Combined influence of rotary inertia and shear coefficient on flexural frequencies of Timoshenko beam: numerical experiments

When dealing with the computation of bending frequencies for Timoshenko beams, it is common practice to assume a constant shear coefficient for a specific cross-sectional shape of the beam within the plane of bending vibration being considered. However, there have been various definitions, interpretations, and values assigned to it, depending on static and dynamic considerations, as well as experimental investigations, regardless of whether a wave approach or a mode approach is utilized. Currently, there is a lack of unanimity regarding the most suitable value for the shear coefficient. This study examines the impact of varying both the shear coefficient κ and the rotary inertia parameter r¯ _g on the first four bending modes of a uniform Timoshenko beam, considering numerous boundary conditions. The present numerical experiments reveal that increasing the shear coefficient tends to increase the frequency parameter values, while increasing the rotary inertia parameter is seen to decrease them. Both effects are very small at lower modes but become more pronounced at higher modes. The combined influence is seen to depress the frequency parameter values for all kinds of end conditions considered in this work.