Nomenclature
L,	length of the beam between two ends;	X,	distance from the left end of the beam;
A,	cross-sectional area;	T,	time coordinate;
ρ,	beam density;	v(x, t),	transverse vibration displacement;
φ (x, t),	angle of rotation of the cross-section;	KL,	spring stiffness coefficient of the vertical
Γ,	axially constant speed;		elastic support at the left end;
P0,	initial axial tension;	KR,	spring stiffness coefficient of the vertical
E,	Young's modulus of a uniform moving		elastic support at the right end;
	beam;	Kt1	torsion spring stiffness coefficient at the
J,	moment of inertial of a uniform moving		left end;
	beam;	Kt2,	torsion spring stiffness coefficient at the
K,	shape factor;		right end;
G,	shearing modulus;	Uf,	bending strain energy;
T,	kinetic energy;	Us,	shear strain energy.

1 Introduction

The objective of the present study is to examine the effects of the generalized boundary conditions on a moving beam. Axially moving continua are common constituent elements in many mechanical systems^[1-9]. One significant problem in these axially moving systems is the occurrence of significant and unwanted resonance. In order to determine when the resonance will occur and the intensity of the resonance, the dynamics of axially moving beams has been widely studied for many decades^[10].

If thick and short axially moving systems are considered, the Timoshenko beam theory is necessary for more accurate results. Lee et al.^[11] studied the transverse vibration and stability of a moving Timoshenko beam. Tang et al.^[12] studied nonlinear vibrations of axially moving Timoshenko beams under excitations. Ghayesh and Amabili^[13] considered the nonlinear dynamics of moving Timoshenko beams with an intermediate spring support. An and Su^[14] studied the forced vibration of axially moving Timoshenko beams with clamped-clamped and simply-supported boundary conditions. Yan et al.^[15] studied the chaotic dynamics of an axially moving viscoelastic Timoshenko beam under double excitations. Yesilce^[16] studied the free vibration characteristics of moving Timoshenko beams with different boundary conditions. All of the above-mentioned works on axially moving Timoshenko beams adopted classical boundary conditions. Therefore, the axially moving beam is either pinned or clamped at the ends. Li et al.^[17] investigated the nonlinear free transverse vibration of moving Timoshenko beams with two free ends. Ding and Chen^[18] studied the vibration characteristics of the axially moving Euler-Bernoulli (EB) beam with one kind of hybrid supports, which can unify the pinned and clamped ends. In this paper, these three classical boundary conditions, i.e., free, pinned, and clamped ends, for the moving beam will be unified by using generalized boundary conditions. Generalized boundary conditions are adopted for studying the vibration characteristics of axially moving beams for the first time by using either an EB beam or a Timoshenko beam model.

It is well known that the dynamic stiffness matrix method is a very accurate solution method for the analysis of the free vibration of structures^[19-24]. Considering static beam structures without the axial speed, the dynamic stiffness matrix of the Timoshenko beam model has been developed by several researchers. The dynamic stiffness matrix has been determined for non-uniform Timoshenko beams^[25], the three-beam system^[26], the beam-column on an elastic foundation^[27], laminated beams^[28], and bi-directional functionally graded Timoshenko beams^[29]. It is well-recognized that the dynamic characteristics of axially moving systems are markedly influenced by the axial speed^[10]. The dynamic stiffness matrix was presented for the free transverse vibration of an axially moving EB beam^[30] or Timoshenko beam^[31] with classical boundary conditions. However, in practical engineering applications, the boundary of an axially moving continuum is less likely to strictly satisfy those classical constraints, such as strictly simply-supported, or completely fixed. More likely the boundary conditions of the moving continuum will be somewhere between these strict constraints.

In the present work, the hybrid constraint is introduced to model the generalized boundary conditions of axially moving materials. This generalized boundary can represent non-strict constraints. Moreover, this proposed constraint can replace various classical boundary conditions. Dynamic stiffness matrices are derived for obtaining natural frequencies of the transverse vibration of the axially moving Timoshenko beams and EB beams. The demand for the Timoshenko theory for the axially moving continua is examined by comparing the results with those of the EB beam theory.

2 Mathematical models

The mechanic model of an axially moving beam is shown in Fig. 1. L is the length of the beam between two ends. v(x, t) is the transverse vibration displacement, where x and t stand for the distance from the left end of the beam and the time coordinate, respectively. Γ and P₀ are the axially constant speed and initial axial tension of the beam, respectively. In order to describe the more general constraint support boundary conditions, the ends of the beam are supported by vertical springs and constrained by torsion springs. K_L and K_R are the spring stiffness coefficients of the vertical elastic support at the left and right ends of the beam, respectively. K_t1 and K_t2 are the torsion spring stiffness coefficients at the ends. By including the effects of rotary inertia and the shear force on the transverse vibration, the Timoshenko theory is applied to the in-plane vibration of the beam. In order to discuss the natural frequencies of free vibration, external excitation and damping are ignored.

The bending strain energy U_f of the axially moving beam can be expressed as

(1)

where the comma preceding a variable stands for partial differentiation. E and J are, respectively, the Young's modulus and the moment of inertial of a uniform moving beam. φ (x, t) is the angle of rotation of the cross-section due to the bending moment. The shear strain energy U_s is described as

(2)

where κ is the shape factor. A and G, respectively, stand for the cross-sectional area and the shearing modulus of the beam. The kinetic energy T for the axially moving beam is written as

(3)

where ρ is the beam density. The generalized Hamilton principle takes the form of^[31-32]

(4)

Performing the variation on these energy terms yields

(5)

The first term of Eq. (5) is expressed as

(6)

The fourth term of Eq. (5) can be expressed as

(7)

Therefore, the transverse vibration of the axially moving Timoshenko beam with the small but finite stretching assumption is governed by^[11]

(8)

with the following boundary conditions:

(9)

Letting K_t1=0 or K_t2=0 with K_L → ∞ or K_R → ∞ yields a pinned end of the axially moving beam. Assuming K_t1 → ∞ or K_t2 → ∞ with K_L → ∞ or K_R → ∞, a clamped end is obtained. If K_t1=0 or K_t2=0 with K_L=0 or K_R=0, a free end is modeled. Therefore, the generalized boundary conditions at both ends are characterized by Eq. (9). More specifically, Eq. (9) can represent all classical boundary conditions, such as simply-supported boundary conditions, fixed ends, the boundary conditions of the cantilever beam, and free boundary conditions at both ends. When the right end of the beam has an elastic support, as shown in Fig. 1, the total tension of the axially moving beam is expressed as the sum of the initial axial force and the force depending on the velocity ηρAΓ², where the constant η can be derived from the static equilibrium position of the beam and support, the beam dynamic displacement, the accompanying beam displacement, and the support displacement. Mote^[33] gave the derivation of the total tension in detail. This paper does not repeat the process but uses the derived results, that is to say, the total tension of the expression is

(10)

where P₀ is the initial axial force, and K_s is the support stiffness coefficient. The constant η is defined as

(11)

Then, the governing equation (8) is replaced by

(12)

Therefore, η is the support rigidity parameter varying between 0 (infinite rigidity) and 1 (no rigidity)^[35]. To avoid round-off due to manipulations with small or large numbers in the following analysis, the following dimensionless variables and parameters are introduced^[34-35]:

(13)

The dimensionless governing equation and generalized boundary conditions are expressed as

(14)

(15)

3 Dynamic stiffness matrix

Assuming simple harmonic vibration of the axially moving Timoshenko beam, the solutions of the dimensionless governing equation (14) are written as^[36-38]

(16)

where ω denotes the natural frequency, and V(ξ) and ψ (ξ) are the natural modes of the transverse vibration. Substituting Eq. (16) into Eq. (14) yields

(17)

where . Substituting the first equation of Eq. (17) into the second one yields

(18)

Assuming V( ξ)=e^{iλ ξ} yields

(19)

where λ_j (j=1, 2, 3, 4) are solved from Eq. (19). Therefore, the solutions of Eq. (17) are expressed as

(20)

Substituting Eq. (20) into Eq. (17) gives

(21)

(22)

Substitution of Eq. (20) into the generalized support boundary conditions (15) yields

(23)

Equation (23) can be written in the following matrix form:

(24)

One can derive the following equation:

(25)

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

(26)

By applying the algorithm of Wittrick and Williams in Ref. [26], the natural frequencies and modes of the moving beam are obtained by setting the determinant of the dynamic stiffness matrix K to be zero.

4 Natural frequencies and modes

Unless otherwise specified, the values of the geometric and physical parameters are set as A=b× h=0.015 5× 0.018 m², L=0.15 m, E=200 MPa, G=68× 10⁶ Pa, ρ =1 200 kg/m³, κ =5/6, and P₀=80 N, where b and h are the width and height of the cross-section of the beam, respectively. Based on Eq. (11), the values of dimensionless parameters are calculated as k₁=197.625, k₂=0.001 2, and k_f=0.914 9.

Figure 2 shows the effects of the axial speed of the moving Timoshenko beam with k_t1= k_t2=0, and k_L=k_R=Inf., where "Inf." denotes an infinite. Figure 2 clearly demonstrates that the natural frequencies of transverse vibration of the moving beam decrease with the increasing axial speed. Furthermore, the finite support at the right end significantly affects the frequencies. Specially, a finite value of the support spring stiffness coefficients K_s can increase substantially the critical speed of the moving Timoshenko beam.

The influence of the torsional spring stiffness at both ends of the moving beam on the natural frequencies is presented in Fig. 3 with η =0, and k_L= k_R=1 000. k_t1= k_t2=0 denotes the pinned-pinned boundary conditions. k_t1= k_t2=Inf. stands for the torsional spring stiffness tending to be infinite. The first four frequencies are all increased with an increasing torsional stiffness at the ends of the beam.

The effects of the torsional spring stiffness at the right end of the axially moving Timoshenko beam on the natural modes are detailed in Fig. 4 with η =0, k_L= k_R=1 000, k_t1=5, and the axial speed γ =1. The natural mode functions are normalized. The natural vibration modes of the symmetrical constraint show an obvious asymmetric feature. Therefore, the axial speed of the beam affects the natural vibration modes by causing asymmetry.

Figure 5 shows the effects of the vertical support spring stiffness on the frequencies of transverse vibration of the moving Timoshenko beam with k_t1= k_t2=0 and η =0. The dot-dash line stands for k_L= k_R=Inf., and means that the moving beam is simply-supported at both ends. Figure 5 demonstrates that the vertical support spring does not change the critical speed of the axially moving beam. The critical speed is analytically derived in Section 7.

Figure 6 presents the effects of the vertical support stiffness on the natural vibration modes of the axially moving Timoshenko beam with k_t1= k_t2=0, η =0, k_L=1 000, and γ =1. The effects of the vertical support spring stiffness on the first order mode are non-significant. For higher order modes, Fig. 6 also shows the significant influence of the stiffness of the vertical spring on vibration modes.

5 Numerical verification

The condition of Eq. (24) to have a nonzero solution is that the determinant of the matrix H is to be zero. Therefore, the following equation can be derived:

(27)

The natural frequencies of the moving beam are numerically determined from Eqs. (27) and (19). By setting k_t1= k_t2=0 and k_L= k_R=Inf., Fig. 7 describes the comparisons of the results of the dynamic stiffness method and numerical calculations. The comparisons clearly show that the frequencies calculated by the dynamic stiffness matrix and the numerical approach almost coincide.

6 Comparisons with EB model

Assuming that the influences of the shear force and the angle of rotation of the cross-section of the moving beam on the transverse vibration could be neglected, the governing equation and generalized boundary conditions of the axially moving EB beam model are derived,

(28)

With k_L= k_R=1 000 and η =0, the natural frequencies calculated from the two kinds of the beam theory are compared in Fig. 8, where ω_EB and ω _TB, respectively, denote the natural frequencies calculated from the EB beam model and the Timoshenko beam model. The relative difference in Figs. 8(c) and 8(d) is defined as (ω_EB-ω_TB)/ω_EB. The results in Fig. 8 clearly show that the differences are more remarkable for the higher axial speed. Moreover, the effects of the torsion spring stiffness on the differences of the two beam models are also presented in Fig. 8. The differences of the natural frequencies based on the two beam models become clearer for a stronger torsional spring, which means that the Timoshenko beam theory will be more appropriate when the system has a larger value of torsion spring stiffness.

The effects of the vertical spring stiffness on the difference of the two beam theories are illustrated in Fig. 9 with k_t1= k_t2=0 and η =0. The difference between the two beam theories increases with the increasing speed. Figure 9(a) also demonstrates that the vertical spring does not change the critical speed of the axially moving beam.

7 Critical speed

In order to ascertain the effect of the vertical spring on the critical speed of the moving beam, the critical speed is derived. The static equations corresponding to the governing equation (15) are

(29)

In order to be able to briefly describe the effect of the vertical support spring, k_t1=k_t2=0 in this section. The generalized boundary conditions of the moving Timoshenko beam are replaced by

(30)

The solution of the second equation of Eq. (30) is written as

(31)

where T_k1 and T_k2 are the coefficients to be determined. Substituting Eq. (31) into Eq. (29) yields

(32)

(33)

Substituting Eqs. (31)-(33) into the first equation of the static equation (29) yields

(34)

Solving Eq. (34) yields the critical speed of the axially moving Timoshenko beams as

(35)

Equation (35) clearly shows that the critical speed of the moving Timoshenko beams with the generalized boundary conditions is not affected by the vertical support spring.

8 Conclusions

The present study focuses on the dynamic stiffness matrix of an axially moving beam with generalized boundary conditions. The classical pinned, clamped, and free boundary conditions can be found as special cases. The natural frequencies and modes of transverse vibration of the moving Timoshenko beam and EB beam are, respectively, determined by utilizing the dynamic stiffness method. These vibration frequencies are further confirmed by a numerical calculation. The influence of the torsion spring and the vertical spring at the ends of the moving beam is determined. The examples demonstrate the significant influence of the axial speed, the torsional spring, and the vertical support stiffness on natural vibration frequencies and modes. The results interestingly show that the critical speed of the axially moving beam is not changed with the stiffness of the vertical spring. Natural frequencies of the moving Timoshenko beam model are compared with those of the EB beam model. Moreover, the effects of the constraint spring and axial speed on the difference between the two kinds of beam model are studied. Thus, we get an interesting conclusion, i.e., the weaker elastic constraints and greater axially moving speed will make the Timoshenko beam theory more needed.