Lecture 33 V-g Method revisited.
Trang 1Lecture 33
V-g Method revisited
Trang 2Final EOM’s for forced response
mb
F B
b
h
θ
2 h
2 h
2 L
µ
−
=
+
−
−
−
2
1 L
x
µ
ω ω
ω is known because we pre-select it
2
Purdue Aeroelasticity
Trang 3Moment equilibrium equation
0
h
÷
+ +
−
−
2
1 L
M x
2 h
2 2
µ
ω µ
ω
2 h
2 2
2 h
2 2
2 2
2
a 2
1 L
a 2
1 L
M
a 2
1 M
r r
E
+
−
+ +
−
+ +
+
−
=
µ
ω µ
ω
µ
ω µ
ω ω
ω
α
α θ
θ θ
Trang 4Theodorsen’s method The system is self-equilibrating
Purdue Aeroelasticity
4
0
o
F mb
h
÷
TM
L
µ
2
µ
Trang 5Moment equilibrium equation
0
h
÷
2
2
2
2
2
2
2
1 2
h
θ
α
ω
= − + + + ÷ −
+ + ÷ − + ÷
Trang 6Eigenvalue Equation of Motion #1
2
0 2
ω
µ
6
Purdue Aeroelasticity
2
2
0 2
h
ω
Divide by ω2
Include structural damping
2
2
h
Trang 7Equation #2, moment equilibrium
2
µ
2
2
1 M
L
a 2
1 M
+ +
+
+
−
θθ
Mθ = − + a L
Divide by ω2
2
1
0
h
θ θ θ θ θ θ θ θθ θ
µ
ω ω
Include structural damping
( )
2
1
0
θ θ ωθ θ θ µ θθθ θ
ω
θ
Trang 8Matrix equations
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8
( )
2
1
0
θ θ ωθ θ θ µ θθθ θ
ω
θ
( )
2
2
h
ig
θ
θ
θ
ω
ω ω
+
2 2
2 2
2 2
0
1
0 1
2
0
1
h
h
r
i
h
b M
g
M
θ θ
θ
α
θ
ω ω
ω
ω
Trang 9The eigenvalue problem
2
2
2 2
2 2
0
1
0 1
2
0
1
h
h
r
i
h
b M
g
M
θ θ
θ
α
θ
ω ω
ω
ω
2 2
2
2 2
2 1
0
h
h
r
θ
α θ
θ
ω ω
µ
Ω
Trang 10Another look at it This should be easy for a 6th grader with MATLAB
Purdue Aeroelasticity
10
2 2
2
2 2
2 1
0
h
h
r
θ
α θ
θ
ω ω
µ
Ω
2
2
2
2
1 1
2
h
θ
÷ ÷ ÷ ÷ ÷
Ω
Trang 11The flutter problem – a complex eigenvalue with flutter frequency and airspeed
unknown
elastic axis location (shear center)
bending-torsion frequency ratio
dimensionless static unbalance dimensionless radius of gyration about SC
=density ratio
frequency
k=reduced frequency
h
a
S
r
θ
θ
θ
θ
ω
ω
µ
ω
=
=
= =
=
=