Modeling and Analyzing the Free Vibration of Simply Supported Functionally Graded Beam

Abstract

Euler, Timoshenko and high shear deformation theories to analyze the free vibration of the functionally graded (FG) beam were developed. The mechanical properties of this beam were assumed to differ in thickness direction according to the model of a power-law distribution. The principle of Hamilton was used to find equations of motion. For free vibration, the analytical solution of these equations was presented using the Navier method. The effect of power index, aspect ratio, modulus ratio, and deformation theories on dimensionless frequency were studied numerically by Ansys software and analytically according to different beam theories using the Fortran program. The obtained results from these programs were compared with each other and with some previous research. Results showed an excellent agreement with the previous research. The numerical and analytical results showed that the use of this new FG beam model especially based on first and high shear deformation theories leads to the reduction of dimensionless frequency. It may be concluded that, the including of shear’s effect leads to a decrease in the dimensionless frequency. From the modeling and analysis of this model, it is possible to know what is the appropriate design for this FG beam model to reduce the vibration.

Keywords

• Dimensionless frequency
• Classical beam theory
• First and high order shear deformation theories
• Power law mode
• Analytical and numerical methods