A Galerkin-type state-space approach for transverse vibrations of slender double-beam systems with viscoelastic inner layer

Abstract

A novel state-space form for studying transverse vibrations of double-beam systems, made of two outer elastic beams continuously joined by an inner viscoelastic layer, is presented and numerically validated. As opposite to other methods available in the literature, the proposed technique enables one to consider (i) inhomogeneous systems, (ii) any boundary conditions and (iii) rate-dependent constitutive law for the inner layer. The formulation is developed by means of Galerkin-type approximations for the fields of transverse displacements in the system. Numerical examples demonstrate that the proposed approach is accurate and versatile, and lends itself to be used for both frequency- and time-domain analyses.

Introduction

Beams are fundamental components in most of the structural systems conceived, designed and constructed in civil, mechanical and aerospace engineering. Hence, free and forced vibrations of single Euler–Bernoulli and Timoshenko beams are covered in hundreds of scientific and technical contributions. On the other hand, relatively few papers have been published on the dynamics of double-beam systems, made of two parallel slender beams continuously connected by a Winkler-type viscoelastic layer.

Despite analytical and numerical difficulties arising in the solution of the coupled partial differential equations governing the motion, this dynamic system is certainly worth of investigation. As an example, a double-beam model can be effective in approximating the dynamic behaviour of sandwich beams, largely used in many engineering situations [1], [2], [3], although shear deformations within the viscoelastic layer may play a crucial role for such members. Motivated by the recent development of the nano-opto-mechanical systems (NOMS) [4], [5], [6], Murmu and Adhikari [7], [8], [9] have considered the dynamics and instability of nanoscale double-beam systems using scale-dependent non-local theory. A continuous dynamic vibration absorber (CDVA) is another important case of double-beam system, where secondary beam and inner layer are designed in order to mitigate the vibration experienced by the primary beam [10]. The double-beam model is also able to capture the dynamic response of floating-slab railway tracks, widely used to control vibration due to underground trains [11].

Several interesting analytical works have been developed in recent years which demonstrate an emerging attention to this subject. Vu et al. [12] formulated a closed-form solution for the vibration of a viscously damped double-beam system subjected to harmonic excitations. Two restrictions, however, limit the practical applicability of this solution: (i) outer beams must be homogeneous and identical; (ii) boundary conditions on the same side of the system must be the same. Oniszczuk [13], [14] presented some analytical expressions for the undamped free and forced vibrations of a simply supported double-beam system. In his formulation the outer beams can be different from each other, but they must be homogeneous and pinned at the ends; moreover, the damping is totally neglected. Abu-Hilal [15], under the same assumptions as in Ref. [12], studied the dynamic response of a double-beam system traversed by a moving force. Kelly and Srinivas [16] tackled the problem of a set of $n\ge 2$ axially loaded Euler–Bernoulli beams connected by layers of elastic Winkler-type springs; in their formulation, beams and springs may have different mechanical properties, while all the beams have the same boundary conditions and experience the same axial load, while damping is not considered.

Several authors have considered distributed parameter systems with viscoelastic damping. In one of the earliest work Banks and Inman [17] have considered viscoelastically damped beam. They have taken four different models of damping: viscous air damping, Kelvin–Voigt damping, time hysteresis damping and spatial hysteresis damping, and used a spline inverse procedure to form a least-square fit to the experimental data. Cortés and Elejabarrieta [18], [19] considered free and forced vibration analysis of axially vibrating rod with viscoelastic damping. Chen [20] considered bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin–Voigt type of damping. Yadav [21] considered the dynamics of a four-layer beam with alternate elastic layer and viscoelastic layer.

The effects of a viscoelastic inner layer on the dynamics of double-beam systems have been addressed by Palmeri and Muscolino [22] by using a component-mode synthesis (CMS) approach, whose practical applicability is limited by the need to solve a fourth-order differential equation for each assumed mode; moreover, the effects of inner transverse vibrations within the viscoelastic core is neglected. In this paper, aimed at overcoming the severe limitations highlighted above, a novel Galerkin-type state-space model for the vibration analysis of double-beam systems is formulated and numerically validated. Based on a convenient choice of assumed modes for the components, the proposed technique allows us to consider inhomogeneous beams and any boundary conditions. Furthermore, since in many engineering applications an elastomeric material is used in the inner layer, the latter is described through the so-called standard linear solid (SLS) model, which is one of the simplest rheological models able to represent the rate-dependent behaviour of viscoelastic solids [23]. The effects of the viscoelastic damping on the double-beam system is represented by generalizing the concept of modal relaxation function, recently suggested by Palmeri and his associates [24], [25].

It is worth noting that, being based on the introduction of a set of additional internal variables, the extension of the proposed technique to more refined rheological models, such as the generalized Maxwell's model or the Laguerre polynomial approximation [26], is quite straightforward. It is also worth mentioning that several works exist in the frequency [27], [28], [29], [30], [31] and also in the time domain [32], [33], [34], [35] for the generalized Maxwell's model in the context of discrete systems, while the proposed approach is specifically tailored to continuous structures, which have received less attention in the past [25], [36].