Exact Closed Form Characteristic Equations for Transverse Vibration of Timoshenko Beams

The dimensionless equations of motion are derived based on the Timoshenko beam theory to study the transverse vibration of beams without further usage of any approximate method. The exact closed form characteristic equations are given within the validity of the Timoshenko beam theory for beams having various boundary conditions. Accurate Eigen frequency parameters are presented for a different length to height ratio for each case. The exact closed form mode shapes related to deflection, slope due to bending and stress resultants are also presented and illustrated for some cases. The modal tests are performed for beams with clamped-Free and Free-Free boundary conditions. Finally, the effect of boundary conditions, length to height ratio on the eigenvalues parameters and vibratory behavior of each distinct case are studied. Validity of the derived closed form characteristic equations are checked through comparison of numerical solutions with the available results. It is believed that in the present work, the exact closed form characteristic equations and their associated Eigen functions, except for the beams with simply supported ends, for the rest of considered cases are obtained for the first time.