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Such expressions can be useful in describing the bottoming out behavior of coil springs, for instance The only constraint is that the function must be a nondecreasing function of displacement in order to capture the desired physical characteristics of a spring A second property that mechanical springs and compliant elements share is the storage of energy It requires work to displace a spring from its free length and, if the spring is assumed ideal, all of that work is stored in the spring As a result, the amount stored (the spring s potential energy) can be found by integrating the work done on the spring For the general nonlinear spring form given above,

Just as the integral may be used to calculate planar area and spatial volume, so this tool may also be used to calculate the arc length of a curve and surface area The basic idea is to approximate the length of a curve by the length of its piecewise linear approximation A similar comment applies to the surface area We begin by describing the basic rubric

(1142)

Be carefbl again Go back and read the question and make sure you understand that the desired time is for the temperature to reach a 10' C difference between the coffee and its surroundings Use the 10' C temperature difference and calculate the time

For the linear spring in Eq 1141, displaced a distance x from its unstretched length, this translates to the familiar expression 1 PE = Khdh = Kd 2 2 0

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Suppose that f (x) is a function on the interval [a, b] Let us see how to calculate the length of the curve consisting of the graph of f over this interval (Figure 828) We partition the interval: a = x0 x1 x2 xk 1 xk = b Look at Figure 829 Corresponding to each pair of points x j 1 , x j in the partition is a segment connecting two points on the curve; the segment has endpoints (x j 1 , f (x j 1 )) and (x j , f (x j )) The length j of this segment is given by the usual planar distance formula: = [x j x j 1 ]2 + [ f (x j ) f (x j 1 )]2

(1143)

y = f (x)

As with the force, the only spring properties needed to determine the amount of energy stored are the spring rate and the unstretched length Spring rates can be determined experimentally by applying a known force and measuring de ection (or de ecting a

known amount and measuring force) but can also be determined analytically for a number of simple geometries Torsional springs (Fig 1110) are completely analogous to the linear springs described above but relate torque and angular de ection instead of torque and linear de ection The corresponding expressions for torsional springs are: T = Kr Dq (1144)

'0= e -004r , -004t=In--10

y = f (x)

where Kr is the torsional spring rate (in units of torque/angle such as in-lbf/rad or Nm/rad) T is the torque exerted by the spring and Dq is angle through which the spring is turned, relative to its equilibrium position The potential energy in the spring is given by:

(xj , f (xj)) (xj _ 1, f (xj _ 1))

(1145)

1141 Coil Springs While many types of manufactured springs exist, coil springs are one of the most commonly used springs in cam systems Coil springs may be used in either extension or compression, depending upon the design of their ends Figure 1111 illustrates a typical compression coil spring such as the type used in automotive suspensions The spring rate for a coil spring in axial tension or compression is given by K= Gd 4 8 D3 N (1146)

We denote the quantity x j x j 1 by tive to obtain x and apply the definition of the deriva-

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where d is the diameter of the spring wire, D is the diameter of the spring coil (measured from the center of the spring wire), G is the shear modulus of the material, and N is the number of active coils N does not need to be an integer and does not include ground ends that are pressed against the mating surface at either end of the spring since these do not participate in the compression or extension of the spring Values of G for various spring materials may be found in engineering handbooks: a typical value used for steel springs when no additional information is available is 110 106 psi Coil springs may be designed in such a way that the diameters d and D change over the length of the spring As a rst order approximation, spring rates in such cases can be estimated by using the average values of these parameters

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