Aeroelastic Stability Analysis of Curved Composite Blades in Hover Using Fully Intrinsic Equations

  • M. R. AmoozgarEmail author
  • H. Shahverdi
Original Paper


In this paper, the aeroelastic stability of a curved composite hingeless rotor blade in hovering condition is investigated. The composite blade is modeled using the geometrically exact fully intrinsic beam equations, and the aerodynamic loads are simulated by using the two-dimensional quasi-steady strip theory combined with the uniform inflow. The nonlinear governing equations are discretized using a time–space scheme, and the stability of the system is determined by inspecting the eigenvalues of the linearized system. The obtained results are compared with those reported in the literature, and an excellent agreement is observed. It is found that the ply angle of the composite layup and the blade precone angle affect the aeroelastic stability of the blade dramatically. Finally, the effect of initial curvatures on the aeroelastic stability of the composite blade is analyzed. The results highlight the importance of the initial curvature combined with ply angle on the aeroelastic stability of composite blades.


Aeroelastic stability Fully intrinsic equations Curved blade Composite blade Precone angle 



  1. 1.
    Johnson W (1980) Helicopter theory. Princeton University Press, PrincetonGoogle Scholar
  2. 2.
    Friedmann PP (1983) Formulation and solution of rotary-wing aeroelastic stability and response problems. Vertica 7(2):101–141Google Scholar
  3. 3.
    Friedmann PP (1990) Helicopter rotor dynamics and aeroelasticity: some key ideas and insights. Vertica 14(1):101–121Google Scholar
  4. 4.
    Hodges DH (1985) Nonlinear equations for the dynamics of pretwisted beams undergoing small strains and large rotations. NASA TP-2470Google Scholar
  5. 5.
    Fulton MV, Hodges DH (1993) Aeroelastic stability of composite hingeless rotor blades in hover—Part II: results. Math Comput Model 18(3):19–35CrossRefGoogle Scholar
  6. 6.
    Cho MH, Lee I (1994) Aeroelastic stability of hingeless rotor blade in hover using large deflection theory. AIAA J 32(7):1472–1477CrossRefzbMATHGoogle Scholar
  7. 7.
    Hodges DH (1990) A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. J Solids Struct 26:1253–1273MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hodges DH (2003) Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams. AIAA J 41(6):1131–1137CrossRefGoogle Scholar
  9. 9.
    Althoff M, Patil MJ, Traugott JP (2012) Nonlinear modeling and control design of active helicopter blades. J Am Helicopter Soc 57(1):1–11CrossRefGoogle Scholar
  10. 10.
    Patil MJ, Hodges DH (2006) Flight dynamics of highly flexible flying wings. J Aircr 43(6):1790–1799CrossRefGoogle Scholar
  11. 11.
    Chang C-S, Hodges DH, Patil MJ (2008) Flight dynamics of highly flexible aircraft. J Aircr 45:538–545CrossRefGoogle Scholar
  12. 12.
    Leamy MJ (2007) Bulk dynamic response modeling of carbon nanotubes using an intrinsic finite element formulation incorporating interatomic potentials. Int J Solids Struct 44(3–4):874–894CrossRefzbMATHGoogle Scholar
  13. 13.
    Chang CS, Hodges DH (2007) Parametric studies on ground vibration test modeling for highly flexible aircraft. AIAA J 44(6):2049–2059Google Scholar
  14. 14.
    Chang CS, Hodges DH (2009) Vibration characteristics of curved beams. J Mater Struct 4(2):675–692CrossRefGoogle Scholar
  15. 15.
    Sotoudeh Z, Hodges DH (2011) Modeling beams with various boundary conditions using fully intrinsic equations. J Appl Mech 78(3):031010-1–031010-9CrossRefGoogle Scholar
  16. 16.
    Patil MJ, Althoff M (2011) Energy-consistent, Galerkin approach for the nonlinear dynamics of beams using intrinsic equations. J Vib Control 17(11):1748–1758MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Patil MJ, Hodges DH (2011) Variable-order finite elements for nonlinear, fully intrinsic beam equations. J Mech Mater Struct 6(1–4):479–493CrossRefGoogle Scholar
  18. 18.
    Mardanpour P, Hodges DH, Neuhart R, Graybeal N (2013) Engine placement effect on nonlinear trim and stability of flying wing aircraft. J Aircr 50(6):1716–1725CrossRefGoogle Scholar
  19. 19.
    Bekhoucha F, Rechak S, Duigou L, Cadou JM (2013) Nonlinear forced vibration of rotating anisotropic beams. Nonlinear Dyn 74(4):1281–1296MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mardanpour P, Hodges DH (2015) On the importance of nonlinear aeroelasticity and energy efficiency in design of flying wing aircraft. Adv Aerosp Eng 2015:1–11CrossRefGoogle Scholar
  21. 21.
    Khaneh Masjedi P, Ovesy HR (2015) Chebyshev collocation method for static intrinsic equations of geometrically exact beams. Int J Solids Struct 54:183–191CrossRefzbMATHGoogle Scholar
  22. 22.
    Khaneh Masjedi P, Ovesy H (2014) Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations. Acta Mech 226(6):1689–1706MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Richards PW, Tao Y, Herd RA, Hodges DH (2016) Effect of inertial and constitutive properties on body-freedom flutter for flying wings. J Aircr 53(3):756–767CrossRefGoogle Scholar
  24. 24.
    Amoozgar MR, Shahverdi H (2016) Analysis of nonlinear fully intrinsic equations of geometrically exact beams using generalized differential quadrature method. Acta Mech 227(5):1265–1277MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sotoudeh Z, Hosking NS (2017) Stability analysis of columns with imperfection. AIAA J 55(4):1417–1424CrossRefGoogle Scholar
  26. 26.
    Amoozgar MR, Shahverdi H, Nobari AS (2017) Aeroelastic stability of hingeless rotor blades in hover using fully intrinsic equations. AIAA J 55(7):2450–2460CrossRefGoogle Scholar
  27. 27.
    Yuan KA, Friedmann PP (1995) Aeroelasticity and structural optimization of composite helicopter rotor blades with swept tips. NASA Report 4665Google Scholar
  28. 28.
    Friedmann PP (1990) Helicopter rotor dynamics and aeroelasticity: some key ideas and insights. Vertica 14(1):101–121Google Scholar
  29. 29.
    Hodges DH (1990) Review of composite rotor blade modeling. AIAA J 28(3):561–564CrossRefGoogle Scholar
  30. 30.
    Hong CH, Chopra I (1985) Aeroelastic stability analysis of a composite rotor blade. J Am Helicopter Soc 30(2):57–67CrossRefGoogle Scholar
  31. 31.
    Panda B, Chopra I (1987) Dynamics of composite rotor blades in forward flight. Vertica 11(1/2):187–209Google Scholar
  32. 32.
    Smith ED, Chopra I (1993) Aeroelastic response, loads, and stability of a composite rotor in forward flight. AIAA J 31(7):1265–1273CrossRefGoogle Scholar
  33. 33.
    Yuan KA, Friedmann PP, Venkatesan C (1992) A new aeroelastic model for composite rotor blades with straight and swept tips. In: Proceedings of the 33rd structures, structural dynamics, and materials conference, Paper No. 92-2259-CP, DallasGoogle Scholar
  34. 34.
    Fulton MV, Hodges DH (1993) Aeroelastic stability of composite hingeless rotor blades in hover-part II: results. Math Comput Model 18(3–4):19–35CrossRefGoogle Scholar
  35. 35.
    Lim IG, Lee I (2009) Aeroelastic analysis of bearingless rotors with a composite flexbeam. Compos Struct 88:570–578CrossRefGoogle Scholar
  36. 36.
    Shang X, Hodges DD, Peters DA (1999) Aeroelastic stability of composite hingeless rotors in hover with finite-state unsteady aerodynamics. J Am Helicopter Soc 44(3):206–221CrossRefGoogle Scholar
  37. 37.
    Amoozgar M, Shahverdi H (2016) Dynamic instability of beams under tip follower forces using geometrically exact, fully intrinsic equations. Lat Am J Solids Struct 13:3022–3038CrossRefzbMATHGoogle Scholar
  38. 38.
    Amoozgar MR, Shaw AD, Zhang J, Friswell MI (2018) Composite blade twist modification by using a moving mass and stiffness tailoring. AIAA J. Google Scholar
  39. 39.
    Greenberg JM (1947) Airfoil in sinusoidal motion in a pulsating stream. NACA Tech. Report TN 1326Google Scholar
  40. 40.
    Hodges DH, Ormiston RA (1976) Stability of elastic bending and torsion of uniform cantilever rotor blades in hover with variable structural coupling. NASA TN D-8192Google Scholar
  41. 41.
    Hodges DH, Atilgan AR, Fulton MV, Rehfield LW (1991) Free-vibration analysis of composite beams. J Am Helicopter Soc 36(3):36–47CrossRefGoogle Scholar
  42. 42.
    De Rosa MA, Franciosi C (2000) Exact and approximate dynamic analysis of circular arches using DQM. Int J Solids Struct 37:1103–1117CrossRefzbMATHGoogle Scholar

Copyright information

© The Korean Society for Aeronautical & Space Sciences 2019

Authors and Affiliations

  1. 1.Swansea UniversitySwanseaUK
  2. 2.Department of Aerospace Engineering, Centre of Excellent in Computational Aerospace EngineeringAmirkabir University of TechnologyTehranIran

Personalised recommendations