AAE556 lecture 17 Typical section vibration

AAE 556
Aeroelasticity
Lecture 17
Typical section vibration

Purdue Aeroelasticity

17-1


Understanding the origins of flutter
Typical section equations of motion - 2 DOF

x
Plunge displacement h is positive
downward & measured at the

restrict to small angle

h(t)

θ(t)

c.g.

shear center
shear
center

xcg
xθ

h ( t ) = plunge freedom (bending )

θ ( t ) = pitch freedom ( twist )
measured at the shear center from static equilibrium position
17-2
Purdue Aeroelasticity


A peek ahead at the final result
coupled equations of motion
dynamically coupled but elastically uncoupled

 m
 mx
 θ

&
mxθ   h&
 Kh
+ 
&


Iθ  θ&
0

0  h 
0 
  =  


KT  θ 
0 
x

mg = weight

restrict to small angle

h(t)

θ(t)

c.g.

shear
center

xcg

xθ
xθ is called static unbalance and is the source of dynamic coupling
17-3
Purdue Aeroelasticity


Lagrange and analytical dynamics
an alternative to FBD’s and Isaac Newton

z(t) is the downward displacement of a small

portion of the airfoil at a position x located
downstream of the shear center

d  ∂ (T − U )  ∂ (T − U )
= Qi

÷−
dt  ∂η&i 
∂ηi

z = h + x sin θ ≅ h + xθ
x

kinetic energy

restrict to small angle

h(t)

θ(t)

c.g.

1 x = xt
T = ∫
( ρ )(h&+ xθ&) 2 dx
2 x =− xl

shear
center

strain energy

xcg

1
1
2
U = K h h + KT θ 2
2
2
17-4
Purdue Aeroelasticity


Expanding the kinetic energy integral

(

)

x = xt
1 x = xt
2
&
&
T = ∫
( ρ )(h + xθ ) dx = ∫
ρ h&2 + 2 ρ xh&
θ&+ ρ x 2θ&2 dx

x =− xl
2 x =− xl

m = ∫ ρ ( x )dx
m is the total mass

Sθ = mxθ = ∫ ρ ( x ) xdx
Sq is called the static unbalance

Iθ = ∫ ρ x 2 dx = I o + mxθ2
Iq is called the airfoil mass moment of inertia – it has 2 parts

1 &2
2
&
&
&
T =
(mh + 2Sθ hθ + Iθ θ )
2
17-5
Purdue Aeroelasticity


Equations of motion for the unforced system (Qi = 0)

∂T
= mh + mxθ θ
∂h

∂T
 + I θ
=
mx
h
θ
θ

∂θ

∂U
= Khh
∂h

∂U
= KT θ
∂θ

EOM in matrix form, as promised

 m
 mx
 θ

&
mxθ   h&
 Kh
+ 
&



Iθ  θ&
0

0  h 
0 
  =  

KT  θ 
0 
17-6

Purdue Aeroelasticity


Differential equation
a trial solution
Goal – frequencies and mode shapes

 h  st
h(t ) 

 =  e
θ (t )
θ 
Substitute this into differential equations

 m
s 
 mxθ

2

mxθ   h  st  K h
 e + 

Iθ  θ 
0

0   h  st
0 
 e =  

K h  θ 
0 
17-7

Purdue Aeroelasticity


There is a characteristic equation here

 m
s 
 mxθ
2

mxθ   h  st  K h
 e + 

Iθ  θ 

0

( s 2 m + K h )

 ( s 2 mxθ )


0   h  st
0 
 e =  

K h  θ 
0 

2
s
( mxθ )   h  st 0
e = 


2
s
( Iθ + K h )  θ  0

17-8
Purdue Aeroelasticity


The time dependence term is factored out

2

2

(s m + K h )
2

( s mxθ )

( s mxθ )

2

( s Iθ + KT )

Determinant of dynamic system matrix

set determinant to zero (characteristic equation)

( s m+K ) ( s I
2

2

h

θ

+ KT ) − ( s mxθ ) ( s mxθ ) = 0
2

2

17-9
Purdue Aeroelasticity


Nondimensionalize by dividing by m and Iθ

( )

2

mx
 2 K h  2 KT 
 − s 2  s 2 θ  = 0
s +
 s +


m 
Iθ 
I

θ 


Define uncoupled frequency parameters

ω h2

(s

2

Kh
=
m

+ ω h2

)(

ωθ2

KT
=
Iθ

) ( )

2

mx
s 2 + ωθ2 − s 2  s 2 θ  = 0


I
θ 


 4  mxθ2  2 2

2
2 2
=
0
 s 1 −
÷+ s ( ω h + ωθ ) + ω h ωθ ÷
÷
I
θ




17-10
Purdue Aeroelasticity


Solution for natural frequencies

(

)

 4  Io  2 2

2
2

2
 s   + s ω h + ωθ + ω h ωθ  = 0
 I 

  θ


(as

4

)

2

+ bs + c = 0
2

− b ± b − 4ac
s =
2a
2

17-11
Purdue Aeroelasticity


Solutions for exponent s
These are complex numbers

−
s2 =

(

ω h2

+ ωθ2

)± (

ω h2

)

2 2
+ ωθ

 Io  2 2
− 4 ω h ωθ
 Iθ 

 Io 
2 
 Iθ 
2

 1 Iθ
 Iθ  2 2
1  Iθ  2

2
2
2 
s = −  ÷( ω h + ωθ ) ± 
ω h + ωθ )  −  ÷ω h ωθ
(
2  Io 
 2 Io
  Io 
2

e =e
st

± iωt
17-12

Purdue Aeroelasticity


solutions for s are complex numbers

Iθ =

2
mrθ

Iθ
=
Io

Io =

and

2
I o + mxθ
2
mro

=

Iθ
= 1+
Io

2
mro

2
mro

2
+ mxθ
2
mro

2
xθ
2

ro
17-13

Purdue Aeroelasticity


Example configuration
2b=c

xθ = 0.10c = 0.20b

and

xθ
= 0.40
ro
 xθ
1 + 
 ro

2


 = 1.16


ro = 0.25c = 0.5b

and

xθ = aro

 xθ

 ro

2


 = 0.16


Iθ
2
= 1+ a
Io

New terms – the radius of gyration
17-14
Purdue Aeroelasticity


Natural frequencies change when the wing c.g. or EA positions change

2

 1 Iθ
 Iθ  2 2
1  Iθ  2
2

2
2
2 
ω = +  ÷( ω h + ωθ ) − or + 
ω h + ωθ )  −  ÷ω h ωθ
(
2  Io 
 2 Io
  Io 
2


 = 1 + a 2

x

restrict to small angle

h(t)

θ(t)

shear
center

xcg

c.g.

40

35
natural frequencies
(rad./sec.)

x
Iθ
= 1 +  θ
Io
 ro

Natural frequencies
vs.
c.g. offset

30
torsion frequency

25
20

fundamental
(plunge) frequency

15
10
5
0
0.00

0.25

0.50

0.75

c.g. offset

c.g. offset in semi-chords
17-15
Purdue Aeroelasticity

1.00


Summary?

17-16
Purdue Aeroelasticity