This document summarizes the content of Chapter 11 (Modal Analysis) from an engineering textbook. It covers modal analysis techniques for various structures: a gearbox, two-story building, compact disk, and guitar string. Key topics discussed include unprestressed and prestressed modal analysis, natural frequencies, resonance, harmonic series, just tuning and equal temperament systems, and beat frequencies. The chapter contains step-by-step modal analysis examples and explanations of engineering and music terminology.

1. Chapter 11 Modal Analysis 1
Chapter 11
Modal Analysis
11.1 Step-by-Step: Gearbox
11.2 Step-by-Step: Two-Story Building
11.3 More Exercise: Compact Disk
11.4 More Exercise: Guitar String
11.5 Review

2. Chapter 11 Modal Analysis Section 11.1 Gearbox 2
Section 11.1
Gearbox

3. Chapter 11 Modal Analysis Section 11.1 Gearbox 3
Techniques/Concepts
• Unprestressed Modal
Analysis
• Prestressed Modal
Analysis
• Create Mode Shape
Results

4. Chapter 11 Modal Analysis Section 11.2 Two-Story Building 4
Section 11.2
Two-Story Building

5. Chapter 11 Modal Analysis Section 11.2 Two-Story Building 5
Original Structure...

6. Chapter 11 Modal Analysis Section 11.2 Two-Story Building 6
After Bracing...

7. Chapter 11 Modal Analysis Section 11.3 Compact Disk 7
Problem Description
• The MythBusters succeeded in shattering CDs at speeds of 23,000 rpm. We want to
find out the maximum stress in the CD due to the centrifugal force to justify if the
shattering is indeed due to the high stress.
• We also want to investigate the possibility of resonant vibrations. We will conclude
that the CD shattering may be due to vibrations rather than centrifugal stress.
Section 11.3
Compact Disk

8. Chapter 11 Modal Analysis Section 11.3 Compact Disk 8
Stresses Due to Centrifugal Force
The maximum tensile
stress is 20.8 MPa, far
less than the material's
tensile strength (65
MPa).

9. Chapter 11 Modal Analysis Section 11.3 Compact Disk 9
0
100
200
300
400
500
0 6000 12000 18000 24000 30000
NaturalFrequencies(Hz)
Rotational Speed (rpm)
Natural Frequencies
[1] The lowest
natural
frequency.
[2] The second
lowest natural
frequency.
[3] This is the "line of
resonance," which has a
slope of unity (note that
30,000 rpm = 500 Hz). If
any frequencies fall on this
line, they will be excited
and resonance will occur.

10. Chapter 11 Modal Analysis Section 11.4 Guitar String 10
Section 11.4
Guitar String
• Harmonic Series
• Just Tuning System
• Twelve-Tone Equally Tempered Tuning System
• Beat Frequency
Terminologies

11. Chapter 11 Modal Analysis Section 11.4 Guitar String 11
Harmonic Series
• A harmonic mode has a
frequency that is an integral
multiplication of the
fundamental frequency.
• If you pluck a string, you will
produce a tone made up of
all harmonic modes, the
harmonic mixes determine
the quality of the note.

12. Chapter 11 Modal Analysis Section 11.4 Guitar String 12
Just Tuning System
C D E F G A B C
do re mi fa sol la ti do
C#
D#
F#
G#
A#
C#
Db
Eb
Gb
Ab
Bb
Db
1
9
8
5
4
4
3
3
2
5
3
15
8
2
• In the early 20th century, the "just
tuning" systems prevail in the music
world.
• If we play the notes do and sol together,
the sound is pleasing to our ears, since
they have the simplest frequency ratio.
• The major cord C consists of the notes
do, me, sol, do, the simplest frequency
ratios.
• The problem of this system is that it is almost
impossible to play in another key.

13. Chapter 11 Modal Analysis Section 11.4 Guitar String 13
Twelve-Tone Equally Tempered Tuning System
• The idea is to compromise on the
frequency ratios between the
notes, so that they can be played in
different keys.
• In this system, an octave is equally
divided into 12 tones (including
semitones) in logarithmic scale.
• In other words, the adjacent tones
have a frequency ratio of 2112
.
250
300
350
400
450
500
550
0 2 4 6 8 10 12
FundamentalFrequency(Hz)
Note
[1] The frequencies in
the just tuning system
are marked by small
circles.
[2] The frequencies of the
notes in the equal
temperament system are
plotted as a solid curve.
C D E F#
G#
A#
C
C
D
E
F
G
A
B
C

14. Chapter 11 Modal Analysis Section 11.4 Guitar String 14
Beat Frequency
• When two waves of different frequencies are combined, they interfere with each other.
The fluctuation in amplitude of the combined wave is called beats, and the frequency is
called the beat frequency.
• The beat frequency is equal to the frequency difference of the two waves.
• When we play C and D together, the beat frequency is 32.03 Hz (293.66 - 261.63), which is
a harsh buzz and unpleasant for our ears.
• When we play C and G together, the beat frequency is 130.37 Hz (392.00 - 261.63), which is
close to a harmonic frequency of the middle C (261.63 Hz).