Download Lecture_18_mode_shapes

AAE 556
Aeroelasticity
Lecture 18
Resonance, Mode shapes
x
restrict to small angle
h(t)
(t)
c.g.
shear
center
xcg
http://paws.kettering.edu/~drussell/Demos/Flexural/bending.html
http://vodpod.com/watch/1799999-sand-vibration-patterns-chladni-plate
Purdue Aeroelasticity
18-1
Summary
• MDOF systems with n degrees of freedom
have n possible “modes of motion”
• Mode of motion means a (natural, resonant)
frequency with a well-defined mode shape
• Eigenvectors – another word for mode
shapes – provide information about node
lines or node points
• Experiments provide node lines and
frequencies to compare with analysis
18-2
Purdue Aeroelasticity
Typical section equations of motion - 2 dof
x
restrict to small angle
h(t)
(t)
c.g.
shear
center
xcg
ht   plunge freedom (bending )
 t   pitch freedom twist 
measured from static equilibrium position
18-3
Purdue Aeroelasticity
Review - Equations of motion for
free vibration
 m
mx
 
mx  h  K h
   

I     0
Trial solution
assume harmonic motion
0  h  0
  

KT    0
ht  h  it
    e
 t   
result
 m
 
mx
2
mx  h  it  K h
 e  

I   
0
0  h  it 0
 e   

KT   
0
18-4
Purdue Aeroelasticity
Harmonic forcing at or near the natural
frequencies
x
mx   h   Kh
  

I     0
 m
 mx
 
Trial solution
harmonic motion
restrict to small angle
0  h 
 1  it
   Fo   e

KT   
d 
h(t)
(t)
c.g.
shear
center
xcg
ht  h  it
    e
 t   
result
 m
 
mx
2
mx  h  it  K h
 e  

I   
0
0  h  it
1  it
 e  Fo  e

KT   
d 
18-5
Purdue Aeroelasticity
Harmonic forcing at or near the natural
frequencies
 m
 
mx
2
mx  h  it  K h
 e  

I   
0
0  h  it
1  it
 e  Fo  e

KT   
d 
2
2


h

 1 
(


I

K
)
(

mx )
F
 

T
o
  
 
2
2
( mx )
( m  K h )  d 

 
 
    m  Kh   I  KT     mx   0
2
2
2
2
At or near resonance, the amplitude of the response is large
(here it is infinite because we have no damping)
18-6
Purdue Aeroelasticity
Defining mode shapes
what will the vibrations look like if we force the
system at natural frequencies?
(1)
(2)
2
(  i2 m  K h )
(  i mx )

2
2
 (  i mx ) (  i I  KT
  h  0 
    
)   0 
eigenvalues & eigenvectors
h


2
( i mx )
( i2 m  K h
)
or
2
( i I  KT )
( i2 mx )
18-7
Purdue Aeroelasticity
A different expression

KT 
2

    i 
2
I 
(  i x )
h


or
Kh 
 

2
2 mx 



 i

   i

m 

I 

h


2
(  i x )
(  i2   h2
)
or
2
(  i
  )
 2 x 
i 2 


r
 

2
18-8
Purdue Aeroelasticity
System mode shapes
x
If  is 1 then how much is h?
restrict to small angle
h(t)
(t)
c.g.
shear
center
xcg
 2 x 
b  i

h
b


 (i2   h2 )
or
b(i2  2 )
 2 x b 2 
 i
2 
b
r
 

18-9
Purdue Aeroelasticity
example
x
restrict to small angle
h(t)
(t)
c.g.

 i2 
m
mx
mx  h  it  K h
 e  
I   
0
0  h  it 0 
 e   
KT   
0 
shear
center
xcg
x  0.10b
r  0.25b
2
Kh
h 
 10 rad / sec.
m
  25 rad / sec .
1  0.3985 
2  1.0245 
2
18-10
Purdue Aeroelasticity
example
reference
 h  10 rad / sec .
actual
1  0.3985 
reference
  25 rad / sec .
actual
2  1.0245 
bending
torsion
18-11
Purdue Aeroelasticity
Mode shape
1  0.3985 
h
b  13.245


2 x 
  0.3985

h
b 
b  
2 



 0.3985 2  h2 



 

When we let h/b=1 then we are
asking about the amount of q in the
plunge mode
1 unit
 
  h
1
b 13.245
18-12
Purdue Aeroelasticity
Shake testing identifies modes and
frequencies
http://macl.caeds.eng.uml.edu/umlspace/feb98.pdf
18-13
Purdue Aeroelasticity
Torsional frequency and mode shape
 2  1.0245 
h
b
1 radian
  0.118 
node point
18-14
Purdue Aeroelasticity
Node point definition
A point in space where there is no displacement,
velocity or acceleration when the structure is
vibrating at a natural frequency
z t   ht   x t   h  x e
it
x
z  t   0   h  xo  e
i t
18-15
Purdue Aeroelasticity
Node point depends on eigenvector
h  x    0
x
o
xo  
h

h
first mode
divide by b
b  13.245

h
xo
 b
b

xo
 13.243
b
18-16
Purdue Aeroelasticity
Node point for second frequency
h
xo
 0.118
b
b  0.118

18-17
Purdue Aeroelasticity
Summary
• MDOF systems have n modes of motion
• Mode of motion means a (natural,
resonant) frequency with a mode shape
• Eigenvectors – mode shapes – provide
node lines or node points
• Experiment provides node lines and
frequencies
18-18
Purdue Aeroelasticity