Maps of dynamic invariants in the estimation of modes of motion of mechanical oscillatory systems

DOI: 10.34759/trd-2023-128-05


Eliseev A. V.1*, Kuznetsov N. K.1**, Mironov A. S.2***

1. Irkutsk National Research Technical University, 83, Lermontov str., Irkutsk, 664074, Russia
2. Irkutsk State Transport University (IrGUPS), 15, Chernyshevsky str., Irkutsk, 664074, Russia



The scientific and methodological foundations for the dynamics problems solving of technological and transport facilities operating under conditions of increased vibrational dynamic loads are being developed. The purpose of the proposed research consists in developing methodological approaches for the assessment, control, formation and management of dynamic states of technical objects (machines, equipment, working bodies of vibrating technological machines), which design schemes of are being displayed in the form of mechanical oscillatory systems with several degrees of freedom.

The studies are based on employing and developing analytical apparatus of system analysis and its applications to the problems of machine dynamics, protection of equipment and devices from vibration effects, which forms the basis of approaches to ensuring safety, reliability of operation of technical means, ensuring the dynamic quality of technological machines.

The article considered the issues of the development of ideas on the generalized states of mechanical oscillatory systems formed by solids under conditions of coherent vibrational loads of a forceful nature, which dynamic state of is being determined based on the dynamic malleability of points distributed over the surface.

The authors suggest considering the so-called dynamic invariant, reflecting the essential features of the mechanical oscillating system dynamic states aggregate in the form of oriented graphs, as a generalized dynamic state. The number of its of vertices and arcs are equal to the number of resonances, frequencies of amplitudes zeroing , as well as positive and negative forms of the elements dynamic interactions.

The article shows that an infinite set of amplitude-frequency characteristics can be juxtaposed with a finite set of dynamic invariants. The general aggregate of dynamic invariants can be constructed based on the zeroing frequency functions, which can be set implicitly by zeroing the transfer function numerator, interpreted as dynamic compliance within the framework of the problem under consideration. The zeroing frequency unction juxtaposes the frequency of external force disturbances with the variation parameters of the system, on which the dynamic compliance is being zeroed, assuming that the zeroing frequency does not coincide with the natural oscillation frequency of the system.

The article demonstrates that the aggregate of dynamic states corresponding to the simultaneous variation of two system parameters may be displayed by the dynamic invariants chart, splitting the plane of two variation parameters into the finite aggregate of non-intersecting areas, boundaries and planes with potentially different dynamic variables. It shows, in particular, that infinite diversity of dynamic states of mechanical oscillations may be represented in the form of finite set of generalized dynamic states.


mechanical oscillatory systems, connected perturbations, dynamic compliance, dynamic invariants


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