Frequencies, modes and orthogonality of vibrating timoshenko beams on viscoelastic foundation

Elastic beams resting on elastic or viscoelastic foundations are extensively studied. The present paper focuses on the free transverse vibration of a Timoshenko beam resting on three-parameter viscoelastic foundation. For the first time, under simply supported boundary conditions, the explicit formulae for natural frequencies and modal functions are derived. Furthermore, a simple and convenient way is provided to obtain analytical solution. Moreover, the orthogonal conditions of modal functions are established for the first time in the state space. In numerical examples, the natural frequencies and decrement coefficients are calculated The results show that the analytical solution of any order natural frequency is obtained with the explicit formulae, avoiding calculation error from numerical methods. The analytical solutions have a good agreement with the numerical data. The effects of the parameters of the Timoshenko beam and the viscoelastic foundation on the free vibration characteristics are also examined.