Abstract
Approximate methods for analyzing the vibrations of an Euler--Bernoulli beam resting on a nonlinear elastic foundation are discussed. The cases of primary resonance (Ω ≈ Ω n ) and subharmonic resonance of order one-half (Ω ≈ 2 Ω n ), where Ω is the excitation frequency and Ω n is the natural frequency of the nth mode of the beam, are investigated. Approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equation and boundary conditions. The amplitude and phase modulation equations show that single-mode discretization leads to erroneous qualitative as well as quantitative predictions. Regions of softening (hardening) behavior of the system, the spatial dependence of the response drift, and frequency-response curves are numerically evaluated and compared using both approaches.
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Nayfeh, A.H., Lacarbonara, W. On the Discretization of Distributed-Parameter Systems with Quadratic and Cubic Nonlinearities. Nonlinear Dynamics 13, 203–220 (1997). https://doi.org/10.1023/A:1008253901255
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DOI: https://doi.org/10.1023/A:1008253901255