AAE556 Lecture_3435_pk_flutter

AAE 556 Aeroelasticity The P-k flutter solution method (also known as the British method) Purdue Aeroelasticity The eigenvalue problem from the Lecture 33 h2 ữ h b + ữ( + ig ) x r Lh + M h h ữ b = h x r x h b r h L + a ữLh b = M x Lh + r M h h L + a ữLh b M Purdue Aeroelasticity Genealogy of the V-g or k method i Equations of motion for harmonic response (next slide) i Forcing frequency and airspeeds are is known parameters Reduced frequency k is determined from and V Equations are correct at all values of and V Take away the harmonic applied forcing function Equations are only true at the flutter point We have an eigenvalue problem Frequency and airspeed are unknowns, but we still need k to define the numbers to compute the elements of the eigenvalue problem We invent ed Theodorsens method or V-g artificial damping to create an iterative approach to finding the flutter point Purdue Aeroelasticity Go back to the original typical section equations of motion, restricted to steady-state harmonic response AEOM + DEOM h BEOM F b = EEOM M SC Purdue Aeroelasticity The coefficients for the EOMs AEOM L = h ữ ữ h ữ BEOM DEOM 1 = x L Lh + a ữữ 1 = x M h Lh + a ữữ ( EEOM = r2 ) Mh + M L Lh + a + + a + a ữ ữ ữ à Purdue Aeroelasticity The eigenvalue problem AEOM DEOM h BEOM b = EEOM AEOM DEOM h BEOM b = EEOM AEOM ( ) DEOM h BEOM b = EEOM 2 Purdue Aeroelasticity Another version of the eigenvalue problem with different coefficents AEOM ( ) DEOM h h BEOM A B b = b = EEOM D E A = h ữ ữ h ữ ( + ig ) ữ+ Lh ữ B = x + L Lh + a ữ Purdue Aeroelasticity Definitions of terms for alternative set-up of eigenvalue equations for k-method h A B b = D E D = x + M h Lh + a ữ E = r + ig ữ M + a + M ( ) h ữ ữ ữ L + a ữ+ Lh + a ữ Purdue Aeroelasticity Return to the EOMs before we assumed harmonic motion Here is what we would like to have &j } + K ij { j } + Aij( 1) { j } + Aij( 2) { &j } + Aij( 3) { & & M ij { & j } = { 0} { } = { } e j j Here is the first step in solving the stability problem pt p = + j p M ij { j } + Kij { j } + Aij( 1) { j } + p Aij( ) { j } + p Aij( ) { j } = { 0} Purdue Aeroelasticity The p-k method will use the harmonic aero results to cast the stability problem in the following form p M ij { } p Bij { } + K ij V Qij ,real { } = { 0} { ( t )} = {} e but first, some preliminaries Purdue Aeroelasticity 10 pt Summary p M ij { } p Bij { } + K ij V Qreal ,ij { } = { 0} i Choose k=b/V arbitrarily i Choose altitude (), and airspeed (V) i Mach number is now known i Compute AICs from Theodorsen formulas or others i Compute aero matrices-B and Q matrices are real Purdue Aeroelasticity 27 Solving for the eigenvalues Convert the p-k equation to first-order state vector form p M ij { } p Bij { } + K ij V Qij ,real { } = { 0} displacement vector = { x j } velocity vector = { v j } = { x&j } State vector = ỡù x j ỹ ùù ù { z j } = ớù v ýù ùợ j ùỵ 28 Purdue Aeroelasticity State vector elements are related x&j } = { v&j } {& { x&j } = { v j } The equation of motion becomes ộM ij ự{ v&j } ỳ ỷ ộBij ự{ v j } + ộK ij ự{ x j } = { 0} ờ ỳ ỷ ỳ ỷ Solve for - { v&j } = - ộờởM ij ựỳỷ ộộ { v&j } = ờờởở { v&j } - ộK ij ự{ x j } + ộM ij ự ờ ỳ ỷ ỳ ỷ ộBij ự{ v j } ỳ ỷ ỡù x j ỹ ùù - - ự ự ộ ự ù ộM ij ự ộK ij ựỳờộM ij ự ộBij ựỳỳớ ý ỳỷởờ ỳ ỷ ỷ ỳ ỷ ỳ ỷỷỷù v j ù ùợ ùỵ 29 Purdue Aeroelasticity State vector eigenvalue equation the plant matrix ộ [ 0] I ] ựỡù x j ỹ ỡù x&j ỹ [ ù ùù ỳ ù ù ù { z&j } = ớù v&ýù = ờộ- M - 1K ự M - 1Bỳớù v ýù = ộờởAij ựỳỷ{ z j } ỳ ù ù ỳ ợù j ỵ ỷ ởờ ỷợù j ỵ Assume a solution Result { } z ( t ) j = { z j } e pt { z&j } = p { z j } = ộờởAij ựỳỷ{ z j } Solve for eigenvalues (p) of the [Aij] matrix (the plant) Plot results as a function of airspeed 30 Purdue Aeroelasticity st order problem i Mass matrix is diagonal if we use modal approach so too is structural stiffness matrix i Compute p roots Roots are either real (positive or negative) Complex conjugate pairs Aij = M K M B I K ij = K ij V Qreal ,ij { z&j } = p { z j } = ộờởAij ựỳỷ{ z j } 31 Purdue Aeroelasticity Eigenvalue roots pi = pi ,real jpi ,imaginary pi = i ( i j ) b k= V i = is the estimated system damping i There are m computed values of at the airspeed V i You chose a value of k=b/V, was it correct? kV = b line up the frequencies to make sure k, and V are consistent Purdue Aeroelasticity 32 p-k computation procedure Input k and V Compute eigenvalues pi = i ( i j ) No, change k ki kinput < ? yes i b ki = V Repeat process for preal = i i = i each pimaginary = i 33 Purdue Aeroelasticity What should we expect? Aij = M K M B I Right half-plane Root locus plot 34 Purdue Aeroelasticity Back-up slides for Problem 9.2 35 Purdue Aeroelasticity A comparison between V-g and p-k h2 ữ h b x h b ữ( + ig ) + x r r Lh L + a ữLh h b + = M h M x x h h2 h b b + 2 r r V Lh ữ ữ ữ V M h 2 h L + a ữLh b = M 36 Purdue Aeroelasticity A comparison between V-g and p-k x x h h2 h b + b 2 r r V Lh ữ ữ ữ V M h 2 x h L + a ữLh b = M x h h2 h b + b 2 r r V 2k b Lh ữ ữ ữ b m M h 2 h L + a ữLh b = M 37 Purdue Aeroelasticity A comparison between V-g and p-k x h h2 h b + b 2 r r x V 2k b Lh ữ ữ ữ b m M h x 2 h L + a ữLh b = M x h h2 h b b + 2 r r V k Lh ữ ữ m M h 2 h L + a ữLh b = M 38 Purdue Aeroelasticity A comparison between V-g and p-k p2 x h x k Lh b p Vb Imag ữ r m M h 2 V k Lh h + ữRe ữ 2 m r M h h L + a ữLh b M h L + a ữLh b = M 39 Purdue Aeroelasticity Flutter in action Accident occurred APR-27-95 at STEVENSON, AL Aircraft: WITTMAN O&O, registration: N41SW Injuries: Fatal REPORTS FROM GROUND WITNESSES, NONE OF WHOM ACTUALLY SAW THE AIRPLANE, VARIED FROM HEARING A HIGH REVVING ENGINE TO AN EXPLOSION EXAMINATION OF THE WRECKAGE REVEALED THAT THE AIRPLANE EXPERIENCED AN IN-FLIGHT BREAKUP DAMAGE AND STRUCTURAL DEFORMATION WAS INDICATIVE OF AILERON-WING FLUTTER WING FABRIC DOPE WAS DISTRESSED OR MISSING ON THE AFT INBOARD PORTION OF THE LEFT WING UPPER SURFACE AND ALONG THE ENTIRE LENGTH OF THE TOP OF THE MAIN SPAR LARGE AREAS OF DOPE WERE ALSO MISSING FROM THE LEFT WING UNDERSURFACE THE ENTIRE FABRIC COVERING ON THE UPPER AND LOWER SURFACES OF THE RIGHT WING HAD DELAMINATED FROM THE WING PLYWOOD SKIN THE DOPED FINISH WAS SEVERELY DISTRESSED AND MOTTLED THE FABRIC COVERING HAD NOT BEEN INSTALLED IN ACCORDANCE WITH THE POLY-FIBER COVERING AND PAINT MANUAL; THE PLYWOOD WAS NOT TREATED WITH THE POLYBRUSH COMPOUND Probable Cause AILERON-WING FLUTTER INDUCED BY SEPARATION AT THE TRAILING EDGE OF AN UNBONDED PORTION OF WING FABRIC AT AN AILERON WING STATION THE DEBONDING OF THE WING FABRIC WAS A RESULT OF IMPROPER INSTALLATION 40 Purdue Aeroelasticity Things you should know Royal Aircraft Establishment The RAE started as HM Balloon Factory From 1911-18 it was called the Royal Aircraft Factory, but was changed its name to Royal Aircraft Establishment to avoid confusion with the newly established Royal Air Force Farnborough was known as a center of excellence for aircraft research Major flutter research was conducted there Famous R&Ms such as the flutter bible came from this facility The RAE played a major role in both World Wars So confident was Hitler that he could occupy England with relative ease that he spared the RAE from bombing in the hope of benefiting from its research Recently the RAE (now known as the Royal Aerospace Establishment) was absorbed into the DRA (Defence Research Agency), itself renamed as DERA (Defence Evaluation and Research Agency) The world famous initials are no more 41 Purdue Aeroelasticity