1 Introduction

This paper presents a profound study for the nonlinear forced vibration of an axially traveling beam with 3:1 internal resonance. The main target of the present paper is to investigate the effects of the viscoelastic properties and the internal resonance on the steady-state response of the traveling beam. Axially traveling beams play an important role in the design of modern machinery. Therefore, the linear and nonlinear dynamics of the axially traveling beams have drawn many researchers' attention^[1-3]. Marynowski and Kapitaniak^[4] have reviewed the research progress of the dynamics of the axially traveling continua in the past sixty years.

Traditionally, the investigations were focused on the dynamics of the traveling elastic beams. Very recently, researchers have paid their attention on the viscoelastic behaviors of the traveling beam. Yao and Zhang^[5] established the equations of motion for the viscoelastic moving belt by use of the Kelvin-type viscoelastic constitutive law. Ozhan and Pakdemirli^[6-7] and Yang and Zhang^[8] adopted the Kelvin model containing the partial time derivative for describing the viscoelastic behavior of beam materials. To study the nonlinear vibration of the traveling viscoelastic beams, Ding and Zu^[9], Yan et al.^[10], and Tang et al.^[11] proved that the material time derivative should be contained in the Kelvin model. The standard linear solid model has been employed in modeling axially traveling viscoelastic beams. Wang et al.^[12] investigated the effect of an arbitrary varying length on the nonlinear free vibration of an axially translating viscoelastic beam. Chen and his co-workers worked on the steady-state responses of the axially traveling strings^[13] and beams^[14] with the standard linear solid model. Saksa and Jeronen^[15] studied the characteristic behaviors of an axially traveling viscoelastic beam by modeling the viscoelasticity with the Poynting Thomson version of the standard linear solid. In the present paper, the effect of the standard linear solid model on the nonlinear dynamics of the traveling beams with internal resonance is firstly investigated.

Generally, internal resonance can complicate the dynamics of the nonlinear system. Therefore, the nonlinear vibration of the traveling beam under internal resonance has been widely concerned. With the Galerkin method, Ghayesh and Amabili^[16] tuned the first two transverse modes to a 3:1 internal resonance, and studied the coupled longitudinal and transverse vibration of an axially traveling beam. By combining the incremental harmonic balance method with the Galerkin method, Chen and his co-workers discovered the planar motion^[17] and the nonlinear vibration to periodic lateral force excitations^[18] for an axially traveling beam. Wang et al.^[19] considered the multimode dynamics of the inextensional beams on the elastic foundation with 2:1 internal resonance based on a similar process. Since the governing equations are discretized into ordinary differential equations, the research techniques in the above-mentioned literatures are called as the indirect perturbation method. The nonlinear dynamics of the traveling beams have also been studied by the direct perturbation method. By use of the direct multi-scale method, Sahoo et al.^[20] focused on the nonlinear transverse vibration of an axially moving beam subjected to two frequency excitations in the presence of internal resonance. By use of the direct multi-scale method to establish the solvability conditions, Tang et al.^[11] investigated the parametric and 3:1 internal resonance of the axially moving viscoelastic beams on the elastic foundation. Without internal resonance, the multi-scale method has been directly used to solve the nonlinear dynamics of the traveling beams. Forced vibrations are studied for traveling beams via the direct multi-scale analysis^[21]. Liu et al.^[22] derived the analytical expression of the first-order uniform expansion of the solution by directly using the multi-scale method. One thing should be mentioned, by use of the direct multi-scale method to study the nonlinear dynamics of traveling beams, all the above-mentioned studies establish the solvability conditions with complex modal functions. The reason is that it is difficult to obtain the real modal functions for the derived equation of the nonlinear continuous Gyro system.

In the present work, the forced periodic response of an axially traveling viscoelastic beam is investigated in the presence of 3:1 internal resonance. The standard linear solid model and the material time derivative are used to model the transverse vibration of the axially traveling beam. The effects of the viscoelastic property on the periodic response are determined. The approximate real modal functions are established based on the corresponding linear derived equation. Therefore, the direct multi-scale method is firstly used to establish the solvable conditions with the real modal functions. The approximate analytical stable steady-state response is confirmed by the Galerkin method.

2 Mathematical model

Consider a slender axially traveling beam with simply supported boundary conditions (see Fig. 1). The traveling beam is modelled as an Euler-Bernoulli viscoelastic beam. The transverse vibration of the traveling beam with the external harmonic excitation is governed by^[23]

(1)

where t is the time, and x is the neutral axis coordinate along the traveling beam. A comma preceding t or x denotes partial derivatives. u(x, t) presents the transverse displacement. Moreover, L and A are, respectively, the length and the cross-section area of the beam. F=bsin (ωt) denotes the transverse load, where b and ω are the excitation amplitude and the frequency, respectively. P is the initial axial load. σ(x, t) presents the disturbed axial stress. For a slender traveling beam, the bending moment M(x, t) is expressed as follows:

(2)

The standard linear solid model with the material time derivative is used to characterize the viscoelastic property of the traveling beam. Therefore, the stress-strain relation is expressed in a differential form as follows:

(3)

where E₁ and E₂ are the stiffness constants, σ denotes the normal stress due to that bending α represents the dynamic viscosity, and ε_L(x, t) is the Lagrangian axial bending strain expressed by

(4)

The total time derivative is given by

(5)

Therefore, substituting Eqs. (4) and (5) into Eq. (3) yields

(6)

In this paper, only small deflections are considered. Therefore, the displacement-strain relationship can be written as follows:

(7)

Therefore, for the standard linear solid, the viscoelastic material of the beam has the following constitution relationship:

(8)

Due to bending and the bending moment, the normal stresses are, respectively, derived from Eqs. (6) and (8). In the following investigation, the simply supported boundary conditions of the traveling beam are considered as follows:

(9)

Incorporate the following dimensionless variables and parameters:

(10)

where I denotes the area moment of inertial, α denotes the dimensionless viscous coefficient, E_a and E_b are, respectively, represent the effects of the flexural stiffness of the traveling beam, and E_c and E_d are the nonlinear coefficients. The governing equation of the transverse vibration of the traveling beam and the boundary conditions are nondimensionalized as follows:

(11)

(12)

3 Natural frequencies and internal resonance condition

To construct the condition of the internal resonance of the traveling beam, the corresponding linear equation can be adopted to abstract the natural frequencies of the traveling beam. Omitting the nonlinear terms and viscosity terms of Eq. (11) yields

(13)

The solution to the linear derived equation (13) can be expressed as follows:

(14)

where c.c. represents the complex conjugate of all preceding terms on the right-hand side of the equation, and the modal function is assumed as follows:

(15)

Substitute Eqs. (14) and (15) into Eq. (13). Then, multiplying both sides of the obtained equation by sin (mπx) (m=1, 2, …) and integrating it with respect to x from 0 to 1 yield a set of second-order ordinary differential equations. The non-triviality of the solutions to the set of the ordinary differential equations requires its determinant of coefficients to be zero. Therefore, the following equation is obtained:

(16)

where M represents the identity matrix, Z and K are, respectively, and the Coriolis acceleration matrix and the stiffness matrix defined by

(17)

(18)

The natural frequencies ω_n can be obtained from Eq. (16).

In the following study, a V-belt is adopted as the prototype of the traveling beam. Figure 2 shows the shape of the cross-sectional area of the V-belt. Table 1 presents the physical parameters of the V-belt.

Based on Table 1, the dimensionless parameters can be determined based on Eq. (10). Therefore,

If there is no special designation, the traveling beam can be considered with these parameter values in all the following examples. The first four natural frequencies are calculated and shown in Table 2 with the dimensionless axial speed γ=0.598 29. As shown in Table 2, the internal resonance condition ω₂:ω₁=3 is satisfied.

4 Scheme of approximate analytical solution

In this section, the multi-scale method is used to solve the steady-state periodic responses of the axially traveling beam. In this work, only weak external excitation is considered. A non-dimensional bookkeeping parameter is introduced to distinguish the different orders of magnitude. Therefore, the following dimensionless variable and parameters are scaled:

(19)

Then, the perturbation solution can be written as follows:

(20)

where T₀=t, T₂=ε²t, and the following relationships are satisfied:

(21)

Substituting Eqs. (19), (20), and (21) into Eq. (11) and equalizing the coefficients of ε⁰ and ε² in the obtained equations yield

(22)

(23)

The solution of Eq. (22) can be written as follows:

(24)

where A₁(T₂) and A₂(T₂) are undetermined functions, c.c. represents the complex conjugate of the two preceding terms on the right hand of the equation, and

(25)

where k=1, 2. Substitute Eqs. (25) and (24) into Eq. (22). Then, p_{k, r} and p_{k, r} (k=1, 2 and r=1, 2, 3, 4) can be solved by the undetermined coefficient method, and the solution of Eq. (23) can be written as follows:

(26)

where

(27)

Introduce the detuning parameters σ₁ and σ₂. Then, the nearnesses of ω to ω₁ and ω₂ to 3ω₁ can be, respectively, represented by

(28)

Substituting Eqs. (26) and (28) into Eq. (23) yields

(29)

Substituting Eqs. (24), (25), and (27) into Eq. (29), multiplying both sides of the obtained equation by sin (mπx) (m=1, 2, 3, 4), integrating with respect to x from 0 to 1, and then abstracting the coefficients of exp (iω₁T₀) and exp (iω₂T₀) yield the following equation:

(30)

where k=1, 2. K₁₁, K₂₂, K₃₃, and K₄₄ are defined in Eq. (18). H_{k, 1}, H_{k, 2}, H_{k, 3}, and H_{k, 4} are defined by

(31)

where

(32)

Therefore, the following solvability condition is derived:

(33)

The solutions of A₁(T₂) and A₂(T₂) are expressed as follows:

(34)

where a₁(T₂) and a₂(T₂) are, respectively, the amplitudes of the primary resonance responses, and θ₁(T₂) and θ₂(T₂) are the phase angles of the corresponding responses, respectively. Then, the following four equations are obtained:

(35)

where Γ₁₁, Γ₁₂, Γ₁₃, Γ ₂₁, Γ₂₂, Γ₂₃, Γ₂₄, Γ ₂₅, Γ₃₁, Γ₃₂, Γ₄₁, Γ ₄₂, Γ₄₃, Γ₄₄, and Γ₄₅ are constant coefficients, and

(36)

The relationships between the detuning parameters and the modal amplitudes are built as follows:

(37)

where Ξ_mk(m=1, 2, 3, 4, 5, and k=1, 2, …, 8) are constant coefficients. The stability of the modal steady-state response amplitudes is determined by the Lyapunov stability theory.

5 Scheme of Galerkin method

The approximate analytic results are verified by the four-term Galerkin truncation. Therefore, the transverse displacement of the traveling beam is assumed as follows^[24-25]:

(38)

where q_m(t) (m=1, 2, 3, 4) are the generalized coordinates. Substituting Eq. (38) into Eq. (11), multiplying the resulting equation by the weighting function sin (gπx) (g=1, 2, 3, 4), and integrating the product from 0 to 1 yield

(39a)

(39b)

(39c)

(39d)

where

(40)

The set of the second-order ordinary differential equation (39) is numerically calculated by the fourth-order Runge-Kutta method. In the following numerical examples, the time step is set as 0.001^[26-27]. Moreover, the initial conditions are set as follows:

(41)

6 Numerical results

Figure 3 illustrates the comparisons of the steady-state periodic responses by use of the multi-scale method and based on the Galerkin method. Similar comparisons with the weaker flexural stiffness of the traveling beam, i.e., E_b=0.03, are shown in Fig. 4. Both Figs. 3 and 4 clearly depict that the second-order mode resonates when the excitation frequency is close to the first-order natural frequency. Therefore, the energy transfers from the second-order mode to the first-order mode. Figures 3 and 4 also demonstrate that the approximate analytical results and the numerical results are almost overlapping. Specially, the amplitude curves of resonance at the first-order mode based on the two approaches overlap completely. Compared with the results without the internal resonance in Refs. ^[10] and ^[21], the amplitude-frequency curves are more complicate in the presence of the internal resonance, specially when the flexural stiffness of the beam is relatively soft.

Figure 5 shows the effects of the amplitude of the excitation on the primary resonance response with the internal resonance. The numerical results show the hysteresis phenomenon of the axially traveling beam. Under the conditions of 3:1 internal resonance between the first two modes and the first-and second-order mode resonates, the amplitude of the resonance response changes with the excitation amplitude. Moreover, the saturated phenomenon is observed in Fig. 5(b). This is an interesting phenomenon since the saturation of the second-order mode happens at the first-order primary resonance.

The effects of the system parameters on the amplitude-frequency curves are presented in Figs. 6, 7, and 8. The results indicate that the amplitudes of the resonance of the first two modes are very sensitive to the system parameters. From Fig. 6, one can easily find that the amplitude-frequency curves bend to the right with the increase in the nonlinear coefficient. The simulations shown in Figs. 7 and 8 depict that the amplitude of the resonance with the internal resonance increases with the decreases in the flexural stiffness and the viscous coefficient of the beam. Moreover, the nonlinear dynamics of the axially traveling beam becomes more complicate when the flexural stiffness and viscous coefficient of the beam become smaller.

7 Conclusions

The purpose of this paper is to investigate the effects of the internal resonance and the viscoelastic behaviors on the primary response of the traveling viscoelastic beam. The standard linear solid model with the material time derivative is adopted to describe the viscoelastic properties of the axially traveling beam. The real modal functions of the linear derived equation are approximately established. Usually, the linear derived system of the nonlinear governing equation is solved by complex modal functions. The steady-state response amplitudes are firstly determined by the direct multi-scale method with the real modal functions. Then, the Galerkin method is used to verify the approximate solutions. The verification of the Galerkin method shows that the approximate analytical results are acceptable. Therefore, the direct multi-scale method with approximate real modal functions is a reliable approach for studying the nonlinear vibration of the traveling beam. The steady-state responses for the internal resonance modes show that the effect of the internal resonance cannot be ignored.