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Nonlinear analysis of the dynamics of articulated composite rotor blades

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Abstract

The general nonlinear intrinsic equations of motion of an elastic composite beam are solved in order to obtain the elasto-dynamic response of a rotating articulated blade. The solution utilizes the linear Variational-Asymptotic Method (VAM) cross-sectional analysis, together with an improved damped nonlinear model for the rigid-body motion analysis of helicopter blades in coupled flap and lead-lag motions. The explicit (direct) integration algorithm implements the perturbation method in order to solve the transient form of the nonlinear intrinsic differential equations of motion and obtain the elasto-dynamic behavior of an accelerating composite blade. The specific problem considered is an accelerating articulated helicopter blade of which its motion is analyzed since it starts rotating from rest until it reaches the steady-state condition. It is observed that the steady-state solution obtained by this method compares very well with other available solutions. The resulting simulation code is a powerful tool for analyzing the nonlinear response of composite rotor blades; and for serving the ultimate aim of efficient noise and vibration control in helicopters.

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Abbreviations

A :

cross-sectional area of the undeformed beam in x 2–x 3 plane

e 1 :

\([\begin{array}{c@{\quad }c@{\quad }c} 1 & 0 & 0 \end{array}]^{T}\)

F i :

elements of the column matrix of internal forces

f :

applied forces per unit length

g :

determinant of the metric tensor in curvilinear coordinates

H :

sectional angular momenta

i 2, i 3 :

cross-sectional mass moment

i 23 :

cross-sectional product of inertia

K.E.:

kinetic energy function

K :

deformed beam curvature vector = \(k + \bar{\boldsymbol{\kappa}}\)

k :

undeformed beam curvature vector

l :

length of the beam

M i :

elements of the column matrix of internal moments

 m :

applied moments per unit length

N :

number of nodes

P :

sectional linear momenta

S :

stiffness matrix

t :

time

V :

velocity field

x i :

global system of coordinates

x 1 :

axis along the beam

x 2 and x 3 :

cross-sectional axes

\(\bar{x}_{2}\) and \(\bar{x}_{3} \) :

offsets from the reference line of the cross-sectional mass center

γ :

\([ \begin{array}{c@{\quad }c@{\quad }c} \gamma_{11} & 2\gamma_{12} &2\gamma_{13} \end{array} ]^{T}\)

Δ:

identity matrix

\(\delta\bar{q} \) :

virtual displacement vector

\(\delta\bar{\psi} \) :

virtual rotation vector

κ 1 :

elastic twist

κ i :

elastic bending curvatures (i=2,3)

μ :

mass per unit length

ρ :

mass density

Ω :

angular velocity

:

perturbations in space

:

perturbations in time

(•)′:

\(\frac{\partial ( \bullet )}{\partial x_{1}}\)

\(( \dot{ \bullet} )\) :

\(\frac{d( \bullet )}{dt}\)

\(( \delta\bar{ \bullet} )\) :

overbar indicates that it need not be the variation of a functional

\(( \hat{ \bullet} ) \) :

the discrete boundary value of quantity (•)

\(( \tilde{ \bullet} )_{ij}\) :

−e ijk (•) k

〈•〉:

∫ A (•) dx 2  dx 3

〈〈•〉〉:

\(\langle(\bullet)\sqrt{g} \rangle =\int_{A} ( \bullet)\sqrt{g} \,dx_{2}\,dx_{3}\), \(\sqrt{g} = 1 - x_{2}k_{3} -x_{3}k_{2}\)

References

  1. Berdichevsky, V.L.: On the energy of an elastic rod. Prikl. Mat. Meh. 45, 518–529 (1981)

    Google Scholar 

  2. Berdichevsky, V.L., Armanios, E.A., Badir, A.M.: Theory of anisotropic thin-walled closed cross section beams. Compos. Eng. 2, 411–432 (1992)

    Article  Google Scholar 

  3. Cesnik, C.E.S., Ortega-Morales, M.: Active composite beam cross-sectional modeling—stiffness and active force constraints. In: 40th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, St. Louis, MO, 12–15 April (1999). Paper No.: AIAA-99-1548

    Google Scholar 

  4. Cesnik, C.E.S., Ortega-Morales, M.: Active beam cross-sectional modeling. J. Intell. Mater. Syst. Struct. 12, 483–496 (2001)

    Article  Google Scholar 

  5. Cesnik, C.E.S., Park, R.S., Palacios, R.: Effective cross-section distribution of anisotropic piezocomposite actuators for wing twist. Proc. SPIE 5056, 21–32 (2003)

    Article  Google Scholar 

  6. Cesnik, C.E.S., Shin, S.J.: Structural analysis for designing rotor blades with integral actuators. In: 39th AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, Long Beach, CA, 20–23 April (1998). Paper No.: AIAA-98-2107

    Google Scholar 

  7. Done, G., Balmford, D.: Bramwell’s Helicopter Dynamics, 2nd edn. AIAA, Washington (2001)

    Google Scholar 

  8. Esmailzadeh, E., Ghorashi, M.: Vibration analysis of a Timoshenko beam subjected to a traveling mass. J. Sound Vib. 199(4), 615–628 (1997)

    Article  Google Scholar 

  9. Ghorashi, M.: Dynamic and vibratory analysis of beams under dynamic loads induced by traveling masses and vehicles. Ph.D. thesis, Mechanical Engineering Department, Sharif University of Technology (1994)

  10. Ghorashi, M., Nitzsche, F.: Steady state nonlinear dynamic response of a composite rotor blade using implicit integration of intrinsic equations of a beam. Int. Rev. Aerosp. Eng. 1, 225–233 (2008)

    Google Scholar 

  11. Ghorashi, M.: Dynamics of elastic nonlinear rotating composite beams with embedded actuators. Ph.D. thesis, Mechanical and Aerospace Engineering Department, Carleton University (2009)

  12. Ghorashi, M., Nitzsche, F.: Nonlinear dynamic response of an accelerating composite rotor blade using perturbations. J. Mech. Mater. Struct. 4, 693–718 (2009)

    Article  Google Scholar 

  13. Hodges, D.H., Atilgan, A.R., Cesnik, C.E.S., Fulton, M.V.: On a simplified strain energy function for geometrically nonlinear behavior of anisotropic beams. Compos. Eng. 2, 513–526 (1992)

    Article  Google Scholar 

  14. Hodges, D.H., Shang, X., Cesnik, C.E.S.: Finite element solution of nonlinear intrinsic equations for curved composite beams. J. Am. Helicopter Soc. 41, 313–321 (1996)

    Article  Google Scholar 

  15. Hodges, D.H.: Nonlinear Composite Beam Theory. AIAA, Washington (2006)

    Google Scholar 

  16. Hopkins, A.S., Ormiston, R.A.: An examination of selected problems in rotor blade structural mechanics and dynamics. In: Proceedings of the 59th American Helicopter Society Forum, Phoenix, AZ (2003)

    Google Scholar 

  17. Johnson, W.: Helicopter Theory. Dover, New York (1994)

    Google Scholar 

  18. Leishman, J.G.: Principles of Helicopter Aerodynamics, 2nd edn. Cambridge University Press, New York (2006)

    Google Scholar 

  19. Shang, X., Hodges, D.H.: Aeroelastic stability of composite rotor blades in hover. In: 36th Structures, Structural Dynamics and Materials Conference, New Orleans (1995). Paper No: AIAA-95-1453-CP

    Google Scholar 

  20. Sharpe, D.L.: An experimental investigation of the flap-lag-torsion aeroelastic stability of a small-scale hingeless helicopter rotor in hover. NASA, TP-2546 (1986)

  21. Yu, W., Volovoi, V.V., Hodges, D.H., Hong, X.: Validation of the variational asymptotic beam sectional analysis. AIAA J. 40, 2105–2112 (2002)

    Article  Google Scholar 

  22. Yu, W., Hodges, D.H.: Elasticity solutions versus asymptotic sectional analysis of homogeneous, isotropic, prismatic beams. J. Appl. Mech. 71, 15–23 (2004)

    Article  MATH  Google Scholar 

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  1. Department of Engineering, University of Southern Maine, Gorham, ME, USA

    Mehrdaad Ghorashi

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  1. Mehrdaad Ghorashi
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Ghorashi, M. Nonlinear analysis of the dynamics of articulated composite rotor blades. Nonlinear Dyn 67, 227–249 (2012). https://doi.org/10.1007/s11071-011-9974-9

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