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Abstract

Stochastic bifurcation and chaos of a rub-impact rotor system with random stiffness under random excitation are studied in this article. Due to the irrational and fractional expressions existing in the denominator of rub-impact force, the integral process is very complicated. Taylor series expansion is used to expand the irrational and fractional expressions into a series of polynomials. Chebyshev polynomial approximation method is applied to reduce the system equations with random parameter to its equivalent deterministic one, and the responses of stochastic system can be obtained by numerical methods. Numerical simulations show that random parameters have a significant effect on the rub-impact rotor system. It may promote the nonlinear response when the rotational speed is near the 1/2 first-order critical speed and may suppress the nonlinear response when the rotational speed is over the first-order critical speed.

Keywords

Chebyshev polynomial approximation rub-impact rotor random stiffness random excitation

Introduction

Rotor bearing system is the core component of rotating machines, and its dynamic characteristics have important influence on the working performance and reliability of rotating machines. Dynamic analysis of rotor bearing system is essential in structure design, maintenance, vibration control, and fault diagnosis of rotating machines. In the past decades, there are a lot of literature works that focus on the deterministic rotor system. With the fast development of uncertainty analysis and processing technology, researchers tend to use uncertainty to explain the rotor systems.¹ In this situation, parameter may submit random distribution which is caused by external environment, manufacture, material, and installation.² Igusa and Kiureghian³ first studied uncertainty and found the uncertainty in the parameters of certain systems may have significant influence on the irreliability, even when the input is a wideband stochastic process. Therefore, it is necessary to study the uncertain model instead of a deterministic one. Especially, the stability of rotor system is an important issue for design, manufacturing, and operation of rotating machinery. Chu and Lu⁴ designed a special structure of stator that can simulate the condition of the full rub and observed very rich forms of periodic and chaotic vibrations. Hou et al.⁵ studied the nonlinear vibration phenomenon caused by aircraft hovering flight in a rub-impact rotor system supported by two general supports with cubic stiffness. The different rubbing forms have significant influences on the dynamic responses of rotor systems, such as fixed-point rubbing,⁶ local rubbing,⁷ and full annular rubbing.⁸ And other activities^9,10 also illustrated the bifurcations and chaos of the rub-impact rotor in the deterministic system. There are many studies that focus on the research of random structures.

Fang and Leng¹¹ first applied Chebyshev polynomial approximation to solve the dynamical response of the random system. Then, this method was introduced to study the dynamical behavior and its control in classical nonlinear systems with random parameters.^2,12 Leng et al.¹³ studied the bifurcation and chaos response of a cracked rotor with random disturbance. Yang et al.^14,15 studied the stochastic bifurcation and chaos of a rub-impact rotor system with random parameter and proposed Taylor series expansion method to fully transfer the random equations to its equivalent deterministic one. From the research above, Chebyshev polynomial approximation method is a good idea to deal with the stochastic nonlinear characteristics in a rub-impact rotor system with random parameters under random excitation. In this article, we focus on the stochastic response of a rub-impact rotor system with random stiffness under random excitation. The Chebyshev polynomial approximation method and Taylor series expansion method are introduced to study the nonlinear response. The complex behavior which affect by the random parameter is shown by the simulation.

Mathematical model

A schematic of a rotor system with rub-impact is shown in Figure 1. Some simplifications are applied to establish the mathematical model: (1) a schematic of a rotor system with nonlinearities which are caused by material, and a cubic term is used to represent the nonlinearity of the shaft; (2) two ends of the rotor are simply supported; (3) this rotor has viscous damping c; and (4) the stator is rigid and its surface is elastic. Then, the normal dynamic equations at the rotor center $(X, Y)$ of a rub-impact rotor are⁵

{\begin{matrix} m \overset{\cdot\cdot}{X} + c \overset{\cdot}{X} + kX + k_{1} (X^{2} + Y^{2}) X = - \frac{k_{r} (r - δ)}{r} (X - fY) + me ω^{2} \cos (ω t) \\ m \overset{\cdot\cdot}{Y} + c \overset{\cdot}{Y} + kY + k_{1} (X^{2} + Y^{2}) Y = - \frac{k_{r} (r - δ)}{r} (fX + Y) + me ω^{2} \sin (ω t) - mg \end{matrix}

(1)

where $r = \sqrt{X^{2} + Y^{2}}$ represents the radial displacement, and the general notations used in equation (1) are shown in Table 1.

Figure 1.

Schematic of rub-impact rotor system and rub-impact forces.

Table 1.

Summary of the general notation used in equation (1).

Symbol	Description
m	Disk mass
k	Linear stiffness of the shaft
k ₁	Nonlinear stiffness of the shaft
k_r	Stator’s surface stiffness
δ	Clearance between rotor and stator
f	Friction coefficient between rotor and stator
e	Mass eccentricity of the disk
ω	Rotating speed
g	Acceleration of gravity

The non-dimensional equations are

{\begin{matrix} \overset{\cdot\cdot}{x} + a \overset{\cdot}{x} + β [1 + b (x^{2} + y^{2})] x = - F_{0 x} + e_{m} \cos τ \\ \overset{\cdot\cdot}{y} + a \overset{\cdot}{y} + β [1 + b (x^{2} + y^{2})] y = - F_{0 y} + e_{m} \sin τ - G \end{matrix}

(2)

where $x = X / δ$ , $y = Y / δ$ , $β = k / m ω^{2}$ , $β_{r} = k_{r} / m ω^{2}$ , $b = k_{1} δ^{2} / k$ , $a = c / m ω$ , $e_{m} = e / δ$ , $G = g / δ ω^{2}$ , and $τ = ω t$ . The non-dimensional rub-impact forces are

{\begin{matrix} F_{0 x} = β_{r} (1 - \frac{1}{\sqrt{x^{2} + y^{2}}}) (x - fy) (\sqrt{x^{2} + y^{2}} \geq 1) \\ F_{0 y} = β_{r} (1 - \frac{1}{\sqrt{x^{2} + y^{2}}}) (fx + y) (\sqrt{x^{2} + y^{2}} \geq 1) \\ F_{0 x} = F_{0 y} = 0 (\sqrt{x^{2} + y^{2}} < 1) \end{matrix}

(3)

Therefore, considering the random stiffness under random excitation of the rub-impact rotor system, the non-dimensional equations can be written as follows

{\begin{matrix} \overset{\cdot\cdot}{x} + a \overset{\cdot}{x} + (\bar{β} + σ_{1} ξ) [1 + b (x^{2} + y^{2})] x = - F_{0 x} + e_{m} \cos τ + σ_{2} ε (τ) \\ \overset{\cdot\cdot}{y} + a \overset{\cdot}{y} + (\bar{β} + σ_{1} ξ) [1 + b (x^{2} + y^{2})] y = - F_{0 y} + e_{m} \sin τ - G + σ_{2} ε (τ) \end{matrix}

(4)

where $\bar{β}$ is the mean value of $β$ , $σ_{1}$ is taken as the random intensity of $β$ , $σ_{2}$ is taken as the random excitation intensity, and $ξ \in [- 1, 1]$ is the random parameter of stiffness which can be described by a given arch-like probability density function and can be expressed by

p (ξ) = {\begin{matrix} \frac{2}{π} \sqrt{1 - ξ^{2}} & | ξ | \leq 1 \\ 0 & | ξ | > 1 \end{matrix}

(5)

Monte Carlo method is a useful method in numerical analysis of nonlinear structure subject to random excitations. Shinozuka¹⁶ presented an efficient method for digital simulation of a general homogeneous process as a series of cosine functions with weighted amplitudes, almost evenly spaced frequencies, and random phase angles. This method can be used in the numerical analysis of wind-induced ocean wave elevation, spatial random variation of material properties, and random surface roughness of airport and highway runways. In this article, we suppose that the random excitation is a standard white noise process which can be simulated as

ε (τ) = 2 \sqrt{\frac{S w_{0}}{N}} \sum_{k = 1}^{N} \cos [k \frac{w_{0}}{N} τ + θ_{k}]

(6)

where N is the total number of modes and N→∞, $ω_{0}$ is the truncation frequency in the simulation of $ε (t)$ , θ_k∈[0, 2π] (k = 1, 2, …, N) is distributed uniformly and randomly, and S = 1.0 is the spectral density of a standard white noise process. If θ_k take a series of deterministic values ${\tilde{θ}}_{k} \in [0, 2 π]$ in equation (6), the corresponding $\tilde{ε} (t)$ will be a sample of the white noise process $ε (t)$ . After numerical tests and the discussion about the minimum requirements in the works by Shinozuka and Jan,¹⁶ we select N = 500 and ω₀ = 25.

Taylor series expansion method

Taylor series expansion method is introduced here to expand the irrational and fractional expressions into a series of polynomials. This method is used to avoid the complex integral operation when we apply Chebyshev polynomial approximation method.

Assume $f (x, y)$ is continuous near a point $(x_{0}, y_{0})$ and has continuous partial derivative until n+1 order. Then

\begin{matrix} f (x_{0} + h, y_{0} + k) = f (x_{0}, y_{0}) \\ + (h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y}) f (x_{0}, y_{0}) + \frac{1}{2!} {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{2} f (x_{0}, y_{0}) \\ + \dots + \frac{1}{n!} {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{n} f (x_{0}, y_{0}) \\ + \frac{1}{(n + 1)!} {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{n + 1} f (x_{0} + θ h, y_{0} + θ k), \\ (0 < θ < 1) \end{matrix}

(7)

where h and k are small perturbation to the point of $(x_{0}, y_{0})$ .¹⁷ Note that

{(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{m} f (x_{0}, y_{0}) = \sum_{p = 0}^{m} C_{m}^{p} h^{p} k^{m - p} \frac{\partial^{m} p}{\partial x^{p} \partial y^{m - p}} |_{(x_{0}, y_{0})}

(8)

When $(x_{0}, y_{0}) = (0, 0)$ , equation (7) can be written as

\begin{matrix} f (x, y) = f (0, 0) + (h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y}) f (0, 0) \\ + \frac{1}{2!} {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{2} f (0, 0) \\ + \dots + \frac{1}{n!} {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{n} f (0, 0) \\ + \frac{1}{(n + 1)!} {(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y})}^{n + 1} f (θ x, θ y), (0 < θ < 1) \end{matrix}

(9)

This is a special case of Taylor series and is known as Maclaurin’s series. The irrational and fractional expressions can be expanded into a series of polynomials using Maclaurin’s series. The non-dimensional polynomial rub-impact forces of equation (3) can be written as

{\begin{matrix} F_{x} = β_{r} (- 0.00207 x + 0.000207 y + 0.115 x^{3} - \\ 0.0115 x^{2} y + 0.115 x y^{2} - 0.0115 y^{3}) (\sqrt{x^{2} + y^{2}} \geq 1) \\ F_{y} = β_{r} (- 0.000207 x + 0.00207 y + 0.0115 x^{3} - \\ 0.115 x^{2} y + 0.0115 x y^{2} - 0.115 y^{3}) (\sqrt{x^{2} + y^{2}} \geq 1) \\ F_{x} = F_{y} = 0 (\sqrt{x^{2} + y^{2}} < 1) \end{matrix}

(10)

Chebyshev orthogonal polynomial

Chebyshev orthogonal polynomial method is used to transfer the random rub-impact rotor equation to its deterministic form. Equation (4) can be approximately expressed using the following series under the convergence in mean square

{\begin{matrix} x (t, ξ) = \sum_{l = 0}^{N} x_{l} (t) H_{l} (ξ) \\ y (t, ξ) = \sum_{l = 0}^{N} y_{l} (t) H_{l} (ξ) \end{matrix}

(11)

where

\begin{matrix} x_{l} (t) = \int_{- \infty}^{\infty} p (ξ) x (t, ξ) H_{l} (ξ) d ξ \\ y_{l} (t) = \int_{- \infty}^{\infty} p (ξ) y (t, ξ) H_{l} (ξ) d ξ \end{matrix}

(12)

$H_{l} (ξ)$ is the lth Chebyshev orthogonal polynomial and N is the greatest order of the polynomials.

Substituting equations (10) and (11) into equation (4), when $N = 4$ we get the following expressions²

{\begin{matrix} (\frac{d^{2}}{d t^{2}} + a \frac{d}{dt} + \bar{β}) x_{0} (t) + \bar{β} b X_{0} (t) + \bar{β} b K_{0} (t) + \frac{σ_{1}}{2} x_{1} (t) + \frac{σ_{1}}{2} X_{1} (t) \\ + \frac{σ_{1}}{2} K_{1} (t) = - f_{x 0} + e_{m} \cos (τ) + σ_{2} ε (t) \\ (\frac{d^{2}}{d t^{2}} + a \frac{d}{dt} + \bar{β}) y_{0} (t) + \bar{β} b S_{0} (t) + \bar{β} b Y_{0} (t) + \frac{σ_{1}}{2} y_{1} (t) + \frac{σ_{1}}{2} S_{1} (t) \\ + \frac{σ_{1}}{2} Y_{1} (t) = - f_{y 0} + e_{m} \sin (τ) - G + σ_{2} ε (t) \\ (\frac{d^{2}}{d t^{2}} + a \frac{d}{dt} + \bar{β}) x_{i} (t) + \bar{β} b X_{i} (t) + \bar{β} b K_{i} (t) + \frac{σ_{1}}{2} [x_{i - 1} (t) + x_{i + 1} (t)] \\ + \frac{σ_{1} b}{2} [x_{i - 1} (t) + x_{i + 1} (t)] + \frac{σ_{1} b}{2} [K_{i - 1} (t) + K_{i + 1} (t)] = - f_{xi} \\ (\frac{d^{2}}{d t^{2}} + a \frac{d}{dt} + \bar{β}) y_{i} (t) + \bar{β} b S_{i} (t) + \bar{β} b Y_{i} (t) + \frac{σ_{1}}{2} [y_{i - 1} (t) + y_{i + 1} (t)] \\ + \frac{σ_{1} b}{2} [S_{i - 1} (t) + S_{i + 1} (t)] + \frac{σ_{1} b}{2} [Y_{i - 1} (t) + Y_{i + 1} (t)] = - f_{yi} \\ \dots \\ (\frac{d^{2}}{d t^{2}} + a \frac{d}{dt} + \bar{β}) x_{4} (t) + \bar{β} b X_{4} (t) + \bar{β} b K_{4} (t) + \frac{σ_{1} b}{2} [X_{3} (t) + X_{5} (t)] \\ + \frac{σ_{1}}{2} x_{3} + \frac{σ_{1} b}{2} [K_{3} (t) + K_{5} (t)] = - f_{x 4} \\ (\frac{d^{2}}{d t^{2}} + a \frac{d}{dt} + \bar{β}) y_{4} (t) + \bar{β} b S_{4} (t) + \bar{β} b Y_{4} (t) + \frac{σ_{1} b}{2} [S_{3} (t) + S_{5} (t)] \\ + \frac{σ_{1}}{2} y_{3} + \frac{σ_{1} b}{2} [Y_{3} (t) + Y_{5} (t)] = - f_{y 4} \end{matrix}

(13)

where $X_{l}$ , $Y_{l}$ , $S_{l}$ , and $K_{l}$ are the functions of $x_{l}$ , $y_{l}$ (l = 0, 1, 2, 3, 4) and the expressions are shown in Appendix 1.

Equation (13) is solved by the fourth-order Runge–Kutta method. Considering the random excitation, the rub-impact rotor can be expressed as

{\begin{matrix} x (t, ξ) \approx \sum_{l = 0}^{4} x_{l} (t) H_{l} (ξ) \\ y (t, ξ) \approx \sum_{l = 0}^{4} y_{l} (t) H_{l} (ξ) \end{matrix}

(14)

The ensemble mean response (EMR) can be obtained as

{\begin{matrix} E [x (t, ξ)] = \sum_{l = 0}^{4} x_{l} (t) E [H_{l} (ξ)] = x_{0} (t) \\ E [y (t, ξ)] = \sum_{l = 0}^{4} y_{l} (t) E [H_{l} (ξ)] = y_{0} (t) \end{matrix}

(15)

From the above equations, the stochastic rub-impact rotor system equation (4) has been transformed into high-dimensional deterministic equation (13). We can study the original stochastic rotor system by equation (15).

Numerical results

The parameters of rub-impact system are as follows: $m = 30 kg$ , $c = 1500 Ns / m$ , $e = 0.1 mm$ , $k_{r} = 6.5 \times 10^{6} N / m$ , $k_{1} = 2 \times 10^{13} N / m$ , $k = 5.6 \times 10^{5} N / m$ , and $δ = 0.2 mm$ . The first-order critical speed is 1304.7 r/min.

Methods validation

The fourth-order Runge–Kutta method is employed to simulate this system. In order to determine whether rub-impact failure occurs or not in equation (4), the friction coefficient f is defined as $f (x) = 1 / 2 [sign (abs (x)) + sign (x)]$ , where sign(x) represents a symbolic function, namely

sign (x) = {\begin{matrix} 1 & (x > 0) \\ 0 & (x = 0) \\ - 1 & (x < 0) \end{matrix}

Equation (4) decays to be a deterministic system with random excitation when $σ_{1} = 0$ . The bifurcation diagrams are shown in Figure 2. The response which calculated by equation (4) directly named Original response (shortly Original). The response which calculated by equation (13) named Chebyshev response (shortly Chebyshev). From bifurcation diagrams, the forms of motion show similar nonlinear characteristic. The phase planes of some typical speeds are shown in Figures 3 –6. The Original responses and Chebyshev responses are almost the same when the rotational speed varies. These results illustrate that Chebyshev polynomial approximation method can be used in the simulation of rub-impact rotor system.

Figure 2.

Bifurcation diagram of the system when $σ_{2} = 0.001$ : (a) original equation, (b) Chebyshev polynomial approximation method, and (c) the largest Lyapunov exponent diagrams of original equation.

Figure 3.

Phase plane when $ω = 400 r / \min$ : (a) original response and (b) Chebyshev response.

Figure 4.

The response when $ω = 600 r / \min$ : (a) original response of phase plane, (b) Chebyshev response of phase plane, (c) original response of Poincare map, and (d) Chebyshev response of Poincare map.

Figure 5.

The response when $ω = 750 r / \min$ : (a) original response of phase plane, (b) Chebyshev response of phase plane, (c) original response of Poincare map, and (d) Chebyshev response of Poincare map.

Figure 6.

The response when $ω = 1190 r / \min$ : (a) original response of phase plane, (b) Chebyshev response of phase plane, (c) original response of Poincare map, and (d) Chebyshev response of Poincare map.

The effect of random stiffness

When $σ_{2} = 0.001$ , choose the random stiffness intensity $σ_{1} = 0.005$ , we can get the bifurcation diagram from equation (13) as shown in Figure 7. By comparing Figures 7 and 2, the forms of motions are almost the same. The phase planes of some typical speeds are shown in Figures 8 –10. By comparing Figures 8 and 4, when $ω = 600 r / \min$ , the two responses are chaotic. But the amplitude range of Figure 8 is less than Figure 4. By comparing Figures 9 and 5, when $ω = 750 r / \min$ , the two responses are quasi-periodic. By comparing Figures 10 and 6, when $ω = 1190 r / \min$ , the response of Figure 6 is quasi-periodic and Figure 10 is periodic-one. All these show that the random stiffness may suppress nonlinear response in most cases.

Figure 7.

Bifurcation diagram of the system when $σ_{1} = 0.005$ and $σ_{2} = 0.001$ .

Figure 8.

Phase planes when $σ_{1} = 0.005$ , $σ_{2} = 0.001$ , and $ω = 600 r / \min$ .

Figure 9.

Phase planes when $σ_{1} = 0.005$ , $σ_{2} = 0.001$ , and $ω = 750 r / \min$ .

Figure 10.

Phase planes when $σ_{1} = 0.005$ , $σ_{2} = 0.001$ , and $ω = 1190 r / \min$ .

Figure 11 shows the rotation speed–amplitude curve for different random type. The random intensity parameters are $σ_{1} = 0.005$ and $σ_{1} = 0.005$ , respectively. A > 1 means the non-dimensional radial displacement is over 1 and the rub-impact occurs. When ω = 100–470 r/min, the amplitudes of the four types of random parameters are the same. When ω = 470–705 r/min, the rotating speed is near the 1/2 first-order critical speed and the max amplitude occurs mainly with the random stiffness. In different ranges, the main factors are different. When ω = 705–900 r/min, the amplitudes of the four types of random parameters are mainly in the same range. After that, for most cases, the max amplitude occurs when $σ_{1} = 0$ . From Figure 7, the nonlinear response near the 1/2 first-order critical speed causes by local rubbing, and the random stiffness may promote nonlinear response. The nonlinear response over the first-order critical speed causes full annular rubbing, and the random stiffness may suppress nonlinear response.

Figure 11.

Rotation speed–amplitude curve of rub-impact rotor: (a) ω = 100–600 r/min and (b) ω = 700–1200 r/min.

Conclusion

Stochastic bifurcation and chaos of a rub-impact rotor system with random stiffness under random excitation are studied by the Taylor series expansion method and the Chebyshev polynomial approximation method in this article. It aims to reveal the influence of random stiffness under random excitation on the nonlinear response of rub-impact rotor. These results are useful for the design and vibration control of rotating machinery rotor. From the above simulation, the following conclusions are made:

Random parameters do not affect the dynamic responses of the rub-impact rotor system when rotational speeds are lower.

Random parameters have a significant effect on the rub-impact rotor system when the rotational speeds become higher. More importantly, the random stiffness may promote the nonlinear response when the rotational speed is near the 1/2 first-order critical speed and may suppress the nonlinear response when the rotational speed is over the first-order critical speed.

Footnotes

Appendix 1

Using the recursion relation of Chebyshev polynomial, the functions $X_{l}$ , $Y_{l}$ , $S_{l}$ , and $K_{l}$ can get the following expressions

\begin{matrix} X_{0} (t) = x_{0}^{3} + x_{2}^{3} + x_{4}^{3} + 3 x_{0} x_{1}^{2} + 3 x_{0} x_{2}^{2} + 3 x_{0} x_{3}^{2} + 3 x_{0} x_{4}^{2} + 3 x_{1}^{2} x_{2} + 3 x_{2} x_{3}^{2} + 3 x_{2}^{2} x_{4} \\ + 3 x_{2} x_{4}^{2} + 3 x_{3}^{2} x_{4} + 6 x_{1} x_{2} x_{3} + 6 x_{1} x_{3} x_{4} \end{matrix}

\begin{matrix} X_{1} (t) = 2 x_{1}^{3} + 2 x_{3}^{3} + 3 x_{0}^{2} x_{1} + 6 x_{1} x_{2}^{2} + 3 x_{1}^{2} x_{3} + 6 x_{1} x_{3}^{2} + 6 x_{1} x_{4}^{2} + 6 x_{2}^{2} x_{3} + 2 x_{3} x_{4}^{2} \\ + 6 x_{0} x_{1} x_{2} + 6 x_{1} x_{2} x_{4} + 12 x_{2} x_{3} x_{4} + 6 x_{0} x_{2} x_{3} + 6 x_{0} x_{3} x_{4} \end{matrix}

\begin{matrix} X_{2} (t) = 3 x_{2}^{3} + 3 x_{4}^{3} + 3 x_{0} x_{1}^{2} + 3 x_{0}^{2} x_{2} + 3 x_{0} x_{2}^{2} + 3 x_{0} x_{3}^{2} + 3 x_{0} x_{4}^{2} + 6 x_{1}^{2} x_{2} + 3 x_{1}^{2} x_{4} + 9 x_{2} x_{3}^{2} \\ + 6 x_{2}^{2} x_{4} + 9 x_{2} x_{4}^{2} + 9 x_{3}^{2} x_{4} + 6 x_{0} x_{1} x_{3} + 12 x_{1} x_{2} x_{3} + 12 x_{1} x_{3} x_{4} + 6 x_{0} x_{2} x_{4} \end{matrix}

\begin{matrix} X_{3} (t) = x_{1}^{3} + 4 x_{3}^{3} + 3 x_{0}^{2} x_{3} + 6 x_{1} x_{2}^{2} + 6 x_{1}^{2} x_{3} + 6 x_{1} x_{3}^{2} + 6 x_{1} x_{4}^{2} + 9 x_{2}^{2} x_{3} + 12 x_{3} x_{4}^{2} \\ + 6 x_{0} x_{1} x_{2} + 6 x_{0} x_{1} x_{4} + 12 x_{1} x_{2} x_{4} + 18 x_{2} x_{3} x_{4} + 6 x_{0} x_{2} x_{3} + 6 x_{0} x_{3} x_{4} \end{matrix}

\begin{matrix} X_{4} (t) = 2 x_{2}^{3} + 5 x_{4}^{3} + 3 x_{0} x_{2}^{2} + 3 x_{0} x_{3}^{2} + 3 x_{0}^{2} x_{4} + 3 x_{0} x_{4}^{2} + 3 x_{1}^{2} x_{2} + 6 x_{1}^{2} x_{4} + 9 x_{2} x_{3}^{2} + 9 x_{2}^{2} x_{4} \\ + 9 x_{2} x_{4}^{2} + 12 x_{3}^{2} x_{4} + 6 x_{0} x_{1} x_{3} + 12 x_{1} x_{2} x_{3} + 12 x_{1} x_{3} x_{4} + 6 x_{0} x_{2} x_{4} \end{matrix}

\begin{matrix} X_{5} (t) = 3 x_{3}^{3} + 3 x_{1} x_{2}^{2} + 3 x_{1}^{2} x_{3} + 6 x_{1} x_{3}^{2} + 6 x_{1} x_{4}^{2} + 6 x_{2}^{2} x_{3} + 12 x_{3} x_{4}^{2} + 6 x_{0} x_{1} x_{4} \\ + 12 x_{1} x_{2} x_{4} + 18 x_{2} x_{3} x_{4} + 6 x_{0} x_{2} x_{3} + 6 x_{0} x_{3} x_{4} \end{matrix}

\begin{matrix} Y_{0} (t) = y_{0}^{3} + y_{2}^{3} + y_{4}^{3} + 3 y_{0} y_{1}^{2} + 3 y_{0} y_{2}^{2} + 3 y_{0} y_{3}^{2} + 3 y_{0} y_{4}^{2} + 3 y_{1}^{2} y_{2} + 3 y_{2} y_{3}^{2} + 3 y_{2}^{2} y_{4} \\ + 3 y_{2} y_{4}^{2} + 3 y_{3}^{2} y_{4} + 6 y_{1} y_{2} y_{3} + 6 y_{1} y_{3} y_{4} \end{matrix}

\begin{matrix} Y_{1} (t) = 2 y_{1}^{3} + 2 y_{3}^{3} + 3 y_{0}^{2} y_{1} + 6 y_{1} y_{2}^{2} + 3 y_{1}^{2} y_{3} + 6 y_{1} y_{3}^{2} + 6 y_{1} y_{4}^{2} + 6 y_{2}^{2} y_{3} + 2 y_{3} y_{4}^{2} \\ + 6 y_{0} y_{1} y_{2} + 6 y_{1} y_{2} y_{4} + 12 y_{2} y_{3} y_{4} + 6 y_{0} y_{2} y_{3} + 6 y_{0} y_{3} y_{4} \end{matrix}

\begin{matrix} Y_{2} (t) = 3 y_{2}^{3} + 3 y_{4}^{3} + 3 y_{0} y_{1}^{2} + 3 y_{0}^{2} y_{2} + 3 y_{0} y_{2}^{2} + 3 y_{0} y_{3}^{2} + 3 y_{0} y_{4}^{2} + 6 y_{1}^{2} y_{2} + 3 y_{1}^{2} y_{4} + 9 y_{2} y_{3}^{2} \\ + 6 y_{2}^{2} y_{4} + 9 y_{2} y_{4}^{2} + 9 y_{3}^{2} y_{4} + 6 y_{0} y_{1} y_{3} + 12 y_{1} y_{2} y_{3} + 12 y_{1} y_{3} y_{4} + 6 y_{0} y_{2} y_{4} \end{matrix}

\begin{matrix} Y_{3} (t) = y_{1}^{3} + 4 y_{3}^{3} + 3 y_{0}^{2} y_{3} + 6 y_{1} y_{2}^{2} + 6 y_{1}^{2} y_{3} + 6 y_{1} y_{3}^{2} + 6 y_{1} y_{4}^{2} + 9 y_{2}^{2} y_{3} + 12 y_{3} y_{4}^{2} \\ + 6 y_{0} y_{1} y_{2} + 6 y_{0} y_{1} y_{4} + 12 y_{1} y_{2} y_{4} + 18 y_{2} y_{3} y_{4} + 6 y_{0} y_{2} y_{3} + 6 y_{0} y_{3} y_{4} \end{matrix}

\begin{matrix} Y_{4} (t) = 2 y_{2}^{3} + 5 y_{4}^{3} + 3 y_{0} y_{2}^{2} + 3 y_{0} y_{3}^{2} + 3 y_{0}^{2} y_{4} + 3 y_{0} y_{4}^{2} + 3 y_{1}^{2} y_{2} + 6 y_{1}^{2} y_{4} + 9 y_{2} y_{3}^{2} + 9 y_{2}^{2} y_{4} \\ + 9 y_{2} y_{4}^{2} + 12 y_{3}^{2} y_{4} + 6 y_{0} y_{1} y_{3} + 12 y_{1} y_{2} y_{3} + 12 y_{1} y_{3} y_{4} + 6 y_{0} y_{2} y_{4} \end{matrix}

\begin{matrix} Y_{5} (t) = 3 y_{3}^{3} + 3 y_{1} y_{2}^{2} + 3 y_{1}^{2} y_{3} + 6 y_{1} y_{3}^{2} + 6 y_{1} y_{4}^{2} + 6 y_{2}^{2} y_{3} + 12 y_{3} y_{4}^{2} + 6 y_{0} y_{1} y_{4} \\ + 12 y_{1} y_{2} y_{4} + 18 y_{2} y_{3} y_{4} + 6 y_{0} y_{2} y_{3} + 6 y_{0} y_{3} y_{4} \end{matrix}

\begin{matrix} S_{0} (t) = 2 x_{3} x_{4} y_{3} + 2 x_{1} x_{2} y_{1} + 2 x_{2} x_{3} y_{3} + 2 x_{0} x_{2} y_{2} + 2 x_{0} x_{3} y_{3} + 2 x_{0} x_{1} y_{1} + x_{1}^{2} y_{2} + 2 x_{3} x_{4} y_{1} \\ + 2 x_{1} x_{2} y_{3} + 2 x_{1} x_{3} y_{2} + 2 x_{2} x_{3} y_{1} + 2 x_{1} x_{4} y_{3} + 2 x_{1} x_{3} y_{4} + x_{3}^{2} y_{4} + x_{3}^{2} y_{0} + 2 x_{2} x_{4} y_{4} \\ + x_{2}^{2} y_{2} + x_{4}^{2} y_{4} + x_{4}^{2} y_{0} + 2 x_{0} x_{4} y_{4} + x_{2}^{2} y_{0} + x_{1}^{2} y_{0} + x_{4}^{2} y_{2} + x_{3}^{2} y_{2} + x_{0}^{2} y_{0} + x_{2}^{2} y_{4} \\ + 2 x_{2} x_{4} y_{2} \end{matrix}

\begin{matrix} S_{1} (t) = 4 x_{1} x_{3} y_{3} + 4 x_{3} x_{4} y_{4} + 2 x_{1} x_{3} y_{1} + 4 x_{1} x_{4} y_{4} + 4 x_{1} x_{2} y_{2} + 4 x_{2} x_{3} y_{2} + x_{0}^{2} y_{1} + 2 x_{4}^{2} y_{3} \\ + 2 x_{0} x_{1} y_{2} + 2 x_{0} x_{2} y_{1} + 2 x_{1} x_{2} y_{0} + 2 x_{2} x_{3} y_{0} + 2 x_{0} x_{3} y_{4} + 2 x_{0} x_{2} y_{3} + 2 x_{0} x_{3} y_{2} \\ + 2 x_{0} x_{4} y_{3} + 2 x_{3} x_{4} y_{0} + 2 x_{1} x_{2} y_{4} + 2 x_{2} x_{4} y_{1} + 2 x_{1} x_{4} y_{2} + 4 x_{2} x_{3} y_{4} + 4 x_{3} x_{4} y_{2} \\ + 4 x_{2} x_{4} y_{3} + 2 x_{4}^{2} y_{1} + 2 x_{3}^{2} y_{1} + 2 x_{2}^{2} y_{1} + 2 x_{0} x_{1} y_{0} + x_{1}^{2} y_{3} + 2 x_{3}^{2} y_{3} + 2 x_{2}^{2} y_{3} + 2 x_{1}^{2} y_{1} \end{matrix}

\begin{matrix} S_{2} (t) = 6 x_{3} x_{4} y_{3} + 4 x_{1} x_{2} y_{1} + 6 x_{2} x_{3} y_{3} + 2 x_{0} x_{2} y_{2} + 2 x_{0} x_{3} y_{3} + 2 x_{0} x_{1} y_{1} + x_{0}^{2} y_{2} + 2 x_{1}^{2} y_{2} \\ + 4 x_{3} x_{4} y_{1} + 2 x_{2} x_{4} y_{0} + 2 x_{0} x_{2} y_{4} + 2 x_{0} x_{4} y_{2} + 2 x_{1} x_{3} y_{0} + 2 x_{0} x_{1} y_{3} + 2 x_{0} x_{3} y_{1} \\ + 4 x_{1} x_{2} y_{3} + 4 x_{1} x_{3} y_{2} + 4 x_{2} x_{3} y_{1} + 4 x_{1} x_{4} y_{3} + 4 x_{1} x_{3} y_{4} + 3 x_{3}^{2} y_{4} + x_{3}^{2} y_{0} + 6 x_{2} x_{4} y_{4} \\ + 3 x_{2}^{2} y_{2} + 3 x_{4}^{2} y_{4} + 2 x_{0} x_{2} y_{0} + x_{4}^{2} y_{0} + 2 x_{0} x_{4} y_{4} + 2 x_{1} x_{4} y_{1} + x_{2}^{2} y_{0} + x_{1}^{2} y_{0} + 3 x_{4}^{2} y_{2} \\ + 3 x_{3}^{2} y_{2} + x_{1}^{2} y_{4} + 2 x_{2}^{2} y_{4} + 4 x_{2} x_{4} y_{2} \end{matrix}

\begin{matrix} S_{3} (t) = 4 x_{1} x_{3} y_{3} + 8 x_{3} x_{4} y_{4} + 4 x_{1} x_{3} y_{1} + 4 x_{1} x_{4} y_{4} + 4 x_{1} x_{2} y_{2} + 6 x_{2} x_{3} y_{2} + x_{0}^{2} y_{3} + 4 x_{4}^{2} y_{3} \\ + 2 x_{0} x_{1} y_{2} + 2 x_{0} x_{2} y_{1} + 2 x_{1} x_{2} y_{0} + 2 x_{0} x_{4} y_{1} + 2 x_{1} x_{4} y_{0} + 2 x_{0} x_{1} y_{4} + 2 x_{2} x_{3} y_{0} \\ + 2 x_{0} x_{3} y_{4} + 2 x_{0} x_{2} y_{3} + 2 x_{0} x_{3} y_{2} + 2 x_{0} x_{4} y_{3} + 2 x_{3} x_{4} y_{0} + 4 x_{1} x_{2} y_{4} + 4 x_{2} x_{4} y_{1} \\ + 4 x_{2} x_{4} y_{1} + 4 x_{1} x_{4} y_{2} + 6 x_{2} x_{3} y_{4} + 6 x_{3} x_{4} y_{2} + 6 x_{2} x_{4} y_{3} + 2 x_{4}^{2} y_{1} + 2 x_{3}^{2} y_{1} + 2 x_{2}^{2} y_{1} \\ + 2 x_{0} x_{3} y_{0} + 2 x_{1}^{2} y_{3} + 4 x_{3}^{2} y_{3} + 3 x_{2}^{2} y_{3} + x_{1}^{2} y_{1} \end{matrix}

\begin{matrix} S_{4} (t) = 8 x_{3} x_{4} y_{3} + 2 x_{1} x_{2} y_{1} + 6 x_{2} x_{3} y_{3} + 2 x_{0} x_{2} y_{2} + 2 x_{0} x_{3} y_{3} + x_{0}^{2} y_{4} + x_{1}^{2} y_{2} + 4 x_{3} x_{4} y_{1} \\ + 2 x_{2} x_{4} y_{0} + 2 x_{0} x_{2} y_{4} + 2 x_{0} x_{4} y_{2} + 2 x_{1} x_{3} y_{0} + 2 x_{0} x_{1} y_{3} + 2 x_{0} x_{3} y_{1} + 4 x_{1} x_{2} y_{3} \\ + 4 x_{1} x_{3} y_{2} + 4 x_{2} x_{3} y_{1} + 2 x_{0} x_{4} y_{0} + 4 x_{1} x_{4} y_{3} + 4 x_{1} x_{3} y_{4} + 4 x_{3}^{2} y_{4} + x_{3}^{2} y_{0} + 6 x_{2} x_{4} y_{4} \\ + x_{2}^{2} y_{2} + 5 x_{4}^{2} y_{4} + x_{4}^{2} y_{0} + 2 x_{0} x_{4} y_{4} + 4 x_{1} x_{4} y_{1} + x_{2}^{2} y_{0} + 3 x_{4}^{2} y_{2} + 3 x_{3}^{2} y_{2} + 2 x_{1}^{2} y_{4} \\ + 3 x_{2}^{2} y_{4} + 6 x_{2} x_{4} y_{2} \end{matrix}

\begin{matrix} S_{5} (t) = 4 x_{1} x_{3} y_{3} + 8 x_{3} x_{4} y_{4} + 2 x_{1} x_{3} y_{1} + 4 x_{1} x_{4} y_{4} + 2 x_{1} x_{2} y_{2} + 4 x_{2} x_{3} y_{2} + 4 x_{4}^{2} y_{3} + 2 x_{0} x_{4} y_{1} \\ + 2 x_{1} x_{4} y_{0} + 2 x_{0} x_{1} y_{4} + 2 x_{2} x_{3} y_{0} + 2 x_{0} x_{3} y_{4} + 2 x_{0} x_{2} y_{3} + 2 x_{0} x_{3} y_{2} + 2 x_{0} x_{4} y_{3} \\ + 2 x_{3} x_{4} y_{0} + 4 x_{1} x_{2} y_{4} + 4 x_{2} x_{4} y_{1} + 4 x_{1} x_{4} y_{2} + 6 x_{2} x_{3} y_{4} + 6 x_{3} x_{4} y_{2} + 6 x_{2} x_{4} y_{3} \\ + 2 x_{4}^{2} y_{1} + 2 x_{3}^{2} y_{1} + x_{2}^{2} y_{1} + x_{1}^{2} y_{3} + 3 x_{3}^{2} y_{3} + 2 x_{2}^{2} y_{3} \end{matrix}

\begin{matrix} K_{0} (t) = 2 x_{3} y_{3} y_{4} + 2 x_{1} y_{1} y_{2} + 2 x_{3} y_{2} y_{3} + 2 x_{2} y_{0} y_{2} + 2 x_{3} y_{0} y_{3} + 2 x_{1} y_{0} y_{1} + x_{2} y_{1}^{2} + 2 x_{1} y_{3} y_{4} \\ + 2 x_{3} y_{1} y_{2} + 2 x_{2} y_{1} y_{3} + 2 x_{1} y_{2} y_{3} + 2 x_{3} y_{1} y_{4} + 2 x_{4} y_{1} y_{3} + x_{4} y_{3}^{2} + x_{0} y_{3}^{2} + 2 x_{4} y_{2} y_{4} \\ + x_{2} y_{2}^{2} + x_{4} y_{4}^{2} + x_{0} y_{4}^{2} + 2 x_{4} y_{0} y_{4} + x_{0} y_{2}^{2} + x_{0} y_{1}^{2} + x_{2} y_{4}^{2} + x_{2} y_{3}^{2} + x_{0} y_{0}^{2} + x_{4} y_{2}^{2} \\ + 2 x_{2} y_{2} y_{4} \end{matrix}

\begin{matrix} K_{1} (t) = 4 x_{3} y_{1} y_{3} + 4 x_{4} y_{3} y_{4} + 2 x_{1} y_{1} y_{3} + 4 x_{4} y_{1} y_{4} + 4 x_{2} y_{1} y_{2} + 4 x_{2} y_{2} y_{3} + x_{1} y_{0}^{2} + 2 x_{3} y_{4}^{2} \\ + 2 x_{2} y_{0} y_{1} + 2 x_{1} y_{0} y_{2} + 2 x_{0} y_{1} y_{2} + 2 x_{0} y_{2} y_{3} + 2 x_{4} y_{0} y_{3} + 2 x_{3} y_{0} y_{2} + 2 x_{2} y_{0} y_{3} \\ + 2 x_{3} y_{0} y_{4} + 2 x_{0} y_{3} y_{4} + 2 x_{4} y_{1} y_{2} + 2 x_{1} y_{2} y_{4} + 2 x_{2} y_{1} y_{4} + 4 x_{4} y_{2} y_{3} + 4 x_{2} y_{3} y_{4} \\ + 4 x_{3} y_{2} y_{4} + 2 x_{1} y_{4}^{2} + 2 x_{1} y_{3}^{2} + 2 x_{1} y_{2}^{2} + 2 x_{0} y_{0} y_{1} + x_{3} y_{1}^{2} + 2 x_{3} y_{3}^{2} + 2 x_{3} y_{2}^{2} + 2 x_{1} y_{1}^{2} \end{matrix}

\begin{matrix} K_{2} (t) = 6 x_{3} y_{3} y_{4} + 4 x_{1} y_{1} y_{2} + 6 x_{3} y_{2} y_{3} + 2 x_{2} y_{0} y_{2} + 2 x_{3} y_{0} y_{3} + 2 x_{1} y_{0} y_{1} + x_{2} y_{0}^{2} + 2 x_{2} y_{1}^{2} \\ + 4 x_{1} y_{3} y_{4} + 2 x_{0} y_{2} y_{4} + 2 x_{4} y_{0} y_{2} + 2 x_{2} y_{0} y_{4} + 2 x_{0} y_{1} y_{3} + 2 x_{3} y_{0} y_{1} + 2 x_{1} y_{0} y_{3} \\ + 4 x_{3} y_{1} y_{2} + 4 x_{2} y_{1} y_{3} + 4 x_{1} y_{2} y_{3} + 4 x_{3} y_{1} y_{4} + 4 x_{4} y_{1} y_{3} + 3 x_{4} y_{3}^{2} + x_{0} y_{3}^{2} + 6 x_{4} y_{2} y_{4} \\ + 3 x_{2} y_{2}^{2} + 3 x_{4} y_{4}^{2} + 2 x_{0} y_{0} y_{2} + x_{0} y_{4}^{2} + 2 x_{4} y_{0} y_{4} + 2 x_{1} y_{1} y_{4} + x_{0} y_{2}^{2} + x_{0} y_{1}^{2} + 3 x_{2} y_{4}^{2} \\ + 3 y_{3}^{2} x_{2} + y_{1}^{2} x_{4} + 2 y_{2}^{2} x_{4} + 4 y_{2} y_{4} x_{2} \end{matrix}

\begin{matrix} K_{3} (t) = 4 x_{3} y_{1} y_{3} + 8 x_{4} y_{3} y_{4} + 4 x_{1} y_{1} y_{3} + 4 x_{4} y_{1} y_{4} + 4 x_{2} y_{1} y_{2} + 6 x_{2} y_{2} y_{3} + x_{3} y_{0}^{2} + 4 x_{3} y_{4}^{2} \\ + 2 x_{2} y_{0} y_{1} + 2 x_{1} y_{0} y_{2} + 2 x_{0} y_{1} y_{2} + 2 x_{1} y_{0} y_{4} + 2 x_{0} y_{1} y_{4} + 2 x_{4} y_{0} y_{1} + 2 x_{0} y_{2} y_{3} \\ + 2 x_{4} y_{0} y_{3} + 2 x_{3} y_{0} y_{2} + 2 x_{2} y_{0} y_{3} + 2 x_{3} y_{0} y_{4} + 2 x_{0} y_{3} y_{4} + 4 x_{4} y_{1} y_{2} + 4 x_{1} y_{2} y_{4} \\ + 4 x_{1} y_{2} y_{4} + 4 x_{2} y_{1} y_{4} + 6 x_{4} y_{2} y_{3} + 6 x_{2} y_{3} y_{4} + 6 x_{3} y_{2} y_{4} + 2 x_{1} y_{4}^{2} + 2 x_{1} y_{3}^{2} + 2 x_{1} y_{2}^{2} \\ + 2 x_{0} y_{0} y_{3} + 2 x_{3} y_{1}^{2} + 4 x_{3} y_{3}^{2} + 3 x_{3} y_{2}^{2} + x_{1} y_{1}^{2} \end{matrix}

\begin{matrix} K_{4} (t) = 8 x_{3} y_{4} y_{3} + 2 x_{1} y_{2} y_{1} + 6 x_{3} y_{2} y_{3} + 2 x_{2} y_{0} y_{2} + 2 x_{3} y_{0} y_{3} + x_{4} y_{0}^{2} + x_{2} y_{1}^{2} + 4 x_{1} y_{3} y_{4} \\ + 2 x_{0} y_{2} y_{4} + 2 x_{4} y_{0} y_{2} + 2 x_{2} y_{0} y_{4} + 2 x_{0} y_{1} y_{3} + 2 x_{3} y_{0} y_{1} + 2 x_{1} y_{0} y_{3} + 4 x_{3} y_{1} y_{2} \\ + 4 x_{2} y_{1} y_{3} + 4 x_{1} y_{2} y_{3} + 2 x_{0} y_{0} y_{4} + 4 x_{3} y_{1} y_{4} + 4 x_{4} y_{1} y_{3} + 4 x_{4} y_{3}^{2} + x_{0} y_{3}^{2} + 6 x_{4} y_{2} y_{4} \\ + x_{2} y_{2}^{2} + 5 x_{4} y_{4}^{2} + x_{0} y_{4}^{2} + 2 x_{4} y_{0} y_{4} + 4 x_{1} y_{1} y_{4} + x_{0} y_{2}^{2} + 3 x_{2} y_{4}^{2} + 3 x_{2} y_{3}^{2} + 2 x_{4} y_{1}^{2} \\ + 3 x_{4} y_{2}^{2} + 6 x_{2} y_{2} y_{4} \end{matrix}

\begin{matrix} K_{5} (t) = 4 x_{3} y_{1} y_{3} + 8 x_{4} y_{3} y_{4} + 2 x_{1} y_{1} y_{3} + 4 x_{4} y_{1} y_{4} + 2 x_{2} y_{1} y_{2} + 4 x_{2} y_{2} y_{3} + 4 x_{3} y_{4}^{2} + 2 x_{1} y_{0} y_{4} \\ + 2 x_{0} y_{1} y_{4} + 2 x_{4} y_{0} y_{1} + 2 x_{0} y_{2} y_{3} + 2 x_{4} y_{0} y_{3} + 2 x_{3} y_{0} y_{2} + 2 x_{2} y_{0} y_{3} + 2 x_{3} y_{0} y_{4} \\ + 2 x_{0} y_{3} y_{4} + 4 x_{4} y_{1} y_{2} + 4 x_{1} y_{2} y_{4} + 4 x_{2} y_{1} y_{4} + 6 x_{4} y_{2} y_{3} + 6 x_{2} y_{3} y_{4} + 6 x_{3} y_{2} y_{4} \\ + 2 x_{1} y_{4}^{2} + 2 x_{1} y_{3}^{2} + x_{1} y_{2}^{2} + x_{3} y_{1}^{2} + 3 x_{3} y_{3}^{2} + 2 x_{3} y_{2}^{2} \end{matrix}

Academic Editor: Davood Younesian

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 11272257) and “Aoxiang New Star” of Northwestern Polytechnical University.

References

Meng

. Effects of boundary conditions on a self-excited vibration system with clearance. Int J Nonlin Sci Num 2007; 8: 571–579.

Zhang

. Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method. Nonlinear Dynam 2011; 65: 77–84.

Igusa

Kiureghian

. Response of uncertain systems to stochastic excitation. J Eng Mech: ASCE 1988; 114: 812–832.

Chu

. Experimental observation of nonlinear vibrations in a rub-impact rotor system. J Sound Vib 2005; 283: 621–643.

Hou

Chen

Cao

. Nonlinear vibration phenomenon of an aircraft rub-impact rotor system due to hovering flight. Commun Nonlinear Sci 2014; 19: 286–297.

Shi

Han

. Fixed-point rubbing fault characteristic analysis of a rotor system based on contact theory. Mech Syst Signal Pr 2013; 38: 137–153.

Lahriri

Weber

Santos

. Rotor-stator contact dynamics using a non-ideal drive-theoretical and experimental aspects. J Sound Vib 2012; 331: 4518–4536.

Zhao

. Dynamic characteristics analysis of a rotor-stator system under different rubbing forms. Appl Math Model 2015; 39: 2392–2408.

Cao

Jiang

. Nonlinear dynamic analysis of fractional order rub-impact rotor system. Commun Nonlinear Sci 2011; 16: 1443–1463.

10.

Khanlo

Ghayour

Ziaei-Rad

. Chaotic vibration analysis of rotating, flexible, continuous shaft-disk system with a rub-impact between the disk and the stator. Commun Nonlinear Sci 2011; 16: 566–582.

11.

Fang

Leng

. Chebyshev polynomial approximation for dynamical response problem of random system. J Sound Vib 2003; 266: 198–206.

12.

Gan

Lei

. Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations. J Sound Vib 2011; 330: 2174–2184.

13.

Leng

Meng

Zhang

. Bifurcation and chaos response of a cracked rotor with random disturbance. J Sound Vib 2007; 299: 621–632.

14.

Yang

Jiang

Gao

. Rationalize the irrational and fractional expressions in nonlinear analysis. Mod Phys Lett B 2016; 30: 1650068.

15.

Yang

Gao

Zhang

. Analysis of stochastic bifurcation and chaos in a rub-impact rotor system with random parameter. In: Proceedings of the ASME turbo expo: turbine technical conference and exposition, Düsseldorf, 16–20 June 2014, vol. 7B, p.V07BT30A005. New York: ASME.

16.

Shinozuka

Jan

. Digital simulation of random processes and its applications. J Sound Vib 1972; 25: 111–128.

17.

Iwashige

Ikeda

. Applicability of a higher-order Taylor-series expansion method for random wave simulations. J Nucl Sci Technol 1995; 32: 257–259.

Nonlinear analysis of a rub-impact rotor with random stiffness under random excitation

Abstract

Keywords

Introduction

Mathematical model

Taylor series expansion method

Chebyshev orthogonal polynomial

Numerical results

Methods validation

The effect of random stiffness

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References