Modeling and analysis of planar multibody systems containing deep groove ball bearing with clearance

Mechanism and Machine Theory 56 (2012) 69–88

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Modeling and analysis of planar multibody systems containing deep groove ball bearing with clearance Xu Li-xin a,⁎, Yang Yu-hu b, Li Yong-gang a, Li Chong-ning a, Wang Shi-yu b a b

School of Mechanical Engineering, Tianjin University of Technology and Education, Tianjin 300222, China School of Mechanical Engineering, Tianjin University, Tianjin 300072, China

a r t i c l e

i n f o

Article history: Received 20 September 2011 Received in revised form 14 May 2012 Accepted 21 May 2012 Available online 20 June 2012 Keywords: Multibody dynamics Deep groove ball bearing Clearance

a b s t r a c t A general methodology for dynamic modeling and analysis of planar multibody systems containing deep groove ball bearing with clearance is presented in this paper. The bearing joint has been modeled by introducing a nonlinear constraint force system, which takes into account the contact stiffness interaction between the rolling elements and the raceways. The evaluation of the contact forces is based on the Hertzian contact deformation theory that accounts for the geometrical and material properties of the contacting bodies. The proposed model has been applied in the dynamic simulations of a planar slider–crank mechanism with a deep groove ball bearing joint. By numerical calculation, the variations of the bearing eccentric trajectory, the contact force on each ball element, the equivalent joint constraint force and the crank moment are discussed. The results indicate that the effects of the bearing clearance and flexibility on the dynamic performance of high-speed mechanisms cannot be ignored. The present methodology can not only be used to analyze the overall dynamic behavior of multibody systems with the deep groove ball bearing, but also to obtain the dynamic load on each ball element in bearing. Furthermore, the simulations of the dynamic loads on ball elements can be used for the strength checking, fatigue life prediction and wear analysis of the deep groove ball bearing in multibody systems. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction It is well known that the rolling element bearings are essential parts of mechanical systems to constitute the revolute joints. As the load carrying members, the rolling element bearings are very critical for safe and efficient operation of the rotating mechanical systems [1,2]. However, it is undeniable that the use of bearings always brings some nonlinearity to the mechanical systems, and the important causes for nonlinearity are the radial clearance [3–5] between rolling elements and raceways, and the nonlinear restoring forces between various curved surfaces with defects in contact [6–15]. Generally, the defects in rolling element bearings can be categorized as localized defects and distributed defects. On the modeling of bearing with localized defects, Choudhury [6] developed a three degrees-of-freedom rotor-bearing system with localized defects on bearing inner and outer raceways. Arslan [7] proposed a shaft-bearing model containing an angular contact ball bearing with localized defects on raceways and ball surface. Rafsanjania [8] investigated the effect of local defects on the stability and the dynamic response of a rolling element bearing rotor system. Patil [9] studied the influence of localized defect size and its position on bearing vibration. Patel [10] modeled the deep groove ball bearing with single and multiple localized defects on inner and outer raceways. When the emphasis is on the modeling of bearing with distributed defects, Tandon [11] developed a theoretical model to predict the vibration response of rolling element bearing with waviness on inner and outer raceways. Sopanen [12,13] proposed a dynamic

⁎ Corresponding author. Tel./fax: + 86 022 88181092. E-mail address: [email protected] (L. Xu). 0094-114X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2012.05.009

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model of a deep groove ball bearing with waviness on raceways. Harsha [14] analyzed the nonlinear dynamic response of a rotorbearing system due to bearing surface waviness. Bai [15] presented a five degrees-of-freedom dynamic model to study the dynamic performance of ball bearings due to the effect of both internal clearance and waviness at high speed, where the centrifugal force and gyroscopic moment from balls were taken into account. In spite of perfect geometry of bearings without clearance and defects, vibrations are commonly generated through the interaction of the rolling elements during motion. Owing to this, the bearings have a significant effect on the dynamic behavior of mechanical systems. Over the last few decades, the dynamic modeling of multibody systems has been recognized as an important tool in the analysis, design, optimization, control, and simulation of complex mechanical systems [16–20]. In the classical dynamic analysis of multibody systems, the joints are assumed to be ideal and formulated by kinematic constraints. That is, the nonlinearity of bearings including radial clearance, defects, restoring forces, local deformations, friction and other phenomena associated with real joints are routinely ignored. Due to the increasing requirement for high-speed machines and mechanisms precision, it becomes imperative to treat the joint in a realistic way, that is, to consider the detailed bearing effect in dynamic analysis of multibody systems. The subject of the modeling and dynamic analysis of multibody systems with real joints draw the attention of a large amount of researchers and many theoretical [21–23,25–34] and experimental [35–37] works have been produced. In these researches, the most representative works which should be mentioned is made by Ravn [21] and Flores [22,23]. They proposed a continuous analysis approach which is combined with a contact force model [24] to describe the joint clearance in planar rigid multibody systems. Performing simulations with this methodology allows not only to quantify the overall dynamic behavior of mechanical systems, but also to provide an in-depth analysis of the characteristics of the contact–impact mechanics in a clearance joint. In some other works [25–29], the approaches to model a clearance joint with lubrication are also presented and discussed. More recently, Tian et al. [30] extended the work of Flores to include the effect of the flexibility of the bodies in the dynamic analysis of multibody systems. The research on the modeling of real joints is not limited to planar mechanisms with only revolute joints. Reports on the dynamic modeling and simulation of multibody systems with spatial clearance joints, including spherical and cylindrical joints, are also available in a number of publications [31–34]. In dealing with the revolute joints with clearance, the above researches have simplified the joint elements as two parts, generally including the bushing and journal. By using the force constraints, model of a revolute joint with clearance is formulated. Based on the model, characteristics of the contact–impact mechanics in a clearance joint, including the phenomenon of continuous contact, impact and rebound, and free flight, can be simulated. However, in the most realistic case of revolute joint, particularly when a deep groove ball bearing is used to constitute the revolute joint, the joint usually consists of a number of parts, including a series of rolling elements, one cage, the inner and the outer raceways. Besides that, the contact and the relative motion still exist among the bearing parts. Therefore, a more detailed model of bearing joint, considering the effects of clearance, restoring forces, local contact deformations and bearing kinematics, is necessary for an in-depth analysis of the dynamic performance of a multibody system. The main goal of this work is focused on the development of a theoretical model that can be used to study the dynamic behavior of planar multibody systems containing deep groove ball bearing with clearance. This paper is organized as follows. In Section 2, a brief discussion of the equations of motion for planar multibody systems is presented. Section 3 introduces the methodology on modeling a revolute joint with a deep groove ball bearing in the multibody system. In Section 4, calculation of stiffness and damping coefficients at contacts in a ball bearing is given. In Section 5, numerical results for a slider–crank mechanism with a typical deep groove ball bearing (SKF BB1B420205) joint between the crank and the connecting rod are obtained and discussed. Finally, the conclusions are presented in Section 6. 2. Equations of motion for planar multibody systems Generally, Cartesian coordinates are used to formulate the equations of motion for the multibody systems. By using Cartesian coordinates, the position and orientation of a body in multibody system can be defined by a set of translational and rotational coordinates. Thus, the generalized coordinates of the body i which is uniquely located in the plane can be expressed as T

qi ¼ ½x; y; θi

ð1Þ

where x, y are the translational coordinates of the body-fixed coordinate system origin, θ is the rotation angle of this system relative to the global coordinate system. For a multibody system a set of generalized coordinates, as shown in Eq. (2), uniquely defines the position and orientation of all bodies in the system. h i T T T T q ¼ ½x; y; θ1 ; ½x; y; θ2 ; …; ½x; y; θi

ð2Þ

The bodies in a multibody system are usually interconnected by joints which impose limitations on their relative motion. Consequently, the generalized coordinates are not independent. When these limitations are expressed as algebraic equations in terms of the generalized coordinates and time t, they are referred to as holonomic kinematic constraints [17] and are expressed as Φðq; t Þ ¼ 0

ð3Þ

L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

71

Due to nonlinearity, Eq. (3) is usually solved by employing the Newton–Raphson method to uniquely determine the position q of the system components. Differentiating Eq. (3) with respect to time t yields the velocity constraint equation. After a second differentiation with respect to time t the acceleration constraint equation is obtained as follows, Φq q_ ¼ −Φt

ð4Þ

  _ _ Φq q€ ¼ − Φq q_ q−2Φ qt q−Φtt

ð5Þ

q

where Φq is the Jacobian matrix of the constraint equations. Eqs. (4) and (5) are linear in terms of q_ and q€ , respectively, and can be solved by any usual method adopted for the solution of systems of linear equations. The kinematic analysis of a multibody system can be carried out by solving Eqs. (3)–(5). When the effects of externally applied loads on the multibody systems are considered, a dynamic analysis is necessary. The equations of motion for a constrained multibody system are written as [16] A

Mq€ ¼ Q þ Q

C

ð6Þ

where M is the mass matrix consisting of masses and moments of inertia for the system components, q€ is the acceleration vector of the constraints, Q A and Q C are the generalized force vector and the constraint reaction force vector, respectively. The joint reaction force can be expressed in terms of the Jacobian matrix of the constraint equations and the vector of Lagrange Multipliers as [16] C

T

Q ¼ −Φq λ

ð7Þ

where λ is a vector of Lagrange Multipliers. Substitution of Eqs. (7) to (6) yields A

1 Pd 4

Di di dm do Do

ri

ro

ð8Þ

D

T

Mq€ þ Φq λ ¼ Q

Fig. 1. The deep groove ball bearing.

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Fig. 2. Modeling of a deep groove ball bearing in a multibody system.

The Eq. (8) is written together with the second time derivatives of the constraint Eq. (5). Then, the set of equations that describe the motion of the multibody system is obtained, " M Φq

T

Φq 0

#    A q€ ¼ Q λ γ

ð9Þ

  _ _ where γ ¼ Φq q€ ¼ − Φq q_ q q−2Φ qt q−Φ tt is vector that groups all the terms of the acceleration constraint equations that depend on the velocities.

Fig. 3. The bearing load distribution in a multibody system.

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3. Modeling of planar multibody systems containing a deep groove ball bearing with clearance Taking the deep groove ball bearing for an example, the bearing consists of a number of parts, including a series of balls, one cage, the inner and the outer raceways, as shown in Fig. 1. Describing each component of the bearing in a multibody system can lead to a simulation model with a large number of degrees of freedom. Furthermore, ball bearings and other radial rolling bearings such as cylindrical roller bearings are designed with clearance. The existence of the clearance also increases the computational complexity of the system. However, the ball bearing calculation should be computationally efficient enough in order for it to be used in simulations of multibody systems. Therefore, the proposed groove ball bearing model has been simplified as follows: 1. The bearing has equispaced balls rolling on the surfaces of the inner and outer raceways. Thus, no slipping or sliding occurs between the components of the bearing, and all the balls move around the raceways with equal velocity. 2. It is assumed that the outer raceway is fixed rigidly to the housing and the inner raceway is fixed rigidly to the journal. 3. There is no bending deformation of the raceways. It is assumed that only nonlinear Hertzian contact deformations are considered at the contacts formed between balls and raceways. 4. For simplifying the complexity of the computational model, only the normal contact force in bearing contact is considered and the friction effect in the bearing contact is neglected. Fig. 2 shows two bodies i and j connected by a deep groove ball bearing. The center of mass of bodies i and j are Oi and Oj, respectively. Body-fixed coordinate systems ξoη are attached to the center of mass of each body, while the XOY coordinate frame represents the global coordinate system. Point Pi indicates the center of the housing or bearing outer raceway, and the center of the journal or bearing inner raceway is denoted by point Pj. Taking into account Fig. 2, the eccentricity vector e connecting the centers of the housing and the journal is calculated as    p p p p e ¼ r j −r i ¼ r j þ Aj sj − ri þ Ai si

ð10Þ

where ri and rj are the vectors linking the global origin and the center or masses of the bodies, siP and sjP are vectors in the local coordinate system that link the center of masses to the housing and journal centers, respectively, and Ai and Aj are matrices that transform the vectors siP and sjP from the local coordinate system to the global system. The velocities of the points Pi and Pj in the global coordinate system is found by differentiating Eq. (10) with respect to time t P r_ k P ¼ r_ k þ A_ k sk ; ðk ¼ i; jÞ

ð11Þ

Thus, the relative velocities between points Pi and Pj can then be computed as υ ¼ r_ jQ −r_ iQ

ð12Þ

y

ωo υo υm ωi di 2 do 2

υi

θj

Fig. 4. Rolling speeds and velocities in groove ball bearing.

x

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L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

Y

φr

ey

X

ex

1 = ex cos φr + e y sin φr − Pd 2 Fig. 5. Radial deflection at a rolling element position.

The magnitude of the eccentricity vector is evaluated as e¼

pffiffiffiffiffiffiffi eT e

ð13Þ

The unit vector n in the direction of eccentricity is n ¼ e=e

ð14Þ

In eccentric direction, the contact point between the housing and the bearing outer raceway is marked as Q i. The contact point between the journal and the bearing inner raceway is indicated as Q j. The locations of contact point Q i and Q j can then be expressed as Q

P

rk ¼ r k þ Ak sk þ Rk n; ðk ¼ i; jÞ

ð15Þ

where Rk,(k = i, j) are the housing and journal radius, respectively.

Fig. 6. Force vectors that act at the points of contact.

L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

75

Fig. 7. Flowchart of computational procedure for dynamic analysis of the multibody system containing deep groove ball bearing with clearance.

When the eccentricity is smaller than the bearing clearance, no stiffness interaction in bearing radial direction is caused. When the eccentricity has a value equal or greater than the clearance, the contact between balls and raceways is established. Thus, when it is greater than clearance, the bearing is subjected to a radial load, as shown in Fig. 3. Obviously, the load distribution of bearing is determined by the eccentric displacement vector e, the internal radial clearance Pd, and the angular positions of the balls ϕr. To determine the angular positions of the balls at any time t, a kinematic analysis of bearing is needed. As shown in Fig. 4, if there is no gross slip at the rolling element–raceway contact, then the velocity of the cage

Fig. 8. The elastic model for a ball bearing.

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and rolling element set is the mean of the inner and outer raceway velocities [1]. The angular velocity of the cage and rolling element can then be computed as ωm ¼

ωi di þ ωo do di þ do

ð16Þ

As mentioned in assumption, the outer raceway is rigidly mounted to the housing and the inner raceway is rigidly mounted to the journal. Thus, the angular velocities of the inner and outer raceways, indicated as ωi and ωo, are determined by the angular velocities of bodys i and j connected by the bearing. The angular position for the rth ball, at any time t can be defined by the following relation ϕr ¼

2π ðr−1Þ þ ωm t; ðr ¼ 1; 2; ⋯; Nb Þ Nb

ð17Þ

where Nb is the number of balls in bearing. According to the local contact Hertzian theory, and taken into account the bearing damping effect, the load-deformation relation for point contact between ball-raceways can be written as follows

1:5   1 F r ¼ K ex cosϕr þ ey sinϕr − P d þ C υx cosϕr þ υy sinϕr ; ðr ¼ 1; 2; ⋯; Nb Þ þ 2 þ

ð18Þ

where K, C are the total stiffness and damping coefficients, involving both the inner and the outer raceways contacts. The calculation of the bearing total stiffness and damping coefficients will be introduced in Section 4. ex and ey are components of the eccentricity vector e in X and Y direction (referring to Fig. 5), respectively. υx and υy are components of the relative eccentricity velocity υ in X and Y direction, respectively. The subscript “+” indicates that when the expression inside the bracket   ex cosϕr þ ey sinϕr − 12 P d is smaller or equal to zero, the ball is not in the load zone, and the restoring force Fr is set to zero. If the expression in the bracket is greater than zero, then the ball at angular location ϕr is loaded giving rise to a restoring force Fr. In other words, the Hertzian forces arise only when there is contact deformation, otherwise the separation between ball and raceways takes place and then the restoring force is set to zero. The equivalent constraint force in bearing is the sum of the restoring forces from each of the rolling elements. Resolving the total restoring force along the X and Y direction, we obtain 

Fx Fy

 ¼

Nb X r¼1

 Fr

 cosϕr ; ðr ¼ 1; 2; ⋯; Nb Þ sinϕr

ð19Þ

After calculation of the equivalent constraint forces Fx and Fy, the contributions to the generalized vector of forces and moments, Q A in the Eq. (9), are founded. These forces that act on the contact points of bodies i and j are transferred to the center of mass of bodies and an equivalent transport moment is applied to the rigid body. Referring to Fig. 6, the forces fi and moment Ti that act on the center of mass of body i due to the eccentricity of the joint element centers can be expressed as follows  fi ¼

Fx Fy

 ð20Þ

Fig. 9. Slider–crank mechanism with a deep groove ball bearing joint.

L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

77

Fig. 10. The initial configuration of the mechanism.

    Q x Q y T i ¼ yi −yi f i − xi −xi f i

ð21Þ

The corresponding forces fj and moment Tj applied to the body j are   F fj ¼ − x Fy

ð22Þ

    Q y Q x T j ¼ − xj −xj f j − yj −yj f j

ð23Þ

For solving the equations of motion of the system, the direct integration method [38] can be used. In each integration time step, the accelerations vector q€ together with velocities vector q_ are integrated in order to obtain the system velocities and positions at the next time step. This procedure is repeated up to final time. But, due to the fact that the direct integration method is prone to integration errors, Baumgarte Stabilization Method [39] is utilized to guarantee the stability of the solution. Fig. 7 shows the flowchart of computational procedure for dynamic analysis of the multibody system containing the deep groove ball bearing with clearance. 4. Stiffness and damping coefficients at contacts in a ball bearing Stiffness and damping coefficients of each component and each contact in a ball bearing system play significant roles in the dynamic study. Thus, precise calculations of values of stiffness and damping are necessary in achieving accuracy in results. In this study, the contacts of balls with the inner and outer raceways are represented by nonlinear contact springs, as shown in Fig. 8. 4.1. Stiffness The stiffness coefficient K depends on the contact geometry between the ball and raceways. According to the literature [1], the stiffness coefficients for the inner race-ball contact and the outer race-ball contact can be expressed as follows, respectively

Ki ¼

Ko ¼

 pffiffiffi 3=2 E 2 2 1−ν 2 1 3ð∑ρi Þ1=2

ð24Þ

δi

 pffiffiffi 3=2 E 2 2 1−ν 2 1 3ð∑ρo Þ1=2

ð25Þ

δo

where, E and ν are the elastic modulus and Poisson's ratio of the ball bearing elements, respectively. δ * is the dimensionless deflection factor and ∑ρo is the curvature sum at the contact point. δ * is given on a graph by Harris [1] as a function of the Table 1 Dimensions and mass parameters for the slider–crank mechanism. Bodies

Length(mm)

Mass(kg)

Moment of inertia (kg•mm2)

Crank Connecting rod Slider

50 120 –

0.30 0.21 0.14

100 250 –

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L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

Table 2 Geometric and material properties for the deep groove ball bearing (SKF BB1B420205). Bore diameter, Di Outer diameter, Do Pitch diameter, dm Inner raceway diameter, di Outer raceway diameter, do Ball diameter, D Radial clearance, Pd Number of balls, Nb Young's modulus, E Poisson's ratio, ν

24 mm 52 mm 37.9 mm 29.2 mm 46.6 mm 8.7 mm 0.02 mm 8 207 GPa 0.3

curvature difference, F(ρ). For computational purpose, stiffness coefficients Ki and Ko can be obtained using [14] 5

K p ¼ 2:15  10 ∑ρ

−1=2   −3=2

δ

; ðp ¼ i; oÞ

ð26Þ

For a single ball element, the total deflection is the sum of the contact deflections between the ball and the inner and outer raceways. δ ¼ δi þ δo

Fig. 11. Trajectories of the journal center with respect to the housing center: (a) 300 rpm; (b) 600 rpm; (c) 900 rpm; (d) 1200 rpm.

ð27Þ

L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

4

x 104

variation of joint force vector

3

Joint force in Y direction(N)

79

2 1 0 -1 -2 -3 -4 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 x 105

Joint force in X direction(N)

Fig. 12. Variation of the constraint force vector for ideal joint at crank rotational velocity 600 rpm.

Using this, we can get the total stiffness for a single ball element contacted with the inner and outer raceways. K¼

1 ð1=K i Þ2=3 þ ð1=K o Þ2=3

3=2

ð28Þ

4.2. Damping Generally, the estimation of damping of a ball bearing is very difficult because of the dominant extraneous damping which swamps the damping of the bearing [3]. Based on the experimental and theoretical study, Dietl [40] has pointed out that the major sources of bearing damping include the lubricant film damping in rolling contacts, the material damping due to the Hertzian deformation of rolling bodies and damping in the interface between the outer ring and bearing housing. In this study, only the lubricant film damping in rolling element contacts is considered.

1

180

ideal joint real bearing joint

Position error(mm)

position(mm)

160 140 120 100 80 60

300 rpm 600 rpm 900 rpm 1200rpm

0.75 0.5 0.25 0 -0.25 -0.5 -0.75

0

0.4

0.8

1.2

1.6

2

-1 0

0.4

0.8

1.2

1.6

Crank angle position( )

Crank angle position( )

(a)

(b)

2

Fig. 13. Position error analysis of the slider: (a) slider position for ideal joint and real bearing joint at crank rotational velocity 1200 rpm; (b) position error of the slider at different crank rotational velocities.

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L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88 4

5

8 x 10

2.4 x 10

ideal joint real bearing joint

1.8 1.2

4

Acceleration(mm/s2)

Acceleration(mm/s2)

6

2 0 -2 -4 -6

-1.2 -1.8

-3 0

6 x 10

0.5

1

1.5

2

2.5

3

3.5

-3.6

4

0.5

1

1.5

2

2.5

3

Crank angle position( )

(a)

(b)

5

1.2 x 10

3.5

4

3.5

4

6

0.9

Acceleration(mm/s2)

2 0 -2 -4 -6 -8 0

0

Crank angle position( )

4

Acceleration(mm/s2)

0 -0.6

-2.4

-8 -10

0.6

0.6 0.3 0 -0.3 -0.6 -0.9 -1.2

0.5

1

1.5

2

2.5

3

3.5

-1.5 0

4

0.5

1

1.5

2

2.5

3

Crank angle position( )

Crank angle position( )

(c)

(d)

Force (N)

Force (N)

Force (N)

Force (N)

Fig. 14. Slider acceleration for ideal joint and real bearing joint: (a) 300 rpm; (b) 600 rpm; (c) 900 rpm; (d) 1200 rpm.

4 6 x 10

4

6 x 10

ball 1

4

4

2

2

0

0 4 6 x 10

0.5

1

1.5

2

ball 3

4

1

1.5

0 2.5 0.75 x 104 6

2 ball 5

1.75

2.25

ball 4

1.25

1.75

2.25

2.75

ball 6

4

2

2

0 1.5

2

2.5 ball 7

4 2 0 1.5

1.25

2

4

1 4 6 x 10

0.75

4

2 0 0.5 4 6 x 10

0 0.25 4 6 x 10

ball 2

3

0 1.25 4 6 x 10

1.75

2.25

2.75

3.25

ball 8

4 2

2

2.5

Ball position angle( )

3

0 3.5 1.75

2.25

2.75

3.25

Ball position angle( )

Fig. 15. Restoring force on each ball element at crank rotational velocity 600 rpm.

3.75

L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

120

90 2000 Force (N) 60 1000

150

120

30

180

240

120

90 50000 Force (N) 60 25000

150

120

30

180

0

240

300

270

240

Crank angle position (deg.) ball 5

0

240

300

270

270

330 240

270

Crank angle position (deg.) ball 8

Fig. 16. Variation of the restoring force on each ball element in one crank rotational cycle.

a

3 x 10

b

4

ideal joint real bearing joint

10 x 10

6

Force(N)

Force(N)

4

8

2 1 0

4 2 0 -2

-1

-4 -2

0

0.5

1

1.5

2

2.5

3

3.5

-6 0

4

0.5

Crank angle position(π)

d

4 x 10

2

2

2.5

3

3.5

4

3.5

4

5

3

1.5

2

Force(N)

Force(N)

1.5

Crank angle position(π)

c 2.5 x 105

1 0.5 0

1 0

-1

-0.5

-2

-1 -1.5 0

1

0.5

1

1.5

2

2.5

3

Crank angle position(π)

3.5

4

-3 0

0.5

1

1.5

2

2.5

3

Crank angle position(π)

30 0

210

300

Crank angle position (deg.) ball 7

Crank angle position (deg.) ball 6

25000

180

330

210

330

90 50000 Force (N) 60

120 150

30

180

0

210

25000

150

30

180

330

210

25000

150

300 270

Crank angle position (deg.) ball 4

90 50000 Force (N) 60

120

330 240

270

Crank angle position (deg.) ball 3

90 50000 Force (N) 60

30 0

210

300

240

Crank angle position (deg.) ball 2

25000

1 80

330

210

90 50000 Force (N) 60

120 150

30 0

300 270

Crank angle position (deg.) ball 1

25000

180

330

210

90 50000 Force (N) 60

120 150

0

300

270

30

180

330 240

10000

150 0

210

90 20000 Force (N) 60

81

Fig. 17. Joint constraint force in X direction: (a) 300 rpm; (b) 600 rpm; (c) 900 rpm; (d) 1200 rpm.

300

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L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

a

1 x 10

b

4

ideal joint real bearing joint

0.5

2

0.25

1

0 -0.25

0 -1

-0.5

-2

-0.75

-3

-1 0

0.5

1

1.5

2

2.5

3

4

3

Force(N)

Force(N)

0.75

4 x 10

3.5

-4 0

4

0.5

Crank angle position(π)

c

8 x 10

1

1.5

2

2.5

3

3.5

4

Crank angle position(π)

d

4

1.5 x 10

6

5

1

2

Force(N)

Force(N)

4

0 -2

0.5 0 -0.5

-4 -1

-6 -8 0

0.5

1

1.5

2

2.5

3

Crank angle position(π)

3.5

4

-1.5 0

0.5

1

1.5

2

2.5

3

3.5

4

Crank angle position(π)

Fig. 18. Joint constraint force in Y direction: (a) 300 rpm; (b) 600 rpm; (c) 900 rpm; (d) 1200 rpm.

Expression for the damping coefficient of balls due to the oil film that builds up during rotation is written as [6,10] 4

cb ¼

3ηah 2h30

ð29Þ

Where ah is the radius of contact area, h0 is the minimum oil film thickness, η is the absolute viscosity. Expressions for ah and h0 for elastohydrodynamic lubrication are well developed [41,42] and listed as follows 1=3

ah ¼ 1:109ðWR=EÞ

0:744

h0 ¼ 1:15Rðυr ηα=RÞ

ð30Þ 

0:78 2 ER =W

ð31Þ

Where W is the load, υr is the relative velocity between ball and raceways, α is the pressure viscosity coefficient. The elemental damping constant that considers contact of the element with both inner and outer raceways is expressed as follows ci co ci þ co where ci and co represent the ball damping coefficient due to contact with the inner and outer raceways, respectively. C¼

ð32Þ

5. Numerical example: slider–crank mechanism The slider–crank mechanism is chosen as an example to demonstrate the application of the methodologies presented in this paper. Fig. 9 shows the configuration of the slider–crank mechanism, which consists of four rigid bodies (ground, crank, connecting rod and slider), two ideal revolute joints and one ideal translational joint. A deep groove ball bearing joint with clearance exists between the crank and the connecting rod. As shown in Fig. 7, the initial values including the positions and velocities of all the bodies in the system should be given accurately for solving the equations of motion of the multibody system. Fig. 10 shows the initial configuration of the mechanism

L. Xu et al. / Mechanism and Machine Theory 56 (2012) 69–88

a

b

c

d

83

Fig. 19. Variation of crank moment: (a) 300 rpm; (b) 600 rpm; (c) 900 rpm; (d) 1200 rpm.

that is defined with the crank and the connecting rod collinear and the housing and journal centers coincident. The initial positions and velocities necessary to start the dynamic analysis are obtained from kinematic simulation of the mechanism in

Fig. 20. Trajectories of the journal center with respect to the housing center at different ball number conditions.

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a 9

x 10

b

5

1.2 8 balls

15 balls

0.8 Crank moment(Nmm)

Acceleration(mm/s2)

7

20 balls

6 3 0 -3 -6

0.4 0 -0.4 -0.8

-9 -12 0

0.5

c 4

x 10

x 10

1 1.5 2 2.5 3 Crank angle position( )

3.5

-1.2 0

4

0.5

1 1.5 2 2.5 3 3.5 Crank angle position( )

4

1 1.5 2 2.5 3 3.5 Crank angle position( )

4

d

5

1.5

3

x 10

5

1 0.5

1

Force(N)

Force(N)

2

0

-0.5

-1

-1

-2 -3 0

0

0.5

1 1.5 2 2.5 3 3.5 Crank angle position( )

4

-1.5 0

0.5

Fig. 21. The dynamic response of the slider–crank mechanism containing the groove ball bearing with different number of balls: (a) slider acceleration; (b) crank moment; (c) the joint constraint force in X direction; (c) the joint constraint force in Y direction.

Fig. 22. The restoring force on ball 1 at different ball number conditions.

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85

Fig. 23. Comparison of the two approaches on modeling a bearing joint with clearance: (a) the deep groove ball bearing joint with clearance; (b) the clearance joint widely used in the literatures.

which all the joints are considered to be ideal. In the numerical example, a deep groove ball bearing (SKF BB1B420205) with eight ball elements is used. The initial angular positions of balls are defined as shown in Fig. 10. The dimension and mass properties for the slider–crank mechanism are shown in Table 1. In addition, geometric and material properties for the deep groove ball bearing used in the model are shown in Table 2. In numerical calculation, the crank rotates with a constant angular velocity of 300, 600, 900, 1200 rpm, respectively. Fig. 11 shows the trajectories of the connecting rod journal center with respect to the crank housing center at different velocity conditions. It is observed that the eccentric displacements at any crank angle position are always different. This is determined by the loading characteristic of the bearing joint in the context of dynamic analysis of mechanical system. The variation of the eccentric trajectory can also be verified by Fig. 12 which shows the variation of the constraint force vector for ideal joint at crank rotational velocity 600 rpm. It is found that the eccentric trajectories of the journal center with respect to the housing center are quite similar to the trajectory of joint constraint force vector. From Fig. 11, it can be observed that the eccentric displacement increases with the increase of the crank rotational velocity. The eccentric distance between the connecting rod journal center and the crank housing center is caused by two aspects, one is the bearing internal radial clearance, and the other one is the contact deformation of the ball bearing under the effect of contact force. Obviously, the dynamic accuracy of the mechanism will be reduced under the influence of the eccentricity. This can be

Fig. 24. Trajectory of the journal center with respect to the bushing center at crank angular velocity 1200 rpm.

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verified by Fig. 13(a) which shows the position response of the slider for ideal joint and real bearing joint at crank rotational velocity 1200 rpm. Fig. 13(b) gives the variation of the slider position errors at different crank rotational velocities. It is observed that the position errors increase with the increase of the crank rotational velocity, and they are usually unequal at different crank angle positions. For the slider–crank mechanism used in this work, the slider position error has a minimum value at crank angle position about 0.4π(72°) and 1.6π(288°). When the mechanism is located at the horizontal position, as shown in Fig. 10, the slider position error has a maximum value. Furthermore, Fig. 14 shows the slider acceleration response for ideal joint and real bearing joint at different crank rotational velocities. When the crank has a lower angular velocity, the slider acceleration curves resemble those obtained for models with ideal joints. But, with the increase of the crank rotational speed, the vibration of the curves is becoming more and more obvious. The above simulation results indicate that the effect of the bearing clearance and flexibility on the dynamic performance of high-speed mechanisms cannot be ignored. Based on the procedure, we can get the normal contact force and its variation law on each ball element in bearing. Taking the simulation at crank rotational velocity 600 rpm for an example, the variation of the contact force on each ball element with respect to the ball angular position is shown in Fig. 15. In this figure, the range of the contact force simulation is an orbital period of ball elements. It is found that the changes of the force on each ball are not the same. At some location, the balls bear a great contact force. On the contrary, in some other location, the balls are not in the load zone and the contact force is zero. Furthermore, Fig. 16 gives the variation of the contact force on each ball in one crank rotational cycle. Generally, due to the change of the ball angular position, the load variation on a ball element is always different at different crank rotational cycles. The simulations of the contact force on each ball element can be used for the strength checking and fatigue life prediction of the bearing. Furthermore, the wear prediction on inner or outer raceways based on the analysis of load distribution in bearing can be performed by the procedure. Based on the analysis of the contact force on each ball elements, the equivalent constraint force for real bearing joint is obtained as shown in Figs. 17 and 18. Fig. 19 shows the variation of crank moment at different crank rotational velocities. The crank moment is the moment necessary to maintain constant the crank angular velocity during the simulation.

a

b

c

d

Fig. 25. The comparisons of the dynamic responses of the slider–crank mechanism at crank angular velocity 1200 rpm: (a) slider acceleration; (b) crank moment; (c) the joint constraint force in X direction; (c) the joint constraint force in Y direction.

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One of the parameters that could be changed in the slider–crank mechanism is the number of balls in the deep groove ball bearing. In this research, the effect of the number of ball elements on the dynamic response of the mechanism and the load characteristics on balls are discussed. Taking the simulation at crank rotational velocity 1200 rpm for an example, Fig. 20 shows the variation of the bearing eccentric trajectory at different ball number conditions. It is observed that the amplitude of the bearing relative eccentricity decreases with the increase of number of balls. This phenomenon can be explained by the fact that the increased stiffness of the bearing, due to the increase of ball elements, suppresses the bearing eccentricity under the action of the inertial force. Obviously, the dynamic accuracy of the mechanism will be improved with the increase of the number of ball elements in bearing. Fig. 21 shows the variation of the slider acceleration, the crank moment and the equivalent constraint force in bearing joint at different ball number conditions. It is found that with the increase of the number of balls the dynamic responses of the slider–crank mechanism with bearing joint tend to be closer to the ideal response. Additionally, it can be seen from Fig. 22, with the increase of the number of balls, the load on ball element is decreasing. As mentioned in the introduction, most of the publications [21–23,38] used a continuous contact force model with hysteresis damping to describe the bearing clearance effect in multibody systems. The detailed modeling approach of the multibody system with the clearance joint and the calculation of the contact stiffness can be found in Flores' literature [38]. In these researches, the bearing elements were simplified as two parts, generally including the bushing and the journal, as shown in Fig. 23(b). In the following work, the dynamic responses of the slider–crank mechanism with a clearance joint described by the continuous contact force model with hysteresis damping have been calculated. The simulation results are used to compare and verify the model with the real bearing joint proposed in this paper. Fig. 24 shows the trajectory of the journal center with respect to the bushing center at crank angular velocity 1200 rpm. Compared with the results shown in Fig. 11, it can be observed that the variation of the eccentric trajectory is quite similar to the simulations with a real bearing joint. As the clearance is very small, both of the results of bearing eccentric trajectories are consistent with variations of inertial forces and no impulsive effects are observed in bearing joint. Furthermore, in Fig. 25 the slider acceleration, the crank moment and the joint equivalent constraint force are used to illustrate the dynamic behavior of the slider–crank mechanism when different approaches on modeling bearing joint are considered. By comparison, it is found that these dynamic responses are roughly the same. The only difference is that the elastic vibration frequency of the curves is slightly different. This is because that the contact stiffness parameters used in these two types of bearing model are different. In references [21–23,38], the contact stiffness between bushing and journal is generally evaluated for sphere to sphere contact. The calculated stiffness is often quite different from that of the real rolling element bearing. 6. Concluding remarks A general methodology for dynamic modeling and simulation of planar multibody systems containing the deep groove ball bearings with clearance was presented and discussed throughout this work. The bearing joint used is modeled by introducing a nonlinear constraint force system, which takes into account the contact stiffness interaction between the rolling elements and the raceways. Calculation of the normal contact force on each rolling element is based on the Hertzian contact theory, which relates the geometrical and material properties of the contacting bodies. A planar slider–crank mechanism, which includes a typical deep groove ball bearing (SKF BB1B420205), was used as a numerical example application, to show the load characteristics of the bearing joint and the global dynamic behavior of the system. From the results obtained, it was demonstrated that the variation of the eccentric trajectory of the journal (or bearing inner raceway) center with respect to the housing (or bearing outer raceway) center is determined by the inertial loads of the bearing joint. In general, the amplitude of the eccentricity which is composed of the clearance and flexible deformation of bearing will gradually increase with the increase of the system speed. Obviously, the dynamic performance of the high-speed mechanisms will be reduced under the influence of the eccentricity. It should be highlighted that, as the radial internal clearance of the deep groove ball bearing is very small, the variations of the slider acceleration, the crank moment and the joint equivalent constraint force are relatively smooth and no impulsive effects are observed in bearing joint. Additionally, the simulations of the normal contact force on each ball element can be used for the strength checking and fatigue life prediction of the deep groove ball bearing. The presented methodology provides a theoretic foundation for further research on wear prediction of the joint with deep groove ball bearing in the context of dynamic analysis of mechanical systems. Acknowledgments The authors would like to express the sincere thanks to the referees for their valuable suggestions. This project is supported by the National High Technology Research and Development Program of China (grant no. 2011AA04A102) and the Key Project of Chinese Ministry of Education (grant no. 210005). These supports are gracefully acknowledged. References [1] T.A. Harris, Rolling Bearing Analysis, Wiley, New York, 2001. [2] J. Brändlein, P. Eschmann, L. Hasbargen, K. Weigand, Ball and Roller Bearings: Theory, Design and Application, Wiley, 1999. [3] M. Tiwari, O. Prakash, K. Gupta, Effect of radial internal clearance of a ball bearing on the dynamics of a balanced, horizontal rotor, Journal of Sound and Vibration 238 (5) (2000) 723–756.

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