Page 407 - Shigley's Mechanical Engineering Design

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382 Mechanical Engineering Design

EXAMPLE 7–4 For the shaft in Ex. 7–3, it was noted that the slope at the right bearing is near the limit
for a cylindrical roller bearing. Determine an appropriate increase in diameters to bring
this slope down to 0.0005 rad.

Solution Applying Eq. (7–17) to the deflection at the right bearing gives

# # 1/4 # # 1/4
# n d slope old # # (1)(0.001095) #
# # = 1.0 # # = 1.216 in
slope all # # (0.0005) #
d new = d old #
Multiplying all diameters by the ratio
d new 1.216
= = 1.216
d old 1.0
gives a new set of diameters,
D 1 = D 7 = 1.216 in
D 2 = D 6 = 1.702 in
D 3 = D 5 = 1.976 in

D 4 = 2.432 in
Repeating the beam deflection analysis of Ex. 7–3 with these new diameters produces
a slope at the right bearing of 0.0005 in, with all other deflections less than their previ-
ous values.

The transverse shear V at a section of a beam in flexure imposes a shearing deflec-
tion, which is superposed on the bending deflection. Usually such shearing deflection
is less than 1 percent of the transverse bending deflection, and it is seldom evaluated.
However, when the shaft length-to-diameter ratio is less than 10, the shear component
of transverse deflection merits attention. There are many short shafts. A tabular method
2
is explained in detail elsewhere , including examples.
For right-circular cylindrical shafts in torsion the angular deflection θ is given in
Eq. (4–5). For a stepped shaft with individual cylinder length l i and torque T i , the
angular deflection can be estimated from

T i l i
$ $
θ = θ i = (7–19)
G i J i
or, for a constant torque throughout homogeneous material, from
T $ l i
θ = (7–20)
G J i
This should be treated only as an estimate, since experimental evidence shows that the
actual θ is larger than given by Eqs. (7–19) and (7–20). 3

2 C.R. Mischke, “Tabular Method for Transverse Shear Deflection,” Sec. 17.3 in Joseph E. Shigley, Charles
R. Mischke, and Thomas H. Brown, Jr. (eds.), Standard Handbook of Machine Design, 3rd ed., McGraw-Hill,
New York, 2004.
3 R. Bruce Hopkins, Design Analysis of Shafts and Beams, McGraw-Hill, New York, 1970, pp. 93–99.