The Lifting System of a PrandtlPlane, Part 2: Preliminary Study on Flutter Characteristics

Abstract

In this part of the paper, a preliminary flutter analysis of the PrandtlPlane 250-seat aircraft is presented. The analysis is performed on the structural solution obtained after that the constraints of maximum static stress, structural instability under compression loads, minimum aileron efficiency, aeroelastic effects on load distribution and flutter conditions were satisfied. A new flutter analysis is performed in order to try to find the main parameters affecting the flutter characteristics of the system. The results show that flutter does not appear to be critical and, also, that this system allows one to position tanks on the tips of the front wing, with a flutter speed improvement.

Appendix

The generalized stiffness and generalized mass being introduced, generalized coordinates can be used in the problem of flutter prediction:

$$ [M]\{\ddot{q}\} + [D]\{\dot{q}\} + [K]\{q\} = Q_A,$$

(12)

where:

[M]::

Generalized mass matrix

[D]::

Structural damping matrix

[K]::

Generalized stiffness matrix

{q}::

Vector of generalized coordinates

Q _A ::

Generalized aerodynamic forces

In Eq. (12), Q _A is determined by an appropriate aerodynamic model.

In general, the flutter equation is solved by assuming a harmonic solution of the form $\{q\} = \{\hat{q}\}e^{pt}$ and finding roots of the characteristic equation for p. It can still be done to solve Eq. (12), provided that the aerodynamic forces Q _A can be explicitly expressed as algebraic functions of {q}. It can be done for simple aerodynamic models, but not when only pure harmonic results are available or when the data is available only in tabular form from experiments.

An intuitive approach to solve this would be to add a harmonic forcing term F(t) to the system and evaluate its response as a function of flight conditions (e.g. the dynamic pressure q of the flight speed U) and the frequency ω of the forcing function. This is actually the method used for flight tests, but it is quite inefficient from a numerical standpoint as it requires harmonic time domain simulation for each point in the U–ω plane.

A number of more efficient methods have been developed and will be described in the following. These methods are less time-consuming than the forcing approach and generate directly approximate flutter diagrams. The two presented methods, the k method [9] and p–k method [5], are both assuming that aerodynamic data is pure harmonic. As we know that at the flutter points, the response is indeed pure harmonic, the solution at these points will be exact.

For the sake of ease and generality, we will manipulate non-dimensional equations. Thus, we will employ a non-dimensional expression for p,

$$ p = \frac{b}{U}(\sigma + i\omega) = \delta + ik,$$

(13)

where b is the half-chord, and k is called the reduced frequency and is a parameter of primary importance in the determination of the aerodynamic forces; it is computed as

$$ k = \frac{\omega b}{U}.$$

(14)

The solutions will be assumed to be of the form $\{q\} =\{\hat{q}\}e^{p\frac{Ut}{b}}$. The equations of motion (12) are then written as

$$ \biggl[\frac{U^2}{b^2}[M]p^2 + \frac{U}{b}[D]p+[K]\biggr]\{\hat{q}\} = \frac{1}{2}\rho U^2 \bigl[A(p)\bigr]\{\hat{q}\}.$$

(15)

Here, A(p) is the aerodynamic matrix and is a function of the Mach number and Reynolds number. It is important to note here that if the aerodynamic forces can be expressed as a sufficient simple function of p, Eq. (15) defines a polynomial in p with real coefficients. In this case, this expression is equivalent to the equations of motion developped in the two-degree-of-freedom model.

The p–k method is based on the assumption that it is possible to approximate the aerodynamics of sinusoidal motions with slowly increasing or decreasing amplitudes using purely harmonic aerodynamic data, i.e. that

$$\bigl[A(p)\bigr] = \bigl[A(\delta + ik, M, \mathit{Re}, \ldots)\bigr]\approx \bigl[A(ik, M, \mathit{Re}, \dots)\bigr]$$

for sufficiently small δ. With this assumed form of aerodynamic terms, the equation of motion is written as

$$ \biggl[\frac{U^2}{b^2}[M]p^2 + \frac{U}{b}[D]p + [K]\biggr]\{\hat{q}\} = \frac{1}{2}\rho U^2\bigl[A(ik)\bigr]\{\hat{q}\},$$

(16)

which we can write

$$ \bigl[F(p,k)\bigr]\{\hat{q}\} = 0.$$

(17)

Once [A(ik)] has been computed from a first estimation of k, one can solve Eq. (16) for p ₁=δ ₁+ik ₁, compute [A(ik ₁)], and again solve equation (16) for p ₂=δ ₂+ik ₂, etc., until the imaginary part k _i of the solution p _i equals the k _i−1 value of the aerodynamics. Hence, this method is called method p–k. An iterative solution method is normally used, the most common one being the “determinant iteration” [5]. For this method to work properly, the initial guesses for p must be reasonably good and are thus determined from the previously analysed flight condition. The assumption on the form of the aerodynamic data proved to be accurate in Ref. [5], as the p–k method is compared with an exact method for damping (the p method), giving a good agreement between the two.

To obtain flutter diagrams, the basic procedure is as follows. First, the mode that will be tracked is chosen as well as a starting value for U, near U=0. The Mach and Reynolds numbers are computed from this U. A first guess, p ₁, can be made using the natural frequency of the chosen mode. Another guess $p_{1}'$ is made by adding a small damping to p ₁. Then the following steps are performed:

• 1. Compute or interpolate for [A(k ₁,M,…)] and $[A(k_{1}', M, \ldots)]$.

• 2. Compute the determinants F ₁=|[F(p ₁,k ₁)]| and $F_{1}' = |[F(p_{1}',k_{1}')]|$.

• 3. Update p using $p_{3} = \frac{p_{2} F_{1} - p_{1} F_{2}}{F_{1} - F_{2}}$.

• 4. Set $p_{1}' = p_{2}$ and $p_{1} = p_{1}'$, and repeat from step 1 until converged.

• 5. Choose the next flight condition (ρ,U,M,Re,…).

• 6. Choose two new values for p based on extrapolation from previous flight conditions.

• 7. Return to step 1 and repeat to find the new p at the next flight condition.

This procedure is repeated for each desired mode. Once a converged solution p _c =δ _c +ik _c has been found, the frequency and damping are computed as

$$ f = \frac{Uk_c}{2 \pi b}\quad\mbox{and}\quad \gamma = \frac{\delta_c}{k_c} = \frac{g}{2}$$

(18)

and used to draw the frequency-damping-velocity diagrams, or flutter diagrams.

The p–k method has two distinct advantages. First, the plots obtained are interpreted much easier. Indeed, the non-physical behaviours that can arise with the k method are not present. Then, no further iteration is required to incorporate Mach or Reynolds number effects, as the density computed already contains this information.