Using a lumped mass model of a single rub-impact rotor system considering the gyroscopic effect, the stability and steady-state response of the rotor system are investigated in this paper. The contact between the rotor and the stator is described by the simple Coulomb friction and piecewise linear spring models. An algorithm combining harmonic balance method with pseudo arc-length continuation is adopted to calculate the steady-state vibration response of a nonlinear system. Meanwhile, Hill’s method is used to analyze the stability of the system. The nonlinear dynamic characteristics of the system are investigated when the gap size, stator stiffness and unbalance are regarded as the control parameters. The results show that the gap size determines the location of the rub-impact; besides, the smaller gap can improve the stability of the system. The unsteady motion can be found as the stator stiffness increases. Moreover, the unbalance directly affects vibration amplitude, which becomes greater with the increasing imbalance.

Stability and steady-state response analysis of a single rub-impact rotor system

Bending-torsional coupled vibration of a rotor-bearing-system due to blade-casing rub in presence of non-uniform initial gap

Aiming at the coupling fault of axial load and radial rub, a geometric nonlinear rotor-casing system is developed in this paper. For the actual case that the aero-engine components (i.e. compressor disc, turbine disc, stator casing, combustor and so on) are sprayed with thermal barrier coatings, the interaction between disc and casing is refined into four stages, including no rub, low rub, rub and high rub. With the increase of invasion depth, the corresponding force models are applied to describe the different impact mechanism. The frictional characteristic between disc and casing is described by the Coulomb model. Then the linear interpolation is used to predict the instantaneous contact of disc-casing and the nonlinear dynamic behaviours of the rotor-casing system subjected to axial load and radial rub are analysed in the form of bifurcation diagram. Moreover, the rub-impact forms, such as full annular rubbing and partial rubbing, are identified by whirl orbit and waveform. On this basis, the dynamic comparison of the rotor system with/without axial load is conducted. The response variations caused by coating hardness and support stiffness of casing are further discussed.

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Dynamics characteristics of a rotor-casing system subjected to axial load and radial rub

This paper focuses on the 1/2 sub-harmonic resonance of an aircraft’s rotor system under hovering flight that can be modeled as a maneuver load G in the equations of motion. The effect on the rotor system is analyzed by using theoretical methods. It is shown that the sub-harmonic resonance may occur due to maneuvering flight conditions. The larger the eccentricity E and the maneuver load G, the greater the sub-harmonic resonance. The effects of nonlinear stiffness, damping of the system, maneuver load, and eccentricity on the sub-harmonic resonance region in parameter planes are also investigated. Bifurcation diagrams of the analytical solutions are in good agreement with that of the numerical simulation solutions. These results will contribute to the understanding of the nonlinear dynamic behaviors of maneuvering rotor systems.

Analysis of 1/2 sub-harmonic resonance in a maneuvering rotor system

In this paper, the dynamical simulation of a nonlinear aircraft rotor system in Herbst maneuvering flight is carried out, though which the load control is discussed for safety purposes. The equatio...

Dynamical simulation and load control of a Jeffcott rotor system in Herbst maneuvering flight

An aero-engine rotor system is simplified as an unsymmetrical-rigid-rotor with nonlinear-elastic-support based on its characteristics. Governing equations of the rubbing system, obtained from the Lagrange equation, are solved by the averaging method to find the bifurcation equations. Then, according to the two-dimensional constraint bifurcation theory, transition sets and bifurcation diagrams of the system with and without rubbing are given to study the influence of system eccentricity and damping on the bifurcation behaviors, respectively. Finally, according to the Lyapunov stability theory, the stability region of the steady-state rubbing solution, the boundary of static bifurcation, and the Hopf bifurcation are determined to discuss the influence of system parameters on the evolution of system motion. The results may provide some references for the designer in aero rotor systems.

Bifurcation on synchronous full annular rub of rigid-rotor elastic-support system

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

Ecological stoichiometry provides a multi-scale approach to study macroscopic phenomena via microscopic lens. A stoichiometric producer-grazer model with maturation delay is proposed and studied in this paper. The interaction between stoichiometry and delay is novel and leads to more interesting insights beyond classical delay-driven periodic solutions. For example, the period doubling route to chaos can occur as the minimal phosphorous:carbon ratio in producer decreases. Mathematically, we establish the conditions for the existence and stability of positive equilibria, and study the occurrence of Hopf bifurcation at positive equilibria. Analytic results show that delay can change the number and stability of positive equilibria through transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation, and it further determines the grazer's extinction. Our model with a small delay behaves like LKE model in terms of light intensity, and Rosenzweig's paradox of enrichment exists in a suitable light intensity. We plot bifurcation diagrams and show rich dynamics driven by delay, light intensity, phosphorous availability, and conversion efficiency, including that a large delay can drive the grazer to go extinct in an intermediate light intensity that is favorable for the survival of the grazer when there is no delay; a limit cycle can appear, then disappear as the delay increases; given the same initial condition, solutions with different delay values can tend to different attractors.

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Hua-biao Zhang

Papers

Bifurcation on synchronous full annular rub of rigid-rotor elastic-support system

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

Dynamics of a stoichiometric producer-grazer model with maturation delay