Aeroelasticity & Experimental Aerodynamics (AERO0032-1)
Lecture 2 Dynamic Aeroelasticity
T. Andrianne
2015-2016 Aeroelastic EOM From Lecture 1: Aeroelastic EOM for a pitching and plunging flat plate with quasi-steady aerodynamics
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 where ! ! c $ $ # 1 # − x f & & ! $ ! $ ! $ m S 2 # " 2 % & 0 0 h A = # &, B = πb , C = # &, q = # & # & # 2 2 & S Iα ! c $ ! c $ b " 0 0 % " α % " % # x x & # # − f & # − f & + & " " 2 % " 2 % 8 % ! !3c $ c $ # 1 x & # − f &+ ! $ # " 4 % 4 & Kh 0 ! 0 1 $ D = cπ , E = # &, F = cπ # & # 2 & # & ! c $ !3c $ c 0 Kα " 0 −ec % # ec x x & " % # − # − f & +# − f & & " " 2 % " 4 % 4 %
3 First order form
The second order equations can be written in first order form:
! 1 1 2 $ ! q!! $ −M− C + ρUD −M− E + ρU F ! q! $ # & = # ( ) ( ) &# & # q! & # &# q & " % " I 0 %" % where M=A+ρB
The first order ODEs are of the form z! = Qz
! 1 1 2 $ ! q! $ −M− C + ρUD −M− E + ρU F z = # &, Q = # ( ) ( ) & where # & # & " q % I 0 " % 4 Analytical solution
First order linear ODEs z ! = Q z have an analytical solution:
Qt z(t) = e z(0) Decomposing the matrix exponential: n λit z(t) = ∑Vie ci i=1 where c = V-1z(0) n is the number of states th Vi is the i eigenvector matrix of Q λ 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 = measures of the amount of damping present in each mode of vibration
Both natural frequencies and damping ratios are functions of airspeed and air density because the matrix Q is a function of these two quantities. " 1 1 2 % −M− C + ρUD −M− E + ρU F Q = $ ( ) ( ) ' $ ' # I 0 & 6 Variation with airspeed Increasing airspeed à Natural frequencies approach each other. à One of the damping ratios increases while the other first increases and then decreases. The critical damping ratio becomes zero and then negative à Instability This phenomenon is called flutter and the zero damping speed is the flutter speed.
7 Subcritical System response
Solution of the EOM from initial conditions α(0) = 5o for U = 30 m/s à Both pitch and plunge decay with time.
8 Critical System Response
Solution of the EOM from initial conditions α(0) = 5o for U = 35.9 m/s à Both pitch and plunge oscillation amplitudes remain constant
9 Supercritical Responses
Solution of the EOM from initial conditions α(0) = 5o for U = 38 m/s à 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
with ζ = - Re(λ)/ωn à The system is stable
• If at least one real eigenvalue is positive
à The system undergoes static divergence
• If at least one pair of complex conjugate eigenvalues has positive real part
à The system has undergone flutter. 11
Routh-Hurwitz (1)
à Obtaining the static divergence and flutter speeds directly from the characteristic polynomial of Q in z! = Qz
The criterion applies to a polynomial of the form
4 3 2 a4λ + a3λ + a2λ + a1λ + a0 = 0
Stable if λ1,2,3,4 are either real negative or complex with negative real parts12 Routh-Hurwitz (2)
à Obtaining the static divergence and flutter speeds directly from the characteristic polynomial of Q in z! = Qz
The criterion applies to a polynomial of the form 4 3 2 a4λ + a3λ + a2λ + a1λ + a0 = 0
The system is unstable if
1. Any of the coefficients ai is zero or negative while at least one is positive 2. There is at least one sign change in the first column of the matrix H
13 Routh-Hurwitz (3)
Any of the coefficients ai is zero or negative while at least one is positive
• Equivalent to a0 < 0 2 2 Kα < ρU ec π à Static divergence condition,
• At the limit :
à a0 = 0 à The characteristic polynomial
14
à Approaching divergence, one of the system eigenvalue tends to zero
Routh-Hurwitz (4)
One of the real eig. val. becomes zero à Divergence
Two pairs of cplx conjugate eigenvalues one pair of cplx conj. à 2 natural frequencies eig. become real
15 Routh-Hurwitz (5)
There is at least one sign change in the first column of the matrix H
• Equivalent to c1 < 0 with a4 = 1, a3 = 0 for U = 0 and d1 = a0
à 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
16 Routh-Hurwitz (6)
Damping jumps from +1 to -1 à Divergence
Zero damping à Flutter
17 Numerical searches
• Routh-Hurwitz can be easily applied to a 2-DOF system
• Aircraft aeroelastic models can have more than 100 DOFs à Routh-Hurwitz is totally impractical
• Numerical methods can be used instead – Directed searches, e.g. Newton-Raphson – Indirect searches, e.g. trial and error
18 Newton-Raphson (1)
= Very widely used method for solving nonlinear problems
• Suppose we need to solve the nonlinear equation f(U)=0
• Start with a first guess Ui. This is a guess so f(Ui) ~ 0
• Calculate a correction ΔU, such that f(Ui+ΔU) = 0
• Expand f(Ui+ΔU) in a Taylor series around Ui: df f (Ui + ΔU) = f (Ui ) + ΔU = 0 dU Ui 19 Newton-Raphson (2)
• Solving for ΔU we get: −1 # df & ΔU = −% ( f (Ui ) % dU ( $ Ui ' • Calculate a better approximation for the solution
of f(U)=0, which is Ui+1=Ui+ΔU
• This value is still not exact à Need to re-apply
the procedure in order to calculate Ui+2, which will be an even better approximation
• We keep iterating until |ΔU|<ε, where ε is the
required tolerance. 20 Flutter test functions
• Need for a suitable function f(U)=0 for flutter determination
• 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.
21 Flutter derivative
−1 # df & ΔU = −% ( f (Ui ) % dU ( $ Ui '
As the calculation of the eigenvalues is numerical, it is not possible to evaluate the derivative analytically.
à We can use a forward difference scheme to calculate the derivative numerically: df f U +δU − f U = ( i ) ( i )
dU Ui δU
where δU is a very small user-defined speed increment.
22 Starting guess
Aeroelastic systems without structural damping flutter at U = 0
à Starting guess Ui should not be close to 0.
Aeroelastic systems with structural damping can flutter at negative airspeeds.
à Choose an airspeed within the flight envelope but far from 0
Some aeroelastic systems may have many flutter airspeeds. à Only the lowest flutter airspeed is of interest.
23 Effect of flexural axis • Significant effect on both flutter and static divergence speeds • For this aeroelastic system the flutter speed is always lower
than the static divergence speed, unless xf / c > 0.75 • Note that placing the flexural axis in front of the aerodynamic center is bad for the flutter speed!
24 Unsteady Aerodynamics
• Remember 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 value of the flutter speed 25 Starting Vortex Remember from your Aerodynamics course (AERO0001-1)
Symmetric airfoil initially @ 0° and impulsively set to 5°
As the starting vortex leaves the airfoil its effect decreases Then the flow becomes steady and lift is constant on the airfoil 26 Effect on lift Symmetric airfoil initially @ 0° à lift coefficient is zero
Impulsively set to 5° à jumps to half its steady-state value
The unsteady lift then asymptotes towards its steady-state value
cl (t) / cl (∞)
27 Wagner Function (1) Wagner function = modeling the effect of the starting vortex on the aerodynamic forces
• At the start of the motion: cl(0)=cl(∞)/2 • The instantaneous lift then slowly increases to reach its steady value as time tends to infinity
The Wagner function is equal to 0.5 when t=0. It increases asymptotically to 1. It can be equally used to describe an impulsive change in angle of attack at constant airspeed Position at time t U
starting point t0=0 28 Wagner Function (2)
• An approximate expression for the Wagner function is
−ε 1Ut / b −ε 2Ut / b Φ(t) = 1− Ψ1e − Ψ2e where Ψ1=0.165, Ψ2=0.335, ε1=0.0455, and ε2=0.3
• The lift€ coefficient variation with time after a step change in incidence is given by
cl (t) = 2παΦ(t) Φ(t) = 0 if t<0
à Lift variation becomes L'(t) = ρπU 2cαΦ(t) = ρπUcwΦ(t)
w is the downwash€ velocity (vertical velocity component of the fluid)
w =U sinα =Uα on the airfoil, because of the flow tangency BC
29 Unsteady Motion (1)
• Unsteady motion modeled as a superposition of many small impulsive changes in angle of attack
• The increment in lift due to a small change in pitch angle at time t0 dw t dL' t = ρπUcΦ t − t dw t = ρπUcΦ t − t ( 0 ) dt ( ) ( 0 ) ( 0 ) ( 0 ) 0 dt0
à Lift variation at all times is obtained by integrating from time -∞ to time t, i.e.
t dw t L' t = ρπUc Φ t − t ( 0 )dt ( ) ∫ -∞ ( 0 ) 0 dt0 meaning = before the very beginning of the motion
30 Unsteady Motion (2)
Downwash velocity w(t) = U α(t) ?
Due to total angle of attack + effective camber
From the thin airfoil theory (Lecture 1)
= αINDUCED @ the ¾ chord
xf c/4 " 3 % ! α l m w(t) =Uαtot (t) =Uα (t) + h(t) +$ c − x f 'α! (t) xf # 4 &
3c/4 Vortices concentrated at ¼ chord BC of flow tangency at ¾ chord 31 Unsteady Motion (3)
For a motion starting at t = 0: à w = 0 for t < 0 à w = w(0) at t = 0
t dw(t0 ) General expression : L' t = ρπUc Φ t − t dt ( ) ∫ -∞ ( 0 ) 0 dt0
$ t dw t ' Becomes L' t = ρπUc w Φ t + Φ t − t ( 0 ) dt w = lim w(t) ( ) & 0 ( ) ∫ 0 ( 0 ) 0 ) 0 tà0+ % dt0 (
" % t " % ! " 3 % !! " 3 % L '(t) = ρπUc$Uα (0) + h(0) +$ c − x f 'α! (0)'Φ(t) + ρπUc Φ(t − t0 )$Uα! (t0 ) + h(t0 ) +$ c − x f 'α!!(t0 )'dt0 # # 4 & & ∫ 0 # # 4 & &
Use integration by parts to get rid of acceleration terms inside the integral " % ! " 3 % L'(t) = ρπUc$Uα (t) + h(t) +$ c − x f 'α! (t)'Φ(0) − # # 4 & & … Result is t ∂Φ t − t " " 3 % % ρπUc ( 0 ) Uα t + h! t + c − x α! t dt ∫ 0 $ ( 0 ) ( 0 ) $ f ' ( 0 )' 0 ∂t0 # # 4 & & 32 Unsteady Motion (4)
" % ! " 3 % L'(t) = ρπUc$Uα (t) + h(t) +$ c − x f 'α! (t)'Φ(0) − # # 4 & &
t ∂Φ t − t " " 3 % % ρπUc ( 0 ) Uα t + h! t + c − x α! t dt ∫ 0 $ ( 0 ) ( 0 ) $ f ' ( 0 )' 0 ∂t0 # # 4 & & = circulatory lift based on the Wagner function
It includes the effect of the entire motion history of the system in the calculation of the current lift force
33 Moment
• Aerodynamic moment around the flexural axis due to the x unsteady lift force is simply f c/4 m (t) = e c l(t) l xf α mxf
• Added mass effects must be superimposed (as was done in the quasi-steady case) • The complete equations of motion become
34 Integro-differential equations
• Integro-differential equations cannot be readily solved in the manner of Ordinary Differential Equations. • A numerical solution can be applied, based on finite differences e.g. Houbolt’s Method • But, numerical solutions are not very good for conducting stability analysis à trial and error à The equations must be transformed to ODEs in order to perform stability analysis 35 Transformation to ODEs (1)
• Use the following substitutions:
• wi variables = Aerodynamic States • Arise from the substitution of the approximate form of the
−ε 1Ut / b −ε 2Ut / b Wagner function, Φ (t ) = 1 − Ψ 1e − Ψ 2e in the equations of motion • Aerodynamic states = mathematical tools to represent history effects€ . 36
Transformation to ODEs (2)
Goal : Deal with the integral term in the lift equation
" % ! " 3 % L'(t) = ρπUc$Uα (t) + h(t) +$ c − x f 'α! (t)'Φ(0) − # # 4 & &
t ∂Φ t − t " " 3 % % ρπUc ( 0 ) Uα t + h! t + c − x α! t dt ∫ 0 $ ( 0 ) ( 0 ) $ f ' ( 0 )' 0 ∂t0 # # 4 & &
• Integrating by parts and substituting for the aerodynamic states
37 Transformation to ODEs (3)
à The integrals have been absorbed by the aerodynamic states.
The full equations of motion are
" 2 2 % m + ρπb S − ρπb (x f − c / 2) " % M $ ' h!! $ 2 '$ '+ 2 2 2 $ α!! ' $ S − ρπb (x f − c / 2) Iα + ρπb (x f − c / 2) + b / 8 '# & # ( ) & " % Φ 0 c / 4 + Φ 0 3c / 4 − x " % C $ ( ) ( )( f ) ' h! πρUc$ '$ '+ $ −ecΦ 0 3c / 4 − x c / 4 − ecΦ 0 '# α! & # ( ) ( f )( ( )) & " % (1) K Uc ! 0 Uc U 0 3c / 4 xf ! 0 $ h + πρ Φ( ) πρ ( Φ( ) +( − )Φ( )) '" h % K $ '$ '+ $ −πρUec2Φ! 0 K − πρUec2 UΦ 0 + 3c / 4 − xf Φ! 0 '# α & # ( ) α ( ( ) ( ) ( )) & " % w1 " 2 2 %$ ' −Ψ ε / b −Ψ ε / b Ψ ε 1−ε 1− 2e Ψ ε 1−ε 1− 2e 3 $ 1 1 2 2 1 1 ( 1 ( )) 2 2 ( 2 ( )) '$ w2 ' W 2πρU $ 2 2 '$ ' $ ecΨ ε / b ecΨ ε / b −ecΨ ε 1−ε 1− 2e −ecΨ ε 1−ε 1− 2e ' w3 # 1 1 2 2 1 1 ( 1 ( )) 2 2 ( 2 ( )) &$ ' $ w ' # 4 & " ! % $ πρUcΦ(t)(h(0) +(3c / 4 − xf )α (0)) + p(t) ' = $ 2 ' $ −πρUec Φ! t h 0 + 3c / 4 − xf α 0 + r t ' 38 # ( )( ( ) ( ) ( )) ( ) & Transformation to ODEs (4)
• There are 2 equations with 6 unknowns à 4 more equations are needed
• These can be obtained by noting that the definitions of w i are of the form
• Differentiating this equation with time:
39 Leibniz Integral Rule
• e.g for w1(t):
• For all wi(t):
(2)
40 Complete Equations (1)
• Equations (1) are 2nd order Ordinary Differential Equations (ODEs). They describe the dynamics of the system states. ! $ w1 # & ! $ ! $ ! $ ! $ πρUcΦ! t h 0 + 3c / 4 − xf α 0 + p t h!! h! h # w2 & # ( )( ( ) ( ) ( )) ( ) & M # &+C# &+ K # &+W = # & # & # & # 2 & " α!! % " α! % " α % w3 # −πρUec Φ! t h 0 + 3c / 4 − xf α 0 + r t & # & " ( )( ( ) ( ) ( )) ( ) % # w & " 4 % • Equations (2) are 1st order ODEs. They describe the dynamics of the aerodynamic states.
à Equations (1) and (2) = complete aeroelastic system of equations 41 Complete Equations (2)
Alternative form of the complete equations u˙ = Qu # h˙ & % ( where % α˙ ( % h ( # −M−1C −M−1K −M−1 W& % ( % ( α Q = I 0 0 , u = % ( % ( % w ( €% ( 1 $ 0 W0 ' % ( % w2 (
% w3 ( % ( $ w4 '
# 1 0 −ε1U /b 0 0 0 & % ( 1 0 0 U /b 0 0 % −ε 2 ( W0 = % 0 1 0 0 −ε1U /b 0 ( % ( $ 0 1 0 0 0 −ε 2U /b' 42
€ Solution of the ODEs
• Unsteady aeroelastic equations are in complete ODE form (6 equations with 6 unknowns)
à Solved as usual, by injecting a harmonic component λt u = u0e à 8th order characteristic polynomial
a λ8 + a λ 7 + a λ 6 + a λ 5 + a λ 4 + a λ 3 + a λ 2 + a λ + a = 0 8 7 € 6 5 4 3 2 1 0
43 Natural Frequencies and damping ratios
Uf QS Uf Unsteady ~ 35m/s ~ 50m/s • Quasi-steady flutter speed prediction is much conservative • Unsteady flutter mechanism is much more abrupt: the damping drops very quickly to zero = HARD FLUTTER 44 Effect of Flexural Axis
• The divergence speed is the same as in the quasi-steady case. • The flutter speeds obtained from Wagner’s method is always higher than that obtained from quasi-steady calculations.
45 Summary
• Solving the EoM using QS aerodynamics – Analysing the sign and type of eigenvalues – Routh-Hurwitz – Newton-Raphson • Adding UNSTEADY aerodynamics à Wagner – Reduction of the aerodynamics forces (time to establish) – Aerodynamic states = history effects
• QS vs US – QS is more conservative – US leads to hard flutter (more dangerous) 46