scholarly journals Thermally affected stiffness matrix of angular contact ball bearings in a high-speed spindle system

2019 ◽  
Vol 11 (11) ◽  
pp. 168781401988975
Author(s):  
Dinh Sy Truong ◽  
Byung-Sub Kim ◽  
Jong-Kweon Park
Keyword(s):  
Thermal Expansion ◽  
Stiffness Matrix ◽  
High Speed ◽  
Structural Model ◽  
Ball Bearings ◽  
Spindle System ◽  
Spindle Shaft ◽  
Ball Surface ◽  
Bearing Stiffness ◽  
Coupling Stiffness

Bearing stiffness directly affects the dynamic characteristics in a high-speed spindle system and plays an important role in terms of manufacturing quality. We developed a new approach for predicting the thermal behavior of a high-speed spindle, calculated the thermal expansion, and generated a bearing stiffness matrix for angular contact ball bearings. The heat convection of spindle housing in air, the balls in lubricant, the spindle shaft in quiescent air, and the bearing inner ring surfaces were determined. Heat sources such as bearing friction, and the heat contributed by the built-in motor, were simulated using an analysis systems (ANSYS) steady-state thermal model. The results were imported into a static ANSYS structural model. Ball thermal expansion was calculated based on changes in the coordinates of nodal points on the ball surface. Finally, a thermally affected bearing stiffness matrix was generated by applying the Newton–Raphson technique. Decreases in the bearing radial, axial, angular, and coupling stiffness values as rotational spindle speed increased were calculated. Also, the stiffness coefficients at a specific speed increased significantly caused by the thermal effects. Finally, for validation, the bearing stiffness was compared to that calculated using an earlier thermal network approach.

scholarly journals Complete Analytical Expression of the Stiffness Matrix of Angular Contact Ball Bearings

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
David Noel ◽  
Mathieu Ritou ◽  
Benoit Furet ◽  
Sebastien Le Loch
Keyword(s):  
Stiffness Matrix ◽  
High Speed ◽  
Computational Cost ◽  
New Method ◽  
Computation Method ◽  
Ball Bearings ◽  
Dynamic Effects ◽  
Analytical Expression ◽  
Bearing Stiffness ◽  
The Impact

Angular contact ball bearings are predominantly used for guiding high speed rotors such as machining spindles. For an accurate modeling, dynamic effects have to be considered, most notably in the bearings model. The paper is based on a dynamic model of angular contact ball bearings. Different kinematic hypotheses are discussed. A new method is proposed for the computation of the stiffness matrix: a complete analytical expression including dynamic effects is presented in order to ensure accuracy at high shaft speed. It is demonstrated that the new method leads to the exact solution, contrary to the previous ones. Besides, the computational cost is similar. The new method is then used to investigate the consequence of the kinematic hypotheses on bearing stiffness values. Last, the relevance of this work is illustrated through the computation of the dynamic behavior of a high speed milling spindle. The impact of this new computation method on the accuracy of a finite element spindle model is quantified.

Dynamics of Machine Tool Spindle/Bearing Systems Under Thermal Growth

1997 ◽  
Vol 119 (4) ◽  
pp. 875-882 ◽  
Author(s):  
Bert R. Jorgensen ◽  
Yung C. Shin
Keyword(s):  
Thermal Expansion ◽  
Dynamic Response ◽  
Heat Generation ◽  
High Speed ◽  
Transfer Model ◽  
Good Prediction ◽  
High Speeds ◽  
Spindle Bearing ◽  
Thermal Growth ◽  
Bearing Stiffness

Increased use of high-speed machining creates the need to predict spindle/bearing performance at high speeds. Spindle dynamic response is a function of the nonlinear bearing stiffness. At high speeds, thermal expansion can play an important role in bearing stiffness. A complete bearing load-deflection analysis including thermal expansion is derived and is coupled with an analysis of spindle dynamic response. Steady-state temperature distribution is found from heat generation at the contact point and from a quasi three-dimensional heat transfer model. Numerical solutions give a good prediction of thermal growth and heat generation in the bearing. Predicted high-speed spindle frequencies show good agreement with experimentation. The effects of loading condition and bearing material type on bearing stiffness are also shown.

Spherical surface generation mechanism in the grinding of balls for ultraprecision ball bearings

2000 ◽  
Vol 214 (4) ◽  
pp. 351-357 ◽  
Author(s):  
B Zhang ◽  
A Nakajima
Keyword(s):  
Kinematic Analysis ◽  
High Speed ◽  
Spherical Surface ◽  
Generation Mechanism ◽  
Surface Generation ◽  
Ball Bearings ◽  
Contact Points ◽  
Ball Surface ◽  
Precision Machines ◽  
Good Agreement

Ultraprecision ball bearings are necessary for high-precision machines and/or high-speed machines since the vibration caused by ball bearings determines the precision of machines as a whole and may make high-speed machines fail to work. To produce ultraprecision ball bearings, it is necessary to clarify spherical surface generation mechanism in the grinding of balls. This paper is the first attempt to investigate the contact trace distribution on the ball surface, which is crucial to spherical surface generation. The kinematic analysis of the contact trace shows that the contact trace is a fixed circle on the ball surface and the contact points are not uniformly distributed on the ball surface. Experimental observation of the contact trace was also carried out. The observation is in good agreement with the analysis. Suggestions on how to distribute the contact trace over the whole ball surface and therefore to improve the precision of balls are given.

Modeling of Flexible Coupling to Connect Misaligned Flexible Rotors Supported on Ball Bearings

2014 ◽  
Author(s):  
T. C. Gupta ◽  
K. Gupta
Keyword(s):  
Dynamic Response ◽  
Stiffness Matrix ◽  
Ball Bearing ◽  
Radial Clearance ◽  
Fe Model ◽  
Ball Bearings ◽  
Flexible Coupling ◽  
Flexible Rotors ◽  
Deep Groove ◽  
Coupling Stiffness

Parallel or/and angular misalignment between rotors connected by flexible coupling causes deformation of coupling elements and rotor shafts both. The forces and moments from flexible coupling act on driver and driven rotors simultaneously and depend upon dynamic response of coupled rotor system. The authors’ aim in the present work is to propose a methodology to incorporate the coupling stiffness matrix and coupling loads into the finite element model of flexibly coupled flexible rotors supported on deep groove ball bearings. The coefficients of coupling stiffness matrix and coupling loads are nonlinear and depend upon the amount of parallel and angular misalignments. To segregate the effect of nonlinear stiffness of coupling on the dynamic response of the system, the stiffness of ball bearing is initially considered to be linear. Thereafter, ball bearing nonlinearities arising from radial clearance, Hertzian deformation of balls and races, and varying compliance effect are included into the FE model of the system. Considering different types of misalignments in succession and combination, the dynamic response is characterized by the presence of 1N, 2N and other super or sub harmonics. The misalignment in one plane is found to affect the dynamic response in both the orthogonal planes. Comparing to the stiffness of flexible coupling, the stiffness of ball bearing is of higher order and therefore, nonlinear coupling forces and moments have dominant influence on the dynamic response of the system.

An Integrated Approach Toward the Dynamic Analysis of High-Speed Spindles: Part I — System Model

1993 ◽  
Author(s):  
C. H. Chen ◽  
K. W. Wang ◽  
Y. C. Shin
Keyword(s):  
High Speed ◽  
Integrated Approach ◽  
Ball Bearings ◽  
Basic Principles ◽  
Gyroscopic Moment ◽  
High Speeds ◽  
Modeling Process ◽  
Radial Stiffness ◽  
Bearing Stiffness ◽  
Made In

Abstract Experimental evidence [Shin, 1992] has shown that the natural frequencies of high speed spindles with angular contact ball bearings decrease with increasing rotational speed. A recent study [Wang, et al., 1991] illustrated that this phenomenon is caused by stiffness change of the bearings. A simplified approximation was used in the analysis to examine the bearing radial stiffness at high speeds. While the investigation explained the experimental observations in a qualitative sense, the analytical results so far are not sufficient to quantitatively describe the spindle behavior under high speed operations due to the approximations made in the modeling process. This paper presents an integrated approach toward the modelling of flexible spindles with angular contact ball bearings from basic principles. The local dynamics of the bearings are coupled with the global shaft motion. The model derived includes both the longitudinal and transverse vibrations of the shaft interacting with the nonlinear bearings. The influences of shaft speed on the bearing stiffness matrix and the system frequencies have been studied. It is shown that the spindle dynamic behavior can vary substantially as speed increases due to the bearing gyroscopic moment and centrifugal force. These effects have been ignored in most of the previous spindle models. Lab tests were conducted to validate the model. The analytical predictions are quantitatively verified by the experimental results.

scholarly journals Time-Varying Total Stiffness Matrix of a Rigid Machine Spindle-Angular Contact Ball Bearings Assembly: Theory and Analytical/Experimental Verifications

2011 ◽  
Vol 18 (5) ◽  
pp. 641-670 ◽  
Author(s):  
Fawzi M.A. El-Saeidy
Keyword(s):  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Time Dependent ◽  
Time Varying ◽  
Ball Bearings ◽  
Rigid Rotor ◽  
Bearing Stiffness ◽  
Machine Spindle ◽  
Matrix Case

A lagrangian formulation is presented for the total dynamic stiffness and damping matrices of a rigid rotor carrying noncentral rigid disk and supported on angular contact ball bearings (ACBBs). The bearing dynamic stiffness/damping marix is derived in terms of the bearing motions (displacements/rotations) and then the principal of virtual work is used to transfer it from the bearing location to the rotor mass center to obtain the total dynamic stiffness/damping matrix. The bearing analyses take into account the bearing nonlinearities, cage rotation and bearing axial preload. The coefficients of these time-dependent matrices are presented analytically. The equations of motion of a rigid rotor-ACBBs assembly are derived using Lagrange's equation. The proposed analyses on deriving the bearing stiffness matrix are verified against existing bearing analyses of SKF researchers that, in turn, were verified using both SKF softwares/experiments and we obtained typical agreements. The presented total stiffness matrix is applied to a typical grinding machine spindle studied experimentally by other researchers and excellent agreements are obtained between our analytical eigenvalues and the experimental ones. The effect of using the total full stiffness matrix versus using the total diagonal stiffness matrix on the natural frequencies and dynamic response of the rigid rotor-bearings system is studied. It is found that using the diagonal matrix affects natural frequencies values (except the axial frequency) and response amplitudes and pattern and causes important vibration tones to be missig from the response spectrum. Therefore it is recommended to use the full total stiffness matrix and not the diagonal matrix in the design/vibration analysis of these rotating machines. For a machine spindle-ACBBs assembly under mass unbalnce and a horizontal force at the spindle cutting nose when the bearing time-varying stiffness matrix (bearing cage rotation is considered) is used, the peak-to-valley variation in time domain of the stiffness matrix elements becomes significant compared to its counterpart when the bearing standard stiffness matrix (bearing cage rotation is neglected) is used. The vibration spectrum of the time-varying matrix case is marked by tones at bearing outer ring ball passing frequency, rotating unbalnce frequency and combination compared to spectrum of the standard stiffness matrix case which is marked by only the rotating unbalnce frequency. Therfore, it is highly recomended to model bearing stiffness matrix to be a time-dependent.

Analysis about Vibration and Noise of Ceramic Electrical Spindle

2011 ◽  
Vol 189-193 ◽  
pp. 1696-1699 ◽  
Author(s):  
Yu Hou Wu ◽  
Li Xiu Zhang
Keyword(s):  
High Speed ◽  
Tetragonal Zirconia ◽  
Major Elements ◽  
High Speed Machining ◽  
Ball Bearings ◽  
High Productivity ◽  
Spindle Shaft ◽  
Ferrous Metals ◽  
High Speed Grinding ◽  
Tetragonal Zirconia Polycrystal

High speed machining (HSM) technology is used in a broad range of applications to machine ferrous metals and nonmetallic material. The electrical spindle is one of the major elements to keep the machine running at high productivity. In recently years, the requirement of rotational speed and rigidity of electrical spindle is getting higher and higher in order to satisfy the high speed processing. So a high speed grinding electrical spindle equipped with hot isostatic pressed silicon nitride (HIPSN) fully-ceramic ball bearings without inner rings and yttria partially stabilized tetragonal zirconia polycrystal (Y-TZP) ceramic spindle shaft is designed for higher speed and rigidity. Because the processing precision is relevant to the vibration of electrical spindle, it is necessary the vibration and noise of the ceramic electrical spindle is detected. This paper presents results of vibration and noise of the ceramic electrical spindle and analysis the reason of phenomenon. The analysis shows the vibration and noise of the ceramic electrical spindle is up to the standard.

scholarly journals Speed Effects on Inherent Rotating Frequency of Motorized Spindle System

2011 ◽  
Vol 2-3 ◽  
pp. 900-905 ◽  
Author(s):  
Bo Wang ◽  
Wei Sun ◽  
Kun Peng Xu ◽  
Bang Сhuan Wen
Keyword(s):  
Centrifugal Force ◽  
High Speed ◽  
Impact Factors ◽  
Motorized Spindle ◽  
Spindle System ◽  
High Speeds ◽  
Radial Stiffness ◽  
Bearing Stiffness ◽  
Coupling Factors ◽  
The Impact

The paper determines the impact factors of high-speed spindle system including the centrifugal force, gyroscopic moments and the bearing stiffness softening, etc, then builds the general spindle-bearing FEM considering high speeds. Taking a motorized spindle as example, the effect of centrifugal force, gyroscopic effect, the radial stiffness and the coupling factors are analyzed qualitatively and quantitatively. Finally the research shows the variations of bearing radial stiffness, centrifugal force and gyroscopic moments have a significant effect on dynamics of spindle system in high speeds, while modeling the high speed spindle system, above factors must be considered.

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 Bearings

1989 ◽  
Vol 111 (1) ◽  
pp. 142-148 ◽  
Author(s):  
J. M. de Mul ◽  
J. M. Vree ◽  
D. A. Maas
Keyword(s):  
General Theory ◽  
Load Distribution ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Contact Analysis ◽  
Hertzian Contact ◽  
Ball Bearings ◽  
Roller Bearings ◽  
Rolling Element ◽  
Bearing Stiffness

A new, general and consistent mathematical model of highly modular character is presented for calculation of the equilibrium and associated load distribution in rolling element bearings. The bearings may be loaded and displaced in five degrees of freedom. High speed rolling element loading is considered, internal friction is neglected, the material is assumed linearly elastic and the bearing rings are modelled as rigid except for local contact deformation. Either classical Hertzian contact analysis or modern non-Hertzian contact analysis of sophisticated or approximate character is used as applicable. The bearing stiffness matrix is computed analytically and used internally in the iterative bearing equilibrium calculation; its final values may be used for other purposes such as (rotor) dynamics analysis. In Part I, the general theory and application to ball bearings is presented. In Part II, application of the general theory to roller bearings and an experimental verification are presented.

scholarly journals Transient Vibration Prediction for Rotors on Ball Bearings Using Load-Dependent Nonlinear Bearing Stiffness

2004 ◽  
Vol 10 (6) ◽  
pp. 489-494 ◽  
Author(s):  
David P. Fleming ◽  
J. V. Poplawski
Keyword(s):  
High Speed ◽  
Roller Bearing ◽  
Transient Behavior ◽  
Rolling Element Bearing ◽  
Ball Bearings ◽  
Transient Vibration ◽  
Vibration Prediction ◽  
Rolling Element ◽  
Bearing Stiffness ◽  
Element Bearing

Rolling-element bearing forces vary nonlinearly with bearing deflection. Thus an accurate rotordynamic transient analysis requires bearing forces to be determined at each step of the transient solution. Analyses have been carried out to show the effect of accurate bearing transient forces (accounting for nonlinear speed and load-dependent bearing stiffness) as compared to conventional use of average rolling-element bearing stiffness. Bearing forces were calculated by COBRA-AHS (Computer Optimized Ball and Roller Bearing Analysis—Advanced High Speed) and supplied to the rotordynamics code ARDS (Analysis of Rotor Dynamic Systems) for accurate simulation of rotor transient behavior. COBRA-AHS is a fast-running five degree-of-freedom computer code able to calculate high speed rolling-element bearing load-displacement data for radial and angular contact ball bearings and also for cylindrical and tapered roller bearings. Results show that use of nonlinear bearing characteristics is essential for accurate prediction of rotordynamic behavior.

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