Page 390 - Marks Calculation for Machine Design

P1: Naresh
January 4, 2005
15:28
Brown.cls
Brown˙C09
APPLICATION TO MACHINES
372
For a linear spring, this relationship is given in Eq. (9.21) as
F s =−kx (9.21)
where (k) is called the spring rate of the spring.
The minus sign (−) is needed in Eq. (9.21) because the spring force (F s ) is a restoring
force, meaning it is always in the opposite direction from the displacement (x). That is, if
the spring is compressed, which is a negative displacement (x), then the spring force (F s )
is positive. Conversely, if the spring is extended, which is a positive displacement (x), then
the spring force (F s ) will be negative.
Based on the linear relationship given in Eq. (9.21), the units of the spring rate (k) are
force per length. If the relationship were not linear, then the units of (k) would be such that
when multiplied by the displacement (x), the units for (F s ) would still be force.
The relationship between the spring force (F s ) and the displacement (x), without the
minus sign (−), given in Eq. (9.21) is shown graphically in Fig. 9.3.
F s
F = kx
s
k

0 x
0
FIGURE 9.3 Spring force versus displacement.
The spring rate (k) is therefore the slope of the straight line representing the linear
relationship between the spring force and the displacement. Note that the zero (0) point on
the horizontal displacement (x) axis does not represent a zero length of the spring. Rather,
it represents the unstretched length of the spring.
Solving for the spring rate (k) in Eq. (9.21), and dropping the minus sign (−),gives
F s
k = (9.22)
x
where any combination of spring force (F s ) and the displacement (x) can be used.
The spring rate (k) given in Eq. (9.22) can be generalized for any spring type, whether
helical, leaf, torsion, or any other type, as
spring force
spring rate (k) = (9.23)
displacement
For cylindrical helical springs, the spring force is the force (F) and the displacement is
the deflection (y) from Eq. (9.19) so that the generalization in Eq. (9.23) gives the spring
rate (k) as
4
spring force F d G
k = = = (9.24)
3
displacement y 8D N a
If the spring rate (k) is known, then Eq. (9.24) can be rearranged to give an expression
for the number of active coils (N a ) as
4
d G
N a = 3 (9.25)
8D k