Introduction – Equations of motion G. Dimitriadis 05

Aeroelasticity Lecture 5: Wagner Function Aerodynamics G Dimitriadis Introduction to Aeroelasticity Wagner Function Aerodynamics •! Theodorsen-based aerodynamics, i.e frequency domain-based aerodynamics is extremely powerful and very practical •! However, transforming the equations of motion to the time domain is not very easy There are several methods but none are exact and they tend to introduce errors •! Solving aeroelastic systems in the time domain can be very useful when looking at nonlinear aeroelasticity •! In this lecture, we will look at a different aerodynamic formulation, which allows the direct development of time domain equations of motion •! This formulation is based on the work by Wagner Introduction to Aeroelasticity Starting Vortex (1) •! The simplest unsteady flow is a flat plate at 0o angle of attack in a steady flow of airspeed U •! At a particular instance in time, t0, the angle of attack is increased impulsively to, say, 5o •! This impulsive change causes the shedding of a strong vortex, 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 U •! As the starting vortex is distances itself from the wing, its effect decreases •! After a while it has no effect at all and the flow becomes steady Introduction to Aeroelasticity Starting Vortex (2) Wake shape of an airfoil whose angle of attack was impulsively increased to o The starting vortex is clearly seen Introduction to Aeroelasticity Effect on lift Initially the angle of attack is zero As the airfoil is symmetrical, its lift coefficient is also zero c l ( t ) /c l (") When the change in angle of attack occurs, the lift jumps to half its steady-state value for the new angle of attack The unsteady lift then asymptotes towards its steady-state value Introduction to Aeroelasticity Wagner Function (1) •! The effect of the starting vortex on the aerodynamic forces around the airfoil can be modeled by the Wagner function •! The Wagner function states that the instantaneous lift at the start of the motion is equal to half the value of the steady lift (i.e the value of the lift if the flow had been steady) •! The instantaneous lift then slowly increases to reach its steady value as time tends to infinity Introduction to Aeroelasticity Wagner Function (2) The Wagner function is equal to 0.5 when t=0 It increases asymptotically to It can be equally used to describe an impulsive change in angle of attack at constant airspeed Introduction to Aeroelasticity Wagner Function (3) •! An approximate expression for the Wagner function is given by "( t ) = 1# $1e #% 1Ut / b # $2e #% 2Ut / b •! Where !1=0.165, !2=0.335, !1=0.0455, and !2=0.3 •! The lift coefficient variation with time after a step change in incidence is given by c l ( t ) = 2"#$( t ) •! So that the lift force variation becomes l( t ) = "#U 2c$%( t ) = "#Ucw%( t ) Introduction to Aeroelasticity w=U" is the downwash velocity Unsteady Motion •! Unsteady motion can be modeled as a superposition of many small impulsive changes in angle of attack •! The increment in lift due to a small change in pitch angle at time t0 dw ( t ) dt dl( t ) = "#Uc$( t % t )dw ( t ) = "#Uc$( t % t ) dt •! So that the lift variation at all times can be obtained by integrating from time -! to time t, i.e t dw ( t ) l( t ) = "#Uc ' $( t % t ) -& Introduction to Aeroelasticity dt 0 dt Unsteady Motion (2) •! Using the thin airfoil theory result obtained in the first lecture, the downwash velocity can be written as $3 & ˙ w ( t ) = U" tot ( t ) = U" ( t ) + h ( t ) + c # x f "˙ ( t ) %4 ' •! For a motion starting at t=0, w=0 for t[...]... Differentiating this equation with time: Introduction to Aeroelasticity Leibniz Integral Rule •! E.g for w1(t): •! For all wi(t): (2) Introduction to Aeroelasticity Complete Equations •! Equations (1) and (2) make up the complete aeroelastic system of equations •! Equations (1) are 2nd order Ordinary Differential Equations (ODEs) They describe the dynamics of the system states •! Equations (2) are 1st order ODEs... aerodynamic states we obtain Introduction to Aeroelasticity Transform to ODEs (3) •! The integrals have been absorbed by the aerodynamic states The full equations of motion are M C (1) K W Introduction to Aeroelasticity Transform to ODEs (4) •! There are two equations with 6 unknowns; 4 more equations are needed •! These can be obtained by noting that the definitions of wi are of the form •! Differentiating... motion become Introduction to Aeroelasticity Unsteady equations of motion This type of equation is known as integro-differential since it contains both integral and differential terms Introduction to Aeroelasticity 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... conducting stability analysis •! The equations must be transformed to ODEs in order to perform stability analysis Introduction to Aeroelasticity Transform to ODEs (1) •! Use the following substitutions: The wi variables are known as the aerodynamic states They arise from the substitution of the approximate form of the Wagner function, !, in the equations of motion Introduction to Aeroelasticity Transform... the basis of Wagner function aerodynamics •! It includes the effect of the entire motion history of the system in the calculation of the current lift force Introduction to Aeroelasticity Fourier Transform Pair •! The Wagner Function and Theodorsen’s Function form a Fourier Transform pair: t ik % "( t # t 0 )e ikt 0 dt 0 = C ( k )e ikt #$ •! Unfortunately, there is a mathematical difficulty of divergence... & Introduction to Aeroelasticity Moment •! The aerodynamic moment around the flexural axis due to the unsteady lift force is simply mxf(t)=ec l(t) •! However, for a complete representation of the aerodynamic force and moment, the added mass effects must be superimposed, exactly as was done in the quasi-steady case •! The complete equations of motion become Introduction to Aeroelasticity Unsteady equations. .. mentioned several times, the aerodynamic forces depend not only the current state of the system but also on the history of the motion •! This history is stored in the aerodynamic states After all they are integrals Introduction to Aeroelasticity Solution of the ODEs •! Now the unsteady aeroelastic equations are in complete ODE form (6 equations with 6 unknowns) and can be solved as usual, by injecting a harmonic... describe the dynamics of the system states •! Equations (2) are 1st order ODEs They describe the dynamics of the aerodynamic states Introduction to Aeroelasticity Complete Equations (2) •! Here is the form of the complete equations •! where u˙ = Qu # "M "1 C "M "1 K % Q =% I 0 % W0 $ 0 #1 % 1 % W0 = %0 % $0 Introduction to Aeroelasticity "M "1 W & ( 0 (, ( ' # h˙ & % ( % )˙ ( %h( % ( ) u =% ( % w1 ( % ( %... responses U=12m/s U=25.4m/s Time responses for pitch-plunge airfoil with Wagner Function aerodynamics for an initial pitch angle of 5o Introduction to Aeroelasticity Discussion •! Wagner function aerodynamics leads directly to time domain equations of motion •! The application of this approach has been mostly limited to simple systems, such as the pitch-plunge airfoil or the pitch-plunge-control airfoil... which are real and negative They vary almost linearly with airspeed Introduction to Aeroelasticity Effect of Flexural Axis The divergence speed is the same as in the quasi-steady case The flutter speeds obtained from Wagner’s and Theodorsen’s methods are identical This is normal, since one method is the Fourier Transform of the other Introduction to Aeroelasticity Time domain responses U=12m/s U=25.4m/s