A comparative study and stiffness analysis of angular contact ball bearings under different preload mechanisms

Mechanism and Machine Theory 115 (2017) 1–17

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Research paper

A comparative study and stiffness analysis of angular contact ball bearings under different preload mechanisms Jinhua Zhang a, Bin Fang a, Yongsheng Zhu b, Jun Hong a,∗ a

State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

b

a r t i c l e

i n f o

Article history: Received 7 December 2016 Revised 5 March 2017 Accepted 25 March 2017

Keyword: Angular contact ball bearing Elastic deformation Preload mechanisms Stiffness

a b s t r a c t The dynamic performance of angular contact ball bearings is greatly affected by preload force and preload mechanisms. In the present research, a comparison model is built to analyze the stiffness of angular contact ball bearings under different preload mechanisms. In the first part, the elastic deformation of inner ring of bearing produced by assembling stress and centrifugal stress is calculated. Then the differences of two different types of preload on the action mechanism of bearings are discussed. Based on these, a comparative model of ball bearing and stiffness with respect to different preload mechanisms is proposed. Finally, the influences of the inner ring interference value, rotating speed and radial load on the bearing characteristics and stiffness are discussed in detail. The results show that under fix-positon preload, the bearing have a better stability of stiffness, and the inner ring interference value and rotating speed also have a significant influence on the bearing dynamic properties and stiffness. In addition, the proposed model is validated by comparing the results obtained by the proposed method with those from the other literatures and experiments. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Angular contact ball bearings are widely used in the spindle of machine tools and various precision instruments due to high precision and high reliability. With the rapidly development of modern machinery industry, rotating machinery is developed towards high speed and high precision. In order to simulate the dynamic characteristics of high-speed bearingrotor system, an accurate mathematical model of rolling bearings is needed. Over the past decades, many scholars have conducted a series of researches on the modeling of action mechanisms of angular contact ball bearings. Palmgren [1] proposed a static model of ball bearing to calculate the bearing deflection and internal load distribution under a combined load. Based on the Hertz contact theory and “Raceway Control Hypothesis”, Jones [2,3] developed a classical bearing dynamic analysis model considering the centrifugal force and gyroscopic moment of bearing balls, Harris [4,5] developed the model in Ref. [2,3] by introducing the Elastohydrodynamic lubrication(EHL) theory to replace “Raceway Control Hypothesis”. The analysis results have a better agreement with the experimental results than those from the “Raceway Control Hypothesis” model. It was known that “Raceway Control Hypothesis” was not sufficiently precise and “Outer Raceway Control” is a special case only occurring in a very limited manner for oil-lubricated ball ∗

Corresponding author. E-mail address: [email protected] (J. Hong).

http://dx.doi.org/10.1016/j.mechmachtheory.2017.03.012 0094-114X/© 2017 Elsevier Ltd. All rights reserved.

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J. Zhang et al. / Mechanism and Machine Theory 115 (2017) 1–17

bearings. Foord [6] calculated the power dissipated by friction within each contact and determined the ball rolling axis using minimum energy dissipation principle in placed of “Raceway Control Hypothesis”. Wang [7] and Ding [8] proposed a new quasi-static analysis model, which also discarded the “Raceway Control Hypothesis” by introducing a controlling equation of ball pitch angle. Gupta [9,10] developed a complete dynamic model to simulate the transient motion of balls and cage, this model could effectively simulate the operation status of ball bearing. However, it is time consuming. Recently, Bovet [11] presents a modelling approach for predicting the internal dynamic behavior of ball bearing under high moment loads. Xu and Yang [12,13] proposed an approach for dynamic modeling of a ball bearing with waviness defects and localized defects in planar multibody systems. Kim [14] proposed a method to design a non-standard angular contact ball bearing for the main shaft of a grinder using design automation and optimization techniques. Yan [15] investigated the heat dissipation characteristic of ball bearing cage and inside cavity at high speed. Most of the aforementioned studies are only available for the bearings that are elastically preloaded. Bearing stiffness as an important dynamic parameter of rolling bearings, is very important to the bearing capacity and vibration of the bearing-rotor system. The accuracy of bearing stiffness value is crucial in the analysis of dynamic behaviors of the bearing-rotor system. Houpert [16] and Hernot [17] proposed a 5-DOF analysis model to compute the bearing stiffness matrix, but the model neglected the dynamic effect of balls. Lim and Singh [18] developed a 6-DOF mathematical model to analyze the cross-coupling terms in the bearing stiffness matrix. Noel [19] proposed a complete analytical method to calculate the bearing stiffness matrix which provides an exact solution. Xia [20] proposed an analytical model based on the implicit differential theory, It was proved that this model could avoid the errors caused by numerical methods. Guo [21] established a FEM model of rolling bearing to calculate the bearing stiffness, but this model took too much time to obtain a high precision solution. Angular contact ball bearings, as supporting parts of the rotor, proper preload is important to improve the rotor stiffness, rotating accuracy and life. There are two main types of preload mechanisms used in practical projects: constant pressure preload and fixed-position preload. Constant pressure preload is achieved by the compression spring applied to the bearing outer ring, Fix-position preload is achieved by restricting the relative axial displacement between the inner ring and the outer ring, it should be noted that few investigations are proposed according to the differences of the dynamic stiffness behaviors under various preload mechanisms. Cao [22] proposed a comparative model to analyze the stiffness and frequency response functions of the spindle under various bearing preload mechanisms. However, no detailed analysis about the changing law and influence factors of the bearing stiffness with respect to two different preload mechanisms. Besides, in the actual running status, the bearing inner ring as rotation component, the radial deformation is produced by the actions of centrifugal and assembling pressure, which might cause the change in bearing internal parameters and then change the internal clearance and the initial contact angle of ball bearings. Also, the change in bearing internal parameters has different influences on the dynamic characteristics for two preload mechanisms. For the constant pressure preload, the springs can absorb the bearing outer ring displacement which is caused by the change in bearing internal parameters, and the magnitude of preload keeps approximately unchanged. For the fix-position preload, the relative axial displacement between the inner ring and the outer ring keeps constant, but bearing preload is changed by the influences of centrifugal force and assembling pressure. In this study, the geometry parameter variations internal the bearing, induced by assembling stress and centrifugal stress, is provided. The process of preload applying is detailed discussed, and the effects of two different preload mechanisms on the internal geometry variations of the bearing are analyzed. In addition, a 5DOF mathematical analytical model is built to calculate ball bearing stiffness matrix. On this basis, the comprehensive comparison of the effect under two different preload mechanisms is given. The bearing contact forces, angles and stiffness matrix under different loads, rotation speed are investigated and compared in detail. In addition, the proposed model is validated by comparing the results obtained by the proposed method with those from the other literatures and experiments.

2. Deformation analysis and bearing structure parameters modification In order to ensure the geometric and kinematic accuracy of the bearing-rotor system, interference connection is usually used for assembling the bearing inner ring and shaft. In the operational condition, especially for the high speed state, the connection condition between shaft and bearing inner ring is changed by centrifugal forces. Meanwhile, the internal geometry parameters of the bearing are changed owing to the bearing inner ring deformation caused by the assembling stress and centrifugal stress. The schematic diagram of the shaft connected with the bearing inner ring is shown in Fig. 1. The inner and outer radii of the shaft are r1 and r2 , and the outer radius of the inner ring is r3 . Due to the width dimension of inner ring is smaller than radius, bearing inner ring, working as a rotation component, is regarded as a hollow disc. According to the theory of elasticity [23], the axisymmetric plane stress theory is applied to resolve the deformation. The equilibrium equation of a cylindrical coordinate system is:

d σr + dr

σr − σθ r

+ ρω2 r = 0

(1)

J. Zhang et al. / Mechanism and Machine Theory 115 (2017) 1–17

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Fig. 1. The schematic diagram of shaft-bearing inner ring connection.

where σ r and σ θ respectively denote the stress component of radial and tangential direction, ρ is the material density, and ω is the rotation speed. The elastic constitutive equations:

⎧ ⎨ εr = 1 (σr − νσθ ) E

(2)

⎩ε = 1 (σ − νσ ) r θ θ E

where ε r and ε θ respectively denote the radial strain and the tangential strain, and E and ν are the material elastic modulus and Poisson’s ratio. The geometric equations are:

⎧ ⎨εr = ds dr

(3)

⎩ε = s θ r

where s denotes the radial displacement. Substituting Eqs. (2) and (3) into Eq. (1), one can get the following expression:

s 1 − ν2 d2 s 1 ds + − 2 =− ρω2 r 2 r dr E dr r

(4)

Solving the above Euler-type differential equation, one can obtain the expressions of radial displacement and stress:

s = C1 r +

σr =

1 − ν2 C2 − ρω2 r3 r 8E



E (1 + ν )C1 + 1 − ν2

(5)

ν −1  r2

C2 −

3+ν ρω2 r2 8E

(6)

For the bearing inner ring, assuming that the assembling pressure between inner ring and shaft is P1 , so the boundary conditions of inner ring can be written as:



r = r2 , σr = −P1 r = r 3 , σr = 0

(7)

Substituting the boundary conditions into Eq. (6) gives:

⎧

r22 (1 − ν ) ( 3 + ν )ω 2 ρ 4 4 ⎪ ⎪ ( r3 − r2 ) ⎪ ⎨ C1 = E (r2 − r2 ) P1 + 8r22 3 2

⎪ (1 + ν )r22 r32 ( 3 + ν )ω 2 ρ 2 2 ⎪ ⎪ P1 + ( r3 − r2 ) ⎩C2 = 8 E (r32 − r22 )

(8)

Substituting Eq. (8) into Eq. (5), one can obtain the radial deformation at random radius r of inner ring. Thus, the deformations at r = r2 and r3 are respectively expressed as:

sr=r2 = C1 r2 +

(1 − v2 )ρ w2 r23 C2 − r2 8E

(9)

sr=r3 = C1 r3 +

(1 − v2 )ρ w2 r33 C2 − r3 8E

(10)

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Compared with the inner ring, the axial dimension of the shaft is much larger than the sizes of its cross section, it can be regarded as a long cylinder. According to the theory of elasticity, the axisymmetric plane strain theory is applied to resolve the deformation, replace the constants E, ν in Eqs. (5) and (6) with E/(1-ν 2 ), ν /(1−ν ), the radial displacement and stress of the shaft can be obtained:

s = C3 r +

σr =

C4 (1 − 2ν )(1 + ν ) ρω2 r3 − r E (1 − ν ) 8



C4 E C3 − 2 1 + v 1−2ν r



−

(11)

ρω2 r2 3 − 2ν 8 1−ν

(12)

Similar to the inner ring, the boundary conditions of shaft can be written as:



r = r2 , σr = −P1 r = r 1 , σr = 0

(13)

Substituting the above boundary conditions into Eq. (12), one can obtain:



⎧ (1 − 2ν )(1 + v )r22 ( 3 − 2ν )ω 2 ρ 4 ⎪ 4 ⎪ −P1 + ( r2 − r1 ) ⎪ ⎨C3 = E (r22 − r12 ) 8(1 − v )r22

⎪ (1 + ν )r12 r22 ( 3 − 2ν )ω 2 ρ 2 2 ⎪ ⎪ −P1 + ( r2 − r1 ) ⎩C4 = 8 (1 − v ) E (r22 − r12 )

(14)

Substituting Eq. (14) into Eq. (11), the deformation of inner shaft at r = r2 can be calculated:

sr=r2 = C3 r2 +

C4 (1 − 2ν )(1 + ν ) ρω2 r23 − r2 E (1 − ν ) 8

(15)

Assuming that the initial value of interference fit between inner ring and shaft is ini , one can obtain the following equation by the deformable coordination condition:

 ini = 2 sr=r2 − sr=r2

(16)

The practical interference fit 1 between inner ring and shaft varies with the rotation speed rise and the assembling pressure P1 can be calculated as following:

P1 = 2r2



1−ν

2

 r22 +r12

E 1

r22 −r12

− 1−ν ν



+



r32 +r22 r32 −r22

+ν



(17)

Combining Eq. (9), (15)–(17), one can get the solutions of 1 and P1 , and sr=r3 can be achieved by substituting P1 into Eq. (10), sr=r3 is used to update the inner raceway contact diameter di for the following analysis. 3. The analysis of preload mechanism Generally, angular contact ball bearings are designed with internal clearance. For both two different preload mechanisms, the clearance is eliminated by external constraints (force or displacement) to improve operation stability and stiffness of the bearings. As shown in Fig. 2, under the static condition, the relative axial displacement between inner ring and outer ring is generated by preload force, assuming that the outer ring is fixed without loss of generality. The axial displacement can be divided into two phases. In the first phase, the axial clearance 0.5Pe is eliminated, and it will generate the free contact angle α o . In the second phase, a new axial displacement δ p will be generated due to the internal contact deformations of bearings, and the magnitude of preload is determined by the axial displacement in the second phase. For the bearings with constant pressure preload, the magnitude of preload basically remains constant while δ p changes caused by the influences of centrifugal force and gyroscopic moment of balls. For the bearings with fix-position preload, due to the constrained relative axial displacement between the inner ring and the outer ring is constrained, the total displacement of the above too phases 0.5Pe +δ p remains constant. Therefore, the magnitude of preload varies with the change of axial clearance 0.5Pe , and the solution procedure of Pe is as follows: The radial clearance can be written as:

Pd = do − di − 2D

(18)

So one can obtain the bearing free contact angle:

α o = arcos

1 2

A − 14 Pd 1 A 2



(19)

J. Zhang et al. / Mechanism and Machine Theory 115 (2017) 1–17

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Fig. 2. The process of preload applied on the rolling bearing.

Fig. 3. The angular speed vectors relation between ball and contact raceway contact.

The axial clearance Pe may be determined by the following expression:

Pe = 2A sin α o

(20)

where A is defined as:

A = ro + ri − D

(21)

In the above equations, ri and ro respectively denotes the inner and outer raceway groove curvature radius, and di , do are respectively the inner and outer raceway contact diameter. Based on the above formulas, the axial clearance Pe can be expressed the bearing internal geometry parameters ri , ro , di , do , D. Through the analysis in Section 2, di should be modified for the influences of assembling stress and centrifugal stress. Usually, the axial clearance is decreased under the effect of above factors, so the magnitude of preload increases with the increased of the axial displacement δ p . 4. Bearing dynamic model 4.1. Ball kinematic analysis Fig. 3 illustrates the angular speed vectors relation between ball and contact raceways of an angular contact ball bearing, and X-axis is collineared with the bearing axis. It is assumed that the outer ring is fixed while inner ring rotates along with the shaft which rotates at angular speed ωi . And the angular speed of ball can be divided into the revolution speed ωm and the spin speed ωb .

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Fig. 4. Analysis of forces applied to the ball.

As shown in Fig. 3, at any point (xi , yi ) in the ball and inner raceway contact elliptical, the linear velocity on the inner raceway in the yi -axis direction is defined as follows:

Vri =

1 dm ωi (1 − γi ) 2

(22)

The linear velocity of the point (xi , yi ) on the ball in the yi -axis can be obtained by the summation of revolution linear speed and spin linear speed:

Vbi =

1 1 dm ωm (1 − γi ) + ωb D cos (αi − β ) 2 2

(23)

Usually, there is no macro sliding between the inner raceway and ball, without loss of generality, (xi , yi ) is selected as the pure rolling point, therefore:

Vri = Vbi

(24)

Thus, the relationship among the angular speed ωi , ωm , ωb can be obtained:

ωb dm (1 − γi ) = ωi − ωm D cos (αi − β )

(25)

A similar analysis may be applied to the ball and outer raceway contact, one can get the following expression:

ωb dm (1 + γo ) = ωm D cos (αo − β )

(26)

Through the above Eqs. (25) and (26), the following expressions can be determined, which is used to the calculation of ball centrifugal force and gyroscopic moment.

ωm (1 − γi ) cos (αo − β ) = ωi (1 + γo ) cos (αi − β ) + (1 − γi ) cos (αo − β )

(27)

ωb dm (1 − γi )(1 + γo ) = ωi D (1 + γo ) cos (αi − β ) + (1 − γi ) cos (αo − β )

(28)

Based on the above analysis, the ball pitch angle β is an important parameter for the angular speed ratios calculation. Jones made a simplified assumption that the ball spins on one of the contact raceways and rolls on the opposing raceway (“Raceway Control Hypothesis”) to solve the ball pitch angle β . As shown in Fig. 4, Wang [7] and Ding [8] calculated the ball pitch angle β by applying the d’Alember’s principle, it can be written as:  =0 Pf + Msi ωsi − Msoωso

(29)

where Ms i and Mso respectively denotes the spinning friction moment of the inner and outer raceway, and Pf is the work of  are the inner and outer spinning angular speeds relative to external forces except for Ms i and Mso . In addition, ωsi and ωso  are the inverse vectors of ω and ω . ball, and ωsi , ωso si so As shown in Fig. 3, ωs i and ωso can be written as:

ωsi = (ωi − ωm )sinαi + ωb sin(αi − β )

(30)

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Fig. 5. Position of ball center and raceway curvature center.

ωso = −ωm sinαo − ωb sin(αo − β )

(31)

Based on the above formulas, ball pitch angle β can be obtained [8]:

tanβ = with

⎧ ⎪ ⎪ ⎪ ⎨

C (S + 1 )sinαi + 2sinαo C (S + 1 )cosαi + 2(αo + D/dm ) + G

Qi ai Li Qo ao Lo G = D/dm ·C [cos(αi − αo ) − S] ⎪ ⎪ 1 + D/dm · cosαo ⎪ ⎩ S= 1 − cosαi

(32)

C=

(33)

where Qi , Qo respectively denotes the ball-inner raceway and ball-outer raceway contact force, ai , ao are semi-major axis of ball-inner raceway and ball-outer raceway contact ellipse, and L is the second kind of elliptic integral function.

4.2. Bearing mechanical analysis To determine the internal load distribution in a high speed angular contact ball bearing, a five-degree mechanical model is established. As shown in Fig. 5, under zero load, the inner and outer raceway groove curvature centers and ball center are collinear and the distance between the centers of inner and outer raceway groove curvature is BD where B = fi +fo -1. When a combined loads is applied for the bearing, the centers of ball and inner raceway groove curvature is changed with the increase of rotation speed. The distances between inner/outer raceway groves and ball center is increased by due to the contact deformations δ i and δ o , as a result, the inner and outer raceway groove curvature centers and ball center are no longer collinear, the contact angles of ball-inner raceway and ball-outer raceway are not equal anymore. Assuming that bearing under the combined loads F = (Fx , Fy , Fz , My , Mz ), the displacements of inner ring relative to outer ring are d = (δ x , δ y , δ z , θ y , θ z ), where δ x , δ y , δ z are translational displacements and θ y , θ z are angular displacements, As shown in Fig. 5, the distances between the inner and outer raceway groove curvature centers at any ball position can be written as:



A1k = BD sin α o + δx − θz i cos ψk + θy i sin ψk A2k = BD cos α o + δy cos ψk + δz sin ψk

(34)

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According to the Pythagorean Theorem, one can obtain:



(A1k − X1k )2 + (A2k − X2k )2 − [( fi − 0.5)D + δik ]2 = 0 2 X12k + X22k − [( fo − 0.5 )D + δok ] = 0

(35)

As shown Fig. 4, the equilibrium equations of forces in the horizontal and vertical directions can be written as:



Qik sinαik − Tik cosαik − Qok sinαok + Tok cosαok = 0 Qik cosαik + Tik sinαik − Qok cosαok − Tok sinαok + Fck = 0

(36)

where the friction forces Ti , To and the contact forces Qi , Qo can be determined by the following expressions:

⎧ ⎪ ⎨ (Tik + Tok )D = Mgk 2

(37)

T T ⎪ ⎩ ik = ok



Qik

Qok

3/2 Qik = Kik δik

(38)

3/2 Qok = Kok δok

In addition, the centrifugal force and gyroscopic moment of bearing ball can be calculated by Eqs. (27) and (28).

⎧  2 ⎪ ⎨Fck = 1 mdm ωi2 ωm 2  ωi ω k ⎪ b ⎩Mgk = Jωi2 ωm sin βk ωi k ωi k

(39)

In the above analysis, the local formulas of ball are obtained, the local variables of Eqs. (35) and (36) are xk = (X1k , X2k ,

δ ik , δ ok ), and the contact angles (α ik , α ok ) and contact loads (Qik , Qok ), can be expressed by the local variables xk . In order to find the values of d = (δ x , δ y , δ z , θ y , θ z ), the global equilibrium equations of bearing inner ring are established by Newton’s second law:

 ⎧ Fx = (Qi sinαi − Ti cosαi ) ⎪ ⎪ Z ⎪  ⎪ ⎪ Fy = (Qi cosαi + Ti sinαi )cosψ ⎪ ⎪ Z ⎪  ⎪ ⎪ ⎨Fz = (Qi cosαi + Ti sinαi )sinψ Z  

  ⎪ ⎪ M =  Q cos α − T sin α cos ψ + T r ( ) y i i i i i i i sin ψ ⎪ ⎪ Z Z ⎪ ⎪  

⎪ ⎪ ⎪   ⎪ ⎩Mz = −i (Qi cosαi − Ti sinαi )cosψ − Ti ri cos ψ Z

(40)

Z

5. Bearing stiffness matrix model Based on the above analysis, for an external load vector F = (Fx , Fy , Fz , My , Mz ) applied on bearing, the displacement vector of inner ring relative to outer ring is d = (δ x , δ y , δ z , θ y , θ z ), the stiffness matrix is described as:

⎡

∂ Fx ⎢ ∂ δx ⎢ ⎢ ∂ Fy ⎢ ⎢ ∂ δx ⎢ ⎢ ∂ Fz [K ] = ⎢ ⎢ ∂ δx ⎢ ∂M ⎢ y ⎢ ∂δ ⎢ x ⎣ ∂ Mz ∂ δx

∂ Fx ∂ δy ∂ Fy ∂ δy ∂ Fz ∂ δy ∂ My ∂ δy ∂ Mz ∂ δy

∂ Fx ∂ δz ∂ Fy ∂ δz ∂ Fz ∂ δz ∂ My ∂ δz ∂ Mz ∂ δz

∂ Fx ∂ θy ∂ Fy ∂ θy ∂ Fz ∂ θy ∂ My ∂ θy ∂ Mz ∂ θy

⎤ ∂ Fx ∂ θz ⎥ ⎥ ∂ Fy ⎥ ⎥ ∂ θz ⎥ ⎥ ∂ Fz ⎥ ∂ θz ⎥ ⎥ ∂ My ⎥ ⎥ ∂ θz ⎥ ⎥ ∂ Mz ⎦ ∂ θz

(41)

The bearing stiffness matrix is obtained by finding the derivatives from the load vector F with respect to the displacement vector d. Through the summary of bearing model in the above chapters, the analytical relationship from d to F is obtained as illustrated in Fig. 6, it can be used to guide the stiffness matrix solution. As shown in Fig. 6, the displacement d is introduced into the global equilibrium equations of bearing by a series of intermediate variables. According to the analysis of bearing model, the expressions of adjacent variables can be divided into

J. Zhang et al. / Mechanism and Machine Theory 115 (2017) 1–17

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Fig. 6. The analytical relationship from the displacement vector to the load vector.

two categories: explicit equation and implicit equation. By substitution and elimination of the partial intermediate variables, the final derivatives of forces about displacements can be written as follow:

∂ Fi  Ki j = = ∂dj Z



∂ Fi ∂ A2k ∂ Fi ∂ A1k + ∂ A1k ∂ d j ∂ A2k ∂ d j





∂ Fi ∂ X2k ∂ Fi ∂ δik ∂ Fi ∂ δok ∂ Fi ∂ X1k + + + + ∂ X1k ∂ d j ∂ X2k ∂ d j ∂ δik ∂ d j ∂ δok ∂ d j



(42)

where the ∂ Fi /∂ A1k , ∂ Fi /∂ A2 k , ∂ Fi /∂ X1k , ∂ Fi /∂ X2k , ∂ Fi /∂ δ ik and ∂ Fi /∂ δ ok terms are obtained by explicit differential of the variable equations. However, the ∂ X1k /∂ dj , ∂ X2k /∂ dj , ∂ δ ik /∂ dj , and ∂ δ ok /∂ dj terms cannot be obtained directly because the relationship between d and xk is determined by the local implicit Eqs. (35) and (36), so these terms should be derived by the implicit differentiation method. Marking the Eqs. (35) and (36) as fk = (F1k , F2k , F3k F4k ), the implicit differentiation ∂ xk /∂ d can be written as:

1 ∂ (F1k , F2k , F3k , F4k ) ∂ X1k

 =− ∂dj J ∂ d j , X2k , δik , δok

(43)

1 ∂ (F1k , F2k , F3k , F4k ) ∂ X2k

 =− ∂dj J ∂ X1k , d j , δik , δok

(44)

1 ∂ (F1k , F2k , F3k , F4k ) ∂ δik

 =− ∂dj J ∂ X1k , X2k , d j , δok

(45)

1 ∂ (F1k , F2k , F3k , F4k ) ∂ δok

 =− ∂dj J ∂ X1k , X2k , δik , d j

(46)

where J is the Jacobian determinant, it can be calculated as follow:

J=

∂ (F1k , F2k , F3k , F4k ) ∂ (X1k , X2k , δik , δok )

(47)

Based on the above analysis, the stiffness matrix calculation includes implicit differentiation of the local equations and explicit differentiation of the global equations. This method can effectively avoid the errors caused by numerical methods. 6. Model solution The calculation flowchart of angular contact ball bearings with respect to different preload mechanisms can be seen in Fig. 7. The relationship between the displacement d and external load f is determined by a series of nonlinear equilibrium equations, which can be solved by Newton-Raphson iteration algorithm. Compared to the tradition bearing model solution, the deformation of bearing inner ring caused by assembling stress and centrifugal stress is calculated, and the results are used to modify the bearing geometry parameters and preload. In addition, for the bearing with constant pressure preload,

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J. Zhang et al. / Mechanism and Machine Theory 115 (2017) 1–17

Fig. 7. The calculation flowchart.

the constant preload force is input as force boundary condition, while for the bearing with constant fix-position preload, the axial preload displacement δ p is input as displacement boundary condition, and δ p needs to keep 0.5Pe +δ p constant. The detailed algorithmic process is shown in Fig. 7. 7. Results and discussions Based on the above analysis, a calculation program of ball bearing with respect to different preload mechanisms is developed. The super precision angular contact ball bearing has been used to study: B7008C/P4, and its geometry parameters can be founded in Table 1. 7.1. The deformation analysis and parameter modification In order to improve the stability of the connection between bearing inner ring and shaft, the interference fit is used. As shown in Fig. 8(a), with the rising of rotating speed, the deformation magnitude of inner ring is much larger than that of the shaft. The actual interference value and assembling pressure between inner ring and shaft gradually decrease as the speed increases. The loosing occurs when the interference value is equal to zero and the corresponding speed is referred to the loosing rotating speed. Loosing has an influence on the stability of bearing-rotor system. Fig. 8(b) gives the loosing rotating speed corresponding to different initial interference and it can be used in the design of the initial interference fit between bearing inner ring and shaft.

J. Zhang et al. / Mechanism and Machine Theory 115 (2017) 1–17

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Table 1 B7008C bearing characteristics. Parameters Bearing inner raceway Bearing outer raceway Bearing inner raceway Bearing outer raceway Number of balls Z Ball diameter D Pitch diameter dm

Values groove curvature groove curvature contact diameter contact diameter

radius ri radius ro di do

4.0 mm 3.79 mm 46.838 mm 61.176 mm 19 7.144 mm 54.007 mm

Fig. 8. The speed-deformation curves and the loosing rotating speed.

Fig. 9. The change curves of bearing internal structure size and initial contact angle.

Based on the above analysis, the radial elastic deformation of bearing inner ring is caused by the assembling stress and centrifugal stress, and it results in the change of bearing internal geometry sizes. As shown in Fig. 9(a), in static state, with the increase of initial interference values, the inner ring raceway contact diameter di also increases gradually, while the initial contact angle decreased gradually. For example, when the initial interference value is equal to 10um, the inner ring raceway contact diameter di increases 8.07um and the initial contact angle reduces to 14.64° Meanwhile, Fig. 9(b) presents the influence of rotating speed on bearing internal geometry size under 10um initial interference value. It can be found that the inner ring raceway contact diameter di continuously increases and the initial contact angle continuously decreases as the rising of rotating speed. And the changes of bearing internal geometry size have a great influence on the dynamic characteristics of bearing under different preload mechanisms.

7.2. The influence analysis of interference under static state Fig. 9 illustrates the changes of bearing internal geometry sizes, induced by the initial interference value and rotating speed under the free condition. When bearing is subjected to a design preload (290 N, medium preload). As shown in Fig. 10, for the constant pressure preload, with the increase of the initial interference value, the static contact angle of bearing with constant pressure preload decreases rapidly and the relative axial displacement between the inner ring and the outer ring increases. For the fix-position preload, with the increase of initial interference value, the axial clearance Pe is decreased, and the magnitude of actual preload is increased due to 0.5Pe +δ p keeps constant. These changes cause the variations of bearing static stiffness. The contrast curves of static stiffness with respect to two different preload mechanisms are shown in Fig. 11. With the increase of initial interference value, the axial stiffness under constant pressure preload decreases while the axial

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Fig. 10. The curves of bearing internal structure changing with initial interference under axial preload.

Fig. 11. The curves of bearing static stiffness changing with initial interference.

Fig. 12. The contrast curves of speed-stiffness under different influencing factors, Initial preload = 290 N.

stiffness under fix-position preload increases and the radial stiffness under both two preload mechanisms increases. The increasing trend is more significant under fix-position preload. 7.3. Comparative analysis of bearing speed-varying stiffness under different factors In previous studies [16,18,19], the calculations of bearing stiffness were mostly focused on the bearing under constant preload. The influences of assembling stress and centrifugal stress on bearing stiffness were usually neglected which may generate a large error between the numerical simulation and practical engineering. As shown in Fig. 12, the speed-varying stiffness under different influencing factors is comparatively calculated, it can be found that the assembling stress and cen-

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Table 2 SKF EEB3-2Z bearing characteristics. Parameters Bearing inner raceway Bearing outer raceway Bearing inner raceway Bearing outer raceway Number of balls Z Ball diameter D Pitch diameter dm

Values groove curvature groove curvature contact diameter contact diameter

radius ri radius ro di do

2.064 mm 2.064 mm 11.78 mm 19.72 mm 7 3.969 mm 15.75 mm

trifugal stress have a great influence on bearing stiffness in both preload mechanisms, especially for the bearing under fix-position preload. So the influences of assembling stress and centrifugal stress are taken into account in the subsequent analysis. 7.4. Bearing dynamic properties and stiffness with rotating speed Theoretical analysis shows that the dynamic properties of ball bearing vary as the rotating speed increases, which is concerned with the change of bearing stiffness. Literature researches [23,24] suggested that the bearing stiffness wound continuously dropped with the increase of the rotating speed, but these studies were aimed at the bearing under constant pressure preload and no comprehensive model of the speed-varying stiffness of the bearing under fix-position preload have been proposed. Fig. 13 shows that the changing regularity of dynamic characteristics and stiffness of bearing under different preload mechanisms and amplitudes. For the bearing under constant pressure preload, with the rotating speed increasing, the relative axial displacement between the inner and outer rings is decreased. At the same time, the ball-inner raceway contact angle is increased and the ball-outer raceway contact angle is decreased. It is apparent that bearing stiffness drops as the rotating speed rising, which agrees well with previous studies [24,25]. While for the bearing under fix-positon preload, the relative axial displacement between the inner and outer rings is constrained, with the increased rotating speed, the actual preload value goes up, compared to the bearing under constant pressure preload. The contact angles of ball-inner raceway and ball-outer raceway have the same variation trend, but the variation is more moderate. Moreover, the stiffness of bearing under fix-positon preload shows no remarkably descending trend compared with that from the bearing under constant pressure preload. However, there is a slight increase in bearing axial stiffness. In summary, the fix-positon preload has a better stability of bearing stiffness, but the fix-positon preload is not recommended at super high speed due to the high temperature rise. 7.5. Bearing dynamic properties and stiffness under radial load Compared to the influence of rotating speed, the external load has a significant effect on bearing stiffness, as shown in Fig. 14. For the bearing under constant pressure preload, with the increased radial load, the relative axial displacement between the inner and outer rings is decreased and bearing axial stiffness is slightly decreased while the radial stiffness is increased. While for the bearing under fix-position preload, with the increased radial load, the actual preload, bearing axial and radial stiffness are both increased. It is noted that radial stiffness has a more remarkable increasing trend. 7.6. Comparison with other theoretical and experiment results To validate the proposed stiffness model, the stiffness of ball bearing SKF EEB3-2Z has been calculated by the proposed method. And the rotating speed for calculating is 10 0 0 rpm, the initial interference value is 2 um, the geometry parameters of bearing SKF EEB3-2Z are listed in Table 2. As shown in Fig. 15, the results calculated by the proposed method are compared against experiments result at 10 0 0 rpm conducted by Kraus [26] and the FEM results [21]. It can be found that the stiffness determined by the proposed method match the experiments well. And the stiffness of the bearing under constant pressure preload is approximately equal to that under fix-position preload due to the low speed and small interference. Therefore, the experiment verification of the stiffness of the bearing under two different preload mechanisms need to be explored in further studies. 8. Conclusion In this paper, the radial elastic deformation of inner ring under the assembling stress and centrifugal stress is calculated. On this basis, the loosing rotating speed corresponding to the different initial interference is obtained. By analyzing the process of preload applying, a comparative model for the calculation of bearing stiffness with respect to different preload mechanisms is established. The dynamic effects on the balls and the bearing internal geometry size variations have been

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Fig. 13. The Speed-varying curves of the bearing dynamic properties and stiffness.

taken into account during the calculation. Besides, the influences of rotating speed and radial load on the dynamic characteristics and stiffness of the bearing under different preload mechanisms have been calculated and discussed in details. From the results, the following conclusions can be drawn: (1) Compared to constant pressure preload, the bearing under fix-positon preload has a better stability of stiffness, so the bearing under fix-position preload is more suitable for the rough machining stage with the heavy cutting load, and the fix-positon preload is not recommended at super high speed due to the high temperature rise.

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Fig. 14. Radial load-varying stiffness at different rotating speed with different preloads.

Fig. 15. Comparison of bearing stiffness between the proposed method with Kraus’ experiment results and Guo’ FEM results.

(2) For both preload mechanisms, the assembling stress and centrifugal stress have a significant influence on the bearing dynamic properties and stiffness; (3) With the increased rotating speed, the stiffness of the bearing under constant pressure preload drops obviously, while the stiffness of the bearing under fix-positon preload shows no remarkably decrease; (4) With the increased radial load, for the bearing under constant pressure preload, the axial stiffness is decreased and the radial stiffness is increased. While for the bearing under fix-positon preload, the bearing axial and radial are both increased with the increased radial load. In addition, the increasing trend of radial stiffness is more remarkable. Compared to the previous models of ball bearing and stiffness, the bearings with respect to two different preload mechanisms are comparatively studied, and the influences of assembling stress and centrifugal stress are fully considered in this paper. A more accurate bearing model is provided in the dynamic characteristic analysis of the bearing-rotor system in the present research.

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Acknowledgements This research was supported by the National Natural Science Foundation of China (51275383) and Henan Key Laboratory of High-performance Bearings, Luo Yang 471003, China (2016ZCKF01). Appendix A. Nomenclature Symbol

σr , σ θ ρ εr , εθ E

ν

P1

1 Pe , Pd

αo

do , di ro , ri D Vri , Vbi γi , γo ωi , ωm , ωb dm αi , αo

β ωsi , ωso

Msi , Mso Qi , Qo ai , ao Z B fi , fo δi , δo F, d A1 , A2 X1 , X2 i

ψk

Ti , To Qi Qo m, J x f Fc , Mg [K]

Description

Unit

Radial and Tangential stress component Material density Radial and tangential strain Material elastic modulus Poisson’s ratio Assembling pressure Practical interference fit Bearing internal axial and radial clearance Bearing free contact angle Bearing inner and outer raceway contact diameter Bearing inner and outer raceway groove curvature radius Ball diameter Linear velocity of the inner raceway and ball in semi-minor axis of ball-raceway contact ellipse D cos αi /dm , D cos αo /dm Bearing inner ring, cage revolution and ball spin angular speed Bearing pitch diameter Ball-inner raceway and ball-outer raceway contact angle Ball pitch angle Angular speed of ball relative to inner raceway and outer raceway Ball spinning friction moment of the inner and outer raceway Ball-inner raceway and ball-outer raceway contact force Semi-major axis of ball-inner raceway and ball-outer raceway contact ellipse Ball number B = fi +fo -1 f i = ri /D, f o = ro /D Ball-inner raceway and ball-outer raceway contact deformation Force vector F = (Fx , Fy , Fz , My , Mz ), and displacement vector d = (δ x , δ y , δ z , θ y , θ z ) The distances between the inner and outer raceway groove curvature centers Auxiliary variable i = dm /2 + (ri − 0.5D )cosα o Angular position of kth ball Ball-inner raceway and ball-outer raceway friction forces Ball-inner raceway and ball-outer raceway contact forces Ball mass and mass moment of inertia Intermediate variable vector x = (X1 , X2 , δ i , δ o ) Local equation vector f = (F1 , F2 , F3 F4 ) Ball centrifugal force and gyroscopic Bearing stiffness matrix

Pa Kg/m3 Pa Pa m mm mm mm mm mm/s rad/s mm

rad/s N.mm N

mm mm mm mm rad N N

N/mm

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