Dunkerley's method
is used in mechanical engineering to determine the critical speed of a shaft-rotor system. Other methods include the
Rayleigh–Ritz method.
Whirling of a shaft
No shaft can ever be perfectly straight or perfectly balanced. When an element of mass is offset from the axis of rotation, centrifugal force will tend to pull the mass outward. The elastic properties of the shaft will act to restore the “straightness”. If the frequency of rotation is equal to one of the resonant frequencies of the shaft, ''whirling'' will occur. In order to save the machine from failure, operation at such whirling speeds must be avoided. Whirling is a complex phenomenon that can include harmonics but we are only going to consider ''synchronous whirl'', where the frequency of whirling is the same as the rotational speed.
Dunkerley’s formula (approximation)
The whirling frequency of a symmetric cross section of a given length between two points is given by:
:
where:
E = Young's modulus,
I =
second moment of area
The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. Th ...
,
m = mass of the shaft,
L = length of the shaft between points.
A shaft with weights added will have an angular velocity of N (RPM) equivalent as follows:
:
See also
*
Vibration
Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...
*
Mechanical resonance
Mechanical resonance is the tendency of a mechanical system to respond at greater amplitude when the frequency of its oscillations matches the system's natural frequency of vibration (its '' resonance frequency'' or ''resonant frequency'') clos ...
Notes and references
Mechanical engineering