On the Timoshenko theory of transverse beam vibrations

where y*(x, t) is the displacement at time t of a point on the central axis. Equation (1) has long been known to be inadequate for the higher modes of vibration. Although Rayleigh [1] introduced the effect of rotary inertia, the one-dimensional theory did not adequately explain the higher modes until Timoshenko [2], [3], extended it to include the effects of transverse shear deformation. The Timoshenko theory is based upon the following equations: