Page 194 - Marks Calculation for Machine Design

P1: Shibu
14:25
January 4, 2005
Brown.cls
Brown˙C04
STRENGTH OF MACHINES
176
As a review of Sec. 3.2.3 on press or shrink fits, at the interface between the spur
gear and the solid shaft, at the radius (R), the gear increases an amount (δ g ) radially
and the inside shaft decreases an amount (δ s ) radially. The difference between the radial
increase (δ g ) of the gear, a positive number, and the radial decrease (δ s ) of the shaft, a
negative number, is called the radial interference (δ) at the interface (R) and is given by
Eq. (4.17),
2 2
2
pR r + R 2 pR R + r i
o
δ = δ g + |δ s | = 2 2 + ν g + 2 − ν s (4.17)
2
E g r − R E s R − r i
o
where (E g ) and (E s ) are the moduli of elasticities, and (ν g ) and (ν s ) are the Poisson ratios
of the spur gear and shaft, respectively.
When the radial interference (δ) is determined from a particular fit specification, and
this is discussed in detail in Sec. 3.2.3, then Eq. (4.17) can be solved for the interference
pressure (P). However, if the spur gear and shaft are made of the same material, then the
modulus of elasticity’s and Poisson’s ratio are equal and so Eq. (4.17) can be rearranged to
give an expression for the interface pressure (P) given in Eq. (4.18).
2
2
Eδ
r − R 2 R − r i 2
o
p = (4.18)
R 2 R 2 r − r 2
2
o i
If the inner shaft is solid, meaning the inside radius (r i ) is zero, then Eq. (4.18) for the
interface pressure (P) simplifies to the expression in Eq. (4.19)

2
Eδ R
p = 1 − (4.19)
2 R r o
Again, just as a review, consider the following calculation for the interface pressure (P)
based on a given, or previously determined, radial interference (δ).
U.S. Customary SI/Metric
Example 8. Calculate the interface pressure Example 8. Calculate the interface pressure
(P) for a solid shaft and spur gear assembly, (P) for a solid shaft and spur gear assembly,
with both parts steel, where with both parts steel, where
δ = 0.0005 in δ = 0.001cm = 0.00001 m
R = 0.75 in R = 2cm = 0.02 m
r o = 4in r o = 10 cm = 0.1 m
9
2
2
6
E = 30 × 10 lb/in (steel) E = 207 × 10 N/m (steel)
solution solution
Step 1. Substitute the radial interface (δ), in- Step 1. Substitute the radial interface (δ), in-
terface radius (R), outside radius (r o ) of the terface radius (R), outside radius (r o ) of the
spur gear, and the modulus of elasticity (E) in spur gear, and the modulus of elasticity (E) in
Eq. (4.19) to give Eq. (4.19) to give