A New Method to Model Spherical Roller Bearings due to Self-Aligning Feature

2015 ◽  
Author(s):  
Yu Xing ◽  
Hua Xu ◽  
Shiyuan Pei ◽  
Xuejing Liu ◽  
Fengcai Wang
Keyword(s):  
Contact Region ◽  
Linear Equations ◽  
Least Square Method ◽  
Least Square ◽  
Hertzian Contact ◽  
Design Parameters ◽  
Roller Bearings ◽  
Proposed Model ◽  
Computing Procedure ◽  
Maximum Contact

This research presents an analytical model to describe the indeterminate contact status and analyze the loaded condition, then acquire the key design parameters so as to improve the carrying capacity of spherical roller bearings. The model based on the non-Hertzian contact theory is applied to reflect the indeterminate contact status due to the self-aligning feature. The loaded condition, which is including the load distribution, the size of the contact region and the maximum contact pressure, is calculated according to the force analysis of spherical roller bearings. The non-linear equations are solved by using secant method and the proposed model is validated by comparing with the published reference. The importance of the indeterminate contact status is illustrated by comparing with the computing results of the Hertzian contact model. The fitted method based on the least square method is used to obtain the equivalent stiffness and the load-deformation exponent, whereby the computing procedure is simplified. In view of the operating condition and the lubricated effect, properly increasing the osculation number or the number of the rollers will obviously improve the carrying ability of spherical roller bearings.

Dynamic Model of Spherical Roller Bearing

2013 ◽  
Author(s):  
Behnam Ghalamchi ◽  
Jussi T. Sopanen ◽  
Aki M. Mikkola
Keyword(s):  
Load Capacity ◽  
Roller Bearing ◽  
Industrial Applications ◽  
Hertzian Contact ◽  
Design Parameters ◽  
Steel Rolling ◽  
Roller Bearings ◽  
Proposed Model ◽  
Rotor Bearing ◽  
Transient Simulations

Rolling element bearings are essential machine elements in the rotating machinery. Extensive research has been conducted to study the dynamics of ball bearings, while studies related to spherical roller bearings are short-shrifted. On the other hand, the number of industrial applications that utilize spherical roller bearings has been increasing constantly. This is due to self-aligning nature and high-load capacity of spherical roller bearings. Typical applications are paper machines, steel rolling, marine equipment, geared transmissions and modern high power wind turbines. This study introduces a three-degree-of-freedom spherical roller bearing model that is computationally efficient, and it is designed to be used in the transient simulations of complete rotor-bearing systems. In the proposed model, the bearing forces are calculated as a function of contact deformation and bearing geometry parameters according to the non-linear Hertzian contact theory. In the numerical results, the important bearing design parameters such as diametral clearance, number of rollers and osculation are studied. Existence of varying compliance (VC) vibrations and the capability of the proposed model in the transient simulations of rotor-bearing systems are demonstrated. The bearing model is verified by using commercial bearing analysis software. Future improvements and model extension possibilities are also discussed.

Prediction of Comprehensive Thermal Error of a Preloaded Ball Screw on a Gantry Milling Machine

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Kuo Liu ◽  
Haibo Liu ◽  
Te Li ◽  
Yongqing Wang ◽  
Mingjia Sun ◽  
...  
Keyword(s):  
Thermal Field ◽  
Thermal Characteristics ◽  
Thermal Error ◽  
Least Square Method ◽  
High Accuracy ◽  
Least Square ◽  
Ball Screw ◽  
Milling Machine ◽  
Thermal Fields ◽  
Proposed Model

The conception of the comprehensive thermal error of servo axes is given. Thermal characteristics of a preloaded ball screw on a gantry milling machine is investigated, and the error and temperature data are obtained. The comprehensive thermal error is divided into two parts: thermal expansion error ((TEE) in the stroke range) and thermal drift error ((TDE) of origin). The thermal mechanism and thermal error variation of preloaded ball screw are expounded. Based on the generation, conduction, and convection theory of heat, the thermal field models of screw caused by friction of screw-nut pairs and bearing blocks are derived. The prediction for TEE is presented based on thermal fields of multiheat sources. Besides, the factors influencing TDE are analyzed, and the model of TDE is established based on the least square method. The predicted thermal field of the screw is analyzed. The simulation and experimental results indicate that high accuracy stability can be obtained using the proposed model. Moreover, high accuracy stability can still be achieved even if the moving state of servo axis changes randomly, the screw is preloaded, and the thermal deformation process is complex. Strong robustness of the model is verified.

scholarly journals Minimization of Deviations of Gear Real Tooth Surfaces Determined by Coordinate Measurements

1993 ◽  
Vol 115 (4) ◽  
pp. 995-1001 ◽  
Author(s):  
F. L. Litvin ◽  
C. Kuan ◽  
J. C. Wang ◽  
R. F. Handschuh ◽  
J. Masseth ◽  
...  
Keyword(s):  
Linear Equations ◽  
Least Square Method ◽  
Least Square ◽  
Tooth Surface ◽  
Hypoid Gear ◽  
Overdetermined System ◽  
Entire Surface ◽  
First Order ◽  
Tooth Surfaces ◽  
Coordinate Measurements

The deviations of a gear’s real tooth surface from the theoretical surface are determined by coordinate measurements at the grid of the surface. A method has been developed to transform the deviations from Cartesian coordinates to those along the normal at the measurement locations. Equations are derived that relate the first order deviations with the adjustment to the manufacturing machine tool settings. The deviations of the entire surface are minimized. The minimization is achieved by application of the least-square method for an overdetermined system of linear equations. The proposed method is illustrated with a numerical example for hypoid gear and pinion.

Mechanical analysis of spherical roller bearings due to surface waviness

2016 ◽  
Vol 232 (4) ◽  
pp. 639-656 ◽  
Author(s):  
Yu Xing ◽  
Hua Xu ◽  
Xuejing Liu ◽  
Hui Xi ◽  
Shibin Wang
Keyword(s):  
Contact Angle ◽  
Mechanical Analysis ◽  
The Self ◽  
Hertzian Contact ◽  
Radial Clearance ◽  
Surface Waviness ◽  
Characteristic Frequencies ◽  
Roller Bearings ◽  
Runge Kutta Method ◽  
Proposed Model

This work presents a theoretical model to research the vibration due to surface waviness of spherical roller bearings (SRBs), taking account of the self-aligning feature and the external axial load. The surface waviness is described by cosinoidal functions. The self-aligning features, including the variation law of the self-aligning contact angle and the interaction with the external loads, are introduced into the non-Hertzian contact model. The nonlinear equations are solved by Runge–Kutta method and the proposed model is validated by comparing with the results of the published references. The results show more characteristic frequencies will be excited under the self-aligning operating condition, whereby the improved equations proposed in this paper are recommended to instead of the previous ones to predict characteristic frequencies of the waviness vibration in an SRB. In addition, these characteristic vibrations caused by waviness are obviously influenced by the magnitude and the direction of the self-aligning contact angle. A proper pretightening load should be chosen according to the self-aligning feature or else it will lead to hidden dangers. The radial clearance and the waviness amplitude can both highlight the effect of waviness. And the vibration caused by a larger radial clearance may be fiercer than the vibration due to waviness.

System Identification of a Class of Second-Order Continuous System

2014 ◽  
Vol 651-653 ◽  
pp. 528-533 ◽  
Author(s):  
Zhi Gang Jia ◽  
Xing Xuan Wang
Keyword(s):  
Linear Equations ◽  
Continuous System ◽  
Least Square Method ◽  
Second Order ◽  
Least Square ◽  
Step Response ◽  
Order System ◽  
Simulation Results ◽  
Order Continuous ◽  
Identification Model

An identification method of a class of second-order continuous system is proposed. This method constructs a discrete-time identification model, forms a set of linear equations. The parameters can be obtained by least square method. Simulation results show that the method is effective for a class of second-order system, and is not only for step response but also for square wave signal.

Thermal Error Robust Modeling for High-Speed Motorized Spindle

2012 ◽  
Vol 466-467 ◽  
pp. 961-965 ◽  
Author(s):  
Chun Li Lei ◽  
Zhi Yuan Rui ◽  
Jun Liu ◽  
Li Na Ren
Keyword(s):  
High Speed ◽  
Thermal Error ◽  
Least Square Method ◽  
Information Criterion ◽  
Least Square ◽  
Proposed Model ◽  
Nc Machine ◽  
Input Variables ◽  
Thermal Error Model ◽  
Multivariate Autoregressive

To improve the manufacturing accuracy of NC machine tool, the thermal error model based on multivariate autoregressive method for a motorized high speed spindle is developed. The proposed model takes into account influences of the previous temperature rise and thermal deformation (input variables) on the thermal error (output variables). The linear trends of observed series are eliminated by the first difference. The order of multivariate autoregressive (MVAR) model is selected by using Akaike information criterion. The coefficients of the MVAR model are determined by the least square method. The established MVAR model is then used to forecast the thermal error and the experimental results have shown the validity and robustness of this model.

scholarly journals DOA Estimation and Self-Calibration under Unknown Mutual Coupling

Sensors ◽  
2019 ◽  
Vol 19 (4) ◽  
pp. 978
Author(s):  
Dong Qi ◽  
Min Tang ◽  
Shiwen Chen ◽  
Zhixin Liu ◽  
Yongjun Zhao
Keyword(s):  
Mutual Coupling ◽  
Linear Equations ◽  
Least Square Method ◽  
Calibration Method ◽  
Least Square ◽  
Frequency Distributions ◽  
Steering Vector ◽  
Practical Applications ◽  
Time Frequency ◽  
Self Calibration

In practical applications, the assumption of omnidirectional elements is not effective in general, which leads to the direction-dependent mutual coupling (MC). Under this condition, the performance of traditional calibration algorithms suffers. This paper proposes a new self-calibration method based on the time-frequency distributions (TFDs) in the presence of direction-dependent MC. Firstly, the time-frequency (TF) transformation is used to calculate the space-time-frequency distributions (STFDs) matrix of received signals. After that, the estimated steering vector and corresponding noise subspace are estimated by the steps of noise removing, single-source TF points extracting and clustering. Then according to the transformation relationship between the MC coefficients, steering vector and MC matrix, we deduce a set of linear equations. Finally, with two-step alternating iteration, the equations are solved by least square method in order to estimate DOA and MC coefficients. Simulations results show that the proposed algorithm can achieve direction-dependent MC self-calibration and outperforms the existing algorithms.

scholarly journals Thermal Analysis of 2D FGM Beam Subjected to Thermal Loading Using Meshless Weighted Least-Square Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
H. M. Zhou ◽  
X. M. Zhang ◽  
Z. Y. Wang
Keyword(s):  
Meshless Method ◽  
Thermal Loading ◽  
Volume Fraction ◽  
Linear Equations ◽  
Least Square Method ◽  
Field Variable ◽  
Least Square ◽  
Power Law Distribution ◽  
Weighted Least Square ◽  
Fgm Beam

The paper analyzed the thermal problem of the 2D FGM beam using meshless weighted least-square (MWLS) method. The MWLS as a meshless method is fully independent of mesh, and an approximate function was used to construct a series of linear equations to solve the unknown field variable, which avoided the troublesome task of numerical integration. The effectiveness and accuracy of the approach were illustrated by a clamped-clamped FGM beam which was subjected with interior heat source. The volume fraction of FGM beam was assumed to be given by a simple power law distribution. The effective material properties of the FGM beam were assumed to be temperature independent and calculated by Mori-Tanaka method. The results showed that a good agreement was achieved between the proposed meshless method and commercial COMSOL Multiphysics.

scholarly journals Forming a Hierarchical Choquet Integral with a GA-Based Heuristic Least Square Method

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1155
Author(s):  
Chen ◽  
Huang
Keyword(s):  
Choquet Integral ◽  
Least Square Method ◽  
Least Square ◽  
Integral Model ◽  
Fuzzy Measures ◽  
Proposed Model ◽  
Choquet Integrals ◽  
The Mean ◽  
Input Variables ◽  
Least Mean Squares Algorithm

: Identifying the fuzzy measures of the Choquet integral model is an important component in resolving complicated multi-criteria decision-making (MCDM) problems. Previous papers solved the above problem by using various mathematical programming models and regression-based methods. However, when considering complicated MCDM problems (e.g., 10 criteria), the presence of too many parameters might result in unavailable or inconsistent solutions. While k-additive or p-symmetric measures are provided to reduce the number of fuzzy measures, they cannot prevent the problem of identifying the fuzzy measures in a high-dimension situation. Therefore, Sugeno and his colleagues proposed a hierarchical Choquet integral model to overcome the problem, but it required the partition information of the criteria, which usually cannot be obtained in practice. In this paper, we proposed a GA-based heuristic least mean-squares algorithm (HLMS) to construct the hierarchical Choquet integral and overcame the above problems. The genetic algorithm (GA) was used here to determine the input variables of the sub-Choquet integrals automatically, according to the objective of the mean square error (MSE), and calculated the fuzzy measures with the HLMS. Then, we summed these sub-Choquet integrals into the final Choquet integral for the purpose of regression or classification. In addition, we tested our method with four datasets and compared these results with the conventional Choquet integral, logit model, and neural network. On the basis of the results, the proposed model was competitive with respect to other models.

Complementary Use of Partial Least-Squares and Artificial Neural Networks for the Annual Electricity Consumption Forecast

2012 ◽  
Vol 512-515 ◽  
pp. 1113-1116 ◽  
Author(s):  
Hui Feng Jiang
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Electricity Consumption ◽  
Least Square Method ◽  
Partial Least Square ◽  
Least Square ◽  
Factors Affecting ◽  
The Neural Network ◽  
Proposed Model ◽  
Partial Least Square Method

A model for predicting annual electricity consumption based on the combination of neural network and partial least square method was proposed. The factors affecting the annual electricity consumption are analyzed by means of partial least square method to extract the most important components so that not only the problem of multi-correlation among variables can be solves but also the amount of input dimensions of the neural network can be reduced. Besides, the application of neural network helps to solve the problem of non-linearity of the model. The application example shows that the proposed model has high precision.

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