Mechanical model development of rolling beari_2018_Mechanical Systems and Si

Mechanical Systems and Signal Processing 102 (2018) 3758Contents lists available at ScienceDirectMechanical Systems and Signal Processingjournal homepage: /locate/ymsspReviewMechanical model development of rolling bearing-rotorsystems: A reviewHongrui Cao a, Linkai Niu a, Songtao Xi a, Xuefeng Chen babKey Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xian Jiaotong University, Xian 710049, ChinaState Key Laboratory for Manufacturing Systems Engineering, Xian Jiaotong University, Xian 710049, Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 6 September 2016Received in revised form 1 July 2017Accepted 15 September 2017Available online 23 September 2017Keywords:Rotor dynamicsRolling bearingCoupled modellingLumped-parameter modelQuasi-static modelQuasi-dynamic modelDynamic modelFinite element methodTransfer matrix methodContentsThe rolling bearing rotor (RBR) system is the kernel of many rotating machines, whichaffects the performance of the whole machine. Over the past decades, extensive researchwork has been carried out to investigate the dynamic behavior of RBR systems.However, to the best of the authors knowledge, no comprehensive review on RBR mod-elling has been reported yet. To address this gap in the literature, this paper reviews andcritically discusses the current progress of mechanical model development of RBR systems,and identies future trends for research. Firstly, ve kinds of rolling bearing models, i.e.,the lumped-parameter model, the quasi-static model, the quasi-dynamic model, thedynamic model, and the nite element (FE) model are summarized. Then, the coupledmodelling between bearing models and various rotor models including De Laval/Jeffcottrotor, rigid rotor, transfer matrix method (TMM) models and FE models are presented.Finally, the paper discusses the key challenges of previous works and provides new insightsinto understanding of RBR systems for their advanced future engineering applications.2017 Elsevier Ltd. All rights reserved.1.2.3.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Rolling bearing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1. Lumped-parameter models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2. Quasi-static models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3. Quasi-dynamic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4. Dynamic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5. Finite element models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Coupled modelling of rolling bearing-rotor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1. De Laval/Jeffcott rotor supported by rolling bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2. Rigid shaft/rotor supported by rolling bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.1. Lumped-parameter bearing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.2. Quasi-static bearing models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.3. Quasi-dynamic bearing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3. Combination of transfer matrix method and bearing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4. Combination of finite element method and bearing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .383839394041434343444445464647 Corresponding author.E-mail address: (H. Cao)./10.1016/j.ymssp.2017.09.0230888-3270/ 2017 Elsevier Ltd. All rights reserved.384.5.H. Cao et al. / Mechanical Systems and Signal Processing 102 (2018) 37583.4.1. Lumped-parameter bearing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.2. Quasi-static bearing models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.3. Dynamic bearing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Discussions on current limitations and future trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .474748495252521. IntroductionSince the invention of the wheel, humans began to use rotors.1 Rotor dynamic analysis plays an important role in designing,operating and troubleshooting rotors. The paper, On the centrifugal force on rotating shafts, published by Rankine in 1869,marked the beginning of rotor dynamics. Since then, the increasing importance and technical difculties have led to a consid-erable growth of the new eld. Nowadays, rotor dynamics is still a eld of very active research. The authors recommend thereaders to read excellent introductions on the history of rotor dynamics 15, and the representative books 618, which havedetailed bibliographies.In order to better understand rotor dynamics, we must turn to those laws of mechanics which determine rotor behav-ior. If we describe a physical system exactly or approximately by a set of equations, we call that set a model of the phys-ical system 18. In general, a rotor consists of shafts whose diameters change depending on their longitudinal position,disks with various shapes, and bearings situated at various positions. There are two primary issues in modelling the rotorsystem. The rst issue focuses on the rotor. Many turbomachines have exible rotors, where the shaft is designed in arelatively long and thin geometry to maximize the available space for components such as impellers and seals. Moreover,machines are operated at high rotating speeds in order to maximize the power output. The second issue is about the bear-ing modelling. The bearings support the rotating components of the system and provide additional damping to stabilizethe system. Both plain bearings (uid lm bearings) and rolling element bearings are widely used in rotor systems. Due tothe high stiffness and a wide range of load, speed, and operating temperature sustainability, rolling bearings applicationshave ranged from simple bicycles to very sophisticated gas turbine engines used in aircraft engines and cryogenic turbop-umps that form critical parts of the space shuttle propulsion system 19. In comparison with plain bearings, it is moredifcult to model the rolling bearing as a whole due to the complicated coupling between the interactions of components(i.e., rolling elements, cages and rings) of rolling bearings. Furthermore, in order to investigate the dynamic behavior of thewhole system, the coupled modelling between the shaft and rolling bearings is another difcult task. Up to now a longseries of methods and studies related to the modelling of rolling bearing rotor (termed RBR by the authors) systems havebeen proposed. However, to the best of the authors knowledge, no comprehensive review on RBR modelling has beenreported yet.To address this gap in the literature, only rotor systems supported by rolling bearings are considered, and the current pro-gress on the mechanical modelling of RBR systems are reviewed and critically discussed in this work. The rest of the paper isorganized as follows. In Section 2, the rolling bearing models are reviewed. Section 3 presents the coupled modelling of roll-ing bearing rotor systems. Discussions on current limitations and future trends are given in Section 4. Finally, the paper isconcluded in Section 5.2. Rolling bearing modelsBearings constitute one of the most critical components in rotating machinery. Early investigators did not recognize theimportance of the bearing properties, so many analytical rotor models (e.g., Jeffcott rotor 20) had rigid supports at thebearing locations (innite stiffness, no moment restraint). In fact, many problems we are facing with in rotating machinerytoday can be attributed to the improper design or application of the bearings. An understanding of how bearings work istherefore essential for making the proper choice for the particular design that best matches the performance requirementsof the machine.Although a rolling bearing consists of only four components (i.e., inner ring, outer ring, cage and ball), the static anddynamic behaviors of rolling bearings are very complicated in reality. The reason lies in the nonlinear contacts between dif-ferent bearing components and the complex tribo-mechanical phenomena that occur during the bearing operation. There-fore, rolling bearing modelling is essential to gain the knowledge of basic principles. Over the past several decades, a numberof models have become available for rolling bearing design and simulation. Rolling bearing models can be classied as thelumped-parameter model, quasi-static model, quasi-dynamic model, dynamic model and nite element (FE) model. A tech-nical review of these models is given as follows.1 In this paper the word rotor” is used to designate the assembly of rotating parts in a rotating machine, including the shaft, disk and other accessories.H. Cao et al. / Mechanical Systems and Signal Processing 102 (2018) 3758 39Fig. 1. The contact between a rolling element and the rings in a lumped-parameter model 25.2.1. Lumped-parameter modelsIn a lumped-parameter model, only the planar motions of bearing components are considered. The contact between therolling element and the raceway, and the effect of the bearing housing are all treated as nonlinear contact springs, as shownin Fig. 1, where mb is the mass of the rolling element, mr represents the total mass of the outer ring and bearing housing (themass of inner ring is not shown). Moreover, the bearing component only has translational motion which is described withthe following differential equation,mx F 1where m is the mass of the element, x is the translational acceleration, and F is the force acting on the bearing component.Besides Newtons law, Lagranges equations have also be used to derive the dynamic equations 2123. In lumped-parameter models, the rotation of the bearing component around the bearing axis is determined based on pure-rollingassumption, which means that the friction is not considered 24.Generally, there are two types of lumped-parameter models depending on whether the motion of the rolling element isconsidered or not. In the rst case where the motion of the rolling element is concerned, each bearing component has its owndynamic equations, which are described by using Lagranges equation. With these lumped-parameter models, bearing defect2528, waviness 26,29 and the structural vibrations 30,31 in rolling element bearings were studied. In the second case,the degree of freedom of the bearing is totally concentrated on a raceway. The force acting on the ring is determined by sum-ming up the force of each rolling element 32. Since the time-varying bearing force acting on the rotor can be convenientlydetermined based on this type of lumped-parameter models, the nonlinear dynamics of RBR systems were studied exten-sively, such as bifurcation 33, subharmonic resonance 34, the effects of unbalance force 33,35,36, radial clearance37, cage-run out 38, rotor exibility 35,36,39, Alford force 40, and rubbing faults 41. Moreover, the structural vibra-tions of rolling element bearings were studied 4246.2.2. Quasi-static modelsIn quasi-static models, force and moment equilibrium equations are established for rollers and rings to investigate themechanical properties of the bearing. The force and moment equilibrium equations are given asPPF 0M 0 2P Pwhere F and M are the total force and moment acting on a rolling element or a ring, which are the function of displace-ment (including translational and rotational displacement). Eq. (2) can be solved by Newton-Raphson method iteratively.Moreover, for the rolling element, since the effect of the centrifugal force and gyroscopic moment are included, high speedeffects can be considered in quasi-static models.ak cosak sin40 H. Cao et al. / Mechanical Systems and Signal Processing 102 (2018) 3758Fig. 2. Forces and moments acting on a ball 47,48.For ball bearings, one of the most famous and representative quasi-static models was proposed by Jones in 1960 47.Fig. 2 shows the forces and moments acting on a ball in Jones model. The force equilibrium equations of the ball are writtenas( Qij sin aijQij cos aijQoj sin aojQoj cos aojMgjD ij ijMgjD ij ijkoj cos aoj 0koj sin aoj Fcj 03where Qij and Qoj are the contact forces at the inner and the outer rings, aij and aoj are the contact angles at the inner and theouter rings, Mgj is the gyroscopic moment, kij and koj are the controlling parameters for inner and outer rings respectively,and Fcj is the centrifugal force. It can be seen from Eq. (3) that, the centrifugal force and gyroscopic moment of the ballare both considered, which shows the capability of Joness model in analyzing high-speed effects of ball bearings.The raceway-control hypothesis is used in Jones model, which assumes that a ball rolls without spinning on one ring(called controlling ring) while it rolls on another ring. On the controlling ring, the gyroscopic moment of a ball can berestricted by the friction force, and thus no gyroscopic slippage occurs and the ball only rolls (pure rolling) on the controllingring. However, sometimes it is difcult to determine which raceway control hypothesis should be used; as a consequence,the types of raceway control are always selected by the experience of designers. Recently, several authors have attempted toimprove the raceway control theory based on different kinematic hypotheses 49,50. Jones model is capable of obtainingthe expression of the stiffness matrix of the bearing, which can be realized by differentiating the force and moment of innerring with respect to the displacement 51,52. Since the expression of the bearing stiffness can be obtained explicitly, quasi-static models have been widely adopted to investigate the mechanical properties of rolling bearings 49,5357. By combin-ing the stiffness matrices of the rotor and bearings, the global stiffness of the RBR system are obtained, and thus the mechan-ical properties of the RBR system can be investigated 58,59, which will be discussed in Section 3 in detail.Besides ball bearings, many researchers developed quasi-static models for cylindrical and tapered roller bearings. Harris60 improved the Jones model by applying a variety of loading conditions and bearings. The slicing method was proposed toaccurately determine the load distribution along the roller contact of roller bearings. Demul et al. 51,52 proposed a generalquasi-static model, which is based on load deection equations in matrix form and the bearing stiffness matrix can be easilygenerated. However, the gyroscopic moment was not considered. Recently, the effects of external loads 61 and angularmisalignment 62 were studied. The roller-ange contact, which largely affects the power loss of a tapered roller bearing,was investigated in 63. Additionally, the high-precision half space theory has been adopted to improve the calculationaccuracy for cylindrical 64 and tapered 65 roller bearings.In some works, friction effects are included to improve the quasi-static models, which are realized by adding frictionforces and moments in Eq. (2). The Colombo friction effect, 50,66,67 elastohydrodynamic lubrication (EHL) 68,69, andthe temperature rise due to friction 70 have been investigated.2.3. Quasi-dynamic modelsIn quasi-dynamic models, the mechanical and dynamic properties of a bearing are investigated with the followingequations.H. Cao et al. / Mechanical Systems and Signal Processing 102 (2018) 3758 41Fig. 3. Calculation procedure in a quasi-dynamic model.( PPF 0M J dxt 4P Pwhere F and M are the total force and moment acting on a bearing component, J is the inertial moment, x is the angularspeed, and