Abstract
The nonlinear problem of the Watt governor dynamics is investigated. It is assumed to be installed on a machine that performs specified harmonic vibrations of small amplitude along the vertical. Viscous friction forces are assumed to arise in regulator hinges of the, and these forces are small. In the main operating mode of the regulator, its rods, carrying massive weights, are deflected from the downward vertical by a constant acute angle. If friction and vertical vibrations of the machine are neglected, an approximate problem is obtained in which the regulator dynamics is described by an autonomous Hamiltonian system with one degree of freedom. On the phase portrait of the approximate problem, the operating mode corresponds to a singular point of the center type. The trajectories encircling this point lie inside the separatrix, which is a homoclinic doubly asymptotic trajectory that passes through the equilibrium position corresponding to the vertical position of the rods with weights. In the phase portrait, this position corresponds to a saddle singular point. The Melnikov method is used to obtain the splitting condition for the unperturbed separatrix in the complete perturbed problem, taking into account dissipation in the hinges and vertical vibrations of the machine.
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Funding
The study, which was supported by the Russian Science Foundation (project no. 19-11-00116) was carried out at the Moscow Aviation Institute (National Research University).
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In loving memory of L.D. Akulenko
Translated by M. Shmatikov
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Markeev, A.P. On Splitting of Separatrices Corresponding to the Operating Mode of the Watt Governor. Mech. Solids 58, 2731–2737 (2023). https://doi.org/10.3103/S0025654423080137
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DOI: https://doi.org/10.3103/S0025654423080137