The relationship between the rotation, and the angular velocity, which allows us to update the orientation of the body with time is one of the governing equations of beams and plates, see [1]. This relationship is strongly non-linear in several parametrizations and very demanding for the numerical integration.

Our research employs three parametrizations of spatial rotations. Two of the parametrization are vector-based parametrizations, which are often used in beam and shell theories: (i) the rotational vector and (ii) the tangential (Argyris) vector [2]. The third parametrization employs a rotational quaternion, which is uncommon in beams and shells. The rotation-angular velocity differential equation represents the system of the first-order differential equations in time. We limit our integration schemes to (i) the midpoint rule, where the averaged rotation matrix is used, as proposed by Simo, Tarnow and Doblare [3], (ii) the new midpoint rule with the averaged quaternion parameter and, (iii) the explicit Runge-Kutta method ode45 as implemented in the commercial program Matlab [4].

The tests of the methods are performed using two characteristic numerical examples. The first example is characterized by a small rotation and a relatively long time interval. The second example combines the spatial rotations, large amplitudes and a stiff regime. In each of the two examples we choose three (analytical) functions for the components of the rotational vector and calculate the angular velocity vector, quaternion and matrix. This was then used in the angular velocity differential equation. After numerically integrating this equation the numerical solution is compared to the analytical one. We introduce a specific error estimates and separately compare the error in the angle (the parametrization norm) and in the direction of rotation.

We found the ode45 method with the quaternion parametrization to be the most suitable regarding both accuracy and efficiency. Our newly proposed midpoint method MP-q based on the quaternion parametrization is shown to be far more accurate than any other combination of the midpoint rule and the parametrization. In numerical experiments the numerical instability has never taken place by any method and in any example, only if the rotational quaternions have been used for the parametrization of rotations.

1

J.C. Simo, L. Vu-Quoc, "On the dynamics in space of rods undergoing large motions - a geometrically exact approach", Comput. Methods Appl. Mech. Engrg., 66, 125-161, 1988. doi:10.1016/0045-7825(88)90073-4

2

J. Argyris, "An excursion into large rotations", Comput. Methods Appl. Mech. Engrg., 32, 85-155, 1982. doi:10.1016/0045-7825(82)90069-X

3

J.C. Simo, N. Tarnow, M. Doblare, "Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms", Int. J. Numer. Methods Engng., 38, 1431-1473, 1995. doi:10.1002/nme.1620380903

4

The MathWorks, Inc., "MATLAB, Using Matlab", Natick, 1999.

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