Temperature Influence on Nonlinear Responses of Timoshenko Beam

Abstract

The goal of this paper is to study large amplitude vibrations of a Timoshenko beam under an influence of the elevated temperature. It is assumed that the beam gets the elevated temperature instantly and the temperature is uniformly distributed along the beam’s length and cross-section. The mathematical model represented by a set of partial differential equations is derived taking into account boundary conditions for a simply supported beam in the both ends. Next, the problem is reduced by the Galerkin method by means of free vibration modes. The influence of the temperature on a resonance localisation and nonlinear oscillations is studied numerically and analytically by the multiple time scale method.

Nonlinear dynamics of a reduced multimodal Timoshenko beam subjected to thermal and mechanical loadings

“…Comparing the one mode reduction given in [19] with the present study based on the three mode reduction we may conclude that for periodic excitation distributed according to the selected mode the involvement of the rest modes is small. For small vibration amplitudes the higher modes involvement can be neglected.…”

“…using expansion (19), after transformation to the trigonometric form, we obtain solutions in the zeroth and the first order approximation…”

“…According to the paper [19] longitudinal displacement can be found from the first equation of the set (1) and the model can be simplified to two partial differential equations, having the dimensionless form: …”

“…The authors found a very reach nonlinear dynamic behaviour of the system including, periodic, quasi-periodic and chaotic oscillations. A thermomechanical model of the vibration of a Timoshenko beam after its one mode reduction is studied by multiple time scale method in [19].…”

Regular and chaotic oscillations of a Timoshenko beam subjected to mechanical and thermal loadings

“…A thermo-mechanical model of a Timoshenko beam subjected to mechanical and thermal loadings, assuming elevated ambient temperature has been proposed in [29]. The set of PDEs has been reduced to the one degree of freedom nonlinear system with thermal components.…”