Page 102 - Electric Drives and Electromechanical Systems

Chapter 3 Power transmission and sizing 95

through the flexing/bending of a membrane (typically designed as a disc, diaphragm,
beam, or bellows), Fig. 3.16B. Those with moving parts generally are less expensive but
need to be lubricated and maintained. Their primary cause of failure in a flexible
metallic coupling is wear, so overloads generally shorten the couplings life through by
increased wear rather than sudden failure. A bellow coupling is torsionally stiff and a
wide temperature operating range. The coupling can only accommodate small
misalignment and cannot damp vibration and absorb shocks, but there is lost motion
in the coupling. One problem that users need to be aware is that the coupling has high
electrical conductivity, which could give rise to unexpected earth loops or related
problems.

3.6 Shafts

A linear rotating shaft supported on bearings can be considered the simplest element in
a drive system, however their static and dynamic characteristics need to be considered.
While it is relatively easy, in principle, to size a shaft, it can pose several challenges to
the designer if the shaft is particularly long or difficult to support. In most systems the
effects of transient behaviour can be neglected for selecting the components of the
mechanical drive train, as the electrical time constants are lower than the mechanical
time constant, and therefore they can be considered independently. While such effects
are not commonly found, they must be considered if a large-inertia load must be driven
by a relatively long shaft, where excitation generated either by the load (for example, by
compressors) or by the drive’s power electronics needs to be considered.
In any shaft, torque is transmitted by the distribution of shear stress over its cross-
section, where the following relationship, commonly termed the Torsion Formula,
holds,
T Gq s
¼ ¼ (3.32)
I o L r
where T is the applied torque, I o is the polar moment of area, G is the shear modulus of
the material, q is the angle of twist, L is the length of the shaft, s is the shear stress and
r the radius of the shaft. In addition, we can use the torsion equation to determine the
stiffness of a circular shaft,
T Gpr 4
K ¼ ¼ (3.33)
q 2L
where the polar moment of area of a circular shaft is given by,
pr 4
I o ¼ (3.34)
2