On the temperature-dependent nonlinear vibrations of functionally graded Timoshenko beams

In this study, the nonlinear free vibrations of an elastically supported beam composed of functionally graded materials (FGMs) are investigated. The beam is in contact with a two-parameter elastic medium and is under a thermal environment. The thermomechanical properties of the beam are dependent on the temperature. Kinetic and strain energy relations for the FGM beam are obtained according to the Timoshenko beam theory and the von Karman nonlinear displacement strain relations. In order to obtain the ratio of nonlinear to linear natural frequencies, the Ritz method is used in cooperation with a direct repetition method. The results are represented in terms of boundary condition length-to-thickness ratio, volume fraction power of the constituents, temperature and elastic foundation parameters. The results show that the increased length-to-thickness ratio for FGM beams with S-S, C-S and C-C supports, decreases the ratio of the first nonlinear to linear natural frequency, while for an FGM beam with C-F support, the mentioned frequency ratio increases. Moreover, increasing the volume fraction power of the constituents for the FGM beam with S-S, C-S and C-C boundary conditions increases the aforesaid frequency ratio initially, and thereafter it decreases. Meanwhile, changing the mentioned power has no effect on the frequency ratio related to a C-F FGM beam.

Keywords:

Nonlinear natural frequency , FGM beams , Elastic foundation , Large amplitude vibrations