Spring-force self-aligned multiball pitch artifact

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Spring-force self-aligned multiball pitch artifact Masaharu Komori a,∗ , Fumi Takeoka a , Takashi Kiten a , Yohan Kondo b , Sonko Osawa b , Osamu Sato b , Toshiyuki Takatsuji b , Ryohei Takeda c a

Department of Mechanical Engineering and Science, Kyoto University, Kyotodaigakukatsura, Nishikyo-ku, Kyoto-shi, Kyoto, Japan Research Institute for Engineering Measurement, National Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology (NMIJ/AIST), Umezono 1-1-1, Tsukuba, Ibaraki, Japan c Osaka Seimitsu Kikai Co., Ltd., 6-5-16, Mikuriya, Higashi-Osaka, Osaka, Japan b

a r t i c l e

i n f o

Article history: Received 30 October 2015 Received in revised form 27 December 2015 Accepted 20 January 2016 Available online xxx Keywords: Pitch Artifact Design Accuracy Angle Measurement Spring

a b s t r a c t Gear noise is influenced by pitch deviation of micrometer order, and, therefore, the pitch of gears is inspected in manufacturing processes using measurement instruments. Master gears or artifacts are used to evaluate the accuracy of the pitch measurement instrument, but their accuracy is not sufficiently high and they are not easy to manufacture. In a previous study, the concept of a novel high-precision pitch artifact composed of simple components was proposed for the evaluation of the accuracy of pitch measurement instruments. Simple components, such as balls, cylinders, and planes, can accomplish an accuracy on the order of several tens of nanometers. Therefore, this artifact can be realized with high accuracy. In the present study, we propose a spring-force self-aligned multiball pitch artifact, in which simple components are assembled using spring force. The design of the artifact is discussed, and the artifact is manufactured. Measurement experiments using a coordinate measurement machine and a pitch measurement instrument are carried out, and the proposed pitch artifact is demonstrated to be fundamentally valid for accuracy evaluation. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Geared power transmission is often used in vehicles. Recently, lowering the vibration/noise of gears for geared power transmissions is becoming increasingly important. The gear vibration/noise is influenced by micrometer-order tooth flank deviation [1–3]. Therefore, the inspection of gears using a measurement instrument is necessary in the gear-manufacturing process. As such, there have been a number of studies on measurement methods for gears [4–7]. In particular, a specialized measurement device for gears [8–10] or a coordinate measuring machine (CMM) [11] is often used for the inspection of gear quality. A high-precision gear-measurement instrument is required in order to accurately evaluate gear quality. In order to verify the accuracy of gear-measurement instruments, a high-accuracy artifact is used in most cases. In principle, the accuracy of the gear measurement instrument does not exceed the accuracy of the artifact used to calibrate the instrument. Thus, an ultrahigh-accuracy artifact is needed.

∗ Corresponding author. Tel.: +81 75 383 3587; fax: +81 75 383 3587. E-mail address: [email protected] (M. Komori).

Measurement of the pitch between teeth is one method of evaluating gear quality [12–17] and is important for reducing gear noise. In many factories, a master gear [18,19], which has a similar shape to that of a common gear, is used to evaluate the pitch measurement accuracy of the instrument. However, the accuracy of the master gear is not sufficiently high because it has a geometrically complicated reference surface and its manufacture is difficult. In order to solve this problem, artifacts specialized for the evaluation of pitch measurement (called pitch artifacts) have been developed [20–27]. It is thought that these pitch artifacts have higher pitch accuracy than master gears, but their practical accuracy would be limited because their manufacture is difficult. Therefore, higher accuracy might be difficult to realize. Moreover, such artifacts are costly to manufacture. In order to solve this problem, the present authors have proposed a multiball artifact, in which elements with simple geometries, such as balls, a cylinder, and a plate, are in contact with each other and produce the desired pitch [28]. This artifact is composed of elements with simple forms that can be manufactured with high accuracy. Therefore, a highly accurate pitch is feasible. In addition, the artifact can be manufactured easily and at low cost. Based on this concept, the authors have proposed a magnetically self-aligned multiball pitch artifact, in which simple-shaped elements are assembled spontaneously and are held in contact with

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each other by magnetic forces [29,30]. However, the strong magnetic force used in this method might affect the measurement results for this artifact, depending on the measurement instrument. Thus, in the present study, a novel method to keep the balls, cylinder, and plane in contact with each other without a magnetic force is discussed. A spring-force self-aligned multiball pitch artifact is proposed, in which the balls, cylinder, and plane are kept in contact with each other by a pushing force generated by springs. The artifact is designed and manufactured, and measurement experiments are performed using a CMM and a pitch measurement instrument in order to evaluate the validity of the artifact.

Fig. 2. Structure of the multiball artifact [28].

2. Background and multiball artifact 2.1. Background In many cases, pitch measurement for a gear is performed using a specialized machine referred to herein as a pitch measurement instrument. A high-accuracy pitch artifact is used to evaluate the accuracy of the target pitch measurement instrument. The pitch artifact is manufactured with high accuracy, and its pitch value is calibrated using a measurement instrument with even higher accuracy. The deviation between this calibrated pitch value and the pitch value measured using the target pitch measurement instrument indicates the error of the target instrument. The target pitch measurement instrument considered herein is calibrated based on this deviation. In industry, an artifact called a master gear is used for the inspection and calibration of the pitch measurement instrument. The shape of the master gear is similar to that of a gear and the master gear is manufactured by grinding. Due to its geometrically complicated reference surface, it is difficult to realize a high-accuracy master gear. In principle, the accuracy of the pitch measurement instrument cannot exceed that of the artifact used to calibrate the instrument. Thus, the accuracy of the pitch measurement instrument is limited when it is inspected and calibrated using a master gear. A pitch artifact with an accuracy higher than that of the master gear is required in order to solve this problem. As such, a specialized pitch artifact was proposed [24] in which the gauge blocks are fixed around the circumference at certain intervals as shown in Fig. 1. The gauge block has highly precise planes, whereas the surface profile of the master gear is complex. Therefore, a specialized pitch artifact can provide more precise reference surfaces than the master gear. However, it is necessary to accurately fix the gauge blocks at the target positions and postures. This requires advanced

Fig. 1. Previously proposed pitch artifact with gauge blocks [24].

Fig. 3. Contact points of the stylus tip in the angular pitch measurement of the multiball artifact [28].

manufacturing techniques and is time consuming. Therefore, it is difficult to obtain ultrahigh precision and, at the same time, lower the manufacturing cost. 2.2. Multiball artifact In a previous report [28], the present authors proposed a novel, highly precise pitch artifact called a multiball artifact, as shown in Fig. 2. The multiball artifact is composed of a combination of balls, a cylinder, and a plane, and the center cylinder is surrounded by multiple balls on a plane. The balls, cylinder, and plane contact the neighboring elements. A cross section of the artifact is shown in Fig. 3. The stylus tip of the pitch measurement instrument contacts the outer balls at reference points A and B when the pitch (angular pitch) is measured. This multiball artifact was demonstrated to realize an arbitrary angular pitch and measurement circle diameter by changing the diameter of the balls and the diameter of the center cylinder. The multiball artifact has the following characteristics: (1) The balls, cylinder, and plane contact each other. Therefore, the positions of these elements are determined automatically. Then, the positions of the reference surfaces for the pitch measurement instrument are also fixed. The multiball artifact is easy to manufacture and assemble because advanced techniques are not necessary. For example, in the case of small gears, it is possible to manufacture this pitch artifact precisely, whereas manufacturing small master gears is difficult. (2) The accuracy of the balls, cylinder, and plane determines the accuracy of the artifact. All of these elements can be manufactured with 10-nm-order accuracy. For instance, the sphericity of the highest grade (grade 3) of steel balls is within 80 nm, according to ISO 3290 [31] and JIS-B 1501 [32]. Moreover, these highly precise products are manufactured as standard products and so are readily available. The high precision of the individual elements results in the high accuracy of the artifact.

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Fig. 5. Experiment to fix three balls with adhesive while maintaining contact between the balls. Fig. 4. Magnetically self-aligned multiball pitch artifact [29].

(3) High-precision products of balls, cylinders, and planes can be obtained easily and at low cost because these components are commercially available, and highly advanced techniques are not necessary in the assembly process. Consequently, it is possible to keep the manufacturing cost low. (4) Highly precise measuring instruments can be used to measure balls, cylinders, and planes. The manufacturers of these elements supply the products with the measurement results for shape and dimension to certify the quality. A pitch value for the multiball artifact can be calculated using the measured data of the composing elements. Therefore, the accuracy of this artifact can be known without a final calibration using other measuring instruments such as a CMM after the assembly of the artifact. Measurement using a CMM needs significant time and labor, and in some cases, its accuracy is not sufficient. The verification of the artifact accuracy without a final calibration after assembly is an advantage of the multiball artifact. 2.3. Magnetically self-aligned multiball pitch artifact and associated problem In previous studies [29,30], a magnetically self-aligned multiball pitch artifact in which the balls, cylinder, and plane are maintained in constant contact with each other using a magnetic force was proposed as shown in Fig. 4. Magnets are placed under the plane of the artifact to apply the magnetic force. The magnetic force causes spontaneous attraction of the balls, cylinder, and plane to each other. As a result, self-alignment of the balls can be achieved. In this structure, the attractive force can only be generated by placing magnets appropriately, and advanced assembly techniques or special tools are unnecessary. In addition, the magnet can generate an attractive non-contact force, and thus machining of individual components is not necessary. Consequently, form deterioration of each component is kept to a minimum and a highly accurate artifact can be obtained. However, a strong magnetic force is used in this artifact. In previous studies [29,30], this artifact was measured experimentally using various measurement instruments, and it was confirmed to be possible to measure the magnetically self-aligned multiball pitch artifact without the influence of a magnetic force. Nevertheless, there are several types of measurement instruments, and some might be affected by the magnetic force of this artifact. Therefore, it is necessary to develop a novel method to keep the balls, cylinder, and plane in contact with each other without the use of a magnetic force. 3. Investigation of a method to fix the balls and the proposal of a spring-force self-aligned multiball pitch artifact In the proposed multiball artifact, the balls must always remain in contact with each other and the relative positions of the balls

must be constant. Thus, a method of fixing the balls is important in the manufacture of this artifact. However, if machining, such as drilling, is performed on the ball in order to fix it, the accuracy of the ball will be deteriorated. Therefore, machining should be avoided. Accordingly, fixing methods for balls using adhesives and springs are discussed in this section.

3.1. Fixing balls using an adhesive Fixing the balls using an adhesive is the simplest fixing method. An experiment using an adhesive is performed in order to investigate whether the balls can be fixed in contact with each other. Three types of cyanoacrylate-based adhesive and three types of epoxy-based adhesive are used in the experiment. Three steel balls with diameters of 12.7 mm (0.5 in.) are bonded using the adhesives, and the adherence condition is investigated. Fig. 5 shows a diagram of the experiment. It is necessary to maintain the contact between the balls. Therefore, during the bonding process, the balls are aligned on the magnet base and the magnetic force is applied so that the balls are attracted to each other. After the adhesive hardens, the balls are removed from the magnet base and the adherence condition is observed. The balls were firmly fixed in the case of each adhesive. However, it is impossible to visually confirm whether the surfaces of the balls are in contact because the part of the balls that must be in contact is covered by the adhesive. As such, contact is confirmed using an electric current. Positive and negative electrodes are attached to two of the three bonded balls, and an electric current is applied in order to determine whether the electric current flows, which was the case for each of the adhesives. Thus, it is confirmed that the balls can be bonded while remaining in contact with each other, as shown in Fig. 5, which validates fixing the balls using adhesive. However, further examination revealed that the following problems are associated with this method. First, it is difficult to confirm that all of the balls are in contact with each other. As previously described, it is impossible to confirm this contact by observation. Although the confirmation method using an electric current is partly valid, even if two of the three balls are not in contact, electricity could flow through the other ball. Thus, it is impossible to confirm contact between all of the balls by means of electric current. Another problem is changes in the adhesive. Even if the balls are in contact with each other immediately after bonding, this may change due to natural drift or thermal transformation of the adhesive. Stable contact between the balls is needed for this artifact, so that the changes in the adhesive are problematic. In conclusion, fixing the balls using adhesive has numerous disadvantages with respect to certification and stability, although it can be used to achieve contact between the balls.

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Fig. 7. Accessing state of the stylus of the measurement device to outer ball of the pitch artifact when the pitch artifact is measured. (a) Cross section AA and (b) position of cross section AA.

Fig. 6. Basic structure of the spring-force self-aligned multiball pitch artifact.

3.2. Proposal of a spring-force self-aligned multiball pitch artifact In this section, a method to maintain the contact between the balls using springs is proposed. Fig. 6 shows the structure of the proposed artifact. The center cylinder and the base plate are fixed to the base, and the component to which the springs are attached (called spring-holding component) is installed on the upper step of the center cylinder. The spring-holding component has beams overhanging radially from the center, and springs are installed on the beams. As shown in the lower side of Fig. 6, both the lateral force toward the center cylinder and vertical force are applied to each ball from the compressed spring. The lateral force toward the center cylinder leads to the self-alignment of the balls. In other words, by applying this force to each ball, contact between neighboring balls and the cylinder is maintained, and the balls are aligned spontaneously. The vertical force maintains the contact between the ball and the base plate. Moreover, the frictional force generated by this vertical force prevents the ball from being displaced on the base plate. It is essential that the ball remains stationary with respect to the contact force applied by the stylus of the measurement instrument during the measurement of this artifact. This is the reason why the vertical force is necessary. The inner balls, as well as the outer balls, are also pushed by the springs. If only the outer balls are pushed toward the inside, the inner balls are pushed from both the center cylinder and the outer balls and might separate from the base plate. In order to avoid such a situation, the inner ball is also pushed by the spring. This structure is assumed to realize constant contact between the balls, cylinder, and base plate. The spring-holding component is not a disc but rather an aggregate of beams, similar to the frame of an umbrella. As shown in Fig. 7,

this structure enables the stylus of the measurement instrument to access the outer balls when the artifact is measured using a typical CMM. The characteristic, manufacturing and assembly method, difficulty of manufacturing, accuracy of the reference surface and positioning accuracy of the reference surface are summarized in Table 1 for master gear, conventional pitch artifact using gauge block, magnetically self-aligned multiball pitch artifact and springforce self-aligned multiball pitch artifact in this paper. Spring-force self-aligned multiball pitch artifact is superior to master gear and conventional pitch artifact using gauge block in the aspect that high accuracy is feasible by easy manufacturing method because the reference surface of this artifact is spherical surface with high accuracy and its positioning is performed spontaneously by self-alignment. Magnetically self-aligned multiball pitch artifact is superior concerning manufacturing; however, spring-force selfaligned multiball pitch artifact is superior in the aspect that it excludes the effect of magnetic force. 4. Design and manufacture of spring-force self-aligned multiball pitch artifact A spring force self-aligned multiball pitch artifact that has the structure shown in Fig. 6 and an angular pitch of 45◦ is designed. 4.1. Design of the reference surface for measurement In the design of the ball and the cylinder, the diameter of the balls is determined first. Subsequently, the diameter of the center cylinder is determined based on geometrical analysis, in which the diameter of the balls and the target angular pitch are considered [28]. 4.1.1. Ball SUJ2 steel balls from Amatsuji Steel Ball Mfg. Co., Ltd. with a diameter of 12.7 mm (0.5 in.) are used for the pitch artifact considering the availability of products with high accuracy. Eight balls are used in each of the inner and outer layers of the pitch artifact. In order to achieve an accurate angular pitch in the artifact, the diameters of the balls must be identical, and the sphericities must be small. JIS standard balls of Grade 3 [31,32] were chosen because these balls are the most accurate among the commercially available JIS standard balls. The sphericity of the balls was measured using a CEJ Johansson AB Mikrokator and the roundness of the balls was measured using a Taylor Hobson TR73. The measurement results for 16 balls used for the pitch artifact are shown in Table 2.

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Table 1 Comparison among four types of artifacts. Artifact type

Master gear

Multiball pitch artifact

Pitch artifact using gauge block

Characteristic

Similar to a gear in shape

Gauge blocks are arranged at certain intervals

Manufacturing and assembly method

Manufactured by grinding

Gauge blocks are fixed precisely in assembly

Difficulty of manufacturing

Difficult

Difficult

Accuracy of the reference surface

Uneveness of approximately 1 ␮m remains due to grinding mark Accuracy of grinding influences Precise positioning requires process with high accuracy

Highly precise plane (ex. flatness of several tens of nanometer)

Positioning accuracy of the reference surface

Positioning accuracy of gauge block influences Precise positioning requires advanced assembling technique

Table 2 Measured dimensions of the balls used for the pitch artifact. No.

Deviation of diameter from 12.7 mm (␮m)

Sphericity (␮m)

Roundness (␮m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.04

0.031 0.035 0.036 0.037 0.040 0.040 0.041 0.041 0.039 0.040 0.035 0.038 0.040 0.031 0.037 0.039

The diameter of the balls was confirmed to be 12.70040 mm. The sphericity ranges from 0.03 to 0.04 ␮m, and the roundness ranges from 0.031 to 0.041 ␮m. 4.1.2. Center cylinder As shown in Fig. 6, the center cylinder is a stepped cylinder and has three cylindrical components. Each of them has each role. The first component is a reference cylinder that is used to estimate the position of the center axis of the artifact when the artifact is measured. The second component is an effective cylinder that contacts the balls of the artifact. The third component is a connecting cylinder that connects the center cylinder with the base plate and the base. When the artifact is placed on the measurement instrument, the reference cylinder is measured and the center axis position of the artifact is estimated. Therefore, high roundness is required. The diameter of the reference cylinder is larger than that of the effective cylinder, and there is a step between these cylinders. The springholding component, which receives a vertical upward force from the springs, is pushed against this step and its position is fixed. The diameter of the reference cylinder is 30 mm, and its length is 15 mm.

Magnetically self-aligned

Spring-force self-aligned

Balls are aligned by magnetic force Disassembly and reassembly is possible Magnetic force might affect the measurement result depending on the measurement instrument Balls are self-aligned by magnetic force and positioned automatically

Balls are aligned by spring force Bad effect of magnetic force does not exist

Extremely easy (it can be assembled in 5 min) Highly precise spherical surface (ex. sphericity of several tens of nanometer)

Balls are self-aligned by spring force and positioned automatically Easy

Balls are self-aligned and positioned automatically Precise positioning is easily feasible because accuracy of the balls is high

The diameter of the effective cylinder is 20.5 mm, and its length is 30 mm. The surface of the effective cylinder plays a role to compose the angular pitch with contacting balls. The diameter of the effective cylinder (20.5 mm) is determined by the number and diameter of the balls composing the artifact. This cylinder requires high accuracy to hold the surrounding balls at the ideal position. In particular, high roundness is needed in the cross section that includes the contact points between this cylinder and the balls. In addition, if the cylinder axis is not perpendicular to the base plate of the artifact, the angular pitch will be affected. Therefore, high perpendicularity between the lower end face (step part) and the side face of the effective cylinder is also required. Moreover, the coaxiality between the reference cylinder and the effective cylinder must be high because the center axis of the reference cylinder is a representative axis of the artifact, although the actual center axis of the artifact is defined by the effective cylinder. The connecting cylinder is inserted into the holes of the base plate and the base, connecting them to the center cylinder, as shown in Fig. 6. The diameter of the connecting cylinder is smaller than that of the effective cylinder, and there is a step between them. In addition, the hole of the base is also stepped. The lower tip of the connecting cylinder is threaded. As shown in Fig. 6, the base plate and the base are interposed between the nut, which is fixed to the tip part of the connecting cylinder, and the step between the connecting cylinder and the effective cylinder. The base plate, base, and effective cylinder are then fixed by fastening the nut. The spring between the nut and the base prevents an excessive force from being applied to the base plate and the base when fastening the nut, which might cause deformation of the base plate. The diameter of the connecting cylinder is 8 mm, and its length is 50 mm. When the cylinder and the balls are made of the same material, the angular pitch of the artifact remains unchanged, despite the thermal expansion of the artifact due to the temperature change. In order to make use of this feature, SUJ2 is also chosen as the material for the center cylinder. The manufactured center cylinder is shown in Fig. 8. The cylinder is measured three times using a CMM (Leitz PMM866) and a roundness measurement instrument from Kosaka Laboratory Ltd. The average cylindricity of the reference cylinder is 0.26 ␮m, and that of the effective cylinder is 0.16 ␮m. The average coaxiality between the reference cylinder and the effective cylinder is 0.07 ␮m. In the

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Fig. 8. Manufactured center cylinder.

Fig. 9. Manufactured base plate.

cross section in which the cylinder and the balls contact, the average roundness of the effective cylinder is 0.17 ␮m. The average perpendicularity between the lower end face (step part) and the side face of the effective cylinder is 0.11 ␮m. The average diameter of the effective cylinder is 20.50670 mm. The diameter of the effective cylinder is larger than the nominal value of 20.5 mm by approximately 7 ␮m. This difference is intentional and generates a gap between the inner balls. The reason for this is explained later in Section 4.3. 4.1.3. Base plate Any unevenness on the base plate can cause a fluctuation in the ball position because the balls and the center cylinder are placed on the upper face of the base plate. This leads to the dispersion of the pitch. In addition, the upper face of the base plate is used as the reference surface in order to accurately place the pitch artifact on the measurement instrument. Namely, high flatness is required for the base plate. The thickness of the base plate is 25 mm, which is determined so that the flatness of the upper face of the base plate can be less than 100 nm. The diameter of the base plate is 110 mm. The base plate is made of CLEARCERAM, which is often used for master optics because it is less prone to thermal expansion and has excellent mechanical characteristics. Fig. 9 shows the manufactured base plate. The base plate has a penetrating hole with a diameter of 8.5 mm at its center for the insertion of the connecting cylinder of the center cylinder. The flatness of the upper face is measured using a flat surface interferometer (Fizeau interferometer). The measured flatness of the upper face of the base plate is 23.49073 nm. 4.2. Parts used to fix the balls 4.2.1. Springs used to fix the balls As shown in Fig. 6, the lateral force toward the center cylinder and the vertical force are applied to the balls from the springs. In order to realize this state, the springs are installed at an incline to the vertical direction. The vertical downward force, which generates the frictional force that prevents the ball from being displaced on the base plate, should be larger than the lateral force required for self alignment. Accordingly, the inclination angle of the spring

with respect to the vertical direction should be smaller than 45◦ . Then, the inclination angle of the spring with respect to the outer balls is selected to be 30◦ . For the inner balls, the lateral force toward the center cylinder is provided from the outer balls. Therefore, the spring force in that direction is not necessary. In contrast, the vertical downward force is necessary in order to preserve the frictional force between the ball and the base plate and to prevent the ball from separating from the base plate. Based on these considerations, the inclination angle of the spring for the inner ball with respect to the vertical direction is set to 5◦ . Such a small inclination angle of the spring for the inner ball is preferable with respect to the accessibility of the stylus during measurement. As shown in Fig. 7, it is necessary to measure the outer ball from upper side in the measurement of the pitch of this artifact. However, if the inclination angle of the spring for the inner ball is large, the stylus cannot access the outer ball due to interference. As such, it is preferable to choose a small inclination angle of the spring for the inner ball. Next, we discuss the required spring force to prevent the ball from being displaced on the base plate. In the measurement of this artifact, the contact force is applied to the outer ball from the stylus of the measurement instrument. Assume that the contact force is 1 N in the lateral direction. When the force is applied to the outer ball, a resistance force is generated from the inner balls and the base plate to obstruct the displacement of the outer ball. In order to simplify the discussion, only the resistance force aroused by the base plate is considered. Assuming that the resistance force is the frictional force and that the coefficient of friction is 0.1, a vertical resistance force of 10 N is required between the ball and the base plate in order to keep the ball stationary under the contact force of 1 N in the lateral direction. In contrast, the inner balls are more difficult to displace than the outer balls because they are surrounded by a number of other balls and the center cylinder. However, a lateral force toward the center cylinder is imposed on the inner balls from the outer balls, which may make the inner balls separate from the base plate. Determining the appropriate pushing force on the inner balls in advance is difficult. Therefore, two vertical resistance forces between the balls and the base plate (8 and 20 N) are assumed, and the effect is evaluated experimentally using two different springs (inner spring 1 and inner spring 2). Considering the inclination of the spring with respect to the vertical direction, the space between the ball and the spring-holding component, and the availability of springs, springs from KS SANGYO Co., Ltd., the specifications of which are listed in Table 3, were selected. When these springs are used, the vertical downward force on the outer ball applied by outer spring is 10.78 N, which satisfies the required vertical downward force. The vertical downward force applied to the inner ball is 8.40 N when inner spring 1 is used and 23.90 N when inner spring 2 is used.

4.2.2. Spring for center cylinder In this section, we discuss the spring for the center cylinder placed in the base. The reaction force generated by the springs pushing the balls is received by the spring for the center cylinder through the spring-holding component. In other words, the sum of the vertical downward forces of the springs for the balls, as explained in Section 4.2.1, is received by the spring for the center cylinder. The sum of the vertical downward forces is 153.44 N when inner spring 1 is used and 277.44 N when inner spring 2 is used. In addition, a force to maintain contact between the center cylinder and the base plate is also needed. Therefore, the spring for the center cylinder should generate a force of approximately 280 N. Based on these conditions, a spring from KS SANGYO Co., Ltd. was selected. This spring is made of piano wire and has an outside diameter of 14.5 mm, a wire diameter of 2 mm, and a free

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Table 3 Specifications of the springs used for the pitch artifact.

Outer spring Inner spring 1 Inner spring 2

Material

Outside diameter (mm)

Wire diameter (mm)

Free height (mm)

Number of total coils

Spring constant (N/mm)

Piano wire SUS 304 Piano wire

6.93 6.5 7.1

0.63 0.7 0.8

17 8.5 10.5

7.5 5.75 5.5

1.166 2.813 4.763

Fig. 11. Spring-holding component with set screws, nuts, springs, and center cylinder installed.

Fig. 12. Base. Fig. 10. Dimensions of the spring-holding component.

height of 22.5 mm. The total number of coils is 5.5, and its spring constant is 23.912 N/mm. 4.2.3. Spring-holding component The spring-holding component is umbrella shaped, as shown in Fig. 6. The dimensions of the spring-holding component are shown in Fig. 10. The spring-holding component is circular and has beams that extend radially. The positions of the eight beams correspond to those of the outer balls. The diameter of the central circular part is 45 mm. The beam outside diameter is 100 mm, and the beam thickness is 6 mm. There is a penetrating hole at the center of the central circular part to insert the effective cylinder of the center cylinder. The diameter of the center hole is 20.7 mm, which is larger than that of the effective cylinder by 200 ␮m in order to avoid scratching the precisely processed surface of the effective cylinder. In the assembly process, the spring-holding component and the center cylinder are bonded using adhesive after the center cylinder is inserted into the hole in order to prevent the relative rotation between the center cylinder and the hole. There is a screw hole inclined by 30◦ at the tip of each radially overhanging beam onto which to install the spring for the outer ball. A set screw is placed in this screw hole. A nut is then placed at the lower tip (ball side tip) of the set screw. The spring for the outer ball is attached to this nut. There are eight screw holes inclined by 5◦ on the central circular part to install the springs for the inner balls. It is supposed that the accuracy of the spring-holding component does not affect the accuracy of the artifact because it holds the balls through the springs. The material of the spring-holding component is SUS304. Fig. 11 shows the

spring-holding component with set screws, nuts, springs, and the center cylinder installed. 4.2.4. Base As shown in Fig. 6, the base is disc-shaped and has a diameter of 110 mm and a thickness of 35 mm, which supports the base plate. The base has a penetrating stepped hole at the center for the insertion of the connecting component of the center cylinder. The upper face of the base might affect the flatness of the base plate because the upper face contacts the base plate. Therefore, the upper face of the base is ground to a flatness of approximately 2 ␮m. The base material is SUS316. Fig. 12 shows the manufactured base. The lower part of this artifact is composed of two parts, i.e., the base plate and the base. Although the lower part may be composed of only the base plate, it would be necessary to form a stepped hole in the base plate. Such a complicated process might be a factor in deteriorating the flatness of the base plate. Thus, the lower part is separated into the base plate and the base so that the only processing required for the base plate is simple drilling, and the complicated stepped hole is formed in the base. 4.3. Assembly of the artifact Table 4 shows the geometry of this artifact, and Fig. 13 shows a schematic diagram of the artifact. The values in Table 4 are calculated values based on the measured dimensions of the elements of the pitch artifact (measured values for the cylinder and ball diameters). An angular pitch of approximately 45◦ is achieved. As shown in Fig. 13, an intentional gap exists between the inner balls at the specified position by using an effective cylinder with a slightly

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Table 4 Geometry of the pitch artifact calculated based on the calibrated value of each element. Angular pitch without gap (p1 through p6 ) Angular pitch with gap (p7 , p8 ) Position of center of outer ball without gap (outer balls 1 through 7) Position of center of outer ball with gap (outer ball 8) Gap between inner balls

44.97234◦ 45.08299◦ 26.34008 mm (distance from center of pitch artifact) 26.31064 mm (distance from center of pitch artifact) 59.24 ␮m

Fig. 14. Rosette washer attached to the spring.

Fig. 13. Schematic diagram of the assembled pitch artifact. (The gap size is exaggerated in this figure and is much smaller in reality.).

larger diameter. If the diameter of the effective cylinder is slightly smaller and this gap does not exist, some of the inner balls might separate from the center cylinder. Under this condition, the inner balls might change their positions due to external forces. Therefore, this condition might be a problem with respect to the stability of this pitch artifact. This is the reason for the intentional gap between the inner balls. Note that the gap size is exaggerated in Fig. 13 for clarification purposes, and the actual gap size is 59.24 ␮m. The pitches related to this gap, i.e., p7 and p8 , have different values from the other six pitches (p1 through p6 ) because of the gap. For the same reason, the distance between the center of the artifact and the center of outer ball 8 related to this gap is different from that of the other seven balls. The parts discussed in the preceding sections were then assembled. First, the assembly trial was performed using inner spring 1 (c.f. Table 3), because the springs for the inner balls generate a smaller force. However, this failed to fix the balls stably. Even after the artifact was completely assembled, the balls were easily dislocated by a small contact force or vibration. This phenomenon is thought to have been caused by the force applied by inner spring 1 (8.43 N in the spring axial direction) being small compared to the force applied by the outer spring (12.45 N in the spring axial direction). In such a case, the outer ball might push the inner ball in the lateral direction, the spring for the inner ball is bent significantly and the ball becomes detached from the spring. Next, the assembly trial was performed using inner spring 2. In this case, the force applied to the inner ball is 24.01 N in the spring axial direction. In addition, in order to hold the ball more stably, a Rosette washer, one side of which has a curved surface, is attached to the ball side end of the spring, as shown in Fig. 14. The Rosette washer was constructed from Duracon. The fixing condition of the ball is shown in Fig. 15. The assembly trial was performed, and the balls were successfully fixed stably. Then, the artifact was completed. The assembly process and the completed artifact are shown in Fig. 16. First, (a) the spring-holding component with set screws and the center cylinder are assembled. Second, (b) the connecting cylinder of the center cylinder is placed through the holes of the base plate and the base, and connected using the spring for the center cylinder and the nut. Next, [(c) and (d)] the inner balls

Fig. 15. Fixing a ball using a Rosette washer.

are aligned. Then, (e) the outer balls are aligned, and (f) the artifact is completed. The assembly process is easy, and no advanced manufacturing techniques or special tools are required. 5. Measurement experiment using the coordinate measuring machine 5.1. Measuring procedure Measurement using a CMM is one method of measuring pitch. Moreover, the pitch of the typical pitch artifact can be also calibrated using a high-precision CMM. Therefore, we investigate whether calibration measurement of the proposed pitch artifact is possible using a CMM. In addition, repeated measurements of the pitch are performed, and the repeatability is investigated. In this experiment, a contact-type CMM Leitz PMM 866 is used, and a ruby ball with a diameter of 5.0 mm is used as the stylus. The measurements are performed under the following process. First, the position of the center axis of the artifact is estimated by measuring the reference cylinder of the center cylinder of the pitch artifact. Then, the pitch measurement is performed on a circle (measurement circle) with a radius of 27.0945 mm centered on the artifact axis. In a plane that is parallel to the base plate and is as high as the centers of the outer balls (6.35 mm above the upper surface of the base plate), the stylus scans the measurement circle, and the coordinate of the center of the stylus is recorded when the stylus comes into contact with each outer ball. Outer ball 1 is measured first, followed by outer balls 2 through 8, in numerical order. The angle between the recorded coordinates of the center of the stylus indicates the angular pitch. Note that outer ball 8 is closer to the center of the artifact than the other outer balls because there is an intentional gap between two specific inner balls. Accordingly, when comparing the measured pitches related to outer ball 8 (seventh and eighth pitches, p7 and p8 ) with the design value, the design value must be compensated for considering this term. The compensated design values are 45.08813◦ for the seventh pitch, p7 , and

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Fig. 16. Process of assembling the pitch artifact. (a) Spring-holding component with set screws installed and the center cylinder are assembled. (b) The threaded part of the center cylinder is placed through the holes of the base plate and the base, which are joined using a spring and the threaded center cylinder and a nut. (c) Inner balls are aligned. (d) Alignment of the inner balls is completed. (e) Outer balls are aligned. (f) Artifact is completed.

Fig. 17. Deviation of the angular pitch measured using the CMM from the design value.

45.07786◦ for the eighth pitch, p8 , in this measurement. This series of measurements is defined as one measurement process, and 10 measurement processes are conducted. 5.2. Measurement results and discussion First, it was verified that this pitch artifact can be measured using a CMM without any problem. Then, the measurement results are analyzed. Fig. 17 shows the deviation of each angular pitch from the design value. The measured pitches p3 and p4 are close to the design value. The measured pitches p1 , p2 , p5 , and p6 are larger than the design value, and the measured pitches p7 and p8 are smaller than the design value. Although pitches p1 through p6 have the same design value, there are differences in the measurement results. The reason for above result may be due to the gap. In the ideal condition, the gap is near outer ball 8. If the gap is narrowed, the distances between the neighboring outer balls 7, 8, and 1 become smaller, and the angular pitches between them also become smaller. At the same time, when the gap near outer ball 8

becomes small, gaps are generated between the other inner balls. When a gap exists between inner balls, the distance between the neighboring outer balls becomes large and the corresponding angular pitches increase. The measured values are smaller than the design value for pitches p7 and p8 , which occurs for the case in which the gap near outer ball 8 becomes small. On the other hand, the measured values are larger than the design value for pitches p1 , p2 , p5 , and p6 , which occurs for the case in which gaps exist between the inner balls near pitches p1 , p2 , p5 , and p6 . Although the artifact was manufactured so that it has only one gap between inner balls at a specified position, other gaps may be generated and dispersed. In the magnetically self-aligned multiball pitch artifact proposed in the previous report [28,29], the attractive force between the balls is generated by the magnetic force. The magnetic attractive force is strong when the distance (gap) between the balls is small, and it is weak when the distance (gap) between the balls is large. Accordingly, even if gaps exist between the inner balls, balls near smaller gaps are more strongly attracted to each other because the magnetic attractive force is strong. Consequently, the smaller gaps tend to disappear. Therefore, in the final state, only one large gap remains, and the artifact is stabilized in that condition. On the other hand, in the method using springs proposed in the present study, three types of lateral forces are applied to the inner balls. The first is the force toward the center cylinder applied from two outer balls and the spring. The second is the force in the opposite direction applied from the center cylinder. The third force is the force in the circumference direction applied from two neighboring inner balls. In this artifact, the first two forces are large. In the ideal condition, force balance can almost be realized by only the first two forces. Accordingly, the circumferential-direction force by two neighboring inner balls is small. In other words, the force to push the inner balls against each other is small. In some cases, the inner ball can balance the forces without contacting the neighboring inner balls

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Table 5 Standard deviation of pitch at each measurement position as measured using CMM. Pitch number

p1

p2

p3

p4

Deviation of pitch (␮rad)

4.843

5.425

3.402

4.675

Pitch number

p5

p6

p7

P8

Deviation of pitch (␮rad)

3.203

4.069

4.167

4.581

Fig. 19. Deviation of measured pitch using the pitch measurement instrument based on the measurement results obtained using the CMM (five measurements).

Fig. 18. Pitch measurement instrument used in the experiment to measure the pitch artifact.

because there is friction between the inner ball and the center cylinder or that between the inner ball and the base plate. Therefore, if a gap is generated between the inner balls, it tends to remain. Consequently, the gap is likely to remain dispersed among the inner balls. Therefore, it is necessary to comprehend this characteristic when using the spring-force self-aligned multiball pitch artifact. In addition, it is preferable to make the gap between the inner balls as small as possible. 5.3. Repeatability Table 5 shows the repeatability (standard deviation) for 10 measurements of the angular pitch. In contrast to the design value of the angular pitch, which is approximately 0.79 rad, the standard deviations of the 10 measurements are less than 6 ␮rad for all angular pitches, which indicates that calibration of the pitch artifact is possible for the case in which the required repeatability is at this level. In addition, it is verified that the displacements of the balls are within the displacement corresponding to the change of the angular pitch value shown in Table 5 at the most. 6. Measurement experiment using the pitch measurement instrument 6.1. Measurement process using the pitch measurement instrument A measurement experiment is performed using this artifact and a pitch measurement instrument, and the results are compared with the measurement results obtained using a CMM. In addition, the measurement repeatability is investigated. Fig. 18 shows the pitch measurement instrument, which is a contact-type computer numerical control (CNC) high-precision fully automatic gear measurement instrument (MGL-26A, Osaka Seimitsu Kikai Co., Ltd.). This measurement instrument has a rotary table on which the object to be measured (the pitch artifact) is placed. The displacement sensor and the stylus are installed on a mechanism that is movable in the X-, Y-, and Z-directions, and the stylus tip contacts the measured surface. In a coordinate system in which the upper face of the base plate is in the XY plane and the central axis of the center cylinder is the Z-axis, the angular pitch of the pitch artifact

is defined as the angle formed by the lines joining the center of the stylus and the center cylinder around the Z-axis (Fig. 3). Hence, in the measurement of this artifact, its position and posture are first adjusted so that the rotary axis of the pitch measurement instrument and the central axis of the center cylinder of the artifact are identical. Second, the stylus tip of the pitch measurement instrument touches the surface of an outer ball. The pitch artifact is then rotated, and the stylus tip touches the surface of the next outer ball. The rotational angle in this process is defined as the angular pitch. The angular pitch formed by the neighboring outer balls is measured in this manner. Thus, in the measurement of the pitch using the pitch measurement instrument, probing is performed by rotating the measured object, and the pitch is calculated based on its rotational angle. 6.2. Measurement results Five pitch measurement experiments are performed. A ruby ball of 3 mm in diameter is used as the stylus of the pitch measurement instrument. It was confirmed that the pitch artifact can be measured by the pitch measurement instrument without any problem. Fig. 19 shows the deviation between the pitch measured using the pitch measurement instrument and the calculated pitch value from the measurement results using the CMM, as described in Section 5. The deviation is less than approximately 120 ␮rad for pitches p1 through p7 . However, the deviation is over 350 ␮rad for the eighth pitch, p8 . As explained in Section 5.2, the condition of the gap between the inner balls is assumed to be different from that in the measurement using the CMM. This indicates that it is necessary to narrow the gap between the inner balls if higher accuracy is needed, as described previously. Table 6 shows the repeatability (standard deviation) calculated for the five measurements. The maximum standard deviation is less than approximately 8 ␮rad. The experimental results confirm that stable measurements are realized at this level when measuring this pitch artifact using the pitch measurement instrument. In addition, it is verified that the displacements of the balls are within the displacement corresponding to the change of the angular pitch value shown in Table 6 at the most. Table 7 summarizes the measurement result of the spring-force self-aligned multiball pitch artifact in this paper and previously reported measurement result of the magnetically self-aligned Table 6 Standard deviation of pitch at each measurement position measured using the pitch measurement instrument. Pitch number

p1

p2

p3

p4

Deviation of pitch (␮rad)

3.796

5.571

3.796

5.470

Pitch number

p5

p6

p7

p8

Deviation of pitch (␮rad)

7.948

6.404

4.341

5.470

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M. Komori et al. / Precision Engineering xxx (2016) xxx–xxx Table 7 Comparison between measurement results of spring-force self-aligned multiball pitch artifact and magnetically self-aligned multiball pitch artifact. Spring-force self-aligned multiball pitch artifact

Magnetically self-aligned multiball pitch artifact

Repetitive measurement result using CMM (standard deviation) Smallest 3.2 ␮rad 1.9 ␮rad Largest 5.4 ␮rad 3.7 ␮rad

11

fundamentally valid as an artifact. In addition, the deviation between the measured angular pitch and the design value was discussed. This pitch artifact is assumed to have a tendency to generate dispersed gaps between the inner balls, which affects the deviation from the design angular pitch.

Acknowledgment

Repetitive measurement result using pitch measurement instrument (standard deviation) Smallest 3.8 ␮rad 2.7 ␮rad Largest 7.9 ␮rad 8.2 ␮rad

The present study was supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.

Difference between the designed pitch value (calculated value from the measurement result of the balls and the cylinder) and the measurement result using CMM −629.1 ␮rad −46.8 ␮rad Smallest Largest 310.0 ␮rad 48.7 ␮rad

References

multiball pitch artifact [30]. The standard deviation for the repetitive measurement using CMM is slightly smaller for magnetically self-aligned artifact. However, the standard deviation for the repetitive measurement using pitch measurement instrument is at the same level for both artifacts. In contrast, the difference between the designed pitch value (calculated value from the measurement result of the balls and the cylinder) and the measurement result using CMM is large in the case of spring-force self-aligned artifact. This is caused by that the gap between the inner balls tends to be dispersed in spring-force self-aligned artifact as described in Section 5.2. 7. Conclusion A pitch artifact with high accuracy is required for highly accurate inspection and calibration of the pitch measurement instrument. The typical pitch artifact, which is referred to as a master gear, has a complicated reference surface and is thus difficult to manufacture. As such, the accuracy of the master gear is limited and its cost tends to be high. In order to address these problems, a spring-force self-aligned multiball pitch artifact was proposed, in which simpleshaped elements such as balls, a cylinder, and a plate are used, and these components are spontaneously aligned by springs. The structure of this artifact was discussed. The artifact was manufactured and experiments were carried out. The following conclusions were obtained. (1) A method to fix the balls while maintaining contact between the balls was discussed, and a method in which a lateral force toward the center cylinder and a vertical force were applied to each ball using springs was proposed. In addition, a method to fix the center cylinder and the base plate using springs was proposed. The structure of the spring force self-aligned multiball pitch artifact was shown utilizing these methods. (2) A spring-force self-aligned multiball pitch artifact was designed. The important aspects in designing the parts, i.e., balls, center cylinder, base plate, base, springs, and springholding component, were clarified. These parts were manufactured and assembled without problem, and no advanced manufacturing techniques or special tools were required. (3) A measurement experiment involving the manufactured artifact was performed using a CMM and a pitch measurement instrument. The ability to measure the artifact using CMM and the pitch measurement instrument without any problem was confirmed. The repeatability (standard deviation) was under 6 ␮rad for the CMM and under 8 ␮rad for the pitch measurement instrument. This result reveals that the pitch artifact is

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