Page 388 - Marks Calculation for Machine Design

P1: Naresh
January 4, 2005
Brown˙C09
Brown.cls
370
then using this spring index (C) in the expression for the maximum shear stress (τ max ) in
Eq. (9.11) gives 15:28 APPLICATION TO MACHINES
8FD
τ max = K s (9.13)
πd 3
where (K s ) is called the shear-stress correction factor given by Eq. (9.14).
1 1 + 2C
K s = + 1 = (9.14)
2C 2C
When springs are subjected to fatigue loading, high localized stresses occur on the inside
surface of the coils. Therefore, the factor (K s ) given in Eq. (9.14) is replaced by either of
the following factors:
4C − 1 0.615
K W = + Wahl factor (9.15)
4C − 4 C
4C + 2
K B = Bergstrasser factor (9.16)
4C − 3
However, as these two factors differ by less than 1 percent, the Bergstr¨asser factor in
Eq. (9.16) is preferred merely on the grounds of mathematical simplicity.
To separate out the curvature effect from the effect of direct shear, a factor (K c ) is used
in the standard fatigue equation, where
K B (2C)(4C + 2)
K c = = (9.17)
K s (1 + 2C)(4C − 3)
therefore the reduced stress-concentration factor (K f ) becomes
1
K f = (9.18)
K c
Consider the following example associated with the helical spring of a flyball governor
used to control the speed of stationary engines. Such a spring is under repeated reversed
dynamic loading, so designing against a fatigue failure is necessary.

U.S. Customary SI/Metric
Example 1. Determine the wire diameter (d) Example 1. Determine the wire diameter (d)
andmeandiameter(D)forahelicalspringusing andmeandiameter(D)forahelicalspringusing
the Bergstr¨asser factor (K B ), where the Bergstr¨asser factor (K B ), where
F = 130 lb F = 585 N
C = 8 C = 8
4
8
τ max = 50 kpsi = 5 × 10 lb/in 2 τ max = 350 MPa = 3.5 × 10 N/m 2
solution solution
Step 1. As there is dynamic loading, calculate Step 1. As there is dynamic loading, calculate
the Bergstr¨asser factor (K B ) using Eq. (9.16). the Bergstr¨asser factor (K B ) using Eq. (9.16).
4C + 2 4(8) + 2 34 4C + 2 4(8) + 2 34
K B = = = = 1.172 K B = = = = 1.172
4C − 3 4(8) − 3 29 4C − 3 4(8) − 3 29