Kjell Simonsson 1

A first glance at rotor dynamics

(last updated 2011-09-06)

Kjell Simonsson 2

Aim

The aim of this presentation is to give a first glance at rotor dynamics, which is a very important topic in many industrial applications.

We will here only introduce the concept of a critical speed of a rotating shaft, and note that it is numerically equal to the eigenfrequency at bending vibration. The presentation will be done by studying a specific example.

For a more comprehensive treatment of the subject, see any book on machine design or rotor dynamics, e.g. Svängningslära med rotordynamik by Karl-Olof Olsson (in Swedish)

Kjell Simonsson 3

A simple example

Let us consider a rotor (rotating shaft) carrying a circular disk at its end. The bearing support can be considered rather stiff, such that that the rotor can be considered clamped when considering its stiffness. Finally, the disk has a small eccentricity, such that the centre of gravity is located a distance e outside the centre of the disk.

LIE

When the rotor spins, the eccentricity will make it bend. By making theassumption that the rotor will spin like a rigid body, we find the followingfree body diagram of the disk, where x is the end deflection of the shaft.

e

m

e

mx

LIE

S S

Kjell Simonsson 4

A simple example

For the radial equation of motion for the disk we now have

As can be seen, the deflections of the shaft will become infinite when the applied speed is equal to the eigenfrequency for bending vibrations. Since there will always exist some eccentricity in reality, we will get resonance at this speed. However, in reality some damping will always be existing, thus making the resonance amplitudes finite.

22bend,e

2

22

k

3

3

2

ex

exmk

xxL

EI3S

EI3SL

x

S)ex(m

2bend,e

e

mx

LIE

S S

Kjell Simonsson 5

Summary

• In order to find the critical speed of a rotor it is sufficient to do a bending analysis of the same system

• In reality the behavior of the bearings and the effect of gyroscopic moments may be of importance. The motion of the shaft may in these cases be much more complex than the found above (where the shaft is not subjected to any time varying loading, i.e. suffers no risk for fatigue failure).