Dynamic analysis of double-row self-aligning ball bearings due to applied loads, internal clearance, surface waviness and number of balls

Journal of Sound and Vibration 333 (2014) 6170–6189

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Dynamic analysis of double-row self-aligning ball bearings due to applied loads, internal clearance, surface waviness and number of balls Yaobin Zhuo a,b, Xiaojun Zhou a,n, Chenlong Yang a a b

State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, PR China College of Technology, Lishui University, Lishui 323000, PR China

a r t i c l e in f o

abstract

Article history: Received 29 October 2013 Received in revised form 21 April 2014 Accepted 27 April 2014 Handling Editor: L.G. Tham Available online 29 July 2014

In this paper, a three degrees of freedom (dof) model was established for a double-row self-aligning ball bearing (SABB) system, and was applied to study the dynamic behavior of the system during starting process and constant speed rotating process. A mathematical model was developed concerning stiffness and damping characteristics of the bearing, as well as three-dimensional applied load, rotor centrifugal force, etc. Balls and races were all considered as nonlinear springs, and the contact force between ball and race was calculated based on classic Hertzian elastic contact deformation theory and deformation compatibility theory. The changes of each ball's contact force and loaded angle of each row were taken into account. In order to solve the nonlinear dynamical equilibrium equations of the system, these equations were rewritten as differential equations and the fourth order Runge–Kutta method was used to solve the equations iteratively. In order to verify accuracy of the dynamical model and correctness of the numerical solution method, a kind of SABB-BRF30 was chosen for case studies. The effects of several important governing parameters, such as radial and axial applied loads, normal internal, inner and outer races waviness, and number of balls were investigated. These parametric studies led to a complete characterization of the shaft-bearing system vibration transmission. The research provided a theoretical reference for new type bearing design, shaft-bearing system kinetic analysis, optimal design, etc. & 2014 Published by Elsevier Ltd.

1. Introduction A SABB generally consists of outer race, inner race, cage, balls and sealing cover, etc. The outer raceway of SABB is a portion of a sphere. This type of bearing is particularly useful in applications where it is difficult to obtain exact parallelism between shaft and housing bores. Because of its self-aligning inherent property, SABB has been extensively employed in vehicles, aircraft, agricultural, and textile machinery applications [1]. Since SABB is an essential component of rotating machinery, its dynamic behavior analysis is important for predicting the shaft-bearing arrangement vibration transmission. Although SABB has been widely utilized, researchers have not presented a comprehensive dynamic model of SABB, largely due to its complex geometric structure. Royston and Basdogan [2] studied

n

Corresponding author. Tel.: þ86 571 87951061. E-mail address: [email protected] (X. Zhou).

http://dx.doi.org/10.1016/j.jsv.2014.04.054 0022-460X/& 2014 Published by Elsevier Ltd.

Y. Zhuo et al. / Journal of Sound and Vibration 333 (2014) 6170–6189

Nomenclature

ðP orw Þij

A

Porw

ðA0 Þij Aij

c cx cy cz Db e f fir for F Fx Fy Fz Fxb Fyb Fzb Fu g kx ky kz K ms mir M Nb Nw Nirw Norw ðOir Þij Oor Os Pn Pirw ðP irw Þij

design distance between the raceway groove curvature centers (mm) initial distance between the raceway groove curvature centers at the jth ball of the ith row (mm) loaded distance between the raceway groove curvature centers at the jth ball of the ith row (mm) equivalent viscous damping factor (N s/mm) linearized bearing damping factor in the x-direction (N s/mm) linearized bearing damping factor in the y-direction (N s/mm) linearized bearing damping factor in the z-direction (N s/mm) ball diameter (mm) rotor mass eccentric distance (mm) r/Db raceway groove curvature radius factor inner raceway groove curvature radius factor outer raceway groove curvature radius factor applied load or force (N) applied load component in the x-direction (N) applied load component in the y-direction (N) applied load component in the z-direction (N) restoring force of all rolling elements in the x-direction (N) restoring force of all rolling elements in the y-direction (N) restoring force of all rolling elements in the z-direction (N) rotor unbalance centrifugal force (N) gravitational constant (mm/s2) linearized bearing stiffness factor in the x-direction (N/mm) linearized bearing stiffness factor in the y-direction (N/mm) linearized bearing stiffness factor in the z-direction (N/mm) load–deflection factor (N/mm3/2) mass of the shaft (kg) mass of the inner race (kg) rotor total mass (kg) number of rolling elements per row number of wave lobes on a circumference number of the inner race wave lobes per circumference number of the outer race wave lobes per circumference inner raceway groove curvature center at the jth ball of the ith row outer raceway groove curvature center rotor geometric center in the x–y plane normal internal clearance (mm) amplitude of the inner raceway surface waviness (mm) amplitude of the inner raceway surface waviness at the azimuth location ψ ij (mm)

Q r rir ror t Δt Σρ F(ρ) α α0 αij δ δij ðδr Þij ðδz Þij σ ψ ψ0 ψ ij εs θs θ0s θ0irw θ0orw ω ωc ωir ωor ωs F Fb f(q,t) q C K M

amplitude of the outer raceway surface waviness at the azimuth location ψ ij (mm) amplitude of the outer raceway surface waviness (mm) normal force between ball and raceway (N) raceway groove curvature radius (mm) inner raceway groove curvature radius (mm) outer raceway groove curvature radius (mm) time (s) time increment (s) curvature sum (mm  1) curvature difference (mm  1) loaded contact angle (rad) unloaded contact angle (rad) loaded contact angle at the jth ball of the ith row (rad) displacement or contact deformation (mm) total elastic deformation at the jth ball position of the ith row (mm) radial distance between ðOir Þij and Oor (mm) axial distance between ðOir Þij and Oor (mm) damping matrix proportionality constant position angle of ball from the y-axis (rad) position angle of the 0th ball of the 1st row from the y-axis (rad) azimuth location of the jth ball of the ith row from the y-axis (rad) angular acceleration of the shaft (rad/s2) angular displacement of the shaft from the y-axis (rad) initial phase angle of the shaft from the y-axis (rad) initial phase angle of the inner race waveform from the y-axis (rad) initial phase angle of the outer race waveform from the y-axis (rad) rotational speed (rad/s) rotational speed of the cage (rad/s) rotational speed of the inner race (rad/s) rotational speed of the outer race (rad/s) rotational speed of the shaft (rad/s) applied load vector (N) total rolling elements restoring force vector (N) alternating resultant shaft force vector (N) three-dimensional displacement vector of the rotor (mm) linearized bearing damping matrix (N  s/mm) linearized bearing stiffness matrix (N/mm) bearing mass matrix (Kg)

Subscripts b c j ir irw n or

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ball cage ball angular position (j¼0, 1, …, Nb 1) inner race or raceway inner race surface wave normal outer race or raceway

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orw r s x y z

outer race surface wave radial direction of the bearing shaft x-axis direction y-axis direction z-axis direction

Superscript 0 i

initial condition bearing row (i¼1, 2)

SABB stiffness properties, but they neglected the effects of damping and rotation speed, and therefore their model did not provide a thorough understanding of vibration transmission through the system. Lim and Singh [3–7] developed a 5 dimensional bearing stiffness matrix for single-row type bearings, and introduced flexural and out-of-plane type motions in their model. Bercea et al. [8,9] solved the relative displacement between the races for various double-row bearing types, but they didn't sufficiently discuss the effect of internal clearance, and their research did not include dynamic analysis. Cao and Xiao [10,11] proposed a dynamic model of double-row spherical roller bearings using energy principles. Gunduz et al. [12] studied vibration transmission through double-row angular contact ball bearings, and analyzed the effect of bearing preloads on the modal characteristics of a shaft-bearing assembly. Many researchers have studied vibration of single-row ball bearings due to the effects of internal clearance, surface waviness, preloads, number of balls, etc [13–20], however, few studies have presented the effects of these parameters on SABB. Furthermore, most researchers are only keen on studying the vibrations of the shaft-bearing system during the constant speed rotating process, but ignore the accelerating period. Therefore, this paper presents a comprehensive dynamical model of shaft-SABB arrangement based on the mathematical model presented by Lim and Singh [3–7]. From the case studies, the effects of several important governing parameters, such as radial and axial applied loads, internal normal clearance, inner and outer races waviness, and number of balls on kinematic behaviors are investigated. Besides, the vibration transmission during the starting procedure is analyzed. 2. Modeling of the system 2.1. Coordinate system and forces analysis Fig. 1a shows a schematic representation of the vibration transmission of a shaft supported by a SABB. The dof of the system is only three translations (x,y,z). Since the outer raceway of SABB is a portion of a sphere, the bearing cannot support any moment in any direction. In this model, shaft and base are assumed to be rigid. The bearing supports the entire three directions external forces which transmitted from the shaft, and the applied loads on the shaft can be represented in terms of a load vector F ¼ fF x ; F y ; F z gT . Assuming that the shaft is fixed rigidly with the inner race (ωs ¼ωir), so do outer race and base (ωor ¼0). The bearing rotational axis is always collinear with the z-axis. Balls and races are considered as nonlinear springs, and therefore these forces cause three relative displacements between the shaft and base, which are represented in terms of a three-dimensional displacement vector q ¼ fδxs ; δys ; δzs gT . The outer raceway groove curvature center Oor is defined as the origin of the coordinate. The kinematic behaviors of inner race, cage, and rolling elements are illustrated in Fig. 1b. Spherical bearings generally cannot work in extremely high rotational speed applications, thus centrifugal forces and gyroscopic moments on the rolling elements due to this operating condition are ignored [1,2,21]. The cage is assumed to be massless and rigid, and hence the angular position of each rolling element relative to one another remains unchanged. Furthermore, supposing there is no

Fig. 1. Coordinate system and a shaft-SABB assembly. (a) System forces and kinematics analysis. (b) Rolling speeds and load distribution.

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gross slip on the ball-inner raceway contact surface, then the revolution speed of the cage and rolling element set can be presented as
 r or  Db ωc ¼ ωir (1) 2r or  Db The load distribution of the rolling element is illustrated in Fig. 1b. In this study, the effect of friction is ignored. The contact force between ball and raceway is calculated utilizing the classic Hertzian elastic contact deformation theory [1]. Since the Hertzian force only arises when there is contact deformation, thus the force is set to zero when ball and raceway separate. The total restoring force of all rolling elements can be represented in terms of a load vector Fb ¼ ½F xb ; F yb ; F zb T . 2.2. Deformation and geometric relations Fig. 2 shows the geometry of the jth ball of the ith row in SABB based on deformation compatibility theory [1], corresponding to the azimuth location ψ ij of the ball from the y-axis which is defined in Fig. 1b. It shows that in the initial position, the normal internal clearance Pn is taken into account. The ball contacts with inner and outer raceways simultaneously under the loaded condition. The total results elastic deformation δðψ ij Þ is derived from the displacement vector q [3,12], which can be calculated as 8 < Aij A for δij 40 i (2) δj ¼ :0 for δij r0 As shown in Fig. 2, Aij and ðA0 Þij are loaded and initial relative distances between inner raceway groove curvature center ðOir Þij and outer raceway groove curvature center Oor, respectively, besides A being its design dimension A ¼ r ir þr or  Db As shown in Fig. 2,

ðA0 Þij

can be expressed in terms of radial distance ðδr Þij

¼ ðA0 Þij

0

cos ðα

(3) ðδr Þij

Þ þ δxs sin ðψ ij Þ þδys

and axial distance

cos ðψ ij Þ

ðδz Þij ¼ ðA0 Þij sin ðα0 Þ þ ð  1Þi δzs Then the total displacement Aij is the vectorial sum of ðδr Þij and ðδz Þij qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Aij ¼ ððδr Þij Þ2 þ ððδz Þij Þ2

ðδz Þij (4) (5)

(6)

2.3. Contact point curvatures and deformation–load relationship According to the Hertzian elastic contact theory, the deformation–load relationship of two contacting bodies is determined by the material properties and curvatures at the contact point [1]. Curvature sum Σρ and curvature difference F(ρ) are two important parameters for designing bearing structure. The calculation formulas of them are shown in Table 1.

Fig. 2. Geometry and deformation in SABB.

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Table 1 Calculation formulas of SABB's contact point curvatures. Curvature parameters

Formulas

Curvature sum of the inner raceway, Σρir

1 Db

Curvature difference of the inner raceway, F(ρ)ir

f ir þ f or  1 4f ir f or  3f ir  f or þ 1
 2 1 2 Db f or

Curvature sum of the outer raceway, Σρor Curvature difference of the outer raceway, F(ρ)or


 1 1 4 þ f ir f or  1

0

The resultant Hertzian contact normal load Q ij , which is between the ψ ij ball and the inner and outer races can be calculated as [1] Q ij ¼ Kðδij Þ3=2

(7)

here K is the load–deflection factor, which is determined by the material properties, Σρ, and F(ρ). 2.4. Races waviness Race surface waviness is an important source of vibration. The global sinusoidal-shaped waviness on the inner and outer raceways is shown in Fig. 3. Waviness imperfections cause variations when the bearing is running. Imperfections with different wave numbers cause vibrations at distinct frequencies, each with a characteristic vibration mode. Therefore, the surface waviness causes additional vibrations [15]. As shown in Fig. 3, wave is represented in terms of two parameters, namely, number of waves per circumference Nw and its amplitude Pw. The magnitudes of inner and outer races' waves at the azimuth location ψ ij can be calculated as ðP irw Þij ¼ P irw sin ðN irw ðψ ij θs Þ þ θ0irw Þ

(8)

ðP orw Þij ¼ P orw sin ðN orw ψ ij þθ0orw Þ

(9)

where θ0irw and θ0orw are the initial phase angles of inner and outer races' waveforms from the y-axis. Because the inner race rotates at the same speed as the shaft, the angular displacement of the inner race θs is Z t ωs ðtÞdt θs ¼ θ0s þ

(10)

0

Since the ball centers rotate synchronously with the cage, the azimuth location ψ ij can be expressed as Z t πð2j þ i 1Þ þ ωc ðtÞdt ψ ij ¼ ψ 0 þ Nb 0

(11)

Taking account of Pn, ðP irw Þij , and ðP orw Þij ; ðA0 Þij can be calculated as ðA0 Þij ¼ A 

Pn  P irw sin ðN irw ðψ ij θs Þ þθ0irw Þ  P orw sin ðN orw ψ ij þθ0orw Þ 2

(12)

2.5. Stiffness matrix Bearing stiffness matrix is a comprehensive representation that combines the effect of each rolling element [3]. In order to obtain the mathematical definition of the bearing stiffness matrix, the resultant bearing load vector Fb is combined with the bearing displacement vector q. Fb is the vectorial sum of all the loaded rolling elements' contact forces 9 8 8 9 > cos ðαij Þ sin ðψ ij Þ > > > > > > F xb = > = < < 2 Nb  1 i i i Þ cos ðψ Þ cos ðα F (13) Fb ¼ ¼ ∑ ∑ Qj yb j j > > > > :F > ; i¼1 j¼0 > > ; : ð 1Þi sin ðαi Þ > zb j

As shown in Fig. 2, cos ðαij Þ and sin ðαij Þ can be given as cos ðαij Þ ¼ sin ðαij Þ ¼

ðδr Þij Aij ðδz Þij Aij

(14) (15)

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Fig. 3. Surface wavinesses of the inner and outer races.

The three-dimensional linearized diagonal bearing stiffness matrix K at the point q can be defined as 2 3 kx 0 0 F 6 7 K ¼ b ¼ 4 0 ky 0 5 q 0 0 kz

(16)

here the explicit expressions of each stiffness term can be calculated as i 3=2

kx ¼

K 2 Nb  1 ðδj Þ ∑ ∑ δxs i ¼ 1 j ¼ 0

ky ¼

K 2 Nb  1 ðδj Þ ∑ ∑ δys i ¼ 1 j ¼ 0

i 3=2

ðδr Þij sin ðψ ij Þ Aij ðδr Þij cos ðψ ij Þ Aij i

kz ¼

i 3=2

K 2 Nb  1 ð 1Þ ðδj Þ ∑ ∑ δzb i ¼ 1 j ¼ 0 Aij

ðδz Þij

(17)

(18)

(19)

2.6. Equations of motion As shown in Fig. 1a, the shaft is subjected to a load vector F, and the resultant movement of the rotor is defined as a displacement vector q. In this 3dof system, the governing equations of the shaft-bearing system can be written in a matrix form as Mq€ þ C q_ þ Kq ¼ fðq; tÞ

(20)

here M, C and f(q,t) are system mass, viscous damping matrix, and alternating resultant shaft vector, respectively. Because balls and cage are assumed to be massless, and therefore the mass of the rotor only consists of the inner race and shaft in this model. The bearing diagonal mass matrix M can be expressed as 2 3 ms þmir 0 0 6 7 0 0 ms þ mir M¼4 (21) 5 0 0 ms þ mir The damping of a ball bearing occurs owing to friction and lubrication. The viscous damping matrix C cannot be easily predicted due to unknown dissipation characteristics of the bearing [3,22]. Kramer [22] provided an estimate of the ball bearing damping. From Refs. [3,22], the amplitude ratio relation of damping and linearized stiffness can be roughly

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Fig. 4. Generalized alternating rotor force.

estimated as C ¼ σK

(22)

2

is the damping matrix proportionality constant. here σ ¼ ð0:25–2:5Þ10 Considering the effects of applied loads, gravity loads, and mass unbalance centrifugal load, the generalized alternating rotor forces are shown in Fig. 4. Here the unbalance load is not only caused by the mass eccentric distance e, but also the movement of the rotor axis center in the x–y plane. In order to simplify the model, rotor geometric center Os, rotor mass center, and outer raceway groove curvature center Oor are supposed to be in one line. Then the centrifugal load Fu of the rotor can be expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F u ¼ ðmir þms Þðeþ δ2xs þδ2ys Þω2s (23) In this model, the gravity load is set to the increasing y-axis direction. In sum, the force vector f(q,t) can be defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 > > F x þ ðmir þ ms Þðe þ δ2xs ðtÞ þ δ2ys ðtÞÞω2s ðtÞ sin ðθs ðtÞÞ > > > > = < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (24) fðq; tÞ ¼ F þðm þ m Þg þ ðm þ m Þðe þ δ2 ðtÞ þ δ2 ðtÞÞω2 ðtÞ cos ðθ ðtÞÞ y s s s ir ir > > xs ys s > > > > ; : Fz

3. Model solving method The dynamical model has been obtained in the previous subsection. The mathematical expression is complex and it is difficult to reach the analytic solution, thus a suitable numerical method should be chosen. Eq. (20) can be rewritten as the following differential equations by introducing a new state vector fδxs ; δys ; δzs ; δ_ xs ; δ_ ys ; δ_ zs gT 9 8 8 ∂δxs 9 > δ_ xs > > > > > > > ∂t > > > > > > > > > > > δ_ ys ∂δys > > > > > > > > > > ∂t > > > > > > > _ > > > > δ zs > > > > ∂δ > > < > > zs = p ffiffiffiffiffiffiffiffiffiffiffiffi = < ∂t 2 2 2 _ F x þ ðmir þ ms Þðe þ δxs þ δys Þωs sin ðθs Þ  kx δxs  cx δxs (25) ¼ ∂δ_ xs mir þ ms > > > > > > > ∂t > > > > > > > F þ ðm þ m Þg þ ðm þ m Þðe þ pffiffiffiffiffiffiffiffiffiffiffiffi > ∂δ_ > 2 2 2 _ > > > > δ δ þ δ Þω cos ðθ Þ  k δ  c s s s y ys y ys > ir ir ys > > s xs ys > > > > y > > > mir þ ms > > > > > ∂t > > > > > > > > ; > > : ∂δ_ zs > F z  kz δzs  cz δ_ zs > > ; : ∂t mir þ ms

_ and accelerations q€ can be obtained by solving the Eq. (25) based on the Three-dimensional displacements q, velocities q, fourth order Runge–Kutta method [23,24]. Initial condition and step value are extremely vital to the numerical solution. An incorrect condition means a sudden change of applied loads or rotor speed at the beginning, which will cause severe vibration and is not permitted in several case studies. A larger vibration at the start should correspond to a smaller time step Δt and consume a longer computer time. Besides, the iterative process will not converge when the initial vibration reaches a certain value. The correct initial condition q0 is related to the static mechanical properties of SABB, and it can be obtained by the computing method from Ref. [25]. A smaller step value can provide a higher accuracy result, but more computing time will be needed. The optimal time step Δt should be chosen according to the value of the shaft rotating speed ωs. A larger ωs means a smaller Δt, we should provide a sufficiently small increment of θs at each iterative step in order to reach an adequate accuracy.

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4. Results and discussion In order to verify accuracy of the dynamical model and correctness of the numerical solution method in this study, as well as to analyze the effect of each parameter on the kinematic behavior of the shaft-SABB system, a kind of SABB-BRF30 manufactured by Zhejiang Zhenhua Bearing Manufacturing Company is chosen for case studies. Its geometrical properties and other system parameters are shown in Table 2. In Table 2, the materials of the bearing components are all supposed to be steel.

4.1. Starting procedure Most researchers only pay heed to the vibration transmission during the steady running condition, but neglect the starting procedure. However, the kinematic behavior during this period is also very important, and it could be the origin of the vibration. Fig. 5 shows the process status of the shaft-bearing system during the starting procedure, for applied loads F ¼ f0; 200; 40gT N, angular acceleration εs ¼ 50 rad=s2 , and initial revolution speed of the rotor ω0s ¼ 0 rad=s. From Fig. 5a–g, it can be seen that the amplitudes of displacements δxs and δys increase with increasing ωs, but displacement δxs increases in the beginning and then decreases. This can be explained that the centrifugal force Fu increases in the radial direction. It also indicates that radial and axial displacements move in opposite directions. From Fig. 5b–e, it can be seen that velocities δ_ xs and δ_ ys , accelerations δ€ xs and δ€ ys and restoring forces Fxb and Fyb increase with increasing ωs. However, δ_ zs ; δ€ zs , and Fzb increase in the first place, then decrease, and finally increase. The above dynamical behaviors can be explained that when Fu is smaller as compared to the radial applied load, the mean resultant force of them is equal to the radial applied load. However, the dynamic part of the resultant force increases with increasing Fu, which raises the vibration of the rotor in the z-axis direction until Fu is equal to the radial applied load. And then, the mean resultant force increases with increasing Fu, which reduces the vibration in z-axis direction. Finally, as Fu increases, it is large enough to arouse the vibrations in every direction. Above all, it can be concluded that there are two optimum working origins as shown in Fig. 5c.

4.2. Effect of applied loads SABB can support radial and axial applied loads simultaneously. Fig. 6 shows the effect of the radial applied load on the shaft-bearing system during the steady running period, for Fx ¼0, Fz ¼40 N, and ωs ¼ 50 rad=s. From Fig. 6a–d it can be seen that δxs and δys decrease with increasing Fy, but δzs increases in the beginning and then decreases, so do δ_ xs ; δ_ ys ; δ_ zs ; δ€ xs ; δ€ ys ; δ€ zs . This phenomenon just corresponds to what is stated above. That is when Fy is equals to Fu, it would cause serious vibrations. From the power spectrums of the accelerations in Fig. 6, it can be noticed that the magnitudes of the three directions' natural frequencies increase with increasing of Fy. This states that a larger value of Fy will enhance the stiffness of the system, and raise the natural frequencies of the system at the same time. Besides, three directions' undamped natural pffiffiffiffiffiffiffiffiffi frequencies can be calculated by k=m=ð2πÞ [26], which are slightly different from the numerical computing results. This can be explained that the analytical formula does not take into consideration the effects of damping, centrifugal force, and so on. These analyzed results are in agreement with the experimental observations by Gunduz et al. [12]. Fig. 7 shows the effect of the axial applied load on the kinematic behavior, for Fx ¼0, Fy ¼ 2000 N, and ωs ¼ 50 rad=s. From Fig. 7a–d it can be seen that δxs decreases with increasing Fz, but δys and δzs increase in the beginning and then decrease, so do δ_ xs ; δ_ ys ; δ_ zs ; δ€ xs ; δ€ ys ; δ€ zs . From the power spectrums of the accelerations in Fig. 7, it can be noticed that the natural radial frequencies increase with increasing Fz, but the natural axial frequency first decreases and then increases. It also indicates that the z-axis natural frequency becomes more and more important with increasing Fz, because it could also affect the vibrations in other directions. This can be explained that the axial stiffness of the system increases in the beginning and then decreases with increasing Fz [25]. When Fz is larger than a certain value, there would be only a row of balls carrying the load, however, SABB is not suggested operating in this condition. During this transition phase, the system is working in an unstable state, and the vibration of the system reaches to the maximum value. Table 2 Geometrical parameters of BRF30 and other system parameters. Geometrical parameters

Values

Ball diameter, Db Inner raceway groove curvature radius factor, fir Outer raceway groove curvature radius factor, for Unloaded contact angle, α0 Number of rolling elements per row, Nb Mass of the rotor, M ¼ mir þ ms Load–deflection factor, K Damping matrix proportionality constant, σ

9.525 mm 0.52 2.965 12.71 11 2 kg 187,458 N/mm3/2 0.0025

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Fig. 5. Process status of the shaft-SABB system during the starting procedure. (a) Displacements δxs ; δys ; δzs , (b) velocities δ_ xs ; δ_ ys ; δ_ zs , (c) accelerations δ€ xs ; δ€ ys ; δ€ zs , (d) restoring forces F xb ; F yb . (e) restoring force Fzb, (f) rotational speed ωs and (g) displacements of the shaft center.

4.3. Effect of internal clearance Most researchers analyze the performance of SABB ignoring the effect of internal clearance, but in practical condition, internal clearance greatly affects operating characteristics. There are three related kinds of internal clearances, which are normal, radial, and axial clearances [25]. This model introduces the effect of the normal internal clearance Pn as shown in Fig. 2. Fig. 8 shows the effect of Pn on the kinematic behavior, for F ¼ f0; 2000; 200gT N and ωs ¼ 50 rad=s. From Fig. 8a–d it can _ and q€ all increase obviously with increasing Pn. From the power spectrums of the accelerations in Fig. 8, it be seen that q, q, can be observed that the natural frequencies all decrease rapidly with increasing Pn. This can be explained that the stiffness of the system decreases with increasing Pn. Altogether, it can be concluded that the internal clearance is a key factor affecting SABB's working quality, and therefore it should be strictly controlled and inspected [25].

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Fig. 6. Effect of the radial applied load on the kinematic behavior of the shaft-SABB system. (a) Velocities, accelerations, and power spectrums at Fy ¼0. (b) Velocities, accelerations, and power spectrums at Fy ¼500. (c) Velocities, accelerations, and power spectrums at Fy ¼ 2000. (d) Displacements of the shaft center at Fy ¼0, 250, 500, 1000, 2000.

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Fig. 7. Effect of the axial applied load on the kinematic behavior of the shaft-SABB system. (a) Velocities, accelerations, and power spectrums at Fy ¼ 50. (b) Velocities, accelerations, and power spectrums at Fz ¼ 700. (c) Velocities, accelerations, and power spectrums at Fz ¼ 1400. (d) Displacements of the shaft center at Fz ¼50, 350, 700, 1050, 1400.

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Fig. 8. Effect of the normal internal clearance on the kinematic behavior of the shaft-SABB system. (a) Velocities, accelerations, and power spectrums at Pn ¼5 μm. (b) Velocities, accelerations, and power spectrums at Pn ¼ 20 μm. (c) Velocities, accelerations, and power spectrums at Pn ¼ 40 μm. (d) Displacements of the shaft center at Pn ¼ 0, 5, 10, 20, 40 (μm).

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Fig. 9. Effect of the amplitude of the inner race surface waviness on the kinematic behavior of the shaft-SABB system. (a) Velocities, accelerations, and power spectrums at P irw ¼ 0:5 μm; N irw ¼ 7. (b) Velocities, accelerations, and power spectrums at P irw ¼ 2 μm; N irw ¼ 7. (c) Velocities, accelerations, and power spectrums at P irw ¼ 4 μm; N irw ¼ 7. (d) Displacements of the shaft center at P irw ¼ 0; 0:5; 1; 2; 4 ðμmÞ; N irw ¼ 7.

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4.4. Effect of inner race surface waviness Fig. 9 shows the effect of Pirw on the kinematic behavior, for F ¼ f0; 8000; 1000gT N and ωs ¼ 50 rad=s. From Fig. 9a–d it _ and q€ all increase obviously with increasing Pirw. From the power spectrums of the accelerations in can be seen that q, q, Fig. 9, it can be observed that the peak amplitudes of the vibrations appear at revolution frequency ωs ¼ 7.96 Hz, ball passage frequency ωcNb ¼34.9 Hz, inner race wave passage frequency ωsNirw ¼55.7 Hz, and other related frequencies ωcNb  ωs ¼ 26.9 Hz, ωcNb þωs ¼42.9 Hz, ωsNirw ωs ¼47.7 Hz and ωsNirw ωs ¼63.7 Hz, respectively. It also can be seen that the inner race wave passage frequency vibration increases with increasing Pirw. This indicates that the inner race surface wave is an important exciting vibration source. Fig. 10 shows the effect of Nirw on the kinematic behavior, for F ¼ f0; 8000; 1000gT N and ωs ¼ 50 rad=s. From Fig. 10a–d it _ and q€ all increase with increasing Nirw. This can be explained that more waves per circumference can be seen that q, q, means a higher stimulus frequency. As shown in the power spectrums of Fig. 10a, where Nirw ¼5, corresponding to that shown in Fig. 9, the peak amplitudes of the vibrations appear at ωs ¼7.96 Hz, ωcNb ¼34.9 Hz and ωsNirw ¼39.8 Hz, respectively. However, as shown in Fig. 10b, where Nirw ¼Nb ¼11, the dominant peak amplitude of δ€ zs appears at (ωs  ωc) Nirw ¼52.6 Hz, and this indicates that the inner race waves would excite resonance in the z-axis direction when Nirw ¼Nb. From Fig. 10c, where Nirw ¼ 2 Nb ¼22, it can be observed that the dominant peak amplitudes of δ€ ys and δ€ zs appear at 2(ωs  ωc) Nirw ¼105.3 Hz, and the peaks of δ€ xs appear at 2(ωs  ωc)Nirw  ωs ¼97.3 Hz and 2(ωs  ωc)Nirw þ ωs ¼113.3 Hz. This can be explained that the arrangement of the two rows of balls is staggered as shown in Fig. 1. Therefore, the inner race waves could excite resonance when Nirw ¼2Nb. However, if the arrangement is aligned, the resonance would appear at Nirw ¼Nb. These results are in agreement with the researches provided by Aktürk et al. [15,16,19,20]. 4.5. Effect of outer race surface waviness Fig. 11 shows the effect of Porw on the kinematic behavior, for F ¼ f0; 8000; 1000gT N and ωs ¼ 50 rad=s. From Fig. 11a–d it can be seen that q increases roughly with increasing Porw as same as the effect of Pirw, however, q_ and q€ increase more slowly. From the power spectrums of the accelerations in Fig. 9, it can be observed that peak amplitudes of the vibrations appear at ωs, ωcNb, 2ωcNb, and other related frequencies ωcNb  ωs,ωcNb þωs, 2ωcNb  ωs, and 2ωcNb þωs, respectively. Comparing to the effect of Pirw, there is no wave passage frequency component. This can be explained that the outer race is fixed (ωor ¼0) in this dynamical model. Fig. 12 shows the effect of Norw on the kinematic behavior, for F ¼ f0; 8000; 1000gT N and ωs ¼ 50 rad=s. From Fig. 12a–d it _ and q€ increase more slowly with increasing Norw. This can be explained can be seen that relative to the effect of Nirw, q, q, that |ωir  ωc| is larger than |ωor ωc| in this mode. Therefore, there would be more stimuli from the inner race than the outer race when Nirw ¼Norw. From the power spectrums in Fig. 12, peaks appear at ωs, ωcageNb, 2ωcageNb, 3ωcageNb, and other related frequencies ωcageNb  ωs, ωcageNb þ ωs, 2ωcageNb  ωs, 2ωcageNb þωs, 3ωcageNb  ωs, and 2ωcageNb þωs, respectively. As discussed above, there is also no wave passage frequency component due to ωor ¼0. As shown in Fig. 12b, where Norw ¼Nb ¼11, the outer race waves excite resonance in the z-axis direction. And as shown in Fig. 12c, where Nirw ¼2Nb ¼22, outer race waves excite resonance in all three directions. 4.6. Effect of number of balls As discussed above, the number of balls Nb would affect the ball passage frequency ωcNb, and when the numbers of inner and outer races waves are equal or multiple to Nb, the wave stimulus would arouse resonance. Fig. 13 shows the effect of Nb on the kinematic behavior, for F ¼ f0; 8000; 1000gT N and ωs ¼ 50 rad=s. When Nb ¼ 12, the ball passage frequency is ωcNb ¼38.0 Hz as shown in Fig. 13a. When Nb ¼13, the ball passage frequency is ωcNb ¼41.2 Hz as shown in Fig. 13b. When Nb ¼14, the ball passage frequency is ωcNb ¼44.4 Hz as shown in Fig. 13c. From Fig. 13f, it can be seen that q decreases with increasing Nb. It is generally accepted that the vibration of the system decreases with increasing Nb, and the natural frequencies of the system increase with increasing Nb. This can be explained that an increasing number of balls means more balls supporting the shaft, which could raise the system stiffness and reduce the vibration amplitude [19]. However, from Fig. 13d, it can be observed that the accelerations at Nb ¼15 are smaller than the accelerations at Nb ¼16. And from Fig. 13e, it can be seen that the averages of the accelerations generally decrease with increasing Nb, but when Nb is larger than a certain value, the accelerations' downtrend will become gentler. Furthermore, in this case, an odd number of Nb is superior to an even number. This may be explained that an odd number of Nb could eliminate the vibration in the asymmetrical arrangement of balls. 5. Conclusions A mathematical model of the shaft-SABB system has been presented concerning stiffness and damping characteristics of the bearing, three-dimensional applied load, rotor centrifugal force, etc. Furthermore, the changes of each ball's contact force and loaded angle of each row have been taken into account. The fourth order Runge–Kutta method has been applied to solve the nonlinear differential equation iteratively. A kind of SABB-BRF30 has been chosen for case studies. The effects of several important governing parameters, such as radial and axial applied loads, normal internal clearance, inner and outer

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Fig. 10. Effect of the number of the inner race surface wave lobes on the kinematic behavior of the shaft-SABB system. (a) Accelerations and power spectrums at P irw ¼ 1 μm N irw ¼ 5. (b) Accelerations and power spectrums at P irw ¼ 1 μm; N irw ¼ 11. (c) Accelerations and power spectrums at P irw ¼ 1 μm; N irw ¼ 22. (d) Displacements of the shaft center at P irw ¼ 1 μm; N irw ¼ 0; 5; 11; 20; 22.

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Fig. 11. Effect of the amplitude of the outer race surface waviness on the kinematic behavior of the shaft-SABB system. (a) Velocities, accelerations and power spectrums at P orw ¼ 0:5 μm; N orw ¼ 7. (b) Velocities, accelerations and power spectrums at P orw ¼ 2 μm; N orw ¼ 7. (c) Velocities, accelerations and power spectrums at P orw ¼ 4 μm; N orw ¼ 7. (d) Displacements of the shaft center at P orw ¼ 0; 0:5; 1; 2; 4ðμmÞ; N orw ¼ 7.

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Fig. 12. Effect of the number of the outer race surface wave lobes on the kinematic behavior of the shaft-SABB system. (a) Accelerations and power spectrums at P orw ¼ 1 μm; N orw ¼ 5. (b) Accelerations and power spectrums at P orw ¼ 1 μm; N orw ¼ 11. (c) Accelerations and power spectrums at P orw ¼ 1 μm; N orw ¼ 22. (d) Displacements of the shaft center at P orw ¼ 1 μm; N orw ¼ 0; 5; 11; 20; 22.

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races waviness, and number of balls have been systematically investigated. The results from the case studies are summarized as: 1. It would arouse severe vibration of the system in the z-axis direction when Fu is equal to the radial applied load. And there are two optimum working origins during the increasing period of ωs. 2. Stiffness and natural frequencies of the system increase with the increasing radial applied load, the vibration of the system decreases with the increasing radial applied load, except for when it is less than or equal to Fu. 3. Radial stiffness and natural radial frequencies of the system increase with the increasing axial applied load, and the radial vibration decreases with the increasing axial applied load. However, the natural axial frequency first decreases and then increases. On the contrary, the axial vibration first increases and then decreases. 4. The natural frequencies of the system decrease rapidly with the increasing Pn. On the contrary, the vibration of the system increases obviously. The internal clearance is a key factor affecting SABB's working quality. 5. The inner race waves would excite resonance when Nirw is equal or multiple to Nb. The vibration of the system generally increases with the increasing Pirw or Nirw. And the inner race waves would cause vibration at the inner race wave passage frequency ωsNirw. 6. The outer race waves would excite resonance when Norw is equal or multiple to Nb. The vibration of the system generally increases more slowly corresponding to the inner race wave with the increasing Porw or Norw. 7. Stiffness and natural frequencies of the system increase with the increasing Nb. The vibration of the system generally decreases with the increasing Nb. However, when Nb is larger than a certain value, a smaller odd number of Nb could be superior to a bigger even number.

Fig. 13. Effect of the number of balls on the kinematic behavior of the shaft-SABB system. (a) Accelerations and power spectrums at Nb ¼ 12. (b) Accelerations and power spectrums at Nb ¼ 13. (c) Accelerations and power spectrums at Nb ¼ 14. (d) Accelerations at Nb ¼ 15,16. (e) Average values of accelerations at Nb ¼7-20. (f) Displacements of the shaft center at Nb ¼10,11,12,13,14.

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Fig. 13. (continued)

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