Page 196 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
14:25
Brown.cls
Brown˙C04
STRENGTH OF MACHINES
178
At the outside radius (r o ) of the gear, the shear stress (τ xy ) is maximum and from
Eq. (4.20) and the polar moment of inertia (J) in Eq. (4.21) becomes Eq. (4.22).
Tr o Tr o 2 Tr o
τ max = = = (4.22)
4
J 1 π r − R 4 π r − R 4
4
2 o o
At the inside radius (R) of the gear, the shear stress (τ xy ) is minimum and from Eq. (4.20)
and the polar moment of inertia (J) in Eq. (4.21) becomes Eq. (4.23).
TR TR 2 TR
τ min = = = (4.23)
4
J 1 π r − R 4 π r − R 4
4
2 o o
Second, the interface pressure (P) between the gear and the shaft, like that determined
in Example 8, causes both a tangential stress (σ t ) given by Eq. (4.24),
pR 2 r o 2
σ t = 1 + (4.24)
2
r − R 2 r
o
and a radial stress (σ r ) given by Eq. (4.25).
pR 2 r o 2
σ r = 2 2 1 − (4.25)
r − R r
o
However, tangential and radial stresses are a maximum at the interface radius (R) where
the shear stress due to the torque would be minimum. Recall that in Sec. 3.2.2 it was shown
that the radial stress (σ r ) at the inside radius of a thick-walled cylinder is the negative of the
internal pressure (p i ), which here is the interface pressure (P). It was also shown that the
minimum radial stress (σ r ) was zero at the outside radius (r o ).
Therefore, at the radial interface (R), the tangential stress (σ t ) given in Eq. (4.24) becomes
a maximum value (σ t max ), with the algebraic steps shown in Eq. (4.26),
2
2
pR 2 2 pR 2
R + r 2

r + R 2
max r o o o
σ t = 1 + = = p (4.26)
2
2
2
r − R 2 R r − R 2 R 2 r − R 2
o o o

Eq. (1.90) with r = R find common denominator rearrange and cancel terms
and the radial stress (σ r ) given in Eq. (1.91) becomes a maximum value (σ r max ) equal to the
negative of the interface pressure (P), with the algebraic steps shown in Eq. (4.27).

2 2
2
pR 2 r o 2 pR 2 R − r 2 o pR 2 − r − R
o
σ max = 1 − = = =−p
r 2 2 2 2 2 2 2 2
r − R R r − R R R r − R
o
o
o

Eq. (1.91) withr=R find common denominator rearrange and cancel terms
(4.27)
Similarly, at the outside radius (r o ), the tangential stress (σ t ) given in Eq. (4.24) becomes
a minimum value (σ t min ), with the algebraic steps shown in Eq. (4.28),

2
pR 2 r o pR 2 2pR 2
min
σ t = 1 + = [1 + 1] = (4.28)
2
2
2
r − R 2 r o r − R 2 r − R 2
o
o
o

simplify bracket terms rearrange
Eq. (1.90) with r = r o