An Efficient Numerical Method for Free and Forced Vibrations of Timoshenko Beams with Variable Cross-Section

In this study, the dynamic behaviour of beams with variable cross-sections subjected to time-dependent loads is analyzed using the Complementary Functions Method (CFM). The material of the rod is assumed to be homogeneous, linear elastic, and isotropic. The governing equation is derived based on the Timoshenko beam theory. The axial, shear deformations and cross-section non-uniformity are also taken into account in the formulation. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method. The obtained solutions are transformed into real space using the modified Durbin's numerical inverse Laplace Transform (LT) method. Free vibration is considered as a special case of forced vibration. Both vibration types of non-uniform beams are calculated for various examples with various Boundary Conditions (BCs). The influence of the non-uniformity parameter in the cross-section on free and forced vibrations is examined, and the obtained results demonstrate good agreement with existing literature ones and ANSYS.