Stability Analysis of a Beam under the Effect of Moving Masses using Homotopy Perturbation Method
Authors
Abstract
In this paper, considering all the linear and nonlinear inertia terms of moving masses on a flexible beam, the dynamic response and dynamic stability of the beam are studied. Homotopy perturbation method is used to perform the analysis and results are provided in a stability map for the different values of mass and velocity of the moving masses. It is concluded that there is a borderline in the diagram that separates the stable and unstable regions. For the first time, this borderline is determined semi-analytically. Results of the stability analysis are validated using the Floquet theory. In addition to this borderline, it is also concluded that the Homotopy perturbation method is capable of evaluating the new critical values for mass and velocity which cause vibration resonance in the beam. The locus of these resonant points, which is totally a new finding in dynamic analysis of beam-moving mass problem, is determined semi-analytically. Finally, the effect of the friction between the beam and the moving mass is studied on the stability of the system and resonant conditions. Accuracy of the results for this case is also evaluated with a numerical simulation.
Keywords
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23. Pirbodaghi, T., Fesanghary, M., and Ahmadian, M. T., “Non-Linear Vibration Analysis of Laminated Composite Plates Resting on Nonlinear Elastic Foundations”, Journal of the Franklin Institute, Vol. 348, pp. 353-368, 2011.
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25. Dehestani, M., Mofid, M. and Vafai, A., “Investigation of Critical Influential Speed for Moving Mass Problems on Beams”, Applied Mathematical Modeling, Vol. 33, pp. 3885-3895, 2009.
26. D’Angelo, H., Linear Time-Varying System: Analysis and Synthesis, Allyn and Bacon, Boston, 1970.
27. Mackertich, S., “Dynamic Stability of a Beam Excited by a Sequence of Moving Mass Particles”, Journal of Acoustical Society of America, Vol. 115, pp. 1416-1419, 2004.
2. Yau, J. D., “Dynamic Response Analysis of Suspended Beams Subjected to Moving Vehicles and Multiple Support Excitations”, Journal of Sound and Vibration, Vol. 325, pp. 907-922, 2009.
3. Garinei, A., and Risitano, G., “Vibration of Railway Bridges for High Speed Trains under Moving Loads Varying in Time”, Engineering Structures, Vol. 30, pp. 724-732, 2008.
4. Ju, S. H., Lin, H. T., and Huang, J. Y., “Dominant Frequencies of Train-Induced Vibrations”, Journal of Sound and Vibration, Vol. 319, pp. 247-259, 2009.
5. Yang, Y. B., Wu, C. M., and Yau, J. D., “Dynamic Responses of a Horizontally Curved Beam Subjected to Vertical and Horizontal Moving Loads”, Journal of Sound and Vibration, Vol. 242, pp. 519-537, 2001.
6. Yang, Y. B., Lin, C. L., Yau, J. D., and Chang, D. W., “Mechanism of Resonance and Cancellation for Train-Induced Vibrations on Bridges with Elastic Bearings”, Journal of Sound and Vibration, Vol. 269, pp. 345-360, 2004.
7. Yau, J. D., and Yang, Y. B., “Vertical Accelerations of Simple Beams Due to Successive Loads Traveling at Resonant Speeds”, Journal of Sound and Vibration, Vol. 289, pp. 210-228, 2006.
8. Ruzzene, M., and Baz, A., “Dynamic Stability of Periodic Shells with Moving Loads”, Journal of Sound and Vibration, Vol. 296, pp. 830-844, 2006.
9. Verichev, S. N., and Metrikine, A. V., “Instability of a Bogie Moving on Flexibly Supported Timoshenko Beam”, Journal of Sound and Vibration, Vol. 253, pp. 635-668, 2002.
10. Verichev, S. N., and Metrikine, A. V., “Instability of Vibrations of Mass That Moves Uniformly Along a Beam on a Periodically Inhomogeneous Foundation”, Journal of Sound and Vibration, Vol. 260, pp. 901-925, 2003.
11. Wang, J. F., Lin, C. C., and Chen, B. L., “Vibration Suppression for High-Speed Railway Bridges Using Tuned Mass Dampers”, International Journal of Solids and Structures, Vol. 40, pp. 465-491, 2003.
12. Visweswara Rao, G., “Linear Dynamics of an Elastic Beam under Moving Loads”, Journal of Vibration and Acoustic, Vol. 122, pp. 281-289, 2000.
13. Wang, Y. M., “The Dynamic Analysis of a Beam-Mass System Due to the Occurrence of Two-Component Parametric Response”, Journal of Sound and Vibration, Vol. 258, pp. 951-967, 2002.
14. Zheng, D. Y., and Fan, S. C., “Instability of Vibration of a Moving-Train-and-Rail Coupling System”, Journal of Sound and Vibration, Vol. 255, pp. 243-259, 2002.
15. Ahmadian, M. T., Mojahedi, M., and Moeenfard, H., “Free Vibration Analysis of a Nonlinear Beam using Homotopy and Modified Lindstedt-Poincare Methods”, Journal of Solid Mechanics, Vol. 1, pp.29-36, 2009.
16. Pirbodaghi, T., Ahmadian, M. T., and Fesanghary, M., “On the Homotopy Analysis Method for Non-Linear Vibration of Beams”, Mechanics Research Communications, Vol. 36, pp. 143–148, 2009.
17. Ganji, S. S., Ganji, D. D., Sfahani, M. G., and Karimpour, S., “Application of AFF and HPM to the Systems of Strongly Nonlinear Oscillation”, Current Applied Physics, Vol. 10, pp. 1317-1325, 2010.
18. Cveticanin, L., “Application of Homotopy-Perturbation to Non-Linear Partial Differential Equations”, Chaos, Solitons and Fractals, Vol. 40, pp. 221-228, 2009.
19. Temimi, H., Ansari, A. R., and Siddiqui, A. M., “An Approximate Solution for the Static Beam Problem and Nonlinear Integro-Differential Equations”, Computers and Mathematics with Applications, Vol. 62, pp. 3132-3139, 2011.
20. Wang, J., Chen, J. K., and Liao, S., “An Explicit Solution of the Large Deformation of a Cantilever Beam under Point Load at the Free Tip”, Journal of Computational and Applied Mathematics, Vol. 212, pp. 320-330, 2008.
21. Hoseini, S. H., Pirbodaghi, T., Ahmadian, M. T., and Farrahi, G. H., “On the Large Amplitude Free Vibrations of Tapered Beams: An Analytical Approach”, Mechanics Research Communications, Vol. 36, pp. 892-897, 2009.
22. Rafiq, A., Malik, M. Y., and Abbasi, T., “Solution of Nonlinear Pull-In Behavior in Electrostatic Micro-Actuators by Using He’s Homotopy Perturbation Method”, Computers and Mathematics with Applications, Vol. 59, pp. 2723-2733, 2010.
23. Pirbodaghi, T., Fesanghary, M., and Ahmadian, M. T., “Non-Linear Vibration Analysis of Laminated Composite Plates Resting on Nonlinear Elastic Foundations”, Journal of the Franklin Institute, Vol. 348, pp. 353-368, 2011.
24. Ozis, T., and Yildirim, A., “A Comparative Study of He’s Homotopy Perturbation Method for Determining Frequency-Amplitude Relation of a Nonlinear Oscillator with Discontinuities”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, pp.243-248, 2007.
25. Dehestani, M., Mofid, M. and Vafai, A., “Investigation of Critical Influential Speed for Moving Mass Problems on Beams”, Applied Mathematical Modeling, Vol. 33, pp. 3885-3895, 2009.
26. D’Angelo, H., Linear Time-Varying System: Analysis and Synthesis, Allyn and Bacon, Boston, 1970.
27. Mackertich, S., “Dynamic Stability of a Beam Excited by a Sequence of Moving Mass Particles”, Journal of Acoustical Society of America, Vol. 115, pp. 1416-1419, 2004.
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