柔性转子-轴承系统非线性动力学行为分析

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基于LabVIEW的滑动轴承转子系统实验设计

第２期
张宏献等：于ＬｂＩＷ的滑动轴承转子系统实验设计基ａＶＥ
了滑动轴承转子系统综合实验台，可实现测量转子系统的转速、轴心轨迹、温度、频谱图、振幅图、相位等动
力学特征，要用于考察滑动轴承圆度误差、主润滑油温粘热效应等因素对滑动轴承转子系统稳定性影响．本实验台的开发对滑动轴承转子动力学的教学科研及工程实际应用都有一定的参考意义．
术界所接受．ＬｂＩＷ软件是一种典型的虚拟仪器开发工具，由于其直观的图形化编程语言及强大的计算功能，ａＶＥ
也越来越多地被应用于轴承转子系统动力学实验．献［］用ＬｂＥ软件编制转子实验台数据采集分文２利ａＶＩＷ析程序。根据转子振幅和相位随转速在临界转速突变的性质，而得到转子系统临界转速．从文献［】［］３、４利用ＬｂｌＷ软件实现对滑动轴承油膜分布情况及油膜压力大小的测量．ａＶＥ文献［］于几种常用的转子动平５基
第２卷第２期２２１０１年６月
广西工学院学报
Ｊ０ＵＲＮＡＬＯＦＧＵＡＮＧ￣ＵＮＩＥＲＳＹＯＥＶｎ’ ＦＴＣＨＮＯＬＧＹＯ
Ｖ０Ｉ２．Ｎｏ２２．
Ｊｎ０ｌｕｅ２１
文章编号１０．４０２１）２０２—５０４６１（００．０８０１

文献翻译-非线性动力学的实验和转子轴承系统支持的行为的数值研究

附录A英文原文Experimental and Numerica Studies on Nonlinear Dynam Behavior of Rotor System Supported by Ball BearingsBall bearings are important mechanical components in high-speed turbomachinery that is liable for severe vibration and noise due to the inherent nonlinearity of ball ing experiments and the numerical approach, the nonlinear dynamic behavior of a ﬂexible rotor supported by ball bearings is investigated in this paper. An experimental ball bearing-rotor test rig is presented in order to investigate the nonlinear dynamic performance of the rotor systems, as the speed is beyond the ﬁrst synchroresonance frequency. The ﬁnite element method and two-degree-of-freedom dynamic model of a ball bearing are employed for modeling the ﬂexible rotor s ystem. The discrete model of a shaft is built with the aid of the ﬁnite element technique, and the ball bearing model includes the nonlinear effects of the Hertzian contact force, bearing internal clearance, and so on.The nonlinear unbalance response is observed by experimental and numerical analysis.All of the predicted results are in good agreement with experimental data, thus validating the proposed model. Numerical and experimental results show that the resonance frequency is provoked when the speed is about twice the synchroresonance frequency, while the subharmonic resonance occurs due to the nonlinearity of ball bearings and causes severe vibration and strong noise. The results show that the effect of a ball bearing on the dynamic behavior is noticeable in optimum design and failure diagnosis of high-speed turbomachinery. [DOI: 10.1115/1.4000586]Keywords: ball bearing, rotor, experiment, nonlinear vibrationA.1 IntroductionBall bearings are one of the essential and important components in sophisticated turbomachinery such as rocket turbopumps, aircraft jet engines, and so on. Because of the requirement of acquiring higher performance in the design and operation of ballbearings-rotor systems, accurate predictions of vibration characteristics of the systems, especially in the high rotational speed condition, have become increasingly important.Inherent nonlinearity of ball bearings is due to Hertzian contact forces and the internal clearance between the ball and the ring.Many researchers have devoted themselves to investigating the dynamiccharacteristics associated with ball bearings. Gustafsson et al. [1] studied the vibrations due to the varying compliance of ball bearings. Saito [2] investigated the effect of radial clearance in an unbalanced Jeffcott rotor supported by ball bearings using the numerical harmonic balance technique. Aktürk et al. [2] used a three-degree-of-freedom system to explore the radial and axial vibrations of a rigid shaft supported by a pair of angular contact ball bearings. Liew et al. [4] summarized four different dynamic models of ball bearings, viz., two or ﬁve degrees of freedom, with or without ball centrifugal force, which could be applied to determine the vibration response of ball bearing-rotor systems. Bai and Xu [5] presented a general dynamic model to predict dynamic properties of rotor systems supported by ball bearings. De Mul et al. [6] presented a ﬁve-degree-of-freedom (5DOF) model for the calculation of the equilibrium and associated load distribution in ball bearings. Mevel and Guyader [7] described different routes to chaos by varying a control parameter. Jang and Jeong [8] proposed an excitation model of ball bearing waviness to investigate the bearing vibration. Then, considering the centrifugal force and gyroscopic moment of ball, they developed an analytical method to calculate the characteristics of the ball bearing under the effect of waviness in Ref. [9]. Tiwari et al. [10,11] employed a two-degree-of-freedom model to analyze the nonlinear behaviors and stability associated with the internal clearance of a ball bearing.Harsha [12-14], taking into account different sources of nonlin-earity, investigated the nonlinear dynamic behavior of ball bearing-rotor systems. Gupta et al. [15] studied the nonlinear dynamic response of an unbalanced horizontal ﬂexible rotor supported by a ball bearing. With the aid of the Floquet theory, Bai et al. [16] investigated the effects of axial preload on nonlinear dynamic characteristics of a ﬂexible rotor supported by angular contact ball bearings. Using the harmonic balance method, Sinou [17]performed a numerical analysis to investigate the nonlinear unbalance response of a ﬂexible rotor supported by ball bearings.In the abovementioned studies, main attention has been paid to the ball bearing modeling and the dynamic properties analysis according to simple bearing-rotor models. With theoretical analysis and experiment, Yamamoto et al. [18] studied a nonlinear forced oscillation at a major critical speed in a rotating shaft,which was supported by ball bearings with angular clearances.Ishida and Yamamoto [19] studied the forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping. They found that a self-excited oscillation appears in the wide range above the major critical speed. A dynamic model was derived, and experiments are carried out with a laboratory test rig for studying the misaligned effect of misaligned rotor-ball bearing systems in Ref. [20]. Tiwari et al. [21] presented an experimental analysis to study the effect of radial internal clearance of a ball bearing on the bearingstiffness of a rigid horizontal rotor. These experimental results validated theoretical results reported in their literatures [10,11]. Recently, Ishida et al. [22] investigated theoretically and experimentally the nonlinear forced vibrations and parametrically excited vibrations of an asymmetrical shaft supported by ball bearings. Mevel and Guyader [23] used an experimental test bench to conﬁrm the predicted routes to chaos in their previous paper [7]. It is noticeable of lack of experiments on nonlinear dynamic behavior of ﬂexible rotor systems supported by ball bearings. In Ref. [24], the ﬁnite element method was used to model a LH2 turbopump rotor system supported by ball bearings. Numerical results show that the subharmonic resonance, as well as synchroresonance, occurs in the start-up process. It is found that the subharmonic resonance is an important dynamic behavior and should be considered in engineering ball bearing-rotor system design. But, the experimental and numerical studies of the subharmonic resonance in ball bearing-rotor systems are very rare.With respect to the above, the present study is intended to cast light on the subharmonic resonance characteristics in ball bearing-rotor systems using experiments and numerical approach. An experiment on an offset-disk rotor supported by ball bearings is carried out, and the ﬁnite element method and two-degree-of-freedom model of a ball bearing are employed for modeling this rotor system. The predicted results are compared with the test data, and an investigation is conducted in the nonlinear dynamic behavior of the ball bearings-rotor system.2 Experimental InvestigationAn experimental rig is employed for studying the nonlinear dynamic behavior of ball bearing-rotor systems, as shown in Fig.1. The horizontal shaft is supported by two ball bearings at both ends, and the diskis mounted unsymmetrically. The shaft is coupled to a motor with a ﬂexible coupling. The motor speed is controlled with a feedback controller, which gets the signals from an eddy current probe. Four eddy current probes, whose resolution is 0.5 m, are mounted close to the disk and bearing at the right end in the horizontal and vertical directions, respectively. The displacement signals, obtained with the help of probes, are input into an oscilloscope to describe the motion orbit, and a data acquisition and processing system were used to analyze the effects of ball bearings on the nonlinear dynamic behavior. The data acquisition and processing system utilizes a full period sampling as the data acquisition method. Its sampling rate is 500 kHz maximum, and sample size is 12 bits. The system provides eight channels for vibratory response acquisition and 1 channel for rotational speed acquisition. All channels are simultaneous.The limitation with the presented experimental setup is that the maximum attainable speed is 12,000 rpm. The ﬁrst critical speed of the rotor system falls in the speed span, as the shaft is ﬂexible and its ﬁst synchroresonance frequency is near 66 Hz (3960rpm).Thus, the dynamic behavior can be studied as the speed is beyond twice the synchroresonance frequency.3 Rotor Dynamic ModelThe bearing-rotor system combines an offset-disk and two ball bearings, which support the rotor at both ends. The sketch map of the system is described in Fig. 2, where the frame oxyz is the inertial frame. The corresponding experiment assembly is shown in Fig. 3.3.1 Equations of Motion . Deﬁne ux and uy as the transverse deﬂections along the ox and oy directions, and x θ and y θ as the corresponding bending angles in the oxz and oyz planes, respectively. When x u 1, y u 1,x 1θ , and y 1θ denote the displacements of the ball bearing center location at the left end, the complex variables 1u and 1θ can be assumed asDenote the displacements of the disk center by 2u and 2θ, and the displacements of the ball bearing center location at the right end by 3u and 3θ. Using the ﬁnite element method, the equations of motion for the rotor system can be written as [25,26]where []M , []C , []K , and []G are the mass, damping, stiffness, and gyroscopic matrix of the rotor system, respectively, ω is the rotational speed, and {}u is the displacement vector{}g F and {}u F are the vectors of gravity load and unbalance forces.{}bF is the vector of nonlinear forces associated with ball bearings.3.2 Ball Bearing Forces. A ball bearing is depicted in a frame of axes oxyz in Fig.4. The contact deformation for the j-th rolling element j δis given aswhere i c and o c are the internal radial clearance between the inner,outer race, and rolling elements, respectively, in the direction of contact, and ubx and uby are the relative displacements of the inner and outer race along the x and y directions, respectively. As shown in Fig. 4, the angular location of the j-th rolling element j ϕ can be obtained fromWhere N , c ω, t , and 0ϕ are the number of rolling elements, cage angular velocity, time, and initial angular location, respectively. The cage angular velocity can be expressed as [27]where b D and p D are the ball diameter and bearing pitch diam- eter,respectively. α is the contact angle, which is concerned with the clearance and can be obtained as follows:Referring to Fig. 4, i r and o r are the inner and outer groove radius,respectively.If the contact deformation j δ is positive, the contact force could be calculated using the Hertzian contact theory; otherwise, no load is transmitted. The contact force j Q between the j-th ball and race can be expressed as follows:where b k is the contact stiffness that can be given bywhere bi k and bo k are the load-deﬂection constants between the inner and outer ball race, respectively[28]. Summing the contact forces for each rolling element, the total bearing reaction fb in a complex form is4 Experimental and Numerical AnalysisAs shown in Fig. 2, the experimental assembly and the ﬁnite element model used in the dynamic analysis represent the ball bearing-rotor system with the following geometrical properties:length between the disk center and left end bearing center mm L 1201=; length between the disk center and right end bearing center mm L 1202=; and the shaft diameter mm D 10=. In addition, the elastic shaft material is steel of density 37950m kg =ρ,Young’s modulus GPa E 211=, and Poisson’s ratio 3.0=v . The ball bearings at both ends are the same model, 7200AC, and its parameters are listed in Table 1.The unbalance load is acted wit h the aid of the mass ﬁxed on the disk. By virtue of this act, the mass eccentricity of the disk can be deﬁnitely ascertained. As the mass eccentricity of the disk is 0.032 mm, the vibratory response at different rotational speed is determined via a numerical integration and Newton –Raphson iterations of the nonlinear differential equation (2). Note that the clearances used to simulate the bearings are measured ones. The horizontal and vertical displacements signals near the disk are acquired at different times, along with the increased rotational speed. Thus, the amplitudes of vibration at different speeds are determined according to the test data, and overall amplitudes are illustrated in Fig. 5, as the rotor system is run from 2000=ω rpm to 10,000 rpm. The prediction results compared with experimental data are shown in Fig. 5. It can be found that all of the predicted results are in good agreement with experimental data, thus validating the proposed model. The ﬁrst predicted resonance peak—the so called forward critical speed in linear theory,located at3960=ω rpm, matches the experimental date near 3960=ω rpm quite well. Moreover, the other amplitude peak appearing in the rotational speedrange7700=ω rpm to 8100 rpm can be found in both experimental and numerical analysis results.The corresponding frequency value of this peak is just the frequency doubling of the system critical speed.The Floquet theory can be used for analyzing the stability and topological properties of the periodic solution of the ball bearingrotor system. If the gained Floquet multipliers are less than unity,the periodic solution of the system is stable. If at least one Floquet multiplier exists with the absolute value higher than unity, the periodic solution is unstable and the topological properties of response alter into nonperiodic motion [29]. The leading Floquet multipliers and its absolute value at 7600=ω rpm, 8029 rpm, and 8200 rpm are shown in Table 2. It is found that the leading Floquet multiplier of the system remains in the unit circle, which indicates a synchronous response, as the rotational speed is less than 7700 rpm. Stability analysis shows that the imaginary part of the two leading Floquet multipliers move in opposite directions along the real axis near 7700=ω rpm. When the speed exceeded 7700=ω rpm, the leading Floquet multiplier crosses the unit circle through -1, as shown in Table 2. The periodic solution loses stability and undergoes a period-doubling bifurcation to a period-2 response, which indicates that a subharmonicresonance occurs. The subharmonic resonance keeps on from 7700=ω rpm to 8100 rpm. At 8100=ω rpm, the leading Floquet multiplier moves inside the unit circle through -1. Imply that the subharmonic resonance vanishes and the synchronous response returns. The synchronous response then continues to exist forspeeds above 8100=ω rpm.The waterfall map of frequency spectrums comparisons for prediction and experiment results are illustrated in Fig. 6. It can be found that agreement between the prediction and the experimental data is remarkable. The frequency component 66.9 Hz, near the forward resonance frequency, emerges and its amplitude rises signiﬁcant when the rotational speed is near 8029 rpm. It is shown that the resonance frequency is provoked when the speed is about twice the critical speed of the ball bearing-rotor system, and the subharmonic resonance occurs. The experimental and numerical analysis indicate that the representative nonlinear behavior and the subharmonic resonance arise from the nonlinearity of ball bearings, Hertzian contact forces, and internal clearance.The orbit and frequency spectrum at 8029=ω rpm are plotted in Fig. 7. Not only the prediction orbit but also the experiment results imply that the response is a period-2 motion, which is illustrated in Fig. 7(a). The predicted frequency components, consisting of 8.133=ω Hz (8029 rpm) and 9.662=ω Hz (4014rpm), coincide with experimental data. It indicates that the periodic response loses stability through a period-doubling bifurcation to a period-2 response. Thus, the subharmonic resonance occurs due to the effects of ball bearings. It can cause severe vibration and strong noise. Moreover, the subharmonic resonance could couple with other destabilizing effects on engineering rotor systems such as Alford forces, internal damping, and so on, and induce the rotor to lose stability and damage.5 ConclusionsAn experimental rig is employed to investigate the nonlinear dynamic behavior of ball bearing-rotor systems. The corresponding dynamic model is established wi th the ﬁnite element method and 2DOF dynamic model of a ball bearing, which includes the nonlinear effects of the Hertzian contact force and bearing internal clearance. All of the predicted results are in good agreement with experimental data, thus validating the proposed model. Numerical and experimental results show that the resonance frequency is provoked, and the subharmonic resonance occurs due to the nonlinearity of ball bearings when the speed is about twice the synchroresonance frequency. The subharmonic resonance cannot only cause severe vibration and strong noise, but also induce the rotor to lose stability and damage, once coupled with other destabilizing effects on high-speed turbomachinery such as Alford forces, internal damping, and so on. It is found that the effect of the Hertzian contact forces could also induce a subharmonic resonance, even if the internal clearance was not present. But, the response amplitude and subharmonic component of the rotor system without internal clearance are less than that with both Hertzian contact forces and internal clearance. Otherwise, the clearance may be unavoidable under high-speed operations, where the bearings are axially preloaded since the effect of unbalanced load is signiﬁcant at high speed. Thus, the nonlinearity of ball bearings,Hertzian contact forces, and internal clearance should be taken into account in ball bearing-rotor system design and failure diagnosis.AcknowledgmentThe authors would like to acknowledgment the support of the National Natural Science Foundation of China (Grant No.10902080) and Natural Science Foundation of Shaanxi Province(Grant Nos. SJ08A19 and 2009JQ1008).References[1] Gutafsson, O., and Tallian, T., 1963, “Resear ch Report on Study of the Vibration Characteristics of Bearings,” SKF Ind. Inc. Technical Report No.AL631023.[2] Saito, S., 1985, “Calculation of Non-Linear Unbalance Response of Horizontal Jeffcott Rotors Supported by Ball Bearings With Radial Clearances,” ASME J.Vib., Acou st., Stress, Reliab. Des., 107(4), pp. 416–420.[3] Aktürk, N., Uneeb, M., and Gohar, R., 1997, “The Effects of Number of Balls and Preload on Vibrations Associated With Ball Bearings,” ASME J. Tribol.,119, pp. 747–753.[4] Liew, A., Feng, N., and Hahn, E., 2002, “Transient Rotordynamic Modeling of Rolling Element Bearing Systems,” ASME J. Eng. Gas Turbines Power,124(4), pp. 984–991.[5] Bai, C. Q., and Xu, Q. Y., 2006, “Dynamic Model of Ball Bearing With Internal Clearance and Waviness,” J. Sound Vib., 294(1-2), pp. 23–48.[6] De Mul, J. M., Vree, J. M., and Maas, D. A., 1989, “Equilibrium and Associated Load Distribution in Ball and Roller Bearings Loaded in Five Degrees of Freedom While Neglecting Friction—Part I: General Theory and Application to Ball Be arings,” ASME J. Tribol., 111, pp. 142–148.[7] Mevel, B., and Guyader, J. L., 1993, “Routes to Chaos in Ball Bearings,” J.Sound Vib., 162, pp. 471–487.[8] Jang, G. H., and Jeong, S. W., 2002, “Nonlinear Excitation Model of Ball Bearing Waviness in a Rigid Rotor Supported by Two or More Ball Bearings Considering Five Degrees of Freedom,” ASME J. Tribol., 124, pp. 82–90.[9] Jang, G. H., and Jeong, S. W., 2003, “Analysis of a Ball Bearing With Waviness Considering the Centrifugal Force and Gyroscopic Moment of the Ball,”ASME J. Tribol., 125, pp. 487–498.[10] Tiwari, M., Gupta, K., and Prakash, O., 2000, “Effect of Radial Internal Clearance of a Ball Bearing on the Dynamics of a Balanced Horizontal Rotor,” J.Sound Vib., 238(5), pp. 723–756.[11] Tiwari, M., Gupta, K., and Prakash, O., 2000, “Dynamic Response of an Unbalanced Rotor Supported on Ball Bearings,” J. Sound Vib., 238(5), pp.757–779.[12] Harsha, S. P., 2005, “Non-Linear Dynamic Response of a Balanced Rotor Supported on Rolling Element Bearings,” Me ch. Syst. Signal Process., 19(3),pp. 551–578.[13] Harsha, S. P., 2006, “Rolling Bearing Vibrations—The Effects of Surface Waviness and Radial Internal Clearance,” Int. J. Computational Methods in Eng Sci. and Mech., 7(2), pp. 91–111.[14] Harsha, S. P., 2006, “Nonlinear Dynamic Analysis of a High-Speed Rotor Supported by Rolling Element Bearings,” J. Sound Vib., 290(1–2), pp. 65–100.[15] Gupta, T. C., Gupta, K., and Sehqal, D. K., 2008, “Nonlinear Vibration Analysis of an Unbalanced Flexible Rotor Supported by Ball Bearings With Radial Internal Clearance,” Proceedings of the ASME Turbo Expo, Vol. 5, pp. 1289–1298.[16] Bai, C. Q., Zhang, H. Y., and Xu, Q. Y., 2008, “Effects of Axial Preload of Ball Bearing on theNonlinear Dynamic Characteristics of a Rotor-Bearing System,” Nonlinear Dyn., 53(3), pp. 173–190. [17] Sinou, J. J., 2009, “Non-Linear Dynamics and Contacts of an Unbalanced Flexible Rotor Supported on Ball Bearings,” Mech. Mach. Theory, 44(9), pp.1713–1732.[18] Yamamoto, T., Ishida, Y., and Ikeda, T., 1984, “Vibrations of a Rotating Shaft With Rotating Nonlinear Restoring Forces at the Major Critical Speed,” Bull.JSME, 27(230), pp. 1728–1736.[19] Ishida, Y., and Yamamoto, T., 1993, “Forced Oscillations of a Rotating Shaft With Nonlinear Spring Characteristics and Internal Damping (1/2 Order Subharmonic Oscillations and Entrainment),” Nonlinear Dyn., 4(5), pp. 413–431.[20] Lee, Y. S., and Lee, C. W., 1999, “Modeling and Vibration Analysis of Misaligned Rotor-Ball Bearing Systems,” J. Sound Vib., 224(1), pp. 17–32.[21] Tiwari, M., Gupta, K., and Prakash, O., 2002, “Experimental Study of a Rotor Supported by Deep Groove Ball Bearing,” Int. J. Rotating Mach., 8(4), pp.243–258.[22] Ishida, Y., Liu, J., Inoue, T., and Suzuki, A., 2008, “Vibrations of an Asymmetrical Shaft With Gravity and Nonlinear Spring Characteristics (IsolatedResonances and Internal Resonances),” ASME J. Vib. Acoust., 130(4),p.041004.[23] Mevel, B., and Guyader, J. L., 2008, “Experiments on Routes to Chaos in Ball Bearings,” J. S ound Vib., 318, pp. 549–564.[24] Bai, C. Q., Xu, Q. Y., and Zhang, X. L., 2006, “Dynamic Properties Analysis of Ball Bearings—Liquid Hydrogen Turbopump Used in Rocket Engine,”ACTA Aeronaut. Astronaut. Sinica, 27(2), pp. 258–261. [25] Nelson, H., 1980, “A Finite Rotating Shaft Element Using Timoshenko Beam Theory,” ASME J. Mech. Des., 102(4), pp. 793–803.[26] Zhang, W., 1999, Basis of Rotordynamic Theory, Science Press, Beijing,China, Chap. 3.[27] Harris, T. A., 1984, Rolling Bearing Analysis, 2nd ed., Wiley, New York.[28] Aktürk, N., 1993, “Dynamics of a Rigid Shaft Supported by Angular Contact Ball Bearings,” Ph.D. thesis, Imperial College of Science, Technology and Medicine, London, UK.[29] Zhou, J. Q., and Zhu, Y. Y., 1998, Nonlinear Vibrations, Xi’an Jioatong University Press, Xi’an, China.附录B英文翻译非线性动力学的实验和转子轴承系统支持的行为的数值研究深沟球轴承在高速流体机械部件承担严重的振动和噪声的固有的非线性是很重要的。

弹性环挤压油膜阻尼器支撑下的柔性转子系统动力学分析

第２２卷第１期２０２４年１月动力学与控制学报J O U R N A L O FD Y N AM I C SA N DC O N T R O LV o l ．２２N o ．１J a n ．２０２４文章编号:１６７２Ｇ６５５３Ｇ２０２４Ｇ２２(１)Ｇ０４３Ｇ００９D O I :１０．６０５２/１６７２Ｇ６５５３Ｇ２０２３Ｇ０２２㊀２０２３Ｇ０２Ｇ０６收到第１稿,２０２３Ｇ０３Ｇ０６收到修改稿．∗国家自然科学基金资助项目(１２１７２３０７,１２１０２４４４),N a t i o n a lN a t u r a l S c i e n c eF o u n d a t i o no fC h i n a (１２１７２３０７,１２１０２４４４)．†通信作者E Ｇm a i l :１８１０４２y y＠１６３．c o m 弹性环挤压油膜阻尼器支撑下的柔性转子系统动力学分析∗赵先锋１㊀杨洋１†㊀王子尧２㊀路宽３㊀曾劲１㊀杨翊仁１(１．西南交通大学力学与航空航天学院,成都㊀６１００３１)(２．中国航空发动机研究院,北京㊀１０１３０４)(３．西北工业大学力学与土木建筑学院,西安㊀７１００７２)摘要㊀弹性环挤压油膜阻尼器(E l a s t i c r i n g s q u e e z e f i l m d a m p e r ,E R S F D )具有良好的支撑作用和减振效果,但由于其结构和流场耦合行为极为复杂,使得已有的物理模型难以完整表现出E R S F D 的力学特性．为了进一步探究E R S F D 的力学机理,本文借助有限元仿真平台,采用双向流固耦合的计算方法,剖析弹性环与油膜之间的相互作用,获取E R S F D 中油膜压力的分布规律．在此基础上,利用最小二乘法进一步拟合出E R S F D 等效刚度㊁等效阻尼与转子轴颈扰动位移的映射关系,并将其分别引入柔性转子系统动力学模型中．通过数值计算研究了E R S F D 支撑下柔性转子系统的振动响应,分别给出了不同转速下转子系统的响应分岔图㊁轴心轨迹等．同时,通过对比分析,进一步揭示了E R S F D 所诱发出的转子系统丰富的非线性动力学行为,有助于对E R S F D 轴承支撑特性的理解．关键词㊀弹性环挤压油膜阻尼器,㊀转子系统,㊀双向流固耦合,㊀动力学特性中图分类号:O ３１３文献标志码:AD y n a m i cC h a r a c t e r i s t i c s o f F l e x i b l eR o t o r S y s t e mS u p p o r t e db yE l a s t i cR i n g S q u e e z eF i l m D a m pe r ∗Z h a oX i a n f e n g １㊀Y a n g Y a n g １†㊀W a n g Z i y a o ２㊀L uk u a n ３㊀Z e n g J i n １㊀Y a n g Yi r e n １(１．S c h o o l o fM e c h a n i c s a n dA e r o s p a c eE n g i n e e r i n g ,S o u t h w e s t J i a o t o n g U n i v e r s i t y ,C h e n g d u ㊀６１００３１,C h i n a )(２．C h i n aR e s e a r c h I n s t i t u t e o fA e r o ＧE n g i n e ,B e i j i n g㊀１０１３０４,C h i n a )(３．N o r t h w e s t e r nP o l y t e c h n i c a lU n i v e r s i t y ,S c h o o l o fM e c h a n i c s ,C i v i l E n g i n e e r i n g an dA r c h i t e c t u r e ,X i a n ㊀７１００７２,C h i n a )A b s t r a c t ㊀E l a s t i cr i n g s q u e e z ef i l m d a m p e r (E R S F D )h a sa g o o ds u p p o r t i n g an dv i b r a t i o nr e d u c t i o n e f f e c t ．H o w e v e r ,d u e t o i t s c o m p l e x s t r u c t u r e a n d f l o wf i e l d c o u p l i n g b e h a v i o r ,e x i s t i n gp h y s i c a lm o d e l s a r ed i f f i c u l t t o f u l l y d e m o n s t r a t e t h em e c h a n i c a l c h a r a c t e r i s t i c so fE R S F D．T of u r t h e re x pl o r e t h e m e Ｇc h a n i c a lm e c h a n i s mo fE R S F D ,t h i s p a p e r a n a l y z e s t h e i n t e r a c t i o nb e t w e e n e l a s t i c r i n g an do i l f i l m w i t h t h e a i do f f i n i t e e l e m e n t s i m u l a t i o n p l a t f o r ma n db i d i r e c t i o n a l f l u i d Ｇs t r u c t u r e c o u p l i n g ca l c u l a t i o n m e t h Ｇo d ,a n dob t a i n s t h e d i s t r i b u t i o n l a wo f o i l f i l m p r e s s u r e i n t h eE R S F D．O n t h i s b a s i s ,t h em a p p i n g r e l a Ｇt i o n s h i p b e t w e e n t h e e q u i v a l e n t s t i f f n e s s a n d e q u i v a l e n t d a m p i n g o f E R S F Da n d t h e d i s t u r b a nc ed i s p l a ce Ｇm e n t of t h e r o t o r j o u r n a l i s f u r t h e r f i t t e db y u s i ng th e l e a s t s q u a r em e t h o d ．T h e n t h e e q ui v a l e n tm o d e l i s f u r t h e r i n t r o d u c e d i n t o t h e d y n a m i cm o d e l o f t h e f l e x i b l e r o t o r s y s t e m．T h e v i b r a t i o n r e s po n s e o f f l e x i b l e r o t o r s y s t e ms u p p o r t e db y E R S F D i s s t u d i e db y n u m e r i c a l c a l c u l a t i o n ,a n d t h e r e s po n s e b i f u r c a t i o nd i a Ｇg r a ma n dw h i r l i n g o r b i to f t h er o t o rs y s t e m u n d e rd i f f e r e n ts pe e d sa r ec o n d u c t e d ．A t t h es a m et i m e ,动㊀力㊀学㊀与㊀控㊀制㊀学㊀报２０２４年第２２卷t h r o u g hc o m p a r a t i v e a n a l y s i s,t h e r i c hn o n l i n e a rd y n a m i cb e h a v i o r so f r o t o r s y s t e mi n d u c e db y E R S F D a r e f u r t h e r r e v e a l e d,w h i c h i s h e l p f u l t ou n d e r s t a n d t h e s u p p o r t i n g c h a r a c t e r i s t i c s o fE R S F D．K e y w o r d s㊀e l a s t i c r i n g s q u e e z e f i l md a m p e r,㊀r o t o r s y s t e m,㊀b i d i r e c t i o n a l f l u i dＧs t r u c t u r e c o u p l i n g,㊀d y n a m i c c h a r a c te r i s t i c s引言弹性环挤压油膜阻尼器(E R S F D)充分结合挤压油膜阻尼器(S F D)减振特性和支承弹性特点,被广泛应用于航空发动机转子系统中[１]．对于传统的挤压油膜阻尼器而言,当转子涡动较为严重时,极易诱发油膜振荡㊁振动突跳等不利现象,对转子系统的平稳运行产生不良影响[２]．相较于此,弹性环挤压油膜阻尼器在油膜间隙中引入了附加的弹性环结构,并且弹性环内外侧均具有交错分布的弧形凸台,能够将轴承外环与轴承座之间的间隙分割成多个独立的油膜区域,有效避免油膜振荡的发生．其中,靠近轴承座的部分称其为外油膜,而与之相反的称其为内油膜．当润滑油受到挤压产生油膜力时,该作用力会传递到弹性环上,继而引起结构变形．同时弹性环变形亦会引起油膜间隙发生变化,导致油膜力发生改变．由此可以发现,弹性环挤压油膜阻尼器中存在典型的双向流固耦合现象．国内外学者对E R S F D进行了广泛研究．周明等[３]基于流体动压理论,提出了弹性环挤压油膜的减振机理．X u等[４]利用有限元法研究了E R S F D渗油孔的分布对油膜阻尼特性的影响,探讨了油膜力与孔口位置在轴向和圆周方向的关系,结果表明:孔口分布可以调节阻尼系数．周海仑等[５]采用双向流固耦合原理及动网格技术,计算了内外层油膜的压力,开展了凸凹台数量㊁几何尺寸和油膜间隙对油膜动力特性的影响规律．李岩等[６]研究了配合关系对油膜阻尼器减振特性的影响,实验结果表明:弹性环内凸台为过盈配合时可能会导致阻尼器减振失效．王震林等[７]基于厚板理论建立了弹性环的运动方程,采用分时迭代方法将弹性环－油膜的控制方程进行耦合求解,结果显示:刚度主要与弹性环厚度有关,阻尼主要取决于凸台高度．江志敏等[８]采用流固耦合技术模拟二维E R S F D,发现在导流孔处流速较大,并探讨了E R S F D的减振机理以及与传统S F D在减振机理上的行为差异．该结果表明:E R S F D油膜压力呈现出与油腔间隔相关的阶梯状分布．C h e n等[９]研究一种带E R S F D的螺旋锥齿轮传动动力学模型,发现了E R S F D支承具有良好的减振效果．此外,围绕E R S F D支撑下的转子系统动力学特性研究亦取得了一定的研究进展．针对组合支撑的转子结构,罗忠等[１０]进行系统性评述,阐明了不同支承的力学特征．P a n g等[１１]利用平均法分析了E R S F D轴承参数与转子系统分岔行为的潜在关联．何洪等[１２]对E R S F D支承的增压转子动力特性进行研究,分析弹性环阻尼器交叉刚度的影响甚小．H a n等[１３]基于半解析法求解E R S F D支承下转子系统动态特征,揭示了油膜特性和突加激励对其影响规律．杨洋等[１４]建立了双盘转子模型,研究不平衡故障下碰摩非线性行为．曹磊等[１５]研究了E R S F D支承下转子的临界转速,证实影响临界转速的最大因素体现在凸台处的接触状态．李兵等[１６]实验探究了弹性环凸台高度㊁供油条件㊁滑油温度和不平衡量等条件下E R S F D的动力学特性,结合转子振动响应,发现弹性环凸台高度较小时,系统的减振特性更为理想．张蕊华等[１７]提出了一种挤压油膜阻尼器的刚度分析方法,采用将油膜刚度和外环进行串联得到其等效刚度．熊万里等[１８]基于N a v i e rＧS t o k e s方程动网格技术,发展了一种计算E R S F D轴承刚度和阻尼的新方法．综上所述,关于E R S F D支撑下柔性转子系统非线性动力学特性的研究尚不充分．针对这一情况,本文首先借助A N S Y S WO R K B E N C H仿真平台对E R S F D进行双向流固耦合分析,辨识出不同轴颈涡动下E R S F D所提供的等效刚度和等效阻尼．在此基础上,将其引入至柔性转子中,进行系统级非线性动力学特性研究,给出不同运行工况下系统的非线性动力学特性．通过对比线性支承和非线性支承,对比分析E R S F D引发的非线性动态特征．研究结果以期为转子系统的结构设计和故障诊断提供一定的技术支持．４４第１期赵先锋等:弹性环挤压油膜阻尼器支撑下的柔性转子系统动力学分析１㊀弹性环挤压油膜阻尼器双向流固耦合分析１．１㊀E R S F D 结构建模根据表１给出的某转子系统中弹性环挤压油膜阻尼器(E R S F D )结构参数,利用S O L I DWO R K 进行精细化实体建模,如图１所示．其中,弹性环上依次分布了内外交错的凸台,将油膜形成错落有致的内外两层,且内外层油膜之间通过导流孔连接．表１㊀E R S F D 结构参数表T a b l e １㊀S t r u c t u r e p a r a m e t e r s o fE R S F D结构参数数值内外凸台数８导流孔数８轴颈半径/mm２１．５弹性环厚度/mm １．５渗油孔直径/mm１阻尼器外圈半径/mm ２３．４阻尼器轴向长度/mm １５弹性环凸台高度/mm ０．２弹性环弹性模量/G P a ２１０弹性环材料密度/k g /m ３７８５０润滑油材料密度/k g /m ３１１００润滑油动力黏度/P a ．S０．０２７图１㊀E R S F D 结构示意图F i g ．１㊀S c h e m a t i c d i a gr a mo fE R S F Ds t r u c t u r e 为获取弹性环挤压油膜阻尼器的支承力学特性,采用双向流固耦合方式进行数值分析,其计算流程图如图２所示．首先将E R S F D 实体模型导入至WO R K B E N C H 中进行切块化网格划分,并结合弹性环结构区域和油膜分布区域进行相关界定,依次定义为S O L I D 和F L U I D 区域．为反映结构和流体之间实时的相互作用,利用T R A N S I E N TS T R U C T U R E 和C F X 进行耦合计算．在当前时间步下,分别对弹性环变形和油膜压力收敛性进行判断,将收敛后结果在耦合系统中进行数据实时交换,并进行总体收敛性判断．倘若结果收敛,则进入下一个时间步计算,否则重复上述计算直至收敛．图２㊀双向流固耦合计算流程图F i g ．２㊀C h a r t o f b i d i r e c t i o n a l f l u i d Ｇs t r u c t u r e c o u p l i n g ca l c u l a t i o n (a)弹性环边界条件(b)油膜边界条件图３㊀E R S F D 流固耦合边界条件F i g ．３㊀E R S F Df l u i d Ｇs t r u c t u r e i n t e r a c t i o nb o u n d a r y co n d i t i o n s 在E R S F D 运行过程中,将弹性环内凸台与转子轴颈进行紧密接触处理,两者接触面上具有相同的运动形式,并且忽略轴颈与内凸台的摩擦效应．５４动㊀力㊀学㊀与㊀控㊀制㊀学㊀报２０２４年第２２卷同时,外凸台与阻尼器外壳之间的摩擦亦不予考虑．弹性环边界条件设置如下:(１)弹性环轴向方向施加远程位移约束,限制其轴向和绕三个轴的转动;(２)流体和固体接触面建立流固耦合面,在该面上进行数据传递;(３)弹性环外凸台处施加固定约束;(４)由于转轴受到不平衡激励的作用,轴颈的运动形式以涡动形式为主,不考虑转轴本身的自转,所以施加周期位移激励,以模拟轴颈涡动,其具体表示形式如下:x i n＝e i n s i n(ωt)y i n＝e i n c o s(ωt)(１)其中,x i n㊁y i n分别为x,y方向位移,e i n表示轴径激励幅度,ω表示转子运行转速．此外,在流体域中边界条件相关设置如下:(１)外层油膜壁面固定;(２)油膜两端进行密封处理; (３)设立相对应的流固耦合面,用于进行流体与固体的数据交换;(４)在内层油膜与轴颈接触处施加相同的位移激励,如图３所示．１．２㊀网格无关性验证本节通过网格无关性来验证所建立的有限元模型的正确性,网格无关性保证网格对结果影响较小．由于润滑油黏度较大,流体模型采用层流模型,残差小于１０－４认为收敛,边界条件如上节所述．轴颈激励幅值为０．０２mm,时间步长为０．０００１s进行计算,得到结果如图４．发现网格数超过２０万时对结果影响较小,因此下面的计算采用此套网格．图４㊀最大内,外层油膜压力随网格数量变化规律F i g．４㊀V a r i a t i o n l a wo fm a x i m u mi n n e r o i l f i l m p r e s s u r ew i t h t h em e s hn u m b e r s１．３㊀E R S F D流场及压力分析基于上述双向流固耦合处理,本节着重关注E R S F D流场及压力分布情况．如图５所示的油膜流动矢量图,其中油膜从挤压处流向非挤压处,且在导流孔处出现了较大流速．(a)油膜流场速度云图(b)油膜流场剖面图图５㊀E R S F D中油膜流场分布图F i g．５㊀O i l f i l mf l o wf i e l dd i s t r i b u t i o n i nE R S FD图６㊀不同偏心量下内层油膜力随时间变化规律F i g．６㊀T i m e v a r y i n g l a wo f i n n e r o i l f i l mf o r c e u n d e rd i f fe r e n t e c c e n t r i c i t i es图７㊀不同偏心量下外层油膜力随时间变化规律F i g．７㊀T i m e v a r y i n g l a wo f o u t e r o i l f i l mf o r c e u n d e rd i f fe r e n t e c c e n t r i c i t i e s根据双向流固耦合系统的稳态响应,进一步分析E R S F D中内外层油膜压力分布和弹性环变形６４第１期赵先锋等:弹性环挤压油膜阻尼器支撑下的柔性转子系统动力学分析情况．对周期内每时刻内外层油膜压力分布进行面上积分,得到油膜力随时间变化．由图６和图７可知,随着轴颈激励幅度的增加,油膜力波动愈发明显,当达到一定程度时容易出现油膜失稳现象．对一个周期内的油膜压力取平均,可以得到不同激励幅度下的油膜力．如图８所示,当偏心量较小时,E R S F D 油膜力与偏心量呈线性关系,而随着偏心量的增加,两者之间的非线性映射关系逐渐显著,这也意味着当转子系统转速提升到一定程度时,转子支承边界不是理想的线性边界而是更为复杂的非线性边界．对比内外层油膜压力,可以发现在小偏心量情况下,内外层油膜压力较接近,而随着偏心量的增加,内外层油膜压力的差别也将凸显．由于在大偏心量下,外层油膜受挤压的面积更大,且弹性环的位移对外层油膜影响更大．图８㊀内外层油膜力在不同偏心量下的变化规律F i g．８㊀V a r i a t i o no f i n n e r a n do u t e r o i l f i l mf o r c eu n d e r d i f f e r e n t e c c e n t r i c i t i es图９㊀不同偏心量下油膜压力分布及弹性环变形程度F i g ．９㊀O i l f i l m p r e s s u r e d i s t r i b u t i o na n d e l a s t i c r i n g de f o r m a t i o nu n d e r d i f f e r e n t e c c e n t r i c i t i e s ㊀㊀为了进一步分析内外层油膜压力分布和弹性环变形随轴颈激励幅度的变化规律,依次令激励幅度分别为:e i n ＝０．０１mm ㊁e i n ＝０．０３mm 和e i n ＝０．０６mm．由图９可知,随着轴颈激励幅度的增加,内外层油膜压力逐渐变大,且最大压力随轴颈位移变化是一种非线性关系．同时,对比内外层油膜压力可以发现,内层油膜的最大压力始终小于外层油膜的最大压力,说明弹性环对内层油膜挤压较大,其次内外层最大压力之间存在一定角度,这是因为弹性环的内外侧凸台交错分布将内外层油膜分隔开来导致．此外,流场采用端封处理,从而内外层油膜压力分布在轴向的分布基本是不变的,这亦说明端封的边界条件是有效的．由于内凸台与轴颈具有相同的涡动位移激励,因此位于内外凸台之间的环７４动㊀力㊀学㊀与㊀控㊀制㊀学㊀报２０２４年第２２卷位移最大且呈非对称分布．１．４㊀E R S F D等效刚度和等效阻尼拟合本节利用最小二乘法,对前节获取的双向流固耦合仿真结果进行拟合处理,以此获取E R S F D等效刚度和等效阻尼随轴颈偏心量变化的表达式．结合E R S F D结构特点,由于外层油膜被弹性环分开,且弹性环与轴颈的接触面积较小,故采用弹性元件和阻尼元件串联的方式,刚度大小是利用力与位移的比值确定．对于阻尼不考虑弹性环阻尼,只考虑油膜的阻尼,利用如下表达式计算:C＝Fe i nω(２)其中C表示油膜阻尼,F表示油膜力．分析不同偏心量下系统的等效刚度和等效阻尼,如表２所示．显然,随着偏心量的增加,E R S F D 等效刚度和等效阻尼均逐渐增大,且呈现非线性变化现象．表２㊀不同偏心量下E R S F D等效刚度和等效阻尼T a b l e２㊀E q u i v a l e n t s t i f f n e s s a n de q u i v a l e n t d a m p i n g o fE R S F D w i t hd i f f e r e n t e c c e n t r i c i t i e se i n(mm)K e(MN/m)C e(N s/m)０．０１３．９７９１３９．４２１０．０２４．１０４１４３．７０１０．０３４．２１９１４７．７８６０．０４４．３５８１５２．６６５０．０５４．５１５１５８．２０４０．０６４．７３７１６４．５１４利用最小二乘法,对表２中的离散数据进行拟合处理．可进一步得到E R S F D等效支撑力表达式为:K e＝－１．４１３ˑ１０２２e４i n＋２．５８４ˑ１０１３e２i n＋㊀３．９８ˑ１０６(３)C e＝４．０９７ˑ１０１３e３i n－１．４５ˑ１０９e２i n＋㊀４．２７ˑ１０５e i n＋１３５．３(４)其中,K e和C e分别表示E R S F D的等效刚度和等效阻尼,e i n表示第i个轴颈的径向位移,可表示为:e i n＝x２i＋y２i(５)其中x i,y i分别是第i个轴承出横向和竖向位移,进一步油膜力可以写为:F x＝K e x i c o sα＋C e x i c o sαF y＝K e y i s i nα＋C e y i s i nα{(６)其中c o sα,s i nα计算表达式为:c o sα＝x i e i ns i nα＝y i e i nìîíïïïï(７)２㊀双盘悬臂转子系统动力学特性分析２．１㊀转子系统动力学建模图１０给出了E R S F D支撑下的双盘悬臂转子系统,其中左右转盘分别表示压气盘和涡轮盘,且压气盘存在不平衡故障．图１０㊀弹性环挤压油膜阻尼器支撑下的双盘悬臂转子F i g．１０㊀S c h e m a t i c d i a g r a mo f d u a lＧd i s c c a n t i l e v e r r o t o rs y s t e ms u p p o r t e db y E R S F D将柔性转轴采用欧拉－伯努利梁单元进行有限元离散[１４],其中每个梁单元包含２节点,且每个节点包含４个自由度．根据结构特性,将结构分为３个转轴单元和２个转盘单元．考虑到转盘刚度远大于转轴刚度,将压气盘和涡轮盘均视为集中质量块,分别安装在转轴对应节点上．因此,转盘质量矩阵和陀螺矩阵分别表示成: M d＝m d００j déëêêùûúú(８) J d＝０００j péëêêùûúú(９)其中,m d表示转盘质量;j d表示转盘赤道转动惯量;j p表示转盘极转动惯量．根据双盘悬臂转子系统节点划分特点,进行整体结构组装．同时,在对应约束位置处,分别引入线性支撑和E R S F D支撑进行分析．同时,将压气盘不平衡激励纳入广义激励中,继而得到转子系统振动方程,如式(１５)所示．M u ＋C u ＋K u＝Q(１０)其中,M表示转子系统质量矩阵;C表示转子系统８４第１期赵先锋等:弹性环挤压油膜阻尼器支撑下的柔性转子系统动力学分析阻尼矩阵,其中包含陀螺矩阵;K表示转子系统刚度矩阵,M,C,K为１６ˑ１６的矩阵;Q表示转子系统广义激励,为１６ˑ１的矩阵．２．２㊀转子系统动力学特性分析本节采用数值仿真的方式得到双盘悬臂转子系统压气盘横向响应分岔图,如图１１所示,其中横轴是转速ω/(r a d/s),纵轴是压气盘的横向振动位移x p(m)．所采用的结构参数如表３所示．表３㊀转子结构参数表T a b l e３㊀S t r u c t u r e p a r a m e t e r s o f r o t o r结构参数数值转轴外径/mm２１．５转轴内径/mm１１．５弹性模量E/G P a２００转盘外径/mm１０６．２转盘厚度/mm２４轴段单元长度L１/mm１４０轴段单元长度L２/mm５０１．１轴段单元长度L３/mm１０１．９转子材料密度/k g/m３７８５０不平衡量/mm０．０３在转速满足ωɪ[３００r a d/s,１０００r a d/s]时,对比分析线性支承和E R S F D支承下转子系统的动态响应差异,其中线性支承下,轴承刚度为３．８３ˑ１０６N/m．由图１１(a)可知,在线性支撑条件下,双盘悬臂转子系统在不同转速下始终呈现规则的周期１运动．同时,双盘悬臂转子系统在ω＝４６０r a d/s时发生一阶共振．采用相同的结构参数,在相同支撑位置处,将线性支撑替换为E R S F D．由此可以进一步得到转子系统横向响应分岔图,如图１１(b)所示．由于E R S F D使用引入了非线性支撑边界,使得转子系统动态响应中出现明显的非线性现象．当转速较低时,转子系统做规则的周期运动．当转速升至ω＝５９０r a d/s时,转子系统进入拟周期运动．随着转速的进一步提高,由于边界非线性的引入,转子系统响应中发生了明显的滞后跳跃现象．当转速进一步增加时,转子系统由复杂的拟周期运动再次回归到规则的周期运动．此外,由于支撑非线性的引入,转子系统一阶临界转速改为ω＝６３２r a d/s．对比图１１(a)和图１１(b),从响应幅值上来说,E R SＧF D能够极大减小转子的振幅,因此E R S F D的使用,可能减轻碰摩的发生．(a)线性支撑边界(b)E R S F D支撑边界图１１㊀双盘悬臂转子系统响应分岔图F i g．１１㊀B i f u r c a t i o nd i a g r a mo f d u a lＧd i s c c a n t i l e v e r r o t o r s y s t e m(a)转子轴心轨迹(b)转子频谱图图１２㊀线性支撑下转子系统在ω＝６００r a d/s时振动响应F i g．１２㊀V i b r a t i o nb e h a v i o r o f r o t o r s y s t e m w i t h l i n e a rs u p p o r t a tω＝６００r a d/s９４动㊀力㊀学㊀与㊀控㊀制㊀学㊀报２０２４年第２２卷(a)ω＝６００r a d/s(b)ω＝７２０r a d/s图１３㊀E R S F D支撑下转子系统在不同转速下轴心轨迹F i g．１３㊀W h i r l i n g o r b i t o f r o t o r s y s t e m w i t hE R S F Ds u p p o r ta t d i f f e r e n t r o t a t i o n a l s p e e d s为了进一步对比分析线性支撑和E R S F D支撑下双盘悬臂转子系统在不同转速下的振动响应差异,选取ω＝６００r a d/s和ω＝７２０r a d/s绘制压气盘轴心轨迹和频谱图,如图１２,１３所示．在线性支撑下,转子系统轴心轨迹呈现出规则的圆形,且频谱图中仅有单一的激励频率．而在E R S F D支撑下,转子系统的轴心轨迹由复杂的花瓣形构成,呈现典型的拟周期特征．此外,在ω＝７２０r a d/s时,转子系统轴心轨迹呈现出非规则的椭圆形状．３㊀结论本文以弹性环挤压油膜阻尼器(E R S F D)为研究对象,采用双向流固耦合的方式数值分析了不同轴颈激励幅度下内外层油膜压力分布情况和弹性环变化规律．随后,通过最小二乘法进一步拟合出E R S F D的等效约束刚度和等效约束阻尼．在此基础上,将其引入至双盘悬臂转子系统中,对比分析线性支撑和E R S F D支撑下系统动力学响应差异．相应地,主要结论可概述如下:(１)通过对E R S F D油膜流场分布分析,发现导流孔处存在明显的高流速集中现象,且从油膜挤压处沿着油膜表面进行内外层流体交换．(２)随着扰动激励幅度的增加,内外层油膜压力均明显提高且存在明显的油膜振荡现象,同时外层油膜刚度始终大于内层油膜刚度．(３)相比于线性支撑条件,E R S F D支撑下双盘悬臂转子系统出现明显的非线性振动现象,如共振滞后和跳跃现象等．同时,对比相同转速下系统的振动幅值,E R S F D起到了明显的振动抑制效果．参考文献[１]Z HA N G W,D I N G Q．E l a s t i c r i n g d e f o r m a t i o na n d p e d e s t a l c o n t a c t s t a t u s a n a l y s i s o f e l a s t i c r i n gs q u e e z e f i l m d a m p e r[J]．J o u r n a l o fS o u n da n d V iＧb r a t i o n,２０１５,３４６:３１４－３２７．[２]崔颖,罗乔丹,邱凯,等．涨圈密封挤压油膜阻尼器流场与阻尼特性[J]．航空动力学报,２０２１,３６(１２):２４７４－２４８１．C U IY,L U O Q D,Q I U K,e ta l．F l o wf i e l da n dd a m p i n g c h a r a c te r i s t i c s of p i s t o n r i ng s e a l e d s q u e e z ef i l m d a m p e r[J]．J o u r n a l o f A e r o s p a c e P o w e r,２０２１,３６(１２):２４７４－２４８１．(i nC h i n e s e) 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e,２０１９,４５(５):２６－３０．(i nC h i n e s e)[９]C H E N W T,C H E NSY,HU Z H,e t a l．An o v e ld y n a m i cm o de lf o r t h e s p i r a l b e v e lg e a r d r i v ew i th eＧl a s t i c r i n g s q u e e z e f i l m d a m p e r s[J]．N o n l i n e a rD yＧn a m i c s,２０１９,９８(２):１０８１－１１０５．[１０]罗忠,王晋雯,韩清凯,等．组合支承转子系统动力学的研究进展[J]．机械工程学报,２０２１,５７(７):４４－６０．L U OZ,WA N GJW,HA N Q K,e t a l．R e v i e wo nd y n a m i c s o f t he c o m b i n e d s u p p o r tＧr o t o r s y s t e m[J]．J o u r n a l o fM e c h a n i c a l E n g i n e e r i n g,２０２１,５７(７):４４－６０．(i nC h i n e s e)[１１]P A N G GY,C A OSQ,C H E N YS,e t a l．S t u d y o n v i b r a t i o na n db i f u r c a t i o no f a na e r o e n g i n e r o t o r s y sＧt e m w i t h e l a s t i c r i n g s q u e e z e f i l md a m p e r[J]．S h o c ka n dV ib r a t i o n,２０２１,２０２１:４６５１３３９．[１２]何洪,王世琥,王连群,等．弹性环阻尼器对涡轮增压器转子动力性能影响研究[J]．车用发动机,２０１６(４):７０－７４＋８１．H E H,WA N GS H,WA N G L Q,e t a l．I n f l u e n c eo f e l a s t i c r i n g d a m p e r o n p o w e r p e r 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滚动轴承-柔性碰摩转子系统非线性动力学响应分析

１．１非线性滚动轴承力模型
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（、５Ｊ），
假设轴承外圈与外支承之间刚性连接，内圈与转子轴之间刚性连接，滚动体等距排列，滚动体与
滚道之间为纯滚动，含接触角的滚动球轴承动力学
则轴承Ａ非线性滚动轴承力可以表示为：
２０１３－０２－２２收到第ｌ稿，２０１３－０４－０２收到修改稿国家高技术研究发展计划项目（２０１２ＡＡ０４０１０４）十通讯作者Ｅ－ｍａｉｌ：ｙｕａｎ－ｈｑ＠１６３．ｃｏｍ
应，建立了滚动轴承一柔性对称碰摩转子系统非线性集中质量模型．通过数值计算与比较，结果表明：低转
速下系续响应主要表现为滚动轴承的变刚度振动，高转速下轴承变刚度振动的影响相对减弱，转子不平衡
和碰摩故障对系统的影响逐渐增强，陀螺效应对高转速下对称转子的响应不容忽略．
图１中转子系统两端由相同的滚动轴承支承，质量为分别为ｍ和ｍ，轴承支反力在，Ｙ方向的
以上对滚动轴承．转子系统动力学的研究多是基于深沟球轴承一刚性转子模型，并且忽略了圆盘陀螺效应对滚动轴承一转子系统非线性响应的影
轴承，对系统的非线性振动进行了数值分析，得到了转子系统的超谐波、亚谐波和混沌运动规律；文
莱截面图和频谱图讨论转速、轴承游隙、碰摩刚度

航空发动机滚动轴承及其双转子系统共振问题研究综述

航空发动机滚动轴承及其双转子系统共振问题研究综述作者：李轩来源：《科技风》2022年第11期摘要：针对航空燃气涡轮发动机滚动轴承及其双转子系统存在的复杂振动问题，综述了近年来国内外该领域的主要研究成果。

首先，概述了双转子系统动力学建模与分析的研究成果。

其次，综述了双转子系统动力学响应分析研究的现状与主要进展。

最后对现有研究工作进行了展望，对该领域的发展趋势进行了说明。

关键词：转子动力学;双转子系统;共振;非线性;滚动轴承滚动轴承及其双转子系统作为航空燃气涡轮发动机的主要结构，存在着大量复杂振动现象，能够引发系统复杂故障甚至灾难性的事故，其产生机理十分复杂。

所以人们针对相关系统进行了大量研究，从不同角度研究并阐述了多种复杂共振现象的触发机制，对进一步改善航空燃气涡轮发动机等相关滚动轴承—双转子系统机械的安全性、稳定性、可靠性具有重要的理论与实际工程意义。

为了缓解航空燃气涡轮发动机滚动轴承及其双转子系统运行时的高频小幅度不规则运动，防止系统在特定运行条件下产生有害共振，并仍能保持良好的动力学性能。

学者们需要深入研究航空发动机滚动轴承—双转子系统的运动学与造成其运动的力学特点，从而分析解决实际系统存在的各种共振问题。

为此，研究创建适合于剖析滚动轴承—双转子系统动力学特性的模型很有必要。

本文对航空发动机滚动轴承—双转子系统动力学建模以及双转子系统的动力学响应特性的研究现状进行了归纳，并对滚动轴承及其双转子系统共振研究的发展趋势进行了预测。

1 航空发动机双转子系统的动力学建模与分析实际双转子航空燃气涡轮发动机工况十分复杂，为了准确研究航空燃气涡轮发动机滚动轴承—双转子系统运行中的动力学行为，航空燃气涡轮发动机双转子系统的动力学建模问题被学者们广泛研究。

路振勇等[1]依据某真实航空发动机的双转子系统，创建了较为复杂的非连续化动力学模型。

并在对该模型进行了降维后，计算了系统发生共振的对应转速，发现依据复杂非连续化动力学模型计算得到的结果与采用传统方法计算得到的结果相比差异极小，证明了降维模型能很好反映双转子系统的实际共振特性。

三自由度齿轮传动系统振动模型及计算【打印】

三自由度齿轮转子轴承系统的间隙非线性模型及方程一个典型的单级齿轮转子一轴承传动系统包括箱体、滚子轴承、支撑轴、齿轮副等零部件，如图2.1所示。

在进行传动分析时，为了使问题简化，箱体被看作是固定的;忽略原动部件惯性载荷的影响，即假设这样的惯性元件是通过柔性的联轴器与齿轮箱联接。

同时假设系统关于齿轮平面对称，故系统的轴向运动可以忽略不计。

该系统的运动微分方程可写成如一般的形式为[]{''()}[]{'()}[]{(())}{()}M x t C x t K f x t F t ++= （2.1）式中[]M 表示时不变的质量矩阵，()x t 表示位移向量[]C 为时不变的阻尼矩阵，即不考虑轮齿分离及时变的啮合特性对啮合阻尼的影响。

刚度矩阵[]K 为周期时变矩阵：[()][(2/)]h K t K t πω=+，h ω为齿轮啮合的基频。

(())f x t 为间隙非线性函数，本文用分段函数如图2所示（包含轴承径向间隙和齿侧间隙），{()}F t 为系统激励力向量。

1. 数学模型的建立：使用集中质量法建立如图1.2所示的单级齿轮传动的动力学模型，认为系统由只有弹性而无质量的弹簧和只有质量而无弹性的质量块组成，则式((1.1)表示的多自由度系统的可简化形式为三自由度非线性齿轮传动系统模型，包括齿轮惯量1I 和2I ，齿轮质量1m 和2m ，基圆直径1d 和2d ，如图3所示。

齿轮啮合由非线性位移函数h f 和时变刚度h k ，线性粘性阻尼h C 描述。

轴承和支撑轴的模型则由等效的阻尼元件和非线性刚度元件表述。

阻尼元件具有线性粘性阻尼系数1C 和2C ，非线性刚度元件由近似分段线性的间隙型非线性力一位移函数1f 和2f ，以及相应的刚度参数1k 和2k 确定。

同时考虑因输入扭矩波动引起的低频外激励和静态传动误差()e t 导致的高频内部激励，忽略输出扭矩的波动，即认为:111()()m a T t T T t =+ 22()m T t T =式中：1()T t 为输入扭矩1m T 为输入扭矩均值1()a T t 为输入扭矩变化部分2m T 为输出扭矩均值并假设在支承上均作用有外径向预载力1F 和2F 。

滚动轴承支承的柔性转子系统的非线性动力学分析

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柔性支承的滑动轴承-刚性转子系统动力学特性分析

柔性支承的滑动轴承-刚性转子系统动力学特性分析王文斌（兰州交通大学机电工程学院，甘肃兰州730070）摘要综合考虑滑动轴承油膜力支撑、柔性支承受外部激励等非线性因素的影响，建立一个两端柔性支承的刚性转子模型，运用4阶变步长Runge-Kutta算法对系统的动力学行为进行数值仿真，得到了刚性转子系统在不同外部激励幅值下随转速变化时的位移分岔图，以及特定参数下的相图、Poincaré映射图，直观揭示了系统的动力学特性。

结果表明，中低速阶段，系统随着转速的提升出现了混沌、多周期、概周期、单周期等复杂动力学行为；在中速和高速阶段，系统主要是多周期和混沌运动的相互转迁，以及阵发性混沌运动现象的出现。

外激励幅值的增大，会使得系统容易形成概周期运动，并导致轴颈-轴瓦处油膜力的增大。

关键词非线性油膜力柔性支承刚性转子Runge-Kutta法混沌Dynamics Characteristic Analysis of Sliding Bearing-rigid RotorSystem with Flexible SupportWang Wenbin（School of Mechatronic Engineering，Lanzhou Jiaotong University，Lanzhou730070，China）Abstract Considering the influence of nonlinear factors such as oil film force support of sliding bearing and external excitation of flexible support，a rigid rotor model with flexible supports at both ends is established. The dynamics behavior of the system is numerically simulated by using the fourth-order variable step size Run⁃ge-Kutta algorithm，the displacement bifurcation diagrams of the rigid rotor system varying with rotational speed with different external excitation amplitudes，and the phase diagram and Poincarémap under specific pa⁃rameters are obtained.These diagrams directly reveal the dynamic characteristics of the system.The results show that the complex dynamic behaviors such as chaos，multi period，almost periodic and single period appear with the increase of rotating speed in the middle and low speed stage，in the middle speed and high speed stage，the system is mainly the mutual transfer of multi period and chaotic motion，and the emergence of paroxysmal chaotic motion.With the increase of the amplitude of external excitation，the system is easy to form almost peri⁃odic motion，and the oil film force at the journal bearing bush increases.Key words Nonlinear oil film force Flexible support Rigid rotor Runge-Kutta method Chaos0引言现代化的工业生产离不开大型旋转机械，而作为旋转机械的主要工作部分，转子系统的动力学行为直接影响着旋转机械的工作平稳性与安全性。

径向滑动轴承—转子系统动力学行为分析

分类号TH133.3UDC密级学号**********硕士学位论文径向滑动轴承-转子系统动力学行为分析刘万万学科名称：机械电子工程学科门类：工学指导教师：吕延军教授张永芳副教授申请日期：2018年6月目录目录1绪论 (1)1.1本文研究目的与意义 (1)1.2滑动轴承-转子系统动力学的研究概况与发展趋势 (1)1.2.1轴承非线性油膜力的研究现状 (1)1.2.2转子动力学的研究现状 (2)1.3本文研究的主要内容 (3)2径向滑动轴承非线性油膜力求解 (5)2.1引言 (5)2.2非线性油膜力模型建立 (5)2.3轴承非线性油膜力的计算方法 (6)2.3.1基于有限差分法的非线性油膜力计算 (6)2.3.2基于近似解析法的非线性油膜力的计算 (9)2.4不同计算方法承载特性的对比 (11)2.5本章小结 (12)3径向滑动轴承-刚性转子动力学分析 (15)3.1引言 (15)3.2轴承-刚性转子系统动力学模型 (15)3.3运动微分方程的计算方法 (16)3.4数值算例 (19)3.4.1轴承-转子随偏心率变化的动力学响应 (20)3.4.2轴承-转子随转速变化的动力学响应 (24)3.5本章小结 (28)4径向滑动轴承-柔性转子动力学分析 (29)4.1引言 (29)4.2转子运动方程 (29)4.3轴承-转子系统组成和矩阵定义方法 (32)4.4数值算例及分析 (35)4.4.1轴承-转子随转速变化的动力学响应 (36)4.4.2扭矩对转子振幅的影响 (40)4.5本章小结 (42)5径向滑动轴承-曲轴系统模态及瞬态动力学分析 (43)5.1引言 (43)5.2曲轴的模态分析 (43)i西安理工大学硕士学位论文ii5.2.1模态分析的求解方法 (43)5.2.2曲轴的模态分析求解过程及结果分析 (43)5.3曲轴的瞬态动力学分析 (46)5.5本章小结 (49)6结论与展望 (51)6.1本文结论 (51)6.2展望 (51)致谢 (53)参考文献 (55)攻读硕士期间发表的论文 (59)绪论1绪论1.1本文研究目的与意义轴承-转子系统是机械设备非常重要的组成部分，在航空、航天、动力等领域有着广泛的应用。

轴承-转子动力学设计-虞烈等

轴承-转子动力学设计-虞烈等轴承—转子系统动力学虞烈刘恒谢友柏摘要介绍了在流体润滑理论和转子动力学基础上发展起来的一门新兴交叉学科——轴承—转子系统动力学的发展历史、所包含的研究内容以及在现代高速旋转机械中开展轴承—转子系统动力学设计的重要性。

关键词轴承转子动力学自激振动润滑理论中国图书资料分类法分类号TH113Dynamics of Bearing-Rotor SystemYu Lie(Xi′an Jiaotong University,Xi′an,China)Liu Heng Xie YoubaiAbstract：This paper introduces dynamics of bearing-rotor system,a new intersecting descip lined branch developed on the basis of the lubrication theory and rotor dynamics .The development history and research contents are discussed.The importance to h igh speed rotating machine design is also explained here.Key words: bearing rotor dynamics oscillation lubricati on theory self-exited1 轴承—转子系统动力学的两大重要组成部分轴承—转子系统动力学是在流体润滑理论和转子动力学基础上发展起来的。

1.1 流体润滑理论人们对于支承技术重要性的认识经历了一个漫长的过程。

1886年,著名的Reynolds方程问世。

Reynolds方程描述了两运动表面间运动速度、表面几何形状、润滑油粘度与油膜压力分布之间的关系,从而奠定了流体润滑理论的基础［1］,也因而带动了一门新兴学科——流体润滑理论及轴承技术的诞生、发展与繁荣。

推力轴承对柔性转子系统的非线性动力影响

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推力轴承对柔性转子系统的非线性动力影响
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叉，高了临界转速和失稳转速，提降低了转子共振振幅，因此推力轴承有助于转子系统的稳定运行．
关键词：推力轴承；柔性转子系统；非线性；叉；定性分稳
中图分类号：ＴＨ１３１０３２文献标志码：Ａ文章编号：０５ —８Ｘ（０１０ —１８０３．；２２３９７２１）９００ —６ＮｏｉａｎｍｉｆｃｆＴｈｒｓａｉｎａＦｌｘｂｅＲｏｏｙｔｍｎｌｎｅｒＤｙａｃＥｆｅｔｏｕｔＢｅｒｎｇｏｅｉｌｔｒＳｓｅ
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鱼雷涡轮转子-滚动轴承系统非线性动力学特性分析

第29卷第6期水下无人系统学报 Vol. 29 No.62021年12月 JOURNAL OF UNMANNED UNDERSEA SYSTEMS Dec. 2021收稿日期: 2021-12-06; 修回日期: 2021-01-04.基金项目: 国家自然科学基金项目(61403306).作者简介: 宋文(1986-), 男, 在读博士, 高级工程师, 主要研究方向为结构动力学设计及振动控制.[引用格式] 宋文, 杨赪石, 陈志伟, 等. 鱼雷涡轮转子-滚动轴承系统非线性动力学特性分析[J]. 水下无人系统学报, 2021,29(6): 690-694.鱼雷涡轮转子-滚动轴承系统非线性动力学特性分析宋文1, 杨赪石1, 陈志伟2, 孙岩2, 任鑫2, 李刚2, 柴晓帅2(1. 中国船舶集团有限公司第705研究所, 陕西西安, 710077; 2. 山西平阳重工机械有限责任公司, 山西临汾, 043003)摘要: 针对鱼雷发动机涡轮转子系统, 考虑轴承径向游隙, 非线性赫兹接触刚度以及转子不平衡量的基础上, 建立了悬臂转子-滚动轴承系统的动力学模型, 并基于变步长的龙格库塔方法对所建立的动力学模型进行了数值仿真。

计算了双参数平面最大Lyapunov 指数谱图, 得到了转子系统随双参数变化时的周期和非周期运动的分布规律, 并进行了系统在特定参数条件下的时域特性、频域特性及周期特性分析。

分析表明, 随着轴承径向游隙增加, 进入混沌区域的最低转速在不断减低。

该研究可为鱼雷涡轮机系统设计提供参考。

关键词: 鱼雷; 发动机; 涡轮转子; 非线性动力学; 接触刚度; 双参数中图分类号: TJ630.32; O194 文献标识码: A 文章编号: 2096-3920(2021)06-0690-05DOI: 10.11993/j.issn.2096-3920.2021.06.007Nonlinear Dynamic Characteristics Analysis of Torpedo TurbineRotor-Rolling Bearing SystemSONG Wen 1, YANG Cheng-shi 1, CHEN Zhi-wei 2, SUN Yan 2, REN Xin 2,LI Gang 2, CHAI Xiao-shuai 2(1. The 705 Research Institute, China State Shipbuilding Corporation Limited, Xi’an 710077, China; 2 ShanXi PingYang Industry Machinery Co.LTD, Linfen 043003, China)Abstract: To investigate the dynamic response of a torpedo turbine rotor, a dynamic model of a cantilever rotor-rolling bearing system is established, and the effect of the radial clearance of the bearing, nonlinear Hertz contact stiffness, and unbalance of the rotor are considered. The numerical simulation of the dynamic model of the system is carried out based on the variable step-size Runge-Kutta method. The maximum Lyapunov exponential spectra of the two-parameter plane are calculated, and the distribution of periodic and aperiodic motions of the rotor system with the change in the two pa-rameters is obtained. The time-domain, frequency-domain, and periodic characteristics of the system under specific pa-rameters are also studied. The analysis results show that the minimum speed in the chaotic region decreases with an increase in the bearing radial clearance. This study provides a useful reference for the design of torpedo-turbine systems. Keywords: torpedo; engine; turbine rotor; nonlinear dynamics; contact stiffness; two-parameter. All Rights Reserved.2021年12月宋文, 等: 鱼雷涡轮转子-滚动轴承系统非线性动力学特性分析第6期 0 引言涡轮机是热动力鱼雷的重要推进形式之一,相对于鱼雷活塞发动机, 鱼雷涡轮发动机工质的焓降大、功率密度大, 能达到活塞发动机无法实现的高航速和远航程[1-2]。

错位安装对多联组配轴承转子系统动力学特性的影响

ｍａｅｈｄｈｏｈｙｔｍｓａｉｔｅｉｎｎｒｏｒｆｈｉｐａｅｅｔａｐａｓｉｉｅ — ｋｓｔｅｗｉｔｆｔｅｓｓｅｉｔｂｌｙｒｇｏａｒｗｅ；ｉｔｅｄｓｌｃｍｎｐｅｒｎＸｄｒｃｎｉｔｏｉｎ，ｔｅａｐｉｄｎＸｄｒｃｉｎｉｓａｌｒｗｉｈｔｅｉｃｅｓｎｎｔｌｄｄｓｌｃｍｅｔｈｍｌｕｅｉｉｅｔｍｌｔｈｎｒａｉｇｉｓａｌｉｐａｅｎ。ｗｈｌｈｔｏｓｅｅｉｔｅｅａｌｕｅｉｉｅｔｎｄｅｏｈｎｅｏｖｏｓｙＴｈｎｌｔｃｒｓｌｓｓｏｔａｈｒｐｒｍｐｉｄｙｄｒｃｉｏｓｎｔｃａｇｂｉｕｌ．ｔｎｏｅａａｙｉｅｕｔｈｗｈｔｔｅｐｏｅｍｉｄｓｌｃｍｅｔｉｂｎｆｉｌｆｒｓａｌｐｒｔｎｏｈｏｏｙｔｍ．ｓｉｐａｅｎｓｅｅｉａｏｔｂｅｏｅａｉｆｔｅｒｔｒｓｓｅｃｏＫｅｗｏｄ：ｍｉｄｓｌｃｍｅｔｏｌｅｒｄｎｍｉｓｃａａｔｒｓｉ；ｓａｉｔｙｒｓｓｉｐａｅｎ；ｎｎｉａｙａｃｈｒｃｅｉｔｃｔｂｌｙｎｉ
ＩｌｅｃｆＭｉｄｉｐａｅｅｎＤｙｍｉｎｆｕｎｅｏｓｓｌｃｍｎｔｏｎａｃＣｈａａｔｒｓｉｓＳｓｅｆＲｏｏｒｃｅｉｔｃｙｔｍｏｔｒＳｐｐｒｅｙＣｏｂｎｄＢａｌＢｅｒｎｓｕｏｔｄｂｍｉｅｌａｉｇ

三种非线性油膜力模型的分析比较1）

三种非线性油膜力模型的分析比较1)王晋麟曹登庆2)王立刚黄文虎（哈尔滨工业大学航天学院，137 信箱，哈尔滨150001）摘要：本文分析比较了三种具有解析表达式的圆轴承非线性油膜力模型，对比了建立油膜力时，Reynolds 方程及其边界条件所采用的假设条件，并以200MW汽轮发电机低压转子为例，比较了不同油膜力模型对系统非线性行为的影响，并分析了产生各种差异的因素。

关键词：转子―轴承；汽轮机；非线性振动；油膜振荡中图分类号：TH133 O3220引言转子―轴承系统非线性动力学行为研究是转子动力学中较为活跃的一个领域。

基于八个油膜动特性系数的线性油膜力模型[1]已经发展得较为完善并获得了广泛的应用。

为了提高发电效率、节约能源、保护环境，汽轮发电机的主力机组从亚临界到超临界、超超临界转型，已经成为必然的选择。

同时，由于发电机转速的提高、结构的轻型化和大柔性使得转子―轴承系统中的非线性因素越来越显著，以小扰动为前提的线性油膜力模型已不再适用。

从20世纪80年代起,转子―轴承系统的非线性油膜失稳问题逐渐引起科学家与工程师们的重视。

建立一个既能较为准确地反映轴承中的油膜力，又简单实用的解析的非线性油膜力模型是研究转子―轴承系统非线性动力学现象的关键。

对圆轴承，从基本的Reynolds方程出发，基于静态Gümbell假设，可以导出无限短轴承和无限长轴承的π油膜力模型的解析表达式[2]。

Muszynska[3]提出用表征流体的周向流速的量来建立非线性的油膜力模型，并据此分析了转子―轴承系统的稳定性。

1991年Capone[4]提出修正的短轴承假设下的非线性油膜力模型，该模型的计算结果表明，它具有较好的精度和收敛性。

张文等[5,6]提出了动态π油膜力模型，它用三个非线性函数描述油膜力，并在短轴承假设下获得了非稳态非线性油膜力的解析表达式。

张文等[7]于2002年进一步提出了非线性油膜力的一般表达式，其瞬态刚度阵和瞬态阻尼阵由三个非线性函数来描述，并通过变分法给出了有限长椭圆轴承的高精度近似解析式。

汽轮机震动的原因及处理分析

汽轮机震动的原因及处理分析摘要：科技不断发展，各领域技术水平逐渐提高，信息技术应用更加广泛的今天，汽轮发电机组是电力系统中重要设备之一，高温高压蒸汽在汽轮机缸体内膨胀做功，推动汽轮机转子及相连的发电机转子同步旋转，由此切割磁感线生产电能。

目前，汽轮发电机组运行中出现的很多震动异常现象仍困扰着机组正常生产工作。

由于汽轮机轴系处在复杂的运行环境，其震动影响因素较多，同时，各运行参数具有极高的耦合性，相关参数不易甄别，因此准确诊断震动故障对机组的安全稳定运行具有重大的现实意义。

关键词：汽轮机；震动；的原因；处理引言大型汽轮发电机组低压转子支撑轴承包含座缸式与落地式。

座缸式轴承因结构紧凑、轴系长度短等优势，广泛应用于汽轮发电机组中，但座缸式轴承位于排汽缸扩压器洼窝中，受缸体变形影响，其支撑结构刚度相对较弱。

工程中，座缸式轴承机组多出现低压缸轴承震动大，轴振小的异常震动问题，给设备的安全运行带来很大隐患。

1汽轮机震动的原因分析1.1转子不对中故障联轴器连接的转子发生对中不良时，转子往往处于有轴线平行位移和轴线角度位移的综合状态，当平行位移和偏角存在时，交变力作用导致转子发生弯曲变形，当主动转子按一定转速旋转时，从动转子的转速会发生周期性变动，使其震动频率为转子转动频率的2倍。

相关研究人员通过实验研究，表明三跨转子的单支撑轴系比双支撑轴系的耦合震动更为强烈，并对比了不同联轴器的震动响应。

相关研究人员利用转子中心线和轴心轨迹组合的特征趋势图描述了不对中与不平衡的响应变化特性。

相关研究人员研究了交角不对中的柔性转子–轴承耦合系统的非线性动力学特性，结果表明：转轴的运动特性在某些参数变化下会出现分叉、跳跃及混沌等非线性特性，使转子失稳，增大横向震动。

1.2由于转子动平衡的偏差产生震动1）运动中汽轮机叶片发生折断、结垢、腐蚀、磨损不均、脱落等情况，使汽轮机转子出现旋转不平衡的现象。

当发电机的转子不平衡或者绕组出现松动情况，导致转子出现不平衡现象。

考虑多间隙的齿轮柔性转子耦合系统非线性动力学分析

ｒｏｔｏｒ — ｂｅｄｉｎｒｇｓｙｓｔｅｍｗａｓｅｓｔａｂｌｉｓｈｅｄｕｎｄｅｒｃｏｍｐｌｅｘｌｏａｄｓ．ＵｓｉｎｇｔｈｅｍｏｄｉｉｅｆｄＦＰＡｍｅｔｈｏｄｔｏｄｅｔｅｒｍｉｎｅｔｈｅｓｏｌｕｔｉｏｎｐｅｒｉｏｄ，ｕｓｉｎｇＲｕｎｇｅ－ＫｕｔｔａｍｅｔｈｏｄａｎｄＮｅｗｔｏｎ— Ｒａｐｈｓｏｎｍｅｔｈｏｄｔｏｓｏｌｖｅｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｅｑｕａｔｉｏｎｓ，ｔｈｅｎｔｈｅｌａｒｇｅｓｔＬｙａｐｕｎｏｖｉｎｄｅｘｗａｓｏｂｔａｉｎｅｄｔｏｄｅｔｅｍｉｒｎｅｔｈｅｄｙｎａｍｉｃａｌｂｅｈａｖｉｏｒｏｆｔｈｅｓｙｓｔｅｍ．Ｔｈｅｄｙｎａｍｉｃｅｑｕａｔｉｏｎｓｗｅｒｅｓｉｍａｌａｔｅｄｎｕｍｅｉｃｒａｌｌｙａｎｄｔｈｅｅｆｆｅｃｔｓｏｆｒｏｔａｔｉｏｎａｌｓｐｅｅｄ，ｂａｃｋｌａｓｈ，ｓｈａｆｔｓｔｉｆｆｎｅｓｓ，ｂｅａｒｉｎｇｒａｄｉａｌｃｌｅａｒａｎｃｅｏｎｔｈｅｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｂｅｈａｖｉｏｒｏｆｔｈｅｓｙｓｔｅｍｗｅｒｅｓｔｕｄｉｅｄ．Ｔｈｅｒｅｓｕｌｔｓｓｈｏｗｅｄｔｈａｔｔｏｏｔｈｏｆａｎｄｂｉｌａｔｅｒａｌｉｍｐａｃｔｐｈｅｎｏｍｅｎｏｎｏｃｃｕｒｗｉｔｈｉｎｃｒｅａｓｅｉｎｂａｃｋｌａｓｈ，ａｎｄｔｈｅｑｕａｓｉ — ｐｅｒｉｏｄｉｃｍｏｔｉｏｎｃｈａｎｇｅｓｉｎｔｏａｃｈａｏｔｉｃｍｏｔｉｏｎａｒｏｕｎｄｔｈｅｃｉｔｒｉｃａｌｓｐｅｅｄ；ｗｉｔｈ

转子动力学分析ppt课件

三、建立转子动力学模型
1、建立模型
当建立转子动力学分析模型时，最重要的是旋转部件和不转动部件分开。
把旋转速度施加到旋转部件上。确保旋转部件是轴对称的结构。无论在ANSYS里建立模型或外部的CAD软件导入模型，需要使用ANSYS中的组件和选择功能来优化分析。这种情况下，要确定转轴、转盘、轴承、支撑结构中哪些需要定义为组件或装配体。
3、常用的术语
（1）陀螺效应所谓陀螺效应，就是旋转着的物体具有像陀螺一
样的效应。陀螺有两个特点：进动性和定轴性。简单来说，陀螺效应就是旋转的物体有保持其旋转方向（旋转轴的方向）的惯性。
对于一个绕轴Δ旋转的结构，如果在垂直于轴Δ施加一个扰动会发生进动且会出现反力矩。这个反力矩就是陀螺力矩。陀螺力矩的轴垂直于旋转轴也垂直于进动轴。这将导致陀螺矩阵耦合了垂直于旋转轴平面上的自由度。这也导致陀螺矩阵为非对称矩阵。
一、概述
➢ 转子动力学是研究轴向对称结构的旋转过程振动行为的一门科学。例如，发动机、转子、光盘驱动器和涡轮机这些设备。
➢ 通过研究惯性对结构的影响可以改进设计并且可以降低失效的概率。像燃气轮机这样的高速旋转设备，必须要考虑旋转件的惯性影响以便准确地预测转子的行为。
➢ 动平衡的理论根据就是转轴的弯曲振动和圆盘的质量以及偏心距的大小的一定确定关系。
所谓的坎贝尔图就是监测点的振动幅值作为转速和频率的函数，将整个转速范围内转子振动的全部分量的变化特征表示出来，在坎贝尔图中横坐标表示转速，纵坐标表示频率，其中强迫振动部分，即与转速有关的频率成分，呈现在以原点引出的射线上，振幅用圆圈来表示，圆圈直径的大小表示信号幅值的大小，而自由振动部分则呈现在固定的频率线上。
KYY（1，0）=0，1000，2000 ！3个旋转速度（rd/s） KYY（1，1）=1E6，2.7E6，3.2E6 ！每一个旋转速度对应的刚度特性

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Ａｂｓｒｔｔａｃ：Ｔｈｅｆｅｉｌｏｏｘｂｅｒｔｒ—ｂｅｒｎｙｔｍｓａｌｚｄｂｙｎｏｉｅｒｄｎｍｉｓｔｏｙ．ｅｅｆｃｆｃｌｕａｉｎｐｅｔｒｌａｉｇｓｓｅｉｎａｙｅｎｌｎａｙａｃｈｅｒＴｈｆｅｔｏａｃｌｔｒｕ－ｏｂｔｏｓｃｎｓｄｅｅｎｃｌｕｌｔｎｉｅｓｅｆＮｅａｉｎｉｏｉｒｄｉａｃａｉｇｔｍｔｐｏｗｍａｋｍｅｈｏｒｔｄ，ｗｈｃｆｅｔｈｃｕｒｃｎｔｂｌｔｆｃｌｕｌ— ｉｈａｆｃｓｔｅａｃａｙａｄｓａｉｉｙｏａｃａｔｏｉｎ，ａｄａｎｗｌｏｔｎｅａｇｒｈｍｓｆｒｅｔｒｕｇｄｆｉｈａｕｓＮｅｉｉｏｍｄｈｏｈｍｏｉｙｎｇｔｅｆｍｏｗｍａｋｍｅｈｄ．ｔｉｈｏｈａｈｅｐｒｐｓｄｍｅｈｒｔｏＩｓｓｗｎｔｔｔｏｏｅｔ — ｏｓｈｇｃｕｒｃｙｃｍｐｒｓｎｏｈｅｒｓｌｓｕｉｄｈａｉｈａｃａｙｂｏａｏｆｔｅｕｔｓｎｇＲｕｎｅ— ＫｕｔｔｄＴｈｔｂｉｉｎｂｆｒａｉｎｏｈｅｉｉｇｔａｍｅｈｏ．ｅｓａｌｔａｄｉｕｃｔｏｆｔｅｐｒ— ｙｏｉｔｏｆｓｓｅｃｎｂｎａｙｅｈｏｅｆｃｔｎｔｏｙＴｈｅｎｍｅｉａｅｕｌｓｉｉａｅｔａｈｙｓｅｄｃｍｏｉｎｏｙｔｍａｅａｌｚｄｂｙｔｅＦｌｑｕｔｂｉｕｒａｉｈｅｒ．ｏｕｒｃｌｒｓｔｎｄｃｔｈｔｔｅｓｔｍｆｔｒｓｎｎｉｅｒｐｈｎｍｅｕｈａｒｏｔｎ，ｑｕｓｐｒｏｔｏｎｄＳｎｅａｕｅｏｌｎａｅｏｎａｓｃｓｐｅｉｄｍｏｉｏａｉ— ｅｉｄｍｏｉｎａＯｏ．
轴承转子系统是一个典型的非线性动力系统，用线性理论已不能解释系统所表现出的某仅些动力行为。由于线性稳定性理论的局限性，以往仅研究了转子系统平衡点解的稳定性，转子而
Ｋｅｏｄ：ｏｒａｂａｎ；ｅｉｌｒｔｓｍ；ｏｌｅｒｙｉｃｉｅｒｅｈｄｂｆｒａｏ；ｔｂｌｙｙｗｒｓｊｎｌｅｒｇｆｘｂｅｏｒｙｔｕｉｌｏｓｅｎｎｉａｔ；ｄｅｔｎｇａｍｔｏ；ｉｃｔｎｓｉｔｎｉｒｔｌｕｉａｉ
ＮｏｉｅｒＤｙａｉｓＡｎａｙｉｎＦｌｘｂｌｔｒ— ｂｅｒｎｇＳｓｅｎｌｎａｎｍｃｌｓｓｏｅｉｅＲｏｏ — ａｉｙｔｍ
ＺＥＧＭｅ一１ＨＩＤＪｉｇ，ＵＹｎ— ｎＨＮｉ３，Ｅｉ，ＩＬａＬａｉｎ１Ａｎ
（．ｌｃｃｌｎｃａｉｌｅａｍｎ，ＳａｎｉａｗｙＩｓｔｔ，ｉｎ７４０，Ｃｉａ１ＥｅｔａａｄＭｅｈｎａＤｐｒｅｔｈａｘＲｉａｎｔｕＷｅａ００ｈｎ；ｉｒｃｔｌｉｅｎ１
了系统周期运动的稳定性及其分岔行为，值结果展现系统具有周期运动、数准周期运动等非线性现象。关键词：动轴承；性转子系统；线性；接积分法；岔；定性滑柔非直分稳中图分类号：Ｈ３．１Ｏ４Ｔ１３３；２１文献标志码：Ａ文章编号：００— ７２２１）０— ０４— ４１０３６（００１０２０
２西安理工大学．机械与精密仪器工程学院，安西７０４）１０８
摘要：用非线性动力学对柔性转子一轴承系统进行了分析，虑了Ｎｗｒ法在计算时间步长内产生的影响运考ｅｍａｋ
计算精度和计算稳定性的计算扰动效应，并将其改进形成一种有效的求解动力学周期响应的方法。将该方法的计算结果与Ｒｎｅｕａ的计算结果进行了对比，ｕｇ —Ｋｔ法ｔ结果表明该方法精度较高。运用Ｆｏｕｔ岔理论分析ｌｑｅ分
２ＳｈｏｏＭｃａｉｌｎｓｕｅｔｎｉｅｉｇ，பைடு நூலகம்ｉａｎｖｒｔｏｅｈｏｇ，ｉａ１０８ｈｎ）．ｃｏｌｆｅｈｎｃｄＩｔｍｎａＥｇｅｒａａｎｒｌｎｎＸ ’ｎＵｉｓｙｆｃｎｌｙＸ ’ ｎ０４，ＣｉａｅｉＴｏ７
！墨
Ｑ二
轴承
２１年１期０００
Ｃ１一ｌ４／ＨＢａｉｇ２１Ｎｏ１Ｎ４１８Ｔｅｒ００，．０ｎ
．试验与分析．Ｉ
柔性转子一轴承系统非线性动力学行为分析
郑美茹黑棣贾良吕延军，，，
（．１陕西铁路３程职业技术学院机电系，７＿陕西渭南７４０；１００