Page 346 - Marks Calculation for Machine Design

P1: Sanjay
January 4, 2005
Brown˙C08
Brown.cls
328
U.S. Customary 15:14 APPLICATION TO MACHINES SI/Metric
Example 2. Determine the stiffness of the Example 2. Determine the stiffness of the
members in a bolted assembly, with no installed members in a bolted assembly, with no installed
washers, where washers, where
d bolt = 0.5 in (nominal) d bolt = 12 mm = 0.012 m (nominal)
D = 1.5 d bolt (washer face) D = 1.5 d bolt (washer face)
◦
◦
α = 30 (cone angle) α = 30 (cone angle)
L grip = 1.75 in L grip = 45 mm = 0.045 m
t 1 = 0.75 in (steel) t 1 = 20 mm = 0.02 m (steel)
t 2 = 1 in (cast iron) t 2 = 25 mm = 0.025 m (cast iron)
6
9
E 1 = 30 × 10 lb/in 2 E 1 = 207 GPa = 207 × 10 N/m 2
6
9
E 2 = 16 × 10 lb/in 2 E 2 = 110 GPa = 110 × 10 N/m 2
solution solution
Step 1. As the thicknesses (t 1 ) and (t 2 ) are not Step 1. As the thicknesses (t 1 ) and (t 2 ) are not
equal, a third thickness (t middle ) is found from equal, a third thickness (t middle ) is found from
Eq. (8.19) as Eq. (8.19) as

L grip L grip
t middle = t 2 − − t washer t middle = t 2 − − t washer
2 2
1.75 in 0.045

= (1in) − − (0in) = (0.025 m) − − (0m)
2 2m
= 0.125 in = 0.0025 m
Step 2. Using the third thickness (t middle ) Step 2. Using the third thickness (t middle )
found in step 1 in Eq. (8.23), calculate the spe- found in step 1 in Eq. (8.23), calculate the spe-
cial diameter (D*) as cial diameter (D*) as
∗
∗
D = 1.5 d + (L grip − 2 t middle ) tan 30 ◦ D = 1.5 d + (L grip − 2 t middle ) tan 30 ◦
= (1.5)(0.5in) = (1.5)(0.012 m)
+(1.75 in − 2 (0.125 in)(0.577) +(0.045 m − 2 (0.0025 m)(0.577)
= (0.75 in) + (1.5in)(0.577) = (0.018 m) + (0.040 m)(0.577)
= 1.62 in = 0.041 m
Step 3. Substitute the special diameter (D*) Step 3. Substitute the special diameter (D*)
found in step 2, the third thickness (t middle ) found in step 2, the third thickness (t middle )
found in step 1, and the given cone angle (α) found in step 1, and the given cone angle (α)
and modulus of elasticity (E 2 ), in Eq. (8.20) to and modulus of elasticity (E 2 ), in Eq. (8.20) to
find the stiffness (k middle ) as find the stiffness (k middle ) as
(π tan α) Ed (π tan α) Ed
k middle = k middle =
(2 t tan α + D − d)(D + d) (2 t tan α + D − d)(D + d)
ln ln
(2 t tan α + D + d)(D − d) (2 t tan α + D + d)(D − d)
◦ ◦
(π tan 30 ) E 2 d bolt (π tan 30 ) E 2 d bolt
= =
C 1 C 1
ln ln
C 2 C 2