Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

the force in a spring with a stiffness ofk2 N/mm, as shown in Figure 18.1. For the linear model

describing the behavior of this spring, constantk2 N/mm represents the slope of the line. The

value of 2 N/mm tells us that each time the spring is stretched or compressed by an additional

1 mm; as a result, the spring force will be changed by 2 N. Moreover, note that forx0, the

spring forceF0. Not all springs exhibit linear behaviors. In fact, you find many springs in

engineering practice whose behaviors are described by nonlinear models.

Temperature Distribution Across a Plane Wall Temperature distribution across a plane wall is

another example where a linear mathematical model describes how temperature varies along the

wall. Under a steady-state assumption, the temperature distribution — how temperature varies

across the thickness of the wall — is given by

(18.2)

where

T(x) temperature distribution ( F or C)

T 2 temperature at surface 2 ( F or C)

T 1 temperature at surface 1 ( F or C)

xdistance from surface 1 (ft or m)

Lwall thickness (ft or m)

For this linear model,Tis the dependent variable, andxis the independent variable. The vari-

ablexis called anindependent variable, because the positionxis not dependent on temperature.

Now, let us consider a situation for whichT 1  68 F,T 2  38 F, a n d L0.5 ft. For these

T 1 x 2  1 T 2 T 12

x

L

T 1

588 Chapter 18 Mathematics in Engineering

F

(N)

45

40

35

30

25

20

15

10

5

0

0 5 10 15 20 25

x (mm)

F  2 x

x(mm) F(N)

00

510

10 20

15 30

20 40

■Figure 18.1 A linear model for spring force–deflection relationship.

x

L

T 1

T 2

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