Dynamics: Free Vibration

Vibration decay and natural frequency

This set of models demonstrates that the higher the natural frequency of a structure, the quicker the decay of its free vibration.

a) Applying initial displacements                     b) Single ruler vibrates for a longer time
Fig. 15-9: The decay of free vibration and the natural frequencies of members

A single steel ruler and a pair of identical steel rulers, bolted firmly together, are placed side by side as cantilevers shown as in Fig. 15-9a. Give the ends of the two cantilevers the same initial displacement and release them suddenly at the same time (Fig. 15-9a). Free vibrations of the rulers follow and it will be observed that the bolted double ruler stops vibrating much more rapidly than the single ruler (Fig. 15-9b).The rate of decay of free vibration is proportional to the product of the damping ratio and the natural frequency of the structure as shown in Eq. 15-15 and Example 15-1. The bolted double ruler has a higher natural frequency since its second moment of area is 8 times of that of the single ruler, making it 8 times as stiff as the single ruler, while its mass is just double of that of the single ruler. In fact the fundamental natural frequency of the bolted double ruler is twice that of the single ruler. The damping ratios for the two cantilevers can be considered to be the same. Hence this demonstration verifies the concept that the higher the natural frequency of a structure, the quicker the decay of its free vibration.