Application of a hybrid numerical method to the nonlinear dynamic analysis of a micro gas bearing system

This paper employs a hybrid numerical method combining the differential transformation method (DTM) and the finite difference method (FDM) to analyze the nonlinear dynamic behavior of a rigid rotor supported by a micro gas bearing system (MGBS). It is shown that the results obtained for the orbits of the rotor center are in good agreement with those obtained using the traditional FDM approach. Moreover, the hybrid method avoids the divergence problem suffered by the FDM scheme at low values of the rotor mass and computational time step. The results presented in this study summarize the changes which take place in the dynamic behavior of the MGBS as the rotor mass and bearing number are increased over the ranges 0.01–0.16 and 1.0–5.0, respectively, and therefore provide a useful insight into the nonlinear dynamics of the bearing system.     

Application of a hybrid numerical method to the bifurcation analysis of a rigid rotor supported by a spherical gas journal bearing system

The nonlinear dynamic behavior of a rigid rotor supported by a spherical gas journal bearing system is analyzed using a hybrid method combining the differential transformation method (DTM) and the finite difference method (FDM). The analytical results reveal that the bearing system has a complex dynamic behavior comprising periodic, subharmonic, and quasi-periodic responses of the rotor center. The evolution in the dynamic behavior of the bearing system is systematically examined as the rotor mass and bearing number are increased. The analytical results are found to be in good agreement with those of other numerical methods. Hence, the validity of the proposed hybrid method as a means of gaining insights into the nonlinear dynamics of spherical gas film rotor-bearing systems is confirmed.     

Bifurcation and nonlinear dynamic analysis of a rigid rotor supported by two-lobe noncircular gas-lubricated journal bearing system

This paper presents the study of the dynamic analysis of a rigid rotor supported by a two-lobe non-circular gas-lubricated journal bearing. A finite element method has been employed to solve the Reynolds equation in static and dynamical states and the dynamical equations have been solved using Runge–Kutta method. To analyze the behavior of the rotor center in horizontal and vertical directions under the different operating conditions, the dynamic trajectory, the power spectra, the Poincare maps, and the bifurcation diagrams are used. Results of this study indicates that by considering bearing number and rotor mass as the parameters of the system, complex dynamic behavior comprising periodic, KT-periodic, and quasi-periodic responses of the rotor center has occurred.     

Vibration characteristics of a hydraulic generator unit rotor system with parallel misalignment and rub-impact

The object of this research aims at the hydraulic generator unit rotor system. According to fault problems of the generator rotor local rubbing caused by the parallel misalignment and mass eccentricity, a dynamic model for the rotor system coupled with misalignment and rub-impact is established. The dynamic behaviors of this system are investigated using numerical integral method, as the parallel misalignment, mass eccentricity and bearing stiffness vary. The nonlinear dynamic responses of the generator rotor and turbine rotor with coupling faults are analyzed by means of bifurcation diagrams, Poincaré maps, axis orbits, time histories and amplitude spectrum diagrams. Various nonlinear phenomena in the system, such as periodic, three-periodic and quasi-periodic motions, are studied with the change of the parallel misalignment. The results reveal that vibration characteristics of the rotor system with coupling faults are extremely complex and there are some low frequencies with large amplitude in the 0.3–0.4× components. As the increase in mass eccentricity, the interval of nonperiodic motions will be continuously moved forward. It suggests that the reduction in mass eccentricity or increase in bearing stiffness could preclude nonlinear vibration. These might provide some important theory references for safety operating and exact identification of the faults in rotating machinery.     

推力主动磁轴承的动特性参数辨识

针对五自由度刚性非对称磁轴承 -转子系统 ,建立了系统动力学模型 ,在此基础上推导出系统动特性参数辨识公式 ,并用多频电流激励法辨识出参数 .与理论计算结果相比表明 :系统所建模型符合实际 ,辨识方法有效 ;径向轴承在 x和 y方向上的力 -刚度和力 -电流系数相同 ,与推力轴承对应系数相比稍大 ;推力轴承对转子横向振动产生的耦合效应明显影响系统的稳定性 ,在设计系统时必须加以考虑     

An optimised short bearing theory for nonlinear dynamic analysis of turbulent journal bearings

An approximate method for nonlinear dynamic analysis of turbulent journal bearings supporting an unbalanced rigid shaft is proposed. The method is based on two assumptions: the separation of variables for pressure and a parabolic pressure distribution in the axial direction of the bearing. To take into account inertia effects, the well-known algebraic turbulent model based on the Prandtl mixing length hypothesis is used. Using the Constantinescu's approach, the pressure equation is modified by introducing two turbulent coefficients which are depending on the local Reynolds number. The nonlinear equations of motion for the rotor-bearing system are solved by means of the Euler's scheme, and the journal centre trajectories are examined for cases with and without unbalance forces. To illustrate the validity of the present study, three cases of journal bearings are analysed. The accuracy of the minimum film thickness, the dynamic transmissibility coefficient and the peak-to-peak displacement amplitudes obtained by the proposed methodology are comparable to the more elaborate and time consuming 2-D finite difference solution, while the turbulent journal centre orbits are comparable to those obtained experimentally.It is concluded that the optimised short bearing theory shows the advantage of minimising the computation time required for nonlinear dynamic analysis of laminar and turbulent journal bearings without any significant loss of accuracy.     

Dynamic characteristics of self-acting gas bearing–flexible rotor coupling system based on the forecasting orbit method

This paper studies the nonlinear dynamic characteristics of a flexible rotor supported by self-acting gas bearings theoretically. The multiple degree freedom model of flexible rotor is established by the finite element method and analyzed coupled with the transient gas lubricated Reynolds equation by employing the forecasting orbit method. The Reynolds equation is solved by the alternating direction implicit method and the dynamic response of the rotor is calculated by the Newmark integral method. To settle the problem that the two kinds of transient solving processes (transient Reynolds equation for bearing and transient equation of motion for rotor) cannot be solved simultaneously, which arises from the fact that they need each other??s results as their initial values, the multi-field coupling algorithm based on the forecasting method is proposed and applied in this paper. By employing the numerical method, the rotor trajectory diagram, phase diagram, frequency spectrum, power spectrum, bifurcation diagram, and vibration mode diagram were obtained. It is to note that the dynamic characteristics of self-acting gas bearing?Crotor system and whirling instability of the system could be depicted successfully. This would establish the foundation for contributing to a further understanding of the gas bearing?Cflexible rotor system.     

Steady-state behaviour of flexible rotordynamic systems with oil journal bearings

This paper deals with the long term behaviour of flexible rotor systems, which are supported by nonlinear bearings. A system consisting of a rotor and a shaft which is supported by one oil journal bearing is investigated numerically. The shaft is modelled using finite elements and reduced using a component mode synthesis method. The bearings are modelled using the finite-length bearing theory. Branches of periodic solutions are calculated for three models of the system with an unbalance at the rotor. Also self-excited oscillations are calculated for the three models if no mass unbalance is present. The results show that a mass unbalance can stabilize rotor oscillations.     

外弹性支承滑动轴承—刚性转子系统非线性动力稳定性的研究

研究外弹性支承滑动轴承－刚性转子系统的动力稳定性问题。建立了弹性支承圆瓦轴承－Jeffcot转子系统的力学模型,针对滑动轴承水平转子,在油膜力非线性的情况下,采用Newmark方法计算其响应。对于一个具体的算例,通过改变其参数,确定其稳定区,并与滑膜力线性化时理论分析的结果进行了比较。     

弹性支承滑动轴承不平衡转子系统非线性动力稳定性和分岔的研究

研究弹性支承滑动轴承不平衡转子系统的稳定性及分岔特性。建立了弹性支承-滑动轴承-转子非线性动力系统的力学模型，在油膜力非线性的情况下，应用数值模拟，采用打靶法计算了刚性转子系统的周期解，并与龙格-库塔法计算的结果进行了对比，依据Floquet理论，分析了周期解的稳定性，再结合龙格-库塔法、Poineare映射法作出了系统运动分岔图。最后，讨论了轴的柔性对转子系统运动稳定性的影响。     

Bifurcation and chaos for porous squeeze film damper mounted rotor–bearing system lubricated with micropolar fluid

This study presents a dynamic analysis of a flexible rotor supported by two porous squeeze micropolar fluid-film journal bearings with nonlinear suspension. The dynamics of the rotor center and bearing center are studied. The analysis of the rotor–bearing system is investigated under the assumptions of non-Newtonian fluid and a short bearing approximation. The spatial displacements in the horizontal and vertical directions are considered for various nondimensional speed ratios. The dynamic equations are solved using the Runge–Kutta method. The methods of analysis employed in this study are inclusive of the dynamic trajectories of the rotor center and bearing center, power spectra, Poincaré maps, and bifurcation diagrams. The maximum Lyapunov exponent analysis is also used to identify the onset of chaotic motion. The numerical results show that the stability of the dynamic system varies with the nondimensional speed ratios, the nondimensional parameter, and permeability. The modeling results obtained by using the method proposed in this paper can be employed to predict the dynamics of the rotor–bearing system, and the undesirable behavior of the rotor and bearing centers can be avoided.     

Experimental and numerical analysis of nonlinear dynamics of rotor–bearing–seal system

Nonlinear dynamics and stability of the rotor–bearing–seal system are investigated both theoretically and experimentally. An experimental rotor–bearing–seal device is designed and corresponding tests are carried out. The experimental rotor system is simplified as the Jeffcott rotor. The nonlinear oil–film forces are obtained under the short bearing theory and Muszynska nonlinear seal force model is used. Numerical method is utilized to solve the nonlinear governing equations. Bifurcation diagrams, waterfall plots, Poincaré maps, spectrum plots and rotor orbits are drawn to analyze various nonlinear phenomena and system unstable processes. Theoretical results from numerical analysis are in good agreement with results from experiments. Conclusions are drawn and prove that this study will contribute to the further understanding of nonlinear dynamics and stability of the rotor system with the fluid-induced forces from oil–film bearings and the seals.     

Nonlinear dynamic analysis of a rotor system with fixed-tilting-pad self-acting gas-lubricated bearings support

Based on the nonlinear theory, the unbalanced responses of a fixed-tilting-pad gas-lubricated journal bearing-rigid rotor system are investigated. A?time-dependent mathematical model is established to describe the pressure distribution of gas-lubricated journal bearing with nonlinearity. The rigid rotor supported by a fixed-tilting-pad self-acting gas-lubricated journal bearing is modeled. The differential transformation method has been employed to solve the time-dependent gas-lubricated Reynolds equation, and the dynamic motion equation has been solved by the direct integral method. The unbalanced responses of the rotor system supported by fixed-tilting-pad gas-lubricated journal bearings are analyzed by the orbit diagram, Poincaré map, time series, and spectrum diagram. The numerical results reveal periodic, period-3, and quasiperiodic motions of nonlinear behaviors of the system. Finally, the effects of pivot ratio and preload coefficient on the nonlinear dynamic characteristics of the fixed-tilting-pad self-acting gas-lubricated journal bearing-rotor system are investigated.     

涡轮泵转子-迷宫密封系统的非线性稳定性和分岔

研究迷宫密封对某一工程涡轮泵转子系统动力特性的影响,迷宫密封力采用Muszynska非线性力模型,应用有限元法建立转子系统的动力学方程,采取系统动力学方程中包含的高阶线性自由度和低阶的非线性自由度进行分块处理的方法,有效地缩短了求解时间。根据Floquet理论,判别系统的临界失稳转速,并由Floquet乘子来确定系统失稳后分岔方式。采用分块-Newmark法数值模拟了转子二涡轮盘的轴心轨迹。最后分析了涡轮泵转子二涡轮盘的质量偏心同相位和存在90度相位差时对转子系统运动特性的影响。     

磁悬浮挠性转子系统模型复合辨识方法

针对磁悬浮挠性转子的纯有限元分析模型精度低的问题,提出一种磁悬浮挠性转子系统模型复合辨识方法。该方法首先采用有限元方法把磁悬浮挠性转子划分为多个Timoshenko梁单元,建立磁悬浮挠性转子系统模型,并对其挠性临界频率、阻尼、刚度和振型等关键参数进行分析,进而得到磁悬浮挠性转子的等效降阶数学模型;然后采用鲁棒自适应方法分析磁悬浮挠性转子系统的动态特性;最后,采用变LEVY方法从动态特性分析数据中对磁悬浮挠性转子进行系统辨识,校正有限元分析得到的降阶数学模型。实验结果表明,本方法可以得到磁悬浮挠性转子较为准确的系统模型。     

非线性轴承转子系统稳定性准则的研究

目前对于非线性轴承转子系统，仍普遍采用其线性化系统的对数衰减率作为系统的稳定性判据，这造成了理论和实验结果相差较大。本文对［１］中提出的ＰＣＭ法作了改进，通过对无限长滑动轴承支承对称刚性单盘转子系统的非线性稳定性规律进行的分析，揭示了上述现象的非线性本质，为建立更适用于非线性轴承转子系统的稳定性准则提供参考。     

滚动轴承-偏置转子系统动力特性数值分析与实验研究

应用有限元法建立偏置转子的计算模型,采用考虑轴承Hertzian接触力和内间隙等非线性因素的二自由度滚动轴承模型,建立了滚动轴承-偏置转子系统的非线性动力学模型.通过数值仿真和实验研究分析了转子系统的非线性动力特性.实验数据和有限元模型计算结果是一致的,证实了所建立滚动轴承-转子系统非线性模型的合理性.发现由于滚动轴承非线性因素的影响,当转速达到系统共振转速的两倍附近时,激起了系统亚谐共振.     

Preload effect on nonlinear dynamic behavior of a rigid rotor supported by noncircular gas-lubricated journal bearing systems

This paper presents the effect of preload, as one of the design parameters, on nonlinear dynamic behavior of a rigid rotor supported by gas-lubricated noncircular journal bearings. A finite element method has been employed to solve the Reynolds equation in static and dynamical states and the dynamical equations are solved using the Runge–Kutta method. To analyze the behavior of the rotor center in horizontal and vertical directions under different operating conditions, dynamic trajectory, power spectra, Poincare maps, and bifurcation diagrams are used. Results of this study reveal how the complex dynamic behavior of two types of noncircular bearing systems comprising periodic, KT-periodic, and quasi-periodic responses of the rotor center varies with changes in preload value.     

Comparison of methods for vibration analysis of electrostatic precipitators

The paper presents two methods for the formulation of free vibration analysis of collecting electrodes of precipitators.The first,called the hybrid finite element method, combines the finit element method used for calculations of spring deformations with the rigid finite element method used to reflect mass and geometrical features,which is called the hybrid finite element method.As a result,a model with a diagonal mass matrix is obtained.Due to a specific geometry of the electrodes,which are long plates of complicated shapes,the second method proposed is the strip method which is a semi-analytical method.The strip method allows us to formulate the equations of motion with a considerably smaller number of generalized coordinates.Results of numerical calculations obtained by both methods are compared with those obtained using commercial software like ANSYS and ABAQUS.Good compatibility of results is achieved.     

Non-linear dynamic analysis of the ultra-short micro gas journal bearing-rotor systems considering viscous friction effects

Due to the micro-fabrication limitations and the low thickness of the silicon wafer, the length-to-diameter ratio (L/D) of the gas journal bearings in Power MEMS is about one order lower than that of the conventional bearings, which suggests that the viscous friction force in the micro-bearing is comparable to the load capacity. The effects of viscous friction force on non-linear dynamic characteristics of the ultra-short micro-bearing-rotor system are studied in this paper. The molecular gas-film lubrication model, which valid for arbitrary Knudsen numbers, is systematically coupled with the rotor kinetic equations and solved simultaneously to investigate the non-linear dynamic behavior of the system. The center orbits, phase portraits, Poincaré maps, and FFT spectra of the system response at different L/D ratio, rotor mass, and bearing number, and the corresponding bifurcation diagrams for cases of ignoring and considering viscous friction force are inspected and compared. The results indicate that, if the viscous friction force is not taken into account in the case of low L/D ratio, the low-frequency large-amplitude self-excited whirl motion will be predicted as the increase of the rotor mass and the bearing number. However, when the viscous friction force is included in the non-linear dynamic model, the rotor motion becomes more stable under the same conditions, as the synchronous motion with smaller amplitude prevails.