In this paper, we used differential transform

method (DTM) to analyse free vibration of a railway bridge,

modelled as an orthotropic rectangular plate, supported by

Pasternak foundation. The effect of damping was considered.

The present method transformed the governing equation to its

algebraic form. Solution form to similar equation was adopted.

The results obtained are in agreement with the ones in

literature, and it shows that the technique introduced is easy to

apply to such differential equation governing the vibration of

such plates

Agarana, M. C.

Ede, A. N.

Covenant University Repository

                                                                                                            Abstract— In this paper, we used differential transform method (DTM) to analyse free vibration of a railway bridge, modelled as an orthotropic rectangular plate, supported by Pasternak foundation. The effect of damping was considered. The present method transformed the governing equation to its algebraic form. Solution form to similar equation was adopted. The results obtained are in agreement with the ones in literature, and it shows that the technique introduced is easy to apply to such differential equation governing the vibration of such plates. Index Terms— Pasternak foundation, Vibration, Damping, Differential transform method   I. INTRODUCTION ll branches of transport have experienced great advances characterized by increasing weight and high speed of vehicles including railway vehicles [9,10]. Railway bridges vibrate as loads move on them [1,10]. In this work, the railway bridge was modelled as a rectangular plate. The dynamic behaviour of Plates, as structural elements, highly influence overall performance of a structure [1,9,10]. The purpose of this paper is to implement the DTM to the fourth order differential equation governing the free vibration of damped orthotropic plates [1,7,8]. Most of the other methods used in solving such problem, are computationally intensive. On the other hand, DTM is relatively simple [2,3,4,5]. It involves the transformation of differential equations to their algebraic forms [3,6].    Figure 1. A moving load on a plate supported by Pasternak foundation  Manuscript received January 16, 2016; revised March 28, 2016. This work was supported by Covenant University.  M. C. Agarana is with the Department of Mathematics, Covenant University,Ota,OgunState,Nigeria.email:michael.agarana@covenantuniver sity.edu.ng.  A.N.  Ede is with the Civil Engineering Department, Covenant  University, Ota, Ogun State, Nigeria. e-mail: Anthony.ede@covenantuniversity.edu.ng  II FORMULATION OF PROBLEM The equation governing the vibration of damped simply supported orthotropic plate resting on Pasternak foundation subject to a moving load is given by [1,9]; 4 4 4 2 2 21 2 3 14 2 2 4 2 2 22 2   (1)w w w w w w wKw m m Gx x y y t t x y                           where w = w(x,y,t) is the deflection of the plate at the point (x, y). t = time in seconds E = Young’s modulus m = mass density per unit area H = thickness of plate v = velocity K, G1 = foundation stiffness   = viscous damping coefficient 1 = flexural rigidity in the x direction  3= flexural rigidity in the y direction  2 = effective torsional rigidity                                                                                                                     A. Initial and boundaryConditions  The following initial and boundary conditions are used [1,9] (x, y,0) w (x, y,0) 0tw  (0,y, t) w(a,y, t) w (0,y, t) w (a,y, t) 0w(x,0, t) w(x,b, t) w (x,0, t) w (x,b, t) 0xx xxyy yyw                          III. METHOD OF SOLUTION     The Taylor series expansion of a function f(x), about x=0, is given by [4]: AApplication of Differential Transform Method to Vibration Analysis of Damped Railway Bridge on Pasternak Foundation under Moving Train    Agarana M.C. and Ede A.N.  Proceedings of the World Congress on Engineering 2016 Vol II WCE 2016, June 29 - July 1, 2016, London, U.K.ISBN: 978-988-14048-0-0 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2016 000( ) ( )                                                                                                           1where                                               !kkxkk kx xf x x x Fd fFk dx                                                               kF  is called the kth order differential transform of f(x) about the point x = x0. Usually, the series is truncated to finite number of terms for practical problems. For this work we used eight terms. Assuming the two opposite edges Y = 0 and Y = 1 to be simply supported, the deflection function can be expressed as [1,4,5] ( ) ( )                                            (2)W w X sin m Y  Substituting equation (4)  into equation (1) leads to [4,5,6] 4 22 2 4 41 2 34 222 21 1220                                        (3)d w d wm m w Kwdx dxd wG G m wdx        Taking the differential transform of equation (5) at x0 = 0 and using the differential transforms of some of the fundamental functions reported in [ 2,3,4,5,6], we have 2 21 4 2 24 4 2 21 2 3 14! 2!(2 )! !2!( ) 0     (4)!k kk kk kw m wk kkG w m G m K wk                 Equation (6) can be written in a more concise form as :          2 21 4 2 1 24 4 2 23 14! 2!(2 )! !( ) 0                          (4)k kkk kw m G wk km G m K w                                          The boundary conditions was also transformed in similar way. Now equation (7) can be rewritten in the form; [4],         4 22 2 4 4 44 22( ) ( ) 0  (5)d w d wm m K wdx dx         where  2 2 121 22 2 24 1 1 13 1            2                                                                                                              K+               (m ) =                 GmK G mand                                                                     The general solution of equation (8) can be expressed as [4,6] 1 1 2 22 212,  k=0,1,2,3...                       (6)!whereR ( )                                                          R (k kkkkd R d Rwkm Km     2 2)                                                          K     and  ab   , 2  is the frequency parameter, a and b are the length and breadth of the plate respectively.  G1and K are the Pasternak foundation moduli,d1 and d2 are parameters.                                                                                         IV. RESULT AND DISCUSSION     For the numerical work, the fourth order differential equation (1) was solved using DTM. The  values of various parameters  used are: m = 7, V = 3.5, 4.5, 5.5, a=1, b=2, d1=1, d2=2,  =  0.02, 0.34, 0.55, G1= 0, 0.09, 0.9, 0.18, E = 2.109 x 107, 2 =101 and K=0,1,2,3,4,5. It can be seen from Figure 2 that as the foundation modulus  increases the deflection of the damped orthotropic plate decreases. This implies that the foundation reduces deflection of the plate. Also Figure 3 - 5 show the effect of damping on the deflection of orthotropic rectangular plate. It can be seen from the Figures that the higher the damping coefficient the lower the maximum amplitude. The damping effect also reduces the deflection of a plate under a moving load.   Figure 2. Deflection of the plate as the foundation modulus increases for various time t.    Figure 3. Deflection of the plate when damping coefficient is 0.02, for various time t Proceedings of the World Congress on Engineering 2016 Vol II WCE 2016, June 29 - July 1, 2016, London, U.K.ISBN: 978-988-14048-0-0 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2016   Figure 4. Deflection of the plate when damping coefficient is 0.34, for various time t   Figure 5. Deflection of the plate when damping coefficient is 0.55, for various time t   V.  CONCLUSION Application of differential transform method to the analysis of transverse vibration of orthotropic damped rectangular railway bridge supported by Pasternak foundation was carried out. The results obtained revealed that both the foundation modulus and damping have effects on the deflection of the railway bridges which was modelled as the plate in this work. These results and others obtained are consistent with the ones in the literature, as evident in some works referenced.  REFERENCES [1] Are E.B., Idowu A.S. and Gbadeyan J.A., Vibration of damped simply supported orthotropic rectangularplates resting on elastic winkler foundation, subjected to moving loads, Advances in Applied Science Research, 2013, 4(5), 387 – 393. [2] K. Yildirim, B.Ibis and M. Bayran, New solutions of the non linear Fisher type equations by the reduced differential transform, Nonlinear Sci. Lett. A., 2012, 3(1), 29-36 [3] M. Matinfar and M. Sacidy, Application of Homotopy analysis method to fourth order parabolic partial differential equation, Appl. Appl. Math. An International journal (AAM,), 2010, 2(1), 76-80 [4] Y. Kumar, Free Vibration analysis of Isotropic Rectangular plates on Winkler foundation using differential transform method, International Journal of Applied Mechanics and Engineering (2013) Vol. 18, no.2, pp 589-597 [5] Y, Kumar, differential transform method to study free transverse vibration of monoclinic rectangular plates restimg on Wimkler foundation, Applied and Computational mechanics, 2013, 7, 145-154 [6] Malik and Allali M., Characteristic equations of rectangular plates by differential transform method, J. Sound Vib., 2000, 233(7), 359-366 [7] M. C. Agarana and O.O.Agbola, Dynamic Analysis of damped driven pendulum using Laplace transform,international journal of Mathematics and computation, 2015, Vol. 26, no 3, 98-107. [8] M. C. Agarana and S. A. Iyase, Analysis of Hermite’s Equation Governing the Motion of Damped Pendulum with Small Displacement, International Journal of Physical Sciences 2015, 10(2), 364 – 370 [9] M. C. Agarana, and J. A. Gbadeyan, "Finite Difference Dynamic Analysis of Railway Bridges Supported by Pasternak Foundation under Uniform Partially Distributed Moving Railway Vehicle," Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering and Computer Science 2015, 21-23 October, 2015, San Francisco, USA, pp996-1000  [10] M. C. Agarana, J. A. Gbadeyan, O. O. Agboola, T. A. Anake, O. J. Adeleke, ‘‘ Dynamic Response of an inclined Railway bridge supported by Winkler Foundation under a Moving Railway Vehicle.’’ Australian Journal of Basic and Applied Sciences, 2015, Vol. 9 No.11, pp 355-361 [11] J.A. Gbadeyan and M.C. Agarana, 2014. “Dynamic analysis of railway bridges supported by Winkler foundation under uniform partially distributed moving railway vehicle”. WIT Transactions on The Built Environment, Vol.135. WIT Press, 2014. Proceedings of the World Congress on Engineering 2016 Vol II WCE 2016, June 29 - July 1, 2016, London, U.K.ISBN: 978-988-14048-0-0 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)WCE 2016

Application of Differential Transform Method to

Vibration Analysis of Damped Railway Bridge on

Pasternak Foundation under Moving Train

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Application of Differential Transform Method to Vibration Analysis of Damped Railway Bridge on Pasternak Foundation under Moving Train

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