Graphical representation of Timoshenko beam modes for clamped-clamped boundary conditions at high frequency: Beyond transverse deflection

Transverse deflection and bending-slope mode shapes for a clamped-clamped Timoshenko beam are presented in combined form and compared with finite element (FE) results. The linkage between the cut-on frequency for multiple phase velocities, and noteworthy change in the character of the mode shapes predicted by Timoshenko beam theory (TBT), observed in recent paper by van Rensburg and van der Merwe, is illustrated. While previous work in bounded systems has employed comparisons of TBT and FE obtained resonance frequencies to demonstrate the limits of validity of TBT, direct comparison of the mode shapes predicted by TBT and FE provides an alternative means of comparison, and unambiguously indicates TBT is not valid above the cut-on frequency for the case considered. Typographical errors in Huang's analysis are also noted.