1 Introduction

Axially moving beams are important engineering elements, e.g., band saw, belt, and crane hoist cable. Due to their wide applications, they have been studied early^[1-3]. Based on the Euler beam theory, Mote^[1] contributed studies of band as an axially moving elastic beam model with the effects of tension and natural frequency. Öz and Pakdemirli^[4] considered that velocity was time-dependent, and investigated the stability situation. They used the complex mode method to obtain the frequency and mode function, and gave the explicit formulation of the mode shapes. Chen and his coworkers contributed a series of studies on parametric stability with the consideration of the effects of viscoelastic material in beam models. Chen and Yang^[5] used directly the multi-scale method to determine the stability formula of beams in simplysupported and clamped boundaries. Yang and Chen^[6] introduced the integral type viscoelastic material into the beam, and used the multi-scale method to investigate the stability of axially accelerating beams. Chen and Yang^[7] used the multi-scale method to investigate the stability of axially moving beams by assuming the simply-supported boundary condition as rotational spring. Chen and Zu^[8] contributed the method of solvability condition in studying axially moving beams via the multi-scale method. Ding and Chen^[9] used the finite difference method to verify the multi-scale analysis for the stability in the principle parametric resonance. Chen and Wang^[10] used the governing equations to investigate the stability by using the differential quadrature method (DQM). Wang and Chen^[11] introduced the three-parameter viscoelastic model into axially moving beams. Wang^[12] investigated the force vibration of axially moving beams based on the three-parameter viscoelastic model. Ding et al.^[13] studied 3:1 internal resonance. Chen and Tang^[14] constructed the equation of motion with energy method, and considered the varying tension. Ding and Chen^[15] focused on the natural frequency of axially moving high speed beams via the Galerkin method. Ghayesh^[16] used the Galerkin method to couple longitudinal and transverse dynamics. Ding et al.^[17] studied the forced vibration of axially moving beams via multi-scale analysis with finite difference confirmation.

Based on the Rayleigh beam theory, Ghayesh and Balar^[18] considered the effect of rotary inertia, and investigated the nonlinear problem on the parametric vibration and stability of axially moving beams with given nonlinear frequencies. Ghayesh and Khadem^[19] constructed the equation of motion of axially moving beams by introducing the rotary inertia and temperature effects to investigate the steady-state response and stability. Chang et al.^[20] used the finite element method to discretize the governing equation of axially moving Rayleigh beams, and studied the vibration and stability.

Based on the Timoshenko beam theory, Lee et al.^[21] used the spectral element method to treat the stability of axially moving beams. Ghayesh and Balar^[22] promoted two dynamic models of axially moving Timoshenko beams, and analyzed the nonlinear parametric vibration and stability. They gave the valid way to calculate the natural frequencies with the complex mode method. Tang et al.^[23] considered the varying tension to construct the axially moving Timoshenko beams with natural frequencies, and investigated the stability. Ghayesh and Amabili^[24] used the Galerking method to analyze the chaotic oscillation of axially moving threedimensional Timoshenko beams. Yan et al.^[25] investigated the dynamic behaviors of axially moving viscoelastic Timoshenko beams with the Galerkin method. The present study focuses on the stability of axially accelerating beams with rotary inertia with the Rayleigh beam model.

In subsequent sections, the extended Hamilton's principle will be used to derive the governing equation of axially accelerating Rayleigh beams with simply-supported boundary conditions. In Section 3, it is of interest to find the approximate solution of the governing equation via the method of multiple scales (MMS), and the process of calculating the natural frequencies is presented. In Section 4, the summation parametric resonance and principle parametric resonance are investigated, the solvability condition is given by using the orthogonal relationships, and the stability boundary is derived. Numerical demonstrations show the effects of various system variables, e.g., viscosity coefficients, mean speed, beam stiffness, and rotary inertia factor in both the summation parametric resonance and principle parametric resonance. In Section 5, the DQM is used to discretize the governing equation, and the numerical results obtained by the DQM are compared with the analytical results obtained by the MMS.

2 Equations of motion

A Rayleigh beam travels with the axial speed γ between two motionless ends separated by the distance l. The axial tension and angle of rotation due to bending are denoted by P and ϕ(x, t), respectively. The transverse and longitudinal displacements are expressed by u(x, t) and v(x, t), respectively. The potential energy and total kinetic energy are given^[14], respectively, as follows:

(1)

(2)

where a comma preceding x or t indicates a partial derivative with respect to x or t. l is the length of the beam, A is the cross area, ρ is the density, I is the moment of inertia, and

The virtual work W due to deformation is given by

(3)

where h is the height, z is any height measured from the plane of the neutral fibers, σ_x is the normal stress component, and ε_x is the normal strain component. The stran-displacement relation is given by

(4)

and the Kelvin constitutive relation is introduced to describe the viscoelastic material of beams with the elastic modulus E and the viscosity α as follows:

(5)

The extended Hamilton's principle is expressed by

(6)

Then, substituting Eqs. (1)–(3) into Eq. (6) gives

(7)

where

(8)

In practice, the longitudinal deflection u(x, t) is much smaller than the transverse deflection v(x, t). Therefore, neglecting the longitudinal deflection u(x, t) leads to the governing equation of the Rayleigh beam from Eq. (7) as follows:

(9a)

(9b)

The simply-supported boundary conditions are selected from Eq. (7) as follows:

(10)

The tension P is given by^[14]

(11)

Introduce the following dimensionless parameters:

(12)

Then, substituting Eq. (12) into Eqs. (9)–(11) yields

(13)

and the boundary conditions

(14)

3 Asymptotic analysis

Suppose that the axial speed is characterized as the simple harmonic variation about the constant mean speed as follows:

(15)

where ω is the axial speed variation frequency, γ₀ is the mean speed, and γ₁ is the disturbed amplitude.

The MMS is used to look for the asymptotic solutions of the following form:

(16)

where T₀ = t, and T₁ = εt.

(17)

Substituting Eqs. (15)–(17) into Eqs. (13) and (14) and equating the coefficients of powers of ε to zero, we arrive at the orders ε⁰ and ε¹.

(ⅰ) ε⁰

(18)

(19)

(ⅱ) ε¹

(20)

(21)

In the above equations, the operators M, G, and K are denoted as follows:

(22)

The assumed solution to Eq. (18) has been given as follows:

(23)

where c.c. represents the complex conjugate of all preceding terms, ϕ_n(x) is the mode shape, and A_n(T₁) is the complex-valued function including the time term. Substituting Eq. (23) into Eq. (18) yields

(24)

where the prime indicates the derivation with respect to x. The solution to Eq. (24) is given by

(25)

Substituting Eq. (25) into Eq. (24) yields

(26)

where j = 1, 2, 3, 4. Meanwhile, Eq. (23) should satisfy Eq. (19). Substituting Eqs. (23) and (25) into Eq. (19) yields^[22-23]

(27)

where

(28)

If Eq. (27) has the non-trivial solution, the determinant of the coefficient matrix should be zero. It then can be rewritten as follows:

(29)

Correspondingly, the nth modal function is given as follows^[4]:

(30)

4 Parametric resonance

To investigate the summation parametric resonance, we define a detuning parameter σ to indicate the nearness of the variation frequency ω to the sum of the natural frequencies ω_n and ω_m, i.e.,

(31)

For summation parametric resonance, we rewrite Eq. (23) as follows:

(32)

Substituting Eqs. (31) and (32) into Eq. (20) yields

(33)

where the dot indicates differentiation with respect to T₁, and N_ST represents the term that will not bring secular terms. The solvability condition should demand the following orthogonal relationships^[16]:

(34a)

(34b)

The application of the distributive law of the inner product to Eq. (34) yields^[8]

(35)

where r, s = m, n, and

(36a)

(36b)

Equation (36) is independent of the viscous damping α and the disturbed amplitude γ₁. It is numerically demonstrated that b_r and c_rs are positive real number and complex number, respectively. Apparently, Eq. (35) has zero solutions. We will determine the stability of the zero solutions of Eq. (35), and introduce the complex-valued function A_n(T₁) into the following real and imaginary parts:

(37)

where p_i and q_i (i = 1, 2) are real functions with respect to T₁.

Substitute Eq. (37) into Eq. (35). Then, the resulting equation can be separated into real and imaginary parts, and we have

(38)

where a comma preceding T₁ indicates a partial derivative with respect to T₁. Evaluating the eigenvalues of the coefficient matrix of the left-hand side of Eq. (38), we have

(39)

where

(40a)

(40b)

(40c)

(40d)

The stability condition of the zero solutions to Eq. (39) is given by the Routh-Hurwitz criterion as follows:

(41)

where

(42a)

(42b)

(42c)

Rewrite Eq. (42) with Eq. (41). Then, the stability condition can be expressed by

(43)

where

(44)

(45)

(46)

The numerical examples of stability conditions for summation parametric resonance are given by Eq. (43). In the σγ₁-plane, the regions inside and outside the boundary of the system is unstable and stable, respectively. The physical parameters of the beam are

The constant mean speed of the beam is γ₀ = 21.081 9 m·s⁻¹. The corresponding dimensionless values are

Figure 1 shows the effects of the viscosity on the stability boundary for the summation parametric resonance at the first and second modes. We specify the system variables of the axially accelerating viscoelastic Rayleigh beam with

The dotted line denotes α = 0.000 2, the dot dashed line denotes α = 0.000 4, and the solid line denotes α = 0.000 0. Obviously, it is illustrated that smaller instability region is obtained with the increase in the viscosity coefficient.

Figure 2 shows the effects of the mean speed on the stability boundary for the summation parametric resonance of the first and second modes. We specify the system variables

The dotted line denotes γ₀ = 1.5, the dot dashed line denotes γ₀ = 2, and the solid line denotes γ₀ = 1. When σ is nearby zero, it is illustrated that smaller instability region is obtained with the increase in the mean speed. Here, the same trend takes place with various viscosity coefficients. But the trend is reversed after arbitrary two boundaries cross σ.

Figure 3 shows the effects of the beam stiffness on the stability boundary for the summation parametric resonance of the first and second modes. We specify the system variables as follows:

The dotted line denotes v_f = 0.7, the dot dashed line denotes v_f = 0.8, and the solid line denotes v_f = 0.6. When σ is nearby zero, it is illustrated that larger instability region is obtained with the increase in the stiffness. But the trend is reversed after arbitrary two boundaries cross σ.

Figure 4 shows the effects of the rotary inertia factor on the stability boundary for the summation parametric resonance of the first and second modes. We specify the system variables as follows:

The dotted line denotes k₃ = 0.006, the dot dashed line denotes k₃ = 0.008, and the solid line denotes k₃ = 0.004. When σ is nearby zero, it is illustrated that larger instability region is obtained when the beam stiffness increases. But the trend is reversed after arbitrary two boundaries cross σ. This phenomenon is the same as the effects of the beam stiffness.

If we investigate the principle parametric resonance, ω can be expressed by

(47)

The stability boundary (35) can be easily revised by specifying m = n = r. Then, according to the Routh-Hurwitz criterion, the stability boundary of the rth order principal parametric resonance can be obtained by

(48)

where r = m, n, and

(49)

(50)

Numerical examples of stability conditions for principle parametric resonance are given by Eq. (48). In the σγ₁-plane, the regions inside and outside the boundary for the system is unstable and stable, respectively.

Figure 5 shows the effects of the viscosity. We specify the system variables of an axially accelerating viscoelastic Rayleigh beam with

The dotted line denotes α = 0.000 2, the dot dashed line denotes α = 0.000 4, and the solid line denotes α = 0.000 0. It is illustrated that smaller instability region is obtained with the increase in the viscosity coefficient in the first two modes. There are the same tendencies with the summation parametric resonance.

Figure 6 shows the effects of the mean speed. We specify the system variables as follows:

The dotted line denotes γ₀ = 1.5, the dot dashed line denotes γ₀ = 2, and the solid line denotes γ₀ = 1. It is illustrated that smaller instability region is obtained with the increase in the mean speed of the first two modes.

Figure 7 shows the effects of the rotary inertia factor. We specify the system variables as follows:

The dotted line denotes k₃ = 0.006, the dot dashed line denotes k₃ = 0.008, and the solid line denotes k₃ = 0.004. It is illustrated that smaller instability region is obtained with the increase in the stiffness.

Figure 8 shows the effects of the beam stiffness. We specify the system variables as follows:

The dotted line denotes v_f = 0.7, the dot dashed line denotes v_f = 0.8, and the solid line denotes v_f = 0.6. It is illustrated that larger instability region is obtained with the increase in the beam stiffness in the first two modes. There are reverse tendencies with the viscoelastic coefficients, mean speed, and rotary inertia factor.

5 DQM

The DQM^[26-28] is used to discretize the governing equation of the axially accelerating beam (13) and the boundary condition (14), where the partial derivatives of a function to a space variable at sampling points are approximately the sums of the weighted functions at the sampling points. The derivatives of the transverse deflection v(x, t) can be expressed as

(51)

where the rth-order weighting coefficients are described by

(52)

and the recurrence relationship is

(53)

(54)

The beam is discretized in the domain of x (0≤x≤1). Here, the δ-technique is used to describe unequally the spaced sampling points as follows:

(55)

Substituting Eq. (51) into Eq. (13) yields

(56)

where δ_ij is Kronecker's delta, and the discretized boundary conditions in Eq. (14) are expressed as follows:

(57)

To investigate the stability boundaries, numerical integration is used to solve the ordinary differential equations (56) and (57) for the transverse deflection v(x_i, t) first, in which δ = 10⁻⁵ and N = 13. We specify the system variables of an axially accelerating viscoelastic Rayleigh beam with η = 0.5, v_f = 0.8, α = 0.000 2, and γ₀ = 2. Figure 9 illustrates the numerical results of the calculation for the stability boundary in the first two principal parametric resonance by the DQM, where the results are compared with the foregoing analytical results by the MMS. The solid and dotted lines represent the stability boundary in the σγ₁-plane calculated by the DQM and the MMS, respectively.

6 Conclusions

The stability of axially accelerating beams in the summation parametric resonance and principle parametric resonance is investigated based on the Rayleigh beam theory. The MMS is used to determine the amplitude frequency equation. The stability boundary is obtained. The DQM is used to verify the validity of the stability boundary. Several conclusions are obtained as follows:

(ⅰ) For summation parametric stability, increasing the viscosity coefficient and the mean speed leads to the decrease in the instability region. However, increasing the beam stiffness and the rotary inertia factor leads to the increase in the instability region. But the trend is reversed after arbitrary two boundaries cross due to the effects of the mean speed, the beam stiffness, and the rotary inertia factor.

(ⅱ) For the principle parametric stability in the first two modes, increasing the viscosity coefficient, the rotary inertia factor, and the mean speed leads to the decrease in the instability region. However, increasing the beam stiffness leads to the increase in the instability region. The same trend occurs for both the first two modes.

(ⅲ) It is illustrated that the stability boundary curves obtained by the MMS argue the results obtained by the DQM.