Stress and Strain Limits on the Attainable Velocity in Mechanical Vibration

The maximum peak velocity which can be developed at the free end of a slender rod driven at resonance in a normal mode of longitudinal vibration is given by $v0 = c0Sxx$⁠, where c₀ is the Young's‐modulus speed of sound $(E/ρ)12$ and S_xx is the limiting value of extensional strain for the rod material. The ratio of allowable stress to the maximum peak velocity is given by $Txx/v0 = ρc0$⁠, where the characteristic specific impedance plays in this case the role of a transfer impedance relating the stress at a node to the velocity at an antinode. The same expressions describe, within half an order of magnitude, the relations between the maximum allowable stress or strain and the normal component of velocity at the antinodes of displacement for thin uniform bars or plates and for wedges or cones vibrating in flexure, and for an exponential solid horn vibrating longitudinally. By inductive extension it is argued that within the indicated precision the same velocity‐strain ratios prevail in the vibration of any elastic body however excited.