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

18.5 Matrix Algebra 605

ln x(ln10)(logx) 2.302585 logx. Using the logarithmic definitions, we can

also prove the following identities, which you will find useful when simplifying

engineering relationships.

Decibel Scale In engineering, the loudness of sound is typically expressed in a

unit called decibel (dB). The threshold of hearing (that is the softest sound that

a healthy human can hear) is 20 mPa or 20  10

 6

Pa. Note this is caused by a

very, very small air pressure change. Amazing! At the other end of hearing range

lies the threshold of hearing pain, which is caused by approximately 100 10

6

mPa

of pressure change. To keep the numbers manageable in the range of hear-

ing, (from 20 mPa to 100,000,000 mPa) the decibel scale is defined by dB

20 log (I/20) mPa, whereIrepresents the pressure change (in mPa) created by the

sound source. For example, a sound created by a moving car (creating a pressure

change of 200,000mPa) has a corresponding decibel rating of 20 log (200000mPa /

20) mPa80 dB.The decibel rating of common sounds is shown in the accom-

panying figure.

Finally, the following are some additional mathematical relationships that you

may find helpful during your engineering education.

18.5 Matrix Algebra

As you will learn later during your engineering education, formulation of many engineering

problems, such as the vibration of machines, airplanes, and structures; joint deflections of struc-

tural systems; current flow through branches of electrical circuits; and the fluid flow in pipe net-

works lead to a set of linear algebraic equations that are solved simultaneously. A good

understanding of matrix algebra is essential in the formulation and solution of these models. As

is the case with any topic, matrix algebra has its own terminology and follows a set of rules. We

will provide a brief overview of matrix terminology and matrix algebra in this section.

Basic Definitions

During your engineering education, you will learn about different types of physical variables. There

are those that are identifiable by a single value or magnitude. For example, time can be described

a

x

y

b

n



x

n

y

n

x

n

x

mx

nm

x

n



1

x

n

x

0

1 x  1   1 x 02

n

2

m

x

nm

1 x y 2

n

x

n

y

n

x

n

x

m

x

nm

log x

n

log n log x

x

y

log xylog xlog y log xlog y

100,000,000

10,000,000

1,000,000

100,000

10,000

1,000

100

20

130

120

110

100

90

80

70

60

50

40

30

20

10

0

140

mPa dB

Threshold of pain

Threshold of hearing

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀圀圀圀⸀夀䄀娀䐀䄀一倀刀䔀匀匀⸀䌀伀䴀