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:

where

moment of inertia around the point P

torsional stiffness around the point Paerodynamic pitching moment around the point Pangle of attack (degree of freedom)

As generally assumed in a classical flutter analysis, the motion of the system is considered to be harmonic as

and the subsequent aerodynamic pitching moment is harmonic too:

where

with

=reduced frequency =

= Mach number of the free airstream

Substituting these expressions into the equation of motion we get the algebraic equation

Introducing the natural frequency of the system at zero airspeed and simplifying

, one obtain

The values of should be assumed to be known for the solution of the previous equation.

The unknown parameters, describing the flight condition, are . These fours unknowns must be determined from the single algebraic previous equation.

The unsteady aerodynamic coefficient is a complex number which may be written as:

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 . Consequently the aerodynamic moment coefficient is only a function of the reduced frequency.The governing equation can be expressed, with these assumptions, as:

which becomes:

from the second equation the reduced frequency at the flutter boundary, , can be solved

and used for the computation of . Therefore from the first equation the value of

can be determined.

Once and are known, the flutter speed results from:

Such flutter speed is determined based on the original assumptions (fixed altitude and

incompressible flow). After the determination of , using the speed of sound at the

specified altitude, the flutter Mach number may be computed from . This value is 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 mechanismA 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):

Torsional motion generates lift force . If is directed in the same direction as , 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 ModelThe 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:

Therefore the moment calculated around the elastic axis is:

Fig. 2.3 - Schematic of an airfoil for flutter analysisThe flutter equation becomes:

or in matrix notation:

with

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:

and or By substitution in the previous equations, it follows that:

Nontrivial solutions of and may be found from determinant (eigenvalue) equation:

which reduces to the characteristic equation:

with:

Such equation has the following four roots:

and therefore the solutions are of type:

,

representing motions for which:

Damping [rad/sec]

Radial frequency [rad/sec]

Mode-shape

Stability of motion is indicated by the value of :

Unstable (increasing amplitude)

Neutral (constant amplitude)

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

>0 >0 >0 <0

>0 >0 <0 or

>0 <0 or or

, , ,

, , ,

Type of motion Harmonic:2 positive freq.2 negative freq.

Aperiodic:2 diverging2 converging

Aperiodic:1 diverging1 convergingHarmonic:1 positive freq.1 negative freq.

Oscillatory:1 div. pos. freq.1 conv. pos. freq.1 div. neg. freq.1 conv. neg. freq.

Type of instability Neutral Divergence Divergence FlutterCategory I III IV 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:

As previously done, the motion is assumed to be harmonic and represented by

The corresponding aerodynamic forces are

substituting these expressions in the previous equations and eliminating the term, the following algebraic equations result:

Recalling the expressions for and

they can be substituted in the previous system of equations to get

or

an homogeneous, linear, algebraic system of equations for and .Introducing the following dimensionless parameters

the following simplification can be obtained:

Since this system of equations is linear and homogeneous, the determinant of the coefficients must be zero for a nontrivial solution. The configuration parameters

are assumed to be known. The unknown remaining quantities

, describe the motion and the flight condition.The previous determinant may be simply rearranged so that a common term explicit in is available:

where .The solution for the flight condition requires the computation of the remaining four unknowns:

The equation available for such solution is that coming from the determinant (second-degree 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 , initially set to zero.

3. Specify a set of values for the reduced frequency, , from 0.001 to 1.0

4. For each value of (and the specified value of ) calculate the complex functions

.

5. Solve the flutter determinant, which is a quadratic equation with complex coefficients, for

the unknowns that correspond to each of the selected values of . 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.

6. Interpolate to find the value of where the imaginary part of one of the roots becomes zero. Plotting the imaginary part of both roots versus shows where the curves cross the

zero axis. This value of has a corresponding real value of which provides the value of .

7. Determine

8. Repeat steps 3 – 7 with the value of 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:

The Theodorsen’s function may be approximed by

Exercise

Consider an incompressible, two-degree-of-freedom flutter problem in which

Compute the flutter speed and the flutter frequency.

Aus.

K-method

To incorporate the damping behaviour the following dissipative terms should be included:

Proceeding as before this equation becomes:

the flutter determinant becomes:

which is a quadratic equation in Z. The two unknowns of this quadratic equation are complex and they may be denoted by:

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 and 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 . 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:

where and are diagonal matrices with elements and

respectively and . The unsteady aerodynamic matrix may be expressed as

As usual for a nontrivial solution the determinant of the coefficients must be zero so that

This determinant, at a fixed speed and altitude, can be solved for p. The result will yield, typically, n solutions represented as

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 . This assumption yields the following flutter determinant

For selected values of the reduced frequency and altitude the flutter determinant can be

solved obtaining, in general, complex roots (remember that is a complex matrix) of the form:

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: