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

18.2 Linear Models

Linear models are the simplest form of equations commonly used to describe a wide range of

engineering situations. In this section, we first discuss some examples of engineering problems

where linear mathematical models are found. We then explain the basic characteristics of lin-

ear models.

A Linear Spring In Chapter 10, we discussed Hooke’s law, which states that, over the elastic

range, the deformation of a spring is directly proportional to the applied force, and conse-

quently to the internal force developed in the spring, according to

(18.1)

where

Fspring force (N or lb)

kspring constant (N/mm or N/cm or lb/in.)

xdeformation of the spring ( mm or cm or in.) (use units that are consistent withk)

It is clear from examining Equation (18.1) that the spring forceFdepends on how much the spring

is stretched or compressed. In mathematics,Fis called adependent variable. The spring force is

called the dependent variable because its value depends on the deformation of the springx. Consider

Fkx

18.2 Linear Models 587

TABLE 18.3 Roman Numerals

I  1 XX  20

II  2 XXX  30

III  3 XL  40

IIIIor IV  4 L  50

V  5 LX  60

VI  6 LXX  70

VII  7 LXXX  80

VIII  8 XC  90

IX  9 C  100

X  10 CC  200

XI  11 CCC  300

XII  12 CCCCor CD  400

XIII  13 D  500

XIV  14 DC  600

XV  15 DCC  700

XVI  16 DCCC  800

XVII  17 CM  900

XVIII  18 M  1000

XIX  19 MM  2000

k

x

F

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