Page 286 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
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10.2.4 Rotating Shaft Dynamics
Shafts are nearly always part of a dynamic system. When a shaft operates at a speed close
to its natural frequency, vibration amplitudes may suddenly increase and may destroy
the system. Shafts may vibrate in transverse, torsional and axial directions. Axial vibra-
tion rarely happens during shaft operation and will not be discussed here.
Due to imperfection in manufacturing and assembly, the mass centre of a shaft
rotating system seldom coincides exactly with the centre of rotation. Consequently,
as the shaft rotates, eccentricity causes a centrifugal force deflection [2]. The greater
the eccentricity, the larger the centrifugal force deflection and the severer the trans-
verse vibration. When the shaft transverse frequency is close to or identical to the
resonance of frequency, transverse resonance happens.
Torsional vibration may be excited when fluctuating torques are encountered due to
periodic changes of transmitted power. If the frequency of torsional input and torsional
natural frequencies of the shaft coincide, torsional resonance may damage the shaft.
Therefore, the operating speeds of shaft should avoid close to the critical speeds to
prevent violent vibration or resonance, preferably below the lowest critical speed by a
factor of 2–3 [6]. For a shaft with an operating speed higher than the lowest critical
speed, it is important to ensure a quick through resonance during the startup and shut
down cycle.

10.2.5 Potential Failure Modes
Considering the rotation of shaft, the bending stress induced by transverse loads from
power transmission elements is completely reversed stress. The transverse shear stress
produced by transverse loads may be completely reversed stress. The torsional shear
stress generated by torques may be steady or fluctuating, depending on applications.
The axial stress arises from helical gears or preloaded bearings, which are usually steady
but can fluctuate sometimes. Obviously, fatigue is an important potential failure mode
for power transmission shafts.
Furthermore, excessive bending or torsional deflections may lead misalignment in
gear meshes and bearings, hamper gear performance and cause undesirable noise [3].
Finally, in high speed mechanical systems, shafts operate close to critical speeds may
excite intolerable vibration. The increased vibration amplitudes may destroy the shaft
system. Therefore, resonance of vibration is also one of the potential failure modes.

10.3 Load Carrying Capacities

10.3.1 Strength Analysis

As discussed before, stresses on a rotating shaft may involve torsional shear stress, trans-
verse shear stress, bending stress or axial stress components, any or all of which may be
fluctuating stresses. In general, shaft design must base on multiaxial states of stress pro-
duced by fluctuating loads [6]. However, in practice, strength analysis is mostly based on
simple loading conditions. If a shaft is or is mainly subjected to a stable torque, that is,
a spindle, a torsional strength calculation is performed. If a shaft is subjected to steady
bending moments only, that is, an axle, a bending strength calculation is used. If a shaft