3.3 Differential equation for Rotating Unbalance

Unbalance in rotating machines is a common source of vibration excitation. The problem of unbalance in a system occurs when the centre of gravity of rotor does not coincide with the axis of rotation. A spring-mass-damper system constrained to move in the vertical direction and excited by a rotating machine that is unbalanced, as shown in Fig. 3.1. Let, x be the displacement of the non-rotating mass ( M-m ) from the static equilibrium position, the displacement of m is

                                                             (3.1)

 

Fig. 3.1: Harmonic disturbing force resulting from rotating unbalance

 

The equation of motion is then

 

                                                 (3.2)

 

Which can be arranged as,

 

                                                  (3.3)

 

The steady-state solution of the equation can be written as,

 

                                                        (3.4)

 

 

And

                                                                 (3.5)

 

This can be further reduced to non-dimensional form

 

                                                              (3.6)

 

The simulated curve for forced vibration with rotating unbalanced at a particular damping ratio of ζ = 0.190909 is shown in Fig. 3.2. The simulated curve shows the variation of non-dimensional quantity Mx/me and phase angle Φ  with frequency ratio ω/ω _nf . The simulated response was obtained using Eq. (3.6) & Eq. (3.7).

 

Fig. 3.2: Response for rotating unbalance at damping ratio of 0.190909

 

Following points can be concluded from the simulated response:

1.  When the value of  ω  is very small as compared to ω _nf , it is known as low speed system. For a low speed system the value of  Mx/me → 0.

2.  Similarly, for a high speed system ω  is very high, then Mx/me  → 1 .

3.  At very high speed the effect of damping seems to be negligible.

4.  Peak amplitude occurs to the right of resonance ( ω/ω _nf ).