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Flutter Analysis

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Submitted By rmarwaha89
Words 833
Pages 4
March 14, 2012

[AEROELASTICITY MIDTERM PROJECT-FLUTTER ANALYSIS ]

Objective:
To conduct flutter analysis of a typical section model of an airfoil using the parameters as discussed in lecture. Consider both the case without the

̇

̇

term and the case with the term in

the aerodynamic model. Create the Eigen value plot for both the cases.
Answer:
Introduction to Flutter:
Flutter is a dangerous phenomenon encountered in flexible structures subjected to aerodynamic forces. This includes aircraft, buildings, telegraph wires, stop signs and bridges. Flutter occurs as a result of interactions between aerodynamics, stiffness and inertial forces on a structure. In an aircraft, as the speed of the wind increases, there may be a point at which the structural damping is insufficient to damp out the motions which are increasing due to aerodynamic energy being added to the structure. This vibration can cause structural failure and therefore considering flutter characteristics is an essential part of designing an aircraft.
̇

Case 1 : Flutter Analysis when there is no additional term ( ) in the aerodynamic damping
For the case of simplicity we consider the following typical section model

Equations of motion for this model are:
̈
̈

̈

(E1)
̈

(E2)

where m=modal mass matrix
=static unbalance
=moment of inertia
=spring constant

Page 1

[AEROELASTICITY MIDTERM PROJECT-FLUTTER ANALYSIS ]

March 14, 2012

=constant of torsional spring
=displacement of typical section (positive down)
=rotation angle of typical section
=distance from the aerodynamic center
=dynamic pressure=

, where =air density and =air speed

= lift force=
=

=Aerodynamic moment and is positive nose up

Similarly assume that the motion is simple harmonic,
Let,

̅

̅

(E3)

where,

The factor ‘
’ represents the frequency of oscillation of the system, whereas ‘
’ represents the growth or decay of the system with time. If the real part of any value of ‘p’ is positive then the motion diverges exponentially with time and the typical section is unstable.
Substituting E3 into E1 and E2 we obtain:
̅

̅

̅

̅

̅

̅

̅

(E4)
̅

(E5)

Equations E4 and E5 can be represented in the following matrix form after eliminating
̅
[ ]{ }
̅
[]

where,

{}

[

]

For non-trivial solution, det [D] = 0:
(

(

)

(

)

Page 2

:

[AEROELASTICITY MIDTERM PROJECT-FLUTTER ANALYSIS ]

March 14, 2012

Rearranging the equation:
(

)

(

)

(

)

(E6)

Given that (unit ommitted): b=1, , c=2, M=1,

5,

Solving E6 with Matlab by assumming,

FIG.1
FIG. 1 shows a representation of the imaginery and real parts of the eigenvalue for case without additional term in the aerodynamic damping. The point at which changes from negative to positive gives us critical value of at which flutter occurs.

Page 3

[AEROELASTICITY MIDTERM PROJECT-FLUTTER ANALYSIS ]

March 14, 2012

̇

Case 2: Flutter Analysis when additional term is persent in the aerodynamic damping ( )
̇

The equations of motion for the damped case (with the term)
̈

̇

̈

̇

̈
̈

(E7)
(E8)

Let us assume the motion to be simple harmonic
̅

̅

(E9)

where, and The factor ‘
’ gives the frequency of oscillation of the system, whereas ‘
’ represents the growth or decay of the system with time. Again, if the real part of any value of ‘p’ is positive then the motion diverges exponentially with time and the typical section is unstable.
Substitute E9 into E7 and E8 to obtain:
̅

̅

̅

̅

̅

̅

̅

̅

(E10)
̅

̅

(E11)

Equations E10 and E11 can be represented in the following matrix form after eliminating
̅
[ ]{ }
̅
Where, [ ]

:

{}

[

]

For a non-trivial solution, det[D]=0:
(

)(

)

Page 4

(

)(

)

[AEROELASTICITY MIDTERM PROJECT-FLUTTER ANALYSIS ]

March 14, 2012

Rearranging the above equation:
(
(

)
)

(

(

)

(

)

)

(E12)

Given that (unit ommitted), b=1, , c=2, M=1,

5,

Solving E12 with Matlab by assumming,

FIG. 2
The red dotted lines in FIG. 2 represent the Torsion branch ( represent the Bending branch (

) and the blue dotted lines

. The Torsion branch represents the imaginary values of p

whereas the Bending branch represent the real values of p for which det[D] = 0

Page 5

March 14, 2012

[AEROELASTICITY MIDTERM PROJECT-FLUTTER ANALYSIS ]

FIG. 3
Flutter condition occurs when the value of real part of p, namely , changes from negative to positive. The red box in FIG. 3 represents the point at this change occurs which gives us the critical value of at which flutter is observed.

FIG. 4
FIG. 4 represents the real and imaginary parts of the Eigen value for case 1 (red and yellow) and case 2 (yellow and green).

Page 6

March 14, 2012

[AEROELASTICITY MIDTERM PROJECT-FLUTTER ANALYSIS ]

Conclusion:
1. With aerodynamic damping omitted the typical section model is neutrally stable until
. At this point the bending and torsion frequencies merge and for the system is unstable.
2. With aerodynamic damping included, for small all values of are stable and flutter occurs at sufficiently large where changes sign from negative to positive. There is a tendency for the frequencies to merge but complete merging does not occur.
3. The addition of aerodynamic damping reduces the flutter velocity, case. Page 7

in this particular

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