Vibrations Driven by the von Kármán Process

This section describes the analytic expressions of PSD responses and cross-PSD ones to seven classes of fractional vibrators driven by the von Kármán process. It shows that there are considerable effects of orders of fractional vibrators on responses driven by the von Kármán process. The responses of seven classes of fractional vibrators driven by the von Kármán process are of short-range dependence.

6.1. Background

The processes that follow the von Kármán spectrum are widely used in wind, ship, and ocean engineering (Faltinsen [117], Holmes [118]). Recently, Li [36] reported the point of view that processes on the von Kármán spectrum can be taken as a type of fractional OU process with the fractal dimension 5/3. Nonetheless, how to analytically represent the PSD and cross-PSD responses to seven classes of fractional vibrators driven by the von Kármán process remains an unsolved problem. This section gives the closed-form analytic expressions of the PSD and cross-PSD responses to seven classes of fractional vibrators driven by the von Kármán process.

The rest of the section is organized as follows. In Section 6.2, Section 6.3, Section 6.4, Section 6.5, Section 6.6, Section 6.7 and Section 6.8, we put forward the analytic expressions of the PSD and cross-PSD responses to seven classes of fractional vibrators excited by the von Kármán process. A summary is given in Section 6.9.

6.2. Responses of Class I Fractional Vibrators Driven by von Kármán Process

6.2.1. Computations

Let f(t) be a process with the von Kármán spectrum and S_ff(ω) be the PSD of the von Kármán type. Then,

$$S_{ff}(\omega )=\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}},$$

(158)

where

$$A_{\mathrm{vk}}=\frac{4{\sigma }_u^2}{{70.8}^{5/6}{\left(\frac{L_u^x}{U}\right)}^{2/3}}$$

(159)

and

$$B_{\mathrm{vk}}=\frac{U}{{70.8}^{1/2}{L}_u^x},$$

(160)

where $L_u^x$ is the turbulence integral scale, U is the mean speed, u_f is the friction velocity (ms⁻¹), b_v is the friction velocity coefficient such that the variance of wind speed is ${\sigma }_u^2=b_v{u}_f^2.$ Figure 79 illustrates the plots of S_ff(ω). In this section, $L_u^x$ = 1 is used simply for facilitating the illustrations. This does not cause the generality to describe the issue of the von Kármán process passing through seven classes of fractional vibration systems to be lost.

Theorem 45 (PSD response I). 

Let x₁(t) be the response to a class I fractional vibrator excited by f(t) of the von Kármán process. Denote with S_xx1(ω) the PSD of x₁(t). Then,

$$S_{xx1}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k^2\left[{\left(1-\frac{{\omega }^{\alpha }}{{\omega }_n^2}\left|\cos \frac{\alpha \pi }{2}\right|\right)}^2+{\left(\frac{{\omega }^{\alpha }}{{\omega }_n^2}\sin \frac{\alpha \pi }{2}\right)}^2\right]}.$$

(161)

Proof. 

Solving the operation in Equation (162)

S_xx1(ω) = S_ff(ω)|H₁(ω)|²

(162)

results in Equation (161). The proof is finished. □

Theorem 46 (cross-PSD response I). 

Let S_fx1(ω) be the cross-PSD between f(t) of the von Kármán process and x₁(t). Then,

$$S_{fx1}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k\left(1-\frac{{\omega }^{\alpha }}{{\omega }_n^2}\left|\cos \frac{\alpha \pi }{2}\right|+i\frac{{\omega }^{\alpha }}{{\omega }_n^2}\sin \frac{\alpha \pi }{2}\right)}.$$

(163)

Proof. 

Solving the operation in Equation (164)

S_fx1(ω) = S_ff(ω)H₁(ω)

(164)

produces Equation (163). The proof ends. □

Denote with r_xx1(τ) the ACF of x₁(t). Let r_ff(τ) be the ACF of f(t). Let h₁(τ) be the impulse response of a class I fractional vibrator. Applying the convolution theory to S_xx1(ω) = S_ff(ω)|H₁(ω)|² yields the ACF response

r_xx1(τ) = r_ff(τ) * h₁(τ) * h₁(−τ).

(165)

Similarly, applying the convolution theory to S_fx1(ω) = S_ff(ω)H₁(ω) produces the cross-correlation response

r_fx1(τ) = r_ff(τ) * h₁(τ).

(166)

6.2.2. Effects of α on Responses

Figure 80 indicates some plots of |H₁(ω)|². Some plots of the PSD response S_xx1(ω) are shown in Figure 81.

When α = 2, a class I fractional vibrator reduces to be a conventional damping-free vibrator. In general, there are three parameters affecting the response S_xx1(ω). From a system perspective, the parameter is the fractional order α. From an engineering perspective, the parameter U is crucial to S_xx1(ω). Figure 82 shows the effect of U on S_xx1(ω). In Figure 83, we illustrate the plots of |S_fx1(ω)|. The plots of the driven signal with the von Kármán spectrum and the response ones are indicated in Figure 84. Figure 82, Figure 83 and Figure 84 show that there are noticeable effects of α on the responses to class I fractional vibration systems driven by the von Kármán process.

6.3. Responses of Class II Fractional Vibration Systems Driven by von Kármán Process

6.3.1. Computation Methods

Theorem 47 (PSD response II). 

Let x₂(t) be the response to a class II fractional vibrator driven by f(t) of the von Kármán process. Let S_xx2(ω) be the PSD of the response x₂(t). Then,

$$S_{xx2}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k^2\left\{{\left[1-{\gamma }^2\left(1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2+{\left(\frac{2\varsigma {\omega }^{\beta }}{{\omega }_n}\sin \frac{\beta \pi }{2}\right)}^2\right\}}.$$

(167)

Proof. 

Using the operation in Equation (168)

S_xx2(ω) = S_ff(ω)|H₂(ω)|²

(168)

yields Equation (167). This finishes the proof. □

Theorem 48 (cross-PSD response II). 

Denote with S_fx2(ω) the cross-PSD between f(t) and x₂(t). Then,

$$S_{fx2}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k\left[1-{\gamma }^2\left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\frac{2\varsigma {\omega }^{\beta }}{{\omega }_n}\sin \frac{\beta \pi }{2}\right]}.$$

(169)

Proof. 

Using the operation in Equation (170)

S_fx2(ω) = S_ff(ω)H₂(ω)

(170)

produces Equation (169). The proof ends. □

Let r_xx2(τ) be the ACF of x₂(t). Denote with r_fx2(τ) the cross-correlation between f(t) of the von Kármán process and x₂(t). Applying the Wiener–Khinchin relation to S_xx2(ω) = S_ff(ω)|H₂(ω)|² results in the ACF response

r_xx2(τ) = r_ff(τ) * h₂(τ) * h₂(−τ).

(171)

Note that r_xx2(τ) in Equation (171) is the ACF response of a class II fractional vibrato driven by the von Kármán process. Similarly, applying the Wiener–Lee relation to S_fx2(ω) = S_ff(ω)H₂(ω) yields the cross-correlation

r_fx2(τ) = r_ff(τ) * h₂(τ).

(172)

In Equation (172), r_fx2(τ) is the cross-correlation response of a class II fractional vibrato driven by the von Kármán process.

6.3.2. Effects of β on Responses

Some plots of |H₂(ω)|² are indicated in Figure 85. Figure 86 shows some plots of the PSD response S_xx2(ω). Figure 87 shows the U effects on S_xx2(ω). Figure 88 illustrates some plots of |S_fx2(ω)|. Figure 87 and Figure 88 exhibit that there are effects of β on the responses to class II fractional vibration systems driven by the von Kármán process. We also use Figure 89 to exhibit the effects of β on the fluctuation range of x₂(t).

6.4. Responses of Class III Fractional Vibrators Driven by von Kármán Process

6.4.1. Computations

Theorem 49 (PSD response III). 

Let x₃(t) be the response to a class III fractional vibrator driven by the von Kármán process. Let S_xx3(ω) be the PSD of x₃(t). Then,

$$S_{xx3}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k^2\left\{\begin{array}{l}{\left[1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\left[\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\right]}^2\end{array}\right\}}.$$

(173)

Proof. 

Considering the operation in Equation (174),

S_xx3(ω) = S_ff(ω)|H₃(ω)|²,

(174)

we have Equation (173). The proof ends. □

Theorem 50 (cross-PSD response III). 

Let S_fx3(ω) be the cross-PSD between f(t) of the von Kármán process and x₃(t). Then,

$$S_{fx3}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{\begin{array}{l}1-{\gamma }^2\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\frac{\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{{\omega }_n\left({\omega }^{\alpha -2}\left|\cos \frac{\alpha \pi }{2}\right|-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}\end{array}}.$$

(175)

Proof. 

Solving the operation in Equation (176)

S_fx3(ω) = S_ff(ω)H₃(ω)

(176)

results in Equation (175). The proof ends. □

Let r_xx3(τ) be the ACF of x₃(t). Denote with r_fx3(τ) the cross-correlation between f(t) and x₃(t). According to the Wiener–Khinchin relation and the Wiener–Lee relation, we have

r_xx3(τ) = r_ff(τ) * h₃(τ) * h₃(−τ).

(177)

In Equation (177), r_xx3(τ) is the ACF response of a class III fractional vibrator driven by the von Kármán process. The cross-correlation response of a class III fractional vibrator driven by the von Kármán process is expressed by

r_fx3(τ) = r_ff(τ) * h₃(τ).

(178)

6.4.2. Effects of α and β on Responses

Some plots of |H₃(ω)|² are shown in Figure 90. Figure 91 indicates some plots of the PSD response S_xx3(ω). Some plots of |S_fx3(ω)| are indicated in Figure 92. Figure 91 and Figure 92 exhibit that the effects of (α, β) on the responses of class III fractional vibrators driven by the von Kármán process are significant.

6.5. Responses of Class IV Fractional Vibration Systems Driven by the von Kármán Process

6.5.1. Computations

Theorem 51 (PSD response IV). 

Let x₄(t) be the response to a class IV fractional vibrator driven by f(t) of the von Kármán process. Let S_xx4(ω) be the PSD of the response x₄(t). Then,

$$S_{xx4}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k^2{\omega }^{2\lambda }{\cos }^2\frac{\lambda \pi }{2}\left[\begin{array}{l}{\left(1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\right)}^2\\ +4{\left(\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2\end{array}\right]}.$$

(179)

Proof. 

Solving the operation in Equation (180)

S_xx4(ω) = S_ff(ω)|H₄(ω)|²

(180)

produces Equation (177). This finishes the proof. □

Theorem 52 (cross-PSD response IV). 

Denote with S_fx4(ω) the cross-PSD between f(t) of the von Kármán process and x₄(t). Then,

$$S_{fx4}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(\begin{array}{l}1-{\gamma }^2\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\\ +i2\gamma \frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\alpha +\lambda -2}\left|\cos \frac{\alpha \pi }{2}\right|\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\end{array}\right)}.$$

(181)

Proof. 

Solving the operation in Equation (182)

S_fx4(ω) = S_ff(ω)H₄(ω)

(182)

produces Equation (181). The proof is finished. □

In the time domain, we have

r_xx4(τ) = r_ff(τ) * h₄(τ) * h₄(−τ).

(183)

In Equation (183), r_xx4(τ) is the ACF of x₄(t). In addition,

r_fx4(τ) = r_ff(τ) * h₄(τ).

(184)

In Equation (184), r_fx4(τ) is the cross-correlation between the excitation f(t) of the von Kármán process and the response x₄(t).

6.5.2. Effects of α and λ on Responses

We illustrate some plots of |H₄(ω)|² in Figure 93. Figure 94 indicates some plots of the PSD response S_xx4(ω). Some plots of |S_fx4(ω)| are indicated in Figure 95. Figure 94 and Figure 95 demonstrate that the effects of (α, λ) on the responses of class VI fractional vibrators driven by the von Kármán process are noticeable.

6.6. Responses of Class V Fractional Vibrators Driven by von Kármán Process

6.6.1. Computation Methods

Theorem 53 (PSD response V). 

Let x₅(t) be the response to a class V fractional vibrator driven by f(t) of the von Kármán process. Denote with S_xx5(ω) the PSD of the response x₅(t). Then,

$$S_{xx5}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k^2{\omega }^{2\lambda }{\cos }^2\frac{\lambda \pi }{2}\left[{\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}\right)}^2+4{\gamma }^2{\left(\frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}^2\right]}.$$

(185)

Proof. 

Solving the operation in Equation (186)

S_xx5(ω) = S_ff(ω)|H₅(ω)|²

(186)

results in Equation (185). The proof ends. □

Theorem 54 (cross-PSD response V). 

Let S_fx5(ω) be the cross-PSD between f(t) of the von Kármán process and x₅(t). Then,

$$S_{fx5}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}\left(1-\frac{{\gamma }^2}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}+i2\gamma \frac{k{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}}{2\sqrt{mk{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\sqrt{\frac{1}{{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}}}\right)}.$$

(187)

Proof. 

Solving the operation in (188) below

S_fx5(ω) = S_ff(ω)H₅(ω)

(188)

yields Equation (187). This finishes the proof. □

In the time domain, we have

r_xx5(τ) = r_ff(τ) * h₅(τ) * h₅(−τ).

(189)

In Equation (189), r_xx5(τ) is the ACF of x₅(t). In addition,

r_fx5(τ) = r_ff(τ) * h₅(τ).

(190)

In Equation (190), r_fx5(τ) is the cross-correlation between the excitation f(t) of the von Kármán process and the response x₅(t).

6.6.2. Effects of λ on Responses

Figure 96 illustrates some plots of |H₅(ω)|². Figure 97 shows some plots of the PSD response S_xx5(ω), and Figure 98 shows some plots of |S_fx5(ω)|. Figure 97 and Figure 98 illustrate the considerable effects of λ on the responses of class V fractional vibrators driven by the von Kármán process.

6.7. Responses of Class VI Fractional Vibrators Driven by von Kármán Process

6.7.1. Computations

Theorem 55 (PSD response VI). 

Let x₆(t) be the response to a class VI fractional vibrator excited by the von Kármán process. Let S_xx6(ω) be the PSD of the response x₆(t). Then,

$$S_{xx6}(\omega )=\frac{1}{k^2}\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{\begin{array}{l}{\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\gamma }^2{\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)}^2\end{array}}.$$

(191)

Proof. 

Considering the operation in Equation (192)

S_xx6(ω) = S_ff(ω)|H₆(ω)|²

(192)

produces Equation (191). The proof ends. □

Theorem 56 (cross-PSD response VI). 

Denote with S_fx6(ω) the cross-PSD between f(t) of the von Kármán process and x₆(t). Then,

$$S_{fx6}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k\left[\begin{array}{l}{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}+{\gamma }^2\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\\ +i\gamma \left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\zeta {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n^2{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\end{array}\right]}.$$

(193)

Proof. 

Solving the operation in Equation (194)

S_fx6(ω) = S_ff(ω)H₆(ω)

(194)

results in Equation (193). The proof is finished. □

Applying the Wiener–Khinchin relation and the Wiener–Lee relation, respectively, to S_xx6(ω) = S_ff(ω)|H₆(ω)|² and S_fx6(ω) = S_ff(ω)H₆(ω), we have

r_xx6(τ) = r_ff(τ) * h₆(τ) * h₆(−τ).

(195)

In Equation (195), r_xx6(τ) is the ACF of x₆(t). In addition,

r_fx6(τ) = r_ff(τ) * h₆(τ).

(196)

In Equation (196), r_fx6(τ) is the cross-correlation between f(t) of von Kármán process and x₆(t).

6.7.2. Effects of α, β, λ on Responses

Figure 99 illustrates some plots of |H₆(ω)|². Figure 100 shows some plots of the PSD response S_xx6(ω). Figure 101 indicates some plots of |S_fx6(ω)|. Figure 100 and Figure 101 show the significant effects of fractional orders (α, β, λ) on the responses to class VI fractional vibration systems under the excitation of the von Kármán process.

6.8. Responses of Class VII Fractional Vibrators Driven by the von Kármán Process

6.8.1. Computations

Theorem 57 (PSD response VII). 

Let x₇(t) be the response to a class VII fractional vibrator driven by the von Kármán process. Denote with S_xx7(ω) the PSD of x₇(t). Then,

$$S_{xx7}(\omega )=\frac{1}{k^2}\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{\begin{array}{l}{\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\right]}^2\\ +{\gamma }^2{\left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)}^2\end{array}}.$$

(197)

Proof. 

Solving the operation in Equation (198)

S_xx7(ω) = S_ff(ω)|H₇(ω)|²

(198)

yields Equation (197). The proof is finished. □

Theorem 58 (cross-PSD response VII). 

Let S_fx7(ω) be the cross-PSD between f(t) of the von Kármán process and x₇(t). Then,

$$S_{fx7}(\omega )=\frac{\frac{A_{\mathrm{vk}}}{{\left[{(B_{\mathrm{vk}})}^2+{\omega }^2\right]}^{5/6}}}{k\left[{\omega }^{\lambda }\cos \frac{\lambda \pi }{2}-\gamma \left(1-2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)+i\gamma \left(2\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}+{\omega }_n{\omega }^{\lambda -1}\sin \frac{\lambda \pi }{2}\right)\right]}.$$

(199)

Proof. 

Solving the operation in Equation (200)

S_fx7(ω) = S_ff(ω)H₇(ω)

(200)

yields Equation (199). The proof ends. □

According to the Wiener–Khinchin relation and the Wiener–Lee relation, we have

r_xx7(τ) = r_ff(τ) * h₇(τ) * h₇(−τ),

(201)

where r_xx7(τ) is the ACF of x₇(t), and

r_fx7(τ) = r_ff(τ) * h₇(τ),

(202)

where r_fx7(τ) is the cross-correlation between f(t) of the von Kármán process and x₇(t).

6.8.2. Effects of β, λ on Responses

Figure 102 indicates some plots of |H₇(ω)|². Figure 103 gives some plots of the PSD response S_xx7(ω). Figure 104 shows some plots of |PSD S_fx7(ω)|. Figure 103 and Figure 104 exhibit the noticeable effects of fractional orders (β, λ) on the responses to class VII fractional vibrators under the excitation of the von Kármán process.

6.9. Summary

We present the closed-form analytic expressions of the PSD and cross-PSD responses to seven classes of fractional vibrators driven by the von Kármán process in Theorems 45–58, respectively. We demonstrate that the orders of fractional vibrators noticeably affect the responses. The responses of seven classes of fractional vibrators are short-range-dependent due to S_xxj(0) < ∞ for j = 1, …, 7.

7. Discussions and Conclusions

Vibrations in random environments are usually called random vibrations (Crandall and Mark [119], Elishakoff and Lyon [120]). Since structures may often suffer from random loading, such as ocean waves, random vibrations are paid close attention (Rothbart and Brown [121], Jensen [122], Harris [123], Lalanne [124]). That is the motivation of studying fractionally random vibrations in this research.

Note that a main aim of studying random vibrations is to study the fatigue damage of structures in random environments (Crandall and Mark [119], Harris [123], Lalanne [125,126], Li [127,128]). Our future work will focus on fatigue damage to structures using fractionally random vibrations.

We introduced the class VII fractional vibrators in Section 2, in addition to six classes of fractional vibrators. The frequency transfer functions and impulse response functions of seven classes of fractional vibrators are given in Section 2. The analytical expressions of the PSD responses and cross-PSD ones to seven classes of fractional vibrators driven by fGn, fBm, the OU processes, and the von Kármán process are proposed in Section 3, Section 4, Section 5 and Section 6, respectively. Our demonstrations exhibit that the effects of fractional orders on the responses of seven classes of fractional vibrators, which are excited by fGn, fBm, the OU processes, and the von Kármán process, are noticeable.