# Free Vibration Analysis of Nonlinear Circular Plates Resting on Winkler and Pasternak Foundations

Document Type: Research Paper

Authors

1 Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria

2 Department of civil and Environmental Engineering, University of Lagos, Akoka, Nigeria

10.22034/jsm.2019.1866236.1423

Abstract

Dynamic behaviour of nonlinear free vibration of circular plate resting on two-parameters foundation is studied. The governing ordinary differential equation is solved analytically using hybrid Laplace Adomian decomposition method. The analytical solutions obtained are verified with existing results in literature. The analytical solutions are used to determine the influence of elastic foundation, radial and circumferential stress on natural frequency of the plate. Also, the radial and circumferential stress determined. From the results, it is observed that, increase in elastic foundation parameter increases the natural frequency of the plate. It is recorded that the modal radial and circumferential stress affect the extrema mode of the plate. It is hoped that the present study will contribute to the existing knowledge in the field of vibration analysis of engineering structures.

Keywords

### References

[1] Andrea R.D., Silva A., Ricardo A.M., Silveira A., Paulo B., Gonc-Balves B., 2001, Numerical methods for analysis of plates on tensionless elastic foundations, International Journal of Solids and Structures-Pergamon 38: 2083-2100.
[2] Lamb H., 2016, On the vibrations of an elastic plate in contact with water, Proceedings of the Royal Society of London , Series A 98: 205-216.
[3] Raoa L.B., Raob C.K., 2018, Vibrations of circular plates resting on elastic foundation with elastically restrained edge against translation, The Journal of Engineering Research 15(1): 14-25.
[4] Haojiang D., Rongqiao X., Weiqiu C., 2000, Exact solutions for free vibration of transversely isotropic piezoelectric circular plates, Acta Mechanica Sinica 16(2): 141-147.
[5] Soni S., Jain N.K., Joshi P.V., 2017, Aanalytical modeling for non-linear vibration analysis of functionally graded plate submerged in fluid, Indian Journal of Science Research 4(2): 229-236.
[6] Kumar S., Prashar P., 2015, Energy method to analysis the vibration of orthotropic circular plate with bi-dimensional thickness variation, International Journal of Applied Research 1(12): 829-835.
[7] Siddiqul A.M., Haroon T., Bhatti S., Ansari A.R., 2011, A comparison of the adomian and homotopy perturbation methods in solving the problem of squeezing flow between two circular plates, Mathematical Modelling and Analysis 15(4): 491-504.
[8] Yin-Shan Y., Temuer C., 2015, Application of the Homotopy perturbation method for the large deď¬‚ection problem of a circular plate, Applied Mathematical Modelling 39: 1308-1316.
[9] Rostamiyan Y., Fereidoon A., Davoudabadi M.R., Yag H., 2010, Analytical approach to investigation of deflection of circular plate under uniform load by Homotopy perturbation method, Mathematical and Computational Applications 15(5): 816-821.
[10] Yalcin H.S., Arikoglu A., Ozkol I., 2009, Free vibration analysis of circular plates by differential transformation method, Applied Mathematics and Computation 212: 377-386.
[11] Xiao-Jing Z., 1987, On exact solution of karman's equations of rigid clamped circular plate and shallow spherical shell under a concentrated load, Applied Mathematics and Mechanics 8(11): 1057-1068.
[12] Shariyat M., Alipour M.M., 2011, Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations, Archive Applied Mechanism 81: 1289-1306.
[13] Leissa A.W., 1970, Vibration of Plates, National Aeronautics and Space Adminstration, Washington.
[14] Wu T.Y., Wung T.Y., Liu G.R., 2002, Free vibration analysis of circular plate using generalize differential quadrature rule, Computer Methods in Applied Mechanics and Engineering 191: 5365-5380.