Chinese Journal of Mathematics

Volume 2017, Article ID 6058035, 30 pages

https://doi.org/10.1155/2017/6058035

## Influence of a Moving Mass on the Dynamic Behaviour of Viscoelastically Connected Prismatic Double-Rayleigh Beam System Having Arbitrary End Supports

^{1}Department of Mathematics, University of Ilorin, Ilorin, Kwara State, Nigeria^{2}Department of Mathematical Sciences, Olabisi Onabanjo University, Ago-Iwoye, Nigeria

Correspondence should be addressed to Jacob Abiodun Gbadeyan; moc.oohay@nayedabga.j

Received 30 August 2016; Accepted 28 November 2016; Published 26 February 2017

Academic Editor: Maria Bruzón

Copyright © 2017 Jacob Abiodun Gbadeyan and Fatai Akangbe Hammed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the lateral vibration of a finite double-Rayleigh beam system having arbitrary classical end conditions and traversed by a concentrated moving mass. The system is made up of two identical parallel uniform Rayleigh beams which are continuously joined together by a viscoelastic Winkler type layer. Of particular interest, however, is the effect of the mass of the moving load on the dynamic response of the system. To this end, a solution technique based on the generalized finite integral transform, modified Struble’s method, and differential transform method (DTM) is developed. Numerical examples are given for the purpose of demonstrating the simplicity and efficiency of the technique. The dynamic responses of the system are presented graphically and found to be in good agreement with those previously obtained in the literature for the case of a moving force. The conditions under which the system reaches a state of resonance and the corresponding critical speeds were established. The effects of variations of the ratio of the mass of the moving load to the mass of the beam on the dynamic response are presented. The effects of other parameters on the dynamic response of the system are also examined.

#### 1. Introduction

The problem of determining the dynamic response of elastic structures traversed by moving loads is of significant technological importance and various researchers (Engineers, Physicists, and Applied Mathematicians) continue to pay considerable attention to studying the various corresponding mathematical models [1–14]. Most of these studies have been carried out for simpler structures such as beams, plates, frames, and shells since such elastic structures form the fundamental components of various modern complex structures and the mathematical analysis involved is relatively less complicated. For instance, the trolleys of overhead travelling cranes which move on their girders, as well as bridges on which trains or vehicles move, may be modeled as moving loads on beam [5]. The theory of vibration of single-beam or single-plate system subjected to moving loads with different boundary conditions has been extensively developed with hundreds of articles on it [1–14]. Frýba [2], in particular, gave a comprehensive survey of some of the techniques for solving various versions of this problem. Some engineering applications of the theory of vibration of a single-beam or single-plate system carrying a moving load include the study of the dynamic behaviour of guided circular saws usually used in the wood products industry, modern high-speed precision machinery processes, design of railway bridges, and the machining processes [14]. However, there exist many problems of notable practical significance in many branches of modern industrial, mechanical, aerospace, and civil engineering for which the theory of vibration of single-beam system under a moving load may not hold and hence one has to resort to the vibration theory of double-beam, triple-beam, or multibeam systems traversed by a moving load. Examples of such problems include the vibration of composite materials which is usually modeled using double-beam system. Elastically connected concentric beams are also being used as continuous system models for carbon nanotubes and a linear model for interatomic Van der Waals forces is usually provided for by the elastic layers connecting the two beams. As a third example, it is remarked that the coupled behaviour of paper translating with the paper cloth (wire screen) during paper making process is usually studied by modeling the system as two axially translating tensioned beams interconnected by an elastic foundation. Some other significant applications of double-beam system are in (i) passive vibration control, (ii) weight reduction, and (iii) strength and stiffness increase [15, 16]. It is, nevertheless, observed from literature that, unlike the single-beam system, relatively few works have been carried out for the non-single-beam system carrying moving loads. This is perhaps due to the difficulties encountered in solving the governing coupled partial differential equations. Dublin and Friedrich [17] studied forced vibration of two elastic Euler beams interconnected by spring-damper system. The free vibration and the impact problem of a double-beam system which is made up of identical beams elastically connected were studied theoretically and experimentally by Seelig and Hoppmann [18] and Seelig and Hoppmann II [19], respectively. Kessel [20] studied the excitation of resonance in an elastically connected double beam system by a cyclic moving load while Kessel and Raske [21] carried out the analysis of the dynamic behavior of the system comprising two parallel simply supported beams which were elastically connected and traversed by a cyclic moving load. There exists other interesting studies which have been conducted on double elastic beams [22–24]. To the authors knowledge, most of these previous works involving double beams under moving loads are acted upon by only moving forces. In other words, the effect of the inertia of the moving load has not been taken into account. Yet problems involving this effect, though relatively more difficult, are more appropriate representations of the realistic problems usually encountered in practice. As a matter of fact the moving force problem is a special case of the moving mass problem and the difficulty in the latter is due to the singularity appearing in the inertia terms. The solution techniques in most of the above existing works have also been suitable only for simply supported end conditions. However, in recent years many authors paid attention to earthquake resistance systems as well as economic construction. This calls for lighter weight structures. Hence, it becomes necessary to investigate the influence of relatively large masses traversing such structure. The dynamic response of such structures to moving loads whose inertia effect is not negligible should therefore be thoroughly analysed for a rational safe design.

In the present paper, attention is focused on the effect of the mass of a moving load of constant magnitude and velocity on the dynamic response of a finite prismatic double-beam system interconnected by a core. Of particular interest is the influence of the mass of the moving load on the dynamic response of two finite prismatic parallel upper and lower Rayleigh beams connected by a viscoelastic core and having various classical end conditions. This has not been accounted for in previous studies [25]. It is also assumed that the effect of noise is negligible. Hence the influence of either Gaussian or non-Gaussian noise as well as the output constraints [26, 27] is not taken into account. To achieve the desired objective, a general versatile solution technique is developed. This technique is based, in the first instance, on reducing the two governing fourth-order coupled partial differential equations to a set of two second-order ordinary differential equations using generalized finite integral transform. The latter is then simplified using modified Struble’s method [1] and solving the resulting set of two coupled ordinary differential equations using a semianalytical method known as differential transform method (DTM). The solution technique is an extended, modified version of the approach developed by the first author (and Oni) in [1] for the dynamic response of (i) a finite Rayleigh beam and (ii) a non-Mindlin rectangular plate under an arbitrary number of concentrated moving masses. The present technique holds for all types of classical end conditions for double-Rayleigh beams acting upon by either moving forces or masses. Its two-dimensional version for double-plate moving load problem can be easily developed. Semianalytical solutions are obtained. The influence of various parameters (especially those of the inertia of the moving load) involved in the problem are presented graphically and discussed qualitatively and quantitatively. The resonance conditions for both the moving force and moving mass problems are also established. Furthermore, the analysis presented is well illustrated using some of the classical end conditions.

The remaining part of this paper is organized as follows: In Section 2, the problem is defined, stating the pertinent governing differential equations as well as the corresponding initial and boundary conditions. The method of analysis is discussed in Section 3 along with the solutions of the moving force and moving mass double-beam problems. Illustrative examples are given in Section 4, followed by the discussion on resonance conditions for the moving force and moving mass double-beam systems in Section 5. Section 6 deals with the numerical analysis of the problem. Finally, concluding remarks are given in Section 7.

#### 2. Mathematical Model

Consider a double-Rayleigh beam system consisting of two finite, prismatic, undamped, parallel upper and lower Rayleigh beams joined together by a viscoelastic layer (core) which is modeled as a set of parallel springs and dashpots as shown in Figure 1. For the sake of brevity and simplicity, the effect of noise on the system is assumed negligible. Thus, the influence of non-Gaussian noises and output constraints [26, 27], in particular, on the system is not considered. The upper beam is subjected to a load having mass and moving with a constant velocity . For simplicity, it is assumed that the two beams are identical having the same length , flexural rigidity , and mass per unit length . For convenience the system is, hereby, referred to as system . The dynamic responses and of the upper and lower Rayleigh beams, respectively, satisfy the following pair of fourth-order, coupled partial differential equations [22, 25].where is the measure of rotatory inertia correction factor, is the Young’s modulus, is the second moment of area, is the spring constant, is the damping coefficient, is the time, is the spatial coordinate, the prime denotes differentiation with respect to and a dot is the differentiation with respect to time , and and are tracing constants, each of which takes on the value unity or zero depending on whether in the subsequent analysis the effects of (i) rotatory inertia and (ii) the joining layer are taken, respectively, into account or not. Furthermore, the concentrated moving load, , is defined as [1]