Terms Used in Vibratory Motion

          The following terms are commonly used in connection with the vibratory motions:

1. Period of vibration or time period: It is the time interval after which the motion is repeated itself. The period of vibration is usually expressed in seconds.

2. Cycle: It is the motion completed during one time period.

3. Frequency: It is the number of cycles described in one second. In S.I. units, the frequency is expressed in Hertz (briefly written in Hz) which is equal to one cycle per second. 

Types of Vibratory Motion

The following types of vibratory motion are important from the subject point of view:

1. Free or Natural vibrations: When no external force acts on the body, after giving it an initial displacement, then the body is said to be under Free or Natural vibrations. The frequency of the free vibrations is called Free natural frequency.

2. Forced vibrations: When the body vibrates under the influence of external force, then the body is said to be under Forced vibrations. The external force applied to the body is a periodic disturbing force created by unbalance. The vibrations have the same frequency as the applied force.

Note: When the frequency of the external force is same as that of the natural vibrations, resonance takes place.

3. Damped vibrations: When there is a reduction in amplitude over every cycle of vibration, the motion is said to be Damped vibration. This is due to the fact that a certain amount of energy possessed by the vibrating system is always dissipated in overcoming frictional resistance to the motion.

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.

Natural Frequency of Free Longitudinal Vibrations

          The natural frequency of the free longitudinal vibrations may be determined by the following three methods:

1. Equilibrium Method:

          Consider a constraint (i.e spring) of negligable mass in an unstained position, as shown in Fig (a).,

          Let       s = Stiffness of the constraint. It is the force required to produce unit displacement in the direction of vibration. It is usually expressed in N/m.

                         m = Mass of the body suspended from the constraint in kg,

                         W = Weight of the body in newtons = m.g,

                  δ = Static deflection of the spring in metres due to weight W newtons, and

                      x = Displacement given to the body by the external force, in metres.

          In the equilibrium position, as shown in Fig (b), the gravitational pull W = m.g, is balanced by a force of spring, such that W = s. δ.

          Since the mass is now displaced from its equilibrium position by a distance 'x' , as shown in Fig. (c), and is then released, therefore after time 't',

      Where      δ = Static deflection i.e. extension or compression of the constraint,

                                  W = Load attached to the free end of constraint,

                                  l = Length of the constraint,

                                  E = Young’s modulus for the constraint, and

                                  A = Cross-sectional area of the constraint.

2.  Energy method:

          We know that the kinetic energy is due to the motion of the body and the potencil energy is with respect to a certain datum position which is equal to the amount of work required to move the body from the datam position. In the case of vibrations, the datam position is the mean or equilibrium position at which th potencial energy of the body or the system is zero.

          In the free vibrations, no energy is transferred to the system or from the system. Therefore the summation of kinetic energy and potencial energy must be a constant quantity which is same at all the times. In other words,

          The time period and the natural frequency may be obtained as discussed in the previous method.

3. Rayleigh’s method:

          In this method, the maximum kinetic energy at the mean position is equal to the maximum potencial energy (or strain energy) at the extreme position. Assuming the motion executed by the vibration to be simple harmonic, then

Note:

In all the expressions, 'ω' is known as natural circular frequency and is generally denoted by 'ω _n' .