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= [ x]2 + [ f (x j ) x]2 = 1 + [ f (x j )]2

A coil spring reaches its maximum compression when all coils have been pushed tightly together For a compression spring with ground ( at) ends, this happens at a length of L = Nt d (1147)

where Nt is the total number of coils in the spring When a spring is compressed to its minimum length, it is said to have bottomed out At the point where a spring bottoms out, the spring rate increases greatly, since further compression requires deformation in the packed spring wires This characteristic is sometimes exploited to create progressive springs in which the spring rate increases as the spring is compressed Figure 1112a illustrates a compression spring with varying pitch Unlike the constant pitch spring, where all coils bottom out at once, the more closely spaced coils on a variable pitch spring bottom out at a lower spring de ection than the more widely spaced coils When a coil bottoms out, the effective number of coils is reduced, so the spring displays a stiffening characteristic in compression, as illustrated in Fig 1112b Note that the bottoming out effect only occurs in compression, since the coils continually move further apart when a coil spring is placed in tension 1142 Compliance of Other Mechanical Elements Spring rates for any element in the system can be obtained through experimental testing of the force-displacement relationship or, for simple geometries, from basic results from mechanics of materials A few common force-de ection relationships for simple geometries and their corresponding spring rates are listed in Table 111; many more can be found in engineering handbooks or textbooks on the mechanics of materials (Popov, 1976, Chen, 1982, Relvas, 1985, Ungar, 1985) If the desired relationship is not readily available, spring

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Summing up the lengths curve:

F Last coil bottoms Next coil bottoms d (compression)

9- 16 A wildlife manager needs to reach a 10,000 population of mule-eared deer in a certain habitat in 6 years There are presently no deer in the habitat The environment is such that the deer can grow without being limited by their environment This means that the growth of the deer population will be proportional to the population, dP/dt = kP In order to detennine the growth equation, 100 deer are introduced into the habitat There are half males and half females, the same ratio as when they reproduce At the end of the year there are 130 deer How many deer need to be introduced to the habitat to acheve the 10,000 goal in the remaining 5 years

d (tension)

1 + [ f (x j )]2

rates can be easily derived from a basic stress analysis First, choose a simpli ed geometry that approximates the shape of the element in question Next, calculate the stress arising in the element as a result of the applied load If it is assumed that the element is acting in the region of elastic deformation (a reasonable assumption for the systems in question here since they must be designed to survive cyclic motion and loading), the stress can then easily be related to strain Finally, the de ection can be calculated from the strain and the spring rate determined from this relationship between force and de ection This procedure is illustrated by a simple example

As the mesh of the partition becomes finer, the approximating sum is ever closer to what we think of as the length of the curve, and it also converges to this integral Thus the integral represents the length of the curve

Suppose a system uses a long rod (Fig 1113) loaded axially to transmit force between members (this is in fact the case in pushrod engines) The rod does not in fact remain perfectly rigid, but de ects as a result of the applied load The axial stress that arises in the rod is given by F s= A

where A is the cross-sectional area of the rod In the linear region, the strain is proportional to the stress through Young s modulus, so e= s F = E EA

Let us calculate the arc length of the graph of f ( x) = 4x 3/2 over the interval [0, 3]

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