Page 389 - Marks Calculation for Machine Design

P1: Naresh
January 4, 2005
15:28
Brown˙C09
Brown.cls
U.S. Customary MACHINE ENERGY SI/Metric 371
Step 2. Substitute the Bergstr¨asser factor (K B ) Step 2. Substitute the Bergstr¨asser factor (K B )
found in step 1 and the other given information found in step 1 and the other given information
in Eq. (9.13). in Eq. (9.13).
8CF 8CF
τ max = K B τ max = K B
πd 2 πd 2
(8)(8)(130 lb) (8)(8)(585 N)
2
2
8
4
5 × 10 lb/in = (1.172) 3.5 × 10 N/m = (1.172)
πd 2 πd 2
Step 3. Solve for the wire diameter (d) from Step 3. Solve for the wire diameter (d) from
step 2. step 2.
(1.172)(8)(8)(130 lb) (1.172)(8)(8)(585 N)
2 2
d = d =
8
2
2
π(50,000 lb/in ) π(3.5 × 10 N/m )
= 0.0621 in 2 = 0.00004 m 2
d = 0.25 in d = 0.00632 m = 6.32 mm
Step 4. Using the definition of the spring Step 4. Using the definition of the spring
index (C) from Eq. (9.12) and the wire diameter index (C) from Eq. (9.12) and the wire diameter
(d) found in step 3, calculate the mean spring (d) found in step 3, calculate the mean spring
diameter (D) diameter (D).
D D
C = C =
d d
D D
8 = 8 =
0.25 in 6.32 mm
D = (8)(0.25 in) D = (8)(6.32 mm)
= 2.0in = 50.6mm = 5.06 cm
Deflection. Without providing the details of its development, the deflection (y) of a
cylindrical helical spring can be determined using strain energy theory to give the expression
in Eq. (9.19)
3
8FD N a
y = 4 (9.19)
d G
where (N a ) is the number of active coils and (G) is the shear modulus of elasticity.
If the deflection (y) is given, then Eq. (9.19) can be rearranged to give the number of
active coils (N a ) as
4
yd G
N a = 3 (9.20)
8FD
The total number of coils (N) will be the number of active coils (N a ) plus any additional
coils that are needed, depending on the type of ends, particularly if either end of the spring
has one of the many common hook designs. The topic of ends and hooks will be discussed
shortly.
9.2.2 Spring Rate
The relationship between the force (F s ) produced by a spring, whether it is extended or
compressed, and the displacement (x), meaning change in length, can be linear or nonlinear.