#### 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]

In (3) is the acceleration due to gravity, the term represents the centrifugal acceleration, the term denotes the Coriolis acceleration, the term represents the local acceleration, and is the Dirac delta function defined asand is the third tracing constant whose value is unity if the effect of inertia of the load is taken into account; otherwise, it is zero. Note that the Dirac delta function is an even function; therefore, it is expressed as a Fourier cosine series and we have [1, 3]

For the system under consideration, the boundary conditions are any of the classical boundary conditions (i.e., any of the simply supported, free, clamped, and sliding boundary conditions) or their combinations. Hence, the conditions can be written as [1]where the subscript indicates that the elements of the vector of linear spatial differential operators and are to be specified at the boundaries of the two beams. Also, without any loss of generality, it is assumed that the boundary conditions on the same side of the system are the same though they can be any of the classical conditions.

Finally, the initial conditions are

#### 3. Method of Solution

To solve the above initial-boundary-value problem made up of (1)–(3) and (5)–(7), the method of solution already alluded to is presented in this section. The method consists of the following three main steps. (i) Reduce the set of two fourth-order coupled partial differential equations (1) and (2) to a set of two coupled ordinary differential equations of order two using the generalized finite integral transform. (ii) Use the modified asymptotic method of Struble [1] to simplify the resulting set of two coupled transformed ordinary differential equations. (iii) Solve the final set of two simplified coupled transformed ordinary differential equations using a semianalytical method known as differential transform method (DTM).

##### 3.1. The Transformed Second-Order Coupled Ordinary Differential Equations

To obtain the solution for the transverse dynamic responses and of the two Rayleigh beams interconnected by a viscoelastic layer, the fourth-order coupled partial differential equations (1) and (2) are, in the first instance, transformed into a set of two second-order coupled ordinary differential equations. To this end, the generalized finite integral transforms which are defined asare introduced.

The corresponding inverse formulae areIn (8) and (9), , are eigenfunctions of a single-Euler beam system which usually have the formThe constants , , , and are usually determined using any of the classical boundary conditions. Also, the constant is defined asand are such that while is the natural circular frequency, defined asTaking the generalized finite integral transform of (1)–(3) using (5), (7), (8), (9), (10), and (11), one obtainswhere

Equations (16)–(25) are the coupled transformed second-order ordinary differential equations governing the lateral behaviour of a double-Rayleigh beam system interconnected by a viscoelastic layer and traversed by a moving mass.

##### 3.2. Simplification of the Coupled Second-Order Differential Equations

In general, it is difficult to get an exact analytical solution to a set of two coupled second-order ordinary differential equations unless certain simplifications are carried out and/or some assumptions are made. Furthermore, it is remarked at this juncture that the difficulty with (16), in particular, is not only that they are highly coupled but that the coefficients of the terms representing the inertia of the moving load are also functions of the independent variable . Hence, it is expedient that this set of coupled, transformed second-order ordinary differential equations are simplified using certain assumptions. Specifically, the method of simplifying these equations involves following these three substeps: (a) Decouple the set of the ordinary differential equations. (b) Obtain modified frequency (I) due to the effect of rotatory inertia. (c) Obtain yet another modified frequency (II) due to the effect of the mass of the moving load.

##### 3.3. Partially Decoupled Transformed Ordinary Differential Equations

In this subsection, we seek to simplify the coupled transformed second-order ordinary differential equations (16). To this end, a system consisting of two unconnected Rayleigh beams is, in the time being, considered. It is also assumed that one of the beams (hereby referred to as first beam) is acted upon by a moving mass while the second vibrates freely. This system is a simplified version of system I and is, for convenience, hereby termed system II. For this type of system, (16) are reduced toNote that (26) and (27) can also be directly obtained from (16) by setting to zero and are still coupled.

##### 3.4. Method of Obtaining the Modified Frequency (I)

Equations (26) and (27) are still difficult to solve. Hence, they have to be further simplified. To achieve this, the first Rayleigh beam of system II is hereby assumed, for the time being, to be acted upon by a moving force as opposed to a moving mass. In other words, the moving force problem of system II is considered. This amounts to setting to zero in (26) so that (26) and (27) becomerespectively.

Furthermore, it is still difficult to obtain an exact analytical solution to (28) and (29). Hence, one resorts to using an approximate analytical technique [1] which is a modification of the asymptotic method due to Struble [1, 3]. This analytic technique involves obtaining a modified frequency (I) of the system due to the presence of the effect of rotatory inertia so that each of the differential operators in (28) and (29) is replaced by an equivalent operator defined by the modified frequency. Hence, following [1], one first denotes the ratio of the rotatory inertia correction factor, , of any of the two beams to its length by and define a small parameter such thatIt follows, therefore, thatConsidering the homogenous part of (28), for instance, one obtainswhereAccording to Struble’s technique [1, 3], the general solution of (28) is of the formwhere and are slowly time varying functions such thatwhile is a finite natural number and “” denotes “is of order.”

For and without loss of generality, we haveand henceSubstituting (36) and (37) into (32) and simplifying the resulting equation, taking into account (30), (31), and (35), we haveNeglecting terms which do not contribute to the variational equations, (38) becomesSetting the coefficients of and in (39) to zero, we have whose solutions arerespectively, while , are constants.

Hence, the desired modified frequency (I) due to the presence of rotatory inertia isand the differential operator which acts on and in (28) is, as earlier alluded to, replaced by the equivalent free system operator defined by the modified frequency, . In other words, (28) which describes the transverse displacement of the first Rayleigh beam traversed by a moving force is reduced towhereUsing arguments similar to those presented thus far in this subsection, (29) is also reduced toThe implication of (45) and (47) is that when the effect of the inertia terms (i.e., the terms involving the tracing constants ) for system II is not neglected, (26) and (27) simplify torespectively. In other words, the set of second-order ordinary differential equations for system II when the first Rayleigh beam is acted upon by a moving mass is now reduced to (48) and (49).

##### 3.5. Method of Obtaining the Modified Frequency (II)

The problem of system II is now reduced to that of seeking the solutions to (48) and (49). It is however remarked at this juncture that while (49) can be easily solved, there is no exact analytical solution of (48). Hence, one again resorts to the approximate analytical method discussed in Section 3.4. The argument is that the problem under consideration involves the effect of both rotatory inertia and the inertia of the moving load. Hence, it is not sufficient to obtain the modified frequency (I) due to the effect of rotatory inertia only [2]. As a matter of fact, having obtained the modified frequency (I), , we now proceed to obtain another frequency (a modified ), say , which is due not only to the presence of rotatory inertia but also to that of the moving mass. Following arguments similar to that of Section 3.4, an equivalent free system operator defined in terms of the new frequency, , for then replaces the terms on the left hand side of (48). To this end, we introduce a small parameter such thatwhere is the mass ratio.

It also follows thatTo obtain the desired modified frequency (II) due to the effect of the inertia of the moving load and that of the rotatory inertia, the homogeneous part of (48) is then considered. The first approximation to the assumed solution of the said homogeneous equation, according to Struble’s technique, isAgain substituting (53) and its first- and second-order time derivatives into the homogeneous part of (48) taking into account (52), we havewhere the terms in and higher power of have been neglected. The corresponding variational equations areSolving (55), we havewhere , and are constants, and hence Equation (57) is the desired modified frequency (II) corresponding to the frequency of the free system involving rotatory inertia and moving mass effect.

Hence, according to Struble’s technique, (48) reduces towhereand the set of second-order transformed ordinary differential equations for system II are now made up of (45) and (58) whose closed form solutions can be obtained without much difficulty.

##### 3.6. Solution of the Two Viscoelastically Connected Rayleigh Beams

Recall that system II is a simplified version of the original system I and it is obtained by assuming that the two Rayleigh beams are not joined by a layer (i.e., ). Now, if the viscoelastic layer is retained, then is not equal to zero. In this case and in view of (45) and (58), the two second-order transformed coupled ordinary differential equations for the double-Rayleigh beam system I are finally simplified toin terms of the two modified frequencies ( and ), respectively. In other words, problem of assessing the dynamic behaviour of a double-Rayleigh beam system I traversed by a moving mass under arbitrary end supports reduces to that of solving (60) subjected to the corresponding transformed initial conditions. To solve the coupled differential equations (60) a semianalytical method known as differential transform method (DTM) is used. To this end, we first state briefly the basic theory of the method as follows. The differential transform of the th derivative of a function is given as [28, 29]The corresponding inverse transformation is defined asHence, (61) and (62) yieldIt is well known that, in application, the series in (62) is finite and usually written assuch that the series is considered unimportantly small.

Furthermore, it can be readily shown [28, 29] that the relationships in Table 1 between the original function and the transformed function , for , hold.

Next, the application of the differential transform on (60), using Table 1, yields the following recurrence relations for Note that, in obtaining (65), (12) had been used. Equations (60) are also to be solved subject to the following transformed initial conditions:Substituting (67) for into recurrence relations (65), (66), using “MAPLE ,” we haveUsing the inverse differential transform of (64) in conjunction with (67), for , we obtain

Equations (76) represent the transverse displacements of the double-Rayleigh beams interconnected by a viscoelastic layer, traversed by a moving mass and having arbitrary end supports.

For the purpose of comparison the moving force problem associated with system I is considered. It is therefore remarked at this juncture that in view of (45), (46), and (47) and retaining the viscoelastic core, the two reduced transformed coupled second-order ordinary differential equations for the corresponding moving force problem of system I having arbitrary boundary conditions are

Solving (77), subject to the corresponding transformed initial conditions, using the differential transform method, one obtains, after inversion, the following:Equations (78) and (79) denote the lateral deflections of the upper and lower Rayleigh beams, respectively, due to concentrated moving force having constant velocity and traversing a viscoelastically connected double-Rayleigh beam system with general end supports.

#### 4. Some Case Studies

Hitherto, the discussion has been for general boundary conditions. In this section, two case studies involving (i) simply supported end conditions and (ii) clamped-clamped end conditions are considered in order to illustrate the theory developed thus far.

##### 4.1. Simply Supported Double-Rayleigh Beam System

The system considered here comprises two finite Rayleigh beams which are simply supported and are interconnected by a viscoelastic layer. The boundary conditions for such simply supported double-Rayleigh beam system are [3, 22, 24]Hence, for the eigenfunction, , we haveas well asIt follows, therefore, that in view of (12), (80), and (81) we haveso thatEquation (15) impliesand the corresponding eigenfunction reduces toThe associated initial conditions are as given in (7). Hence the generalized finite transforms, (8) and (9) and their inverses, (10) and (11) as well as (13) reduce torespectively.

Substituting (86) into (22)–(25) and the resulting expressions into (17)–(21) while the results of the latter are in turn substituted into the transformed equations (16), one obtainswhere