Fundamental frequency for large amplitude vibrations of uniform Timoshenko beams with central point concentrated mass using coupled displacement field method

Complex structures used in many fields of engineering are made up of simple structural elements like beams, plates, etc. These structural elements, sometimes carry concentrated point masses at discrete points, and when subjected to severe dynamic environment tend to vibrate with large amplitudes. Both the continuum and the finite-element solutions are available in the open literature to tackle this coupled nonlinear problem, without concentrated point masses with particular emphasis on the fundamental linear and nonlinear frequencies. However, for short beams and moderately thick plates, one has to consider the effects of shear deformation and rotary inertia to evaluate their fundamental linear and nonlinear frequencies. A commonly used method for obtaining the same is the energy method, or a finite-element analogue of the same. In this paper the authors used a coupled displacement field method where in the number of undetermined coefficients ‘2n’ existing in the classical energy method are reduced to ‘n’, which significantly simplifies the procedure to obtain the analytical solution. The large amplitude free vibration behaviour of the most commonly used uniform shear flexible hinged–hinged and clamped–clamped beams with central point concentrated masses is studied here. This study reveals some interesting aspects concerned with the problem considered. The numerical results in terms of the linear frequency parameter and the ratios of nonlinear to linear radian frequencies for the uniform with a central point concentrated mass are given in the digital form.