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The first part of the paper contains a short review of a series of published papers in the area of system dynamics with coupled rotations as well as of a series of author's various published research results in the area of vector method based on the mass inertia moment vectors and corresponding deviational vector components and vector rotators for the pole and oriented axis, introduced and defined by K. Hedrih in 1991. The vector with principal importance is vector of the rigid body mass inertia moment at the point and for the axis oriented by the unit vector, and with a corresponding component vector of the rigid body mass deviational moment for the rotation axis through the same pole. Second part presents the vector method based on mass moment vectors and vector rotators coupled for pole and oriented nonintersecting axes in application for obtaining vector expressions for kinetic parameters of a rigid body dynamics with multi-coupled rotations around nonintersecting axes. A complete analysis of obtained vector expressions for linear momentum and angular momentum and their corresponding derivatives gives us a series of the kinematical vector rotators around the corresponding directions determined by axes of the rigid body coupled multi-rotations around nonintersecting axes. Series of the theorems are defined. [ABSTRACT FROM AUTHOR]

Multi body system contains finite number of rigid bodies that are placed on each axis, which do not intersect each other. Derivatives of the linear momentum and angular momentum with respect to time by use mass moment vectors and vector rotators for pole and axis, are determined. A number of theorems are defined. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Under external loads trees exhibit complex oscillatory behaviour: their canopies twist and band. The great complexity of this oscillatory behaviour consists to an important degree of torsional oscillations. Using a system of ordinary differential fractional-order equations, free and forced main eigen-modes of fractional-type torsional oscillations of a hybrid discrete biodynamical system of complex structures were done. The biodynamical system considered here corresponds to a tree trunk with branches and is in the form of a visco-elastic cantilever of complex structure. Visco-elasticity corresponds to different ages of a tree. We set up a new model of torsional oscillations of a complex discrete, biodynamical system, using the Kelvin-Voigt visco-elastic model involving a fractional-order time derivative. The analytical expressions describing the characteristic properties of its fractional-type oscillations are determined. Based on mathematical and qualitative analogies, this concept represents a new model of torsional oscillations of a light cantilever that takes into account visco-elastic, dissipative properties of the material. Rigid discs are attached to the cantilever. Expressions for kinetic energy, deformation work and a generalized function of the fractional-type energy dissipation of this biodynamical system are defined. Independent main eigen-modes of the fractional type for free and forced torsional oscillations were determined for a special class of such systems, using formulas for the transformation of independent generalized angle coordinates to the principal main eigen-coordinates of the system. The forms of their approximate analytical solutions are shown. In the general case for inhomogeneous biodynamical systems of fractional type, there are no independent main fractional-type eigen-modes of torsional oscillations. The system behaves as a nonlinear system. A new constitutive relation between coupling of torsion loading to a visco-elastic fractional-type cantilever with fractional-type dissipation of cantilever mechanical energy and angle of torsion deformation is determined using a fractional-order derivative. The main advantages of the proposed model are the possibility to analyse torsional oscillations of more complex structures and the possibility to analyse complex cantilevers with different cross-sectional diameters., Competing Interests: Conflict of interestThe authors have no conflicts of interest (financial or non-financial) to declare that are relevant to the content of this article., (© The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2023, Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.)