Page 417 - Analysis and Design of Machine Elements

(a) (b) (c) (d) Springs 395
Figure 14.9 Typical spring characteristic curves.

While for a torsion spring, the applied load is torque and deflection is angular deflec-
tion and the spring rate is

dT
k = (14.2)

d
Spring rate reflects the stiffness of spring. Under the same loads, a stiff spring has small
deflection, while a flexible spring has a large deflection. The variation of spring stiffness
is revealed clearly by characteristic curves. It is an important information for the design,
manufacturing and testing of spring.
Depending on the structure of spring, a spring may have constant, progress, degres-
sive or combined stiffness. The characteristic curve of a constant stiffness spring is a line,
as shown in Figure 14.9a. The deflection is directly proportional to the applied load, fol-
lowing Hooke’s law. It is a fundamental relationship in spring design. Cylindrical helical
compression, extension and torsion springs have constant stiffness characteristic curves.
Nonlinear characteristic relations are desirable for some special applications.
Figure 14.9b shows a progressive characteristic curve, exhibiting an exponential
increase in resistance as deflection increases [9]. This kind of characteristic is useful in
damping harmful vibrations. Conical springs possess this characteristic and are often
used in vehicle suspension systems to control the maximum deflection. Figure 14.9c
present a degressive spring characteristic, that is, the load maintained is nearly constant
as deflection increases. Figure 14.9d shows combined stiffness characteristic curves,
that is, the loading and unloading characteristic curves do not coincide. They are repre-
sented by various types of springs, more specifically illustrated from Figures 14.5–14.7.
There is partial energy release during unloading.

14.2.3 Storage and Dissipation of Energy
When a spring deflects under an external load, the work associated with the load is
stored as strain energy. The input work is equal to the area under the given characteristic
curve. The strain energy for a compression or extension spring is defined as

U = F( )d (14.3)
∫
0
While for a torsion spring, the strain energy is defined as

U = T( )d (14.4)
∫
0