Exact closed form solution for the analysis of the transverse vibration modes of a Timoshenko beam with multiple concentrated masses

Concentrated masses on the beams have many industrial applications such as gears on a gearbox shafts, blades and disks on gas and steam turbine shafts, and mounting engines and motors on structures. Transverse vibration of the beam carrying a point mass was studied in many cases by both Euler–Bernoulli and Timoshenko beam theory for a limited number of concentrated masses mounted on a specific place on the beam. This was also investigated for a beam carrying multiple concentrated masses, yet they were solved by numerical methods such as Differential Quadrature (DQ) method. The present study investigated an exact solution for free transverse vibrations of a Timoshenko beam carrying multiple arbitrary concentrated masses anywhere on the beam with various boundary conditions. Using Dirac’s delta in governing equations, the effects of concentrated masses were imposed. After extracting a closed form solution, basic functions were used to reduce the amount of computations. Standard symmetric and asymmetric boundary conditions were enforced for beam; in addition, the effects of value, position, and number of concentrated masses were examined. Generally, while the existence of concentrated masses reduces the natural frequencies, the reduction depends on the parameters of concentrated masses. Finally, there were acquired mode shapes for different boundary conditions and different value, position, and number of concentrated masses.