1 Introduction

Axially moving beams can represent many engineering devices such as power transmission belts, magnetic tapes, paper sheets, and textile fibers. Many of them involve the foundation effect. For example, in the galvanizing line, to restrict the transverse vibration which leads to poor surface quality, moving steel strips are supported by an air layer from a sequence of jets along the axis of the sheet, and the magnetic tape is supported by the air bearing to lubricate relative motion between the tape and the recording head. Therefore, the vibration of axially moving continua with foundation urgently needs to be understood.

Bhat et al. ^[1] examined the nonlinear free response of a translating string completely supported along its length by a uniform linear elastic foundation. Perkins ^[2] obtained the closed form of free and forced response of a string translating across elastic foundation. Wickert ^[3] determined the natural frequencies and the eigenfunctions for the string traveling on a complete uniform elastic foundation. The closed form expressions of the responses to arbitrary excitation and initial conditions were obtained. Xiong and Hutton ^[4] regarded the translating strings as a special case of rotating circular string constrained by pointed or distributed springs. Tan and Zhang ^[5] and Riedel and Tan ^[6] obtained the exact free and harmonic end excited response of an axially moving string constrained by an intermediate elastic support or an elastic foundation. Saeed and Festroni ^[7] determined the natural frequencies and the mode shapes of an axially moving string constrained by discrete springs. Parker ^[8] investigated the stability of an axially moving string constrained by a discrete or distributed elastic foundation. Jha and Parker ^[9] considered spatial discretization of an axially moving string on the completely distributed elastic foundation. By adopting the modal analysis, Kartik and Wickert ^[10] examined the steady-state-forced vibration of a moving medium that was guided by a partial elastic foundation. Yurddas et al. ^[11] investigated nonlinear vibrations of an axially moving string with nonideal multi-support conditions. Yang et al. ^[12] investigated the free vibration of an axially moving elastic beam with simple supports resting on elastic foundation. Ba\v{g}atli et al. ^[13-14] examined the nonlinear vibration of the accelerating beam with single-spring support and multi-spring support. Kural and Özkaya ^[15] investigated the transverse vibrations of an axially accelerating flexible beam resting on multiple supports. Ghayesh ^[16] derived the stability and bifurcation of the axially moving beam with an intermediate spring support in one dimension.

It should be noted that in all the above studies, the viscoelastic foundation was not accounted for the vibration of axially moving continua. Nevertheless, in some industrial processes, the effect of the viscoelastic foundation cannot be neglected easily. Yang et al. ^[17] investigated the dynamic response of finite Timoshenko beams resting on a viscoelastic foundation subjected to a moving load. Sun ^[18] solved the problem of the steady state response of a beam on a viscoelastic foundation subjected to a harmonic line load. Ghayesh ^[19] investigated the transversal nonlinear vibration of an axially moving viscoelastic string supported by a partial viscoelastic guide. Ahmadian et al. ^[20] considered the nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to a mono-frequency excitation. However, considering the linear vibration of the axially moving beam, to the authors' best knowledge, there is no published work involving the viscoelastic foundation.

To investigate the free and forced response of an axially moving beam with viscoelastic foundation, the modal analysis method is adopted. Modal analysis is a powerful tool to obtain the analytical solution to the axially moving continua ^[21-22]. Nevertheless, due to the damping effect of the viscoelastic foundation, the classical modal analysis cannot be directly applied to the axially moving beam system as before ^{[3, 10]}. The motion equation cannot be converted into the canonical form with one symmetric and one skew-symmetric matrix differential operators. Thus, the orthogonality cannot be used to decouple the damped system. It is no longer a self-adjoint but a non-self-adjoint system mathematically. To solve this problem, Chung and Kao ^[23] applied the classical modal analysis to derive the analytical solution of a damped axially moving wire. They achieved the similar orthogonal condition and the analytical solution to initial conditions, but their method was not available to arbitrary external excitation but only to initial conditions.

In view of the lack of research for the free and forced response of an axially moving beam with damping, that is, the beam translates on a viscoelastic foundation, this paper develops a complex modal analysis method for damped gyroscopic continua. The adjoint eigenvalues and adjoint eigenfunctions are introduced to use the orthogonal relationship. The response of an axially moving beam on a viscoelastic foundation is derived to not only initial conditions but also general external excitations.

2 Equations of motion

A uniform axially moving elastic beam of the linear density $\rho$, the cross-sectional area $A$, the bending stiffness $EI$, and the initial tension $P$, which is supported by a uniform viscoelastic foundation, travels at a constant axial transport speed $\Gamma(t)$ between two eyelets separated by the distance $l$. The stiffness and the damping factor per unit length of the foundation are $k$ and $\zeta $, respectively. The beam is subjected to an external transverse excitation $F(x, t)$ which depends on the axial coordinate $x$ and the time $t$. The physical model is shown in Fig. 1.

Consider the transversal displacement only. The potential energy of the system includes three components, i.e., the strain energy of the beam $U_{\rm s} $, the energy of the foundation stiffness $U_k $, and the energy due to the pretension $U_{\rm p}$. The strain energy of the beam is

(1)

The potential energy due to the tension $P$ and the foundation stiffness $k$ is

(2)

where d$s$ is the deformation of d$x$, $v(x, t)$ is the beam transverse displacement at the $x$-axis and the time $t$, and the comma preceding $x$ or $t$ denotes the partial derivative with respect to $x$ or $t$.

The kinetic energy of the beam can be written as

(3)

The virtual work of the damping force due to the foundation viscosity is

(4)

The virtual work of the external force is

(5)

The generalized Hamilton's principle takes the form of

(6)

Substitution of (1)-(5) into (6) leads to the governing equation ^[24-25],

(7)

the variation boundary condition,

(8)

and the variation initial condition,

(9)

Introduce the dimensionless transformations and parameters as follows:

(10)

and simplify (7). Then, one has

(11)

3 Eigenvalues and eigenfunctions of moving beam on viscoelastic foundation 3.1 Eigenvalue problem

To obtain the eigenvalues and eigenfunctions of the system, consider a homogeneous problem without external excitations,

(12)

With the differential operators

(13)

rewrite (12) in the differential operator form,

(14)

Assume a solution in the form of

(15)

where $\lambda _n$ and $\varphi _n(x)$ are the $n$th eigenvalue and eigenfunction, respectively.

Substituting (15) into (12) yields

(16)

Assume $\varphi _n{(x)}={\rm e}^{\gamma _n x}$ and substitute it into (15). Then, one has

(17)

Thus, the solution of the eigenfunction can be written as

(18)

Using the motionless boundary conditions

(19)

which satisfy (18), one has

(20)

which must yield nontrivial solutions of $c_i $. Hence, the determinant of the coefficient matrix vanishes, i.e.,

(21)

Solving (17) and (21), $\gamma _{1n} $, $\gamma _{2n} $, $\gamma _{3n} $, $\gamma _{4n} $, and $\lambda _n\, (n=1, 2, 3, \cdots)$ can be obtained numerically. Introducing them into (20), the constants $c_{1n}$, $c_{2n}$, $c_{3n}$, and $c_{4n}$ can be determined. Thus, according to (18), the eigenfunctions of the system are expressed as

(22)

where the constant $c_{1n}$ can be confirmed by normalization conditions.

The first two natural frequencies as functions of velocity are plotted in Figs. 2 and 3 for different values of foundation stiffness and damping with $\mu =0.8$. It is observed that the increasing foundation stiffness and the decreasing damping lead to higher natural frequencies.

3.2 Adjoint eigenvalue problem

The definition of the adjoint differential operator is

(23)

where $u$ and $v$ are functions. If $L=L^\ast $, the differential operator is self-adjoint.

The adjoint equation of (14) is

(24)

with the adjoint boundary conditions

(25)

According to the adjoint differential operator definition and boundary condition, $M$, $C$, $K$, $D$, and $T$ are self-adjoint operators but $H$ and $G$ are not, that is,

(26)

Hence, (24) can be expressed as

(27)

Using the same method presented in Subsection 3.1, the analytic-numerical adjoint eigenvalues $\lambda _n^\ast $ and eigenfunctions $\varphi _n^\ast ( x )$ can also be obtained.

According to Subsections 3.1 and 3.2, the original system has $n$ pair conjugated eigenvalues, and the corresponding eigenfunctions can be written as $ {\lambda _1 }$, $\lambda _2$, $\cdots$, $\lambda _n$, $\overline{\lambda }_1$, $\overline{\lambda}_2$, $\cdots$, $\overline{\lambda}_{n}$, $\varphi _1 (x)$, $\varphi _2 (x)$, $\cdots$, $\varphi _n (x)$, and $\overline{\varphi }_1 (x)$, $\overline {\varphi }_2 (x)$, $\cdots$, $\overline{\varphi}_n (x)$, where the short line in $\overline {\lambda }$ and $\overline {\varphi }(x)$ represents conjugate. The adjoint system has $n$ pair conjugated eigenvalues and corresponding eigenfunctions which can be written as $ {\lambda^{\ast} _1 }$, $\lambda^{\ast} _2$, $\cdots$, $\lambda^{\ast} _n$, $\overline{\lambda}^{\ast}_1$, $\overline{\lambda}^{\ast}_2$, $\cdots$, $\overline{\lambda}^{\ast}_{n}$, $\varphi^{\ast}_1 (x)$, $\varphi^{\ast}_2 (x)$, $\cdots$, $\varphi^{\ast}_n (x)$, and $\overline{\varphi }^{\ast}_1 (x)$, $\overline {\varphi }^{\ast}_2 (x)$, $\cdots$, $\overline{\varphi}^{\ast}_n (x)$.

The original system and the adjoint system share conjugated eigenvalues $\lambda _n =\overline {\lambda }_n^\ast $ but the same eigenfunctions $\varphi _n =\varphi _n^\ast $.

4 Complex modal analysis

The closed-form analytical solution of linear vibration of the system can be obtained with the complex modal analysis method using the orthogonality property.

With the matrix differential operators and state vector,

(28)

the governing equation (14) can be written as

(29)

With the external excitation, the governing equation of the system can be written as

(30)

where $f=\left[ \begin{array}{*{35}{l}} 0 \\ f \\ \end{array} \right]$. The eigenvalue problem associated with (29) is

(31)

where

(32)

is the $r$th state space eigenfunction, and $\lambda _r $ and $\varphi _r $ are the $r$th eigenvalue and eigenfunction of the governing equation (14), respectively.

The adjoint eigenvalue problem to (27) is

(33)

where the adjoint matrix differential operators ${{A}^{*}}$ and ${{B}^{*}}$ can be expressed as

(34)

The $r$th adjoint state space eigenfunction ${\rm {\bf \Psi }}_r ^\ast $ is

(35)

where $\lambda _r^\ast $ and $\varphi _r^\ast $ are the $r$th eigenvalue and eigenfunction of (27), respectively.

The orthogonal relationships of the eigenfunctions on the axially moving beam with a viscoelastic foundation are ^[26]

(36)

The dynamic response of damped gyroscopic continuous system is expressed as

(37)

where

(38)

with the initial condition

(39)

where $U( {0, x} )$ can be obtained with given $x( 0 )$.

5 Numerical applications/simulation and discussion

The aim of the numerical simulation reported here is to illustrate the main results derived in this paper. An example is adopted which shows the forced response of an axially moving beam with the viscoelastic foundation under a uniform excitation. This procedure demonstrates the effectiveness of the complex modal analysis method to this specified problem. Besides, the foundation effect on the response is discussed. A graphical representation of dynamic response with different $\kappa $ and different $\zeta $ is reported. The dimensional parameters of the physical model are listed in Table 1. All the non-dimensional parameters used in the following numerical simulation can be obtained according to (10).

5.1 Examples

Applying the initial conditions $v(x, 0)=0.01$ sin($\pi x$) and the uniform excitation $F(x, t)=F_{0}$cos($\Omega t$) on the axially moving beam with the viscoelastic foundation, where $F(x, t)$ can be non-dimensionalized as $f=F_{0}l$cos($\omega t$)/$P$ and $\omega =\Omega \sqrt {{\rho l^2}/P}$. The forced vibration response of the system can be obtained with $\mu=0.8$, $\kappa =3.26$, $\zeta =3.5$, $c=0.2$, $F_{0}=0.1$, and $\omega =7$, as shown in Fig. 4. Figure 4(a) illustrates the forced response of the entire axially moving beam in 6 s, and Fig. 4(b) presents the forced response of the beam at the specific position $x=0.6$ in 6 s. Obviously, the response consists of transient response and steady state response. The transient response disappears with time due to the existence of damping, and only the steady response is left whose frequency is identical to the excitation frequency.

5.2 Foundation effect 5.2.1 Foundation effect on free vibration response

With $\zeta =3.5$, $c=0.2$, $\mu =0.8$, and the initial condition $v(x, 0)=0.01$sin($\pi x$), the free vibration responses of an axially moving beam with the viscoelastic foundation at $x=0.6$ to different stiffness factors $\kappa $ are displayed in Fig. 5. Similarly, with $\kappa =3.26$, $c=0.2$, $\mu =0.8$, and the initial condition $v(x, 0)= 0.01$sin($\pi x$), the free vibration responses at $x=0.6$ to different damping factors $\zeta $ are displayed in Fig. 6. As expected, the free vibration response gets smaller at a certain frequency with time elapsing. The effect of the stiffness factor $\kappa$ is reflected in the frequency. As $\kappa$ gets small, the frequency decreases. The effect of the damping factor $\zeta $ is reflected in the amplitude. Under the same conditions, the larger damping factor $\zeta $, the more amplitude reduction.

5.2.2 Foundation effect on forced response

Applying the uniform excitation $F(x, t)=F_{0 }$cos($\Omega t$) on the axially moving beam with the viscoelastic foundation, the forced vibration response of the axially moving beam can be obtained ^[27-28]. Figure 7 shows the response at $x=0.6$ with $\zeta=3.5$, $c=0.2$, $\omega =7$, $f =0.1$, $\mu =0.8$, and different stiffness $\kappa $. Figure 8 shows the response at $x=0.6$ with $\kappa =3.26$, $c=0.2$, $\omega =6$, $f =0.1$, $\mu =0.8$, and different damping $\zeta $. From Figs. 7 and 8, we can conclude that the amplitude of the vibration decreases when the damping factor increases or the stiffness factor increases. The phase difference between the force and the response increases as the damping factor increases or the stiffness factor decreases. The frequency of the vibration is identical to the excitation frequency.

6 Conclusions

This paper investigates transverse vibration of an axially moving beam supported by a viscoelastic foundation. The complex modal analysis is applied to the non-self-adjoint gyroscopic system. The eigenvalues and the eigenfunctions are determined semi-analytically. The responses of the non-self-adjoint system to arbitrary external excitations and initial conditions are derived from the modal expansion expression using the orthogonal relationship of the eigenfunctions and adjoint eigenfunctions. Numerical examples are presented to illustrate the effectiveness of the approach and the effects of the foundation parameters on the free vibration and forced vibration. In the free vibration, the response dies out with the elapsing time. Under the same conditions, the larger damping factor, the more amplitude reduction. In the forced vibration, the response amplitude decreases with the increasing damping factor and stiffness. Moreover, the phase difference between the excitation and the response increases with the increasing damping and the decreasing stiffness.