Kinematics and trajectory of both-sides cylindrical lapping process in planetary motion type

International Journal of Machine Tools & Manufacture 92 (2015) 60–71

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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool

Kinematics and trajectory of both-sides cylindrical lapping process in planetary motion type Julong Yuan n, Weifeng Yao, Ping Zhao, Binghai Lyu, Zhixiang Chen, Meipeng Zhong Key Laboratory of Special Purpose Equipment and Advanced Processing Technology (Zhejiang University of Technology), Ministry of Education & Zhejiang province, Hangzhou, Zhejiang 310014, China

art ic l e i nf o

a b s t r a c t

Article history: Received 14 March 2014 Received in revised form 27 December 2014 Accepted 5 February 2015 Available online 23 February 2015

The both-sides lapping process in planetary motion type is proposed in this paper to lap and polish the cylindrical surface of bearing roller. The rolling speed of roller is the key kinematical parameter that affects the generation of cylindrical surface of roller. Through analysis of friction forces and pure rolling motion, it was discovered that the rolling speed of roller only depends on the rotation speed of lower plate rather than upper plate. Based on the above result, the geometry and kinematics of workpiece in this method is described, and the functions are proved valid by an experiment in which the rolling speed of roller is observed in video. By theoretical tests under different speed conditions, it is indicated that the roller's rolling speed varies with respect to time along a nonstandard cosine curve with an offset, its amplitude depends on the speeds of lower plate rotation and carrier circulation, and its offset depends on the speeds of lower plate rotation and carrier rotation. Based on geometry and kinematics, the trajectories on the cylindrical surface of roller and on the flat surface of plate are both described and simulated. The standard deviation and the range of path curve length distribution density are applied to numerically evaluate and analyze the trajectory distribution. The effect of speed parameters on the trajectory distribution, and the generation of trajectory with respect to time are investigated. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Kinematics Trajectory Cylindrical surface Both-sides lapping Bearing roller

1. Introduction The cylindrical roller, as the rolling element of rolling bearing, has a large advantage of being loaded high pressure, due to its line contact with the bearing raceway. The dimensional accuracy, the geometrical accuracy and the surface quality of cylindrical roller play a key role in the service life and the mechanical performance of bearing. For processing the cylindrical surface of roller, the centerless grinding method, of which the present configuration was born in 1917 [1], has been applied widely in various fields of industry, especially automotive and bearings, providing high precision and high efficiency in production [2]. In throughfeed centerless grinding of precision rollers (ϕ15
20 mm2), the advanced technology makes possible the production of rollers with roundness accuracy of 0.1–0.3 mm [3]. At present, in order to produce high efficiency hybrid or electric cars, the centerless grinding is demanded to provide nanometer order processing accuracy. However, there is a shortage that the method is very sensitive to setup conditions. When the machines are not properly set, various n

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

http://dx.doi.org/10.1016/j.ijmachtools.2015.02.004 0890-6955/& 2015 Elsevier Ltd. All rights reserved.

problems may appear, such as out-of-roundness, chatter vibration or workpiece. Therefore, the new generation of centerless grinding machines is required to develop [2]. The both-sides cylindrical lapping, which was developed through applying the both-sides lapping method in processing the cylindrical surface of roller, is completely different from the centerless grinding [4,5]. Usually in the tradition, the both-sides lapping method was applied in the planarization and finish processing of plane parts, such as silicon wafers, gauging block and mechanical seals, etc. [6,7]. With its development, it was successfully applied by John Indge to process the cylindrical surface of roller, and the high accuracies were achieved, such as accuracy of size and straightness up to 0.125 μm, surface finish of 0.025 μm, and roundness of up to 0.125 μm [5]. John Indge's machine mainly employs two parallel plates and a disc-shaped annular carrier as the most critical elements. The rotation axis of carrier positioned eccentrically is driven to revolute around the rotation axis of plates. In this machine and process, the nanoscale surface finish can be easily obtained, and the multidirectional surface texture is typically different from the traditional centerless grinding [4]. In this paper, the both-sides cylindrical lapping process in planetary motion type is developed to lap and polish the cylindrical surface of roller, by applying the planetary both-sides machine that is different from the John Indge's one in kinematics.

J. Yuan et al. / International Journal of Machine Tools & Manufacture 92 (2015) 60–71

Sufficient researches were carried out to analyze the geometry and kinematics of workpiece in both-sides lapping process of plane parts in planetary motion type [8–16]. Compared to that, the cylindrical workpiece makes the additional rolling motion particularly in the both-sides cylindrical lapping process in planetary motion type, which greatly affects the generation of cylindrical surface of roller. Hence, the rolling motion of roller during processing is analyzed in this paper, by establishing the geometrical and kinematical functions. And then, the effect of speed parameters on workpiece's rolling motion is also investigated. The analysis of trajectory is a standard method to study the generation of workpiece's geometrical profile and to predict the processing accuracy. Much research [8–19] has been done in an effort to understand the trajectory on the flat surface in the process of plane part, and another research [20–28] for the trajectory on the spherical surface in the process of bearing ball. In this paper, referring to the above relevant analytical solutions, the trajectories on the cylindrical surface of roller and on the flat surface of plate are both investigated in the both-sides cylindrical lapping process in planetary motion type, based on the geometry and kinematics of workpiece. In addition, the standard deviation and the range of path curve length distribution density are applied to numerically evaluate and analyze the trajectory distribution.

2. Both-sides cylindrical lapping process in planetary motion type Basically, the both-sides cylindrical lapping process is aimed to lap or polish the outer diameter of cylindrical parts between two facing parallel and flat plates [5]. The principle of both-sides cylindrical lapping process in planetary motion type is shown in Fig. 1. The machine employs such major components as upper plate, lower plate, carrier, inner pin ring and outer pin ring. The upper and lower plates that are positioned concentrically rotate at the different speeds. The disc shaped carrier, which is driven by the inner pin ring and the outer pin ring simultaneously, makes the rotation around its own center and the circulation around plates' rotation axes. Before processing, at least three slots are designed and distributed radially on each carrier around carrier’s center, at least three carriers are set on the lower plate, and then one cylindrical roller is set in the slot of each carrier between the two facing plates. The pressure is loaded on the upper plate. During processing, the cylindrical rollers that are restricted by the carrier make the translational cycloid movement and the rolling motion simultaneously, and the materials on the cylindrical surface of roller

Fig. 1. Schematic figure of main components of both-sides cylindrical lapping process in planetary motion type, where ω1, ω2 , ω3 and ω 4 are respectively rotation speeds of lower plate, upper plate, inner pin ring and outer pin ring.

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are removed by the abrasive particles in the slurry that are pumped through the holes of upper plate into the contact region between the roller and the plates. Nomenclature 1 A, B Contact points of roll- V ′ Ap , V ′Bp er between lower and upper plates, respectively G

gravity

V ′ Ar , V ′Br

μ

Friction coefficient

FOr

VOr

Line speed of roller’s center Or

FSA, FSB

VAp, VBp Line speeds of lower FNA, FNB plate and upper plate, respectively, in the ground coordinate system VAr , VBr Line speeds of roller at MS contact point A and B, respectively, in the ground coordinate system

Line speeds of lower plate and upper plate, respectively, in the roller’s center coordinate system Line speed of roller at contact point A and B, respectively, in the roller’s center coordinate system Normal force acted on roller center between carrier and roller Friction forces acted on roller at contact point A and B, respectively Normal forces acted on roller at contact point A and B, respectively Friction moments acted on roller

3. Analysis of pure rolling motion In the process, the roller contacts both of the lower plate and the upper plate, and is restricted by the carrier simultaneously, thus it is under the over-constraint. Hence, the pure rolling motion only occurs at one position where the roller contacts with the upper plate or the lower plate. At this pure rolling position, the velocity vector on the cylindrical surface of roller will equals the one on the flat surface of plate. This analysis is necessary and will contribute to the analysis of geometry and kinematics of workpiece in the both-sides cylindrical lapping process. Fig. 2 shows the forces and the relative movement between each component on the plane perpendicular to the central axis of roller. Nomenclature 1 defines the relevant parameters. To find the position where the pure rolling motion occurs, the velocity vectors VAp , VBp , VAr , VBr defined in the ground coordinate system in Fig. 2 (a) are transferred to the ones V ′ Ap , V ′Bp , V ′ Ar , V ′Br defined in the roller central coordinate system in Fig. 2(b), and their relationships are expressed as ′ = VAp − VOr V Ap

(1)

′ = VBp − VOr V Bp

(2)

′ = VAr − VOr V Ar

(3)

′ = VBr − VOr . V Br

(4)

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Fig. 2. Schematic diagram of friction forces and relative velocities on contact points.

If there is a difference between V ′ Ap and V ′ Ar , the friction force FSA is generated at contact point A, similarly, another one FSB at point B. Assuming that the friction coefficients of lower plate and upper plate are equal, the relationship between FSA and FSB is expressed as

F SA = F NA ⋅μ = (F NB + G) ⋅μ > F SB.

(5)

Further, a friction moment MS which drives the roller to roll is generated by FSA and FSB , expressed as

MS = (F SA + F SB ) ⋅r > 0.

(6)

Therefore, the direction of V ′ Ar is made same to one of MS . The value of V ′ Ar on the roller is accelerated until it finally reaches one of V ′ Ap on the plate at contact point A. Differently, there is a difference between V ′Br and V ′Bp generated at contact point B. Based on the above, it is concluded that the pure rolling motion occurs at contact point A between the roller and the lower plate, further, the rolling direction and speed of roller depends on the direction and speed of rotation of lower plate rather than upper plate, assuming that the friction coefficients of lower plate and upper plate are equal.

4. Geometry and kinematics of workpiece 4.1. Coordinate system and parameters It is assumed that the workpiece is an ideal cylinder, its sliding motion in tribology is ignored, and the facing surfaces of plates are both ideally flat and parallel to the ground surface. Because the rolling motion of roller depends on the lower plate rather than upper plate as analyzed in the above, the rotation speed of upper plate is not taken into account here. Fig. 3 shows the coordinate system, the geometrical and kinematical parameters in the both-sides cylindrical lapping system in planetary motion type. Nomenclature 2 defines the relevant parameters. The relative movement between the cylindrical roller, the carrier and the lapping plates can be appropriately described in a cartesian global coordinate system fixed on the ground. Its origin point O is positioned at the rotational axis of plates. The Zaxis is positioned vertical to the ground surface. The X- and Y-axes define a plane fixed on the ground, and the geometrical centers of roller and carrier move on this plane. The roller's center Or and contact point A of roller with lower

Fig. 3. Geometrical and kinematical parameters in planetary both-sides cylindrical lapping system.

plate are defined, which coincide on the XOY plane in the view perpendicular to the ground. The N-axis is defined along the rolling’s central axis at an angle of α from the line Oc Or , and the Maxis is defined perpendicular to the N-axis. The translational cycloid movement of roller's center Or is defined by distances L co , L rc , and angles φco , φrc . The rolling motion of roller is defined by the angular speed ωrc . At initial time t = 0, the points Oc and Or are both positioned on the X-axis. 4.2. Analysis of velocity vectors The relationships of all velocity vectors defined in Fig. 3 are shown in Fig. 4. Here, in order to derive the functions with respect to velocity vectors and geometrical parameters, another 2D cartesian coordinate system X′O′Y′ is defined. Because the roller's rolling motion is determined by the rotation of carrier and lower plate rather than the circulation of carrier, the X′-axis is fixed on the velocity vector Vco , and the plate center O′ is fixed on the starting point of velocity vector Vco . The Y′-axis is perpendicular to X′-axis. The M–N axes defined in Fig. 3 are also transformed to this coordinate system as shown in Fig. 4. Nomenclature 2 M axis Direction of roller’s

ωco

circulation angular

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Because of the pure rolling motion between the roller and the lower plate at contact point A, the following equation is obtained, expressed as ′ = V Ar ′ + V ro ′ V Ao

(7)

where the velocity vectors V ′ Ao and V ′ro are the components respectively of velocity vectors VAo and Vro projected on the M-axis. So, the following equations are established, expressed as ′ V Ao = VAo cos φ3

(8)

′ V ro = Vro cos φ5.

(9)

By combining the Eqs. (7)–(9), the velocity vector V ′ Ar is obtained, expressed as ′ V Ar = VAo cos φ3 − Vro cos φ5

Fig. 4. Velocity vectors of lower plate, carrier and roller.

N axis

rolling motion, perpendicular to roller’s central axis Direction of roller's ωrc central axis

O, Oc , Or Centers of plates, carrier ωroll and roller, respectively Distance from carrier L co VAo center Oc to plate center O L rc

L ro

φco

φrc

φ1 φ2

φ3 φ4

φ5 ωa , ω b

Distance from roller center Or to carrier center Oc Distance from roller center Or to plate center O Angle from carrier center Oc to X-axis around plate center O Angle from roller center Or to line OOc around carrier center Oc

Angle between lines Oc Or and Oc O Angle between lines OOc and OOr or between vectors VAo and Vco Angle between vectors VAo and V ′ Ar Angle between vectors Vro and Vco Angle between vectors

Vco

Vrc

Vro

VAr

V ′ Ar V ′ Ao

speed of Carrier around plate center O rotation angular speed of Carrier around its own center Oc rolling angular speed of roller Velocity vector of contact point A fixed on lower plate relative to plate center O Velocity vector of carrier's center Oc relative to plate center O Velocity vector of roller's center Or relative to carrier's center Oc Velocity vector of roller's center Or relative to plate center O Velocity vector of contact point A fixed on roller's cylindrical surface relative to carrier's center Or Projection of velocity vector VAr on M axis Projection of velocity vector VAo on M axis

V ′ro Projection of velocity vector Vro on M axis α Angle from N axis to line Or Oc r Roller's radius

′ Vro and V Ar rotation angular speeds t of Lower and upper plates, respectively

Time

(10)

where VAo , Vro , φ3 and φ5 are obtained by the following derivation of formulas. In Fig. 3, the angles φco , φrc and φ1 are expressed as

φco = ωco t

(11)

φrc = ωrc t

(12)

φ1 = 180° − ωco t .

(13)

Thus, the geometrical parameters of a triangle ΔOOc Or are obtained in the following equations

L ro =

L co2 + L rc 2 + 2L co L rc cos (ωrc t)

⎛ L rc sin (ωrc t) ⎞ φ2 = arcsin ⎜ ⎟. L ro ⎝ ⎠

(14)

(15)

The coordinate of velocity vector Vco is defined as below.

Vco = (ωco L co, 0).

(16)

The modules of velocity vectors Vrc and VAo are obtained as below.

Vrc = ωrc L rc

(17)

VAo = ω Ao L co.

(18)

The coordinates of velocity vectors VAo , Vrc and Vro are obtained in the following equations

VAo = ( VAo cos φ2, VAo sin φ2 )

(19)

Vrc = ( Vrc cos φrc , Vrc sin φrc )

(20)

Vro = Vrc + Vco = ( Vco + Vrc cos φrc , Vrc sin φrc ).

(21)

Therefore, the modules of velocity vectors VAo and Vro are obtained, expressed as

VAo = ω a L co2 + L rc 2 + 2L co L rc cos (ωrc t) Vro =

(ωco L co )2 + 2ωco L co ωrc L rc cos (ωrc t) + (ωrc L rc )2 .

(22) (23)

In addition, according to the geometry in Fig. 4, the angles φ3, φ4 and φ5 are obtained in the following equations.

φ3 = α + φrc − φ2

(24)

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⎛ ⎞ Vrc sin φrc ⎟⎟ φ4 = arctan ⎜⎜ ⎝ Vco + Vrc cos φrc ⎠

(25)

φ5 = φrc + α − φ4 .

(26)

Table 1 Experimental and theoretical tests design and conditions. (a) Speeds (rpm)

Further, the angles φ3 and φ5 are expressed as

⎛ L rc sin (ωrc t) φ3 = α + ωrc t − arcsin ⎜ ⎜ 2 + L 2 + 2L L cos (ω t) L rc co rc rc ⎝ co

⎞ ⎟ ⎟ ⎠

(27)

Test no.

1

2

3

Lower plate ωa Upper plate ωb Carrier revolution ωco Carrier rotation ωrc

8.4  2.8 2.8 1.2

17.7  5.8 5.1 1.8

26.1 -9.1 8.3 2.8

Parameters

Value

Parameters

Value

Cylindrical roller radius r Cylindrical roller length l Plates diameter

10 30 380

Carrier diameter Carrier thickness

140 12 130

(b) Dimensions (mm)

⎛ ⎞ ωrc L rc sin (ωrc t) φ5 = α + ωrc t − arctan ⎜ ⎟. ω ω ω L L cos ( t ) + ⎝ co co rc rc rc ⎠

(28)

Finally, the rolling angular speed ωroll of roller related to the time can be obtained in the following equation. ′ /r ω roll = V Ar

(29)

where the module V ′ Ar of velocity vector V ′ Ar is obtained by combining the Eqs. (10), (22), (23), (27) and (28). 4.3. Experimental validation In order to validate the above obtained theoretical functions, an experiment was carried out, in which the rolling motion of roller during processing was observed in video. Fig. 5 shows the exterior views of the both-sides cylindrical lapping machine in planetary motion type as the experimental device, and the principle of experimental observation. To prepare the experiment, the white papers were attached on the two end surfaces of each cylindrical roller, and the colorful lines were marked shown in Fig. 5. A camera was fixed on the machine and pointed towards the roller between upper and lower plates. In the experiment, the slurry was not used in order to observe clearly. Thus, when the marked rollers were driven to move and roll, the video was captured by the camera to record the rolling motion of roller. The angle θ related to time is defined in Fig. 5 to describe the rolling motion of marked roller, and its actual value can be obtained by measuring the various position of colorful mark in the videos at every moment. Further, the actual value

Distance L co Distance L rc

40

of rolling angular speed ω′roll of roller can be obtained in the equation ′ = ω roll

θi + 1 − θi Δt

(30)

where i is the serial number of values of angle θ , and Δt is the time interval between θ i + 1 and θ i . The experimental tests were carried out according to the above method under the conditions as shown in Table 1, and the theoretical tests were carried out based on the geometrical and kinematical functions of workpiece under the same conditions. The experimental and theoretical results of rolling angular speed ωroll of roller were obtained and compared in Fig. 6. It is indicated that there is a relative error of 22% which is acceptable between the experimental and theoretical results. Therefore, the geometrical and kinematical functions of workpiece are proved valid.

5. Analysis of trajectory In this paper, the trajectories on the cylindrical surface of roller and the flat surface of plate are both studied. The analysis steps are shown as follows: 1. Geometry of trajectory. The trajectories on the cylindrical surface of roller and on the flat surface of plate are simulated, based on the theoretical geometry and kinematics of workpiece. 2. Mesh generation. The cylindrical surface of roller is meshed to lots of rectangle units, and the flat surface of plate is meshed to sectorial units.

Fig. 5. Exterior views of experimental device and principle of experimental observation.

Fig. 6. Theoretical and experimental results of roller's rolling speed ωroll .

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3. Evaluation of trajectory distribution. The standard deviation of path curve length distribution density is applied to evaluate the trajectory distribution respectively on the cylindrical surface of roller and on the flat surface of plate, and the range of path curve length distribution density is applied to evaluate the circular profile of roller.

Nomenclature 3 Contacts points between circle of plate and cylindrical T1, T2 surface of roller Distance from point T to center O of plate Lto

Ltc Ltr Lr θ roll x r , yr , zr φro

φa φra x p , yp

Distance from point T to center Oc of carrier Distance from point T to center Or of carrier Length of roller Angle of roller's rolling motion around its axis 3D cartesian coordinates of point T related to time relative to center Or of roller Angle from roller center Or to X-axis around plate center O Angle of plate's rotation around its axis Angle of roller's circulation around plate's axis relative to plate Coordinates of roller's center Or relative to plate's center O on the X–Y plane

Because the trajectory on the cylindrical surface of roller generated by the upper plate is approximate to one generated by the lower plate, the former one is ignored. Similarly, the trajectory on the flat surface of upper plate generated by the roller is also ignored. 5.1. Path curve on cylindrical surface of roller In the both-sides cylindrical lapping process, there is the line contact between the cylindrical roller and the plate, so the actual trajectory on the cylindrical surface is the area. However, it is difficult for this area to analyze and interpret the relative movement between the roller and the plate. Therefore, a special method is applied in this paper to simulate the trajectory on the cylindrical surface of roller. As shown in Fig. 7, the working flat surface of plate is simplified and divided to lots of concentric circles which are distributed at

Fig. 8. Geometry and kinematics of contact point when contact occurs between roller and circle.

equal interval in the redial direction. Thus, the contact line between cylindrical and flat surfaces is divided to lots of contact points. The trajectory on the cylindrical surface of roller is defined as a set of path curves of contact points, and can be simulated by calculating the coordinate of each contact point relative to the center of roller. Fig. 8 shows the kinematics and geometry of contact point in the same coordinate system defined in Fig. 3. Nomenclature 3 defines the relevant parameters. When the cylindrical surface of roller contacts the circle of plate, the central axis of roller intersects with the circle of plate at the contact point T1 or T2, and the circle's radius equals the distance Lto from the point T1 or T2 to the center O of plate. Whereas, this case cannot continue during the whole process. When the contact occurs, it must be satisfied that the rotation angle φrc of carrier is within a range that is obtained in the equation

⎡ L to L to ⎤ ∠TOc O ∈ ⎢−arcsin , arcsin ⎥, L co L co ⎦ ⎣

(31)

The angle ∠TOc O is obtained in the equation

∠TOc O = π − φrc = π − ωrc t.

(32)

Based on the cosine theorem of triangle ΔTOr O , a quadratic equation is established, expressed as 2 2 L tc2 + L co − L to − 2L tc L co cos ∠TOc O = 0.

(33)

By solving the above equation, the distance Ltc is obtained, expressed as

L tc = L co cos ∠TOc O ±

2 2 L to − L co sin2∠TOc O .

(34)

Further, the distance Ltr is obtained in the equation

L tr = L tc − L rc .

(35)

Considering that the roller has a length along its axis, the value of Ltr should be compared with one of roller's length L r . If Ltr > L r /2, the contact point T beyond the cylindrical surface of roller is invalid and its value is deleted, otherwise it is valid and its value is saved. In addition, by applying the Eq. (29), the angle θ roll of roller's rolling motion related to the time is obtained in the equation

θ roll (t) = − 90° + Fig. 7. Schematic figure of trajectory simulation on cylindrical surface.

∫ ωroll (t)dt

(36)

where the initial value of θ roll equals −90° when the time t = 0.

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x r (t) = r⋅ cos θ roll (t)

(37)

yr (t) = L tr (t)

(38)

the sectorial meshing method is applied for the flat surface of plate by separating its circle and radius, as shown in Fig. 9(b). Next, the total length l sum of trajectory as well as its length li distributed in the ith mesh region on one surface is calculated according to the coordinate of trajectory point [9,15]. Then, the density ρi of trajectory distributed in the ith mesh region is calculated, expressed as

z r (t) = r⋅ sin θ roll (t).

(39)

ρi = li /l sum × 1000, (i = 1, 2, ⋯⋯, N).

Therefore, the coordinates of each contact point relative to center Oc of roller are described in the 3D cartesian coordinate system, expressed as

Thus, the trajectory on the cylindrical surface of roller can be simulated due to a set of values of the above coordinates related to time. 5.2. Path curve on flat surface of plate

x p = L ro cos (φro − φa )

(40)

yp = L ro sin (φro − φa )

(41)

where φro is obtained in the equation

φro = φco + φ2,

(42)

in which φco and φ2 have been obtained in the Eqs. (11) and (15), and φa is obtained in the equation

φa = ωa t,

At last, the standard deviation (SD) of all the values of density ρi is applied to evaluate the distribution of trajectory on the cylindrical surface, expressed as

SD=

The flatness of plate which is a critical factor of forming good cylindrical surface is largely affected by the trajectory on the flat surface of plate, therefore its distribution is analyzed according to its geometrical profiles. Because the process trajectory generated by the roller on the upper plate is approximate to one on the lower plate, only the latter one is analyzed in this paper, according to the coordinates of roller's center Tr relative to plate's center O on the X–Y plane of Fig. 3. The coordinates are defined as

(43)

(44)

∑iN (ρi − ρ¯ )2

N−1

(i = 1, 2, ⋯⋯, N)

(45)

where ρ¯ is the average of all the values of density ρi and N is the total number of all the meshed regions on the cylindrical surface. SDr is defined as the standard deviation for the trajectory on the cylindrical surface of roller, and SDp is defined as the standard deviation for the trajectory on the flat surface of plate. In addition, in order to evaluate the circular profile of cylindrical surface of roller, the range between all the density ρi on the circular section of roller is applied, expressed as

R circle = max (ρi ) − min (ρi ) .

(46)

The smaller value of SD means that the better distribution trajectory on the surface is obtained, which is more helpful improve the geometrical accuracy of surface. The smaller value Rcircle means that it is more helpful to improve the roundness roller's circular section.

of to of of

6. Results and discussion 6.1. Rolling angular speed ωroll of roller

as well as L ro is obtained by the Eq. (14). 5.3. Evaluation of trajectory distribution An evaluation method of trajectory is applied in this paper, because it is difficult to accurately judge whether the trajectory is well-distributed or not on the cylindrical surface or the flat surface of plate only according to the geometrical profile of trajectory. Firstly, the surface on which the trajectory is distributed is meshed to lots of unit regions with the specific shape. The rectangle meshing method is applied for the cylindrical surface of roller by separating its circle and axis, as shown in Fig. 9(a). And,

Due to the Preston's formula, the material removal stock is mostly affected by the relative speed in the lapping process [29]. Therefore, in the both-sides cylindrical lapping process, assuming that the pressure is kept constant, the material removal stock of every point on the cylindrical surface of roller is also affected linearly by the relative speed between roller and plate. It means that the smaller difference between each other the relative speed at every point on the circular section of roller has, the lower the roundness deviation of circular section of roller is. Besides, when the speed of plate is kept constant, the relative speed is linearly related to the rolling angular speed ωroll of roller. Hence,

Fig. 9. Rectangle mesh type for cylindrical surface of roller (a) and sectorial mesh type for flat surface of plate (b). (a) Rectangle and (b) sectorial.

J. Yuan et al. / International Journal of Machine Tools & Manufacture 92 (2015) 60–71

Table 2 Test run design. Test no.

1 2 3 4

Parameters Kca

Kra

0.25 0.5 2 4

{0.25, 0.5, 2, 4}

decreasing the amplitude of ωroll is helpful to lower the roundness deviation. In this paper, ωroll is firstly analyzed, which is considered as the most important kinematic parameter. In the following analysis, the effect of upper plate is ignored according to the analysis result of Section 3. Based on the above-mentioned fundamental model, a series of simulation tests are carried out in order to investigate the effects of lower plate's rotation speed ωa , carrier's circulation speed ωco and rotation speed ωrc on the roller's rolling speed ωroll . Table 2 shows the test run design. The two parameters are identified, one is the ratio Kca of carrier circulation speed ωco to lower plate rotation speed ωa , and the other one is the ratio Kra of carrier rotation speed ωrc to lower plate rotation speed ωa . They are respectively expressed as

Kca = ωco/ωa

(47)

Kra = ωrc /ωa.

(48)

The values of Kra and Kca are both designed at four levels chosen as {0.25, 0.5, 2, 4}. In the tests, each level of Kra is chosen to investigate at each level of Kca . Besides, the values of dimensional

67

parameters in Table 1(b) are applied in this simulation tests, and the speed of lower plate is chosen as 20 rpm. Fig. 10 shows the simulation results of ωroll in the above tests. It is clearly discovered that the value of roller’s rolling speed ωroll in relation of the time varies along a cosine curve with unique cycle period, amplitude and offset in the different test. However, the knowledge is not enough only due to the diagrams of simulation results. Therefore, the cycle period, amplitude and offset of result curve are analyzed theoretically in addition, by further deriving the formulas of fundamental model, to obtain the scientific knowledge correctly and essentially. The formula to obtain the roller's rolling speed ωroll is a combination of sine and cosine functions with respect to carrier’s circulation speed ωrc and time t . So, the cycle period of result curve is expressed as

Tcycle = 2π /ωrc .

(49)

It is clearly seen that the high peak appears only when

t = 2nπ /ωrc (n = 0, 1, 2, ⋯, k).

(50)

So, by applying the above in the Eq. (29), the high peak is obtained, expressed as

ω r − high = [(ωa − ωco ) L co + (ωa − ωrc ) L rc ]/r .

(51)

Similarly, the low peak appears only when

t = (2n + 1) π /ωrc (n = 0, 1, 2, ⋯, k),

(52)

expressed as

ω r − low = [ − (ωa − ωco ) L co + (ωa − ωrc ) L rc ]/r .

(53)

The amplitude of result curve is obtained by the difference between high peak and low peak, expressed as

Fig. 10. Theoretical results of roller's rolling angular speed ωroll under conditions of different speed parameters. (a) Kca ¼0.25, (b) Kca ¼ 0.5, (a) Kca ¼2 and (b) Kca ¼4.

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J. Yuan et al. / International Journal of Machine Tools & Manufacture 92 (2015) 60–71

ω r − amplitude = (ω r − high − ω r − low )/2 = (1 − Kca ) ωa L co/r .

(54)

The offset of cosine curve is obtained by the average of high peak and low peak, expressed as

ω r − offset = (ω r − high + ω r − low )/2 = (1 − Kra ) ωa L rc /r .

(55)

As above-mentioned, in order to achieve the lower roundness deviation, ωr − amplitude should be obtained in a smaller value. When Kca tends to 1, ωr − offset approximately tends to the ideal value of 0. In addition, it is seen from Fig. 10(a)–(d) that there is a value of zero for the curves. It means that the roller’s rolling direction changes during the lapping process, which is harmful to process the cylindrical surface. To avoid this case occurring, the offset of curve should be obtained larger than the amplitude of curve. The mathematic relation is expressed as

ω r − offset > ω r − amplitude

(56)

1 − Kra L rc > 1 − Kca L co

(57)

which includes two cases. When ωr − offset ≥ 0, it is obtained

Kra < 1 −

L co 1 − Kca , L rc

(58)

and when ωr − offset < 0, it is obtained

Kra > 1 +

L co 1 − Kca . L rc

(59)

From the above, it is concluded under conditions of definite ωa , L co , L rc and r that Tcycle depends on ωrc , and ωr − amplitude depends on Kca , as well as ωr − offset depends on Kra . In order to form good cylindrical surface and promote the processing efficiency, ωr − amplitude should be obtained in the low value, and ωr − offset should be obtained in the high value. Therefore, the value of Kca is advised to be chosen as approximate to 1 as possible, and meanwhile the relationship of Kra and Kca satisfies the formula (59). Fig. 11. Results of simulation tests with increase of ratio Kra from 1.2 to 10 when Kca is 0.9. (a) Standard deviation SDr , (b) standard deviation SDp and (c) range Rcircle .

6.2. Effects of speeds combination on trajectories In order to investigate the effect of rotating speeds on the trajectory, the simulation tests are carried out under the conditions as below. According to the analysis result as represented in Section 6.1, the ratio Kca is chosen as 0.9. Because the rotating speeds cannot be set extreme large in the practice, the ratio Kra is chosen within a range from 1.2 to 10 that satisfies the formula (59). The rotation speed of lower plate is chosen as ωa ¼10 rpm. The flat surface of plate is defined as an annular surface which has an inner diameter of 75 mm and an outer diameter of 185 mm, and it is divided to 51 circles at the interval of 2 mm in radius. The values of other dimension parameters are chosen as shown in Table 1. The simulation time is chosen as 60 s. The results of simulation tests are obtained as shown in Fig. 11. It is seen from Fig. 11(a) that the result of SDr is a wave curve, and its fitting curve is obtained due to the equation expressed as

y = 6 × 10−5x 4 − 0.0028x 3 + 0.0462x2 − 0.3091x + 0.8164.

(60)

When the ratio Kra increases from 1.2 to 2.5, the SDr value decreases sharply; when the ratio Kra increases from 2.5 to 10, the SDr value varies fluctuantly around 0.1. The lowest value of SDr is obtained as 0.0630 when Kra equals 7.0. It is seen from Fig. 11(b) that the result of SDp is a wave curve, and its fitting curve is obtained due to the equation expressed as

y = − 5 × 10−6x5 + 0.0003x 4 − 0.0092x 3 + 0.122x2 − 0.784x + 2.17.

(61)

When the ratio Kra increases from 1.2 to 3.0, the SDp value decreases sharply; when the ratio Kra increases from 3.0 to 10, the SDp value decreases slowly and fluctuantly. The lowest value of SDp is obtained as 0.0006 when Kra equals 8.0. It is seen from Fig. 11(c) that the result of Rcircle is a wave curve, and its fitting curve is obtained due to the equation expressed as

y = 0.0329x + 0.5574.

(62)

Particularly, when the ratio Kra increases from 7 to 9, there is a great sharp peak on the result curve of Rcircle . The lowest value of Rcircle is obtained as 0.3386 when Kra equals 1.5. As represented in Section 5.3, the low value of SDr is beneficial to obtain the cylindrical surface of roller with high accuracy, and the low value of SDp is beneficial to ensure the flat working surface of plate during processing, as well as the low value of Rcircle is beneficial to obtain the circular profile of roller with low roundness. Therefore, in order to improve the processing accuracy, SDr , SDp and Rcircle should be all ensured in the low value. Considering the balance of SDr , SDp and Rcircle , the ratio Kra is advised to be chosen as 3.0 for the best under the unique conditions given in this paper. The values of SDr , SDp and Rcircle are respectively

J. Yuan et al. / International Journal of Machine Tools & Manufacture 92 (2015) 60–71

Fig. 12. Rolling angular speed ωroll during the period of 4 s when Kra ¼ 3.0 and Kca ¼0.9.

obtained as 0.1002, 0.1444 and 0.7012‰. 6.3. Path pattern of trajectory Based on the above analysis result, the path pattern of processing trajectory is further investigated when Kra ¼3.0 and Kca ¼0.9. The other conditions are also the same as the above. In this case, the cycle period Tcycle is 2 s. Fig. 12 shows the result of rolling angular speed ωroll of roller during the period of 4s. The amplitude of ωroll equals 13 rpm, and the offset of ωroll equals 80 rpm. Fig. 13 shows the trajectory on the cylindrical surface of roller at the time of 4s in 3D and 2D view. It is seen that it is composed of lots of interlaced curves.

Z-axis (mm)

10 5 0 -5 -10 15

10 7.5 Y-a

5 0

0 xis (mm -7.5 -15 )

-5 -10

) (mm xis a X

Circular angle (degree)

360

270

180

90

0 -15

-7.5

0

7.5

15

Cylindrical axis (mm) Fig. 13. Trajectory on cylindrical surface of roller at time of 4 s when Kra ¼ 3.0 and Kca ¼0.9 in 3D view (a) and in 2D view (b).

69

Fig. 14. Trajectory on flat surface of plate at time of 60 s when Kra ¼3.0 and Kca ¼ 0.9.

Fig. 14 shows the trajectory on the flat surface of plate at the time of 60 s. It is seen that it is the interlaced hypocycloid curve. In order to investigate the formation of trajectory on the cylindrical surface, the five ones of concentric circles of plate defined in Section 6.2 are chosen, and their path curves on the cylindrical surface of roller are observed and analyzed. The radiuses of five circles are respectively chosen as 80 mm, 105 mm, 130 mm, 155 mm, and 180 mm. Fig. 15 shows the path curves that are generated by the chosen five circles of plate at time evolution 1s (a), 2s (b) and 4s (c) on the cylindrical surface in 2D view. Fig. 16 shows the position of each path curve point on the cylindrical surface of roller with respect to the roller rolling turn which is related to the time and also describes the position of path curve point on the circular section of roller. It is seen from Figs. 15 and 16 that each curve is nonlinear which intersects with another sometimes, and it is completely different for each curve in path pattern, path length and distribution region. When the flat surface of plate is simplified and divided to 51 circles, the trajectory is generated as shown in Fig. 13, composed of lots of path curves similar to the ones in Fig. 15.

7. Conclusions In the both-sides cylindrical lapping process of planetary motion, the rolling angular speed ωroll of roller is considered as the most important kinematic parameter to analyze the generation of cylindrical surface of roller. Through the analysis of friction force and movement pattern, the pure rolling motion occurs in the contact region between the roller and the lower plate, and then the rolling angular speed ωroll of roller only depends on the rotation speed of lower plate rather than the upper plate. Based on the above result, the roller’s geometry and kinematics in the process is described. The rolling angular speed ωroll of roller is obtained by the geometrical and kinematical functions. An experiment is carried out to observe the rolling motion of marked roller in the process by the video. For the rolling angular speed ωroll of marked roller, the experimental results and the theoretical ones are obtained in agreement with the error of 22%. It is proved that the functions to calculate the rolling angular speed ωroll are correct but need to be further improved. The effects of various speeds of roller's circulation and rotation on the rolling angular speed ωroll are investigated in the theoretical tests where the two ratios Kra and Kca are defined. The low value of ωr − amplitude is beneficial for improving the processing accuracy, and

70

J. Yuan et al. / International Journal of Machine Tools & Manufacture 92 (2015) 60–71

15

360

180

Cylindrical axis (mm)

Circle radius 105 270

80 155

180

130

90

9

-3

-7.5

0

7.5

15

0

90

-7.5

80

0

7.5

3

4

5

Acknowledgments The authors would like to thank the financial support of the projects from the National Natural Science Foundation of China (No. 51175468, 51375455 and 51228501) and the Zhejiang Provincial Natural Science Foundation of China (No. Z11110794 and LY14E050021).

360

Circle radius 155

180

2

15

Cylindrical axis (mm)

270

1

the comparison of SDr , SDp and Rcircle . The trajectories on the cylindrical surface of roller (3D and 2D) and on the flat surface of plate are simulated and shown under the conditions of Kca ¼0.9 and Kra ¼3.0, and the path pattern is investigated. The above studies are still not enough to understand the bothsides cylindrical lapping process of planetary motion. It needs to be further studied in geometry, kinematics, dynamics and tribology, considering the sliding motion, contact deformation and stress, material removal and others.

130

0 -15

130

180

270

155

155

Fig. 16. Position of each path curve point on cylindrical surface of roller with respect to roller rolling turn, respectively in circle radiuses of 80, 105, 130, 155 and 180 mm.

360

180

105

Roller rolling turn (round)

Cylindrical axis (mm)

Circle radius 105

180

-9 -15

0 -15

80 3

105

80 References

90

0 -15

130 -7.5

180 0

7.5

15

Cylindrical axis (mm) Fig. 15. 2D view of path curves on cylindrical surface generated by 5 circles whose radiuses are 80, 105, 130, 155 and 180 mm respectively, at time evolution 1s (a), 2s (b) and 4s (c).

the high value of ωr − offset for the processing efficiency. In order to achieve that, Kca is advised to be chosen as approximate to 1 as possible, meanwhile Kra and Kca satisfy the relationship expressed as

Kra > 1 +

L co 1 − Kca . L rc

The trajectories on the cylindrical surface of roller and on the flat surface of plate are respectively defined and described in geometry. Specially, the effect of rotation speed of upper plate on the trajectory is ignored. The trajectory distribution is evaluated by applying the parameters SDr , SDp and Rcircle . When Kca is chosen as 0.9, the effect of various Kra on the trajectory distribution is investigated. As a result, Kra is chosen as 3.0 for the best according to

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