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Precision Engineering 30 (2006) 373–380

The development of pivot bearing with multiple degrees of freedom1st report: Prototype of pivot bearing with two degrees of freedom

Shoji Noguchi a,1, Hirotoshi Aramaki b,2, Tohru Kanada c,∗a Department of Mechanical Engineering, Tokyo University of Science, 2641 Yamasaki, Noda-city, Chiba 278-8510, Japan

b NSK Ltd., 1-5-50 Kugenuma Shinmei, Fujisawa-city, Kanagawa 251-0021, Japanc Department of Mechanical Engineering, Kanto Gakuin University, 1-50-1 Mutsuura-higashi, Kanazawa-ku, Yokohama-city, Kanagawa 236-8501, Japan

Received 19 April 2005; received in revised form 19 October 2005; accepted 15 November 2005Available online 23 February 2006

bstract

This paper deals with the prototype of a new pivot bearing having two degrees of freedom. The idea of the pivot bearing is based on a continuouselocity joint (CVJ). The experimental axial stiffness and contact pressure are compared with those determined by theoretical analysis. Then, it isonfirmed that the pivot bearing swings smoothly with a range of ±25◦. Furthermore, the stiffness of the bearing increases as the swinging angleecomes larger. Therefore, this newly developed pivot bearing may be applied to a parallel mechanism, a joint of robot and so on.

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2006 Elsevier Inc. All rights reserved.

eywords: Pivot bearing; CVJ; Parallel mechanism; Stiffness; Two degrees of

. Introduction

The possibility of parallel mechanism as shown in Fig. 1 [1]or a new driving mechanism is increasing in the area of machineools. The parallel mechanism controls the attitude or movementf a tool and a stage by expanding or contracting several links.igh speed, high stiffness and high-precision positioning cane achieved by the parallel mechanism alongside of the con-entional mechanism, which is constructed by a combination ofeveral linear motions [2,3]. In order to realize such a perfor-ance, the most important thing is to develop the supporting

lement (bearing).High stiffness, high accuracy, wide swinging angle and low

orque are expected for such a supporting element. To real-ze multiple degrees of freedom for the swinging movement,ball joint as shown in Fig. 2 [4] or a spherical sliding bear-

ng has been applied. Furthermore, the combination of rolling

earings, such as a universal joint, as shown in Fig. 3 [5] haslso been applied in the case in which a sufficient setting spacean be secured. However, due to the clearance gap in the ball

∗ Corresponding author. Tel.: +81 45 786 7111; fax: +81 45 786 7111.E-mail addresses: nog@rs.noda.tus.ac.jp (S. Noguchi),

anada@kanto-gakuin.ac.jp (T. Kanada).1 Tel.: +81 4 7122 9596; fax: +81 4 7123 9814.2 Tel: +81 466 21 3226; fax: +81 466 27 9766.

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141-6359/$ – see front matter © 2006 Elsevier Inc. All rights reserved.oi:10.1016/j.precisioneng.2005.11.007

m; Prototype

oint or spherical bearing, stiffness and swinging accuracy arensatisfactory. Moreover, in the combined mechanism of rollingearings, stiffness markedly decreases as the size of the rollingearing decreases.

In order to solve these problems, a multi-degree-of-freedomearing has been developed [6,7]. In such a bearing as shownn Fig. 4, a swinging element (referred to as a sun ball) is cir-umscribed by several rolling elements (referred to as planetaryalls) and its stiffness is upgraded by adding a preload. However,he movement of the sun ball may be locked by the position ofhe ball cage, therefore its structural problem is unsolvable.

The scope of this research is to develop a practical pivotearing with multiple degrees of freedom. The basic idea of theivot bearing is derived from a constant velocity joint (CVJ)or vehicles. Then, two prototypes of pivot bearing with twoegrees of freedom, which has rolling and slipping contact, wereroduced. This paper deals with their summary, design methodnd evaluation of performance.

. Design of pivot bearing with two degrees of freedom

.1. Design of mechanism

Fig. 4 shows a small spherical rolling bearing (joint) withultiple degrees of freedom, which is available in the market-

lace. It is used in some test equipment [8]. The required torque

374 S. Noguchi et al. / Precision Engineering 30 (2006) 373–380

Fig. 1. Example of hexapod parallel link (obtained from: http://ww

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(1) The mechanism can restrain the motion of the ball cage(including the rolling element);

Fig. 2. Example of ball joint.

s small and smooth motion can be realized. However, the spher-cal rolling bearing has a defect arising from the nonrestrictivelanetary balls in the ball cage. That is, in the case that the initialosition of the ball cage is misaligned by the motion of the sun

Fig. 3. Example of universal joint.

w.physikinstrumente.de/products/prdetail.php?secid=7-16).

all, the bearing motion may be locked due to such structuralifferences. Moreover, in such a case, the thicknesses of the topnd bottom housings are different, the number of planetary balls,hich are in contact with the housing, varies. Furthermore, the

ensile stiffness becomes lower than the compressive one.Therefore, the following requirements were set for the pro-

otype:

Fig. 4. Example of spherical rolling joint.

S. Noguchi et al. / Precision Engineering 30 (2006) 373–380 375

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Fig. 5. Schematic view of constant velocity joint (CVJ).

2) The tensile stiffness and the compressive one are almost thesame.

In consideration of the above conditions, a prototype of theivot bearing was designed.

Fig. 5 shows a simplified mechanism of the constant velocityoint (CVJ), which is an example of the prototype to be designed.

hen the two parts, which have some circular-arced grooves, arerought together upon contact with balls, the rotating motion cane carried in constant velocity between the two noncoaxial axes.riginally, it is used for transmission of uniform motion. If oneart is fixed, the other part can swing in two degrees of freedom.he balls are rolling with differential slip motion in time ofwinging. By applying this mechanism, the swinging motionith two degrees of freedom can be realized with restriction on

he rolling direction of balls.Constant velocity joints for vehicles have been mass-

roduced and the cost per piece is low. If the pivot bearingtructure is similar to the CVJ for vehicles, the manufacturingosts incurred can be reduced.

However, if the CVJ is not modified, it cannot be a “bearing”.hen, the following views were checked in the design procedure:

1) If the balls are monostichous, the fixing of the position ofrolling element and loading of the axial force cannot beguaranteed. Therefore, the balls were drawn up in two linesand the rolling element was interleaved between the two-lined balls.

2) If the balls were drawn up in two lines and both parts hadconcave grooves, the swinging motion was not realized.Then, concave groove was worked on one part, and the sur-face of the other part was to be spherical as shown in Fig. 6.

3) Considering (1) and (2), if the housing is integrated withsuch a mechanism, it is hard to assemble. Thus, the housingshould be separated into two bodies. Herewith, a preloadbecame possible within the bearing structure.

4) To hinder free movement of the balls, a ball cage, the sameas that used in a rolling bearing was fixed between the sun

ball and the housing.

Specifications of the bearing will be described later and Fig. 6hows the structures of the pivot bearing with two degrees of

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ig. 6. Prototype of pivot bearing with two degrees of freedom. (a) When sunall (swinging element) has grooves; (b) when housings have grooves.

reedom. One structure has eight grooves in the rolling elementnd the other has eight grooves in the housing.

.2. Stiffness design

Although dynamic stiffness should be investigated in thepplication to the parallel mechanism, the calculations of axialisplacement and contact surface pressure should be performeds static stiffness is important in practical use. Under the assump-ion that the material is bearing steel, the attributes of bearingteel are used, such as the modulus of longitudinal elasticityYoung’s modulus) and Poisson’s ratio. The same design methods the rolling bearing can be applied directly to the structure [9].o, the calculation of contact surface pressure and stiffness isimple.

The cross-sections of the swinging element, rolling elementnd housing are presented in Fig. 7. When contact surface pres-

ure was calculated by the Hertzian elastic contacting theory, itas necessary that the axial force of the rolling element Q bea/(Z sin α). Here, α denoted the contact angle, Z denotes theumber of rolling elements and Fa denotes the axial force. The

376 S. Noguchi et al. / Precision Engineering 30 (2006) 373–380

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Although a large swinging angle was desirable, it was set to−25◦ to +25◦ for the primary purpose of this research, namelyconfirmation of the swinging structure. The contacting angle of

Table 1Specifications of prototype pivot bearing with two-degree-of-freedom

Dimension φ 60 mm, height 30 mmAngle of swing (◦) −25 to +25Diameter of sun ball 25.4 mm (1′′)Diameter of swinging shaft (mm) 10Diameter of rolling element (ball) 4.7625 mm (3/16′′)Diameter of concave curvature (mm) 34.925Disposition of balls 8 balls every 45◦Arrangement of balls in orthogonal

direction to disposition of balls2 (contacting angle from −30◦ to+30◦)

Fig. 7. Schematic view of structure inside.

xial displacement δα, can be approximated by the followingquation:

α = δ

sin α(1)

ere, δ is the total displacement along the contact angle.Figs. 8 and 9 show the calculation results for axial displace-

ent, the maximum contact pressure with changing numberf grooves and the diameter ratio between the swinging ele-ent and the rolling element provided that the inner diameter of

he housing is fixed to 34.925 mm. In Fig. 8(a), d1 denotes theiameter of the sun ball (swinging element) and d2 denotes theiameter of the rolling element. If the housing has grooves, it cane said that the stiffness becomes larger with increasing numberf grooves and approximating the swinging element radius tohat of the rolling element. However, since the contact betweenhe swinging element and the rolling element became convex-o-convex, the contact pressure became higher overall. In themallest swinging element bearing with eight grooves, the stiff-ess was approximately 120 N/�m by measuring the gradient ofhe tangential line in the consecutive curve of 200 N in the axiaload.

From the viewpoint of grooves, having grooves in the swing-ng element was advantageous to the stiffness and contact pres-ure. In particular, the contact condition between the surfacesecame concave-to-convex. Therefore, the contacting pressureas moderated to a large extent. In the smallest rolling ele-ent bearing with eight grooves, a stiffness of 200 N/�m could

e obtained. Although the diameter of the swinging elementecame smaller, the number and area of grooves were reducedonsidering the necessity of a flat area for attaching a shaft.

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ig. 8. Axial stiffness of prototype pivot bearing. (a) Influence of number ofrooves; (b) influence of diameter ratio.

The axial stiffness of the prototype pivot bearing was assumedo be 100 N/�m. Furthermore, considering the accuracy, thevailability of sphere balls and the workability of the bearing, thepecifications of the prototype pivot bearing were determined as

iameter of groove 5 mm (depth 1 mm), by thedesign basis for a deep-grooveball bearing

S. Noguchi et al. / Precision Engineering 30 (2006) 373–380 377

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Table 2Design parameters for bearing

Static performance Dynamic performance

(1) Position of grooves (6) Depth of groove(2) Dimension of curvature (7) Ball cage(3) Numbers of balls and grooves((

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ig. 9. Maximum contacting surface pressure of prototype pivot bearing. (a)nfluence of number of grooves; (b) influence of diameter ratio.

he rolling element was set to 30◦ considering the assembly andonscrambling of the clear aperture of the upper housing. Theimensions (width and depth) of the groove were determined onhe design basis of the deep-groove ball bearings.

.3. Design of ball cage

The ball cage was made of engineering plastics. It has lad-ered pocket holes along a thin-walled taper with a diameterf 5.5 mm, which is larger than that of the installed ball. Therototype bearing is the same as a rolling bearing with a rotat-ng inner ring, considering the cross-section of the swingingirection. Letting the rotating angle of the swinging element be, the orbital angle of the rolling element is as small as 0.4.hen, the movement of the rolling element becomes relativelymall.

The prototype bearing with grooves in the housing had noroblems, but the other one with grooves in the swinging element

as different. That is, the rolling elements were positioned in airection different to the swinging direction and pressed againsthe ball cage. Thus, if the pocket hole was narrow, the ball cageended to move in the same motion angle as the swinging ele-

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ent. Therefore, the shape of the pocket hole became that of aandscape as shown in Fig. 10.

Table 2 shows design parameters, which will influence theerformance of the pivot bearing to be developed. In Table 2,erms (1)–(5) are concerned with static performances suchs surface pressure, displacement and stiffness. Terms (6)nd (7) are concerned with dynamic performances such asorque.

. Prototype and evaluation of bearing

.1. Techniques for working process and assembly ofearing

The most important factor in the swinging performance is theorking process of the grooves for the rolling elements (balls).he dimension and accuracy of the groove influence the capa-ility of assembly and dynamic performance. In the prototype

f this research, a ball-end mill with a diameter of 5 mm wassed to make the grooves by a three-dimensional CNC millingrocess. Since grinding was not performed, surface roughnessnd form accuracy were not sufficient. However, in investigating

378 S. Noguchi et al. / Precision Engineering 30 (2006) 373–380

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the horizontal center of swinging element in line with the ver-tical rotation center of the driving motor. The load cell waspreloaded, then the oscillation of the force (torque) could bemeasured around the preloaded force using only one load cell.

Fig. 11. Schematic view of torque measurement apparatus.

he capability of swinging, it was considered that the ball-endilling was sufficient.Techniques for assembly and adjustment of the bearing were

erformed in the following procedure:

1) Preparing the rolling elements (balls) having slightly differ-ent diameters (from −10 to +10 �m at integrals of 2 �m, onthe basis of a reference diameter);

2) Choosing adequate spheres (balls) in order to preload lightlyand not to form a clearance in the bearing;

3) Tightening up the screws for coupling the upper and lowerhousings with constant torque.

Thus, the accuracy of the groove dimensions should be withinhe limits of −10 to +10 �m.

.2. Measurement of torque for swinging

The torque required for swinging was examined quantita-ively. Fig. 11 shows the measuring apparatus. By evening uphe pivot bearing, the pivot bearing was allocated by bringing

Fig. 12. Example of oscillation in measured torque.FW

Fig. 13. Schematic view of stiffness measurement apparatus.

ig. 14. Experimental relationship between axial load and displacement. (a)hen sun ball (swinging element) has grooves; (b) when housings have grooves.

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he torque could be calculated by multiplying the measuredorce loaded on the swinging shaft by the radius of load cellosition.

Fig. 12 shows an example of the calculated torque. Here,orque = 0 means the average torque. The maximum torque towing is determined by the magnitude of the preload in assem-ling the bearing. Therefore, the absolute magnitude of torqueannot be argued due to the measuring technique applied here,ut it can be said that the oscillation of torque to swing wasithin −0.7 to +0.7 Nm. The magnitude of oscillation wille reduced in order to enhance the dimensional accuracy ofrooves.

.3. Experimental verification of stiffness

In order to investigate the stiffness of the bearing, the axialisplacement of the pivot bearing was measured as shown inig. 13. To keep the swinging shaft vertical, the housing waseclined by 0◦, 10◦, 20◦ and 30◦. Then, when the axial forceas loaded, the axial displacement was measured using a dis-lacement sensor. The declinations of housing were given byurpose-built pedestals, which had predefined angles. In thisay, the influence of displacement other than the pivot bearing

ould be eliminated.

Fig. 14 shows the relationship between axial load and dis-

lacement. Comparing the results with the analytical theory,amely Figs. 8 and 9 in the previous chapter, displacementecame slightly larger and the stiffness, smaller. This may arise

Fig. 15. Geometrical situation of declined bearing.

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neering 30 (2006) 373–380 379

rom the accuracy in process and measuring errors and theseactors will be examined hereafter.

Moreover, by increasing the declination, the axial displace-ent became smaller and the stiffness became larger. This

eason is believed to be as follows. The axial displacementα of the bearing is expressed by Eq. (1). If the housing isositioned horizontally, the force is distributed uniformly. How-ver, when the housing is inclined, the rolling elements tendo move. Then, the contact angles of the rolling elements tohe loading direction (vertical) differ from each other and thexial displacement, which comes under the influence of theontact angle and the partial charging force, also differs. Thengle of declination increased with increasing ratio of mag-itude for the partial charging force. This is because the con-act angle becomes larger. That is, α′ > α as shown in Fig. 15.herefore, this phenomenon results in the increase in contactngle.

Hence, in the stiffness design, the lowest stiffness is obtainednder the horizontal condition. Therefore, the value under theorizontal condition should exceed a design target value. Sincehe axial displacement under the horizontal condition can be cal-ulated on the basis of the Hertzian contact theory, it is confirmedhat the design calculation can be simplified.

. Conclusions

In this study, pivot bearings with two degrees of freedom wereroduced by applying the idea of constant velocity joint (CVJ)or a vehicle. Then, some performances were examined. A briefummary follows:

1) Both prototypes, that is, one having grooves in housings, theother having grooves in the swinging element, can swingsmoothly. In other words, the variation of torque to swingwas between −0.7 and +0.7 Nm.

2) The prototype having grooves in the swinging element issuperior from the viewpoints of stiffness and contact sur-face pressure. However, in order to obtain a larger swingingangle, the shape of the pocket holes of ball cage should bethat of a landscape.

3) It is confirmed that the stiffness becomes the smallest whenthe swinging shaft is vertical. This vertical situation makesstiffness design calculation simple.

Swinging motion with two degrees of freedom can be realizedy the structure in this research. In the future: (a) measurementor the process accuracy of parts, (b) optimum design of grooveimensions and (c) evolution to a bearing with three degrees ofreedom will be investigated.

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4] THK Co. Ltd. Total catalogue for translatory system, no. 401.

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7] Ozaki K, Takeda M. Prototype of rolling bearing mechanism with three-

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