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Nonlinear aeroelasticity of high-aspect-ratio wings with laminated composite spar

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Abstract

The aeroelastic instability of a high-aspect-ratio wing including anisotropic composite wing–spar in an incompressible flow is investigated. Combining a nonlinear Euler–Bernoulli beam theory and a composite laminate theory, the third-order expansion of nonlinear structural equations of motion and associated boundary conditions are obtained for vertical, forward/afterward, and torsional motion of the high-aspect-ratio composite wing undergoing large deformations and small strains, neglecting warping, shear deformation, and small Poisson effects. The unsteady aerodynamic strip theory based on Wagner’s function is used for determining the aerodynamic loading of the wing. Combining these two sets of equations gives a set of nonlinear integro-differential aeroelastic equations of motion. The governing partial differential equations are discretized using Galerkin’s method, and the obtained equations are solved with a numerical method with no need to add any aerodynamic state-space degrees of freedom. Some test cases are analyzed and the results are evaluated based on the results given in other references. Also, a study is conducted to show the effects of fiber orientation variations on nonlinear aeroelastic instability speed and nonlinear aeroelastic instability frequency of the composite wing. The study shows that fiber orientation strongly affects the aeroelastic characteristics of a non-isotropic wing where the aeroelastic instability speed dominantly decreases for the fiber orientation between − 90° and  − 45° or 0° and  + 45°. It is also shown that in the fiber orientation between − 45° and  + 45°, the bending–bending stiffness and in the other fiber orientation, the bending–torsion coupling stiffness have a significant role in decreasing or increasing the nonlinear instability speed of the wing.

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Abbreviations

\((e_{X} ,e_{Y} ,e_{Z} )\) :

Global coordinates system

\((e_{x} ,e_{y} ,e_{z} )\)::

Local inertial coordinates system

\((e_{\xi } ,e_{\eta } ,e_{\zeta } )\) :

Local curvilinear coordinates system

\(x,y,z\) :

Undeformed coordinates system

\(\xi ,\eta ,\zeta\) :

Deformed coordinates system

\(\rho_{\xi } ,\rho_{\eta } ,\rho_{\zeta }\) :

Curvature vector components

\(\omega_{\xi } ,\omega_{\eta } ,\omega_{\zeta }\) :

Angular velocity vector components

\(D,L,M_{e.a}\) :

Drag, lift, and pitching moment distribution about elastic axis

\(Q_{u} ,\,Q_{v} ,\,Q_{w} ,Q_{\theta }\) :

Generalized force components

\(u,v,w,\theta\) :

Degrees of freedom

K, U :

Kinetic, potential energy

W :

Work done by non-conservative forces

\(\overrightarrow {V}\) :

Velocity vector

\(EI,GJ\) :

Bending, torsional rigidity

\(a\) :

Distance coefficient of mid-chord to elastic axis

l, c, b :

Length, chord, and half chord of wing

\(h_{\left( k \right)}\) :

Layer thickness

m :

Layer number

\(\overline{m}\) :

Mass per unit length

\(\rho_{m}\) :

Mass density

\(U,\rho_{air}\) :

Airspeed and density of free stream

\(\rho_{m}\) :

Mass density

U1-LCO, U2-LCO :

First and second LCO boundary speed

\(w_{0}\) :

Initial displacement of wing tip in w direction

\(\overline{Q}_{ij}\) :

Reduced stiffness matrix

\(\nu_{ij}\) :

Poisson ratio

\(\sigma_{ij} ,\varepsilon_{ij}\) :

Stress, strain tensor

\(\psi ,\alpha ,\theta\) :

Euler angles

\(\theta_{L}\) :

Ply angle in lamina

\(\varphi \left( t \right)\) :

Wagner’s function

\(\left[ D \right]\) :

Bending stiffness matrix

\([T]\) :

Transformation matrix

References

  1. Bisplinghoff RL, Asheley H, Halfman RL (1955) Aeroelasticity. Addison-Wesley, Cambridge, Chaps. 2 and 3

  2. Hollowell SJ, Dugundji J (1984) Aeroelastic flutter and divergence of stiffness coupled graphite/epoxy cantilevered plates. J Aircr 21(1):69–76

    Article  Google Scholar 

  3. Weisshaar TA, Foist BL (1985) Vibration tailoring of advanced composite lifting surfaces. J Aircr 22:141–147

    Article  Google Scholar 

  4. Chandra R, Stemple AD, Chopra I (1990) Thin-walled composite beams under bending, torsional, and extensional loads. J Aircr 27:619–626

    Article  Google Scholar 

  5. Banerjee JR, Williams FW (1995) Free vibration of composite beams: an exact method using symbolic computation. J Aircr 32(3):636–642

    Article  Google Scholar 

  6. Crawley EF, Dugundji J (1980) Frequency determination and nondimensionalization for composite cantilever plates. J Sound Vib 72(1):1–10

    Article  Google Scholar 

  7. Lottati I (1985) Flutter and divergence aeroelastic characteristics for composite forward swept cantilevered wing. J Aircr 22(11):1–10

    Article  Google Scholar 

  8. Librescu L, Khdeir AA (1988) Aeroelastic divergence of swept-forward composite wings including warping restraint effect. AIAA J 26(11):1373–1377

    Article  Google Scholar 

  9. Librescu L, Song O (1992) On the static aeroelastic tailoring of composite aircraft swept wings modeled as thin-walled beams structures. Compos Eng 2(5–7):497–512

    Article  Google Scholar 

  10. Volovoi VV, Hodges DH (2002) Single- and multicelled composite thin-walled beams. AIAA J 40(5):960–965

    Article  Google Scholar 

  11. Volovoi VV, Hodges DH (2000) Theory of anisotropic thin-walled beams. J Appl Mech 67(3):453–459

    Article  Google Scholar 

  12. Yu W, Volovoi VV, Hodges DH, Hong X (2002) Validation of the variational asymptotic beam sectional analysis. AIAA J 40(10):2105–2112

    Article  Google Scholar 

  13. Yu W, Hodges DH, Volovoi VV, Cesnik CES (2002) On Timoshenko-like modeling of initially curved and twisted composite beams. Int J Solids Struct 39:5101–5121

    Article  Google Scholar 

  14. Patil MJ, Hodges DH (2004) On the importance of aerodynamic and structural geometrical nonlinearities in aeroelastic behavior of high-aspect-ratio wings. J Fluids Struct 19:905–915

    Article  Google Scholar 

  15. da Silva MRMC (1988) Non-linear flexural-flexural-torsional-extensional dynamics of beams, I-formulation: response analysis. Int J Solids Struct 24(12):1225–1234

    Article  Google Scholar 

  16. Pai PF, Nayfeh AH (1990) Three-dimensional nonlinear vibrations of composite beams, I: equations of motion. Nonlinear Dyn 1:477–502

    Article  Google Scholar 

  17. Pai PF, Nayfeh AH (1991) Three-dimensional nonlinear vibrations of composite beams-II. Flapwise excitations. Nonlinear Dyn 2:1–34

    Article  Google Scholar 

  18. Pai PF, Nayfeh AH (1992) A nonlinear composite beam theory. Nonlinear Dyn 3:273–303

    Article  Google Scholar 

  19. Pai PF, Nayfeh AH (1994) A fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three-dimensional stress effects. Int J Solids Struct 31:1309–1340

    Article  Google Scholar 

  20. Dunn P, Dugundji J (1992) Nonlinear stall flutter and divergence analysis of cantilevered graphite/epoxy wings. AIAA J 30(1):153–162

    Article  Google Scholar 

  21. Patil MJ (1999) Nonlinear aeroelastic analysis, flight dynamics, and control of a complete aircraft. Ph.D thesis, Georgia Institute of Technology

  22. Patil MJ, Hodges DH, Cesnik CES (2000) Nonlinear aeroelastic analysis of complete aircraft in subsonic flow. J Aircr 37(5):753–760

    Article  Google Scholar 

  23. Patil MJ, Hodges DH, Cesnik CES (2001) Nonlinear aeroelasticity and flight dynamics of high-altitude long-endurance aircraft. J Aircr 38(1):88–94

    Article  Google Scholar 

  24. Patil MJ, Hodges DH, Cesnik CES (2001) Limit-cycle oscillations in high-aspect-ratio wings. J Fluids Struct 15(1):107–132

    Article  Google Scholar 

  25. Guo SJ, Bannerjee JR, Cheung CW (2003) The effect of laminate lay-up on the flutter speed of composite wings. J Aerosp Eng 217(3):115–122

    Google Scholar 

  26. Liu XN, Xiang JW (2006) Stall flutter analysis of high aspect-ratio composite wing. Chin J Aeronaut 19(1):36–43

    Article  Google Scholar 

  27. Xie CC, Leng JZ, Yang C (2008) Geometrical nonlinear aeroelastic stability analysis of a composite high-aspect-ratio wing. Shock Vib J 15:325–333

    Article  Google Scholar 

  28. Jian Z, Jinwu X (2009) Nonlinear aeroelastic response of high-aspect-ratio flexible wings. Chin J Aeronaut 22:355–363

    Article  Google Scholar 

  29. Petrolo M (2013) Flutter analysis of composite lifting surfaces by the 1D carrera unified formulation and the doublet lattice method. Compos Struct 95:539–546

    Article  Google Scholar 

  30. Stodieck O, Cooper JE, Paul M, Weaver PM, Kealy P (2013) Improved aeroelastic tailoring using tow-steered composites. Compos Struct 106:703–715

    Article  Google Scholar 

  31. Koohi R, Shahverdi H, Haddadpour H (2014) Nonlinear aeroelastic analysis of a composite wing by finite element method. Compos Struct 113:118–126

    Article  Google Scholar 

  32. Koohi R, Shahverdi H, Haddadpour H (2015) Modal and aeroelastic analysis of a high-aspect-ratio wing with large deflection capability. Int J Adv Des Manuf Technol 8(1):45–54

    Google Scholar 

  33. Shams Sh, Sadr MH, Haddadpour H (2012) An efficient method for nonlinear aeroelasticity of slender wings. Nonlinear Dyn J 67:659–681

    Article  MathSciNet  Google Scholar 

  34. Badiei D, Sadr MH, Shams Sh (2014) Static stall model in aeroelastic analysis of flexible wing with geometrical nonlinearity. J Aerosp Eng 27(2):378–389

    Article  Google Scholar 

  35. Georg Staab H (1999) Laminar composite, Butterworth-Heinemann, ISBN 0–7506–7124–6

  36. Meirovitch L (1967) Analytical methods in dynamics. Macmillan, New York

    MATH  Google Scholar 

  37. Jones RT (1940) The unsteady lift of a wing of finite aspect ratio, NACA report 681

  38. Shams Sh, Lahidjani MHS, Haddadpour H (2008) Nonlinear aeroelastic response of slender wings based on Wagner function. Thin-Walled Struct 46:1192–1203

    Article  Google Scholar 

  39. Gelfand IM, Fomin SV (1963) Calculus of variations. Prentice-Hall, INC, Englewood Cliffs

    MATH  Google Scholar 

  40. Courant R, Hilbert D (1953) Methods of Mathematical Physics, Vol. I (First English ed.). New York: Interscience Publishers Inc. ISBN 978–0471504474

  41. Taha HE, Hajj MR, Beran PS (2014) State-space representation of the unsteady aerodynamics of flapping flight. Aerosp Sci Technol. https://doi.org/10.1016/j.ast.2014.01.011

    Article  Google Scholar 

  42. dos Santos CR, Marques FD (2017) Lift prediction including stall, using vortex lattice method with Kirchhoff-based correction. J Aircr. https://doi.org/10.2514/1.C034451

    Article  Google Scholar 

  43. de Marqui Jr C, Tan D, Erturk A (2018) On the electrode segmentation for piezoelectric energy harvesting from nonlinear limit cycle oscillations in axial flow. J Fluids Struct. https://doi.org/10.1016/j.jfluidstructs.2018.07.020

    Article  Google Scholar 

  44. dos Santos CR, Pacheco DRQ, Taha HE, Zakaria MY (2021) Nonlinear modeling of electro-aeroelastic dynamics of composite beams with piezoelectric coupling. Compos Struct. https://doi.org/10.1016/j.compstruct.2020.112968

    Article  Google Scholar 

  45. Patil MJ, Hodges DH, Cesnik CES (2001) Limit cycle oscillations in high aspect ratio wings. J Fluids Struct 15:107–132

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, Faculty of New Sciences and Technologies, University of Tehran, North Kargar Street, Po Box: 14395-1561, 14399-55941, Tehran, Iran

    Sh. Shams

  2. Department of Aerospace Engineering, Amirkabir University of Technology, Hafez Avenue, Tehran, Iran

    M. H. Sadr & D. Badiei

Authors
  1. Sh. Shams
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  3. D. Badiei
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Correspondence to M. H. Sadr.

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Technical Editor: André Cavalieri.

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Shams, S., Sadr, M.H. & Badiei, D. Nonlinear aeroelasticity of high-aspect-ratio wings with laminated composite spar. J Braz. Soc. Mech. Sci. Eng. 43, 334 (2021). https://doi.org/10.1007/s40430-021-02993-8

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  • DOI: https://doi.org/10.1007/s40430-021-02993-8

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