Theodorsen's and Garrick's Computational

https://ntrs.nasa.gov/search.jsp?R=20200002621 2020-05-24T04:40:06+00:00Z

THEODORSEN’S AND GARRICK’S COMPUTATIONAL AEROELASTICITY, REVISITED

Boyd Perry, III Distinguished Research Associate Aeroelasticity Branch NASA Langley Research Center Hampton, Virginia, USA

International Forum on Aeroelasticity and Structural Dynamics Savannah, Georgia June 10-13, 2019 “Results of Theodorsen and Garrick Revisited” by Thomas A. Zeiler Journal of Aircraft Vol. 37, No. 5, Sep-Oct 2000, pp. 918-920

Made known that – • Some plots in the foundational trilogy of NACA reports on aeroelastic flutter by Theodore Theodorsen and I. E. Garrick are in error • Some of these erroneous plots appear in classic texts on aeroelasticity

Recommended that – • All of the plots in the foundational trilogy be recomputed and published

2 “Results of Theodorsen and Garrick Revisited” by Thomas A. Zeiler Journal of Aircraft Vol. 37, No. 5, Sep-Oct 2000, pp. 918-920

Recommended that – • All of the plots in the foundational trilogy be recomputed and published Cautioned that – • “One does not set about lightly to correct the masters.” 3 Works Containing Erroneous Plots

1. Theodorsen, T.: General Theory of Aerodynamic Instability and the Mechanism of Flutter. NACA Report No. 496, 1934.

2. Theodorsen, T. and Garrick, I. E.: Mechanism of Flutter, a Theoretical and Experimental Investigation of the Flutter Problem. NACA Report No. 685, 1940.

3. Theodorsen, T. and Garrick, I. E.: Flutter Calculations in Three Degrees of Freedom. NACA Report No. 741, 1942.

4. Bisplinghoff, R. L., Ashley, H., and Halfman, R. L.: Aeroelasticity, Addison-Wesley- Longman, Reading, MA, 1955, pp. 539-543.

5. Bisplinghoff, R. L. and Ashley, H.: Principles of Aeroelasticity, Dover, New York, 1975, pp. 247-249.

4 Purpose of This Presentation

• Make known a multi-year effort to re-compute all of the example problems in the foundational trilogy of NACA reports • Re-computations performed using the solution method specific to each NACA report • Re-computations checked and re-checked using modern flutter solution methods (“One does not set about lightly to correct the masters.”) • NASA TP has been / will be published for each report in the trilogy • Present outlines of Theodorsen’s and Garrick’s – • Equations of motion • Solution methods • Present representative re-computations and comparisons with the originals

5 Publications of Re-Computed Results

NASA/TP-2019-xxxxxx

685

Summer or Fall 2019

Available on NASA Technical Report Server https://ntrs.nasa.gov/

6 Outline

• Background and Purpose • Brief History • Equations of Motion • Solution Methods • Re-Computations and Comparisons • Concluding Remarks

7 1930’s and 40’s NACA Computing Environment

8 1930’s and 40’s NACA Computing Environment

• “Computers”

9 1930’s and 40’s NACA Computing Environment

• “Computers” Employees whose job function was to perform computations

10 1930’s and 40’s NACA Computing Environment

• “Computers” Employees whose job function was to perform computations • Tools

11 1930’s and 40’s NACA Computing Environment

• “Computers” Employees whose job function was to perform computations • Tools – Pencil and paper

12 1930’s and 40’s NACA Computing Environment

• “Computers” Employees whose job function was to perform computations • Tools – Pencil and paper

– Slide rules

13 1930’s and 40’s NACA Computing Environment

• “Computers” Employees whose job function was to perform computations • Tools – Pencil and paper

– Slide rules

1930’s 1940’s – Mechanical calculators (comptometers – patented 1887)

14 1930’s and 40’s NACA Computing Environment

• “Computers” Employees whose job function was to performStrong motivation computations to minimize • Tools human time and effort required to – Pencil solveand paper equations: • Recast equations to eliminate - - solution steps – Slide rules - complex arithmetic

1930’s 1940’s – Mechanical calculators (comptometers – patented 1887)

15 1930’s and 40’s NACA Computing Environment

• “Computers” Employees whose job function was to perform computations • Tools Unfortunately – – Pencil and paperHuman computers … … are prone to error

– Slide rules

1930’s 1940’s – Mechanical calculators (comptometers – patented 1887)

16 Outline

• Background and Purpose • Brief History • Equations of Motion • Solution Methods • Re-Computations and Comparisons • Concluding Remarks

17 Assumptions Made in NACA 496

• Flow is potential, unsteady, incompressible • “Wing” is two-dimensional typical section • Three degrees of freedom – Torsion – α – Aileron deflection – β – Vertical deflection (flexure) – ℎ • Wing motions are sinusoidal and infinitesimal • Wing has no internal or solid friction, resulting in no internal damping

18 Assumptions Made in NACA 496

Assumption removed in NACA 685 and NACA 741

19 Equations of Motion Collected

20 Equations of Motion Collected

(A) = Sum of moments about the elastic axis

21 Equations of Motion Collected

(A) = Sum of moments about the elastic axis (B) = Sum of moments about the aileron hinge

22 Equations of Motion Collected

(A) = Sum of moments about the elastic axis (B) = Sum of moments about the aileron hinge (C) = Sum of forces in the vertical direction

23 Equations of Motion Collected

24 Equations of Motion Collected

25 Equations of Motion Collected

Steps taken to obtain “final” equations of motion: 푣 푖푘 푡 (1) Make substitutions 훼 = 훼0푒 푏 푣 푖(푘 푡+휑1) 훽 = 훽0푒 푏 푣 푖(푘 푡+휑2) ℎ = ℎ0푒 푏 26 Equations of Motion Collected

Steps taken to obtain “final” equations of motion: 푣 푖푘 푡 푣 (1) Make substitutions 훼 = 훼0푒 푏 훼ሶ = 𝑖푘 훼 푣 푏 2 푖(푘 푡+휑1) 푣 훽 = 훽0푒 푏 훼ሷ = − 푘 훼 푣 푏 푖(푘 푡+휑2) ℎ = ℎ0푒 푏 etc. 27 Equations of Motion Collected

After substitutions all terms, except these terms, contain 푣2 Steps taken to obtain “final” equations of motion: 푣 푖푘 푡 푣 (1) Make substitutions 훼 = 훼0푒 푏 훼ሶ = 𝑖푘 훼 푣 푏 2 푖(푘 푡+휑1) 푣 훽 = 훽0푒 푏 훼ሷ = − 푘 훼 푣 푏 푖(푘 푡+휑2) ℎ = ℎ0푒 푏 etc. 28 Equations of Motion Collected

Steps taken to obtain “final” equations of motion: 푣 푖푘 푡 푣 (1) Make substitutions 훼 = 훼0푒 푏 훼ሶ = 𝑖푘 훼 푣 푏 2 푖(푘 푡+휑1) 푣 훽 = 훽0푒 푏 훼ሷ = − 푘 훼 푣 푏 푖(푘 푡+휑2) ℎ = ℎ0푒 푏 etc. 29 Equations of Motion Collected

Steps taken to obtain “final” equations of motion: 2 (2) Normalize all equations by 푣 푏푘 휅

30 Equations of Motion Collected

After normalization only these terms 1 contain Τ푣2 Steps taken to obtain “final” equations of motion: 2 (2) Normalize all equations by 푣 푏푘 휅

31 Final 3DOF Equations of Motion

(A) (퐴푎훼 + Ω훼푋)훼 + 퐴푎훽훽 + 퐴푎ℎℎ = 0

(B) 퐴푏훼훼 + (퐴푏훽 + Ω훽푋)훽 + 퐴푏ℎℎ = 0

where 퐴푎훼 = 푅푎훼 + 𝑖퐼푎훼 , etc.

32 Final 3DOF Equations of Motion

(A) (퐴푎훼 + Ω훼푋)훼 + 퐴푎훽훽 + 퐴푎ℎℎ = 0

(B) 퐴푏훼훼 + (퐴푏훽 + Ω훽푋)훽 + 퐴푏ℎℎ = 0

2 2 휔훼푟훼 1 푏휔푟푟푟 휔푟푟푟 휅 푣푘

Wa 푋 33 Final 3DOF Equations of Motion

(A) (퐴푎훼 + Ω훼푋)훼 + 퐴푎훽훽 + 퐴푎ℎℎ = 0

(B) 퐴푏훼훼 + (퐴푏훽 + Ω훽푋)훽 + 퐴푏ℎℎ = 0

2 2 휔훼푟훼 1 푏휔푟푟푟 Ω 푋 The s and are central to 휔 푟 휅 푣푘 Theodorsen’s solution methods 푟 푟

Wa 푋 34 Final 3DOF Equations of Motion

-- with addition of structural damping terms --

(A) (퐴푎훼 + Ω훼(1 + 𝑖𝑔훼)푋)훼 + 퐴푎훽훽 + 퐴푎ℎℎ = 0

(B) 퐴푏훼훼 + (퐴푏훽 + Ω훽(1 + 𝑖𝑔훽)푋)훽 + 퐴푏ℎℎ = 0

35 Final 3DOF Equations of Motion

-- with addition of structural damping terms --

(A) (퐴푎훼 + Ω훼(1 + 𝑖𝑔훼)푋)훼 + 퐴푎훽훽 + 퐴푎ℎℎ = 0

(B) 퐴푏훼훼 + (퐴푏훽 + Ω훽(1 + 𝑖𝑔훽)푋)훽 + 퐴푏ℎℎ = 0

For all equations of motion, 2DOF and 3DOF, solution is obtained when their determinant is zero

36 Outline

• Background and Purpose • Brief History • Equations of Motion • Solution Methods • Re-Computations and Comparisons • Concluding Remarks

37 Solution Methods

Solution Method 1 Solution Method 2 Solution Method 3

Employed in NACA 496 Employed in NACA 685 Employed in NACA 741 2DOF only 2DOF or 3DOF 2DOF or 3DOF No 𝑔, allows 휉 Allows 𝑔 and 휉 Allows 𝑔 and 휉

38 Solution Methods

Solution Method 1 Solution Method 2 Solution Method 3

Employed in NACA 496 Employed in NACA 685 Employed in NACA 741 2DOF only 2DOF or 3DOF 2DOF or 3DOF No 𝑔, allows 휉 Allows 𝑔 and 휉 Allows 𝑔 and 휉 Expand complex determinant Separate into real and imaginary equations; set both to zero

39 Solution Methods

Solution Method 1 Solution Method 2 Solution Method 3

2DOF Example – Torsion-Aileron (훼, 훽)

퐴푎훼 + 훺훼 푋 퐴푎훽 = 0 where 퐴푖푗 = 푅푖푗 + 𝑖퐼푖푗 퐴푏훼 퐴푏훽 + 훺훽 푋

Real equation 2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Imaginary equation

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0 40 Solution Methods

Solution Method 1 Solution Method 2 Solution Method 3

2DOF Example – Torsion-Aileron (훼, 훽)

Solution퐴푎훼 + 훺is훼 obtained푋 퐴푎 훽when real and imaginary equations = 0 where 퐴푖푗 = 푅푖푗 + 𝑖퐼푖푗 are퐴 both푏훼 satisfied퐴푏훽 + 훺for훽 푋 the same values of 푋 and 푘

Real equation 2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Imaginary equation

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0 41 Solution Methods

Solution Method 1 Solution Method 2 Solution Method 3

2DOF Example – Torsion-Aileron (훼, 훽)

Solution퐴푎훼 + 훺is훼 obtained푋 퐴푎 훽when real and imaginary equations = 0 where 퐴푖푗 = 푅푖푗 + 𝑖퐼푖푗 퐴푏훼 퐴푏훽 + 훺훽 푋 Do are both satisfied for the same values of 푋 and 푘 this for Real equation 2 many Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0 values of 푘 Imaginary equation

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0 42 Solution Methods

Solution Method 1 Solution Method 2 Solution Method 3

Employed in NACA 496 Employed in NACA 685 Employed in NACA 741 2DOF only 2DOF or 3DOF Straightforward2DOF or 3DOF No 𝑔, allows 휉 Allows 𝑔 and 휉 Allows 𝑔 and 휉 Expand complex determinant Separate into real and imaginary equations; set both to zero

2DOF Example – Torsion-Aileron (훼, 훽)

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0 43 Solution Methods

Solution Method 1 Solution Method 2 Solution Method 3

Employed in NACA 496 Employed in NACA 685 Employed in NACA 741 2DOF only Ingenious, 2DOF or 3DOF Straightforward2DOF or 3DOF No 𝑔, allows 휉 but complicatedAllows 𝑔 and 휉 Allows 𝑔 and 휉 Expand complex determinant Separate into real and imaginary equations; set both to zero

2DOF Example – Torsion-Aileron (훼, 훽)

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0 44 Solution Methods

2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0

Solution Method 1

• Treat Ω훼 and 푋 as parameters • Solve 2 equations in 2 unknowns, Ω훼 and 푋

45 Solution Methods

2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0 휴휶 vs 1/풌 Solution Method 1

• Treat Ω훼 and 푋 as parameters • Solve 2 equations in 2 unknowns, Ω훼 and 푋

푭 vs 휴휶

46 Solution Methods

2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0 휴휶 vs 1/풌 Identify 1 푘 = Solution Method 1 problem-specific 푓 1 ൗ푘 value of 휴휶 푓 • Treat Ω훼 and 푋 as parameters • Solve 2 equations in 2 unknowns, Ω훼 and 푋

ퟐ 흎휶풓휶 푭 vs 휴휶 휴휶 = 흎휷풓휷 푏푟훽휔훽 푣 = 퐹 푓 휅

Identify problem-specific value of 휴휶

47 Solution Methods

2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0

Solution Method 1

• Treat Ω훼 and 푋 as parameters • Solve 2 equations in 2 unknowns, Ω훼 and 푋

Solution Method 2

• Define Ω훼 and Ω훽 • Treat 푋 as a parameter • Solve polynomial equations for 푋1 and 푋2

48 Solution Methods

2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0

Solution Method 1

• Treat Ω훼 and 푋 as parameters • Solve 2 equations in 2 unknowns, Ω훼 and 푋

Solution Method 2

• Define Ω훼 and Ω훽 • Treat 푋 as a parameter • Solve polynomial equations for 푋1 and 푋2

49 Solution Methods

2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0

Solution Method 1

• Treat Ω훼 and 푋 as parameters • Solve 2 equations in 2 unknowns, Ω훼 and 푋

Solution Method 2

• Define Ω훼 and Ω훽 • Treat 푋 as a parameter • Solve polynomial equations for 푋1 and 푋2

Solution Method 3 • Define Ω and Ω 훼 훽 푎 푋 + 푎 = 0 • Treat 푋 as a parameter 1 0 푏 푋 + 푏 = 0 • Employ method of elimination 1 0 • Solve linear equations for 푋1 and 푋2 50 Solution Methods

2 Ω훼Ω훽푋 + Ω훼푅푏훽 + Ω훽푅푎훼 푋 + 푅푎훼푅푏훽 − 퐼푎훼퐼푏훽 − 푅푎훽푅푏훼 + 퐼푎훽퐼푏훼 = 0

Ω훼퐼푏훽 + Ω훽퐼푎훼 푋 + 푅푎훼퐼푏훽 + 푅푏훽퐼푎훼 − 푅푎훽퐼푏훼 − 퐼푎훽푅푏훼 = 0

Solution Method 1

• Treat Ω훼 and 푋 as parameters • Solve 2 equations in 2 unknowns, Ω훼 and 푋

Solution Method 2

• Define Ω훼 and Ω훽 • Treat 푋 as a parameter • Solve polynomial equations for 푋1 and 푋2

Solution Method 3 • Define Ω and Ω 훼 훽 푎 푋 + 푎 = 0 • Treat 푋 as a parameter 1 0 푏 푋 + 푏 = 0 • Employ method of elimination 1 0 • Solve linear equations for 푋1 and 푋2 51 Outline

• Background and Purpose • Brief History • Equations of Motion • Solution Methods • Re-Computations and Comparisons • Concluding Remarks

52 Comparisons for Solution Method 1 From NACA 496; Case 1

Effect of 휔ℎ on Flutter Velocity, 푣 휔훼

53 Comparisons for Solution Method 1 From NACA 496; Case 1

Effect of 휔ℎ on Flutter Velocity, 푣 휔훼

54 Comparisons for Solution Method 1 From NACA 496; Case 1

Effect of 푥훼 on 퐹

55 Comparisons for Solution Method 1 From NACA 496; Case 1

Effect of 푥훼 on 퐹

56 Comparisons for Solution Method 1 From NACA 496; Case 1

Effect of 푥훼 on 퐹

57 Comparisons for Solution Method 1 From NACA 496; Case 1

Effect of 푥훼 on 퐹

58 Comparisons for Solution Method 2 From NACA 685; 2DOF Case 1 Case 2

Single flutter mode

59 Comparisons for Solution Method 2 From NACA 685; Case 1

Effect of 휔ℎ on 푣 for various 𝑔 and 𝑔 휔훼 푏휔훼 ℎ 훼

60 Comparisons for Solution Method 2 From NACA 685; Case 2

휔 Effect of 훽 on 푣 for various quantities 휔ℎ 푏휔ℎ

61 Comparisons for Solution Method 2 From NACA 685; Case 2

휔 Effect of 훽 on 푣 for various quantities 휔ℎ 푏휔ℎ In parts (b) and (d) ퟏ ퟐ ퟏ 풙휷 = ퟔퟎ 풓휷 = ퟏퟐퟎ

62 Comparisons for Solution Method 3 From NACA 741; Case 2

휔 Effect of 훽 on 푣 for various 푥 휔훼 푏휔푎 훽

63 Comparisons for Solution Method 3 From NACA 741; Case 2

휔 Effect of 훽 on 푣 for various 푥 휔훼 푏휔푎 훽

64 Outline

• Background and Purpose • Brief History • Equations of Motion • Solution Methods • Re-Computations and Comparisons • Concluding Remarks

65 Concluding Remarks

• In an AIAA Engineering Note Thomas A. Zeiler – • Made known that numerical errors exist in three foundational reports on aeroelastic flutter and on early aeroelasticity texts • Recommended that all of the plots in NACA 496, NACA 685, and NACA 741 be re-computed and published • Current work is following Zeiler’s recommendation by – • Re-computing and checking all numerical examples in these foundational reports • Comparing original and re-computed results • Publishing and making known the existence of the re-computations • This paper has presented – • Theodorsen’s and Garrick’s equations and solution methods • Representative examples of re-computations and comparisons • Overall good agreement between original and re-computed results (with some notable discrepancies) 66