C Vibration Analysis of a Beam on Elastic Foundation with Elastically Restrained Ends Using Spectral Element Method
In this study, a spectral element method (SEM) is employed to predict the free vibration of a Euler-Bernoulli beam resting on a Winkler foundation with elastically restrained ends. The formulation of the dynamic stiffness matrix has been established by solving the differential equation of motion which was transformed to frequency domain. Non-dimensional natural frequencies and shape modes are obtained by solving the partial differential equations, numerically. Numerical comparisons and examples are performed to show the effectiveness of the SEM and to investigate the effects of various parameters, such as the springs at the boundaries and the elastic foundation parameter on the vibration frequencies. The obtained results demonstrate that the present method can also be applied to solve the more general problem of the dynamic analysis of structures with higher order precision.
 Mustafa Özgür Yayli, Murat Aras, and Süleyman Aksoy:’’ An Efficient Analytical Method for Vibration Analysis of a Beam on Elastic Foundation with Elastically Restrained Ends’’. Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 159213, 7 pages.
 Alev Kacar , H. Tugba Tan , Metin O. Kaya’’ Free vibration analysis of beams on variable Winkler elastic foundation by using the differential transform method’’. Mathematical and Computational Applications, Vol. 16, No. 3, pp. 773-783, 2011.
 D. Zhou, A General solution to vibrations of beams on variable Winkler elastic foundation. Computers & Structures 47 (1993), 83-90.
 Amin Ghannadiasl and Massood Mofid’’An Analytical Solution for Free Vibration of Elastically Restrained Timoshenko Beam on an Arbitrary Variable Winkler Foundation and Under Axial Load’’ Latin American Journal of Solids and Structures 12 (2015) 2417-2438
 G. V. Rao and N. R. Naidu, “Free vibration and stability behaviour of uniform beams and columns with non-linear elastic end rotational restraints,” Journal of Sound andVibration, vol. 176, no. 1, pp. 130–135, 1994.
 H. K. Kim and M. S. Kim, “Vibration of beams with generally restrained boundary conditions using fourier series,” Journal of Sound and Vibration, vol. 245, no. 5, pp. 771–784, 2001.
 B. Balkaya, M.O. Kaya, Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method. Meccanica 79 (2009), 135-146.
 Alima Tazabekova, Desmond Adair, Askar Ibrayev, and Jong Kim’’Free Vibration Calculations of an Euler- Bernoulli Beam on an Elastic Foundation Using He’s Variational Iteration Method’’. W. Wu and H.-S. Yu (Eds.): Proceedings of China-Europe Conference on Geotechnical Engineering, SSGG, pp. 1022–1026, 2018.