See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/284178215 Worm Gear Drives With Adjustable Backlash Article in Journal of Mechanisms and Robotics · August 2015 DOI: 10.1115/1.4030164 CITATIONS READS 18 4,510 3 authors: Wojciech Kacalak Maciej Majewski Koszalin University of Technology Koszalin University of Technology 351 PUBLICATIONS 1,052 CITATIONS 95 PUBLICATIONS 541 CITATIONS SEE PROFILE SEE PROFILE Zbigniew Budniak Koszalin University of Technology 50 PUBLICATIONS 119 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Innovative technologies and tools for micromachining of surfaces with abrasive foils View project Innovative Speech Interaction System of Lifting Devices and their Human Operators View project All content following this page was uploaded by Maciej Majewski on 20 November 2015. The user has requested enhancement of the downloaded file. Worm Gear Drives With Adjustable Backlash Wojciech Kacalak Professor Faculty of Mechanical Engineering, Koszalin University of Technology, Koszalin 75-620, Poland e-mail: [email protected] Maciej Majewski1 Associate Professor Faculty of Mechanical Engineering, Koszalin University of Technology, Koszalin 75-620, Poland e-mail: [email protected] 2 The Known Constructional Ways of Backlash Reduction Even the most accurate conventional gear-based drives do not provide nonbacklash operation, due to manufacturing imperfections of its elements [10], installation inaccuracy [2,9], operating conditions, or degradation over time. Drive’s elements’ geometrical imperfection comprises dimensional inaccuracy and shape distortions. Backlash, which is a defect of precise machinery and devices systems increases along with abrasion of interacting surfaces. Zbigniew Budniak Assistant Professor Faculty of Mechanical Engineering, Koszalin University of Technology, Koszalin 75-620, Poland e-mail: [email protected] This article presents the design of patented worm gear drives, which allow backlash adjustment or elimination by using specially designed worms and worm wheels. Many of the presented solutions allow backlash adjustment without disassembling the drive. The proposed solutions allow reduction of both backlash and its standard deviation to as little as 7.5% and 5% of their initial values, respectively. The presented solutions are a good alternative to harmonic drives and conventional precise drives, which are currently in use. The presented drives can successfully find their application in mechanisms for precise positioning of test benches, as well as in precise technological equipment, technological instrumentation, and in the case of miniaturization—in mechanisms resistant to severe working conditions. [DOI: 10.1115/1.4030164] 1 Introduction When it comes to linear or angular alignment systems, the automotive and machine-tool industries, robotics, automation systems, and other applications, it is demanded that their mechanisms are nonbacklash [1–6]. The carried out research shows that backlash is nonlinear and is present in every mechanical system [7,8]. There are many factors impacting meshing and circumferential backlash of a worm gear drive. These can be divided into two groups. The first one are factors related to the spatial, mutual position of interacting components [2,9]. These factors have the most substantial impact on meshing and, for the most part, on circumferential backlash. The other group are factors related to a worm gear drive components’ manufacturing accuracy [2,10]. These have impact on the pressure values [11], the condition of surfaces, the directions and values of the friction forces between the worm and the worm wheel [1]. Understanding of the impact of some of the factors on meshing [1,7,10], e.g., the line of action [9], the teeth contact line [12], the teeth surface’s topography, the estimated durability of a worm gear drive [13] may lead to aware 1 Corresponding author. Manuscript received November 18, 2014; final manuscript received March 15, 2015; published online August 18, 2015. Assoc. Editor: James Schmiedeler. Journal of Mechanisms and Robotics modification and optimization of worm gear drives [1] in order to improve their operating parameters. Backlash adjustment mechanisms for conventional drives are quite complex, and therefore expensive. Such complex backlash adjustment mechanisms are also present in worm gear drives; however, simpler solutions for backlash elimination are available as well. The solution most commonly discussed in publications is a drive with a double-lead worm [1]. Some other solutions are also mentioned: worm gear drive with a split worm wheel [14] or with a conical worm or a nonbacklash double-roller enveloping hourglass worm gear [8]. Scientific publications are usually descriptions of phenomena connected with the typical worm gear drives or the technological ways of backlash adjustment. This article presents new types of worm gear drives, the construction of which allows reduction or elimination of backlash. Their advantages and disadvantages related to operation and durability have been identified. 2.1 Worm Gear Drive With a Double-Lead Worm. One of the most common design approaches—aiming at worm gear drive backlash reduction—is to substitute the conventional worm with a double-lead worm. Drives with double-lead worms are used in indexing tables and also as dividing heads in lathes. As a result of implementation of two different pitch values of both sides of the thread, the worm’s teeth thickness is variable. The difference between thicknesses of teeth is linear and depends on the pitch difference of both helical surfaces [1]. Another result of using two different pitch values in doublelead worms is that each tooth’s profile differs between its sides. For this reason, in the case of double-lead worms it is advised to use different profile angles, so that teeth’s working surface is not limited on one side, which also helps avoid shortening of worm wheel’s teeth [15]. A disadvantage of such drives is the necessity for the worm to have considerable axial travel in order to reduce small backlash. This can result in drive’s increased size. Such a worm gear drive has a decreased contact area, which reduces its load ability or causes quick degradation under high load [1]. In addition, for each full turn of the worm wheel, there is only one nonbacklash position. Other positions have backlash, the amount of which depends on the pitch difference between both sides of the worm’s thread. 2.2 Worm Gear Drive With a Split Worm Wheel. It is a solution comprising a worm wheel split with a symmetry plane perpendicular to the shaft’s axis. As a result, it consists of two symmetrical parts mounted on one hub. The width of teeth can be increased with angular displacement of the parts [14]. The width increase causes reduction of backlash of the meshing teeth. Backlash elimination may be carried out automatically, periodically, or by using elastic elements. Periodical adjustment is the most difficult one to carry out and, moreover, it requires the drive to be partially disassembled or to use removable covers facilitating adjustment components access. Unfortunately, the disadvantages of this solution are low load ability and poor durability of the drive, which are caused by discontinuity of the worm wheel teeth’s profile. It means that the worm wheel rim’s width is meshed only half-width, which is the fundamental constraint in application of such drives. It is C 2016 by ASME Copyright V FEBRUARY 2016, Vol. 8 / 014504-1 Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 08/27/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use 3 Fig. 1 The designed adaptive worm wheel solutions: (a) with cut-outs and (b) with a joining wall Fig. 2 The design of a worm wheel with its rim split with a circumferential cut-out: (a) bare worm wheel and (b) fully assembled worm wheel technologically quite difficult to manufacture such a worm wheel, due to the necessity of maintaining high accuracy of that process for both of its halves. The advantages of this solution are compactness and its ability to bear varying loads. The angle of a worm wheel’s teeth profile can be slightly decreased in order to lower the disadvantageous impact of the aforementioned discontinuity. New Worm Gear Drive Design Ideas 3.1 Worm Gear Drive With an Adaptive Worm Wheel. One of the solutions, which uses a modified worm wheel is a drive described in the patent [16]. Its distinctive feature is a worm wheel with deep, concentric to the hub hole, annular cutouts in its side areas (Fig. 1(a)). Their depth is greater than half of the wheel’s width, and so their diameters are different and when looked at in a cross section, they look like two oppositely situated cut-outs. The aim of such a modification is to introduce a diametrical and axial adaptivity of the worm wheel’s rim. Such an adaptivity can be also achieved by using two components to make a worm wheel—a rim and a hub—and joining them together with a thin wall (Fig. 1(b)). The wall should have at least one wavy bending, concentric to the wheel’s rim [16]. One of the difficulties related to this solution is achieving concentricity of the rim and hub during the process of manufacturing such a worm wheel. 3.2 Worm Gear Drive With a Worm Wheel’s Rim Split With a Circumferential Cut-Out. In a nonbacklash worm gear drive [17], worm wheel’s rim is split symmetrically with a circumferential cut-out, deep enough to reach the narrowing’s smaller diameter. As a result, the width of the rim is split into two equal parts, which can be brought closer together by applying pressure onto the rim with an annular pressing element. One of the solutions—described in the aforementioned patent—is that the pressing element has the shape of an annular screw plug, and it is mounted on the threaded surface of the worm wheel’s hub (Fig. 2(a)). The advantages of this solution are easy both manufacturing and backlash adjustment. Figure 2 presents a solid model of this design. Bringing the worm wheel’s halves closer together leads to a decrease of backlash and also causes the meshing zone to move toward the vertices of the worm wheel’s teeth. This can be taken into account during the design stage, by decreasing tilt angle of the teeth by the value appropriate to the chosen range of adjustment. However, this solution entails a disadvantage being the result of the worm wheel adaptive tooth’s discontinuity [18]. The authors have designed a more effective solution, where one part of the worm wheel has an adaptive rim (Fig. 3). Its deformation (adaptation) is achieved with a pressure disk, which has a conical pressure surface with advantageously adaptive working surface. Fig. 3 The design of a worm wheel, one part of which has an adaptive rim: 1—rim, 2—worm wheel’s circumferential cut-out, 3—circumferential narrowing of the worm wheel’s cross section, 4—narrowing concentric to the worm wheel’s shaft hole, 5—pressure element, 6—worm wheel’s hub, 7—a screw, 8—a nut, 9—pressure ring’s circumferential cut-out, 10—conical pressure area, and 11—a washer 014504-2 / Vol. 8, FEBRUARY 2016 Transactions of the ASME Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 08/27/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 4 The design of a worm gear drive with worm wheel’s teeth situated on a thin-walled sleeve: 1—teeth, 2—shaft, 3—thin-walled sleeve, 4—bearing, 5—worm, 6—thin-walled sleeve’s bottom wall, 7—collar, 8—case, 9—eccentric pressure mechanism, 10—screw, 11—circumferentially prolonged holes, and 12—hook spanner hole [18] 3.3 Worm Gear Drive With Its Worm Wheel’s Rim Situated on a Thin-Walled Sleeve. Another solution of backlash reduction is to situate the worm wheel’s teeth on a thin-walled sleeve. This can be done in a variety of ways [19–22]. Figures 4 and 5 show a worm gear drive with adjustable backlash characterized by a worm wheel—with an elastic thin-walled sleeve—joined to a driven shaft. It is equipped with an adjustable eccentric pressure mechanism, which allows to displace the driven shaft’s axis along with the thin-walled sleeve and the wormwheel’s teeth perpendicularly to the worm’s axis. As a result of the adjustable displacement of the entire unit along with the worm wheel’s teeth, the worm wheel comes closer to the worm’s axis and circumferential backlash is continuously reduced until the worm wheel teeth’s surface circumferentially rests on the worm’s teeth [18,19]. Continuously bringing the interacting elements closer causes elastic deformation of the thin-walled sleeve and an increase of surface pressure of teeth proportional to the stress of the sleeve, Fig. 5 A photorealistic rendering of the worm gear drive with worm wheel’s teeth situated on a thin-walled sleeve [18] Journal of Mechanisms and Robotics which allow to fully reduce the backlash. One of the disadvantages of this type of drive is the requirement that the worm wheel be of the shape of a thin-walled sleeve, which results in the drive’s increased size. 3.4 Worm Gear Drive With a Locally Axially Adaptive Worm. Another solution with a modified worm is a nonbacklash worm gear drive [23] presented in Figs. 6 and 7. Worm with an advantageous cone-derived helical surface can be abraded with an abrasive disk characterized by a working surface consisting of straight lines in an axial cross section. The worm has an axial hole with the diameter equal to or slightly larger than the diameter of the tooth space’s bottom. The most important element of this solution is the cut between the threads along the helical line present only in the middle part of the worm. The helical cut can be produced using an end mill with the diameter smaller than the width of the of the tooth space, or with an erosion-based metalworking, such as using a laser. As a result, the worm—in its middle part— becomes a type of a spring with fixed endings, and characterized by high axial stiffness. The cut results in the worm being locally adaptive in the axial direction and it can be compressed or expanded. Such a worm is slidable, it is situated on an arbor, and it has one of its ends axially fixed in place and it is secured against rotation around the arbor. This allows the other (loose) end’s position to be altered with a shift adjustment mechanism. This mechanism impacts the coreless part of the worm. Compressing the worm results in a change of pitch, thereby impacting the amount of backlash. The length of the worm’s part with its tooth space’s bottom cut along the helical line depends on the length of the meshing area as well as the working conditions and it has to be shorter than the length of the threaded part of the worm [18]. The advantage of such a solution is the ease of backlash adjustment, which can be done even when the drive is in operation. It has good lubricating conditions in the meshing area between the sides of teeth being under load, regardless of the direction in which the drive is working. The adaptivity of the coreless part of the worm (Figs. 6 and 7) has positive impact on reduction of dynamical excess and also helps suppress vibrations. The latter can be also impacted by the introduction of an elastic material into the cut between the threads. FEBRUARY 2016, Vol. 8 / 014504-3 Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 08/27/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 6 The design of the worm gear drive with a locally axially adaptive worm: 1—worm, 2— worm wheel’s rim, 3—arbor, 4—pressure nut, 5—housing, 6—fitted worm situation surface, 7—screw cuts, and 8—worm wheel The disadvantage of this solution is the limited maximal torque it can transfer for the chosen requirements regarding its kinematic accuracy when under load. However, by choosing appropriate structural parameters of the worm, a sufficiently high maximal torque value can be made possible. In order to improve the working properties of such a drive, a modification to the worm teeth’s profile is introduced [18]. The radius of the teeth side curvature is decreased, which causes them to be more protruding (Fig. 8(a)). The worm tooth outline’s curvature can be increased using a grinding wheel of an appropriate diameter or one with a modified outline. It performs rotary motion at the peripheral speed of vs (Fig. 8(b)). The worm rotates at the peripheral speed of vw, which is combined with the axial feed rate fa. The worm outline’s curvature also depends on the angle, at which the grinding wheel’s axis is set, which usually corresponds to the angle of the helical line’s lift in the tooth reference cylinder. Such a modification reduces local shifts of the actual area of meshing. The initial compression of the worm, causing a decrease of its axial pitch, does not increase pressure forces in the meshing zone until backlash is reduced to zero (in one of the worm and worm wheel’s possible positions). Further compression is not advised, as it would lead to increased pressing forces, due to excessive pitch decrease. Only during the initial breaking-in phase it is an acceptable practice to use increased axial compression of the worm. In yet another solution [24], which aims at reducing the possibility of one of the worm’s ends excessive axial travel, which would result in too great pitch shortening in the middle part, an option of radial deformation in that area was introduced. It is the result of situating the worm on an arbor, which in its middle part has a slightly shorter diameter (Figs. 6 and 7). 4 Research on the Worm Gear Drive With a Locally Axially Adaptive Worm 4.1 Analysis of Changes of Drive’s Teeth Meshing Resulting From the Introduction of Pitch Adjustment. The necessary quantities for the determination of a drive’s backlash Fig. 7 A photorealistic rendering of the worm gear drive with a locally axially adaptive worm 014504-4 / Vol. 8, FEBRUARY 2016 Fig. 8 The modified outline of the locally axially adaptive worm’s tooth, where Rc—the tooth curvature’s radius, Rm—the modified curvature radius, R 5 Rc or R 5 Rm, Rm < Rc, vs— peripheral speed of the grinding wheel, vw—peripheral speed of the workpiece (of the worm), and fa—axial feed rate of the worm Transactions of the ASME Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 08/27/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 9 A diagram illustrating backlash reduction as a result of worm axial contraction in a worm gear drive with a locally axially adaptive worm, where (a) state of drive defined as initial, (b) boundary contraction, and (c) full contraction (Fig. 9) are worm’s thread thickness t1 and worm wheel teeth’s thickness t2, as well as the worm wheel teeth’s transverse pitch’s deviation and the worm thread’s axial pitch’s deviation measured for both sides of the teeth. When the worm wheel rotates counterclockwise, as another tooth takes the load, backlash values for pairs of teeth with the (R) index increase, and for the ones with the (L) index they decrease. When the direction of rotation is changing, backlash values for the (R) indexed teeth decrease, and for the (L) indexed ones they increase. In order to determine the amount of a drive’s backlash after the adjustment has been made DB, it is not necessary to measure the backlash for pairs on both sides of the worm’s teeth. When the amount of backlash in a drive is being reduced, the backlash amount of biR and bjL for pairs indexed iR, i < 0 and jL, j 0 becomes smaller, however, for pairs indexed iR, i > 0 and jL, j < 0 it becomes larger. This is the reason why backlash amount for specific pairs of the worm’s thread and the worm wheel’s tooth, which become larger during the adjustment process, can be omitted. Thus, the quantities important for a drive’s backlash determination are the following: b2R ; b1R ; b0L ; b1L ; b2L . Such a practice is acceptable, because already during the design process the engineer should ensure the minimal backlash of a drive, and, moreover, after assembly, drives are broken in, which further increases their initial backlash value. Thus, brand new drives are already characterized by some amount of backlash, which is aimed at to be reduced by means of adjustment. The main assumption of the analysis of teeth meshing is that the initial state of a drive is the one in which in the meshing zone its worm and worm wheel’s teeth are in contact (Fig. 9(a)). These are the teeth, which the measurements were started with, and were used to reset the readings of the measuring device. The outcome of the adjustment DB, which is worm axial compression, are changes in the meshing zone. First is backlash reduction until a drive reaches a nonbacklash state in only one angular position. Depending on the position of the pairs remaining meshed, the following phases can be observed: (1) Worm compression phase until reaching nonbacklash state for only one angular position of the worm (Fig. 9(a)). (2) Range increase of the worm’s angle of turn, until the boundary compression state is reached (Fig. 9(b)), where backlash b0 ¼ 0 (for u ¼ u1 ), which is an acceptable phenomenon only during the breaking-in process. Fig. 10 The decrease of axial pitch along the worm’s thread for different amounts of compression DB [25] Journal of Mechanisms and Robotics This article talks about the state, in which a worm wheel’s tooth in the meshing zone is in contact with a worm’s tooth. As backlash decreases, by means of worm axial compression, the axial pitch also decreases. Worm’s thread is most adaptive in the meshing zone, and this is where backlash should decrease and reach FEBRUARY 2016, Vol. 8 / 014504-5 Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 08/27/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 11 A chart presenting the dependency between backlash and angular position of the worm wheel with different drive’s adjustments (amounts of axial compression) [25] zero value for a certain angular position of the worm (Fig. 9(b)). It is due to the fact that the worm with a cut tooth space becomes a spring with fixed endings. This makes it a specific type of spring, characterized by a nonlinear adaptivity of threads along the cut. The characteristics of its adaptivity are nonlinear and the maximal pitch decrease is present in the middle of the worm’s adaptive part (Fig. 10). In the case of considerable deviations of geometrical parameters of meshing, it may occur that two distant teeth pairs mesh together—away from the middle of the meshing zone (Figs. 9(b) and 9(c)). Simultaneous contact of two pairs causes a change of worm’s stiffness. Since that moment the worm can be described as comprising at least two elastic systems, which are characterized by increased stiffness [25]. Such a state of axial compression is a boundary one for a drive’s proper operation. Further worm contraction would lead to increased surface pressure of the already meshed tooth pairs. 4.2 Research Results. As for measurement of teeth’s profile, precise coordinates measurement devices or the method described in the [26] patent may be used. The backlash research results for one of the drives [24] with an axially and radially adaptive worm with different amounts of introduced adjustment are depicted in Fig. 11. The amount of backlash for the adjustment number zero was on average 40 lm, and for the biggest one DB ¼ 150 lm it decreased to about 5 lm; however, the range of backlash values decreased from 30 lm to 2 lm. 5 Conclusions Difficulties connected with worm and worm wheel teeth profile modification were the subject of the papers [1,8,12,27], their application [7,13], also in tool manufacturing [2,10], and the technological process of installation [9]. The presented designs of patented worm gear drives allow to adjust or eliminate backlash by introducing innovative designs of worms and worm wheels. These solutions present a worthwhile alternative for harmonic drives or precise conventional drives. Simulations and research in operational conditions have been carried out for many of the presented worm gear drives and improvement has been shown possible. Based on experimental research and numerical analyses the following conclusions have been formulated: (1) Analysis of dimensions and shape deviations of interacting components and deviations in their mutual positions allows to predict drive’s backlash amount before installation. 014504-6 / Vol. 8, FEBRUARY 2016 (2) The new worm gear drive designs, including the one with a locally axially and radially adaptive worm, allow effective and considerable backlash reduction as well as dispersion of its values. Axial adaptivity in this type of drives results in reduction of average backlash amount and leads to dispersion of its values. Radial adaptivity facilitates automatic reduction of local variations of backlash amount. (3) The carried out research has shown it is possible for precise drives with modules m ¼ 2 4 to have average backlash amount reduced to 7.5% of its initial value (from 40 lm to 3 lm) and even achieve twenty times smaller standard deviation (from 13 lm to 0.6 lm). (4) Local radial bending of the worm’s thread does not result in disadvantageous impact on the drive’s kinematic precision. Also, increasing axial stiffness of a worm with locally cut tooth space, by assuming a greater module value, allows to achieve a sufficient load ability of the drive. Worm’s boundary adaptivity may be determined with the knowledge of the drive’s load amount and by assuming the drive’s required kinematic precision. Increased stiffness of the worm requires usage of a pressure mechanism, which is optimized for such a structural property of the worm. (5) The designed drives may be successfully applied in mechanisms for precise test bench positioning, in precise technological devices, technological instrumentation, and in miniaturization, also in mechanisms resistant to severe working conditions. Acknowledgment This project has been funded by the National Science Center on the basis of the decision No. DEC-2012/2005/B/ST8/02802. Nomenclature biR ¼ backlash amount for specific pairs of the worm’s thread and the worm wheel’s tooth with the (R) index—backlash on the worm thread’s right side (lm) bjL ¼ above-mentioned backlash amount with the (L) index—backlash on the worm thread’s left side (lm) t1 ¼ worm’s thread thickness (mm) t2 ¼ worm wheel teeth’s thickness (mm) DB ¼ worm axial compression—(pitch) adjustment (lm) u ¼ angular position of the worm wheel (deg) References [1] Chen, T. Y., Perng, J. N., and Chiou, S. J., 2003, “Two-Stage Optimum Design of the Dual-Lead Worm,” Eng. Optim., 35(5), pp. 561–572. [2] Dudas, L., 2013, “Modelling and Simulation of a New Worm Gear Drive Having Point-Like Contact,” Eng. Comput., 29(3), pp. 251–272. [3] Jawale, H. P., and Thorat, H. T., 2013, “Positional Error Estimation in Serial Link Manipulator Under Joint Clearances and Backlash,” ASME J. Mech. Rob., 5(2), p. 021003. 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[10] Deng, X., Wang, J., and Horstemeyer, M. F., 2013, “Modification Design Method for an Enveloping Hourglass Worm Gear With Consideration of Machining and Misalignment Errors,” Chin. J. Mech. Eng., 26(5), pp. 948–956. [11] Jiaxing, Z., and Ilie, K., 2014, “Static Shear Strength Calculation of Plastic Helical Gears Mating With Steel Worm,” Int. J. Precis. Eng. Manuf., 15(2), pp. 235–239. Transactions of the ASME Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 08/27/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use [12] Brauer, J., 2005, “Transmission Error in Anti-Backlash Conical Involute Gear Transmissions: A Global-Local FE Approach,” Finite Elem. Anal. Des., 41(5), pp. 431–457. [13] Kim, S. H., Shin, M. C., Byun, J. W., Hwan, K. O., and Chu, C. N., 2012, “Efficiency Prediction of Worm Gear With Plastic Worm Wheel,” Int. J. Precis. Eng. Manuf., 13(2), pp. 167–174. [14] Marinkovic, S. V., 1999, “Split Gear Assembly for Use in a Worm Gear Drive,” U.S. Patent No. 5934144. [15] Hsu, R. H., and Su, H. H., 2014, “Tooth Contact Analysis for Helical Gear Pairs Generated by a Modified Hob With Variable Tooth Thickness,” Mech. Mach. Theory, 71, pp. 40–51. [16] Kacalak, W., 1993, “A Non-Backlash Worm Gear Drive,” Patent application 301669. [17] Kacalak, W., 1992, “A Non-Backlash Worm Gear Drive,” Patent No. 169114. [18] Kacalak, W., 1995, “Selected Problems of Precise Worm Gear Drives’ Construction and Technology,” Mechanical Engineering Department, Koszalin University of Technology, Koszalin, Poland, Monograph No. 51. [19] Kacalak, W., Ryckiewicz, J., and Ziolkowski, S., 1990, “Worm Gear Drive With Adjustable Backlash,” Patent No. 164102. Journal of Mechanisms and Robotics [20] Kacalak, W., and Ryckiewicz, J., 1990, “Precise Worm Gear Drive,” Patent No. 164104. [21] Kacalak, W., Ryckiewicz, J., and Ziolkowski, S., 1990, “A Worm Gear Drive for Non-Backlash Torque Transmission,” Patent No. 164105. [22] Kacalak, W., Ryckiewicz, J., and Ziolkowski, S., 1990, “A Non-Backlash Worm Gear Drive,” Patent No. 163445. [23] Kacalak, W., 1988, “A Non-Backlash Worm Gear Drive,” Patent No. 137131. [24] Kacalak, W., and Biedny, D., 2005, “A Non-Backlash Worm Gear Drive,” Patent No. 207801. [25] Biedny, D., 2007, “Fundamental Principles of Backlash Adjustment and Geometrical Parameters Selection in a Worm Gear Drive With a Locally Axially and Radially Adaptive Worm,” Doctoral dissertation, Koszalin University of Technology, Koszalin, Poland. [26] Kacalak, W., Lewkowicz, R., and Lechowski, T., 1986, “A Method of Worm Helical Surface’s Profile Inaccuracy Measurement and a Device for Its Realization,” Patent No. 137523. [27] Simon, V., 2007, “The Influence of Gear Hobbing on Worm Gear Characteristics,” ASME J. Manuf. Sci. Eng., 129(5), pp. 919–925. 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