Introduction – Equations of motion G. Dimitriadis 07

Aeroelasticity Lecture 7: Three-Dimensional Wings G Dimitriadis Introduction to Aeroelasticity Wings are 3D •! All the methods described until now concern 2D wing sections •! These results must now be extended to 3D wings because all wings are 3D •! There are two methods for 3D wing aeroelasticity: –! Strip theory –! Panel methods Introduction to Aeroelasticity Strip theory •! Strip theory breaks the wing into spanwise small strips •! The instantaneous lift and moment acting on each strip are given by the 2D sectional lift and moment theories (quasi-steady, unsteady etc) Introduction to Aeroelasticity dy S y Panel methods Introduction to Aeroelasticity s Wake Panels i,j+1 i+1,j+1 y i,j i+1,j z •! The wing is replaced by its camber surface •! The surface itself is replaced by panels of mathematical singularities, solutions of Laplace’s equation x c Hancock Model •! A simple 3D wing model is used to introduce 3D aeroelasticity ! 1*+,- A rigid flat plate of span s, chord c and thickness t, suspended through an axis xf by two torsional springs, one in roll (K!) and one in pitch (K") !!"$ !!"# " $ !"' ! !"& The wing has two degrees of freedom, roll (!) and pitch (") ) ( !"% !"# !"# ! !"$ ! /*+,!!"# !!"$ !!"# )*+,- Introduction to Aeroelasticity Hancock model assumptions •! The plate thickness is very small compared to its other dimensions •! The wing is infinitely rigid (in other words it does not flex or change shape) •! The displacement angles ! and " are always small •! The z-position of any point on the wing is ( ) z = y" + x # x f $ Introduction to Aeroelasticity Equations of motion •! As with the 2D pitch plunge wing, the equations of motion are derived using energy considerations •! The kinetic energy of a small mass element dm of the wing is given by 1 dT = z˙ dm = dm y"˙ + x # x f $˙ 2 ( ( )) •! The total kinetic energy of the wing is: m 2 2 ˙2 ˙ ˙ ˙ T= 2s " + 3s c # 2x f "$ + c # 3x f c + 3x f $ 12 ( Introduction to Aeroelasticity ( ) ( ) ) Structural equations •! The potential energy of the wing is simply 1 V = K" " + K# # 2 •! The full structural equations of motion are then: $ I" &I % "# I"# ' *"˙˙- $ K" + ˙˙ + & ) I# ( ,# / % ( ) ' *" - * M1 )+ = + K# ( ,# / , M / ( ) I" = ms2 /3, I"# = m c $ 2x f s /4, I# = m c $ 3x f c + 3x 2f /3 Introduction to Aeroelasticity Strip theory •! The quasi-steady or unsteady approximations for the lift and moment around the flexural axis are applied to infinitesimal strips of wing •! The lift and moment on these strips are integrated over the entire span of the wing •! The result is a quasi-steady pseudo-3D lift and moment acting on the Hancock wing s M1 = " # yl( y )dy s M = " # mx f ( y )dy Introduction to Aeroelasticity Quasi-steady strip theory •! Denote, "=# and h=y! Then l mxf •! Carrying out the strip theory integrations will yield the total moments around the y=0 and x=xf axes Introduction to Aeroelasticity Panelling up and solving •! The process of dividing a wing planform into panels •! The wing can be swept, tapered and twisted •! It cannot have thickness •! The wake must also be panelled up •! The object of the VLM is to calculated the values of the vorticities ! on each wing panel at each instant in time •! The vorticity of the wake panels does not change in time Only the vorticities on the wing panels are unknowns Introduction to Aeroelasticity Calculating forces •! Once the vorticities on the wing panels are known, the lift and moment acting on the wing can be calculated •! These are calculated from the pressure difference acting on each panel •! Summing the pressure differences of the entire wing yields the total forces and moments Introduction to Aeroelasticity Panels for static wing Even if the wing is not moving, the wake must still be modelled because it describes the downwash induced on the wing’s surface and, hence, the induced drag Introduction to Aeroelasticity Unsteady wake panels •! At each time instance a new wake panel is shed in the wake •! The previous wake panels are propagated downstream at the local flow airspeeds Introduction to Aeroelasticity Wake shapes Wake shape behind a rectangular wing that underwent an impulsive start from rest The aspect ratio of the wing is and the angle of attack degrees Introduction to Aeroelasticity Effect of Aspect Ratio on lift coefficient Lift coefficient variation with time for an impulsively started rectangular wing of varying Aspect Ratio It can be clearly seen that the 3D results approach Wagner’s function (2D result) as the Aspect Ratio increases Introduction to Aeroelasticity Wake shapes Wake shape behind a rectangular wing undergoing sinusoidal rolling motion The aspect ratio of the wing is and the roll amplitude is degrees The airspeed is 50m/s and the frequency 30 Hz Introduction to Aeroelasticity Wake shapes Wake shape behind a rectangular wing undergoing sinusoidal pitching motion The aspect ratio of the wing is and the pitch amplitude is degrees The airspeed is 50m/s and the frequency 30 Hz Introduction to Aeroelasticity Wake shapes Wake shape behind a rectangular wing undergoing combined sinusoidal rolling and pitching motion The aspect ratio of the wing is 4, the pitch amplitude is degrees and the roll amplitude is degrees The airspeed is 50m/s and the frequency 30 Hz Introduction to Aeroelasticity All planforms can be treated Wake shape behind a bird-like wing performing roll oscillations at 10Hz with amplitude degrees Introduction to Aeroelasticity Unsteady lift and drag Unsteady lift and drag coefficients for a birdlike wing performing roll oscillations at 10Hz with amplitude degrees Drag coefficient Introduction to Aeroelasticity Lift coefficient Industrial use •! Unsteady wakes are beautiful but expensive to calculate •! For practical purposes, a fixed wake is used with unsteady vorticity, just like Theodorsen’s method •! The wake propagates at the free stream airspeed and in the free stream direction •! Only a short length of wake is simulated (a few chord-lengths) •! The result is a linearized aerodynamic model Introduction to Aeroelasticity Aerodynamic influence coefficient matrices •! 3D aerodynamic calculations can be further speeded up by calculating everything in terms of the mode shapes of the structure •! This treatment allows the expression of the aerodynamic forces as modal aerodynamic forces, written in terms of aerodynamic influence coefficients matrices •! These are square matrices with dimensions equal to the number of retained modes They also depend on response frequency •! Therefore, the complete aeroelastic system can be written as a set of linear ODEs with frequency-dependent matrices, to be solved using the p-k method Introduction to Aeroelasticity Commercial packages •! There are two major commercial packages that can calculate 3D unsteady aerodynamics using panel methods: –! MSC.Nastran –! ZAERO (ZONA Technology) •! They can both deal with complete, although idealized geometries Introduction to Aeroelasticity ZAERO AFA example •! The examples manual of ZAERO features an Advanced Fighter Aircraft model Aerodynamic model Structural model Introduction to Aeroelasticity [...]... strip becomes l Introduction to Aeroelasticity Incremental Moment •! The incremental moment on each strip becomes mxf Introduction to Aeroelasticity Full equations of motion Introduction to Aeroelasticity Aerodynamic state equations of motion •! As in the 2D case, the unsteady equations of motion need to be completed by four extra equations •! These are obtained from the aerodynamic states Introduction. .. Quasi-steady equations of motion •! The full 3D quasi-steady equations of motion are given by •! They can be solved as usual Introduction to Aeroelasticity Natural frequencies and damping ratios Introduction to Aeroelasticity Unsteady aerodynamics •! Wagner function unsteady aerodynamics can be implemented on the Hancock model in exactly the same way •! The aerodynamic states need to be redefined as Introduction. .. moderate and small aspect ratios (less than 10) Introduction to Aeroelasticity Comparison of 3D and strip theory, static case The 3D lift distribution is completely different to the strip theory result! Introduction to Aeroelasticity Vortex lattice method Introduction to Aeroelasticity w v i,j+1 u P i+1,j+1 ni,j z •! The basis of the VLM is the division of the wing planform to panels on which lie vortex... velocity induced is 0 Introduction to Aeroelasticity Panelling up and solving •! The process of dividing a wing planform into panels •! The wing can be swept, tapered and twisted •! It cannot have thickness •! The wake must also be panelled up •! The object of the VLM is to calculated the values of the vorticities ! on each wing panel at each instant in time •! The vorticity of the wake panels does... induced drag Introduction to Aeroelasticity Unsteady wake panels •! At each time instance a new wake panel is shed in the wake •! The previous wake panels are propagated downstream at the local flow airspeeds Introduction to Aeroelasticity Wake shapes Wake shape behind a rectangular wing that underwent an impulsive start from rest The aspect ratio of the wing is 4 and the angle of attack 5 degrees Introduction. .. Effect of Aspect Ratio on lift coefficient Lift coefficient variation with time for an impulsively started rectangular wing of varying Aspect Ratio It can be clearly seen that the 3D results approach Wagner’s function (2D result) as the Aspect Ratio increases Introduction to Aeroelasticity Wake shapes Wake shape behind a rectangular wing undergoing sinusoidal rolling motion The aspect ratio of the... 4 and the roll amplitude is 2 degrees The airspeed is 50m/s and the frequency 30 Hz Introduction to Aeroelasticity Wake shapes Wake shape behind a rectangular wing undergoing sinusoidal pitching motion The aspect ratio of the wing is 4 and the pitch amplitude is 5 degrees The airspeed is 50m/s and the frequency 30 Hz Introduction to Aeroelasticity ... vorticities on the wing panels are unknowns Introduction to Aeroelasticity Calculating forces •! Once the vorticities on the wing panels are known, the lift and moment acting on the wing can be calculated •! These are calculated from the pressure difference acting on each panel •! Summing the pressure differences of the entire wing yields the total forces and moments Introduction to Aeroelasticity Panels... damping ratios Introduction to Aeroelasticity Theodorsen function aerodynamics •! Again, Theodorsen function aerodynamics can (unsteady frequency domain) can be implemented directly using strip theory: l mxf Introduction to Aeroelasticity Flutter determinant •! The flutter determinant for the Hancock model is given by •! And is solved in exactly the same was as for the 2D pitch-plunge model Introduction. .. panels on which lie vortex rings, usually called a vortex lattice •! A vortex ring is a rectangle made up of four straight line vortex segments i,j y ri,j x i+1,j Characteristics of a vortex ring •! The vector nij is a unit vector normal to the ring and positioned at its midpoint (the intersection of the two diagonals - also termed the collocation point) •! The vorticity, !, is constant over all the