Vibrational Analysis of Cables Using the Non Standard Finite Difference Method

Document Type : Research Paper

Authors

1 Ferdowsi University of Mashhad

2 Mechanical Engineering Department, Ferdowsi University of Mashhad

Abstract

In continuous systems such as rods, cables, and shafts, the differential equation governing the behaviour of the structure can be obtained by applying Newton's second law. Only a limited group of these equations have an analytic solution; hence the provision of efficient and stable numerical methods is of particular importance. The method used in this study is a nonstandard finite difference (NSFD) scheme, which is an effective numerical method for solving the above-mentioned equations. One of the important features of this method is its relatively good high accuracy. Results obtained by nonstandard finite difference methods are more compatible with the exact behaviour of the problem than that of the standard finite difference method. In this research, the equations governing the vibrations of cables with three different boundary conditions are solved. Three solutions including the analytical solution, the standard finite difference method (SFD) and the NSFD method are compared. Results show that the error of the NSFD is significantly less than of the SFD.

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[1] Hirota, R., "The Direct Method in Solution Theory", Cambridge University Press, (2004).
 
[2] Olver, P.J., and Rosenau, P., "The Construction of Special Solutions to Partial Differential Equations", Physics Letters A, Vol. 114, pp. 107-112, (1986).
 
[3] Matinfar, M., and Mahdavi, M. , "Application of Variational Homotopy Perturbation Method for Solving the Cauchy Reaction-diffusion Problem", World Applied Sciences Journal, Vol. 9, pp. 1328-1335, (2010).
 
[4] Abassy, T.A., El-Tawil, M.A., and El Zoheiry, H., "Toward a Modified Variational Iteration Method", Journal of Computational and Applied Mathematics, Vol. 207, pp. 137-147, (2007).
 
[5] Ascher, U., "On Symmetric Schemes and Differential-algebraic Equations", SIAM Journal on Scientific and Statistical Computing, Vol. 10, pp. 937-949, (1989).
 
[6] Butcher, J., and Chan, R., "Efficient Runge-Kutta Integrators for Index-2 Differential Algebraic Equations", Mathematics of Computation of the American Mathematical Society, Vol. 67, pp. 1001-1021, (1998).
 
[7]Çelik, E., and Bayram, M., "On the Numerical Solution of Differential-algebraic Equations by Padé Series", Applied Mathematics and Computation, Vol. 137, pp. 151-160, (2003).
 
[8] Hosseini, M.M., "Numerical Solution of Linear Differential-algebraic Equations", Applied Mathematics and Computation, Vol. 162, Issue. 1, pp. 7-14, (2005).
 
[9] Hosseini, M.M., "An Index Reduction Method for Linear Hessenberg Systems", Applied Mathematics and Computation, Vol. 171, No. 1, pp. 596-603, (2005).
 
[10] Qureshi, M.I., Hashmi, M., Uddin, Z., and Iqbal, S., "A Study of Obstacle Problems using Homotopy Perturbation Method", Journal of the National Science Foundation of Sri Lanka, Vol. 45, pp. 32-45, (2017).
 
[11] Somathilake, L., and Wedagedera, J., "Investigation of Branching Structure Formation by Solutions of a Mathematical Model of Pattern Formation in Coral Reefs", Journal of the National Science Foundation of Sri Lanka, Vol. 44, pp. 15-28, (2016).
 
[12] Majeed, A., Zeeshan, A., and Gorla, R., "Convective Heat Transfer in a Dusty Ferromagnetic Fluid over a Stretching Surface with Prescribed Surface Temperature/heat Flux Including Heat Source/sink", Journal of the National Science Foundation of Sri Lanka, Vol. 46, pp. 56-60, (2018).
 
 [13] Naguleswaran, S., "Transverse Vibration of an Uniform Euler–Bernoulli Beam under Linearly Varying Axial Force", Journal of Sound and Vibration, Vol. 275, pp. 47-57, (2004).
 
[14] Shakeri, F., Dehghan, M., "Solution of a Model Describing Biological Species Living Together using the Variational Iteration Method", Mathematical and Computer Modelling, Vol. 48, pp. 685-699, (2008).
 [15] Biazar, J., Porshokuhi, M.G., and Ghanbari, B., "Extracting a General Iterative Method from an Adomian Decomposition Method and Comparing it to the Variational Iteration Method", Computers & Mathematics with Applications, Vol. 59, pp. 622-628, (2010).
 
[16] Hsu, J. C., Lai, H. Y., and Chen, C. K., "Free Vibration of Non-uniform Euler–Bernoulli Beams with General Elastically End Constraints using Adomian Modified Decomposition Method", Journal of Sound and Vibration, Vol. 318, pp. 965-981, (2008).
 
[17]Mickens, R.E., "Calculation of Denominator Functions for Nonstandard Finite Difference Schemes for Differential Equations Satisfying a Positivity Condition", Numerical Methods for Partial Differential Equations: An International Journal, Vol. 23, pp. 672-691, (2007).
 
[18] Mickens, R.E., and Washington, T.M., "NSFD Discretizations of Interacting Population Models Satisfying Conservation Laws", Computers & Mathematics with Applications, Vol. 66, pp. 2307-2316, (2013).
 
[19] Roeger, L. I.W., and Mickens, R.E., "Exact Finite-difference Schemes for First Order Differential Equations having Three Distinct Fixed-points", Journal of Difference Equations and Applications, Vol. 13, pp. 1179-1185, (2007).
 
[20]González-Parra, G., Arenas, A.J., and Chen-Charpentier, B.M., "Combination of Nonstandard Schemes and Richardson’s Extrapolation to Improve the Numerical Solution of Population Models", Mathematical and Computer Modelling, Vol. 52, pp. 1030-1036, (2010).
 
[21] Moaddy, K., Momani, S., and Hashim, I., "The Non-standard Finite Difference Scheme for Linear Fractional PDEs in Fluid Mechanics", Computers & Mathematics with Applications, Vol. 61, pp. 1209-1216, (2011).
 
[22] Ongun, M.Y., and Turhan, I., "A Numerical Comparison for a Discrete HIV Infection of CD4+ T-Cell Model Derived from Nonstandard Numerical Scheme", Journal of Applied Mathematics, Vol. 3, pp. 61-84, (2012).
 
[23]Cresson, J., and Pierret, F., "Non Standard Finite Difference Scheme Preserving Dynamical Properties", Journal of Computational and Applied Mathematics, Vol. 303, pp. 15-30, (2016).
 
[24] Zhang, L., Wang, L., and Ding, X., "Exact Finite Difference Scheme and Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher Equations", Journal of Applied Mathematics, Vol. 5, pp. 79-92, (2014).
 
[25] Mickens, R.E., Oyedeji, K., and Rucker, S., "Exact Finite Difference Scheme for Second-order, Linear ODEs Having Constant Coefficients", Journal of Sound and Vibration, Vol. 287, pp. 1052-1056, (2005).
 
[26] Roeger, L. I.W., "Exact Nonstandard Finite-difference Methods for a Linear System–the Case of Centers", Journal of Difference Equations and Applications, Vol. 14, pp. 381-389, (2008).
 
[27] Roeger, L. I.W., "Exact Finite-difference Schemes for Two-dimensional Linear Systems with Constant Coefficients", Journal of Computational and Applied Mathematics, Vol. 219, pp. 102-109, (2008).
 
[28] Zibaei, S., and Namjoo, M., "A Nonstandard Finite Difference Scheme for Solving‎‎ Fractional-order Model of HIV-1 Infection of‎‎ CD4^{+} t-cells", Iranian Journal of Mathematical Chemistry, Vol. 6, pp. 169-184, (2015).