The exact characteristic equation of frequency and mode shape for transverse vibrations of non-uniform and non-homogeneous Euler Bernoulli beam with general non-classical boundary conditions at both ends

In this research, the frequency characteristic equation for some non-uniform and homogeneous beams based on Euler-Bernoulli beam theory is presented in closed analytical form. At both ends, the beam carries objects with deviations from the center, mass moment of inertia and linear and rotational elastic constraints. Therefore, the closed analytical characteristic equation has the ability to provide frequency parameters for a wide range of non-classical boundary conditions. Solving the governing differential equation and applying boundary conditions leads to the solution of an eigenvalue problem. Since the beam is non-uniform, the exact solution of the governing differential equation depends on finding a closed analytical solution for the beam deflection. Therefore, a limited type of non-uniform beams can be accurately solved. In order to verify the validity and accuracy of the presented relationships, the results of the method presented in this research have been compared with the existing results for uniform beams. Also, the mode shape of the deflection are given for a beam sample.