Nonlinear Oil-film Force Research Articles

Nonlinear dynamic response of a rub-impact rod fastening rotor considering nonlinear contact characteristic

The nonlinear dynamic characteristics of a rub-impact rod fastening rotor have been investigated in this paper. A model of the rod fastening rotor bearing system under rub-impact condition is proposed considering nonlinear contact characteristic between disks, nonlinear oil-film force, unbalance mass, etc. The equation of motion of the system has been derived by D’Alembert principle. The contact effects between disks are modeled as a flexural spring with nonlinear stiffness. The dynamic equations of motion are solved using fourth-order Runge–Kutta method. Bifurcation diagram, vibration waveform, frequency spectrum, shaft orbit and Poincare map are used to illustrate the rich diversity of the system response with complicated dynamics. The studies indicate that it is unsuitable to take the rod fastening rotor as an integral rotor in analyzing the rub-impact dynamic response of the system. The subharmonic periodic motion, multiple periodic motion, quasi-periodic motion and chaotic motion are observed in this study. Larger radial stiffness of stator will simplify the system motion and make the oil whirl weaker or even disappear at a certain rotating speed. The corresponding results can provide the guidance for the fault diagnosis of a rub-impact rod fastening rotor; meanwhile, the study may contribute to the further understanding of the nonlinear dynamic characteristics of a rub-impact rod fastening rotor.

Global Dynamics Analysis of a Continuum Rotor through G-Function and LFunction Method

The global dynamics of a continuum rotor excited by nonlinear oilfilm force and electromagnetic force is investigated. The governing equation of the system is obtained using the Galerkin’s method. The dynamical systems are reduced into two 2-dimensional systems in Yand Z-directions. The analytical conditions for global transversality and tangency to the separatrix are presented. The global and local dynamics of the system is determined through the G-function and G(1)-function and is compared to numerical solution. The periodicity of periodic flow of the rotor is determined by the L-function. The complexity of the nonlinear continuum rotor system is demonstrated through global transversality and tangency of the periodic motions to separatrix.

Nonlinear dynamic behaviors of circumferential rod fastening rotor under unbalanced pre-tightening force

A nonlinear dynamic model of the rod fastening rotor bearing system is established considering nonlinear oil-film force, unbalanced mass, unbalanced rod pre-tightening force, etc. The motion equation of the system has been deduced from Lagrange’s equations. The nonlinear dynamic and bifurcation characteristic is investigated using fourth-order Runge–Kutta method. Bifurcation diagram, vibration waveform, frequency spectrum, phase trajectory and Poincare map are applied to analyze the nonlinear dynamic phenomena of the rod fastening rotor. The numerical results indicated that the initial deflection caused by the unbalanced pre-tightening force, nonlinear oil-film force and rotational speed has a great influence on the nonlinear dynamic characteristics of the rod fastening rotor bearing system. The corresponding results can provide the guidance for the fault diagnose of a rod fastening rotor with unbalanced pre-tightening force; meanwhile, the study may contribute to the further understanding of the nonlinear dynamic characteristics of a rod fastening rotor with unbalanced pre-tightening force.

Nonlinear Dynamics Analysis of a Continuum Rotor through Generalized Harmonic Balance Method

The dynamics of a continuum rotor system excited by nonlinear oilfilm force and electromagnetic force is investigated through the Generalized Harmonic Balance Method. The governing differential equation of the continuum rotor system is reduced through the Galerkin’s approximation method. Firstly, the motion in Y-direction is considered as the response of a planar continuum rotor. The analytical periodic motion is obtained and the bifurcation and stability is determined. Secondly, the dynamical model of a spatial continuum rotor is studied. The stability and bifurcation of the analytical periodic motion is obtained and compared to numerical results to show the accuracy of the Generalized Harmonic Balance Method.

スクイズフィルムダンパのモデル化と実験的検証

Squeeze film damper is used in high-speed rotating machinery such as aircraft engines and gas turbine to suppress the vibration amplitudes of rotor response. However, non-linear characteristics of SFD has a large influence on the vibration characteristics of the system. This paper deals with the vibration analysis of a flexible shaft supported by SFD. Particularly, the case that the oil supply system is oil bath type is considered. In such case, the existing range of the oil film of SFD is limited to the bottom 120 degree of the SFD. This paper calculated the nonlinear oil film force of this type of SFD, and analyzed the rotor vibration by numerical simulation. The obtained results were compared with experimental results, and discussed.

Effects of unbalance location on dynamic characteristics of high-speed gasoline engine turbocharger with floating ring bearings

For the high-speed gasoline engine turbocharger rotor, due to the heterogeneity of multiple parts material, manufacturing and assembly errors, running wear in impeller and uneven carbon of turbine, the random unbalance usually can be developed which will induce excessive rotor vibration, and even lead to nonlinear vibration accidents. However, the investigation of unbalance location on the nonlinear high-speed turbocharger rotordynamic characteristics is less. In order to discuss the rotor unbalance location effects of turbocharger with nonlinear floating ring bearings(FRBs), the realistic turbocharger of gasoline engine is taken as a research object. The rotordynamic equations of motion under the condition of unbalance are derived by applied unbalance force and nonlinear oil film force of FRBs. The FE model of turbocharger rotor-bearing system is modeled which includes the unbalance excitation and nonlinear FRBs. Under the conditions of four different applied locations of unbalance, the nonlinear transient analyses are performed based on the rotor FEM. The differences of dynamic behavior are obvious to the turbocharger rotor systems for four conditions, and the bifurcation phenomena are different. From the results of waterfall and transient response analysis, the speed for the appearance of fractional frequency is not identical and the amplitude magnitude is different from the different unbalance locations, and the non-synchronous vibration does not occur in the turbocharger and the amplitude is relative stable and minimum under the condition 4. The turbocharger vibration and non-synchronous components could be reduced or suppressed by controlling the applied location of unbalance, which is helpful for the dynamic design, fault diagnosis and vibration control of the high-speed gasoline engine turbochargers.

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.

Dynamic simulation and experimental study of an asymmetric double-disk rotor-bearing system with rub-impact and oil-film instability

This study deals with numerical analysis and experimental investigations where the nonlinear dynamic behaviors of an asymmetric double-disk rotor-bearing system are reproduced considering rub-impact and oil-film instability. A dedicated experimental setup was designed to validate the interaction between rubs and oil-film vibration. A model considering nonlinear rub-impact forces and nonlinear oil-film forces is put forward to analyze the complicated nonlinear vibrations of the rotor-bearing system. The experimental study focuses on the interaction of oil-whirl/whip and rub-impact faults, and the results show that the oil-whirl is restrained or even removed, but the whip is uninfluenced when the rub becomes heavy contact. The numerical simulation focuses on the effect of eccentricity and the stator stiffness on the onset of instability and nonlinear responses of the rotor-bearing system by using bifurcation diagrams, Poincare maps, frequency spectrum and three-dimension spectral plots. Agreeing with the experimental results, the results from simulation indicate that the motion of system becomes simple and the oil-whirl gets diminished or even disappeared, but the whip keeps unchanged when the eccentricity or the stator stiffness increases. The research discloses the worthy energy exchange phenomenon of multi-fault system and presents the nonlinear dynamical characteristics of the interaction between rub-impact and oil-film instability for such a rotor-bearing system.

Nonlinear model identification of oil-lubricated tilting pad bearings

A new method of modeling and identification of weak nonlinear behavior of tilting pad journal bearings (TPJBs) is proposed. A novel nonlinear model of a TPJB based on a Taylor series expansion of oil film forces, consistent with the transient form of Reynolds differential equation (the original model), is developed. The least mean square technique in time domain is employed to identify Taylor series coefficients of the nonlinear model. The terms with dominant effects on the nonlinear oil film forces are chosen using the subset selection method. Good agreement is achieved between the predicated response obtained from the original model and the one evaluated through using the identified nonlinear model.

Nonlinear analysis of rotor system supported by oil lubricated bearings subjected to base movements

Response analysis of rotor system subjected to the base movements in a large rotational machinery is a complex high-dimensional nonlinear problem. In this paper, rotor system of a centrifugal pump with a pair of oil lubricated bearings is modeled as a six-degrees-of-freedom nonlinear system by the Lagrange method. The Runge–Kutta–Felburg method with variable step size is applied to solve the equations based both on nonlinear bearing model and on linearized bearing model. Poincaré maps, phase trajectories, bifurcation diagrams, and cascade diagrams are used to illustrate the dynamics of the rotor with the two bearing models. The responses of these two kinds of systems under El Centro seismic wave are obtained. The acceleration data of the seismic wave are used as the input of base movements. The short bearing model is applied to express the nonlinear oil film force. Compared to the system with model of linearized bearing, the nonlinear bearing–rotor system can accurately reflect the dynamics of the system. Furthermore, trigonometric functions are employed to express the base movements, and the responses of nonlinear bearing–rotor system subject to the base movements are obtained. The fast Fourier transform is used to analyze the nonlinear bearing–rotor system with effects of the base movements. The results reveal not only the dynamic behavior of the rotor of the centrifugal pump supported by two nonlinear bearings, but also the responses of the nonlinear bearing–rotor system with the combined effects of centrifugal force and base movements.

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 $$f_{\mathrm{r}}$$ – $$2f_{\mathrm{n}2}$$ .

Nonlinear dynamic behaviors of rod fastening rotor-hydrodynamic journal bearing system

The model of a rod fastening rotor-finite length journal bearing system is established in this paper. The contact surfaces between the disks and rods are modeled as bending resistance springs with nonlinear stiffness. The gyroscopic effect is considered in the 12 degrees-of-freedom model of the rod fastening rotor-bearing system. Based on the variational principle, an approximate analytical solution of nonlinear oil film forces of finite length journal bearing is proposed by the separation of variables method. By taking rotating speed, eccentricity and bending stiffness as control parameters, the nonlinear behaviors of the system are investigated by the improved Newmark method. The numerical results reveal periodic, period-doubling, period-4, quasi-periodic solutions etc. of rich and complex nonlinear behaviors of the system.

Study on Rub-Impact Dynamical Behavior of An Asymmetrical Rotor-Bearing with Variable Ratio of Stiffness

A rub-impact mathematic model of an Asymmetrical rotor-bearing with nonlinear oil film force is built in the paper. The bifurcation diagrams of the response were given following the changing of ratio of stiffness. We analyzed the bifurcation and the chaos character of an Asymmetrical rotor-bearing with operating rotor stiffness changing, then we get the dynamical character and the law when the rubbing happen. The nonlinear dynamic behaviors of the system were studied by using the numerical value integral and Poincare mapping methods. By analysis the process of rub-impact, it is so complex and many low frequencies with large amplitude are in the range of 0.3~0.6 X. The way of period-chaos-period and the phenomena of diverging backward whirl are discovered when the rubbing happen. These results provide important theoretical references for the safety operation of generating sets and the exact identification of the faults in rotating machinery.

Dynamic Analysis on Nonlinear Fluid-Structure Interaction Forces of Rub-Impact Rotor System

Based on the coupling model of nonlinear oil-film force and nonlinear seal fluid force, a nonlinear dynamic model of rotor system with rub-impact fault is set up. The dynamic characteristics of the system were studied with numerical simulation and the effects of airflow excited force, rubbing gap and stiffness parameters on movement characteristics of the rotor were analyzed. The results indicate that the airflow excited force can significantly restrain the stability and amplitude of rubbing rotor. The less rubbing gap and larger rubbing stiffness are in favor of the stability of the system.

Approximate solution of oil film load-carrying capacity of turbulent journal bearing with couple stress flow

The instability of the rotor dynamic system supported by oil journal bearing is encountered frequently, such as the half-speed whirl of the rotor, which is caused by oil film lubricant with nonlinearity. Currently, more attention is paid to the physical characteristics of oil film due to an oil-lubricated journal bearing being the important supporting component of the bearing-rotor systems and its nonlinear nature. In order to analyze the lubrication characteristics of journal bearings efficiently and save computational efforts, an approximate solution of nonlinear oil film forces of a finite length turbulent journal bearing with couple stress flow is proposed based on Sommerfeld and Ocvirk numbers. Reynolds equation in lubrication of a finite length turbulent journal bearing is solved based on multi-parametric principle. Load-carrying capacity of nonlinear oil film is obtained, and the results obtained by different methods are compared. The validation of the proposed method is verified, meanwhile, the relationships of load-carrying capacity versus eccentricity ratio and width-to-diameter ratio under turbulent and couple stress working conditions are analyzed. The numerical results show that both couple stress flow and eccentricity ratio have obvious influence on oil film pressure distribution, and the proposed method approximates the load-carrying capacity of turbulent journal bearings efficiently with various width-to-diameter ratios. This research proposes an approximate solution of oil film load-carrying capacity of turbulent journal bearings with different width-to-diameter ratios, which are suitable for high eccentricity ratios and heavy loads.

Nonlinear dynamic behaviors of a rod fastening rotor supported by fixed–tilting pad journal bearings

A rod fastening rotor with the fixed–tilting pad journal bearings support is modeled. The rotor is connected by rods, where the rods and connection surfaces of disks are considered as resistance bending springs with nonlinearity, and the inertia of the tilting pads is considered in the modeling. The nonlinear oil film forces of the fixed–tilting pad journal bearings are obtained by the database method. The nonlinear dynamic behaviors and bifurcation of the rotor-bearing system is investigated by the orbit diagrams, the time series, the frequency spectrum diagrams, and the Poincaré maps. The comparison of the orbits between the rod fastening rotor and integral rotor system is implemented, and the results reveal that the rod fastening rotor system is more stable than the integral rotor system. The orbits of the rotor system and the pendulum angles of pads are also studied at different preloads and pivot ratios.

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.

A new nonlinear dynamic analysis method of rotor system supported by oil-film journal bearings

The strong nonlinear behavior usually exists in rotor systems supported by oil-film journal bearings. In this paper, the partial derivative method is extended to the second-order approximate extent to predict the nonlinear dynamic stiffness and damping coefficients of finite-long journal bearings. And the nonlinear oil-film forces approximately represented by dynamic coefficients are used to analyze nonlinear dynamic performance of a symmetrical flexible rotor-bearing system via the journal orbit, phase portrait and Poincaré map. The effects of mass eccentricity on dynamic behaviors of rotor system are mainly investigated. Moreover, the computational method of nonlinear dynamic coefficients of infinite-short bearing is presented. The nonlinear oil-film forces model of finite-long bearing is validated by comparing the numerical results with those obtained by an infinite-short bearing-rotor system model. The results show that the representation method of nonlinear oil-film forces by dynamic coefficients has universal applicability and allows one easily to conduct the nonlinear dynamic analysis of rotor systems.

Nonlinear behavior analysis of geared rotor bearing system featuring confluence transmission

In this paper, the nonlinear vibration characteristics of geared rotor bearing system and the interactions among gears, shafts, and plain journal bearings were studied. First, with the consideration of backlash, transmission error, time-varying mesh stiffness, and layout parameters, the dynamic model of geared rotor bearing system featuring confluence transmission was proposed. The nonlinear oil-film forces were computed with the Reynolds equation for finite-length journal bearings. Second, the responses of meshing vibration and bearing vibration were discussed. The numerical results revealed that the system exhibited a diverse range of periodic, sub-harmonic, and chaotic behaviors. Under different ranges of rolling frequency, the system got into chaos state through different roads. Moreover, in lower frequency, meshing vibration showed coexist of different periodic motions. Lastly, couplings of nonlinear oil-film force and nonlinear gear mesh force were discussed through a range of rolling frequencies. Gear-bearing dynamic interactions were demonstrated through the analysis of dynamic gear loads and dynamic bearing loads, and the coupling effect behaved different when rolling frequency changed.

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.