Moderately large amplitude vibrations of uniform beams and square plates - proving intuitive formulas through rigorous analysis

Simple intuitive formulas, to calculate the large amplitude vibration frequencies, have been recently developed for the beams and square plates. The basis for development of these formulas is the master formula, obtained for the initially loaded structural members. These basic formulas, for the initially loaded beams and square plates, are modified to obtain the ratios of the nonlinear frequency (undergoing moderately large amplitude vibrations), for a given amplitude ratio, to the linear frequency parameters. Though, these intuitive formulas give a quick and accurate estimation for large amplitude vibrations, it is necessary to show that these formulas can be obtained through the rigorous formulation such as, the energy formulation. This is the main objective of the present study, where in it is demonstrated that the intuitive formulas, for the large amplitude vibrations of beams and square plates, are exactly the same with those derived by the rigorous energy formulation, for the s-s-s-s and c-c-c-c square plates, where s and c stand for the simply supported (hinged for beams) and clamped end/edge conditions. This study is further extended to beams and square plates with different boundary conditions, other than h-h and c-c boundary conditions for beams and s-s-s-s and c-c-c-c boundary conditions for square plates. The numerical results presented give the required confidence to the structural engineers and researchers, to use the simple intuitive formulas, whenever required. © 2013 © 2013 The Institution of Engineers, Singapore.