Page 284 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
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10.2 Working Condition Analysis
10.2.1 Force Analysis
The loads on a shaft arise from its mounted elements. To simplify force analysis, the
distributed loads on mounted elements, such as gears, belt pulleys, chain sprockets and
bearings, are treated as concentrated forces acting at the midpoint of element width, as
the width of the element is small compared to the total length of the shaft. The weight
of the shaft and the attached elements are also neglected, as they are small compared to
the applied loads.
In a typical transmission shaft shown in Figure 10.3a, power transmission elements,
like gears, apply tangential, radial and axial forces F , F , F , on the shaft. The forces are
t
r
a
generally not all in the same plane and bending moment diagrams are in mutually per-
pendicular planes, as shown in Figure 10.3b,c. Their vector sum produces the resultant
bending moment in Figure 10.3d. Since the shaft rotates, the phase angle of moments is
not important [2].
While transmitting power, a shaft is inherently subjected to a torsional moment
or torque. The torque developed from one power transmission element must balance
the torque from other elements, usually through a portion of a shaft, as illustrated by
the torque distribution diagram in Figure 10.3e. Torque keeps a constant value at stable
operation.
Apparently, shafts are subjected to the combination of axial, transverse shear, bending
and torsional loads. These loads may be static or fluctuating during operation, depending
on the specific application. After visualizing the forces, torques and bending moments in
the shaft, the critical sections, as well as stresses and strength at these critical locations,
canbedetermined.

10.2.2 Stress Analysis
When a rotating shaft transmits a constant unidirectional torque, the produced tor-
sional shear stress is greatest on the shaft outer surface. These torsional shear stresses
are usually steady, but may sometimes fluctuate, depending on applications.
Transverse loads from power transmission elements generate bending moments,
result in completely reversed cyclic bending stresses. The bending stress is greatest on
the outer surfaces. Transverse loads may also result in transverse shear stresses, which
are normally small and can be neglected in stress analysis [5].
In addition, axial loads, such as those generate from helical gears, may produce either
normal tensile or compressive stresses on the shaft. However, they are usually negligibly
constant small stresses, and can be neglected when bending stresses are present in a
shaft [2].
These stresses may exist simultaneously in a shaft. Therefore, the maximum shear
stress theory or maximum distortion energy theory [5] are used to appraise stresses
for the shaft analysis and design.

10.2.3 Deflection and Rigidity
A shaft deforms due to the loads applied by the elements it carries. Excessive bending or
torsional deflections may cause misalignments in gear meshes, leading to an uneven load