Comparison between different time integrators when dealing with fundamental problem in multibody dynamics – the reconstruction of rotations from angular velocities is crucial for choosing the most suitable time integrators to be employed in more complex software environment for solving mere demanding problems. Development of a new integrator and its comparison to the existing represents an important knowledge and basis for further development and application of time integrators on nonlinear configuration spaces.
COBISS.SI-ID: 5478497
The classical concept of parametrizing the rotation matrix by the rotational vector is completely abandoned so that the only rotational parameters are the rotational quaternions representing both rotations and rotational strains in the beam. Because the quaternions are the elements of a four dimensional linear space, their use is an advantage compared to the elements of the special orthogonal group. This makes possible, e.g. to interpolate the rotational quaternions in a standard additive way and to apply standard Runge-Kutta time integration methods. Thus we benefit from theoretically well based adaptive time-step algorithms and avoid the analytical linearization of the governing equations.
COBISS.SI-ID: 5751393
With the use of the quaternion parametrization of rotations we completely avoid the need for use of the rotational vector and the rotation matrix. The equations are therefore presented in terms of quaternion algebra. An adaptation of the Newmark integration method for use with quaternions is presented. This work brings a firm theoretical tools and knowledge for the use of the quaternion parametrization of rotation in any finite-element implementation of the rotation-based theories. The proposed numerical model was verified by numerical examples. Present elements were shown to be particularly accurate and computationally efficient when a higher-order interpolation is used. The interesting property of the present time integrator is that the increased numerical stability of the total mechanical energy can be observed.
COBISS.SI-ID: 5825377
In the paper we present the Reissner-Simo beam theory in which the rotations are represented by quaternions. From the generalized virtual work principle, where the unity constraint of the rotational quaternion is properly considered and the consistent energy complements of the rotational quaternions are employed, we derive the weak kinematic equations in the quaternion-based description. These equations are then employed in the extended virtual work principle to obtain the consistent governing equations of the three-dimensional beam in terms of the quaternion algebra. The quaternion moment equilibrium equation is analysed, discussed and interpreted. In numerical implementation the standard Galerkin discretization is used to obtain the quaternion-based finite element formulation. Various examples prove the suitability of the formulation.
COBISS.SI-ID: 6211681
In the paper we study the integration of angular velocities with a special attention given to exact and higher order methods. Both exact and approximate results are expressed in terms of rotational quaternions. Analytical solution is found using the theory of analytic differential systems. This exact solution serves as a suitable basis for derivation of various numerical methods. Approximative approaches based on Taylor series and several maps from pure to unit quaternions are presented. A special care is taken in describing the higher order approximations. The computational performance and comparison of numerical methods is demonstrated by examples.
COBISS.SI-ID: 6429793