Vibration of Machine Having Rotating/Reciprocating Parts:

Rotating/reciprocating machine is illustrated in Figure 5. The machine is mounted on the elastic foundation, the stiffness of which is defined by k. Assume 'c' be the damping coefficient.

Assume m = rotating mass,

M = mass of the complete machine, and

e = eccentricity.

The differential equation shall be

 (M - m)  x' + md ²/ dt ² ( x + e sin ωt ) + cx'+ kx = 0

 or           (M - m)x'+ m x'- m e ω² sin ωt + cx'  + kx = 0

or            Mx'+ cx'  + kx = m e ω² sin ωt

(i)                                                 (ii)

Equation (10) is same to Equation (1). The similar analysis might be carried out and the steady solution of the above differential equation may be obtained that is

             x = X sin (ωt - φ)

where

= DM ×( me ω²/k) = m / M (M/k)  e ω²

or            X = DM (me/ M                )^k (ω/ ω_n )²

In the diagram of the phase there is modify and is similar as that in Figure.

The frequency response diagram is also illustrated in Figure. The ratio (XM / me) is plotted against frequency ratio (ω/ ω _n). The curve begins from zero and it attain maximum value close the resonance and after that this becomes asymptotic at XM/ me   = 1.0.

As discussed in the previous section if machine operates close the resonance adequate damping must be provided in the system. If we desire that vibration must not occur, damping equivalent to the critical damping must be provided.

In case of reciprocating machinery eccentricity 'e' is replaced through the crank radius 'r' in the expressions and rest of the expressions and formulae are similar.