Vibration analysis of heterogeneous rods and

In the first part, the longitudinal motion of a rod with varying cross-section and elastic properties will be examined. The longitudinal vibration of non-uniform rods is a subject of considerable scientific and practical interest that has been studied extensively. The methods of formulation and solution of an axially loaded rod modeled as a continuous system will be analyzed. The motions of continuous systems will be obtained from the superposition of the modes. In superposition analysis, Duhamel?s integral equation will used to evaluate the response of the system to any form of dynamic loading. The transient response of an axially-loaded bar will also be studied by performing the formulation in the Laplace transform space. The solution in the time domain will be obtained by an appropriate numerical inverse Laplace transformation method. The results will be compared with superposition analysis.In the second part, free vibration analysis of functionally graded (FG) beam will be performed. Young?s modulus of the beam varies in thickness direction according to an exponential law. Using plane strain equation of elasticity, governing equation of axial and transverse motion are solved and exact natural frequencies are obtained. The results are compared with those of various beam theories.