flutter speed determination using p

7/27/2019 Flutter Speed Determination Using p

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AS5970 STRUCTURAL DYNAMICS AND AEROELASTICITY (Jan-May 2013)

ASSIGNMENT TITLE : FLUTTER SPEED DETERMINATION USING P METHOD

Submitted by

NAME: S VIJAY KISHORE

DEPARTMENT: AEROSPACE ENGINEERING

DEGREE: M Tech

REG.NO AE12M019


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FLUTTER SPEED DETERMINATION USING P-METHOD

Consider a typical airfoil section where the linear spring provides the plunge displacement

and torsional spring provides the twist (torsional stiffness) as shown in the figure below.

The reference point would represent elastic axis. P, C, Q, T represents reference point,

Centre of mass, aerodynamic center and three-quarter chord respectively. Thedimensionless parameter e and a represents determine the location of C and P.The chord

offset of center of mass from the reference point is made dimensionless by airfoil semi-

chord b and denoted by .The Langrangeans equation is used to find the kineticenergy and the potential energy.

In this method the following assumptions are made

1. The flow is steady.

2. A simple aerodynamic theory (thin airfoil theory) is used to express the lift and moment

expression.

( )

where where

The langrangean is given by


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The generalized coordinates are h and On solving we obtain a two system of equations

( ) Lift is given by

The governing equation for the flutter using steady flow theory in matrix form is given by

2q h

2 2q p q

m mbx K 0 h h 00 2U bqh+ + =

mbx I 0 K q q 00 -2U b (0.5+a)q

We assume the solution of the equation in exponential form

On substitution we get

2 2 2 2 2 2 2 2 2 2

h

2 2 2 2 2 2 2 2

p p

hmb s +mb mb s x +2b U 0

=bmb s x I s +I -2 0.5+a b U 0


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Now we define non-dimensional parameters

- Mass ratio

- Ratio of uncoupled bending to torsional frequencies

r - Radius of gyration

V - Dimensionless free stream speed of air also known as reduced

velocity

On further simplification

22 2 2

22 2 2 2

Vs + s x +2 h

0=b

0Vs x s r +r -2 0.5+a

For non-trivial solution to exist, the determinant of the matrix is made zero

The roots of the determinant are complex conjugates

Limitations of p-method

The unsteady effects were completely neglected and lift and moment equations were

obtained from simple aerodynamic theory.

Matlab code

The flutter determinant is solved using a MATLAB code to find the flutter speeds.

.The real and imaginary roots of the solution of the determinant are plotted versus reduced

speed. The negative of real part of root () corresponds to modal damping and the


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imaginary part () corresponds to modal frequency. The various non-dimensional

parameters are changed and the flutter speed variation is studied.

Baseline system of parameters considered is as follows[1]

a = -0.2 e = -0.1

= 20 = 0.4

r2 = 0.24

The mass ratio, ratio of plunge frequency to pitch frequency and radius of gyration is varied

independently and the corresponding nature of variation of flutter speed is studied.

1.Effect of change of mass ratio

For a constant , x and r2(square of radius of gyration) as the mass ratio() is

varied between 20 to 100 corresponding to density variation from sea level to 10000 ft the

flutter speed also increases.

Table 1.1 Flutter speeds for different mass ratios

(mass ratio) Flutter speed

20 1.9

40 2.7

60 3.280 3.7

100 4.2

Fig 1.1 Plot of flutter speed versus mass ratios

0

1

2

3

4

5

0 20 40 60 80 100 120

flutter

speed


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Fig.1.2 Plot of imaginary part and real part of root versus Reduced speed for =20

Fig1.3Plot of imaginary and real part of roots versus Reduced speed for =40


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Fig 1.4 Plot of imaginary and real part versus the reduced speed for =60

Fig 1.5 Plot of imaginary and real roots versus Reduced speed for =80


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Fig 1.6 Plot of imaginary and real roots versus Reduced speed for =100

2.Effect of change of

For a constant mass ratio , x and r2 (square of radius of gyration) as the ratio of

plunge to pitch frequency() is varied from 0.2 to 0.8 the flutter speed decreases.

Table 2.1 Flutter speed at different

Flutter speed at=20 Flutter speed at=40 Flutter speed at=60

0.2 2.2 3.1 3.8

0.4 1.9 2.7 3.2

0.6 1.5 2.1 2.6

0.8 1.1 1.6 1.9


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Fig 2.1 Plot of reduced speed versus

Fig. 2.1 plot of imaginary and real part versus reduced speed at=0.2

0

0.5

1

1.5

2

2.53

3.5

4

0 0.2 0.4 0.6 0.8 1

Reducedspeed

mu=20

mu=40

mu=60


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Fig 2.2 Plot of imaginary and real part of roots versus reduced speed at=0.4

Fig 2.3 Plot of imaginary and real parts of roots versus Reduced speed at=0.6


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Fig. 2.4 Plot of imaginary and real part of roots versus reduced speed at=0.8

3.Effect of change of r(radius of gyration)

For a constant mass ratio , and x .as the radius of gyration is varied from 0.24 to

0.36 the flutter speed increases .Flutter speed also increases with increase in mass ratios

corresponding to constant radius of gyration.

r^2 mu=20 mu=40 mu=60

0.24 1.9 2.6 3.2

0.28 2 2.9 3.5

0.32 2.2 3 3.7

0.36 2.3 3.2 3.9


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Fig 3.1 Plot of Flutter speed versus square of radius of gyration (r2)

Results

1. As the mass ratio was varied from 20 to 100 the flutter speed increased.2. As the ratio of plunge to pitch frequency is varied 0.2 to 0.8 the flutter speed

decreased.

3. As the radius of gyration (square) varied from 0.24 to 0.36 the flutter speeddecreased.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.1 0.2 0.3 0.4

Flutterspeed

Radius of gyration(r2)

mu=20

mu=40

mu=60


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APPENDIX

Matlab code for determining flutter speed using p-method

clear all;

clc;

clf;

% mu is mass ratio

mu=20;

% R is square of radius of gyration

R=0.24;

e=-0.1;a=-0.2;

%x_theta is offset distance between reference point and center of mass

x_theta=e-a;

% ratio of pitch frequency to plunge frequency

sigma=0.4;

%semi chord

b=1;

% V is reduced speed

V_start=0;

V_inc=0.1;

V_end=3;

% Flutter determinant

syms s;

icount=0;

for V=V_start:V_inc:V_end

icount=icount+1;

f11=s^2 + sigma^2;

f12=s^2*x_theta+2*(V^2)/mu;

f21=s^2*x_theta;f22=s^2*R+R-(2*V^2/mu)*(0.5+a);

FD=[f11 f12;f21 f22];

W=det(FD);

roots=solve(W);

for jj = 1:4

im(jj,icount) = imag(roots(jj));


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re(jj,icount) = real(roots(jj));

end

vel(icount)=V;

end

subplot(2,1,1);

plot(vel,abs(im(2:3,1:end)),'k');

title('Imaginary part Vs Non-dimensional velocity(Reduced speed)');

xlabel('V(Reduced speed)');

ylabel('Imaginary part');grid

subplot(2,1,2);

plot(vel,re,'r');

title('Real part Vs Non-dimensional velocity(Reduced speed)');

xlabel('V(Reduced speed)');

ylabel('Real part');grid

flutter speed determination using p

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