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 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 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 • 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 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 = 푘 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 = 푘 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 = 푘 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 (C) 퐴푐훼훼 + 퐴푐훽훽 + (퐴푐ℎ + Ωℎ 푋)ℎ = 0 where 퐴푎훼 = 푅푎훼 + 퐼푎훼 , etc. 32 Final 3DOF Equations of Motion (A) (퐴푎훼 + Ω훼푋)훼 + 퐴푎훽훽 + 퐴푎ℎℎ = 0 (B) 퐴푏훼훼 + (퐴푏훽 + Ω훽푋)훽 + 퐴푏ℎℎ = 0 (C) 퐴푐훼훼 + 퐴푐훽훽 + (퐴푐ℎ + Ωℎ 푋)ℎ = 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 (C) 퐴푐훼훼 + 퐴푐훽훽 + (퐴푐ℎ + Ωℎ 푋)ℎ = 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 + 훼)푋)훼 + 퐴푎훽훽 + 퐴푎ℎℎ = 0 (B) 퐴푏훼훼 + (퐴푏훽 + Ω훽(1 + 훽)푋)훽 + 퐴푏ℎℎ = 0 (C) 퐴푐훼훼 + 퐴푐훽훽 + (퐴푐ℎ + Ωℎ (1 + ℎ)푋)ℎ = 0 35 Final 3DOF Equations of Motion -- with addition of structural damping terms -- (A) (퐴푎훼 + Ω훼(1 + 훼)푋)훼 + 퐴푎훽훽 + 퐴푎ℎℎ = 0 (B) 퐴푏훼훼 + (퐴푏훽 + Ω훽(1 + 훽)푋)훽 + 퐴푏ℎℎ = 0 (C) 퐴푐훼훼 + 퐴푐훽훽 + (퐴푐ℎ + Ωℎ (1 + ℎ)푋)ℎ = 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 , allows 휉 Allows and 휉 Allows 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 , allows 휉 Allows and 휉 Allows 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 Employed in NACA 496 Employed in NACA 685 Employed in NACA 741 2DOF only 2DOF or 3DOF 2DOF or 3DOF No , allows 휉 Allows and 휉 Allows and 휉 Expand complex determinant Separate into real and imaginary equations; set both to zero 2DOF Example – Torsion-Aileron (훼, 훽) 퐴푎훼 + 훺훼 푋 퐴푎훽 = 0 where 퐴푖푗 = 푅푖푗 + 퐼푖푗 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 Employed in NACA 496 Employed in NACA 685 Employed in NACA 741 2DOF only 2DOF or 3DOF 2DOF or 3DOF No , allows 휉 Allows and 휉 Allows and 휉 Expand complex determinant Separate into real and imaginary equations; set both to zero 2DOF Example – Torsion-Aileron (훼, 훽) Solution퐴푎훼 + 훺is훼 obtained푋 퐴푎 훽when real and imaginary equations =