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Frontiers of Mechanical Engineering

Front. Mech. Eng.
RESEARCH ARTICLE |
New analysis model for rotor-bearing systems based on plate theory
Zhinan ZHANG1, Mingdong ZHOU2(), Weimin DING3, Huifang MA4
1. School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
2. State Key Laboratory of Mechanical System and Vibration, Shanghai Key Laboratory of Digital Manufacture for Thin-walled Structures, Shanghai Jiao Tong University, Shanghai 200240, China
3. Ningbo Donly Co., Ltd., Ningbo 315000, China
4. AECC Commercial Aircraft Engine Co., Ltd. Shanghai 200240, China,
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Abstract

The purpose of this work is to develop a new analysis model for angular-contact, ball-bearing systems on the basis of plate theory instead of commonly known approaches that utilize spring elements. Axial and radial stiffness on an annular plate are developed based on plate, Timoshenko beam, and plasticity theories. The model is developed using theoretical and inductive methods and validated through a numerical simulation with the finite element method. The new analysis model is suitable for static and modal analyses of rotor-bearing systems. Numerical examples are presented to reveal the effectiveness and applicability of the proposed approach.

Keywords rotor-bearing system      rolling element bearing      plate theory      finite element analysis     
Corresponding Authors: Mingdong ZHOU   
Online First Date: 09 October 2018   
 Cite this article:   
Zhinan ZHANG,Mingdong ZHOU,Weimin DING, et al. New analysis model for rotor-bearing systems based on plate theory[J]. Front. Mech. Eng., 09 October 2018. [Epub ahead of print] doi: 10.1007/s11465-019-0525-2.
 URL:  
http://journal.hep.com.cn/fme/EN/10.1007/s11465-019-0525-2
http://journal.hep.com.cn/fme/EN/Y/V/I/0
Fig.1  Illustration of concentrated load applied to an annular plate [27]. (a) Concentrated load is applied to the rigid shaft; (b) equivalent axial load; (c) equivalent radial load
Fig.2  (a) Illustration of the deformation of an annular plate; (b) illustration of an infinitesimal element
Fig.3  Deformation of the annular plate under (a) axial load and (b) radial load
Test Modulus of elasticity, E/GPa Axial or radial load, Fa or Fr/N Maximum axial displacement, wmax/mm Radial displacement, wr/mm a/mm b/mm
1 200 1000 4.892×102 7.712×104 9.784 0.1542
2 100 200 1.957×102 3.805×104 9.785 0.1543
3 150 400 2.609×102 4.113×104 9.784 0.1542
4 300 2000 6.523×102 1.028×103 9.784 0.1542
Tab.1  Calculation of axial and radial deformation parameters
Fig.4  Application procedure for the proposed method [27]
Fig.5  3D model of the rotor-bearing system in a water ring vacuum pump
Fig.6  Results of comparison of the first bending mode for the rotor bearing system. (a) Using the spring element-based approach; (b) using the plate theory-based approach
Results Maximum displacement/mm Frequency/Hz
Spring element-based method 1.422 51.326
Plate theory-based method 1.431 49.602
Relative error 0.009 1.724
Tab.2  spring element-based approach and plate theory-based approach
Fig.7  Calculating displacement during acceleration
Fig.8  Comparison of the results of displacement during acceleration. (a) Using the spring element-based method; (b) using the plate theory-based method
h/mm a /mm b/mm a/b
1.0 9.78400 0.15420 63.430
1.5 3.01400 0.10270 29.360
2.0 1.34100 0.07686 17.450
3.0 0.45700 0.05104 8.954
4.0 0.22820 0.03808 5.993
5.0 0.14010 0.03036 4.615
6.0 0.09758 0.02524 3.866
7.0 0.07368 0.02156 3.417
8.0 0.05880 0.01880 3.128
9.0 0.04880 0.01665 2.932
10.0 0.04166 0.01493 2.791
Tab.3  Datasheet for displacement parameters α and β
Fig.9  Flowchart for database development
Fig.10  Plate theory-based model and potential new models
α Axial deflection parameter
β Radial deflection parameter
E Modulus of elasticity
Kr Radial stiffness
Ka Axial stiffness
wmax Maximum axial displacement
a Outer radius of a bearing/annular plate
b Inner radius of a bearing/annular plate
h Thickness of a bearing/annular plate
μ Poisson’s ratio
  
  Schematic of the coordinate system
Description Diagram and maximum axial displacement wmax
A circular plate is subjected to evenly distributed load q with a simply supported boundary wmax?=qE316 a4 h 3( 1μ) (5+μ)
(In the formula, E is the elastic modulus of the slab, mis the Poisson’s ratio, and h is the thickness of the plate. The same applies to the formulas below.)
A circular plate is subjected to evenly distributed load q with a fixed boundary wmax?=qE316 a4 h 3( 1 μ 2 )
A circular plate is subjected to evenly distributed load q of radius b with a simply supported boundary wmax?=qE34 a2 b 2 h3[3 2μ μ2 b2 1 μ 2 a2ln?ab b24 a2(7 4μ+ μ2)]
A circular plate is subjected to evenly distributed load q of radius b with a fixed boundary w max?= qE 34 1 h3( 1 μ 2)( a2+ b22ln?ab+ 5 b24)
A circular plate is subjected to concentrated force P on the center of the plate with a simply supported boundary w max=PE34π a 2 h3(1 μ) (3+μ)
A circular plate is subjected to concentrated force P on the center of the plate with a fixed boundary w max?= PE 34π a2h3 (1μ2)
A circular plate is subjected to evenly distributed bending moment M0 on its periphery with a simply supported boundary w max?= M0E6 a 2 h3(1 μ)
A circular plate with a hole is subjected to evenly distributed bending moment M1 acting on the internal hole with a simply supported boundary w max?= M1E 6 b 2 h3(2 a 21+μ a 2b2ln? ba1+μ )
A circular plate with a hole is subjected to evenly distributed bending moment M1 acting on the periphery with a simply supported boundary w max?= M1E 6 b 2 h3(1 μ2 b2 1+μ a2b2ln? ba)
A circular plate with a hole is subjected to evenly distributed load Q0 acting on the inter hole with a simply supported boundary wmax?= Q0E3b h 3[ ( μ 2 1)( a2 +b2)+ 2a2b2 a2 b2 (1+μ)2 (ln? ba)2+ a2b22 (1μ) 2 ]
  Maximum axial displacement under different boundary conditions or loads
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