A flexible rod finite element with separate storage of cumulated and extra rotations for large displacements of aircraft structural parts modeling

Deformable body mechanics


Аuthors

Popov V. V.1*, Sorokin F. D.1**, Ivannikov V. V.2***

1. Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia
2. Scientific and technical centre of rotor dynamic «Alfa-Tranzit», 1, Leningradskaya str., Khimky, Moscow region, 141400, Russia

*e-mail: vvpopov@bmstu.ru
**e-mail: sorokinfd@bmstu.ru
***e-mail: vvivannikov@alfatran.com

Abstract

A great number of aircrafts parts, modeled numerically, can be treated as a flexible rod, which exhibits large displacements and rotations, but which strains remain small. Spars wing panels, fuselage stringers, screw blades are among these parts.

The paper presents derivation of tangent stiffness matrix and deformation loads vector of a geometrically nonlinear flexible rod finite element with large rotations and small increments stored separately. The tangent stiffness matrix and the deformation loads vector are both written in the closed forms and can be programmed relatively easily. The element is derived with conventional finite rotations theory, based on the Euler vector and rotation tensor. Following the Updated Lagrangian formulation, the element rotations are decomposed into to the accumulated part and the small incremental one. This approach allows avoid possible singularities and reach rotation angle magnitudes as large as 2p and even more.

The proposed algorithms for the stiffness matrix and deformation loads vector computation are verified by some classical benchmarks. The thorough comparison of the obtained numerical results with existing publications, presenting alternative nonlinear beam formulations, reveals the high accuracy and numerical stability of the developed finite element.

It is worth noting, that in the majority of works in this area the authors derive their nonlinear beam models departing from general expressions of the elasticity theory, which are subjected to certain assumptions regarding the dimensions and kinematics of the analyzed object. Such approach though being general (can be applied not only to beams), suffers from excessive complexity for one’s understanding and can hardly be implemented straightforwardly. In the current contribution an alternative paradigm is proposed: the finite element is constructed on the basis of a well-known linear ancestor, which operates with simple strength of materials hypotheses. The classical beam stiffness matrix is wrapped by specific mathematics responsible for the proper description of large rotations and displacements. In this aspect, the developed finite element reminds the corotational FEA concept, described in [28].

Another important advantage of the presented nonlinear beam formulation is the symmetry of its tangent matrix, which is particularly important for numerical implementation, since it permits the more efficient linear equations solvers to be invoked and significantly improves convergence of Newton-Raphson iterations [29]. Not all the geometrically nonlinear beam models (see, for instance, [30]) possess this property.

Keywords:

flexible rod, finite elements, Euler vector, rotation tensor, large displacements, large rotations

References

  1. Bratukhina A.I. Trudy MAI, 2001, no.4: http://www.mai.ru/science/trudy/eng/published.php?ID=34669

  2. Bratukhina A.I. Trudy MAI, 2001, no.4: http://www.mai.ru/science/trudy/eng/published.php?ID=34668

  3. Zagordan A.A. Trudy MAI, 2010, no.38: http://www.mai.ru/science/trudy/eng/published.php?ID=14145

  4. Komarov V.A., Kuznetsov A.S., Lapteva M.Yu. Trudy MAI, 2011, no.43: http://www.mai.ru/science/trudy/eng/published.php?ID=24759

  5. Svetlitskii V.A. Mekhanika sterzhnei. Statika (Mechanics of rods. Statics.), Moscow, Vysshaja shkola, 1987, 320 p.

  6. Eliseev V.V., Zinov'eva T.V. Mekhanika tonkostennykh konstruktsii. Teoriya sterzhnei. (Mechanics of thin-walled structures. The theory of rods), St. Petersburg, Izdatel'stvo SPbGTU, 2008, 95 p.

  7. Zhilin P.A. Prikladnaja mekhanika. Teorija tonkih uprugih sterzhnej (Applied mechanics. Theory of thin elastic rods), St. Petersburg, Izdatel'stvo SPbGTU, 2007, 101 p.

  8. Badikov R.N. Raschetno-eksperimental'noe issledovanie napryazhenno-deformirovannogo sostoyaniya i rezonansnykh rezhimov vrashcheniya vintovykh pruzhin v pruzhinnykh mekhanizmakh (Calculation-experimental study of stress-strain state and the resonant modes of rotation of helical springs in spring mechanisms), Doctor’s thesis, Moscow, 2009, 166 p.

  9. Sorokin F.D. Izvestija RAN. Mekhanika tverdogo tela, 1994, no. 1, pp. 164-168.

  10. Levin V.E., Pustovoj N.V. Mekhanika deformirovaniya krivolineinykh sterzhnei (Mechanics of curved bars deformation), Novosibirsk, Izdatel'stvo NGTU, 2008, 208 p.

  11. Geradin M., Cardona A. Flexible Multibody Dynamics – A Finite Element Approach, Wiley, New York, 2000, 327 p.

  12. Lalin V.V., Javarov A.V. Zhilishchnoe stroitel'stvo, Moscow, 2013, no. 5, pp. 51-55.

  13. Semenov P.Ju. Trudy II mezhdunarodnoi konferentsii “Problemy nelineinoi mekhaniki deformiruemogo tverdogo tela”, Kazan, 2009, pp. 24-29.

  14. Levjakov S.V. Prikladnaya mekhanika i tekhnicheskaya fizika, 2012, vol. 53, no. 2, pp. 128-136.

  15. Pimenta P.M., Yojo T. Geometrically exact analysis of spatial frames. // Applied Mechanics Reviews. 1993. V. 46. N. 1. P. 118-128.

  16. Ibrahimbegović A. On finite element implementation of geometrically nonlinear Reissner’s beam theory: three–dimensional curved beam elements // Computer Methods in Applied Mechanics and Engineering. 1995. V. 122. N. 1-2. P. 11-26.

  17. Branec V.N., Shmyglevskij I.P. Vvedenie v teoriyu besplatformennykh inertsial'nykh navigatsionnykh sistem (Introduction to the theory of strapdown inertial navigation systems), Moscow, Nauka, 1992, 280 p.

  18. Bremer H. Elastic multibody dynamics: a direct Ritz approach, Springer, 2008, 449 р.

  19. Zhilin P.A. Vektory i tenzory vtorogo ranga v trekhmernom prostranstve (Vectors and tensors of the second rank in the three-dimensional space), St. Petersburg, Nestor, 2001, 276 p.

  20. Rankin C.C., Brogan F.A. An Element Independent Corotational Procedure for the Treatment of Large Rotation, Journal of Pressure Vessel Technology-Transactions of The ASME, 1986, vol. 108, no. 2, pp. 165-174.

  21. Crisfield M.A. Nonlinear Finite Element Analysis of Solid and Structures, John Wiley & Sons, Chichester, 1996, vol. 2, 493 p.

  22. Felippa C.A. A Systematic Approach to the Element-Independent Corotational Dynamics of Finite Elements, Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado, 2000, 42 p.

  23. Belytschko T., Glaum L.W. Application of Higher Order Corotational Stretch Theories to Nonlinear Finite Element Analysis, Computers & Structures, 1979, vol. 10, no. 1-2, pp. 175-182.

  24. Battini J.M., Pacoste C. Co-rotational beam elements with warping effects in instability problems, Comput. Methods Appl. Mech. Engng, 2002, vol. 191, pp. 1755-1789.

  25. Hsiao K.M., Horng H.J., Chen Y.R. A corotational procedure that handles large rotations of spatial beam structures, Comput. Structures, 1987, vol. 27, no. 6, pp. 769-781.

  26. Crisfield M.A. A consistent co-rotational formulation for non-linear three-dimensional beam elements, Comput. Methods Appl. Mech. Engng, 1990, vol. 81, pp. 131-150.

  27. Meiera C., Wall W., Popp A. Geometrically Exact Finite Element Formulations for Curved Slender Beams: Kirchhoff-Love Theory vs. Simo-Reissner Theory, Cornell University Library, 2016, available at: https://arxiv.org/abs/1609.00119 (accessed 19.10.2016)

  28. Felippa C.A., Haugen, B.A unified formulation of small-strain corotational finite elements: I. Theory. // Computer Methods in Applied Mechanics and Engineering. 1994. V. 194. N. 21. P. 2285-2335.

  29. Simo J.C., Vu-Quoc L.A three-dimensional finite strain rod model. Part II: Computational aspects. // Computer Methods in Applied Mechanics and Engineering. 1996. V. 58. N. 1. P. 79-116.

  30. Simo J.C. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. // Computer Methods in Applied Mechanics and Engineering. 1995. V. 49. N. 1. P. 55–70.


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