For solving stability of motion in a nonlinear relative rotation shafting, a sort of periodic variable coefficient linear ordinary differential equations was studied. Movement stabilization criterion of nonlinear relative rotation system was obtained by transferring the fundamental matrix of the solution of the ordinary differential equation to Jordan canonical. Using the Lyapunov function method, the stabile balance position of relative rotation motion of driving system in a variety of mechanical equipments was studied, and the analytic expression of the stable region was obtained. The method supplies theoretical laws, similarity analog criterion and technique means of selecting working parameter of stable working interval of a type of mechanical driving systems. The result of the research can be used for analysis and evaluation of the torsional vibration stability of the main driving system of heavy and complex rotary machines.

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To clarify the action mechanism of parameter change on system stability, the chaotic property of BTA deep-hole machining system under the effect of inner cutting fluid was analyzed. According to the kinematic characteristics of the internal cutting fluid and the equation of the moment of momentum of the system, the kinematic equation of the boring bar considering the effect of the internal fluid was established. The critical conditions of chaos were deduced according to the Hamiltonian function and Melnikov function of the plane near-Hamilton system. The mechanism of the liquid filling ratio, cutting fluid flow velocity, and frequency ratio parameters on the system’s critical instability surface is investigated. The correlation and sensitivity of influencing factors, such as filling ratio and frequency ratio, and cutting fluid flow velocity to the sensitivity of system chaos are explored. The results show that in precision machining, the change of liquid filling ratio is positively related to the stability of the system, the change of cutting fluid flow velocity is negatively correlated with the stability of the system, and the change of frequency ratio has no monotonicity effect on the stability of the system. The sensitivity of the chaotic characteristics of the system to each parameter is bounded by the filling liquid ratio h = 0.58. When 0 ≤ h ≤ 0.58, frequency ratio ω ¯  > filling ratio h > cutting fluid flow velocity V0; when 0.58 < h ≤ 1, filling ratio h > frequency ratio ω ¯  > cutting fluid flow velocity V0. These research conclusions can lay a certain theoretical foundation for the analysis, control, and optimization of the complex mechanical behavior of BTA deep-hole machining systems in engineering practice.

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