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

A reduced model is proposed and analyzed for the simulation of vortex-induced vibrations (VIVs) for turbine blades. A rotating blade is modelled as a uniform cantilever beam, while a van der Pol oscillator is used to represent the time-varying characteristics of the vortex shedding, which interacts with the equations of motion for the blade to simulate the fluid-structure interaction. The action for the structural motion on the fluid is considered as a linear inertia coupling. The nonlinear characteristics for the dynamic responses are investigated with the multiple scale method, and the modulation equations are derived. The transition set consisting of the bifurcation set and the hysteresis set is constructed by the singularity theory and the effects of the system parameters, such as the van der Pol damping. The coupling parameter on the equilibrium solutions is analyzed. The frequency-response curves are obtained, and the stabilities are determined by the Routh-Hurwitz criterion. The phenomena including the saddle-node and Hopf bifurcations are found to occur under certain parameter values. A direct numerical method is used to analyze the dynamic characteristics for the original system and verify the validity of the multiple scale method. The results indicate that the new coupled model is useful in explaining the rich dynamic response characteristics such as possible bifurcation phenomena in the VIVs.

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Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

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

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

A two-degree-of-freedom bifurcation system for an elastic cable with 1:1 internal resonance is investigated in this paper. The transition set of the system is obtained with the singularity theory for three cases. The whole parametric plane is divided into several different persistent regions by the transition set. The bifurcation diagrams in different persistent regions are obtained.

Singularity Analysis of a Two-Dimensional Elastic Cable With 1∶1 Internal Resonance

Bifurcation of periodic solutions widely exists in nonlinear dynamical systems. In this paper, categories of bifurcations of systems with two state variables with different types of constraints are discussed, where some new types of transition sets are added. Additionally, the bifurcation properties of two-dimensional systems without constraints are compared with the ones with constraints. The results obtained in this paper can be used by engineers for the choice of the structural parameters of the systems.

Transition sets of bifurcations of dynamical systems with two state variables with constraints

A nonlinear model of a low pressure cylinder-generator rotor system is presented to study sub-synchronous resonance and combined resonance. Analytical results are obtained by an averaging method. Transition sets and bifurcation diagrams are obtained based on the singularity theory for the two-state variable system. The bifurcation characteristics are analyzed to provide a basis for the optimal design and fault diagnosis of the rotor system. Finally, the theoretical results are verified with the numerical results.

Jun Li

Papers

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

Singularity Analysis of a Two-Dimensional Elastic Cable With 1∶1 Internal Resonance

Transition sets of bifurcations of dynamical systems with two state variables with constraints

Combined resonance of low pressure cylinder-generator rotor system with bending-torsion coupling