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' .