Euler vector application specifics for large turns description while flying vehicles structural elements modeling on the example of a rod finite element

Deformable body mechanics


Nizametdinov F. R.*, Sorokin F. D.**

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia



Structural elements in mechanic engineering are often may be reduced to a rod model, characterized by large generalized displacements with small deformations These elements are, for example, propeller blades, outer wing spars, etc. [1-4].

Many works have been devoted to the problem of describing the large displacements of structural elements represented by rod models. For example, a wide class of methods is based on the use of systems of differential equations [5-10]. At the very beginning of computational mechanics development, the method of representing elastic bodies by a set of solid elements was used, and the elastic properties of deformed bodies were modeled by elastic bonds [11-13]. Attempts were made to combine the solid-state modeling with the finite element method (FEM) [14]. According to this method, the total displacement is a sum of independent large rigid displacement and small elastic displacement due to deformations [15]. The next step was the method of successive approximations, which was modified to the method of the attached coordinate system. But this method was not widely used due to the appearance of more advanced approaches, such as the method of absolute nodal coordinates [16-18]. This formulation is the most complete, but at the same time is very complicated for performing out practical calculations [19]. In [20], the asymmetric Piola-Kirchhoff tensor is used. In [21], the position of the FE cross-sections is given by the radius vector and two ortas. Such approach increases the number of kinematic parameters, but allows avoiding singular points.

In [22], the FE is constructed using an independent description of displacements and rotations by Hermite polynomials, which leads to the problem of shear locking. In [23], the quadratic approximation of the tensor elements ro`tation is used. In [24], the vector-tensor theory of large rotations is used, but the magnitude of the rotations is restricted by the angle of 2π radians. In [19], the Craig-Bampton method is used.

A widespread approach to the geometrically nonlinear beam FE construction is the co-rotation approach [25]. In [26], a co-rotational FE is considered, in which the hypothesis of the relative strain smallness is built-in. The disadvantage of this element is the presence of singular points at the rotation angle of 2π. The model [27] had no such shortcoming due to introduction of an intermediate position. But it is considerably more complicated due to application of the nonlinear theory of elasticity apparatus. An incremental description of the rotations was also used in the element proposed in [28] in application to the problems of multi-body dynamics. Such a variety of approaches is explained by the presence of more than a dozen methods of large rotations describing [29-31].

The most common and natural way to describe finite rotations is the Euler vector [6, 32-34]. However, the Euler vector has a disadvantage, namely, critical value of the rotation angle (2π). At this value of the rotation angle, the tensors connecting small physical rotations with derivatives of the kinematic parameters become degenerate. The article considers two modifications to overcome this problem. These modifications allow describe infinitely large rotations. Their comparison is performed on the example of the rod FE. The features of both modifications are described in the paper. The tangential stiffness matrices and the node-force vector of the rod FE were obtained in a closed analytical form, with account for these features. The solution of test tasks by two approaches, analysis of the program code allowed reveal the advantages and disadvantages of both modifications.

The modification based on the adjustment of the Euler vector when the critical rotation value is reached is extremely simple for implementation, and correction is not required at every step of the solution. This modification can be integrated into the FEM package without significant changes in its architecture. Disadvantages of this modification are as follows: sudden change in the Euler vector when the critical value is reached, and asymmetric stiffness matrix.

The incremental way of description is not free from such shortcomings as the sudden change in the Euler vector, an asymmetric tangent stiffness matrix. But this modification is much more difficult for implementation and requires completely different system of storing the results. In addition, the architecture should provide the possibility of combining incremental and non-incremental elements.

Modifications are considered on the example of a rod element, but without significant changes they can be extended to other two-node FEs, such as FE farms, FE springs, FE rigid constraint, etc.


Euler vector, rotation tensor, large displacements, large turns, tangent stiffness matrix, finite element, shadow element


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