Aeroelasticity Lecture 2: Dynamic Aeroelasticity G. Dimitriadis 1 Aeroelastic EOM • In the previous lecture we developed the aeroelastic equations of motion for a pitching and plunging flat plate: 2 2nd Order ODEs • The equations are 2nd order linear ODEs of the form (A + ρB)q!! +(C + ρUD)q! +(E + ρU 2F)q = 0 ! ! $ $ # c & • where 1 # − x f & ! $ # " % & ! $ ! $ m S 2 2 0 0 h A = # &, B = πb , C = # &, q = # & # & # 2 2 & S Iα ! $ ! $ " 0 0 % " α % " % # c − c − + b & # # x f & # x f & & " " 2 % " 2 % 8 % ! $ !3c $ c # 1 # − x &+ & " f % ! $ ! $ # 4 4 & Kh 0 0 1 D = cπ , E = # &, F = cπ # & # 2 & # & ! $ ! $ 0 Kα " 0 −ec % # − c − + 3c − c & " % # ec # x f & # x f & & " " 2 % " 4 % 4 % 3 First order form • The second order equations can be easily written in first order form: ! $ ! −1 −1 2 $! $ q!! −M C + ρUD −M E + ρU F q! # & = # ( ) ( ) &# & # ! & # &# & " q % " I 0 %" q % • where M=A+ρB • The first order ODEs are of the form z! = Qz • where ! $ ! −1 −1 2 $ q! −M (C + ρUD) −M E + ρU F z = # &, Q = # ( ) & # & # & " q % " I 0 % 4 Analytical solution • Recall from last year’s Flight Mechanics course that first order linear ODEs have an analytical solution: z(t) = eQtz(0) • or, after decomposing the matrix exponential: n λit z(t) = ∑vie ci i=1 • where c=V-1z(0), n is the number of states, V is the eigenvector matrix of Q and l are the eigenvalues of Q. 5 Frequency and Damping • The absolute values of the eigenvalues are the natural frequencies, ωn=|λ| • The damping ratios are defined as: ζ=-Re(λ)/ωn • The damping ratios are measures of the amount of damping present in each mode of vibration • It must be kept in mind that both natural frequencies and damping ratios are functions of airspeed and air density because the matrix Q is a function of these two quantities. 6 Variation with airspeed As the airspeed increases, the two natural frequencies approach each other. One of the damping ratios increases while the other first increases and then decreases. The critical damping ratio becomes zero and then negative. Instability ensues. This phenomenon is called flutter and the zero damping speed is the flutter speed. 7 Subcritical System response Solve the equations of motion for the time responses of the system from initial conditions (α(0)=5o). Time responses for U=30m/s. Both pitch and plunge decay with time. 8 Critical System Response Solve the equations of motion for the time responses of the system from initial conditions (α(0)=5o). Time responses for U=35.9m/s. Both pitch and plunge oscillation amplitudes remain constant. 9 Supercritical Responses Solve the equations of motion for the time responses of the system from initial conditions (α(0)=5o). Time responses for U=38m/s. Both pitch and plunge oscillation amplitudes increase with time. 10 Stability criteria • The stability of the system can be estimated directly from the eigenvalues of the system matrix: – If all eigenvalues have negative real parts, the system is stable – If at least one real eigenvalue is positive, the system has undergone static divergence – If at least one pair of complex conjugate eigenvalues has positive real part, the system has undergone flutter. 11 Determining the flutter speed • The flutter speed can be determined by trial and error: – Choose an air density (i.e. flight altitude) – Calculate the system eigenvalues for a starting airspeed – Keep increasing the airspeed until at least one pair of complex eigenvalues has positive real part – Continue to try different airspeeds until the real part is almost zero. 12 Routh-Hurwitz (1) • The static divergence and flutter speeds can also be obtained directly from the characteristic polynomial • This can be achieved using the Routh- Hurwitz stability criterion. • The criterion applies to a polynomial of the form 4 3 2 a4λ + a3λ + a2λ + a1λ + a0 = 0 13 Routh-Hurwitz (2) • The system is unstable if – any of the coefficients ai is zero or negative while at least one is positive – There is at least one sign change in the first column of the matrix H • The matrix H is given by 14 Routh-Hurwitz (3) • The condition a0<0 gives the static divergence 2 2 condition, Kα<ρU ec π • The condition c1<0 yields • Which, when expanded, yields a 4th order polynomial in U. • Two of the solutions are U=+0 and U=-0 • The other two solutions are U=+UF and U=- U=-UF 15 Numerical searches • Routh-Hurwitz can be easily applied to a 2- DOF system. • Aircraft aeroelastic models can have more than 100 DOFs. Routh-Hurwitz is totally impractical for such large systems. • Numerical methods can be used instead. • These are generally divided into two categories – Directed searches, e.g. Newton-Raphson – Indirect searches, e.g. trial and error 16 Newton-Raphson • Newton-Raphson is a very widely used method for solving nonlinear problems. • Suppose we need to solve the nonlinear equation f(U)=0. • We start with a first guess Ui. This is a guess so f(Ui)=0. However, we want to calculate a correction DU, such that f(Ui+DU)=0. • We expand f(Ui+DU) in a Taylor series around Ui: df f (Ui + ΔU) = f (Ui ) + ΔU = 0 dU Ui 17 Newton-Raphson • Solving for DU we get: −1 # df & ΔU = −% ( f (Ui ) $ dU Ui ' • Now we can calculate a better approximation for the solution of f(U)=0, which is Ui+1=Ui+DU. • This value is still not exact. We need to re-apply the procedure in order to calculate Ui+2, which will be an even better approximation. • We keep iterating until |DU|<e, where e is the required tolerance. 18 Flutter test functions • For flutter determination we need to define a suitable function f(U)=0. • Several different test functions work well. The simplest is: n f (U) = ∏ℜ(λ j (U)) j=1 • Where n is the number of states. • This test function is equal to 0 when the real part of any of the eigenvalues is equal to 0. • If we want to detect only flutter and not static divergence, then we can choose to include only the complex eigenvalues in the product. 19 Flutter derivative • As the calculation of the eigenvalues is numerical, it is not possible to evaluate the derivative analytically. • We can use a forward difference scheme to calculate the derivative numerically: f U +δU − f U df = ( i ) ( i ) δ dU Ui U • Where dU is a very small user-defined speed increment. 20 Starting guess • The starting guess for the flutter speed should not be close to 0. Aeroelastic systems without structural damping flutter at U=0. • Aeroelastic systems with structural damping can flutter at negative airspeeds. • Choose an airspeed within the flight envelope but far from 0. • Some aeroelastic systems may have many flutter airspeeds. Only the lowest flutter airspeed is of interest. 21 Effect of flexural axis The position of the flexural axis has a significant effect on both flutter and static divergence. For this aeroelastic system the flutter speed is always lower than the static divergence speed, unless xf/c>0.75. Also note that placing the flexural axis in front of the aerodynamic center is bad for the flutter speed! 22 Unsteady Aerodynamics • As mentioned in the first lecture, quasi-steady aerodynamics ignores the effect of the wake on the flow around the airfoil. • The effect of the wake can be quite significant, it effectively reduces the magnitude of the aerodynamic forces acting on the airfoil. • This reduction can have a significant effect on the values of the flutter. 23 Kelvin’s theorem • One of the bases of unsteady aerodynamics is Kelvin’s theorem. • It states that the total circulation in a flow cannot change in time; this includes circulation on the wing, , and in the wake, . • If the circulation over a wing increases at a particular Γtime" instance (for exampleΓ# because the angle of attack increases) then equal and opposite circulation must be shed into the wake. • In equation form: ' Γ" + Γ# = 0 • so that any change in'( bound circulation must be accompanied by a change in wake circulation . ΔΓ" # ΔΓ = 24 − ΔΓ" Starting Vortex (1) • The simplest unsteady flow is a flat plate at 0o angle of attack in a steady flow of airspeed . • At a particular instance in time, , the angle of attack is increased impulsively to, say, 5o. ! • This impulsive change causes the"# shedding of a strong vortex in the wake, known as the starting vortex. • The starting vortex induces a significant amount of local velocity around the airfoil. However, it travels downstream because of the steady flow . • As the starting vortex distances itself from the wing,! its effect decreases • After a while it has no effect at all and the flow becomes steady 25 Starting Vortex (2) Wake shape of an airfoil whose angle of attack was impulsively increased to 5o. The starting vortex is clearly seen 26 Wagner’s model • The first solution of the impulsively started flat plate problem was obtained by Wagner in the 1920s. • He modelled the wake as a flat horizontal line containing a continuous vorticity distribution. • The flat plate is also horizontal while the free stream is inclined to the angle by a small angle . ! " / Δ% Δ% " 0 % ΔΓ+ ΔΓ- ΔΓ, ΔΓ. !• The vortices are ejected at the th time instance and then travel at the free 0stream airspeed,0 such that . • Their strength does not change# as they travel downstream. • The vorticity distribution becomes continuousΔ% = !Δ' as . Δ' → 0 27 Vortex summary • Recall from the Aerodynamics course that a point vortex induces a potential around it.