Page 394 - Marks Calculation for Machine Design

P1: Naresh
January 4, 2005
15:28
Brown.cls
Brown˙C09
APPLICATION TO MACHINES
376
In the absence of friction, the energy stored in or released from a spring is conservative,
meaning no energy is lost if the spring is repeatedly loaded and unloaded. When energy is
conservative it is called potential energy (PE), and is equal to the work (Work) done on the
spring given in Eq. (9.26). 1→2
1 2
PE spring = kx 1 (9.27)
2
Expanding these principles to a spring that is compressed from one displacement (x 1 ) to
another displacement (x 2 ), or released from these same displacements, the work done on
the spring to compress it, or the energy given up by the spring when released, is shown in
Fig. 9.5 as the shaded trapezoidal area.
F s F = kx
s
kx 1
k
kx 2 kx 1
0 x
x x
0 2 1
FIGURE 9.5 Work done or energy stored by a
spring (two displacements).

The area of the trapezoid in Fig. 9.5 is the difference between the areas of two triangles
as given in Eq. (9.28) as
1 2 1 2 1 2 2
Work = kx 1 − kx 2 = k x − x 2 (9.28)
1
1→2 2 2 2

x 1 triangle x 2 triangle
where the displacements are the differences between the final and unstretched lengths.
If the unstretched length of the spring is denoted (L o ) and the initial and final lengths are
denoted (L i ) and (L f ), respectively, then the displacements (x 1 ) and (x 2 ) are given by the
following two relationships:

x 1 = L i − L o
(9.29)
x 2 = L f − L o
where (x 1 ) or (x 2 ) are either both positive or both negative. Negative values are not a
problem, as the displacements are squared in Eq. (9.28).
Also, if the work done comes out positive, then the spring is doing work on the system,
and if it comes out negative, then work is being done by the system on the spring.
As mentioned earlier, in the absence of friction, the cyclic loading and unloading of
the spring is conservative, meaning no energy is lost, therefore, the work done given in
Eq. (9.28) is equal to the stored potential energy and given as

1
2 2
PE spring = k x − x 2 (9.30)
1
2