Finite element of a flexible rod with separate storage of accumulated and additional rotations for the problems of nonlinear dynamics of aerial vehicles structures

Dynamics, strength of machines, instruments and equipment


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



Dynamical behavior of various aircraft parts, such as wing spars, fuselage stringers, propeller blades etc. [1–4], can be accurately described by a flexible rod model, as though these structures may undergo large displacements and rotations, while their strains remain small. A large set of numerical techniques is available for flexible rods dynamics analysis. However, FEM is the most appropriate in the present context.

A number of finite elements has been developed for solving flexible rods nonlinear dynamics problem. Reference [5] is one of the pioneering work in this area. Treating a rod as a three dimensional body, which dimensions and kinematics are specifically constrained, the author proposes a powerful model, where both displacements and strains can be arbitrarily large. This contribution gave rise to an entire family of the so-called geometrically exact approaches for statics and dynamics analysis of flexible beams [6, 7].

Another popular alternative to constructing geometrically nonlinear beam models is the so-called corotational approach [8]. A simple corotational finite element based on the small strain hypothesis was proposed in [9]. Being quite simple to derive and implement, this model suffers fr om a serious issue, namely, the total Lagrangian formulation is chosen for storing both displacements and rotations. For the latter, it leads to numerical problems when approaching the rotation angle of 2π value. The model from [10], which in fact combines the two aforementioned fundamental approaches, eliminates this limitation introducing the updated reference configuration, though the underlying math is much more complicated as nonlinear elasticity theory is applied. The similar incremental description of rotations is also used in [11] in the context of constructing a specific rod finite element adapted for multi-body dynamics problems. However, the obtained tangent stiffness matrix revealed to be non-symmetric even for the simple conservative loading cases. This, obviously, restricts the selection of linear solvers to be used in association with this model, and, thus, compromises the overall performance of the numerical solution.

The Absolute Nodal Coordinates [12, 13] formulation is another reliable technique for building nonlinear rod finite elements based on the choice of high order interpolation functions for the rod geometry and deformation description.

The majority of existing approaches (along with the currently presented) employ the Euler vector for rotations description. But alternative quantities, such as the Rodrigues vector [14], can be introduced as well. A particular attention to exact conservation of momentum and total energy is given in this article. This aspect was carefully studied also in [15] in the context of nonlinear of rods dynamics.

The present contribution is the continuation of the authors’ previous work [16], wh ere a flexible rod finite element for static analysis has been proposed. The nonlinear rod FE, which large motion kinematics is based on the updated Lagrangian formulation for the rotations increments, is currently adapted for transient dynamics problems. The rotations data is separated into two parts. The accumulated rotation is stored as the rotation matrix, while the incremental part is described by the Euler vector [17]. Unlike the existing approaches, the design of the developed element is beyond the scope of the well-known stiffness and mass matrices of the conventional rod FE that significantly simplifies the derivations. This allows embed practically any linear rod model, whether it is of Timoshenko or Euler-Bernoulli type.

The article presents the closed-form expressions for the element generalized mass and gyroscopic matrices, and for the right hand side inertia loads vector. Owing to the underlying update Lagrangian formulation for rotations description, the proposed element does not exhibit any numerical instability related issues for any rotation magnitudes during the structure evolution. Besides, as the total rotation matrix is updated after each Newton-Raphson iteration [18], nearly quadratic convergence has been achieved.

To demonstrate the element workability and robustness two numerical examples were analyzed. The results were compared with the existing reference data [19, 20, 21] and with the third party FEM packages.


flexible rod, finite elements, Euler vector, rotation tensor, Zhilin’s tensor, large displacements, large rotations, mass matrix, gyroscopic matrix


  1. Bratuhina A.I. Trudy MAI, 2001, no. 4, available at:

  2. Bratuhina A.I. Trudy MAI, 2001, no. 4, available at:

  3. Zagordan A.A. Trudy MAI, 2010, no. 38, available at:

  4. Komarov V.A., Kuznecov A.S., Lapteva M.Ju. Trudy MAI, 2011, no. 43, available at:

  5. Simo J.C. A finite strain beam formulation. The three-dimensional dynamic problem. Part I, Computer Methods in Applied Mechanics and Engineering, 1985, vol. 49, no. 1, pp. 55 – 70.

  6. 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, vol. 122, no. 1–2, pp. 11 – 26.

  7. Ignacio R. The interpolation of rotations and its application to finite element models of geometrically exact rods, Computational Mechanics, 2004, vol. 34, no. 2. pp. 121 – 133.

  8. Felippa C.A., Haugen B. A unified formulation of small-strain corotational finite elements: I. Theory, Computer Methods in Applied Mechanics and Engineering, 1994, vol. 194, no. 21, pp. 2285 – 2335.

  9. Crisfield M.A., Galvanetto U., Jelenić G. Dynamics of 3-D co-rotational beams, Computational Mechanics, 1997, vol. 20, no. 6, pp. 507 – 519.

  10. Jelenić G., Crisfield M.A. Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics, Computer Methods in Applied Mechanics and Engineering, 1999, vol. 171, no. 1, pp. 141 – 171.

  11. Cardona A., Geradin M. A beam finite element non-linear theory with finite rotations, International Journal for Numerical Methods in Engineering, 1988, vol. 26, no. 11, pp. 2403 – 2438.

  12. Shabana A.A., Yakoub R.Y. Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory, Journal of Mechanical Design, 2001, vol. 123, no. 4, pp. 606 – 613.

  13. Yakoub R.Y., Shabana A.A. Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications, Journal of Mechanical Design, 2001, vol. 123, no. 4, pp. 614 – 623.

  14. Pimenta P.M., Campello E.M.B., Wriggers P. An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods, Computational Mechanics, 2008, vol. 42, no. 5, pp. 715 – 732.

  15. Simo J.C., Tarnow N., Doblare M. Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms, International Journal for Numerical Methods in Engineering, 1995, vol. 38, no. 9, pp. 1431 – 1473.

  16. Popov V.V., Sorokin F.D., Ivannikov V.V. Trudy MAI, 2017, no. 92, available at:

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

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

  19. Simo J. C., Vu-Quoc L. On the dynamics in space of rods undergoing large motions – a geometrically exact approach, Computer Methods in Applied Mechanics and Engineering, 1988, vol. 66, no.2, pp. 125 – 161.

  20. Greco M., Coda H.B. Positional FEM formulation for flexible multi-body dynamic analysis, Journal of Sound and Vibration, 2006, vol. 290, no.3-5, pp. 1141 – 1174.

  21. Fotouhi R. Dynamic analysis of very flexible beams, Journal of Sound and Vibration, 2007, vol. 305, no. 3, pp. 521 – 533.

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