Vibratory characteristics of flexural non-uniform Euler-Bernoulli beams carrying an arbitrary number of spring-mass systems

A new exact method for the analysis of free flexural vibrations of non-uniform multi-step Euler-Bernoulli beams carrying an arbitrary number of single-degree-of-freedom and two-degree-of-freedom spring-mass systems is presented in this paper. The closed-form solutions for free vibrations of non-uniform Euler-Bernoulli beams are derived for five important cases. Then, using the massless equivalent springs to replace the spring-mass systems and the fundamental solutions developed in this paper, the frequency equation for free flexural vibrations of a multi-step non-uniform beam with any kind of support configurations and carrying an arbitrary number of spring-mass systems can be conveniently established from a second-order determinant. The proposed method is computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures.