Vibration Of a Beam on a Variable Winkler Elastic Foundation Using The Spectral Element Method
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Abstract
This study investigates the free vibration problem of Euler-Bernoulli beams resting on variable Winkler elastic foundations using the spectral element method. The soil stiffness under the foundation is assumed to vary as constant, linear, or parabolic along the beam’s longitudinal axis. Based on Euler-Bernoulli beam theory, the governing partial differential equation is derived and solved in the frequency domain. The spatial solution is obtained by solving the fourth-order linear homogeneous differential equation, while an integral formulation is employed to determine the particular solution. The obtained solution is then used to examine the effects of the variable elastic foundation, beam-soil interaction, and the intensity of variation of the Winkler foundation on the nonlinear natural frequencies of a cantilevered beam. Finally, the computed results are compared with available reference data, demonstrating the accuracy and robustness of the proposed numerical approach.
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