Free Vibration of Beams Carrying Spring-Mass Systems: A Dynamic Stiffness Approach

Free vibration analysis of beams carrying spring-mass systems is carried out by using the dynamic stiffness method. The eigenvalue problem for the free vibration study is formulated by assembling the dynamic stiffness matrices of beam and spring-mass elements. The Wittrick-Williams algorithm is then applied to yield the required natural frequencies and mode shapes of the combined system. Numerical examples are given for a cantilever beam carrying a spring-mass system at the tip. A parametric study is then carried out by varying the mass and stiffness properties of the spring-mass system and the subsequent effects on the natural frequencies and mode shapes are illustrated. The proposed theory can be applied for other boundary conditions of the beam and can be extended to complex structures carrying spring-mass systems. The results are discussed and validated against published literature.