We now take a quick look at a few representative physical systems, so that we can appreciate the great variety of circumstances in which the mathematics of harmonic motion can be applied. The model system has taught us what properties we must look for if we wish to know whether vibration is possible in any given real system. The mathematics of the prototype can be used in a production-line spirit once we have isolated the ingredients (inertia and stiffness) necessary for vibration.

As well as appreciating the essential similarity of different kinds of vibration, we shall in this chapter be vitally interested in the numerical sizes of the physical quantities involved, and particularly in the values of the free vibration frequencies of the various systems. One of the most striking things we shall discover is the very wide spread of frequencies to be found.

Angular vibrations

The mass in the model system vibrates to and fro along a straight line. Some of the most familiar vibrations involve massive objects rotating back and forth under the influence of a return torque. We shall consider two kinds of return torque: first an elastic torque provided by a stiff suspension, and then one due to gravity.

Torsional vibrations. The system shown in fig. 2.1 is the exact rotational analogue of the model system. The coordinate ψ now measures the angular displacement of the mass from its equilibrium position.