Introduction – Equations of motion G. Dimitriadis 08

Aeroelasticity Lecture 8: Supersonic Aeroelasticity G Dimitriadis Introduction to Aeroelasticity Introduction A  All the material presented up to now concerned incompressible flow A  In this lecture we will present a simple treatment of 2D supersonic flow aeroelasticity A  The discussion will concern the same pitch-plunge airfoil treated in the 2D incompressible case Introduction to Aeroelasticity Pitch-Plunge airfoil A  Flat plate airfoil with pitch and plunge degrees of freedom A  Equations of motion: # m S & ) h

, # K h % (* - + % $ S Iα ' +α

$ Introduction to Aeroelasticity & ) h , ) −l(t) , ( * - = *m (t)Kα ' +α + xf Supersonic flow A  In order to complete the model, we need to derive expressions for the lift and moment around the flexural axis, in the presence of a supersonic free stream A  The supersonic free stream is defined by: –  –  –  –  –  Airspeed U∞, Pressure P∞, Temperature T∞, Density ρ∞, Speed of sound a∞, A  Furthermore, the air is described by the ratio of specific heats at constant pressure and constant volume, i.e cp γ= = 1.4 cv Introduction to Aeroelasticity Steady Potential equation A  For supersonic flow, the steady potential equation, in terms of perturbation potential, is given by 2 ∂ φ ∂ φ + 1− M ( ∞ ) ∂x ∂y = ∂φ ∂φ = u, =v ∂x ∂y A  Where A  And u, and v are small local velocity perturbations from the free stream Introduction to Aeroelasticity Linearized Small Disturbance Equation A  For unsteady flows, the potential equation includes unsteady terms A  The Linerized Small Disturbance Equation is given by: 2 2 (1 − M ∞2 ) M∞ ∂ φ ∂φ ∂φ ∂φ + − − 2 =0 a∞ ∂x∂t a∞ ∂t ∂x ∂y A  Where, again, the potential represents a small perturbation A  This equation is difficult to solve As a first approximation, a 1D method known as piston theory can be used Introduction to Aeroelasticity Piston theory A  The aerodynamics of the moving wing are calculated with the piston theory assumption: –  Flow disturbances spread in a direction normal to the wing’s surface The wing’s movement is equivalent to the movement of a piston in a column of air –  All disturbances are isentropic A  Under this assumption, the pressure on the surface of the wing is given by 2γ % γ − w(x,t) ( γ −1 p( x,t ) = p∞ '1 + * a∞ ) & A  Where w(x,t) is the downwash velocity Introduction to Aeroelasticity Downwash A  The downwash velocity of the wing is given by %'− U α ( t ) + h
( t ) + x − x α
( t ) ∞ f w ( x,t ) = & '( U ∞α ( t ) + h
( t ) + x − x f α
( t ) ( ( ( ) ) ) 
   
 Surface discretization Source and doublet panels are placed on the surface of the aircraft The geometry can be significantly simplified by this process The wake can also be discretized Drawing from ZAERO manual Introduction to Aeroelasticity Boundary conditions A  The source and doublet strengths are obtained from the application of the nonpenetration boundary condition over the complete surface of the aircraft A  This boundary condition is unsteady, since the aircraft structure is vibrating A  It can be written in terms of the modal displacements of the structure In this case, the complete unsteady aerodynamic forces can be written in modal space Introduction to Aeroelasticity ZAERO Example Trapezoidal wing with wingtip tank and pylon-store Introduction to Aeroelasticity Discussion A  This full aircraft approach is satisfactory for both subsonic and supersonic aeroelastic problems A  However, at transonic flight conditions, the aerodynamics become very complicated Shock waves can oscillate on the wing surfaces, introducing nonlinearity and causing Limit Cycle Oscillations A  Furthermore, the oscillating shock waves can interact with the boundary layer, forcing its separation Even higher levels of nonlinearity can be generated A  Under these circumstances, linearized methods cannot be applied and higher fidelity aerodynamic modelling is required Introduction to Aeroelasticity Example: Goland Wing A  The Goland wing is a straight rectangular wing with a store Its aeroelastic response is modeled using a coupling of CFD and FE methods The CFD is an Euler equation solver while the Finite Element analysis retains only the first four modes of vibration Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.85 Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.90 Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.91 Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.915 Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.92 Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.93 Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.94 Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.945 Introduction to Aeroelasticity Bifurcation Goland Wing A  Goland wing aeroelastic response at M=0.95 Introduction to Aeroelasticity LCO simulation videos Goland+ wing, Euler aerodynamics, aeroelastic responses M=0.910 Introduction to Aeroelasticity M=0.915 [...]... into the aeroelastic equations of motion gives: ! m S # # S Iα " $ ') &( &) %* h α +) ! K h ,+# )- #" 0 ' ! S $  ) − #U∞α + h + α & $ 0 ' h + 2 ρ∞U∞ λ c ) " m % &( ( ,= & M∞ ) ! S Kα % * α S  Iα $ − U α + h + α & ) #" ∞ m m m % * A  i.e the complete supersonic aeroelastic model Introduction to Aeroelasticity + ) ) , ) ) - In matrix form A  In matrix form the equations of motion can be written... " $ ') &( &) %* h α +) ! K h ,+# )- #" 0 0 $' h + &( , & Kα % * α - A  These are quasi-steady, small disturbance equations They are valid for M∞>1.2 Introduction to Aeroelasticity Solution A  The equations of motion are 2nd order linear ODEs and can be solved as usual A  At each value of the Mach number and airspeed, the eigenvalues, χi, i=1,…,4, can be evaluated A  From the eigenvalues, natural frequencies,... Consider the case where the wing is flying at sea level and the atmospheric pressure is 1bar: – ρ∞=1.225kg/m3 – p∞=101325Pa A  Then the speed of sound is 340m/s Therefore, the flutter speed of 607m/s corresponds to a Mach number of 1.8 A  But the Mach number used for the simulation is 1.5 This case is an example of an unmatched flutter speed The system can flutter but not at an attainable Mach number... to a Mach number of 1.2 The flight condition is very unsafe Introduction to Aeroelasticity International Standard Atmosphere A  According to the equations of motion, the aerodynamic forces depend on the Mach number, flight speed and air density A  The air density is a function of the flight altitude A  The altitude also defines the speed of sound Therefore, the aerodynamic forces only depend on flight... α ( ' $2 ' 3 M∞ $ $ 2 ' Introduction to Aeroelasticity Lift and moment A  Remembering from lecture 1 that m 2 #c % S = m − x f , Iα = c − 3cx f + 3x 2f $2 & 3 ( ) A  We can simplify the lift and moment expressions such that: 2 ρ∞U∞ λ c " S %  l= $U∞α + h + α ' m & M∞ # 2 ρ∞U∞ λ c " S S  Iα % mxf = − $U∞ α + h + α ' m M∞ # m m & Introduction to Aeroelasticity Equations of motion A  Substituting the... altitude and flight Mach number A  The International Standard Atmosphere determines the variation of density and speed of sound with altitude from sea level Introduction to Aeroelasticity ISA Graph Air density, speed of sound, pressure and viscosity ratios with respect to their values at 0m Altitude range: 0-21000m Introduction to Aeroelasticity Mach-Airspeed diagrams A  For each Mach number and altitude,... this calculation difficult Introduction to Aeroelasticity Binomial series A  The binomial series is a special case of a Taylor series A  For |x|