Identification of Eighteen Flutter DerivativesIdentification of Eighteen Flutter Derivatives  

Arindam Gan Chowdhury Arindam Gan Chowdhury a a and Partha P. Sarkarand Partha P. Sarkar b b

  a a Graduate Research Assistant, Department of Aerospace Engineering, Iowa State University, Ames, Iowa, USAGraduate Research Assistant, Department of Aerospace Engineering, Iowa State University, Ames, Iowa, USA

b b Associate Professor/Wilson Chair, Departments of Aerospace Engineering and Civil, Construction and Environmental Engrg., Iowa State University, Ames, Iowa, USAAssociate Professor/Wilson Chair, Departments of Aerospace Engineering and Civil, Construction and Environmental Engrg., Iowa State University, Ames, Iowa, USA

INTRODUCTIONINTRODUCTION

AEROELASTICITY:AEROELASTICITY: Interaction between aerodynamic forces and structural motion.Interaction between aerodynamic forces and structural motion.

FLUTTER INSTABILITY: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 ANALYSIS:

Flutter speed is calculated using frequency-dependant Flutter Derivatives that are experimentally obtainedFlutter speed is calculated using frequency-dependant Flutter Derivatives that are experimentally obtained from wind from wind

tunnel testing of section models.tunnel testing of section models.

FLUTTER DERIVATIVE FORMULATIONFLUTTER DERIVATIVE FORMULATION

SECTION MODEL FOR WIND TUNNEL TESTING:SECTION MODEL FOR WIND TUNNEL TESTING:

, M, MUU

h, Lh, L

p, Dp, D

AEROELASTIC FORCE VECTOR:AEROELASTIC FORCE VECTOR:

ae

ae

ae

ae

D

M

L

F *

5.000

05.00

005.0

2

22

2

BU

BU

BU

p

h

p

h

BPKPKBPKUKPUBKPUKP

BAKAKBAKUKAUBKAUKA

BHKHKBHKUKHUBKHUKH

/////

/////

/////

**

42*

32*

62*

1*

2*

5

*6

2*3

2*4

2*5

*2

*1

*6

2*3

2*4

2*5

*2

*1

where, where, is air density; U is the mean wind velocity; is air density; U is the mean wind velocity; BB is the width of section model; is the width of section model; K = BK = B / /U is the reduced frequencyU is the reduced frequency

Non-dimensional aerodynamic coefficients Non-dimensional aerodynamic coefficients HiHi, Ai, Ai and and PiPi, i = 1-6,, i = 1-6, are called the Flutter Derivatives. are called the Flutter Derivatives.

Flutter Derivatives evolve as functions of reduced velocity, U Flutter Derivatives evolve as functions of reduced velocity, U / n B/ n B = 2 = 2//KK ( (nn is frequency; is frequency; is circular frequency).is circular frequency).

AEROELASTICALLY MODIFIED EQUATIONS OF MOTION:AEROELASTICALLY MODIFIED EQUATIONS OF MOTION:

,0 yKyCy effeff Tphy

,XAX ,

y

yX

effeff CK

IA

0

STATE-SPACE FORMULATIONSTATE-SPACE FORMULATION::

CCeffeff and and KKeffeff are the aeroelastically modified effective damping and stiffness matrices. are the aeroelastically modified effective damping and stiffness matrices.

Flutter Derivatives are extracted by identifying elements of effective damping & Flutter Derivatives are extracted by identifying elements of effective damping &

stiffness matrices at zero & various non-zero wind speeds.stiffness matrices at zero & various non-zero wind speeds.

NEW SYSTEM IDENTIFICATION (SID) TECHNIQUENEW SYSTEM IDENTIFICATION (SID) TECHNIQUE

SAMPLE DISPLACEMENT TIME HISTORY WITH NOISE/SIGNALSAMPLE DISPLACEMENT TIME HISTORY WITH NOISE/SIGNAL RATIO OF 20% OBTAINED NUMERICALLY TO TEST THE NEW RATIO OF 20% OBTAINED NUMERICALLY TO TEST THE NEW SID SID TECHNIQUE:TECHNIQUE:

ITERATIVE LEAST SQUARE METHOD (ILS METHOD)ITERATIVE LEAST SQUARE METHOD (ILS METHOD)::

A new SID technique was developed for the extraction of flutter derivatives from free vibration A new SID technique was developed for the extraction of flutter derivatives from free vibration

displacement time histories obtained from a section model testing.displacement time histories obtained from a section model testing.

MOTIVATIONMOTIVATION::

Extraction of all eighteen flutter derivatives required development of a robust SID technique 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 that will efficiently work with noisy signal outputs from a three-degree-of-freedom dynamic

system. system.

ALGORITHM FOR ILS METHODALGORITHM FOR ILS METHOD:: EXPERIMENTAL SETUP (WiST Laboratory, ISU)EXPERIMENTAL SETUP (WiST Laboratory, ISU)

Vertical & Horizontal Force TransducersVertical & Horizontal Force Transducers

Three-DOF Elastic Suspension SystemThree-DOF Elastic Suspension System

Torsional DOF Assembly & Torque SensorTorsional DOF Assembly & Torque Sensor

RESULTSRESULTS

Average Percentage Errors for Numerical Simulations Average Percentage Errors for Numerical Simulations

DOF Combinations and Corresponding Flutter Derivatives ObtainedDOF Combinations and Corresponding Flutter Derivatives Obtained

CaseCase Noise-to-Signal Noise-to-Signal RatioRatio

Diagonal Diagonal Stiffness TermsStiffness Terms

Non-Diagonal Non-Diagonal Stiffness TermsStiffness Terms

Diagonal Diagonal Damping TermsDamping Terms

Non-Diagonal Non-Diagonal Damping TermsDamping Terms

1-DOF (ILS)1-DOF (ILS) 20%20% 0.020.02 -- 1.671.67 --

2-DOF (MITD)2-DOF (MITD) 5%5% 0.190.19 2.222.22 0.810.81 2.022.02

2-DOF (MITD)2-DOF (MITD) 10%10% 0.370.37 4.474.47 1.601.60 2.922.92

2-DOF (ILS)2-DOF (ILS) 10%10% 0.060.06 0.820.82 0.560.56 1.411.41

2-DOF (ILS)2-DOF (ILS) 20%20% 0.130.13 0.960.96 2.012.01 5.045.04

3-DOF (ILS)3-DOF (ILS) 5%5% 0.440.44 1.511.51 2.552.55 5.995.99

3-DOF (ILS)3-DOF (ILS) 10%10% 0.890.89 2.342.34 4.834.83 8.438.43

CaseCase DOF CombinationDOF Combination Flutter-Derivatives ExtractedFlutter-Derivatives Extracted

11 1-DOF Vertical (V)1-DOF Vertical (V) H1*, H4*H1*, H4*

22 1-DOF Torsional (T)1-DOF Torsional (T) A2*, A3*A2*, A3*

33 1-DOF Lateral (L)1-DOF Lateral (L) P1*, P4*P1*, P4*

44 2-DOF Vertical+Torsional (V&T)2-DOF Vertical+Torsional (V&T) H1*, H2*, H3*, H4*, A1*, A2*, A3*, A4*H1*, H2*, H3*, H4*, A1*, A2*, A3*, A4*

55 2-DOF Vertical+Lateral (V&L)2-DOF Vertical+Lateral (V&L) H1*, H4*, H5*, H6*, P1*, P4*, P5*, P6*H1*, H4*, H5*, H6*, P1*, P4*, P5*, P6*

66 2-DOF Lateral+Torsional (L&T)2-DOF Lateral+Torsional (L&T) P1*, P4*, P2*, P3*, A2*, A3*, A5*, A6*P1*, P4*, P2*, P3*, A2*, A3*, A5*, A6*

77 3-DOF 3-DOF All the 18 flutter derivatives All the 18 flutter derivatives

EIGHTEEN FLUTTER DERIVATIVES OF NACA 0020 AIRFOIL:EIGHTEEN FLUTTER DERIVATIVES OF NACA 0020 AIRFOIL:

-25

-20

-15

-10

-5

0

0 5 10 15

U/nB

H3*

H3* (2 DOF,V&T)

H3* (3 DOF)

-5

-3

-1

1

3

5

0 5 10 15

U/nB

H4*

H4* (1 DOF)H4* (2 DOF,V&T)H4* (2 DOF,V&L)H4* (3 DOF)

-5

-3

-1

1

3

5

0 5 10 15

U/nB

H5*

H5* (2 DOF,V&L)

H5* (3 DOF)-5

-3

-1

1

3

5

0 5 10 15

U/nB

H6*

H6* (2 DOF,V&L)

H6* (3 DOF)

0

0.5

1

1.5

2

2.5

3

0 5 10 15U/nB

A1*

A1* (2 DOF,V&T)A1* (3 DOF)

-2

-1

0

1

2

3

4

5

6

0 5 10 15

U/nB

A3*

A3* (1 DOF)A3* (2 DOF,V&T)A3* (2 DOF, L&T)A3* (3 DOF)

-3

-2

-1

0

1

2

3

0 5 10 15

U/nB

A4*

A4* (2 DOF,V&T)

A4* (3 DOF)

-1

-0.5

0

0.5

1

0 5 10 15

U/nB

A5*

A5* (2 DOF, L&T)

A5* (3 DOF)

-1

-0.5

0

0.5

1

0 5 10 15

U/nB

A6*

A6* (2 DOF, L&T)

A6* (3 DOF)

-3

-2

-1

0

1

2

3

0 5 10 15

U/nB

P3*

P3* (2 DOF,L&T)

P3* (3 DOF)-3

-2

-1

0

1

2

3

0 5 10 15

U/nB

P4*

P4* (1 DOF)P4* (2 DOF,V&L)P4* (2 DOF,L&T)P4* (3 DOF)

-3

-2

-1

0

1

2

3

0 5 10 15

U/nB

P5*

P5* (2 DOF,V&L)

P5* (3 DOF)

-3

-1

1

3

5

7

0 5 10 15

U/nB

P6*

P6* (2 DOF,V&L)

P6* (3 DOF)

-15

-13

-11

-9

-7

-5

-3

-1 0 5 10 15

U/nB

H1*

H1* (1 DOF)H1* (2 DOF,V&T)H1* (2 DOF,V&L)H1* (3 DOF)

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 5 10 15

U/nB

H2*

H2* (2 DOF,V&T)

H2* (3 DOF)

-3

-2

-1

0

1

2

3

0 5 10 15

U/nB

P2*

P2* (2 DOF,L&T)

P2* (3 DOF)

-3

-2

-1

0

1

2

3

0 5 10 15

U/nB

P1*

P1* (1 DOF)P1* (2 DOF,V&L)P1* (2 DOF,L&T)P1* (3 DOF)

-6

-5

-4

-3

-2

-1

0

1

2

0 5 10 15

U/nB

A2*

A2* (1 DOF)A2* (2 DOF,V&T)A2* (2 DOF,L&T)A2* (3 DOF)

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

),( ity ,)1( titi 22,......,2,1 Ni

XX ,

1 TT XXXXA

01 XeX tA

1111

TT

XXXXA

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)(Note: Modified Ibrahim Time Domain (MITD) method was developed by Sarkar, Jones, and Scanlan in1994)

REFERENCES:REFERENCES:

Sarkar, P.P., Jones, N.P., Scanlan, R.H. (1994). “Identification Sarkar, P.P., Jones, N.P., Scanlan, R.H. (1994). “Identification of Aeroelastic Parameters of Flexible Bridges”. J. of Engineering of Aeroelastic Parameters of Flexible Bridges”. J. of Engineering Mechanics, ASCE 1994, 120 (8), pp. 1718-1742.Mechanics, ASCE 1994, 120 (8), pp. 1718-1742.

Gan Chowdhury, A., Sarkar, P.P. (2003). “A New Technique Gan Chowdhury, A., Sarkar, P.P. (2003). “A New Technique for Identification of Eighteen Flutter Derivatives using Three-for Identification of Eighteen Flutter Derivatives using Three-Degrees-of-Freedom Section Model”. Accepted 21 July 2003, Degrees-of-Freedom Section Model”. Accepted 21 July 2003, Engineering Structures.Engineering Structures.

Sarkar, P.P., Gan Chowdhury, A., Gardner, T. B. (2003). “A Sarkar, P.P., Gan Chowdhury, A., Gardner, T. B. (2003). “A Novel Elastic Suspension System for Wind Tunnel Section Model Novel Elastic Suspension System for Wind Tunnel Section Model Studies”. Accepted 12 September 2003, J. of Wind Engineering Studies”. Accepted 12 September 2003, J. of Wind Engineering and Industrial Aerodynamics.and Industrial Aerodynamics.

Gan Chowdhury, A., Sarkar, P.P. (2003). “Identification of Gan Chowdhury, A., Sarkar, P.P. (2003). “Identification of Eighteen Flutter Derivatives”. Proceedings of the 11th Eighteen Flutter Derivatives”. Proceedings of the 11th International Conference on Wind Engineering , Lubbock, International Conference on Wind Engineering , Lubbock, Texas, USA, pp. 365-372.Texas, USA, pp. 365-372.

DISPLACEMENT TIME HISTORY AS ABOVE WITHOUT DISPLACEMENT TIME HISTORY AS ABOVE WITHOUT NOISENOISE MATCHES WELL WITH THE FILTERED ONE (SEE MATCHES WELL WITH THE FILTERED ONE (SEE ALGORITHM):ALGORITHM):