Vibrations of Continuous Systems

1
Vibrations of Continuous Systems

• Normal mode
• Displacement pattern or shape of a system which
  is capable of free vibration at some particular
  natural frequency
• Higher order modes have increasingly more nodes
  and higher natural frequencies.
• Each normal mode can be considered as a degree of
  freedom of the system
• Continuous systems have an infinite number of
  degrees of freedom and hence an infinite number
  of normal modes

2
Cool!
Guitar String
y (x,t)
y
x
L

• Length L
• Mass / Length
• Tension T

• Equilibrium position is x axis,
• displacement is in y direction
• y is a continuous function of x and t

3
y (x,t)
Fy T?R -??T?L
?R ?(x dx,t) ?L ?(x,t)
Fy T ?(xdx,t)???(x,t)?
4
T
dx
?(xdx)
?(x)
T
xdx
For small displacements
x
y force on element
5
y-force on element
Fy
m
Mass of element
dx
Newtons Law
xdx
x
Equation of Motion
6
Wave Equation
Equation of motion
Define the wave speed
Wave Equation
7
One Solution
wave going to the right
wave going to the left
where f and g are completely arbitrary functions
Most general solution
Not the kind of solution were looking for
8
Normal Modes
y (x,t)
y
x
L
wave equation
(1)
look for solutions in the form
(2)
substitute (2) in (1)
9
Normal Modes
y
x
L
mode shape satisfies Helmholtz equation
solution
B 0
boundary conditions
10
Normal Modes
Natural frequencies
Normal modes
Note Natural frequencies are harmonics of f1
11
Normal Modes
1
T
n 1
f
?
1
2L
?

L
1
T
f
n 2
?
2
L
?
L
n 3
3
T
f
?
3
2L
?
L
2
T
f
?
n 4
4
L
?
L
12
Initial Condition (Plucked String)
x
13
Initial Condition (Plucked String)
x
14
Subsequent motion?
x
15
Subsequent motion?
x
16
Subsequent motion?
x
17
Subsequent motion?
x
18
Subsequent motion?
x
19
Subsequent motion?
x
20
Subsequent motion?
x
21
Subsequent motion?
x
22
Subsequent motion?
x
23
Subsequent motion?
x
24
Subsequent motion?
x
25
Normal Mode Motion
y
x
26
Normal Mode Motion
y
x
27
Normal Mode Motion
y
x
28
Normal Mode Motion
y
x
29
Normal Mode Motion
y
x
30
Normal Mode Motion
y
x
31
Normal Mode Motion
y
x
32
Normal Mode Motion
y
x
33
Normal Mode Motion
y
x
34
Normal Mode Motion
y
x
35
Normal Mode Motion
y
x
36
Normal Mode Motion
y
x
37
Normal Mode Motion
y
x
38
Actual Motion
x
39
Actual Motion
x
40
Actual Motion
x
41
Actual Motion
x
42
Actual Motion
x
43
Actual Motion
x
44
Actual Motion
x
45
Actual Motion
x
46
Actual Motion
x
47
Actual Motion
A
x
L
48
Actual Motion
A
x
49
Actual Motion
A
x
50
Actual Motion
A
x
51
Actual Motion
A
x
52
Normal Modes
1
T
n 1
f
?
1
2L
?

L
1
T
f
n 2
?
2
L
?
L
n 3
3
T
f
?
3
2L
?
L
2
T
f
?
n 4
4
L
?
L
53
Fundamental Frequency of a Guitar String
Pitch Frequency

• Depends on
• length
• density
• tension

54

• Tune guitar by adjusting tension T
• Play guitar by changing length L
• Use higher density strings for low notes

55
Free-Free Beam
y
w(x,t)
x
b
L
Euler equation of motion for thin beam
Look for modal solution
56
Free-Free Beam
y
w(x,t)
x
b
L
Equation for wn(x)
General solution
57
Boundary Conditions

• Four boundary conditions
• Zero shear and zero moment at x 0 and x L

Moment
Shear
Boundary Condition Free ends
58
Other Possible Boundary Conditions
Clamped
Zero deflection and zero slope at x 0 and x L
Pinned
Zero deflection and zero moment at x 0 and x L
Shear
59
Boundary Conditions

• Four boundary conditions
• Zero shear and zero moment at x 0 and x L

Moment
Shear
Boundary Condition Free ends
60
y
w(x,t)
x
b
L
Characteristic equation
Characteristic equation
wn
Normal modes
61
Free-Free Natural Frequencies
Characteristic Equation cosx1/cosh(x)
62
y
w(x,t)
x
b
L
Characteristic equation
Characteristic equation
wn
Normal modes
63
Modes 1, 2 and 3
Normalized Displacement
x/L
64
Example
For first mode
hence
65
What about the Glockenspiel?
Are the overtones harmonic?
No, but overtones are not strongly excited and
decay quickly
66
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67
Forced Drive
Sinusoidal force applied at xa

F
1
å
-
W

f
t
x
w
a
w
D
t
x
w
)
cos(
)
(
)
(
)
,
(
n
n
n
n
2
w
m

n
1
n
ö
æ
x
W
r
2
1

ç

F


n
arctan
r
D
n
,
,

ç
n
n
n
2
-
w
1
r
2
ø

-
x
2
2
è
r
r
)
2
(
)
1
(
n
n
n
n
n
-
W

f
x
t
x
w
t
x
w
))
(
cos(
)
(
)
,
(
A
A
x
w
)
,
(
W
Force -Displacement Transfer Function

W
A
T
)
(
F
68
Reciprocity
Reciprocity
xb
xb
xa
xa

Accelerometer
Accelerometer
Shaker
Shaker
69
Force-Displacement Transfer Function (Method 1 -
Random Excitation)
70
Force-Displacement Transfer Function (Method 2 -
Sinusoidal Excitation)
71
Force-Displacement Transfer Function
X/F Transfer Function (dB)
Frequency (Hz)