Page:Influence of a Moving Mass on the Dynamic Behaviour of Viscoelastically Connected Prismatic Double-Rayleigh Beam System Having Arbitrary End Supports.pdf/2

2 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 𝑃$1$ (𝑥, 𝑡) having mass 𝑀$𝐿$ and moving with a constant velocity V. For simplicity, it is assumed that the two beams are identical having the same length 𝐿, flexural rigidity 𝐸𝐼, and mass per unit length 𝜇. For convenience the system is, hereby, referred to as system 𝐼. The dynamic responses 𝑊$1$ (𝑥, 𝑡) and 𝑊$2$ (𝑥, 𝑡) of the upper and lower Rayleigh beams, respectively,