Variational principles for a double Rayleigh beam system undergoing vibrations and connected by a nonlinear Winkler–Pasternak elastic layer

1 Introduction

Double beams have a number of properties which make them more suitable to use in some applications as compared to single beams. This is due to the fact that, compared to single beams, double beams exhibit higher strength and stiffness as well as better vibration absorption properties for the same weight as noted in the literature [1,2,3,4]. The areas of applications of double beams include cranes, bridges, aircraft wing spars, railway tracks, truss structures and pipelines as noted by Mirzabeigy et al. [2]. Further application areas of double beams are discussed by Nguyen [5], involving the use of double beams for crack detection, and by Zhang et al. [6] on their use in (Microelectromechanical system applications. The use of double beam system to model floating-slab tracks has been reported by Hussein and Hunt [7] with the upper beam representing the rails and the lower beam accounting for the slab. Another area of application of double beam systems is their use as dynamic vibration absorbers [3,8,9]. As noted by Kim et al. [9], the elastic layer between the beams can dampen the vibrations of another beam subject to some form of dynamic loads.

Double beams are often connected to each other by an elastic layer which could be of a Winkler or Pasternak type or a combination of both. An extensive study of several beam systems is given in the book by Stojanovic and Kozic [10] which covers the vibrations of Euler–Bernoulli, Rayleigh, Timoshenko and Reddy–Bickford beam systems. There have been several studies on the vibrations of double Euler–Bernoulli beams and the most recent studies can be found in refs [10–14], where solutions for both free and forced vibrations were given using numerical, approximate and analytical methods. These studies included solutions under general boundary conditions [11,12], the effect of tip mass on free and forced vibrations [13] and double beams under a moving load [14]. Euler–Bernoulli beam model does not take the effect of rotational inertia on vibrations and, in particular, on vibration frequencies, into account. The Rayleigh beam theory improves on the Euler–Bernoulli model by accounting for the rotary inertia which affects the frequencies. The basic theory of Rayleigh beams can be found in a number of studies and, in particular, in refs [15,16].

The effect of rotatory inertia on the frequencies was studied in recent publications [17–20] based on single Rayleigh beams. Extending the results for single Rayleigh beam to double Rayleigh beams has been the subject of a number of studies. Vibrations of double Rayleigh beams under various dynamic loads have been studied previously [21–25]. In these studies, the effect of moving loads on the dynamic response of the double beams was investigated. Forced vibrations of double Rayleigh beams subject to axial loads were studied [26,27]. In previous studies [21,25], the layer between the two beams were taken as viscoelastic and in the study by Mohammadi and Nasirshoaibi [27], the middle layer was modelled as a Pasternak type. Pasternak foundations improve the accuracy of the results over Winkler foundations as noted by Numanoglu et al. [28].

The layer between the double beams, in most cases, is taken as an elastic Winkler layer as noted in the book by Stojanovic and Kozic [10]. However, Winkler model of the elastic layer can be considered as a first approximation of the interaction between the double beams and may lead to discontinuities in the deflection as noted by Kerr [29], which, in turn, can lead to inaccurate results. The accuracy of elastic layer between the beams can be improved by using a two-parameter model by including the Pasternak foundation in addition to a nonlinear Winkler foundation, leading to a combination of linear and nonlinear components.

Several works involved the study of beams and plates on nonlinear Winkler-linear Pasternak foundations [30–38]. Dynamic analysis of beams on nonlinear foundation was the subject of few earlier studies [30–34] and of plates in few other studies [35–37]. The effect of nonlinear foundation on column buckling was studied by Abumandour et al. [38]. In the case of double Rayleigh beams subject to forced vibrations, the elastic layer between the beams was taken as a linear Winkler layer by Stojanovic and Kozic [26] and as a Pasternak layer by Mohammadi and Nasirshoaibi [39]. In the bending study of a double Euler–Bernoulli beam, the middle layer was taken as a Winkler–Pasternak foundation by Brito et al. [40]. In the studies [26,39,40], the layer between the double beams was taken as a linear elastic layer, while in the studies [41,42], on the vibrations of double Euler–Bernoulli beams, the elastic layer between the beams was taken as a nonlinear Winkler layer by including the cubic terms in the modelling of the inner layer.

The objective of the present study is to develop the variational formulation of a double Rayleigh beam system and to determine the variationally consistent boundary conditions applicable to double Rayleigh beams connected by a nonlinear Winkler–Pasternak elastic layer. Variationally consistent boundary conditions were derived for sixth-order shear deformable beams by Shi and Voyiadjis [43]. Variational consistent formulations and the corresponding boundary conditions were derived for a number of cases involving glassy polymers [44] and beams and plates [45].

In the present study, the variational principle for double Rayleigh beams is formulated using a semi-inverse method. The semi-inverse method has been applied to several problems in mechanics and physics and a detailed treatment of the subject is given by Esmailzadeh [46]. Recent studies on the derivation of variational principles using this approach include the study by Xu and Deng [47] and a recent application of the semi-inverse method is given by Tekiyeh et al. [48]. Semi-inverse method was employed to derive the variational principles and the variationally consistent boundary conditions for multi-walled carbon nanotubes connected by Van der Waals forces in a number of studies [49–53].

In the present study, the layer connecting the double Rayleigh beams is specified as a Winkler–Pasternak type in which the Winkler layer consists of linear and nonlinear parts and the Pasternak layer is expressed in terms of linear second derivatives. In the first step, the variational formulation of the problem is derived for the forced vibration case with the beams subject to axial forces. Next the Hamilton’s principle for this case is presented. For the freely vibrating double beam case, the Rayleigh quotients are given for the vibration frequency and the buckling load. The variationally consistent boundary conditions are derived next using the variational formulation of the problem. It is observed that the boundary conditions are not coupled for clamped and simply supported boundary conditions, but are coupled for the case when the shear force has to be specified due to the presence of Pasternak layer between the beams. It is noted that formulations presented in the present study would be useful in the implementation of approximate and numerical methods solutions for double Rayleigh beams subject to natural and geometric boundary conditions.

2 Governing equations

Double Rayleigh beam system under consideration is subject to compressive loads P 1 and P 2 and undergoing transverse vibrations. The beams are connected by an elastic layer between them which is modelled as a combination of a linear Winkler layer with an elastic modulus of k 0 , a nonlinear Winkler layer with an elastic modulus of k nl and a Pasternak layer with an elastic modulus of G 0 . The beams have the same length which is denoted as L . However, the stiffnesses E i I i , the cross-sectional areas A i and the densities ρ i of the beams could be different with i = 1,2 and with E i denoting the Young’s modulus and I i denoting the moment of inertia of the i th beam. Displacements of the beams are denoted as w 1 ( x , t ) and w 2 ( x , t ) , and the equations governing the forced vibrations of double Rayleigh beams are given as follows [23,24,26,39]:

where the differential operators L i ( w i ) are given as follows:

and the coupling operator K ( w 1 , w 2 ) is as follows:

where the first two terms of Eq. (5) correspond to a nonlinear Winkler foundation with elastic constants k 0 and k nl and the third term to Pasternak foundation with an elastic constant G 0 . In Eqs. (1) and (2), f i ( x , t ) is the external force acting on the beams with t 1 ≤ t ≤ t 2 .

3 Variational formulation

In order to formulate the variational principle applicable to the system of Eqs. (1) and (2), we introduce the functionals V 1 ( w 1 , w 2 ) and V 2 ( w 1 , w 2 ) such that

where V ( w 1 , w 2 ) is the variational functional to be determined. We first introduce the variational functionals V 1 ( w 1 , w 2 ) and V 2 ( w 1 , w 2 ) as follows:

where Φ i ( w i ) is given by

which corresponds to the variational forms of Eqs. (3) and (4) for i = 1,2 . In Eq. (7), F ( w 1 , w 2 ) is an unknown function of w 1 and w 2 and their derivatives, and has to be determined such that the Euler–Lagrange equations of V ( w 1 , w 2 ) correspond to governing Eqs. (1) and (2). We first determine the Euler–Lagrange equations of the variational functional (6) as follows:

In Eqs. (10) and (11), δ F δ w i is the variational derivative of F ( w 1 , w 2 ) with respect to w i . The expression for the variational derivative of F ( w 1 , w 2 ) is given by

We now compare Eqs. (10) and (11) with Eqs. (1) and (2) and note that the variational derivatives of F ( w 1 , w 2 ) have to satisfy the following equations:

so that Euler–Lagrange equations of the variational functional V ( w 1 , w 2 ) in Eq. (6) correspond to the governing equations of the double Rayleigh beam system given by Eqs. (1) and (2). Comparing Eqs. (1) and (2) with Eqs. (13) and (14), we determine the function F ( w 1 , w 2 ) as follows:

It is observed that, with F ( w 1 , w 2 ) given by Eq. (15), the Euler–Lagrange equations of the variational functional V ( w 1 , w 2 ) in Eq. (6) corresponds to the governing equations of the double Rayleigh beam given by Eqs. (1) and (2). The derivation of the variational expressions (9) and (15) and the corresponding boundary conditions are given in the Appendix.

4 Hamilton’s principle

The Hamilton’ principle can be expressed as follows:

In the present problem, the functionals K E ( t ) , W E ( t ) , PE 1 ( t ) and PE 2 ( t ) are given by

In Eqs. (17)–(20), KE ( t ) is the kinetic energy, W E ( t ) is the work done by external forces, PE 1 ( t ) is the potential energy of deformation and PE 2 ( t ) is the potential energy due to the elastic layer between the beams.

5 Free vibrations

Variational principle for a freely vibrating double beam system is formulated next. For freely vibrating beams, the deflection can be expressed as follows:

where ω is the vibration frequency. In this case, the differential equations for the free vibrations of the double beams become

The differential operators L FV i ( W i ) and K ( W 1 , W 2 ) are given by

For the freely vibrating case, the variational functionals Φ FV i ( W i ) and F FV ( W 1 , W 2 ) are given by

The variational formulation of the freely vibrating system can be expressed as follows:

where

Variational expressions (27) and (28) can be verified by noting that Euler–Lagrange equations of (27) and (28) correspond to the differential Eqs. (23) and (24). Next Rayleigh quotient is given for the freely vibrating double beam system. Rayleigh quotient for the vibration frequency ω is given by the following expression:

We introduce the functional

Rayleigh quotient for the buckling load can be expressed as follows:

where F FV ( W 1 , W 2 ) is given by Eq. (26) and Z FV i ( W i ) by Eq. (30).

6 Boundary conditions

In this section, natural and geometric boundary conditions are derived using the variational formulation of the freely vibrating double beam system. The first variation of V FV ( W 1 , W 2 ) in Eq. (27) with respect to W i , denoted as δ W i V FV , can be obtained by integration by parts. We note that

where ∂ Ω i ( 0 , L ) denotes the boundary terms and

with ∂ Ω i ( 0 , L ) given by

.

We note that the boundary term ∂ Ω i ( 0 , L ) can be expressed as

where Q i is the shear and M i is the moment expression. Thus, the boundary conditions at x = 0 and x = L are given by

with i = 1,2 . It is observed that due to the presence of Pasternak layer between the beams, the boundary conditions are coupled if the shear force is specified at x = 0 or at x = L as given by Eq. (37). Derivations of the variational Eqs. (25) and (26) and the boundary conditions (37) and (38) are given next. We note that

where

Similarly,

where

The variational form of the last two terms of Eq. (23) can be obtained by noting that

where

Finally, we have

where

Eqs. (39), (41), (43), and (45) indicate the variational expressions in Eqs. (25) and (26). Similarly, Eqs. (40), (42), (44), and (46) have the boundary terms which appear in the boundary conditions shown in Eqs. (37) and (38).

7 Conclusion

Variational formulations for double Rayleigh beams undergoing forced and free vibrations and connected by a nonlinear Winkler and linear Pasternak elastic layer are given. In the previous studies [21–27] on double Rayleigh beams, the connecting layer between the two beams was taken as linear. In the present study, the connecting layer is taken as a combination of linear Winkler and Pasternak layers and a nonlinear Winkler layer. Inclusion of the nonlinear Winkler layer improves the accuracy of the modeling by providing another parameter to model the connection between the two beams. Variational principles applicable to forced vibration and freely vibrating cases are derived by a semi-inverse approach. Based on the forced vibration case, the expression for Hamilton’s principle is given. Derivations of the variational expressions and the resulting boundary conditions are given in the Appendix for the double beam system undergoing forced vibrations. The corresponding derivations for the freely vibrating beam system are also given. These derivations determine the variationally consistent expressions for the shear force and moment which can act at the boundaries. As such, solutions for the double beam system can be obtained for a combination of boundary conditions. For the freely vibrating beam system, the expressions for Rayleigh quotients are obtained for the vibration frequency and the buckling load. The variational expression developed for the present problem can be used in the implementation of approximate and numerical methods of solution and in the implementation of different boundary conditions.