Page 424 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
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where the length of wire is approximate L = Dn when pitch angle is less than 10 ,
and the polar moment of inertia of round wire is
d 4
J =
32
These two forms of energy are equal. Therefore
1 1
F = T
2 2
By substitution and rearrangement, we have
( 1 ) 2 L
F = T = FD
2 GJ
Therefore, the linear deflection is
8nFD 3 8nFC 3
= 4 = (14.21)
Gd Gd
where n is the number of active coils, G is shear modulus of spring material and is
4
4
usually 8 × 10 Nmm −2 for steel, and 4 × 10 Nmm −2 for bronze [10]. Again, wire
diameter is noticed to have strong effect on spring performance.
2) Spring rate
Incorporating Eq. (14.21), the spring rate can be derived as
F Gd Gd 4
k = = = (14.22)
8nC 3 8nD 3
Therefore, spring rate is inversely proportional to the cube of spring index C. Small
spring index increases spring rate k and lead to a stiff spring, while large spring index
C reduces spring rate and produces a flexible spring. Hence, proper selection of
spring index C can control the spring flexibility. In addition, the effects of G, d, n
on the spring rate k also need to be considered.
3) Number of active coils
For a compression spring or an extension spring without preload, the number of
active coils of cylindrical helical spring can be obtained from Eq. (14.21) as

G d 4 G d Gd
n = = = (14.23)
3
8FD 3 8FC 3 8C k
For extension springs with preload, the number of active coils of a cylindrical helical
spring is
Gd
n = (14.24)
8(F − F )C 3
0
If n < 15, n is selected as a multiple of 0.5. If n > 15, n is selected as an integer [4].

14.3.1.5 Buckling Analysis
Elastic instability, or buckling, occurs when a long, slender cylindrical helical spring is
loaded by a compressive force and produces large axial deflections. Buckling is affected
by spring free length, mean diameter and spring end constraint [2, 5].
The ratio of spring free length H to the mean diameter D is defined as slenderness
f
ratio b. Buckling could be prevented by selecting proper slenderness ratios. Spring end