Symplectic Method for Natural Modes of Beams Resting on Elastic Foundations

This paper proposes a new symplectic method for obtaining the natural frequencies and modes of beams resting on elastic foundations. A generalized Hamiltonian functional is first derived via the Lagrange multiplier method, and the first-order dual equation of motion in the time domain and its corresponding boundary conditions are obtained. The time coordinate is separated from the dual equation to yield the natural-frequency-related eigenvalue equation and the first-order dual equation in the frequency domain. Then, the spatial coordinate is separated from the newly derived dual equation to yield the natural-mode-related eigenvalue equation. According to the eigenvalue analyses, this paper obtains a series of eigenvalue spectra: the continuous and discrete eigenvalue spectra that represent the relationships of the two types of dimensionless eigenvalues for infinite and finite beams, respectively; these eigenvalue spectra help to understand the connection between structural vibration and wave propagation. For a finite beam with a specific boundary condition, its mode vectors are composed of transverse deflection, bending rotation, shear force, and bending moment, and they are called the full mode shape vectors (FMSVs) in this paper. These FMSVs were verified to be orthogonal according to the property of the Hamilton matrix. A simply supported and a cantilever beam were used as examples to validate the accuracy and applicability of the symplectic method. Their vibration properties, mainly the discrete eigenvalue spectra and FMSVs, are comprehensively analyzed, and some significant conclusions are drawn.