Forced vibration analysis of inhomogeneous rods with non-uniform cross-section

Forced vibration analysis of cantilever rods is presented that have material properties and cross-section areas that arbitrarily vary in the axial direction, solved using Laplace transform in time domain and complementary functions method (CFM) in the spatial domain. Under the Laplace transformation, the partial differential equation is transformed into time-independent boundary value problem in the axial direction, which is solved by CFM. Then, inverse transform is taken by modified Diubin's method into the time domain. In the end, the non-dimensional displacement results are compared with both benchmark and finite element method (FEM) solutions available in the literature. In addition to satisfying a fair amount of accuracy with small computational costs, the approach presented in this study is well-structured, simple, and efficient.