Dynamic Instability Analysis of Transverse Vibrations of Functionally Graded Rectangular Plates under Moving Masses
Author
Abstract
In this paper, dynamic instability due to parametric and external resonances of moderately thick functionally graded rectangular plates, under successive moving masses, is examined. Plate mass per unit volume and Young’s modulus are assumed to vary continuously through the thickness of the plate and obey a power-law distribution of the volume fraction of the constituent. The considered rectangular plates have two opposite simply supported edges while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The governing coupled partial differential equations of the plate are derived based on the first-order shear deformation theory with consideration of the rotational inertial effects and the transverse shear stresses. All inertial components of the moving masses are considered in the dynamic formulation. Using the Galerkin procedure, the partial differential equations are transformed into a set of ordinary differential equations with time-dependent coefficients. The Homotopy Analysis Method (HAM) is implemented as a semi-analytical method to obtain stable and unstable zones and external resonance curves in a parameters space. The effects of the index of volume fraction, thickness to length ratio, and different combinations of the boundary conditions on the dynamic stability of the system are also investigated. The results indicate that decreasing the index of volume fraction, increasing thickness to length ratio, and higher degree of edge constraints (respectively from free to simply-supported to clamped) applied to the other two edges of the plate shift up the instability region and resonance curves in the parameters plane and, from a physical point of view, the system becomes more stable. In addition to using numerical simulations of the plate midpoint displacement, Floquet theory is also employed to validate the HAM results. Finally, the results of this study, in a particular case, are compared and validated with the results of other works.
Keywords
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15. Liao, S. J., “Comparison Between the Homotopy Analysis Method and Homotopy Perturbation Method”, Applied Mathematics and Computation, Vol. 169, pp. 1186-1194, 2005.
16. Yousefzadeh, Sh., Akbari, A., and Najafi, M., “Hydro-Elastic Vibration Analysis of Functionally Graded Rectangular Plate in Contact with Stationary Fluid”, European Journal of Computational Mechanics, Vol. 27, pp. 229-246, 2018.
17. Hosseini-Hashemi, Sh., Rokni Damavandi Taher, H., Akhavan, H., and Omidi, M., “Free Vibration of Functionally Graded Rectangular Plates using First-Order Shear Deformation Plate Theory”, Applied Mathematical Modelling, Vol. 34, pp. 1276-1291, 2010.
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19. Qian, Y., and Chen, S., “Accurate Approximate Analytical Solutions for Multi-Degree-of-Freedom Coupled Vander Pol-Duffing Oscillators by Homotopy Analysis Method”, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, pp. 3113–3130, 2010.
20. D᾿Angelo, H., Linear Time-Varying System: Analysis and Synthesis, Allyn and Bacon, Boston, 1970.
21. Torkan, E., and Pirmoradian, M., and Hashemian, M., “On the Parametric and External Resonances of Rectangular Plates on an Elastic Foundation Traversed by Sequential Masses”, Archive Applied Mechanics, Vol. 88, pp. 1411-1428, 2018
2. Malekzadeh, P., and Darai, M., “Dynamic Analysis of Functionally Graded Truncated Conical Shells Subjected to Asymmetric Moving Loads”, Thin-Wall Structures, Vol. 84, pp. 1-13, 2014.
3. Song, Q., Shi, J., and Liu, Z., “Vibration Analysis of a Functionally Graded Plate with a Moving Mass”, Applied Mathematical Modeling, Vol. 46, pp. 141-160, 2017.
4. Yousezadeh, S., Akbari, A., and Najafi, M., “Dynamic Response of FG Rectangular Plate in Contact with Stationary Fluid under Moving Load”, Journal of Science and Technology of Composites, Vol. 6, pp. 213-224, 2019 (In Persian).
5. Nelson, H. D., and Conover, R. A., “Dynamic Stability of a Beam Carrying Moving Masses”, Applied Mechanics, Vol. 38, No. 4, pp.1003-1006, 1971.
6. Mackertich, S., “Dynamic Stability of a Beam Excited by a Sequence of Moving Mass Particles”, Acoustical Society of America, Vol. 115, No. 4, pp. 1416-1419, 2004.
7. Aldraihem, O. J., and Baz, A., “Dynamic Stability of Stepped Beams under Moving Loads”, Journal of Sound and Vibration, Vol. 250, No. 5, pp. 835-848, 2002.
8. Rao, G. V., “Linear Dynamics of an Elastic Beam under Moving Loads”, Journal of Vibration and Acoustics, Vol. 122, pp. 281-289, 2000.
9. Pirmoradian, M., and Karimpour, H., “Nonlinear Effects on Parametric Resonance of a Beam Subjected to Periodic Mass Transition”, Modares Mechanical Engineering, Vol. 17, No. 1, pp. 284-292, 2017 (In Persian).
10. Torkan, E., Pirmoradian, M., and Hashemian, M., “Stability Analysis of Transverse Vibrations of Rectangular Plate under Periodic Passage of Moving Masses”, Journal of Mechanical Engineering and Vibration, Vol. 8, No. 3, pp. 18-26, 2017 (In Persian).
11. Torkan, E., Pirmoradian, M., and Hashemian, M., “Instability Inspection of Parametric Vibrating Rectangular Mindlin Plates Lying on Winkler Foundations under Periodic Loading of Moving Masses”, Acta Mechanica Sinica, Vol.35, pp. 242-263, 2019.
12. Torkan, E., Pirmoradian, M., and Hashemian, M., “Dynamic Instability Analysis of Moderately Thick Rectangular Plates Influenced by an Orbiting Mass Based on the First-Order Shear Deformation Theory”, Modares Mechanical Engineering, Vol. 19, No. 9, pp. 2203-2213, 2019 (In Persian).
13. Ghomeshi Bozorg, M., and Keshmiri, M., “Stability Analysis of a Beam under the Effect of Moving Masses Using Homotopy Perturbation Method”, Journal of Computational Methods in Engineering, Vol. 34, No. 1, pp. 79-95, 2015 (In Persian).
14. Hassanabadi, M. E., Attari, N. K. A., Nikkhoo, A., and Mariani, S., “Resonance of a Rectangular Plate Influenced by Sequential Moving Mass”, Coupled Systems Mechanics, Vol. 5, pp.87-100, 2016.
15. Liao, S. J., “Comparison Between the Homotopy Analysis Method and Homotopy Perturbation Method”, Applied Mathematics and Computation, Vol. 169, pp. 1186-1194, 2005.
16. Yousefzadeh, Sh., Akbari, A., and Najafi, M., “Hydro-Elastic Vibration Analysis of Functionally Graded Rectangular Plate in Contact with Stationary Fluid”, European Journal of Computational Mechanics, Vol. 27, pp. 229-246, 2018.
17. Hosseini-Hashemi, Sh., Rokni Damavandi Taher, H., Akhavan, H., and Omidi, M., “Free Vibration of Functionally Graded Rectangular Plates using First-Order Shear Deformation Plate Theory”, Applied Mathematical Modelling, Vol. 34, pp. 1276-1291, 2010.
18. Wang, C., Wu, Y. Y., and Wu, W., “Solving the Nonlinear Periodic Wave Problems with the Homotopy Analysis Method”, Wave Motion, Vol. 4, pp. 329-337, 2005.
19. Qian, Y., and Chen, S., “Accurate Approximate Analytical Solutions for Multi-Degree-of-Freedom Coupled Vander Pol-Duffing Oscillators by Homotopy Analysis Method”, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, pp. 3113–3130, 2010.
20. D᾿Angelo, H., Linear Time-Varying System: Analysis and Synthesis, Allyn and Bacon, Boston, 1970.
21. Torkan, E., and Pirmoradian, M., and Hashemian, M., “On the Parametric and External Resonances of Rectangular Plates on an Elastic Foundation Traversed by Sequential Masses”, Archive Applied Mechanics, Vol. 88, pp. 1411-1428, 2018
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