Hysterically damped free and forced vibrations of axial and torsional bars by a closed form exact method

Hysterically damped free and forced vibrations of axial and torsional bars are investigated using a closed form exact method. The method is exact and yields closed form expressions for the vibratory displacements. This is in contrast with the well known eigenfunction superposition method which requires expressing the distributed forcing functions and the displacement response functions as infinite sums of free vibration eigenfunctions. The hysterically damped free vibration frequencies and corresponding damped mode shapes are calculated and plotted instead of undamped free vibration and mode shapes which is typically computed and applied in vibration problems. The hysterically damped natural frequency equations are exactly derived. Accurate axial or torsional amplitude vs. forcing frequency curves showing the forced response due to distributed loading are displayed with various hysteretic damping parameters.