AAE556 lecture 17 Typical section vibration

AAE 556 Aeroelasticity Lecture 17 Typical section vibration... restrict to small angle 17-2 Purdue Aeroelasticity... restrict to small angle mg = weight x is called static unbalance and

AAE 556 Aeroelasticity Lecture 17

Typical section vibration

measured at the shear center from static

equilibrium position

h t  plunge freedom bending

  t pitch freedom twist  

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

Plunge

displacement h is

positive downward

& measured at the

shear center

h(t)

 (t)

x cg

shear center

x

c.g.

restrict to small angle

17-2

Purdue Aeroelasticity

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

h

T

mx mx





 

   

   





h(t)

 (t)

x cg

shear center

x

c.g.

restrict to small angle

mg = weight

x is called static unbalance and is the source of dynamic

Lagrange and analytical dynamics

an alternative to FBD’s and Isaac Newton



x h

z   sin  

kinetic energy

2

1

2

t

l

x x

x x



strain energy

U  K h  K 

i

d

Q dt

     

  

z(t) is the downward

displacement of a small

portion of the airfoil at a

position x located

downstream of the shear

center

17-4

Purdue Aeroelasticity

h(t)

 (t)

x cg

shear center

x

c.g.

restrict to small angle

Expanding the kinetic energy integral

1

2

T  mh   S h     I 

  x dx

m    S   mx       x xdx

o

I    x dx I   mx

m is the total mass S q is called the static unbalance

Iq is called the airfoil mass moment of inertia – it has 2 parts

1

2

T     h  h   d

Equations of motion for the unforced

system (Qi = 0)



 



 m h mx h

T















 mx h I

T









h

K h

U

h









 KT

U







EOM in matrix form, as promised

h

T



 

   

   





17-6

Purdue Aeroelasticity

Differential equation

a trial solution

st

e

h t

t h





























 ( )

) (

Substitute this into differential equations

h

h

s



 

Goal – frequencies and mode shapes

There is a characteristic equation

here

0 0

st h

h





h

h

s



 

The time dependence term is factored out

) (

) (

) (

)

(

2 2

2 2

T

h

K I

s mx

s

mx s

K m

s











Determinant of dynamic system matrix

set determinant to zero (characteristic equation)

 s m K2  h   s I2   KT    s mx2    s mx2    0

Nondimensionalize by dividing by m

and I

2 2

2 2

2

























































mx s

s I

K s

m

K

Define uncoupled frequency parameters

m

Kh

h2 









I

K T

 2

2 2

2 2

2 2

2



































I

mx s

s s

s h

2

4 1 mx 2 h2 2 h2 2 0

I





17-10

Purdue Aeroelasticity

Solution for natural frequencies

 2 2  2 2 0

2 4





















































I

I

s

 as 4  bs 2  c   0

a

ac b

b s

2

4

2



Solutions for exponent s

These are complex numbers

















































I I

I

I

s

o

h

o h

h

2

2 2 2

2 2

2

2

s

17-12

Purdue Aeroelasticity

st i t



solutions for s are complex

numbers

2



I  and I  o mr o 2

2

2 2

2

2

o

o o

o

mx

mr mr

mx

I I







2

2

1

o

x I





Example configuration

b c

x   0 10  0 20 and r o  0 25 c  0 5 b

40 0



o

r

x 

16 0

2



















o

r

x

and

16 1 1

2





















o

r

x

o

ar

I

I

o







2b=c

New terms – the radius of gyration

17-14

Purdue Aeroelasticity

Natural frequencies change when the wing

c.g or EA positions change

2

or

                 

2

2

1

r

x I

I

o o

























1.00 0.75

0.50 0.25

0.00 0 5 10 15 20 25 30 35 40

Natural frequencies vs.

c.g offset

fundamental (plunge) frequency

torsion frequency

h(t)

 (t)

x cg

shear

center

x

c.g.

restrict to small angle

Summary?