Non-linear forced vibration study of axially functionally graded non-uniform beams by using Broyden method

Abstract

In the forced vibration case due to external loading deflection tends to higher value so beam element undergoes the non-linearity behavior. Basically non linearity is categorized into two parts material and geometric non-linearity. Due to large amplitude we have to take care of both in-plane and out-plane displacement fields. With the help of variation form of energy principle all displacement fields calculated. These displacement fields are the combination of admissible orthogonal function. Admissible functions satisfies the both flexural and Membrane boundary conditions. By considering non-linear strain displacement relationship geometric nonlinearity is introduced in this thesis. The resulting nonlinear set of governing equations is solved through a numerical procedure involving direct substitution method using relaxation parameter. So calculation of natural frequency under both free and forced vibration decides the working condition of beam. Now a day in structural field tremendous growth is taking place for which we required a material which attains a good property and it became possible by functionally graded materials. Functionally graded material is non-homogenous and anisotropic material whose both structural and material property varies along the element. In FGM effect of residual stress and stress concentration is minimum in between two dissimilar materials which increase the strength and toughness of that structural element. So in this paper we are dealing with forced vibration analysis of FGM. For the static analysis we are using minimum potential energy principle and for the dynamic analysis we are using Hamilton’s principle. Further non-linearity of beam can be calculated by using Broyden method. Through this research we obtained some results which are validated with some previous papers and then we submitted our further research results.