Aeroelastic Problem: Formulations of the Aeroelastic Stability Problem Flashcards
What is the Equivalent Air Speed (EAS)?
The EAS is the speed that will give the same dynamic pressure at zero altitude, so that UEAS = sqrt(ρh/ρ0)*Uinf.
How can the density/altitude of flutter be found?
By assigning Minf, we find the qinf or UEAS for flutter and then we detecxt at which ρh or altitude the flutter is happening.
Discuss the flight envelopes.
The flight envelopes are expressed in two planes:
1. Altitude (h) - EAS
*The lines at constant Mach are hyperboles.
*The flight envelope is limited by hmax, the altitude.
*There is a maximum dynamic pressure.
*There is a maximum Mach number.
2. EAS - Mach
*There is a maximum dynamic pressure.
*There is a maximum Mach number.
In both flight envelopes the regulations require the structure to be fluter free at +15% of the maximum values.
How is the assembly of the aeroelastic system done?
Consider a structural model where a set of generalized coordinates q together with the associated shape functions are used to describe the displacements of any structural node η. Then ηs = Nq.
The structural model is written as:
Mq_dot_dot + Cq_dot + Kq = qFa, where qFa represents the unsteady aerodynamic forces.
Using the PVW for the external forces, it is equal to δWa = qS(δηa^TnαCp(xα)dsa), where As is the aerodynamic surface where the aerodynamic grid xα is defined. The vector ηa is the vector of the displacements of the aerodynamic nodes. Afterwards, the defined interface scheme is used that exresses the aerodynamic nodes in terms of structural nodes: ηa = Hηs. Ultimately, the PVW for the external forces is equal to qδq^TDCp, where D is [nqxna]. Then, using Morinos panel method, Cp is expressed in terms of structural nodes, and the final expression for the external forces is Fa(k) = Ham(k,M)q, where Ham is the frequency response matrix of the aerodynamic forces and depends on the M number and the reduced frequency.
Ending, by moving the external forces to the time domain, the original equation becomes:
Mq_dot_dot + Cq_dot + Kq = q S ham(t-τ)q(τ)dτ
And by moving to Laplace domain:
(s^2M + sC + K -qHam(p,M))q = 0, where p is the nondimensional Laplace variable p=sb/U.
In general, we do not have the full transfer function Ham(p,M) but we only have available the frequency response, i.e. Ham(k,M). Ham is only known in the imaginary axis as Ham(k,M).
How can the eigenvalues of the aeroelastic equation be calculated with the quasi-steady approximation of the frequency response?
It is possible to express the transfer function using the 2nd order Mclaurin series around s=0. Then H is approximated as H(s) = H(s=0) + s*dH/ds(s=0) + s^2/2 * d^2H/ds^2(s=0) and is transformed to the time domain using the inverse Laplace transformation. Then we approximate the aerodynamic stiffness, damping and mass matrices as Ka = H(0), Ca = dH(0)/dp, Ma = 1/2d^2H(0)/dp^2. This is a good approximation whenever we have an eigenvalue that is close to s=0, so a small real and imaginary part. Usually we do not use an approximation higher than 2nd order, because the structural nodes are usually up to 2nd order.
How can the eigenvalues of the aeroelastic equation be calculated with the direction solution?
Most of the times the structural frequencies and therefore the damping is not small, so the McLaurin series may not be accurate. If we take the hypothesis that we are at flutter, where the eigenvalues pass from the negative to positive real party, then:
s = 0 + jωf
p = 0 + jkf
qf = 1/2ρhUf^2
And the equation becomes:
det(-ωf^2M + jωfC + K - qFHam(Kf,Mf)) = 0.
This is a nonlinear imaginary equation that can be solved if we impose Mf and look for ωf and qf. kf = f(ωf). When qf is known, the corresponding density and so the altitude can be computed.
The con with this method is that not all the eigenvalues and damping is calculated, which is useful for verification/regulations.
How can the eigenvalues of the aeroelastic equation be calculated with the flutter p-k method?
In order to obtain the V-f and V-g diagrams, it is necessary to evaluate the eigenvalues for different speeds. Indeed, the eigenvalues are not obtainable through the quasisteady approximation, being far from the origin, while the direct solution calculates only the flutter frequency.
The p-k method uses a Taylor approximation for point k and linearizes with respect to jk. The goal is to use the knowledge of the transfer function on point B (the one known on the imaginary axis) and approximate the value at another point A. The approximation is exact when A belongs to the imaginary axis, i.e. at flutter speed. Usually, an approximation of order 0 is used, but higher order approximations are possible.
The algorithm works as followed:
For an initial given set of parameters Uinf, ρ, h, Μ
Compute a preliminary guess for one of the eigenvalue λi0
Compute ωi = Im(λi0) and ki = ωib/Uinf.
*Solve the equation: det(λ^2M + λC -qλb/UinfIm(Ham(k))/k + K - qRe(Ham(k)) = 0.
*Identify the closest eigenvalue λi to λi0.
*If |λi-λi0| < ε stop, otherwise λi0 = λi and iterate
The algorithm is repeated for every eigenvalue of interest. As a starting guess, it is possible to take the structural frequencies at zero speed.
What are the flutter V-f and V-g diagrams.
In practice, it is more convenient to compute all the desired eigenvalues for a range of EAS (or dynamic pressures) starting from zero and evaluating their real part. Flutter is detected as the lowest airstream speed at which the real part of at least one eigenvalue crosses the zero axis from negative to positive.
The knowledge of all eigenvalues from zero to the desired airstream velocity not only allows the analyst to check for flutter clearance, but also to determine how the eigenvalues evolve with the EAS velocity allowing an experimental verification of the aircraft aeroelastic stability.
How can the flutter speed be calculated using the continuation approach?
The eigenvalue problem could be seen as a nonlinear algebraic problem:
f(s,q) = (s^2M + sC + K - qHam(p,M)) = 0. There are n+1 complex uknowns or 2(n+1) real uknowns. However, the eigenvector are defined but for a constant, so the uknowns are 2n+1 and there are 2n equations. We add the normalization equation: q*q = 1.
This nonlinear problem can be solved using Newton’s methods, q=qi + Δq and s = si + Δs. The problem iterates until Δq, Δs is lower than a threshold in order to find the solutions si, qi. The algorithm could be applied for any eigenvalue/eigenvector of interest starting from a set of initial guesses and at each speed of interest.
To obtain an initial guess at a new speed, it is possible to use the value at the previous speed or take a better guess by trying to estimate the change of the eigenvalue/eigenvector with speed.
