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

18.7 Differential Equations 625

exact solution of these equations renders detailed behavior of the system under the given set of

conditions. Examples of governing equations, boundary conditions, initial conditions, and solu-

tions are shown in Table 18.8.

In Example 1, the functionYrepresents the deflection of the beam at the location denoted

by the position variableX. As shown in Table 18.8, the variableXvaries from zero toL, the

position of the beam. NoteXis measured from the left-support point. The load or the force acting

TABLE 18.8 Examples of Governing Differential Equations, Boundary Conditions, Initial Conditions,

and Exact Solutions for Some Engineering Problems

Governing Solution,

Boundary Conditions

Problem Type or Initial Conditions Solution

Example 1: The deflection of a beam. Deflection of the beamY, as the function

Boundary conditions:

of distance X:

atX0,Y0 and

atXL,Y 0

Example 2: The oscillation of an The position of the massy, as the function

elastic system.

where

of time:y(t) y 0 cos vnt

Initial conditions:

at timet0,yy 0 and

at timet0,

Example 3: The temperature Temperature distribution along the

distribution along a long fin.

Boundary conditions:

fin as the function ofX:

at X0,TTbase

asLSq,TTair

TTair 1 TbaseTair 2 e



B

hp

kAc

X

d

2

T

dX

2 

hp

kAc

1 TTair 2  0

dy

dt

 0

v^2 n

k

m

d

2

y

dt

2 v

2

n^ y^0

Y

w

24 EI

1 X^4  2 LX^3 L^3 X 2

EI

d

2

Y

dX

2 

wX 1 LX 2

2

W

L

E,I

X

Y

m

k

y

y 0

X

L

Tbase

Tair, h

AC

p = Perimeter

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