Singularity-Free Lie Group Integration and Geometrically Consistent Evaluation of Multibody System Models described in terms of Standard Absolute Coordinates

A classical approach to the MBS modeling is to use absolute coordinates, i.e. a set of (possibly redundant) coordinates that describe the absolute position and orientation of the individual bodies w.r.t. to an inertial frame (IFR). A well-known problem for the time integration of the equations of motion (EOM) is the lack of a singularity-free parameterization of spatial motions, which is usually tackled by using unit quaternions. Lie group integration methods were proposed as alternative approach to the singularity-free time integration. Lie group integration methods, operating directly on the configuration space Lie group, are incompatible with standard formulations of the EOM, and cannot be implemented in existing MBS simulation codes without a major restructuring. A framework for interfacing Lie group integrators to standard EOM formulations is presented in this paper. It allows describing MBS in terms of various absolute coordinates and at the same using Lie group integration schemes. The direct product group SO(3)xR3; and the semidirect product group SE(3) are use for representing rigid body motions. The key element of this method is the local-global transitions (LGT) transition map, which facilitates the update of (global) absolute coordinates in terms of the (local) coordinates on the Lie group. This LGT map is specific to the absolute coordinates, the local coordinates on the Lie group, and the Lie group used to represent rigid body configurations. This embedding of Lie group integration methods allows for interfacing with standard vector space integration methods.

Singularity-Free Lie Group Integration of Multibody System Models Described in Absolute Coordinates

A classical approach to the modeling of multibody systems (MBS) is to use absolute coordinates, i.e. a set of (redundant) coordinates that describe the absolute position and orientation of the individual bodies w.r.t. to an inertial frame (IFR). A well-known problem for the time integration of the equations of motion (EOM) is the lack of a singularity-free parameterization of spatial motions, which is usually tackled by using unit quaternions. Lie group integration methods were proposed as alternative approach to the singularity-free time integration. Lie group methods are inherently coordinate free and thus incompatible with any absolute coordinate description. In this paper, an integration scheme is proposed that allows describing MBS in terms of arbitrary absolute coordinates and at the same using Lie group integration schemes, which allows for singularity-free time integration. Moreover, the direct product group SO (3) × ℝ3 as well as the semidirect product group SE (3) can be use for representing rigid body motions, which is beneficial for constraint satisfaction. The crucial step of this method, which renders the underlying Lie group integration scheme applicable to EOM in absolute coordinates, is the update of the (global) absolute coordinates in terms of the (local) coordinates on the Lie group by means of a local-global transitions (LGT) transition map. This LGT map depends on the used absolute coordinates and the local coordinates on the Lie group, but also on the Lie group itself used to represent rigid body configurations (respectively the deformation field of flexible bodies), i.e. the geometry of spatial frame motions. The Lie group formulation is thus embedded, which allows interfacing with standard vector space integration methods.

EQUATIONS OF PLANETARY SYSTEMS MOTION

The study of the dynamically evolution of planetary systems is very actually in relation with findings of exoplanet systems. free spherical bodies problem is considered in this paper, mutually gravitating according to Newton's law, with isotropically variable masses as a celestial-mechanical model of non-stationary exoplanetary systems. The dynamic evolution of planetary systems is learned, when evolution's leading factor is the masses' variability of gravitating bodies themselves. The laws of the bodies' masses varying are assumed to be known arbitrary functions of time. When doing so the rate of varying of bodies' masses is different. The planets' location is such that the orbits of planets don't intersect. Let us treat this position of planets is preserve in the evolution course. The motions are researched in the relative coordinates system with beginning in the center of the parent star, axes that are parallel to corresponding axes of the absolute coordinates system. The canonical perturbation theory is used on the base aperiodic motion over the quasi-canonical cross-section. The bodies evolution is studied in the osculating analogues of the second system of canonical Poincare elements. The canonical equations of perturbed motion in analogues of the second system of canonical Poincare elements are convenient for describing the planetary systems dynamic evolution, when analogues of eccentricities and analogues of inclinations of orbital plane are sufficiently small. It is noted that to obtain an analytical expression of the perturbing function main part through canonical osculating Poincare elements using computer algebra is preferably. If in expansions of the main part of perturbing function is constrained with precision to second orders including relatively small quantities, then the equations of secular perturbations will obtained as linear non-autonomous system. This circumstance meaningful makes much easier to study the non-autonomous canonical system of differential equations of secular perturbations of considering problem.

ACCURACY OF MEASURING SYSTEMS BASED ON LASER MODULES

Measuring systems using the design of laser module beams on the surface of the object under study are considered. A technique is proposed for experimental studies of the brightness structure of the study of laser modules for their subsequent testing. Adaptive algorithms for determining the type of module and distance have been developed for determining the coordinates of light marks on the surface of controlled products, ensuring the accuracy and reliability of the measurement. The need for high-precision measuring systems to carry out their preliminary selection and calibration of laser modules according to the proposed method, taking into account the range of design of light marks, is shown. It is shown in the work that the accuracy of determining the relative coordinates in the trajectory of the light marks of laser modules at a distance of 5 m for plain surfaces of the observed objects can be several times higher (0,2…0,3 mm) of the accuracy of determining their absolute coordinates (»1 mm).

DIFFERENTIAL EQUATIONS OF PLANETARY SYSTEMS

In this article will be considered many spherical bodies problem with variable masses, varying non-isotropic at different rates as celestial-mechanical model of non-stationary planetary systems. In this article were obtained differential equations of motions of spherical bodies with variable masses to reach purpose exploration of evolution planetary systems. The scientific importance of the work is exploration to the effects of masses’ variability of the dynamic evolution of the planetary system for a long period of time. According to equation of Mescherskiy, we obtained differential equations of motions of planetary systems in the absolute coordinates system and the relative coordinates system. On the basis of obtained differential equations in the relative coordinates system, we derived equations of motions in osculating elements in form of Lagrange's equations and canonically equations in osculating analogs second systems of Poincare's elements on the base aperiodic motion over the quasi-canonical cross- section.

The measuring accuracy study of the light mark coordinates of laser modules

Measuring systems using the design of laser module beams on the surface of the object under study are considered. A technique is proposed for experimental researches of the brightness structure of the study of laser modules. Adaptive algorithms on the type of module and distance have been developed for determining the coordinates of light marks on the surface of controlled products. The need for high-precision measuring systems to carry out their pre-liminary selection and calibration of laser modules according to the proposed method, taking into account the range of design of light marks, is shown. It is shown in the work that the accu-racy of determining the relative coordinates in the trajectory of the light marks of laser mod-ules for monochromatic targets can be several times higher (0.2 - 0.3 mm) than the accuracy of determining their absolute coordinates (≈ 1 mm).

Effectiveness of the segment method in absolute and joint coordinates when modelling risers

This paper presents two formulations of the segment method: one with absolute coordinates and the second with joint coordinates. The nonlinear equations of motion of slender links are derived from the Lagrange equations by means of the methods used in multibody systems. Values of forces and moments acting in the connections between the segments are defined using a new and unique procedure which enables the mutual interaction of bending and torsion to be considered. The models take into account the influence of the velocity of the internal fluid flow on the riser’s dynamics. The dynamic analysis of a riser with fluid flow requires calculation of the curvature by approximation of the Euler angles with polynomials of the second order. The influence of the sea environment, such as added mass of water, drag and buoyancy forces as well as sea current, is considered. In addition, the influence of torsion is discussed. Validation is carried out for both models by comparing the authors’ own results with those obtained from experimental measurements presented in the literature and from COMSOL, Riflex and Abaqus software. The validation is concerned with vibrations of cables and the riser with internal fluid flow as well as with frequencies of free and forced vibrations of a riser fully or partially submerged in water. The numerical effectiveness of both formulations is examined for dynamic analysis of the riser, whose top end is moving in a horizontal plane. Conclusions concerned with the effectiveness of both formulations of the segment method and the influence of torsional vibrations on numerical results are formulated.