On (Non) Applicability of a Mode-Truncation of a Damped Traveling String

This study investigates a linear homogeneous initial-boundary value problem for a traveling string under linear viscous damping. The string is assumed to be traveling with constant speed, while it is fixed at both ends. Physically, this problem represents the vertical (lateral) vibrations of damped axially moving materials. The axial belt speed is taken to be positive, constant and small in comparison with a wave speed, and the damping is also considered relatively small. A two timescale perturbation method together with the characteristic coordinate’s method will be employed to establish the approximateanalytic solutions. The damped amplitude-response of the system will be computed under specific initial conditions. The obtained results are compared with the finite difference numerical technique for justification. It turned out that the introduced damping has a significant effect on the amplitude-response. Additionally, it is proven that the mode-truncation is applicable for the damped axially traveling string system on a timescale of order .


INTRODUCTION
Vibrations (mainly lateral) in such devices have limited their applications. Over the last six decades, there has been vast research on the lateral vibrations of axially traveling strings or beams. Miranker [1] was the first who developed the mathematical equations of motion for laterals oscillations of axially accelerating string. For further studies on vibrations of string and beams, the reader is referred the papers [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. To suppress the vibrations and noise in structures and machines, damping of string material is widely taken into consideration [17]. Gaiko [8] recently developed the equations of motion for viscous damping of accelerating string with the help of extended Hamilton principle.
Various analytical and numerical techniques were employed to investigate elastic and viscoelastic strings or beams under different operating conditions. Sandilo et. al. [10] examined the viscous damping via two timescales perturbation method. Gaiko [14] investigated the damped axially moving system via perturbation method together with the Fourier-series expansion method. However, only few modes were taken in [13] to examine the behavior of the amplitude-response of the damped traveling string system.
In continuation of [14], this study will compute the all mode-responses without any truncation via the combination of the method of characteristic coordinates and two timescales perturbation method. In addition, the (non)applicability of mode-truncation for damped traveling string systems will also be discussed. The results obtained through the proposed technique will also be validated via finite difference numerical technique.

EQUATIONS OF MOTION
Consider a uniform axially moving string of mass density, initial tension T 0 , the viscous damping parameter and uniform axial transport speed V that travels between a pair of pulleys separated by a distance L and is assumed to be fixed at both ends, i.e. at x = 0 and x = L as shown in Fig with the boundary conditions:

FIG. 1. SCHEMATIC MODEL OF A TRAVELING STRING
and initial conditions: where u 0 is the initial displacement and u 1 is the initial speed of the system.
To put the Equations (1-3) in a dimensionless form, the following non-dimensional quantities are taken into consideration: Thus, the Equations (1-3) in dimensionless form becomes, The hats describing the non-dimensional quantities are neglected in Equations (5-7), and from now on. In the subsequent sections, the Equations (5-7) will be solved through the combination of two timescales perturbation method and the method of characteristics.

CONSTRUCTION OF FORMAL APPROXIMATIONS
In this section, the formal asymptotic approximations for the homogeneous linear initial-boundary value problem Equations (5-7) will be developed via the combination of the method of two timescales perturbation and the method of characteristic coordinates. For further details of this technique, the reader is referred to [2,[6][7][22][23]. The low axial belt speed Vof the string is considered in comparison to wave speed cand 0() that is, V = 0(), where (0  1) is a book keeping parameter. It is also presumed that L is small compared to c and 0(), i.e.  = 0(). Based on these assumptions, Equations (5-7) can be written as follows: where the boundary and initial conditions are given as in Equations (6-7). In order to investigate the initial boundary value problem (8), a two timescales perturbation technique will be employed. However, we encounter several computational complexities whenever we employ a Fourier expansion method of the unperturbed solution as were noticed in [13]; the equations of motion (8) is alternatively investigated by means of the characteristic variables  = x -t and =x+t.
In this method, we will replace the initial-boundary value problem (8) by an initial-value problem. This replacement requires the extension of the dependent variable u and its derivatives and the initial values in x to two-periodic odd functions. Since the terms u xt and u x in Equation (8) are not odd; in order to make them odd, we multiply these terms with the Fourier-sine series: With the multiplication of H(x) in Equation (8), the equation becomes u tt -u xx = -(2Vu xt H(x) = u t )- 2 (V 2 u xx + Vu x H(x)), 0x1, t0 (10) By applying the perturbation method to Equation (10), the solution is assumed in the form where  = x -t,  = x + t, and  = t.
The introduction of  and  leads to the following transformation: Plugging Equations (11)(12) into Equation (10), we obtain: Further, we assume that the function v() can be Plugging Equation (14) into Equation (13), and comparing the terms of order 0 and  1 , we obtain the O(1) and O()problem below: It should be noticed that the term v 1 in Equation (14) is assumed to be small in comparison to the term v o and that the expansion is uniform in the sense that the ratio v where  appears only parametrically to this order of approximation, and the functions f 0 and g 0 satisfy the initial conditions: Integration of Equation (18) with respect to  yields: where the function F() is a constant of integration and to be determined by the condition on v 1 . In Equation (19), it can be observed that the term  - = 2t is of order  -1 . So, v 1 will be of O( -1 ) unless the terms linear in (that is, of order  -1 are eliminated. It turns out that both v 1 and v 1 is of O(1) on a timescale of O( -1 ) if the functions f 0 and g 0 satisfy the following conditions: and -2g 0 -g 0 = 0 The Equation (20) and Equation (21) are equivalent.
The solution of Equation (20) subject to the initial conditions: The integration of Equation (23) with respect to yields

CONCLUSION
In this study, we examined the transverse (lateral) computedand shown as damped out. However, in this paper, the amplitude-response of the system in explicitform has been computed and shown to be damped out.
It is concluded that the mode-truncation is also possible for a damped traveling string.

ACKNOWLEDGMENT
The authors wish to express their gratitude towards the unknown reviewers for their valuable comments and suggestions, which indeed helped us to improve our manuscript.