1 Introduction

The main purpose of this paper is to present an exhaustive study of complex modes of axially moving Timoshenko beams and corresponding traveling wave phenomena. The dynamics obtained in this study has been verified with the numerical method and compared with the results in the literature.

Axially moving Timoshenko beams are the most widely-used structures in engineering fields, such as band saw blades, steel strips, and power transmission belts. The axially moving velocity induces a lot of practical and theoretical problems. With the increase in the axially moving velocity, a series of bifurcations, from stability to instability due to divergence and then from the second stability to instability due to flutter, have been revealed ^[1-3], which constructed the key routine to the study of the axially moving continua. Advanced investigations of axially moving structures include the nonlinear analysis ^[4-8], parametric excitations ^[9-14], internal resonances ^[15-18], supercritical region phenomena ^[19-21], vibration control ^[22-24] and so on.

In the study of the axially moving beam, if the slender beam is relatively thin compared with its length, the Euler-Bernoulli beam theory has been traditionally used. However, if the beam is thick, the shear deformation and rotary inertia involved in the Timoshenko beam theory should be needed in modeling the axially moving beam. Lee et al.^[3] used the spectral element method to study free transverse vibrations of the moving Timoshenko beam. Ghayesh and Balar^[25] and Ghayesh and Amabili^[17] investigated the effects of shear deformation and rotary inertia on the nonlinear parametric vibration of the axially moving beam. Tang et al. ^[26-27] examined the free and forced vibration of the axially moving Timoshenko beam. Chen et al.^[28] and Yan et al.^{[9, 29]} investigated stability in parametric resonance of the axially accelerating Timoshenko beam. On the other hand, Ding et al.^[30] explored the equilibrium of the moving Timoshenko beam in the super-critical regime.

Complex modes and traveling waves are investigated based on the axially moving Euler-Bernoulli model. However, such research on the Timoshenko beam has not been conducted to the authors' best knowledge. In this study, we focus on the complex modes and corresponding traveling waves in the axially moving Timoshenko beam. The analytical results are verified with the numerical method.

2 Governing equations of motion

A uniform axially moving Timoshenko beam with Young's modulus E, the density ρ, the shearing modulus G, the initial axial tension P₀, the area moment of inertia of the cross-section about the neutral axis I, the cross-sectional area A, the neutral axis coordinate x, the time t, and the distance L between two simple supports, travels with the constant axial speed γ. Assume that the deformation of the Timoshenko beam is confined to the vertical plane. The Timoshenko beam with the in-plane motion is specified by the slope of the deflection curve due to the bending deformation alone φ(x, t) related to the spatial frame, the transverse displacement v(x, t), and the longitudinal displacement u(x, t) related to the coordinate moving at the speed γ. The physical model is given in Fig 1.

The total kinetic energy T for the axially moving Timoshenko beam is

(1)

where a comma preceding to x or t denotes partial derivatives with respect to x or t. The three terms in Eq. (1) represent the kinetic energies associated with the longitudinal axial motion, the transverse motion, and the rotation, respectively. The transverse vibrations play the dominant role since the frequencies of longitudinal vibrations are much higher in small but finite stretching problems in the literature of oscillations. Based on the assumption, the longitudinal displacement is ignored in this study.

From the theory of elasticity, the potential energy U is

(2)

where ε_x is the normal strain component, γ_zx is the shear strain component, h is the height, and z is any height measured from the plane of the neutral fibers. The three terms in Eq. (2) represent the potential energies due to the axial force P=P₀+ηρAγ², the normal stress component σ_x, and the shear stress component τ_zx, respectively. η=1/(1+k_r/(2EA/L)) is the axial support rigidity parameter varying between 0 (infinite rigidity) and 1 (no rigidity)^[31], where k_r is the axial support rigidity. The strain-displacement relation is

(3)

The constitutive equations are as follows:

(4)

where k is the shape factor. The value of k is directly related to the shape of the cross-section.

Hamilton's variational principle for the dynamics takes the form of

(5)

Performing the variation on the energy terms given by Eqs. (1) and (2), and then substituting the results with Eqs. (2) and (4) into Eq. (5), one obtains

(6a)

(6b)

the boundary conditions

(7a)

(7b)

and the initial conditions

(8a)

(8b)

The following variables and parameters in the dimensionless form are introduced:

(9)

where ε is a dimensionless parameter. It is accounting for the fact that the transverse displacement and the slope of the deflection curve are small. k₁ accounts for the effects of the shear deformation, k₂ represents the effects of the rotary inertia, and k_f denotes the stiffness of the beam. The dimensionless coupled governing equations of the axially moving Timoshenko beam are

(10)

(11)

The simple boundary conditions are

(12)

Differentiating Eq. (11) with respect to x, we have

(13)

Solving φ_{, x} from Eq. (13) and substituting the result into Eq. (10) yield

(14)

Similarly, we can obtain

(15)

Equations (14) and (15) are the fourth-order differential equations in the same form, which describe the transverse dynamics and the rotational dynamics of the Timoshenko beam. Since Eqs. (14) and (15) are decoupled, we can investigate the two equations independently.

3 Complex mode approach

In this section, the complex mode approach is performed. The solutions to Eqs. (14) and (15) can be expressed as

(16)

where ϕ_n and ϑ_n are the nth complex mode functions, A_n and B_n denote the degrees of involvement of the corresponding mode, and "c.c." denotes the complex conjugate of all the proceeding terms on the right-hand side. The complex frequency is purely imaginary in the subcritical regime. However, the real part may appear when the axial moving velocity is increased due to the self-excited vibration. It is assumed that λ_n=δ_n+iω_n, where the value of the real part presents the energy dissipation due to damping, and the imaginary part presents the actual frequency.

The mode functions ϕ_n and ϑ_n are real functions for the case of the static Timoshenko beam. If the beam is traveling in the axial direction, the mode functions become complex due to the gyroscopic effect.

Substituting Eq. (16) into Eqs. (14) and (15) leads to

(17)

(18)

The solutions to the ordinary differential equations (17) and (18) can be expressed as

(19)

where A_jn and B_jn (j=1, 2, 3, 4) are complex constants to be determined. Substitution of Eq. (19) into Eq. (17) leads to

(20)

where

(21)

The solutions to Eq. (20) are

(22)

where

(23)

Substitution of Eqs. (16) and (19) into the boundary conditions (12) leads to

(24)

where

(25)

For the non-trivial solution to Eq. (24), the determinant of the coefficient matrix is zero, that is,

(26)

We can determine A_jn(j=1, 2, 3, 4) by considering Eqs. (24) and (26). Similarly, B_jn(j=1, 2, 3, 4) can be obtained by the same procedure. Hence, the nth responses can be calculated as

(27)

where the nth modal functions can be redefined

(28a)

(28b)

and

(29)

Further, the frequencies can be obtained by solving λ_n with (20). Consider the axially moving Timoshenko beam with E=169× 10⁹ Pa, G=66× 10⁹ Pa, ρ =7 850 kg/m³, A=0.133 5× 0.067 412 m², L=0.3 m, k=5/6, and P₀=10⁷ N. The nth energy varying rate δ_n and the nth natural frequency ω_n can be calculated numerically for η =0.5, k₁=71.28, k₂=0.004 2, and k_f=0.8. Figure 2 shows the variation of the first five-order dimensionless energy varying rates and natural frequencies changing with the axial speeds. The lines in Fig. 2 denote the results with the analytical complex mode method, and the discrete symbols denote the results with the numerical method discussed in the next section. The first five linear natural frequencies vary with variation of the axial speeds. It is easily found that the energy varying rates of all orders keep zero while the natural frequencies decrease from γ =0 to γ =3.56, and consequently, the system is stable. Beyond the critical speed γ=3.56, the first natural frequency vanishes, and the first energy varying rate begins to be positive where divergence happens, and the system is unstable about the zero equilibrium. Beyond γ =5.92, it is noticed that the first two natural frequencies coincide, and so do both positive energy varying rate and natural frequency. Then, the flutter phenomenon occurs. The system loses stability for the second time after a short regime of stability. After that, the system becomes more complex and has no stable regions.

The value of A_nA_1n only affects the amplitude of the nth modal functions. In the next analysis, it is equal to 1. Curves in Fig. 3 show the first four-order complex mode functions for η =0.5, k₁=71.28, k₂=0.004 2, k_f=0.8, and γ =1, where lines denote the results of the current analytical method, and symbols denote the results with the numerical method discussed in the next section. It is noticed that the shapes of the complex mode functions are time dependent, which leads to the morphing mode shape during a period of time. The complex mode functions yield traveling waves instead of standing waves for the static beams^[32]. Plots in Fig. 4 illustrate the morphing process of the first four-order mode shapes with the time, which is a typical phenomenon of gyroscopic materials.

4 Differential quadrature scheme

The first four-order complex frequencies and the complex modes can be calculated by solving the coupled motions (10) and (11) with the differential quadrature scheme^[33].

Consider the constrained domain 06 ≤ x ≤ 1 of the Timoshenko beam in the x-direction. For the generalized coordinates v and φ, the numbers of sampling points are N_v and N_φ , respectively. In this study, we choose the sampling points as follows:

(30)

The partial derivatives of the generalized coordinates v and φ at any sampling point x_i as the weighted linear sum of the functions v_i and φ_i at all the sampling points are chosen in the solution domain of the spatial variable. The partial derivatives are described as follows:

(31)

where the weight coefficients are

(32a)

(32b)

and in the case of r=2, 3, …, N_v-1 and N_φ -1,

(33a)

(33b)

Following the above definitions and using

(34)

the differential quadrature analogue of the Timoshenko beam can be obtained as

(35a)

(35b)

(35c)

The matrix form of Eqs. (35a)-(35c) can be written as

(36)

where M, G, and K denote the mass matrix, the gyroscopic matrix, and the stiffness matrix, respectively. Their dimensions are (N_v+N_φ-2) × (N_v+N_φ-2). S presents the generalized displacement matrix, and dimensions are (N_v+N_φ-2)× 1.

In Table 1, the convergence of the differential quadrature method has been verified. It can be concluded that the sampling points N=N_v=N_φ =15 are an appropriate option to guarantee satisfactory results and endurable computer time.

Table 2 shows the comparison of the first five natural frequencies of the beam with the precise boundary (present) and the approximate boundary^{[25, 34]}. The comparison demonstrates that the results with the approximate boundary are slightly larger than the results with the precise boundary. The trend is more obvious with the increasing axial speeds.

In Fig. 2, the symbols denote that they are calculated with the differential quadrature scheme, and the lines denote the first five energy varying rates and natural frequencies calculated with the complex mode approach. It is noticed that the numerical results agree well with the analytical results.

The comparison of mode shapes between the numerical integrations and the analytical results has been shown in Fig 3. The results are in good agreement, which provides mutual verifications of both analytical and numerical methods.

5 Conclusions

In this study, the complex modes and traveling waves in axially moving Timoshenko beams have been investigated analytically, and the results have been verified with the numerical method. The complex modes and corresponding traveling waves emerge in the axially moving material instead of real value modes and standing waves in the classical static structures. The morphing mode shapes have been illustrated to show the gyroscopic feature of the axially moving Timoshenko beam.