Types of Free Vibrations

          The following three types of free vibrations are important from the subject point of view:

1. Longitudinal vibrations, 2. Transverse vibrations, and 3. Torsional vibrations.

          Consider a weightless constraint (spring or shaft) whose one end is fixed and the other end carrying a heavy disc as shown in Fig., This system may execute one of the three above mentioned types of vibrations.

1. Longitudinal Vibrations: When the particles of the shaft or disc moves parallel to the axis of the shaft, as shown in Fig (a), then the vibrations are known as longitudinal vibrations. In this case, the shaft is elongated and shortened alternatively and thus the tensile and compressive stresses are induced alternatively in the shaft.

2. Transverse Vibrations: When the particles of the shaft or disc move approximately perpendicular to the axis of the shaft, as shown in Fig (b), then the vibrations are known as Transverse Vibrations. In this case the shaft is straightened and bent alternatively and bending stresses are induced in the shaft. 

3. Torsional Vibrations: When the particles of the shaft or disc move in a circle about the axis of the shaft, as shown in Fig (c), then the vibrations are known as Torsional vibrations. In this case, the shaft is twisted and untwisted alternatively and the torsional shear stresses are induced in the shaft.

Note: If the limit of proportionality (i.e., stress proportional to strain) is not exceeded in the three types of vibrations, then the restoring force in longitudinal and transverse vibrations or the restoring couple in torsional vibrations which is extended on the disc by the shaft ( due to the stiffness of the shaft) is directly proportional to the displacement or mean position. Hence it follows that the acceleration towards the equilibrium position is directly proportional to the displacement from that position and the vibration is, therefore, simple harmonic.