Nonlinear free vibrations of quintic inextensional beams lying on Winkler elastic substrate based on three-mode assumptions

The vibrational behavior of quintic nonlinear inextensional beam on two-parameter elastic substrate based on the three-mode assumptions is investigated in this paper. Using the Galerkin decomposition principle by considering three-mode assumptions, the nonlinear governing equations are derived. An asymptotic approach called parameter expansion method is employed to obtain the approximate expressions of nonlinear frequency–amplitude relationship for the first, second, and third modes of vibrations. Afterwards, it is clearly shown that the first term in series expansions gives excellent results in contrast with numerical calculations. Finally, the effect of the initial configuration of the beam as well as system parameters on the natural frequencies of quintic nonlinear beams is studied.