Page 426 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
404
−3
where is density and for steel is 7700 kg m , for beryllium bronze is
−3
8100 kg m .
14.3.2 Analysis of Helical Extension Springs
Extension springs may have initial tension or without initial tension. Most extension
springs are made with adjacent coils contact tightly, imposing an initial preload on the
spring. Only when the applied load overcomes the initial tension, that is, F > F ,do
i
spring coils begin to separate and the spring deflects in a linear relationship at a uniform
rate, as shown in Figure 14.14. The characteristic curve of an extension spring without
initial tension is exactly the same as that of a compression spring.
The initial tension in an extension spring is created during winding process by twisting
wire. The value of initial tension depends on the material, wire diameter, spring index
and manufacturing process, which is typically 10–25% of the maximum design force [2].
It can be estimated from Eq. (14.12), as
3
d i
F = (14.27)
i
8K D
w
where is initial torsional stress, and can be selected by [3]
i
b
=(0.4 ∼ 0.8) (14.28)
i
C
Except for the initial tension consideration, the stresses for an extension spring can
be computed by the same formula as that for a compression spring. The critical stresses
in an extension spring often occur in the end loops or hooks. Therefore, bending and
torsion in the hook must be included in the analysis while designing a spring with a
hook end. Interested readers can refer to references [1, 2] for detailed stress analysis.
The deflections for an extension spring can be computed by the same formula as that
for a compression spring. Referring to Eq. (14.21), the maximum deflection for a com-
pression spring or extension spring without preloading is
3
8F max C n
max = (14.29)
Gd
while for an extension spring with preloading, the maximum deflection is
3
8(F max − F )C n
i
= (14.30)
max
Gd
Contrary to a compression spring, all coils in an extension spring are active.
Figure 14.14 Characteristic curve of a helical extension
spring with initial tension.