Oil-film Force Model Research Articles

Stability analysis of multi-leaf oil-lubricated foil bearings with back springs based on nonlinear oil film force model

Stability analysis of multi-leaf oil-lubricated foil bearings with back springs based on nonlinear oil film force model

Static Eccentricity Fault of Elastic Ring Squeeze Film Damper for Aeroengine

The complete reduced-order model of an aeroengine and diagnosis recognition of the static eccentricity fault of the elastic ring squeeze film damper (ERSFD) is the research topic in this study. The finite element method was to set up the complete dynamic model of the “ERSFD–rolling bearing–double rotor–casing” by considering nonlinear bearing and oil film forces. Based on the Karhunen–Loève orthogonal decomposition reduced theories, the reduced-order model of the complete double-rotor systems is obtained. The improved short-time Fourier transform (STFT) + pseudocolor technology and lightweight network mode EfficientNetV2 realize the static eccentricity fault diagnosis and automatic recognition of the ERSFD. The general expression of oil film force and effective reduced-order model are given. The numerical comparison shows that the degree of freedom was reduced to , and the computational running time was shortened to of the full-order model by analyzing the amplitudes and bifurcations at the front and back supports of the casing. It has the highest recognition accuracy and the shorter automatic recognition time of EfficientNetV2 than other network models. The theories and methods used are of guiding significance for effective vibration modeling and order reduction solution of the complete aeroengine system, as well as dynamic simulation and fault mechanism research.

Semi-analytical prediction of the periodic vibration in a sliding bearing–rotor​ system

As for studying the sub-harmonic and super-harmonic periodic motions, eigenvalue dynamics and period-doubling bifurcation tree of the nonlinear rotor system, an implicit discrete mapping method is applied to a nonlinear sliding bearing–rotor system. The nonlinear term of the rotor system is triggered by the oil film forces of the sliding bearing at both ends. In order to accurately reveal the dynamic characteristics of the system, in this paper, a thirteen-parameter polynomial oil film force model is proposed. It considers the effects of displacement and velocity on the oil film force. And, the discrete mapping method used in this paper is a semi-analytical tool for solving equations with high precision. Consequently, continuous eigenvalue diagrams and bifurcation trees of periodic solutions from Period-1 to Period-4 motions of the nonlinear rotor system are obtained. For verification, numerical simulations are completed. Numerical and semi-analytical results are compared in time–history displacement, velocity and trajectories. Furthermore, the active feedback controller is carried out to improve the stability of the system.

Vibration Modelling and Verification for Rotor-Support-Casing Coupling System with Misalignment Fault

Misalignment is one of the common malfunctions that occur in rotating machines. Effects of misalignment on the casing vibration response of a rotor-support-casing (RSC) coupling system is investigated in detail. The model of an RSC coupling system is established using the lumped mass method. The coupling effects between the rotor, support and casing are fully considered. A misalignment model is proposed, and equivalent misalignment force is applied on corresponding lumped mass points. Nonlinear factors of bearing, such as the clearance of bearing, oil film force, nonlinear Hertzian contact force, and the varying compliance vibration are developed. The influences of oil film thickness and bearing size are considered in the nonlinear oil film force model. By using a numerical method, the governing equations of the system are solved to obtain the steady-state vibration. The simulation results from a coupling model are compared with the experimental results and the effectiveness of the new model is verified. It has been found that spectrums and orbit plots are effectively used to reveal the unique nature of misalignment faults, leading to reliable misalignment diagnostic information.

Dynamic Behavior Analysis of a Rotor System Based on a Nonlinear Labyrinth-Seal Forces Model

Taking the flow field features of labyrinth seal into consideration, the fluid force generated from the seal cavity, which is spatially separated into two regions, is modeled with the perturbation method. The rotor orbit defined in the perturbation analysis is spatio-temporal varied, which is quite different from the usually preconditioned elliptical track. Meanwhile, the nonlinear fluid force originating from the seal clearance is delineated by the Muszynska's model. Based on the short bearings assumption, a nonlinear oil-film force model is employed. The rotating shaft is simulated by Timoshenko beam finite element with the consideration of geometric asymmetry. Applying the Lagrange's equations, the motion equations of the rotor-bearing-labyrinth seal system are derived. By means of spectrum cascades, bifurcation diagrams, Poincaré maps, etc., the numerical analysis of the system dynamic characteristics is conducted. The results show that abundant nonlinear behaviors can be triggered in the speed-up. The instability threshold and the vibration amplitude of the rotor system are, respectively, enhanced and reduced by the increasing inlet pressure. With shorter seal length, the sealing effect is decreased, whereas the system stability is improved. The fluid-whip phenomenon can be eliminated by increasing the mass unbalance eccentricity at a certain rotational speed.

Nonlinear Characteristics Analysis of a Rotor-Bearing-Brush Seal System

The vibration response and nonlinear dynamic behavior of a rotor-bearing-brush seal system were investigated with a new seal force model of the brush seal. The nonlinear oil–film force model was adopted based on a short bearing assumption. The dimensionless equation of motion was solved using the fourth order Runge–Kutta method. The effects of key parameters including rotor speed, installation spacing of the brush seal, disk eccentricity, disk mass, and journal mass on the nonlinear dynamic characteristics of rotor-bearing-brush seal system were determined and compared under different operating conditions with a bifurcation diagram, time history, axis orbit, poincaré map, frequency spectrum, and spectrum cascade. The results showed that the system response contained various nonlinear phenomena, such as periodic motion, multi-periodic motion, and quasi-periodic motion. The interaction of the rotor speed, installation spacing of the brush seal, disk eccentricity, disk mass, and journal mass could seriously affect the stability and working condition of the system. This study provides a theoretical support for the selection of key design parameters and further understanding of the nonlinear characteristics of rotor-bearing-seal systems with a brush seal.

Dynamic modeling and response of a spur planetary gear system with journal bearings under gravity effects

This paper focuses on the modeling method and the gravity-induced dynamic response of a spur planetary gear system with journal bearings. The lumped-parameter model of a planetary gear system with journal bearings is established. Both contact on drive-side and back-side of the tooth are considered simultaneously. Linear and nonlinear bearing force models are introduced into the system model separately to take the planet bearing oil-film forces into account. A demonstration is given to show the adopted nonlinear oil-film force model is still valid for the lubrication of support for planet gears. Equilibrium positions of the planet gear are depicted under different input rotational speeds and input torques. Under gravity effect, system responses at different rotational speeds are calculated by employing Newmark integration; tooth wedging at ring-planet meshes is examined with different backlashes. The system responses are presented as vibration spectra, planet bearing forces, orbits of members, tooth forces, and the percentage of tooth wedging in one carrier cycle. The results show that the gravity effect dominates the response at low rotational speeds. The linear bearing force model is not valid in some cases. The fluctuation of the bearing force and the enlargement of the planet orbits are induced by gravity effect. Tooth wedging is the combined effect of gravity, centrifugal force, and planet bearing clearance.

Dynamics response of an on-board rotor supported on modified oil-film force considering base motion

In an on-board rotor journal bearing system, hydrodynamic bearing forces are affected by the coupling motion between rotor and base. To study the system dynamic response, a model of the on-board rotor supported on hydrodynamic bearing force is proposed in this paper. Motion equations of rigid disk and elastic shaft are derived by applying the Lagrange principle and finite element method. Based on the short bearing assumption and Hahn boundary condition, a modified oil-film force model of the hydrodynamic bearing is derived by introducing the base motion velocity. Under sinusoidal excitations of the base translational motion, dynamic responses of the on-board rotor bearing system are calculated by a Newmark time-step integration algorithm. Results are presented by time history, fast Fourier transforms, orbits and Poincare maps, and are compared with the response dominated by Capone's oil-film force model. The influences of amplitude and frequency of base motion on system dynamic behavior are investigated. Differences after introducing base motion velocity into the oil-film force model show that the convective velocity of the bearings should not be ignored in some extreme cases.

Nonlinear coupled dynamics of an asymmetric double-disc rotor-bearing system under rub-impact and oil-film forces

The nonlinear dynamic behavior of an asymmetric double-disc rotor-bearing system with interaction between rub-impact and oil-film forces is addressed in this paper. Using dynamics theory, the mathematical model of an asymmetric double-disc rotor-bearing system is established, considering nonlinear oil-film force and rub-impact force. The nonlinear oil-film force model is presented in Reynolds equation, and the rub-impact is assumed with a Hertz contact and a Coulomb friction. The dynamic equations with coupled rub-impact and oil-film forces are numerically solved using the Runge–Kutta method. Bifurcation diagrams, largest Lyapunov exponent, Poincaré maps, and three-dimension spectral plots are employed to analyze the dynamic behavior of the system. The sub-harmonic, multiple periodic, quasi-periodic and chaotic motions are observed in this study. A special phenomenon is occurring that the motion of system becomes simple and the oil-whirl is restrained or even removed with an increasing imbalance by magnifying the eccentricity. Another special phenomenon is also occurring that the oil-whirl gets diminished or even disappeared with increasing stator stiffness, but the oil-whip is uninfluenced. The discoveries will have a considerable value as diagnostic tools in settling oil-film instability. The numerical results show that the nonlinear dynamic behavior of the system varies with the rotational speed and model parameters.

Oil-film instability simulation in an overhung rotor system with flexible coupling misalignment

Aiming at the oil-film instability of the sliding bearings at high speeds, this paper systematically investigates oil-film instability laws of an overhung rotor system with parallel and angular misalignments in the run-up and run-down processes. A finite element (FE) model of the overhung rotor system considering the gyroscopic effect is established, and the sliding bearings are simulated by a nonlinear oil-film force model based on the assumption of short-length bearings. Moreover, the effectiveness of the FE model is also verified by comparing our simulation results with the experimental results in the published literature. In the run-up and run-down processes with constant angular acceleration, the effects of parallel misalignment (PM) and angular misalignment (AM) on oil-film instability laws are simulated. The results show that under the perfectly aligned condition, the onsets of the first and second vibration mode instability in the run-down process are less than those in the run-up process due to the hysteresis effect. Under the misalignment conditions, the misalignment of the coupling can delay the onset of the first vibration mode instability and decrease its vibration amplitude. In comparison with the PM, the amplitudes of multiple frequency components are more obvious under the given AM conditions. Moreover, in the run-up and run-down processes with different misalignment conditions, the variation of the dominant vibration energy was observed according to the rotating frequency $$f_{\mathrm{r}}$$ , the first-mode whirl/whip frequency $$f_{\mathrm{n}1}$$ , the second-mode whirl/whip frequency $$f_{\mathrm{n}2}$$ , or the their combinations, such as $$f_{\mathrm{r}}$$ – $$2f_{\mathrm{n}2}$$ .

Establishment of Approximate Analytical Model of Oil Film Force for Finite Length Tilting Pad Journal Bearings

Tilting pad bearings offer unique dynamic stability enabling successful deployment of high-speed rotating machinery. The model of dynamic stiffness, damping, and added mass coefficients is often used for rotordynamic analyses, and this method does not suffice to describe the dynamic behaviour due to the nonlinear effects of oil film force under larger shaft vibration or vertical rotor conditions. The objective of this paper is to present a nonlinear oil force model for finite length tilting pad journal bearings. An approximate analytic oil film force model was established by analysing the dynamic characteristic of oil film of a single pad journal bearing using variable separation method under the dynamicπoil film boundary condition. And an oil film force model of a four-tilting-pad journal bearing was established by using the pad assembly technique and considering pad tilting angle. The validity of the model established was proved by analyzing the distribution of oil film pressure and the locus of journal centre for tilting pad journal bearings and by comparing the model established in this paper with the model established using finite difference method.

Modified oil film force model for investigating motion characteristics of rotor-bearing system

The oil film force induced from bearings has a direct influence on the stable operation of a rotor system because the change in motion characteristics will lead to variation of the oil film force. An accurate oil film force model is therefore critical to predict the motion characteristic state of a rotor system. The linear oil film force model and the Muszynska model are employed by many scholars to predict linear force and nonlinear oil film force respectively. When the rotor system has a larger eccentricity ratio, the changing law of the rotor system is not predicted by the Muszynska model. In this paper an improved oil film force model is developed for the motion changing law. This model is based on short bearing assumption with the turbulence effect and rotor motion equation included. The motion law is well described by this model. In particular the dynamic characteristics can be described well by the modified model. To verify the simulation results of the modified model, an experiment is carried out. Finally, there is a comparison between theoretical and experimental results and it is found that the theoretical results are in good agreement with the experimental results. This paper provides a reference for further research of the motion changing law of rotor systems.

Numerical and experimental analysis of the first-and second-mode instability in a rotor-bearing system

Aiming at the oil film instability of the sliding bearing at high speeds, a rotor test rig is built to study the non-linear dynamic behaviours caused by the first- and second-mode instability. A lumped mass model (LMM) of the rotor system considering the gyroscopic effect is established, in which the graphite self-lubricating bearing and the sliding bearing are simulated by a spring–damping model and a nonlinear oil film force model based on the assumption of short bearings, respectively. Moreover, a finite element model is also established to verify the validity of the LMM. The researches focus on the effects of two loading conditions (the first- and second-mode imbalance excitation) on the onset of instability and nonlinear responses of the rotor-bearing system by using the amplitude–frequency response, spectrum cascade, vibration waveform, orbit, and Poincare map. Finally, experiments are carried out on the test rig. Simulation and experiment all show that oil film instability can excite complicated combination frequency components about the rotating frequency and the first-/second-mode whirl/whip frequency.

Nonlinear dynamic analysis of a rotor-bearing-seal system under two loading conditions

The operating speed of the rotating machinery often exceeds the second or even higher order critical speeds to pursue higher efficiency. Thus, how to restrain the higher order mode instability caused by the nonlinear oil-film force and seal force at high speed as far as possible has become more and more important. In this study, a lumped mass model of a rotor-bearing-seal system considering the gyroscopic effect is established. The graphite self-lubricating bearing and the sliding bearing are simulated by a spring-damping model and a nonlinear oil-film force model based on the assumption of short bearings, respectively. The seal is simulated by Muszynska nonlinear seal force model. Effects of the seal force and oil-film force on the first and second mode instabilities are investigated under two loading conditions which are determined by API Standard 617 (Axial and Centrifugal Compressors and Expander-compressors for Petroleum, Chemical and Gas Industry Services, Seventh Edition). The research focuses on the effects of exciting force forms and their magnitudes on the first and second mode whips in a rotor-bearing-seal system by using the spectrum cascades, vibration waveforms, orbits and Poincaré maps. The first and second mode instability laws are compared by including and excluding the seal effect in a rotor system with single-diameter shaft and two same discs. Meanwhile, the instability laws are also verified in a rotor system with multi-diameter shaft and two different discs. The results show that the second loading condition (out-of-phase unbalances of two discs) and the nonlinear seal force can mainly restrain the first mode instability and have slight effects on the second mode instability. This study may contribute to a further understanding about the higher order mode instability of such a rotor system with fluid-induced forces from the oil-film bearings and seals.

Effects of eccentric phase difference between two discs on oil-film instability in a rotor–bearing system

The operating speed of the rotating machinery often exceeds the second order or even higher order critical speeds to pursue higher efficiency. Thus, the second mode whirl/whip can appear when the operating speeds approach or exceed twice the second order critical speed according to the reference A. Muszynska (2005) [1]. In this study, we investigate how the eccentric phase difference between two discs influence the oil-film instability (the first/second mode whirl/whip) in a rotor–bearing system. Firstly, a lumped mass model with 20 degrees of freedom (DOFs) of a rotor system with two discs considering the gyroscopic effect is developed. The graphite bearing and sliding bearing are simulated by a spring-damping model and a nonlinear oil-film force model based on the assumption of short bearings, respectively. The research focuses on the effect of eccentric phase differences of two discs on the onset of instability and nonlinear responses of the rotor–bearing system by using the bifurcation diagrams, spectrum cascades, vibration waveforms, orbits and Poincaré maps. The results show that the instability speed increases when the eccentric phase difference becomes larger and it increases almost linearly when the eccentric phase difference is greater than 20°. Moreover, complicated combination frequency components related to the first and second mode whirl/whip frequencies are also excited and the transfer of the self-excited vibration energy between the first and second mode whips can be observed under a larger eccentric phase difference. The study may contribute to a further understanding of the oil-film instability of such a rotor–bearing system.

Eccentricity and Rotational Speed Effect on the Rotor-Bearing

Bearing dimensionless nonlinear oil film force model is deduced based on Capone theory of cylindrical bearings in this paper. Jeffcot rigid rotor-bearing system dynamic equations are built based on nonlinear dynamics, bifurcation, chaos theory. Eccentricity increases with the speed of the system by writing MATLAB codes. It appears the periodic motion, times of periodic motion and a series of non-linear kinetics. The system eccentricity increases with a series of emergence of non-linear dynamics when speed conditions is fixed, which is the actual system design’s basis. The finite element model of gas turbine rotor-bearing system is built by ANSYS software platform in this paper. The radial bearing deformation relationship are obtained by deformation theory of centrifugal force at high speed bearing radial deformation.

Coupled Bending-Torsional Vibration Analysis of Rotor System with Two Asymmetric Disks

The dynamic equations derived based on the actual rotor system with two asymmetric disks. In the analysis, the eccentric, rubbing fault characteristics and internal damping effects is considered, and all the analysis is established based on nonlinear oil film force model and coupled bending-torsional differential equations. The Rugge-Kutta method is used to solve numerical model, the torsional displacement response, torsion angle and Poincare map are obtained. The results show torsion amplitudes with initial phase difference π / 2 is larger than initial phase difference of π and 0. In order to eliminate the rigid rolling component the relative torsional angle must be considered.

Approximate analytical model for fluid film force of finite length plain journal bearing

Non-linear dynamic performance of rotor–bearing systems supported by plain journal bearings strongly depends on the mathematical oil film force model. In this article, the analytical solution of oil film pressure for finite length plain journal bearing is obtained by employing the separation of variables method to analytically solve the Reynolds equation based on dynamic Gümbel boundary conditions. The analytical expression of oil film force is then derived by applying the integral method. The expression of the pressure is analysed to investigate the pressure distribution. The oil film force of the analytical model is compared with the results from other methods, namely, long bearing approximation, short bearing approximation, as well as the finite difference method. The results clearly validate the current model. The proposed model also proved to be efficient for analysing the dynamic characteristics of a rigid rotor supported by plain journal bearings.

A novel nonlinear model of rotor/bearing/seal system and numerical analysis

Abstract A novel nonlinear model of rotor/bearing/seal system based on the Hamilton principle is proposed for steam turbine systems in power plants. The Musznyska model and unsteady bearing oil-film force model were applied to describe the nonlinear steam excitation force and oil-film force. The Runge–Kutta method was used to solve the motion equation of the rotor/seal/bearing system. The dynamic characteristics of the rotor/bearing/seal system were analyzed with bifurcation diagrams, time-history diagrams, trajectory diagrams, Poincare maps and frequency spectrums. The numerical analysis indicates that the seal force and the oil-film force influence the nonlinear dynamic characteristics of the rotor system. With the increase of rotation speed, the rotor system exhibits rich forms of periodic, double-periodic, multi-periodic, quasi-periodic and chaotic motion. The combined impact of steam excitation force and bearing oil-film force may cause a severe vibration which seriously affects the safety and stability of the rotor. The presented model provides a theoretic foundation for further research on the ultra-supercritical steam turbines.

Analysis of motion stability of the flexible rotor-bearing system with two unbalanced disks

The complicated dynamical behavior of a flexible rotor-bearing system is studied in this paper. The unsteady oil-film force model described by three functions is considered. The bifurcation and chaos behaviors were revealed by calculating the maximum Lyapunov exponent of the system. Two new phenomena were found in this system: first, the chaos with two attracting areas which cannot be distinguished from the stable period doubling motion on Poincarè section; second, for the flexible rotor system with two unbalanced disks, the response varies in a large extent when the phase angle between the eccentricities of disks is different. The experiments were also carried out. Comparison between experimental and calculated results shows that the significant use of the max Lyapunov exponent in revealing the bifurcation and chaos characteristics of the rotor-bearing system.