Calculation of Ball Bearing Speed-Varying Stiffness

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Calculation of ball bearing speed-varying stiffness Article in Mechanism and Machine Theory · November 2014 DOI: 10.1016/j.mechmachtheory.2014.07.003

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Mechanism and Machine Theory 81 (2014) 166–180

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Calculation of ball bearing speed-varying stiffness☆ Xia Sheng, Beizhi Li ⁎, Zhouping Wu, Huyan Li Center of Advanced Manufacturing Technology, School of Mechanical Engineering, Donghua University, North Renmin Road, Songjiang District, Shanghai, China 201620

a r t i c l e

i n f o

Article history: Received 18 October 2013 Received in revised form 8 July 2014 Accepted 9 July 2014 Available online xxxx Keywords: Ball bearing Speed-varying stiffness Implicit function differentiation

a b s t r a c t Dynamic properties of ball bearing (or any rolling bearing) are varying during operation, especially the stiffness which mostly depends on rotating speed and loads applied. Therefore, in this paper, the notion of rolling bearing speed-varying stiffnesses introduced and explained by studying the relations of load-deflection through the bearing dynamic model which is based on Jones & Harris's efforts. After the acquisition of load-deflection relations, two common numerical (linearization) methods are provided to calculate bearing stiffness, which turned out to be too unstable to get satisfied results. To improve the method of calculating speed-varying stiffness, an analytic model based on the differentiation of implicit function is proposed to calculate the speedvarying stiffness. The comparisons between the results from the numerical method and the proposed method have been made to validate the proposed method. In addition, the results calculated by the proposed method are compared with those which appeared in other literatures or experiments. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Rolling bearings can be seen in all kinds of rotational machineries for their low-cost, high precision and high reliability. As support parts of a rotating system, the rolling bearings play a crucial role in the transmission of the loads and vibrations. Thus, it is needed to study the dynamic properties of the rolling bearing, particularly the high speed rolling bearing. When a bearing is operating at high speed, centrifugal and gyroscopic effects induced by the rotating rolling elements in the bearing will have great influence on the entire dynamic properties of the bearing, including the bearing stiffness which is very important to the analysis of dynamic behavior of the rotor-bearing system. Most previous research focused on the static bearing stiffness, considering it a constant and adopting it in dynamic analysis. For a lower speed rotor-bearing system, using static bearing stiffness in dynamic analysis may have a conclusion close to the real one. However in a high speed situation, bearing stiffness will change a lot due to the effect of speed and variations of bearing internal load distribution. Ball bearings, as a kind of rolling bearing, are one of the most important parts of spindle-bearing system. Many researchers [1–3] found that spindle performance changes dynamically due to the nonlinear effect of stiffness of the bearings. M. Matsubara [4] studied this nonlinear effect and proposed a piecewise-linear model to simulate the nonlinearity of bearing stiffness. But Matsubara's paper didn't include the influence of speed on bearings. J. Jedrzejewski [5] gave a view about relations between bearing running speed and axial stiffness. It was found that axial stiffness would continue dropping if the rotational speed grew. Matti Rantatalo et al. [6] had the same point, they predicted that the speed-varying radial stiffness would fall down to 40% of the static radial stiffness when the

☆ This document is a subsidized program of the National 863 Project (2012AA041309). ⁎ Corresponding author. E-mail addresses: warterfl[email protected] (X. Sheng), [email protected] (B. Li).

http://dx.doi.org/10.1016/j.mechmachtheory.2014.07.003 0094-114X/© 2014 Elsevier Ltd. All rights reserved.

X. Sheng et al. / Mechanism and Machine Theory 81 (2014) 166–180

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spindle runs at 20,000 rpm. However, both of them didn't give a detailed model of the calculation of speed-varying stiffness which should depend on the analysis of bearing dynamic characteristics. Early studies on rolling bearing dynamic characteristics were started by A.B. Jones [7,8] who studied the ball motion and internal contact mechanism and their relations. Later on, T.A. Harris reviewed Jones's theory and further concluded a model which was suitable for the calculation of bearing internal load distribution and bearing deflection [9,10], which was widely known as Jones & Harris model. Although Jones & Harris model did well in studying the dynamic characteristics in rolling bearings, their equations were complex in calculation. There were some other meaningful studies concentrated on the bearing deflection vs. applied load [11,12], though they were analyzed or measured statically. Gargiulo [13] formed empirical formulae for the load-deflection displacement relations by assuming rigid bearing races, whose conclusion was regarded as a benchmark for bearing stiffness. It is available to calculate bearing stiffness based on the theories listed above, but these theories either were proposed in static state or they cannot determine tilting and cross-coupling stiffness between the radial, axial, and tilting deflections of bearings which are necessary to form the stiffness matrix. T.C. Lim and R. Singh [14] proposed a comprehensive bearing stiffness matrix of six dimensions which demonstrated a coupling between the shaft bending motion and the flexural motion on the casing plate. Hoon–Voon Liew [15] analyzed time-varying rolling element bearing characteristics based on T.C. Lim's bearing stiffness model. Yi Guo [16] created a FEM model to numerically calculate the bearing stiffness matrix which covered a wide range of bearing types and was also in six dimensions, this method, however, was a time-consuming job as the numerical results were obtained by FEM which would certainly take much time. Besides, their models were on the assumptions that didn't consider the dynamic behaviors in bearings. That means the models proposed above didn't realized influence of rotational speed on bearing stiffness. G.D. Hagiu [17] developed a theoretical research concerning the rigidity and damping characteristics of high speed angular contact ball bearings. It studied deformations of bearings by adopting Hertzian contact theory, but the definition of bearing stiffness was not referred. Many researchers also estimated bearing stiffness by carrying out the experiments. Walford and Stone [18] developed a rotor-bearing test rig to estimate the bearing's radial stiffness and damping by measuring the response of the rotor. It also studied the effect of temperature and rotating speed on the stiffness of the bearings. Then Stone published a review of the measurements of bearing stiffness [19]. In his review, various kinds of method in measuring the stiffness were introduced. Similarly, Kraus et al. [20] measured stiffness and damping characteristics of a radial ball bearing by performing experimental modal tests on a transmission test stand. Their results indicated that static bearing stiffness is very close to the stiffness measured when the bearing is running. However their conclusion is not convincing enough to prove that speed-varying stiffness can be estimated by static stiffness because their testing speed was only up to 1000 rpm which now seems to be a very low rotational speed. Rajiv Tiwari and Nalinaksh S. Vyas [21] performed a series of experiments on a multi-disk rotor-bearing system to estimate the bearing stiffness through examining random responses. In this research, they measured the rotor displacement and velocity signals and set up a relation between rotor displacement and bearing stiffness rather than the relation between bearing displacement and bearing stiffness. In this paper, a mathematical model for calculating bearing speed-varying stiffness is proposed by analyzing the equations in the bearing dynamic model which is based on Jones & Harris's efforts [10]. This mathematical model aims to give a comprehensive consideration on the nonlinear stiffness of ball bearing and the influence of speed on the bearing stiffness that affects the load vs. deflection displacement characteristic of bearing which also affects the dynamics of ball bearing-supported rotor system. Extending Jones & Harris's model to an analytical system of five degrees of freedom, it also provides a stiffness matrix of five dimensions. Moreover, the comparisons have been made with other views on bearing stiffness to see how the effect of speed affects bearing stiffness.

2. Starting assumption Some assumptions are necessary due to the complicated interactions between bearing elements and parts. These complexities include indeterminate bearing internal geometry, nonlinear interaction between the rolling bearing elements and the motions of internal rolling elements. Therefore, to establish a practical mathematical model, the following assumptions are proposed: – The bearing analyzed is ideally shaped and is free from faults. – Bearing outer ring is fixed in space while the inner ring will deflect in radial or axial direction. – The outer ring and inner ring are assumed not to be bent. Small deformations occurring at the contact areas of rolling elements and raceways do not affect the entire shape of both rings. – Rolling elements are separated by a constant angle by the cage. The dynamic behavior of the cage is neglected. – The bearing operates under isothermal conditions. – The external loads act centrically upon the rings, which will not cause angular misalignment of the bearing rings. – Tribological issues are beyond the scope of this study, thus, lubrication is ignored. These assumptions above are generally accepted in most analyses of rolling bearings. Some further assumptions will be introduced and discussed to serve this work in special cases.

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3. Bearing stiffness matrix Bearing stiffness matrix which is wildly used in rotor dynamics formulations contains useful information of bearing stiffness like radial stiffness, axial stiffness and tilting stiffness. It can be presented as Eq. (1): 0

kxx Bk B yx B ½K  ¼ B B kzx Bk @ θx x kθy x

kxy kyy kzy kθx y kθy y

kxz kyz kzz kθx z kθy z

kxθx kyθx kzθx kθx θx kθy θx

1 kxθy kyθy C C C kzθy C C kθx θy C A kθy θy

ð1Þ

where x and y respectively stand for radial direction which are axes in the plane of the bearing, z is the axial direction of the bearing. θx, θy are the angular deflections around the x and y axes respectively. Diagonal terms in the stiffness matrix include radial stiffness kxx, kyy, axial stiffness kzz, and tilting stiffness kθx θx ; kθy θy . The rest of the ingredients of the stiffness matrix are cross-coupling terms that are coupling between radial and axial deflections, between radial and angular deflections, between axial and angular deflections or between different angular deflections. For an external load vector applied on the bearing's inner ring: {F} = {Fx, Fy, Fz, Mx, My} and the deflection vector of the inner ring which was caused by the load: {δ} = {δ x, δ y, δ z, θ x , θ y}, the stiffness matrix under the given load {F} is described as Eq. (2): 0

∂F rx B B ∂δrx B B ∂F ry B B ∂δ B rx B B ∂F z ½K  ¼ B B ∂δrx B B ∂M θx B B B ∂δrx B B ∂M θ @ y ∂δrx

∂F rx ∂δry ∂F ry

∂F rx ∂δz ∂F ry

∂F rx ∂θx ∂F ry

∂δry ∂F z ∂δry ∂Mθx

∂δz ∂F z ∂δz ∂Mθx

∂θx ∂F z ∂θx ∂Mθx

∂δry ∂M θy

∂δz ∂Mθy

∂θx ∂Mθy

∂δry

∂δz

∂θx

∂F rx ∂θy ∂F ry

1

C C C C C ∂θy C C C ∂F z C C: ∂θy C C ∂Mθx C C C ∂θy C C ∂Mθy C A ∂θy

ð2Þ

4. Speed-varying stiffness and the numerical calculation of it Based on Jones' model, T. A. Harris [9] had built a high-speed bearing model that can describe rolling element bearing loaddeflection relations. In his model, within these equations, he concluded that clear load-deflection relations will change due to the change of bearing rotating speed, which consequently leads to the variation of stiffness. 4.1. A brief summary of Harris's high-speed bearing dynamic model When a ball bearing is operating, the internal geometric relations between balls and each inner and outer raceways change a lot due to the remarkable centrifugal and gyroscopic effects induced by the orbiting rolling elements. Meanwhile, the contact forces generated by centrifugal forces and gyroscopic moments will reach a significant magnitude, forcing the inner raceway contact angles to increase and outer raceway contact angles to decrease. To determine the load distribution and loaddeflection displacement characteristics of a high speed ball bearing, Harris [10] established the geometry and compatibility equations for the ball bearing. According to his study, several simultaneous equations were used to solve load distribution and load-deflection, they include equations of load distribution and equations of equilibrating applied loads. The equations of internal load distribution:  2  2 h i2 A1q −X 1q þ A2q −X 2q − ð f i −0:5ÞD þ δiq ¼ 0

ð3Þ

h i2 2 2 X 1q þ X 2q − ð f o −0:5ÞD þ δoq ¼ 0

ð4Þ

Q iq sinα iq −Q oq sinα oq þ

 Mgq  2cosα oq ¼ 0 D

ð5Þ

X. Sheng et al. / Mechanism and Machine Theory 81 (2014) 166–180

Q iq cosα iq −Q oq cosα oq −

 Mgq  2sinα oq þ F cq ¼ 0 D

169

ð6Þ

where the meaning of symbols can be found in Appendix A. The equations of applied loads equilibrating inner raceway contact load: q¼Z   X Q iq cosα iq cosφq ¼ 0 Fx−

ð7Þ

q¼1

q¼Z   X Fy− Q iq cosα iq sinφq ¼ 0

ð8Þ

q¼1

q¼Z   X Fz− Q iq sinα iq ¼ 0

ð9Þ

q¼1

and Mx ¼

q¼Z h X

 i Q iq sinα iq ℜi sinφq

ð10Þ

q¼1

My ¼

q¼Z h X

 i Q iq sinα iq ℜi cosφq

ð11Þ

q¼1

where Fx/Fy, Fz and Mx/My stand for applied radial load, axial load and applied moment respectively, and subscripts like x and y stand for coordinate directions. The meanings of the rest of the symbols can be found in Appendix A. Fig. 1 describes the distribution of the rolling elements and the coordinate system of a bearing which is used in the equations above. The total number of rolling elements is Z. These elements are separated by cage by an angle of Δφ, Δφ = 2π/Z. Every rolling element is named by order, and the azimuth of a given element is φq. The internal geometry of a ball and raceways is shown in Fig. 2. If the bearing is load-free, the centers of inner raceway groove curvature Oi, outer raceway groove curvature Oo and that of balls Ob are collinear. Once the bearing is operating, the ball center Ob will shift to the position of Ob due to centrifugal force. With those balls shifted, the inner raceway groove center Oi shifts to Oi. The changes of geometry, as shown in Fig. 2 and presented by Eqs. (3)–(6), will lead to a variation of the entire bearing dynamics. It should be noted that both the load vector {F} = {Fx, Fy, Fz, Mx, My} and deflection vector {δ} = {δx, δy, δz, θx, θy} are all involved in Eqs. (3)–(11). Thus, there is no definite specification of what vector should be inputs and what should be outputs. Both of them are applicable for inputs or outputs. If let Fx, Fy, Fz, Mx, My be the inputs and then δx, δy, δz, θx, θy are the outputs and vice versa. However, it is found that using {δ} as inputs calculates much faster than using {F} as inputs. Therefore, most calculation cases in this paper use {δ} as inputs.

Fig. 1. The bearing coordinate system.

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Fig. 2. Positions of shifted ball center and raceway groove curvature centers for the qth ball.

4.2. Numerical methods for calculating speed-varying stiffness From J–H's (Jones-Harris) work, it is possible to calculate speed-varying stiffness of ball bearings with the numerical methods. The load-deflection can be established according to J–H's high-speed bearing model. After that, two possible methods of calculating the bearing speed-varying stiffness numerically can be used: 1. It can be calculated numerically through finite differences:

K¼

∂F F ðδ þ ΔδÞ− F ðδÞ ≈ Δδ ∂δ

ð12Þ

2. Or even with higher order like second order:

K¼

∂F F ðδ þ ΔδÞ− F ðδ−ΔÞ ≈ : 2Δδ ∂δ

ð13Þ

where δ is the bearing deflection vector at given load {F}, and {Δδ} is the small disturbance vector at {δ}, fΔδg ¼   Δδx ; Δδy ; Δδz ; Δδθx ; Δδθy . Obviously the smaller Δδ is, the better solution of K it will be. 3. The other possible way to obtain speed-varying stiffness of rolling bearing is to measure slope of the load-deflection curves at the required deflection location(s) (Fig. 3 presents this method, load F and δ are the two elements in establishing load-deflection relations). To determinate the required speed-varying stiffness at a certain deflection location, different loads F and δ calculated by Harris's model at different speeds are required to form load-deflection curves. In the process of forming load-deflection curves, a number of load-deflection points should be calculated which would cost much time. These solutions, however, are not always practically feasible because the results calculated by the bearing model are based on iteration of multiple nonlinear equations. Therefore, in order to have an accurate calculation of bearing stiffness, the disturbance δ should be as small as possible. That means the load-deflection locations (δ, F) calculated by Harris's load-deflection equations must be very close to each other. For those complex nonlinear simultaneous equations (Eqs. (3), (4), (5), (6)), some variables calculated through the equations are sensitive to the small disturbance of input and then cause the calculation results unstable, which is called, in mathematics, numerical sensitivities. Thus, it is nearly impossible to obtain the entire range of stiffness of all speeds. To make this point clear, an example bearing is studied to show the shortages of numerical calculation. The example bearing is B7005C/P4 which is high-speed angular contact ball bearing for machine tool spindle. The essential design parameters of this bearing are given in Table 1:

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171

Fig. 3. Rolling bearing load-deflection curve.

By providing δ (or F) as inputs and setting the reasonable initial iteration values X1q, X2q, δiq and δoq, the applied load F (or δ) can be determined by solving the formulas. Fig. 4 shows that the bearing radial stiffness results calculated numerically are quite unstable. They fluctuate with large amplitude. As was said, to ensure accuracy, the disturbance Δδ should be small enough. But when it is set too small, accuracy can no longer be guaranteed as the calculation results are not even stable. It is because in Eqs. ((3), (4), (5), (6)), nonlinearity is strong and besides, there are several coupling parameters among those equations. Otherwise, the unstable situation can be a relief if disturbance is set to a relative larger value. However in this case, the calculation accuracy is not guaranteed. It is a certain truncation error that cannot be ignored when the disturbance Δδ is large. In Table. 2 two different values of Δδx and their numerical calculation results of stiffness are presented. It is clear that there will be differences between both ends, and it is estimated that the results with smaller disturbance will likely be better. Fig. 5 shows that the load-deflection curve plotted with numerical calculation data cannot form a continuous curve, making the determination of stiffness by measuring the slope of load-deflection curve unrealizable. Similarly, this failure is caused by unstable results. In addressing this issue, in this paper, a new analytic method based on differentiation of implicit function is developed to calculate ball bearing stiffness. An example bearing has been calculated on speed-varying stiffness characteristics. 5. Analytic method for calculating speed-varying stiffness 5.1. Theoretical explanation In the dynamic model of high-speed ball bearings, the applied load equilibrium in Eqs. (3)–(11) shows no direct relation between the applied loads or moments (F = {Fx, Fy, Fz, Mx, My}) and the deflection displacements (δ = {δx, δy, δz, θx, θy}). To calculate the bearing stiffness, the relations between loads and deflection displacements should be found. If the variables Qiq, cos αiq and sin αiq, in Eqs. (3)–(11), are substituted with X1q, X2q, δiq and δoq, then it yields:  2  3 1:5 q¼Z K X iq A2q −X 2q δiq 4 5cos φ Fx ¼ q ð f i −0:5ÞD þ δiq q¼1  2  3 1:5 q¼Z K X iq A2q −X 2q δiq 4 5sin φ Fy ¼ q ð f i −0:5ÞD þ δiq q¼1

ð14Þ

Table 1 B7005C/P4 bearing parameters. SKF Explorer6205 Parameters Number of rolling elements Z Nominal contact angle α0 Pitch diameter dm Ball diameter D outer raceway groove curvature radius coefficient fo inner raceway groove curvature radius coefficient fi

Values 15 15° 36 mm 5.5 mm 0.54 mm 0.52 mm

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Fig. 4. Unstable results of bearing radial stiffness with calculated numerically, the small disturbance Δδxx = 0.0002.

Fz ¼

q¼Z  X q¼1

Q iq sin α iq



 2  3 1:5 q¼Z K X iq A1q −X 1q δiq 4 5 ¼ ð f i −0:5ÞD þ δiq q¼1

ð15Þ

and  2  3 1:5 q¼Z K X iq A1q −X 1q δiq 4 5ℜi sin φ Mx ¼ q ð f i −0:5ÞD þ δiq q¼1

ð16Þ

 2  3 1:5 q¼Z K X iq A1q −X 1q δiq 4 5ℜ i cos φ : My ¼ q ð f i −0:5ÞD þ δiq q¼1

It was previously mentioned that the variables X1q, X2q, δiq and δoq are calculated by the inputs δx, δy, δz, θx and θy through Eqs. ((3), (4), (5), (6)). It is obvious that different inputs lead to the variation of X1q, X2q, δiq and δoq. Therefore, in this paper, each variable X1q, X2q, δiq and δoq is considered as a function of δx, δy, δz, θx and θy:   X 1q ¼ f x1 δx ; δy ; δz ; θx ; θy   X 2q ¼ f x2 δx ; δy ; δz ; θx ; θy   δiq ¼ f δi δx ; δy ; δz ; θx ; θy   δoq ¼ f δo δx ; δy ; δz ; θx ; θy :

ð17Þ

Table 2 Comparison of different disturbance δxx used in numerical calculation. Deflection mm

0.001 0.002 0.003 0.004 0.005 0.006

Stiffness calculated with Δδ

Absolute differences %

0.0005

0.00002

41,745.73 53,845.27 56,833.21 53,513.82 67,500.27 69,736.72

45,327.12 56,684.02 57,880.42 54,879.65 67,937.93 73,890.53

8.6 5.3 1.8 2.6 6.5 6.0

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173

Fig. 5. Load-deflection curve with bad smoothness.

Through this, the applied loads Fx, Fy, Fz and moments Mx, My are the functions of δx, δy, δz, θx and θy. Then relations between the load and deflection have been established. Thus, the equations which describe the load–displacement relations are:  2  3 1:5 q¼Z K X iq A2q −X 2q f δi 4 5cos φ Fx ¼ q ð f i −0:5ÞD þ f δi q¼1  2  3 1:5 q¼Z K X iq A2q −X 2q f δi 4 5sin φ Fy ¼ q ð f i −0:5ÞD þ f δi q¼1  2  3 1:5 q¼Z K X iq A1q −X 1q f δi 4 5 Fz ¼ ð f i −0:5ÞD þ f δi q¼1

ð18Þ

ð19Þ

ð20Þ

and  2  3 1:5 q¼Z K X iq A1q −f x1 f δi 4 5ℜi sin φ Mx ¼ q ð f i −0:5ÞD þ f δi q¼1  2  3 1:5 q¼Z K X iq A1q −f x1 f δi 4 5ℜi cos φ My ¼ q ð f i −0:5ÞD þ f δi q¼1

ð21Þ

ð22Þ

where A1q = BD sin α0 + δz + ℜi(θx sin φq + θycos φq), A2q = BD cos α0 + δx cos φq + δysin φq. From Eqs. (18), (19) and (20), the radial stiffness in x and y directions and axial stiffness of the bearing can be expressed respectively by the partial derivatives with respect to δx, δy and δz as: Kxx = ∂Fr/∂δx, Kyy = ∂Fy/∂δy, and Kzz = ∂Fz/∂δz by chain rule: K xx ¼

K yy ¼

" # q¼Z ∂F x X ∂F x ∂A2q ∂F x ∂f x2 ∂F x ∂f δi ¼  þ  þ  ∂δx ∂A2q ∂δx ∂f x2 ∂δx ∂f δi ∂δx q¼1

ð23Þ

" # q¼Z X ∂F y ∂A2q ∂F y ∂f x2 ∂F y ∂f δi  þ  þ  ∂A2q ∂δy ∂f x2 ∂δy ∂f δi ∂δy q¼1

ð24Þ

∂F y ∂δy

¼

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X. Sheng et al. / Mechanism and Machine Theory 81 (2014) 166–180

"

K zz

# q¼Z ∂F z X ∂F a ∂A1q ∂F z ∂f x1 ∂F a ∂f δi : ¼ ¼  þ  þ  ∂δz ∂A1q ∂δz ∂f x1 ∂δz ∂f δi ∂δz q¼1

ð25Þ

Similarly, the expressions of tilting stiffness (or angular stiffness) are: K θxθx ¼

K θy θy ¼

" # q¼Z ∂M x X ∂Mx ∂A1q ∂M x ∂f x1 ∂Mx ∂f δi ¼    þ þ ∂θx ∂A1q ∂θx ∂f x1 ∂0x ∂f δi ∂θx q¼1 ∂My ∂θy

¼

" # q¼Z X ∂My ∂A1q ∂My ∂f x1 ∂My ∂f δi :    þ þ ∂A1q ∂θy ∂f x1 ∂θy ∂f δi ∂θy q¼1

ð26Þ

ð27Þ

However, in Eq. (23), for example, ∂fx2 /∂δrx or ∂f δi =∂δrx cannot be determined directly because the functions fx2 and f δi previously defined in the text do not have the explicit expressions between fx2 and δx . These partial derivatives should be derived from the implicit differentiation of simultaneous functions (3, 4, 5, 6). Let F1,F2,F3 and F4 stand for the left-side hand of Eqs. ((3), (4), (5), (6)) respectively. According to the implicit function differentiation theory: ∂f x2 1 ∂ð F ; F ; F ; F Þ ¼−  1 2 3 4 J ∂ X ;δ ;δ ;δ ∂δx 1q

∂f δi ∂δx

x

iq

ð28Þ

oq

1 ∂ð F 1 ; F 2 ; F 3 ; F 4 Þ   J ∂ X ;X δ ;δ

¼−

1q

2q x

ð29Þ

oq

where J is the Jacobian determinant:   ∂F 1  X  1q   ∂F 2  X ∂ð F 1 ; F 2 ; F 3 ; F 4 Þ  ¼  1q J¼   ∂F 3 ∂ X 1q ; X 2q ; δiq ; δoq X  1q   ∂F 4   X 1q ∂fδ

∂ f x2 ∂δry

∂F 1 X 2q ∂F 2 X 2q ∂F 3 X 2q ∂F 4 X 2q

∂F 1 δiq ∂F 2 δiq ∂F 3 δiq ∂F 4 δiq

 ∂F 1  δoq   ∂F 2   δoq  ∂F 3  δoq   ∂F 4   δoq 

ð30Þ

∂fδ

and ∂δ or ∂∂δf and ∂δ or the rest of the partial derivatives can be solved in the same way. From this analytic method, we can get much more exact solutions of Kxx,Kyy, Kzz and K θx θx , K θy θy . This method can also provide coupling stiffness like: x1 a

i

a

" # q¼Z ∂F x X ∂F x ∂A2q ∂F x ∂f x2 ∂F x ∂f δi ¼  þ  þ  ∂δy ∂A2q ∂δy ∂f x2 ∂δy ∂f δi ∂δy q¼1

ð31Þ

" # q¼Z ∂M x X ∂M x ∂A1q ∂Mx ∂f x1 ∂M x ∂f δi : ¼ ¼  þ   þ ∂δx ∂A1q ∂δx ∂f x1 ∂δx ∂f δi ∂δx q¼1

ð32Þ

K xx ¼

K θx x

i

ry

As the speed-varying stiffness calculation is accumulative, it is unavoidable that the calculation procedure takes many loops of which the number is equal to the number of rolling elements in the bearing (the value of Z). It should firstly solve every accumulation components in stiffness expression, then sum all the components to obtain the stiffness value. Flowchart Fig. 6 briefly shows how the calculation procedure works. 5.2. Calculation examples of the proposed method The proposed speed-varying stiffness calculation method is based on the Jones & Harris bearing model equations, it can firstly be verified by the stiffness calculated by Jones & Harris's model through numerical method. The object bearing is still B7005C/P4. Table. 3 shows the comparison of the results calculated by the numerical method and the analytic method (proposed method). From Table. 3 it can be seen that the results are close to each other. However, a set of results does not mean the numerical method is applicable. Figs. 7 and 8 illustrate the comparisons between the proposed method and numerical difference method. From these figures, even though the results of difference method are not stable and cannot be plotted as a smooth curve of stiffness, it is clear that they still

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175

Start Input bearing design parameters D, dm, fi, fo, 0 and operating parameters ,{F} or { } Set initial iteration values, let q=1

Calculate the variables X1q, X2q, iq, oq on the qth rolling element Iteration continues N

If the iteration convergent or not?

Saving obtained data , q=q+1

Y

Save the data of solved X1q, X2q, iq, oq

Solve the expression of implicit partial derivatives required fx2/ x, f i/ x etc... Substitute the values of X1q, X2q, iq, oq to the partial derivatives

Solve the required qth component of stiffness expressions and substitute the values of implicit partial derivatives into them

q