Identification of Eighteen Flutter Derivatives Arindam Gan Chowdhury a and Partha P. Sarkar b a Graduate Research Assistant, Department of Aerospace Engineering, Iowa State University, Ames, Iowa, USA b Associate Professor/Wilson Chair, Departments of Aerospace Engineering and Civil, Construction and Environmental Engrg., Iowa State University, Ames, Iowa, USA INTRODUCTION  AEROELASTICITY: Interaction between aerodynamic forces and structural motion. Interaction between aerodynamic forces and structural motion.  FLUTTER INSTABILITY: Self-excited oscillation of a structural system (e.g., flutter-induced failure of the Tacoma Narrows Bridge in 1940). Self-excited oscillation of a structural system (e.g., flutter-induced failure of the Tacoma Narrows Bridge in 1940).  FLUTTER ANALYSIS: Flutter speed is calculated using frequency-dependant Flutter Derivatives that are experimentally obtainedfrom wind Flutter speed is calculated using frequency-dependant Flutter Derivatives that are experimentally obtained from wind tunnel testing of section models. tunnel testing of section models. FLUTTER DERIVATIVE FORMULATION  SECTION MODEL FOR WIND TUNNEL TESTING: , M U h, L p, D  AEROELASTIC FORCE VECTOR: where,  is air density; U is the mean wind velocity; B is the width of section model; K = B  /U is the reduced frequency Non-dimensional aerodynamic coefficients Hi , Ai  and Pi , i = 1-6, are called the Flutter Derivatives. Flutter Derivatives evolve as functions of reduced velocity, U / n B = 2  /K (n is frequency;  is circular frequency).  AEROELASTICALLY MODIFIED EQUATIONS OF MOTION:  STATE-SPACE FORMULATION : C eff and K eff are the aeroelastically modified effective damping and stiffness matrices. Flutter Derivatives are extracted by identifying elements of effective damping & stiffness matrices at zero & various non-zero wind speeds. NEW SYSTEM IDENTIFICATION (SID) TECHNIQUE  SAMPLE DISPLACEMENT TIME HISTORY WITH NOISE/SIGNAL RATIO OF 20% OBTAINED NUMERICALLY TO TEST THE NEW SID RATIO OF 20% OBTAINED NUMERICALLY TO TEST THE NEW SID TECHNIQUE: TECHNIQUE:  ITERATIVE LEAST SQUARE METHOD (ILS METHOD) : A new SID technique was developed for the extraction of flutter derivatives from free vibration displacement time histories obtained from a section model testing.  MOTIVATION : Extraction of all eighteen flutter derivatives required development of a robust SID technique that will efficiently work with noisy signal outputs from a three-degree-of-freedom dynamic system.  ALGORITHM FOR ILS METHOD : EXPERIMENTAL SETUP (WiST Laboratory, ISU) Vertical & Horizontal Force Transducers Three-DOF Elastic Suspension System Torsional DOF Assembly & Torque Sensor RESULTS Average Percentage Errors for Numerical Simulations DOF Combinations and Corresponding Flutter Derivatives Obtained Case Noise-to-Signal Ratio Diagonal Stiffness Terms Non-Diagonal Stiffness Terms Diagonal Damping Terms Non-Diagonal Damping Terms 1-DOF (ILS) 20% DOF (MITD) 5% % DOF (ILS) 10% % DOF (ILS) 5% % Case DOF Combination Flutter-Derivatives Extracted 1 1-DOF Vertical (V) H1*, H4* 2 1-DOF Torsional (T) A2*, A3* 3 1-DOF Lateral (L) P1*, P4* 4 2-DOF Vertical+Torsional (V&T) H1*, H2*, H3*, H4*, A1*, A2*, A3*, A4* 5 2-DOF Vertical+Lateral (V&L) H1*, H4*, H5*, H6*, P1*, P4*, P5*, P6* 6 2-DOF Lateral+Torsional (L&T) P1*, P4*, P2*, P3*, A2*, A3*, A5*, A6* 73-DOF All the 18 flutter derivatives  EIGHTEEN FLUTTER DERIVATIVES OF NACA 0020 AIRFOIL: OBTAIN NOISY DISPLACEMENT TIME HISTORIES [SIZE n x (2N+2) ] BUILD LOW PASS ‘BUTTERWORTH’ FILTER PERFORM ZERO-PHASE DIGITAL FILTERING OF DISPLACEMENTS OBTAIN VELOCITY AND ACCELERATION TIME HISTORIES BY FINITE DIFFERENCE FORMULATION (EACH HAVING SIZE n x 2N ) PERFORM ‘WINDOWING’ TO OBTAIN NEW SETS OF DISPLACEMENT, VELOCITY, ACCELERATION TIME HISTORIES (EACH HAVING SIZE n x N ) CONSTRUCT (EACH HAVING SIZE 2n x N ) GENERATE A MATRIX BY LEAST SQUARES (SIZE 2n x 2n ) USING INITIAL CONDITIONS SIMULATE, UPDATE A MATRIX BY LEAST SQUARES (SIZE 2n x 2n ): ITERATE TILL THE CONVERGENCE OF A MATRIX CALCULATE FLUTTER DERIVATIVES FROM ELEMENTS OF A MATRIX EXTRACTED AT ZERO AND VARIOUS NON-ZERO WIND SPEEDS (Note: Modified Ibrahim Time Domain (MITD) method was developed by Sarkar, Jones, and Scanlan in1994) REFERENCES:  Sarkar, P.P., Jones, N.P., Scanlan, R.H. (1994). “Identification of Aeroelastic Parameters of Flexible Bridges”. J. of Engineering Mechanics, ASCE 1994, 120 (8), pp  Gan Chowdhury, A., Sarkar, P.P. (2003). “A New Technique for Identification of Eighteen Flutter Derivatives using Three- Degrees-of-Freedom Section Model”. Accepted 21 July 2003, Engineering Structures.  Sarkar, P.P., Gan Chowdhury, A., Gardner, T. B. (2003). “A Novel Elastic Suspension System for Wind Tunnel Section Model Studies”. Accepted 12 September 2003, J. of Wind Engineering and Industrial Aerodynamics.  Gan Chowdhury, A., Sarkar, P.P. (2003). “Identification of Eighteen Flutter Derivatives”. Proceedings of the 11th International Conference on Wind Engineering, Lubbock, Texas, USA, pp  DISPLACEMENT TIME HISTORY AS ABOVE WITHOUT NOISE MATCHES WELL WITH THE FILTERED ONE (SEE ALGORITHM): MATCHES WELL WITH THE FILTERED ONE (SEE ALGORITHM):