One degree of Freedom Flutter
To discuss some simple flutter analysis calculations we will present the cases of one and
two degrees of freedom.
Both examples are referred to a rigid two-dimensional airfoil which in the first case (SDOF)
is permitted only to rotate in pitch about a specific reference point, and the plunge degree
of freedom is equal to zero as shown in figure:
Fig. 2.1 – Geometry of the wing section showing the pitch degree of freedom
The system equations of motion reduce to one equation which can be written as:
Ip   K   M
where
Ip 
moment of inertia around the point P
K  torsional stiffness around the point P
M  aerodynamic pitching moment around the point P
  angle of attack (degree of freedom)
As generally assumed in a classical flutter analysis, the motion of the system is considered
to be harmonic as
  0 eit
and the subsequent aerodynamic pitching moment is harmonic too:
M  M0 eit
where


M 0    b4 2 m k, M   0
with
k
M
b
V
= Mach number of the free airstream
= reduced frequency =
Substituting these expressions into the equation of motion we get the algebraic equation


K  0   2 I p  0    b4  2 m k, M   0
Introducing the natural frequency of the system at zero airspeed  
K
and simplifying
Ip
 0 , one obtain
  2 
 1
  m k, M   0
 b4   2 
Ip


The values of I p ,  and b should be assumed to be known for the solution of the previous
equation.
The unknown parameters, describing the flight condition, are  ,  ,k, M . These fours
unknowns must be determined from the single algebraic previous equation.
The unsteady aerodynamic coefficient m k, M  is a complex number which may be


written as:





m k, M   Re  m   i Im  m   M R k, M   i M I k, M 

If for the solution of the previous equation we assume a fixed altitude, the value of   is
known. Additionally, we may start with the further assumption of incompressible flow,
therefore M   0 . Consequently the aerodynamic moment coefficient is only a function of
the reduced frequency.
The governing equation can be expressed, with these assumptions, as:
 2 
 1
  M R k,0  i M I k,0  0
 b4   2 
Ip
 
 
which becomes:
 2
 b4
   1   M R k,0
2
Ip
 

M I k,0  0

 
 
from the second equation the reduced frequency at the flutter boundary, k F , can be solved


and used for the computation of M R k F ,0 . Therefore from the first equation the value of
 F can be determined.
Once K F and  F are known, the flutter speed results from:
VF 
b F
KF
Such flutter speed is determined based on the original assumptions (fixed altitude and
incompressible flow). After the determination of VF , using the speed of sound c at the
V
specified altitude, the flutter Mach number may be computed from M F  F . This value is
c
realistically greater than zero. At this point the process can be iterated with the new value
of the Mach number to find convergence between the assumption of the Mach number and
the resulting flutter Mach number.
Two Degree of Freedom Flutter
Ohysical explanation of Flutter mechanism
A simple physical explanation of the mechanism of flutter is given in Fig. 2.2.
Fig. 2.2 – Qualitative calculation of the work per cycle for a wing section with pitch and
plunge degree of freedom
For such model only lift can perform net work over one cycle (mass and stiffness forces
are conservative and cannot perform net work):
T


T
W   Re L d Re h   Lh&dt
0
0
Torsional motion  generates lift force L  . If L   is directed in the same direction as
h , then positive work is done, and flutter instability occur.
The flow acts as energy reservoir. When aerodynamic forces perform positive work,
energy flows from airflow into wing system. If work is negative, energy is extracted from
wing system.
Steady Aerodynamic Model
The use of steady aerodynamic theory strongly simplifies the presentation of such
aeroelastic instability, but does not detract from essential matters and therefore it becomes
a pleasant introduction to the flutter problem.
Steady aerodynamic behaviour implies that the flow follows the angle of attack
instantaneously, which means non phase difference between the angle of attack variation
and the associated aerodynamic forces. Further simplifications adopted in the following
presentation are:
 Ideal and incompressible flow
 Airfoil thickness neglected (flat plate approximation)
Let’s consider a typical airfoil as shown in fig. 2.3. with bending and torsional motion only
(binary system). Only torsional motion induces aerodynamic forces:
L t  qSC L  t


M t  0

ac
Therefore the moment calculated around the elastic axis is:
M EA  2Leb  M AC  2qSebCL  t


Fig. 2.3 - Schematic of an airfoil for flutter analysis
The flutter equation becomes:
mh&& S&& K h h  qSCL   0

S h&& I&& K  2qSebCL   0

or in matrix notation:
 
  
 M  x&&   K   q  A0  x  0
with
 m S 


 M   
 S I 


 K
0 

 K    h
 0 K 


 0 SC
L
 A0   
 0 2SebC
L

Mass or inertial matrix
Structural stiffness matrix




Aerodynamic stiffness matrix


Degrees of freedom (binary system)



It is important to note the asymmetry of the aerodynamic stiffness matrix, essential for all
flutter characteristics.
The solution of the flutter problem, in order to get a discussion on the flutter characteristics
is performed, following the usual procedure, by putting:
φ pt and   φe pt or x  xφ e pt
h  he
x  h
 
By substitution in the previous equations, it follows that:
mp2  Kh hφ S p2  qSCL φ  0
 

S p hφ I
2





p2  K  2qSebCL φ  0

Nontrivial solutions of hφ and φ may be found from determinant (eigenvalue) equation:


 M  p 2   K   q  A0   0
which reduces to the characteristic equation:
a4 p4  a2 p2  a0  0
with:
a4  mI  S2

a  K K

 2qSebC 


a2  m K  2qSebCL  I Kh  qSS CL  mK  I Kh  2meb  S qSCL
0
h




L
Such equation has the following four roots:

p1K 4    i

1K 4


1
a2  a22  4a4 a0
2a4
and therefore the solutions are of type:
x  xφ e
representing motions for which:
j

  xφ e t ei t , j  1K 4
j
 j  i j t
j
j
j

j
Damping [rad/sec]
j
Radial frequency [rad/sec]
xφ 
j
Mode-shape
Stability of motion is indicated by the value of  :
 0
Unstable (increasing amplitude)
 0
Neutral (constant amplitude)
 0
Stable (decreasing amplitude)
Some possible flutter solutions are given in the next table, showing:



Condition of coefficients
Type of solution and physical meaning
Classification into stability categories
a22  4a4 a0
>0
>0
>0
<0
a0
>0
>0
<0
 0 or  0
a2
>0
<0
 0 or  0
 0 or  0
p2
p
12 ,  22
12 ,  22
 2 ,  2
 , i
g  ih
Type of motion
Type of instability
Category
i 1 , i 2
 1 ,   2
Harmonic:
Aperiodic:
2 positive freq. 2 diverging
2
negative 2 converging
freq.
Neutral
I
Divergence
III
Aperiodic:
1 diverging
1 converging
Harmonic:
1 positive freq.
1 negative freq.
Divergence
IV
 i
Oscillatory:
1 div. pos. freq.
1 conv. pos. freq.
1 div. neg. freq.
1 conv. neg. freq.
Flutter
II
Types of instabilities of a typical airfoil (steady aerodynamic model)
The analysis of a two degree of freedom system yields to a more realistic situation and
includes additional parameters which are essential for a flutter boundary discussion.
Unsteady Aerodynamic Model
The equations of motions are, in this case, written as:
 m
 mbx


mbx   h   K h
 
I p     0
0  h  L 
  
K     M 
As previously done, the motion is assumed to be harmonic and represented by
h  h0 eit ;  0 eit
The corresponding aerodynamic forces are
L  L0 eit ; M  M0 eit
substituting these expressions in the previous equations and eliminating the eit term, the
following algebraic equations result:
 2 m
  

mbx

mbx  mh2

I p   0
0    h0   L0 
    
I p2   0  M 0 
Recalling the expressions for L0 and M 0


h
L0   b3 2  lh k, M  0  l k, M   
b




h
M 0   b4 2  mh k, M  0  m k, M   
b










they can be substituted in the previous system of equations to get
 2 m
  

mbx

mbx  mh2
0    h0 
 b2lh b3l   h0 
2

       3
 
4
I p   0
I p2   0  
b mh b m  0 
or
 mh2

 0

m
0 
2




I p2 
mbx
mbx 
blm    h0 
 l
b22  h

     ...
2
Ip 
bmh b m   0 
an homogeneous, linear, algebraic system of equations for h0 and  0 .
Introducing the following dimensionless parameters

m
 mass ratio
b 2
 
Ip
mb 2
 radius of gyration about P
the following simplification can be obtained:
    2 
  1   h    lh
     


 x  mh




 h0 / b  0
 

2
 0  0







 2 1       m 
    

 x  l
Since this system of equations is linear and homogeneous, the determinant of the
coefficients must be zero for a nontrivial solution. The configuration parameters
m, e, a, I p, h,  and b are assumed to be known. The unknown remaining quantities
h0 ,  0 ,  ,  , M  and k , describe the motion and the flight condition.
The previous determinant may be simply rearranged so that a common term explicit in 
is available:

2
   
   lh
   
 1   2 

 x  mh
 xx  l
   2 
 1      m
    
0
2
where   h /  .
The solution for the flight condition requires the computation of the remaining four
unknowns:

b
m
, 
,
M
and
k



v
 b2
The equation available for such solution is that coming from the determinant (seconddegree polynomial equation). Since the aerodynamic coefficients are complex quantities,
the determinant equation represents two real equations, for which both the real and the
imaginary parts must be identically zero for a solution to be obtained.
Again this translates in the fact that two of the four unknowns must be specified.
A possible procedure for the computation of the flutter determinant is:
1. Fix an altitude which defines the value of 
2. Specify M  , initially set to zero.
3. Specify a set of values for the reduced frequency, k , from 0.001 to 1.0
4. For each value of k (and the specified value of M  ) calculate the complex functions
lh , l , mh and m .
5. Solve the flutter determinant, which is a quadratic equation with complex coefficients, for
2
the unknowns  /   that correspond to each of the selected values of k . These roots
will be, generally, complex numbers with the real part an approximation of  /   and
the imaginary part related to the damping of the mode.
2
6. Interpolate to find the value of K where the imaginary part of one of the roots becomes
zero. Plotting the imaginary part of both roots versus K shows where the curves cross the
zero axis. This value of K has a corresponding real value of  /   which provides the
value of  .
2
7. Determine V   b / k
M   V / c
and
8. Repeat steps 3 – 7 with the value of M  obtained in step 7 until convergence is reached
at a given value of  .
9. Repeat the whole process for various values of  (variation with the altitude for a given
aircraft).
The aerodynamic coefficient used herein, based on the Theodorsen’s theory are:
lh  1  i
l  a 

2C k
k
2C k
  i  1  2  1  a C k 
k2

k
 2



k 
1
C k
mh  a  i 2   a 
2
 k
1
m   a 2 
8
1

2   a C k
2


k2



1
i  1
2
  a   2   a  C k 
k  2

4



The Theodorsen’s function may be approximed by
k2
2
C k 
0.01365  0.3455 i k  k 2

0.01365  0.2808 i k 
Exercise
Consider an incompressible, two-degree-of-freedom flutter problem in which
1
1
6
2
a   ; e   ;   20;  2 
and  
5
10
25
5
Compute the flutter speed and the flutter frequency.
Aus. VF  2.170 b and F  0.6443 
K-method
 m
mbx
 
mbx  h  Kh 0  h  L  Dh 
 
      
I p    0 K    M  D 
To incorporate the damping behaviour the following dissipative terms should be included:
Dh  Dh eit  ig h mh2 h eit
D  D eit  ig I p2  eit
Proceeding as before this equation becomes:
     2
 h

h

1

1

ig

l
   x  l   0





h
h
 

    
 b
2


h  2    
  x  mh    1    1  ig   m   0
b 
   



 
  h ; Z     1  ig 

 
2
the flutter determinant becomes:
 1   2 Z   lh
 x  l
 x  mh
 2 1  Z   m
0
which is a quadratic equation in Z. The two unknowns of this quadratic equation are
complex and they may be denoted by:
2
Z1,2
 
    1  ig1,2 
 
 1,2 
The computational strategy for solving the flutter determinant proceeds in a manner similar
to the one previously outlined. The main difference is that the numerical results consists of
two pairs of real numbers 1, g1  and 2 , g2  which can be plotted versus airspeed.
The aerodynamic unsteady forces are computed assuming harmonic motion, and we have,
with this method, harmonic motion only when g  0 . The numerical values of g obtained
during the calculations of the flutter margins can be interpreted as the necessary damping
for assuring harmonic motion at that specific frequency.
The plots of the frequency versus airspeed, in conjunction with the damping plots can, in
many cases, help in understanding the physical mechanism which drives the flutter
instability. At zero airspeed the frequencies are those coming from the real eigenvalue
extraction (or from a ground measured vibration test). As the airspeed increases the
aerodynamic effect on the structure behaviour becomes evident and can indicate which
modes is able to transfer energy from another, identifying those which may be responsible
of the onset of the instability.
P-K method
In its original version the K method, providing an engineering tool for the flutter calculation,
remains a mathematically improper formulation.
To illustrate the p method let’s write the equation of motion in this following matrix form:
 2

b2
2
 p  M   2  M       A p     0
V


   
M 
and  2  are diagonal matrices with elements M 1 , M 2 , ..., M n and
12 , 22 , ..., n2 respectively and p    i . The unsteady aerodynamic matrix may be
expressed as
where
A p  p2 a  pb c
As usual for a nontrivial solution the determinant of the coefficients must be zero so that
p2  M  
b2
M  2     A p  0
V2
This determinant, at a fixed speed and altitude, can be solved for p. The result will yield,
typically, n solutions represented as
p   k  ik
where K is the reduced frequency and  the decay rate at that specific speed for each
mode-shape.
The K method can be easily achieved with this procedure by considering p  ik . This
assumption yields the following flutter determinant
k 2  M  

b2
 M   2     A ik   0
V 2   
For selected values of the reduced frequency and altitude the flutter determinant can be
solved obtaining, in general, complex roots (remember that A  iK  is a complex matrix) of
the form:
  idi 
b2
1 ig 
V2
where g is the structured damping required to sustain simple harmonic motion.
Severed authors have used both the p and the K method for the flutter calculation
obtaining results which are in some extent not correlated. The differences are, usually, not
found for the value of the flutter speed, but for the analysis of the instability (modal
coupling, unstable mode). As result of such investigations the p-K method …
It is a compromise for both the p and the K methods.
It is based on conducting a p method type of analysis with the restriction that the unsteady
aerodynamics matrix is for simple harmonic motion. The flutter determinant in this case
becomes:
 
b2
p  M   2  M   2     A ik   0
V
2