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 ).