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A generalization of the method of lines for the numerical solution of coupled, forced vibration of beams

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  • Sarker, Pratik
  • Chakravarty, Uttam K

Abstract

Beam vibrations are encountered in real life and are important to investigate for proper monitoring of the structural health. Closed-form solutions to one-dimensional beam vibration problems are not always available, especially, if the governing equations are nonlinear or have strongly coupled multiple degrees-of-freedom for which, the numerical method is the only solution technique. The method of lines is a numerical technique for solving the initial boundary value problems; however, in the literature, the application is mostly based on simple or lower order linear governing equations with only one degree-of-freedom. Therefore, in this paper, the generalized solution of one-dimensional, axially loaded, coupled, forced beam vibration having multiple degrees-of-freedom is developed by the method of lines which is applicable to any other similar initial boundary value problems. Four different case studies featuring coupled/uncoupled, linear/nonlinear governing equations having single/multiple degree(s)-of-freedom with different types of boundary conditions are presented to demonstrate the applicability of the generalized theory. The models are validated either by theoretical solutions, simulated results, or by published results.

Suggested Citation

  • Sarker, Pratik & Chakravarty, Uttam K, 2020. "A generalization of the method of lines for the numerical solution of coupled, forced vibration of beams," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 170(C), pages 115-142.
    • Handle: RePEc:eee:matcom:v:170:y:2020:i:c:p:115-142
    DOI: 10.1016/j.matcom.2019.10.011
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    1. A. Bouhamidi & K. Jbilou, 2013. "A Fast Block Krylov Implicit Runge–Kutta Method for Solving Large-Scale Ordinary Differential Equations," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & E. N. Pistikopoulos (ed.), Optimization, Simulation, and Control, edition 127, pages 319-330, Springer.
    2. Berardi, Marco & Vurro, Michele, 2016. "The numerical solution of Richards’ equation by means of method of lines and ensemble Kalman filter," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 125(C), pages 38-47.
    3. Younes, A. & Konz, M. & Fahs, M. & Zidane, A. & Huggenberger, P., 2011. "Modelling variable density flow problems in heterogeneous porous media using the method of lines and advanced spatial discretization methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(10), pages 2346-2355.
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