### Modal Testing and Analysis - Saeed Ziaei-Rad

```Modal Testing and Analysis
1
Single Degree-of-Freedom (SDOF)
Undamped
Viscously Damped
Hysterically (Structurally) Damped
Modal Analysis and Testing
2
Undamped Systems (Theory)
F(t)
Spatial Model (Free vibration)
m
m x  kx  0
x ( t )  Xe
i t
k
 m  X  kX  0
2
(k  m  ) X  0
2
 0  (k / m )
1/ 2
Modal Analysis and Testing
3
Undamped Systems (Forced vibration)
1
0.9
m x  kx  f ( t )
0.7
i t
0.6
H()
x ( t )  Xe
0.8
0.5
0.4
f ( t )  Fe
i t
0.3
0.2
0.1
(k  m  ) X  F
2
H ( )   ( ) 
0
0
2
4
6
8
10
12
14
16
18
20
X
F

1
k  m
2
Modal Analysis and Testing
FRF=Frequency
Response Function
4
Viscous Damping (Free Vibration)
m x  c x  kx  0
x ( t )  Xe
st
ms  cs  k  0
2
s1 , 2    0   i  0 1  
Oscillatory solution
2
 0  k / m ,   c /( 2 km )
2
x ( t )  Xe
 t
e
i d t
   0 ,  d   0 1  
2
Modal Analysis and Testing
5
Viscous Damping (Forced Vibration)
m x  c x  kx  f ( t )
0.1
0.09
i t
(   m  i  c  k ) Xe
2
 Fe
0.08
i t
0.07
H ( ) 
X
F

H()
0.06
1
0.05
0.04
( k   m )  i c
2
0.03
0.02
0.01
H ( ) 
1
0
0
2
4
6
8
10
12
14
16
18
20
18
20
180
( k   m )  ( c )
2
2
2
160
140
  c 
 H ( )  Arc tan g 

2
 k  m 
 (Degree)
120
100
80
60
40
20
0
0
2
4
6
8
10
Modal Analysis and Testing
12
14
16
6
Structural Damping
# Viscous damping is not a good representative of real
structures.
# Damping in real structures is frequency-dependent.
# A damper whose rate varies with frequency.
f
f
x
Viscous damper
f
x
x
Dry friction
Structural damping
Modal Analysis and Testing
7
Structural Damping
Equivalent Viscous damping
ce  d / 
m x  c e x  kx  f ( t )
(  m   ic e  k ) X  F
2
(  m   id  k ) X  F
2
H ( ) 
H ( ) 
1
( k  m  )  id
2
1/ k
(1  ( /  0 ) )  i 
2
 =Structural damping
loss factor
Modal Analysis and Testing
8
Alternative Forms of FRF
H ( ) 
Xe
Fe
Y ( ) 
Ve
Fe
A ( ) 
Ae
Fe
i t
i t


V
Mobility
F
i t
i t
Receptance
F
i t
i t
X

A
F
Inertance or accelerance
Modal Analysis and Testing
9
Relation between receptance and
mobility
x ( t )  Xe
i t
i t
i t

v ( t )  x ( t )  i  Xe
 Ve
Y ( ) 
V

F
i X
 i  H ( )
F
Y ( )   H ( )
 Y   H  90
Modal Analysis and Testing
10
Relation between receptance and
Inertance
x ( t )  Xe
i t
2
i t
i t
a ( t )  x( t )    Xe
 Ae
A ( ) 
A
 X
2

F
   H ( )
2
F
A ( )   H ( )
2
 A   H  180
Modal Analysis and Testing
11
Definition of FRFs
Response
Parameter: R
Standard FRF: R/F
Inverse FRF:
F/R
Displacement
Receptance
Dynamic Flexibility
Dynamic Compliance
Dynamic
Stiffness
Velocity
Mobility
Mechanical
Impedance
Acceleration
Inertance
Accelerance
Apparent Mass
Modal Analysis and Testing
12
Graphical Display of FRFs
Modulus of FRF vs. frequency and phase vs.
frequency (Bode type of plot)
Real part of FRF vs. frequency and imaginary part
vs. frequency
Real part of inverse FRF vs. frequency (or
frequency^2) and imaginary part of inverse FRF
vs. frequency (or frequency^2)
Real part of FRF vs. imaginary part of FRF
(Nyquist type of plot)
Modal Analysis and Testing
13
Modulus vs. Frequency
Receptance FRF
Inertance FRF
Mobility FRF
Modal Analysis and Testing
14
Modulus vs. Frequency
K=100000
K=1000000
Receptance FRF
Mobility FRF
M=10
M=100
Inertance FRF
Modal Analysis and Testing
15
Modulus vs. Frequency
A low Frequency straight-line (correspond to
stiffness)
A high frequency straight-line (correspond to
mass)
The resonant region with its abrupt magnitude and
phase variation
Modal Analysis and Testing
16
Frequency Response of Mass and
stiffness Elements
FRF
Mass
Stiffness
H ( )
1/ m
1/ k
log H ( )
 log m  2 log 
 log k
Y ( )
 i / m
i / k
log Y ( )
 log m  log 
log   log k
A ( )
1/ m
 /k
log A ( )
 log m
2 log   log k
2
Modal Analysis and Testing
2
17
Real and Imaginary vs. Frequency
Receptancd FRF
Mobility FRF
Inertance FRF
Modal Analysis and Testing
18
Real vs. Imaginary
(Viscous Damping)
Modal Analysis and Testing
19
Real vs. Imaginary
Y ( )  i  H ( ) 
i
(Viscous Damping)
 c  i ( k   m )
2
k  m   i c
2

2
( k  m  )  ( c )
2
2
2
 c
2
Re( Y ) 
( k  m  )  ( c )
2
2
2
 (k   m )
2
Im( Y ) 
( k  m  )  ( c )
2
2
2
Let
U  Re( Y )
V  Im( Y )
(U  1 / 2 c )  V
2
2
 (1 / 2 c )
2
Modal Analysis and Testing
20
Real vs. Imaginary
(Structural Damping)
Modal Analysis and Testing
21
Real vs. Imaginary
H ( ) 
(Structural Damping)
( k   m )  id
2
1
k  m   id
2

(k  m  )  (d )
2
2
2
(k   m )
2
Re( Y ) 
Im( Y ) 
(k  m  )  (d )
2
2
2
d
(k  m  )  (d )
2
2
2
Let
U  Re( Y )
V  Im( Y )
U
2
 (V  1 / 2 d )  (1 / 2 d )
2
2
Modal Analysis and Testing
22
Conclusions
Close inspection of real structures suggests that
viscous damping is not a good representative for
MDOF systems.
All structures show a degree of structural
damping.
Structural damping acts like an imaginary
stiffness in frequency domain.
Modulus vs. Frequency and Nyquist type plots for
FRFs are more common.
Modal Analysis and Testing
23
```