Manufacturability and Viability of Different C-Gear Types: a Comparative Study

Proceedings of the ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2012 August 12-15, 2012, Chicago, IL, USA

DETC2012-71030

MANUFACTURABILITY AND VIABILITY OF DIFFERENT C-GEAR TYPES: A COMPARATIVE STUDY

Hani A. Arafa Mostafa Bedewy† Professor Assistant Lecturer Department of Mechanical Design and Production Department of Mechanical Design and Production Engineering, Cairo University, Engineering, Cairo University Cairo, Egypt † Currently at the University of Michigan, e-mail: [email protected] Ann Arbor, MI, USA e-mail: [email protected]

ABSTRACT ratio introduced by the helix angle results in the simultaneous The family of cylindrical parallel-axis involute gearing meshing of more teeth (higher overall contact ratio). Moreover, currently include spur, helical and double-helical gears, all helical teeth are characterized by uniform wear as opposed to having straight tooth traces in the developed pitch plane. spur gears. This is attributed to their meshing characteristics However, gears with curved tooth traces have also been that dictate an inclination of the progressive contact lines proposed. One of the obvious merits of this configuration is the between the involute helicoidal flanks in mesh. This feature insensitivity to shaft misalignment. Although this and other also renders helical gears favorable running-in characteristics. merits of gears with lengthwise curved teeth (C-gears) were However, due to the helix angle of helical gear teeth, an highlighted, they have never been mass-manufactured. Many “unwanted” axial thrust force component is developed in types and shapes of C-gears have been envisioned, a dozen or addition to the tangential and radial force components that so, but the particular merits and demerits of each type were normally act on the tooth surface. never put together in a comparative study aiming at stipulating which type of gear can be manufactured simplest of all or which type is most appropriate for use in specific applications. In this paper, a comprehensive comparative study is carried out for all C-gear types in the repertoire. Finally, the promising ones are singled out for detailed scrutiny; and prospective applications are pointed out for these types.

1. INTRODUCTION Cylindrical parallel-axis gears can be classified, as shown in Fig. 1, according to their tooth profile into two categories: conjugate (such as cycloid and involute gears), and conformal (such as gears with concave and convex circular arc profiles or Wildhaber-Novikov gears). The family of parallel-axis involute gearing (Fig. 2), which belongs to the first category, currently includes spur, helical and double-helical gears, all having straight tooth traces in the developed pitch plane. The main difference between spur and helical tooth traces being the Fig. 1. C-gears as members of the family of cylindrical inclination between the direction of their tooth trace and the parallel-axis involute gears gear blank axis. Helical gears run quieter than spur gears and have a higher load carrying capacity because the axial contact

1 Copyright © 2012 by ASME in Fig. 4, represent a new type of involute parallel-axis gearing, complementing the other widely used members of this family; spur, helical and double-helical gears. The advantages of curved-tooth gears will be summarized later on, but their major feature is indicated in Fig. 4; the pair offers a unique 4-DOF interface, which no other type of gearing in full-face line contact possesses. Since, in any kinematic pair, the sum of DOF and constraints is six, then curved-tooth gears have two constraints: one being against axial freedom of the pinion relative to the gear (not really essential) and the other against

the tooth flanks penetrating each other; that the gears do drive Fig. 2. Types of parallel-axis involute gearing: (a) spur, (b) helical, (c) double-helical gears one another!

Double-helical gears and herringbone gears rectify the latter problem of axial thrust. Nevertheless, the “true herringbone” gear, shown in Fig. 2 (c), can only be manufactured by a low-speed cutting process such as gear shaping, and it cannot be finish-ground after hardening. Accordingly, such gears are confined to relatively low speeds and load carrying capacities. Hardened and ground double- helical gears can only be realized if there is a central recess (apex gap) between the right hand and the left hand helical halves for tool relief, as shown in Fig. 3. This groove results in the introduction of a weight and volume penalty. Moreover, a problem with double-helical gears arises from the possibility of having different amounts of backlash between the two helical halves, which normally ensues as a result of finish-grinding each half separately. In cases of deceleration or torque reversal, this difference in backlash causes the axially free pinion to move jerkily back and forth in the axial direction. Failure caused by this problem of axial shuttling was previously encountered in practical applications [1].

Fig. 4. Longitudinally curved tooth gears

2.1 Historical background The curved tooth configuration was first suggested by Semple [2] in the first half of the nineteenth century. Since then, it captured the interest of many mechanical engineers and inventors. This is evidenced by the wealth of patents filed on this issue, some of which turned out to be mere re-inventions rather than genuine inventions. Despite this wealth, and despite the fact that it is still growing—owing to the on-going issuance of patents accruing up till recently [3]—limited research work Fig. 3: Double-helical gears: (a) without central recess; was documented. Because it was treated as a new invention, herringbone, (b) with central recess almost each time it was proposed, many names were given to describe the curvature of the tooth trace. This fact made it even 2. LONGITUDINALLY CURVED TOOTH GEARS more difficult for researchers to find and cite previous related Although all the parallel-axis involute gears discussed so work. Table 1 summarizes most of these names that are too far (spur, helical and double-helical gears) have straight tooth many to be listed in the list of keywords of any publication. traces in the developed pitch plane, gears with curved tooth The simplest and most straightforward name is “Gears with traces traces have also been suggested several times in the past longitudinally/lengthwise curved teeth” (abridged to “C- 160 years, or so. Cylindrical gears with curved tooth traces in gears”). This short name was recently coined by Arafa [4] for the form of circular arcs or other closely similar curves, shown

2 Copyright © 2012 by ASME collectively designating all gear types belonging to this Table 1: Previously proposed names for C-gears category. Name Reference Comments By closely inspecting the literature on C-gears published Gears With Böttcher This name refers to the shape of the so far, one can categorize the available references into three Curved Teeth [5] teeth as curved; however, it is rather main groups. The first category includes those references ambiguous, as it does not state whether (mostly patents) having a more descriptive nature with little or this curved shape is the tooth profile, or no detailing for the gear generation possibilities or the it is in the longitudinal direction. respective gear tooth geometry. A recent example for this Cylindrical Gears Stepanov et This name is more specific, but it may category is the patent granted to Yamada et al. in 2002 [17]. As With Circular al. [6] confuse the reader with circular arc Teeth (conformal) gears. a result of their vagueness, most of these patents are not even holding any intellectual property rights, as they are just Curved Tooth Boor [7] This name is ambiguous as well. Gear documenting the century-and-a-half-old idea of making gears with lengthwise curved tooth and merely highlighting their Radius Toothed Cantrell [8] This name is ambiguous as well. Gear merits. The second category comprises references with elaboration on the machining details and kinematics without Cylindrical Gears Tseng and This name is ambiguous as well. With Curvilinear Tsay [9] correlation with the resulting gear tooth geometry and their Shaped Teeth meshing characteristics such as the patent awarded to Cylindrical Gear Sidorenko This name is ambiguous as well. Zablonskij et al. [18]. In these documents either new machine With Arched et al. [10] tools are designed, or modifications to existing machines are Teeth proposed. The third category, which is the scarcest of all, Circular-Arc- Ishibashi This name is descriptive enough, but consists of detailed research papers/patents in which geometric Toothed [11] may confuse the reader with circular features of the proposed gear tooth are discussed in connection Cylindrical Gears arc (conformal) gears with their fabrication methods such as the work of Ishibashi Cylindrical Gear Dai et al. This name is linguistically accurate for [11]. With Curved [12] describing this type of gears, but this The early work of Ishibashi [11] published in 1966 can be Tooth Traces name refers to the shape of the tooth trace in the developed pitch plane. considered the oldest research paper found on C-gears. Research papers on C-gears became more abundant with the Gear With Koga [13] This name also refers to the shape of Arcuate Tooth the tooth trace in the developed pitch beginning of the third millennium, when several research Traces plane. groups from Japan [12], the Republic of China [15], Romania Gear With Waguri This name is more specific, but it also and the United Kingdom [16], and Egypt [4] conducted Circular Tooth [14] refers to the shape of the tooth trace in research on C-gears and started publishing their work. This Traces the developed pitch plane. indicates that the topic is becoming a rather appealing research Cylindrical Gears Lee and This name refers to the shape of the topic. Owing to the contemporary dynamicity and advancement With Convex- Chen [15] tooth trace in the developed pitch of scientific directories and search engines, cross-citing Concave Tooth plane, although a curved line cannot be between the above mentioned scholarly publications is a recent Traces referred to as being “convex” or trend in the amassed literature on C-gears. In spite of all these “concave”. references, C-gears have not yet been actually incorporated in Curved Face Andrei et This name is not accurate because, by commercial products. Width Gears al. [16] definition, the face width of a cylindrical gear is a distance measured C-gear is a generic manifestation for any gear geometry along the axis of the cylindrical gear involving a curved tooth trace. Several variations were blank suggested throughout the years, with the only common attribute Circoid Gear Yamada et The word circoid cannot be found as an of having lengthwise curved tooth. The first research paper to al. [17] entry to any dictionary. Most probably, bring all these gear forms together and categorize them was The authors just joined together the published in 2005 by Arafa [4]. In this reference, C-gears were prefix circ- with the suffix -oid to create a new word to describe this type classified, according to the variation of pressure angle across of gear. (They published their patent in the face width. The name CV-gears was suggested for Japanese and in German, nevertheless, designating gears with variable pressure angle, and CC-gears the English title and abstract are found for designating gears with constant pressure angle. In that on the website of the European patent office article, eleven gear forms were scrutinized, discussing their < http://worldwide.espacenet.com >) machining methods, in connection with their consequential C-Gears Arafa [4] This name was suggested as a short geometric features, and with correlation to their meshing name for all cylindrical gears with characteristics. longitudinally (or lengthwise) curved teeth

3 Copyright © 2012 by ASME 2.2 Tooth load spread characteristics of C-Gears in apices formed by each two-halves having opposite hand helices comparison with spur and helical Gears (due to slight differences in backlash of the right- and the left- Heavily loaded, straight-tooth-trace gearing suffers a hand parts of a double-helical gear pair in mesh). chronic problem of being unable to spread the tooth load Despite the complexity of both the gear generation evenly across their face, highly overloading one set of the tooth kinematics and the design of machines involved in C-gear edges. This edge-loading problem is due to two main fabrication, the cutters are—in most cases—relatively simple, phenomena. Firstly, the torsional deflection (windup) of slender as compared with hobs and gear shaper cutters. Also, high pinions; secondly, and more importantly, the uneven elastic speed finish cutting can be implemented due to the absence of a deformation of the gearbox itself, which is the supporting reciprocating tool ram motion. structure of the system of bearings that carry the gears. This problem is only partly dealt with in helical gears by either 2.4 Disadvantages of C-gears manufacturing the two meshing gears with slightly different Several reasons hindered the implementation of that novel helix angles, or by longitudinally crowning the teeth. The first gear type, the most important of which is the requirement of solution leads to a good load distribution only at the rated load, dedicated machining and finishing processes for their i.e. the difference in helix angle is designed to be fabrication; C-gears cannot be manufactured by conventional commensurate with the torsional deflection corresponding to gear cutting and grinding machines. In addition, the complexity the rated load. The second solution places one more of the tooth geometry of several C-gear types caused many manufacturing step; thus, the difficulty and machining time researchers to refrain from further assessing their potential for associated there-with add to the cost of gear production. application when compared to double-helical gears, for Contrary to this, C-gears readily accommodate these instance, as there was no application that justified delving deformation phenomena since the lengthwise curvature of their deeper into this complexity. Other issues of concern include teeth makes them conform to each other, with a commensurate gear metrology, tooth form identification, center distance amount of axial self-adjustment. adjustment, and interchangeability. Crowning of involute gear teeth in the profile direction results in smoother operation because of the compensation it 2.5 Limitations of C-gears offers for elastic deformations. Finite Element simulations for The use of C-gears is limited to external gears only. Thus, the stress distribution in doubly crowned helical pinion teeth C-gears cannot be employed in planetary gear systems. They previously reported an improved stress distribution (without also cannot be used in the particular application where a long edge loading) in cases of misalignment errors [19]. pinion is designed to mesh with two half-width gears. Nevertheless, the doubly crowned helical gear tooth surface naturally requires elaborate manufacturing techniques. 3. COMPARISON BETWEEN THE DIFFERENT C- Crowning (if needed) is much easier to impart to C-gears by GEAR TYPES just providing a small mismatch between the radii of curvature The literature abounds with publications on C-gears, of the convex and the concave tooth surfaces. spanning the twenty-first, the twentieth, and even the nineteenth century, but many of these publications are of a 2.3 Advantages of C-gears descriptive nature and do not even describe an exact geometric Like helical gears, C-gears should run smoother than spur configuration for the lengthwise tooth curvature. In the rest of gears because of the introduced axial contact ratio, i.e. the publications, several alternative tooth geometries that result number of tooth pairs in mesh at any given point in time is from various cutting kinematics and/or cutter geometries were larger than in spur gears, which also results in an improved proposed. In addition to the eleven types of C-gears that were load carrying capacity. The curved teeth also render C-gears grouped together recently [4], two more are added in this study, their inherent self-aligning capability. In case of shaft skewing, one of which is suggested for the first time. Thirteen the contact between the convex and concave surfaces of the technically viable types are juxtaposed and the most promising meshing teeth can be compared to the contact between ones are singled out for further study. spherical rollers and the outer race of a self-aligning spherical Serious appreciation for C-gears and the assessment of bearing, and thus the misalignment-induced tooth edge loading their potential application in any field have to be based on is avoided. rigorous research grounds. Owing to the lack of research work The curved tooth flanks are deemed to conform to one in the subject, a unified approach for evaluating the feasibility another better than helical gears during meshing, leading to of each type of C-gears in the context of comparing all types higher bending strength as a result of the more uniform load has not been carried out before. Since the geometric variations distribution. In addition, the continuous curve of the tooth trace between the numerous types of C-gears would be difficult to helps evading two of the inherent disadvantages of double- recognize by the naked eye, the identification of both the tooth helical gears: the first is the presence of the center recess that profile and the longitudinal curvature requires accurate can lead to substantial weight penalty, the other pertains to metrological measurement. In addition, the manufacturing of solving the axial shuttling problem that arises from errors in the some of these types is inherently problematic, meaning that

4 Copyright © 2012 by ASME they will be discarded. Thus, a comprehensive comparative study is needed in order to single out the promising C-gear types for further study and discard other impractical types before carrying out any further scientific research. This can be considered as a first step towards standardization and dissemination.

3.1 Methodology of comparison Owing to the large number of C-gear types proposed so far, and the absence of applications for them hitherto, a juxtaposition of the various particulars of all C-gear types is made herein in an attempt to highlight the merits and demerits of each of the thirteen types for comparative purposes. Points of comparison span four different categories: tooth geometry, meshing characteristics, manufacturing details, as well as inherent merits and demerits. In fact, the borders between these categories are blurry; for example, tooth contact is a geometric feature and can be considered a meshing characteristic in the same time. Also, the possibility of crowning or finish-grinding is both a manufacturing detail and a merit. Some points of comparison may even be positioned under three different categories. The self-complementarity of the generating racks across the tooth face width is a geometric feature that characterizes the meshing and stems from the cutting tool kinematics (manufacturing detail). The category of tooth geometry includes the shape, Fig. 5. C-rack nomenclature (drawing with a tooth trace thickness, and symmetry of the tooth trace in the developed inclination at the side planes of 31˚ and its radius of pitch plane; the cutting rack flank surface, and its profile in the curvature to module ratio of 15) side planes; and the tooth profile, its whole depth, and conjugacy. On the other hand, the category of meshing 3.2 Comparison results characteristics combined with the category of manufacturing The results of this comparative study are presented herein details comprises the base surface, the surface of action, tooth in a tabulated format. The abbreviations in reference [4] are contact, the complementarity of the generating racks, indexing, adopted, viz. CV for gears with a pressure angle that varies number of cutter heads and the number of cutting cycles, the across the face, and CC for gears with a constant pressure generating rolling surface, and the cutter head inclination to the angle. Table 2 presents a detailed comparison of the different cutter head axis. The final category highlights the merits such geometric features of each type of the seven CV-gears and six as the possibility of finish-grinding, crowning, and profile CC-gears. It is noteworthy that the first three columns in this shifting; favorable cutting conditions, and the maximum Table refer to the tooth trace geometry in the developed pitch number of teeth to be cut; insensitivity to center distance plane. In addition, further gear meshing characteristics are variation; tolerance to misalignment; and interchangeability. compared in Table 3 alongside some manufacturing details. The essential nomenclature needed for this comparison is Finally, all thirteen types are judged in a merit-based shown on a C-rack in Fig. 5. comparison, which is given in Table 4.

Tooth Side-plane Oldest Rack tooth Profile adhering to Whole Name trace Tooth trace(s) Tooth thickness rack Conjugacy reference surface(s) involute depth symmetry profile

Böttcher Identical circular Constant in all transverse In all transverse CV1 YES Identical cones Hyperbola Only in the midplane Constant [5] arcs planes planes

Circular arcs with Decreases towards side Only in the CV2 Shurr [21] YES radii that differ by a planes (constant normal Dissimilar cones Hyperbola Only in the midplane Constant midplane half-pitch tooth space)

Circular arcs with radii that differ by a Decreases towards side Farnum half-pitch In all transverse CV3 YES planes in one gear; constant Dissimilar cones Hyperbola Only in the midplane Constant [22] (interchanged planes in the other gear between the two meshing gears

Only in the Decreases towards side midplane; deviations In all transverse CV4 Koga [13] YES Slightly elliptical planes in one gear; constant Dissimilar cones Hyperbola Variable at side planes are planes in the other gear larger than in CV3

Wingqvist Prolate trochoidal Constant in all transverse Identical Only in one plane In all transverse CV5 NO Curved Constant [23] arcs planes trochoidal cones close to the midplane planes

Prolate trochoidal Decreases towards side Dissimilar Only in one plane Only in the CV6(a) Suggested NO Curved Constant arcs planes trochoidal cones close to the midplane midplane

Decreases towards side Dai et al. Prolate trochoidal Dissimilar Only in one plane In all transverse CV6 NO planes in one gear; Curved Constant [12] arcs trochoidal cones close to the midplane planes increases in the other gear

Identical circular Constant in all transverse In all transverse In all transverse CC1 Cantrell [8] YES Oblique cylinder Straight Constant arcs planes planes planes

Only in the Andrei et Identical circular Nearly constant in the midplane, Only in the CC2(b) YES Oblique cylinder Hyperbola Variable al.[16] arcs transverse planes approximate in all midplane others; pseudo-inv.

Only in the Sidorenko Nearly identical Approximately constant in midplane, Only in the CC2 YES Oblique cylinder Straight Variable et al. [10] circular arcs all transverse planes approximate in all midplane others; pseudo-inv.

Only in the Sidorenko Circular arcs with Decreases towards side midplane, Only in the CC3 YES Oblique cylinder Curved Variable et al. [24] different radii planes approximate in all midplane others; pseudo-inv.

Identical slightly Constant in all transverse Cylinder (simplest In all transverse In all transverse CC4 Lewis [25] YES Straight Variable elliptical arcs planes of all) planes planes

Mammano Nearly prolate Constant in all transverse Trochoidal In all transverse In all transverse CC5 NO Curved Constant [26] trochoidal arcs planes cylinder planes planes

Generating No. of cutter Cutter edge Tooth Cutting/ generating Generating Name Base surface Surface of action process heads / No. of inclination to cutter contact racks rolling surface (Indexing) cycles head axis Oppositely warped, symmetrical ruled Line Single- Midplane pressure CV1 Barrel-shaped surface Fully self-complementary Two / Two Pitch cylinder surface, inflects about contact indexing angle (φm) the pitch line Two barrel-shaped surfaces: the smaller for convex Point Self-complementary only Single- Midplane pressure CV2 N.A One / One Pitch cylinder flanks, the other for concave contact in the midplane indexing angle (φm) flanks. Two barrel-shaped surfaces: One / One Oppositely warped, Self-complementary the smaller for convex (male cutter) symmetrical ruled Line only in the midplane (for Single- Midplane pressure CV3 flanks, the other for concave and Pitch cylinder surface, inflects about contact each of the two non- indexing angle (φ ) flanks (opposite for the One / One m the pitch line identical racks) mating gear). (female cutter) One / One Self-complementary (φ -θ) for the outside (male cutter) m Line only in the midplane (for Single- male and female CV4 Two barrel-shaped surfaces Warped and Pitch cylinder contact each of the two non- indexing cutting edges; (φ +θ) One / One m identical racks) for other two edges (female cutter) Oppositely warped, Fully self-complementary Two / Two Unsymmetrical barrel- un-symmetrical ruled Line Continuous Midplane pressure CV5 (for each of the two or Pitch cylinder shaped surface surface, inflects about contact indexing angle (φ ) opposite hand racks) Two / One m the pitch line Two unsymmetrical barrel- Self-complementary shaped surfaces: the smaller Point only in the midplane (for Continuous Midplane pressure CV6(a) N.A. One / One Pitch cylinder for convex flanks, the other contact each of the two opposite indexing angle (φm) for concave flanks. hand racks) Two unsymmetrical barrel- One / One Oppositely warped, Self-complementary shaped surfaces: the smaller (male cutter) un-symmetrical ruled Line only in the midplane (for Continuous Midplane pressure CV6 for convex flanks, the other and Pitch cylinder surface, inflects about contact each of the two opposite indexing angle (φ ) for concave flanks (opposite One / One m the pitch line hand racks) for the mating gear). (female cutter) Line Single- CC1 Cylinder Plane Fully self-complementary One / One Pitch cylinder Pressure angle (φ) contact indexing Point-- Self-complementary only Single- Midplane base CC2(b) Barrel-shaped surface N.A line Two / Two Single point; not edge in the midplane indexing circle contact Cylinder with a diameter Point-- Self-complementary only Single- CC2 appropriately smaller than N.A. line Two / Two Base cylinder Single point; not edge in the midplane indexing the root diameter contact Cylinder with a diameter Point Self-complementary only Single- CC3 appropriately smaller than N.A. One / One Base cylinder Single point; not edge contact in the midplane indexing the root diameter Cylinder that may be larger Line Single- CC4 than the root diameter for Plane Fully self-complementary Two / Two Base cylinder Parallel contact indexing small N Fully self-complementary Line Continuous CC5 Cylinder Plane (each of the two opposite One / One Base cylinder Parallel contact indexing hand racks) Remarks 1. The generating rack is, by definition, complementary to 2. Single indexing (also called face milling) means that the a rack being cut; hence, if the generating rack is self- cutting process is discrete and repeated for each tooth after complementary, then the rack being cut is also self- indexing. On the other hand, continuous indexing (also called complementary. However, a generating rack is said to be fully face hobbing) means that gear cutting is done continuously; no self-complementary if the rack profiles in all transverse plane individually repeated indexing. complement themselves.

Can be Insensitivity to Crowning Profile shifting Favorable cutting Max. no. of teeth to Tolerance to Name finish center-distance Interchangeability possibility possibility conditions be cut misalignment ground variation

CV1 YES To be investigated YES YES Limited Good Sensitive YES Compulsory crowning by CV2 To be investigated YES YES Limited Excellent Sensitive YES a substantial amount More limited than in CV3 YES To be investigated YES YES Good Sensitive NO CV1 CV4 YES To be investigated YES YES Unlimited Good Sensitive NO CV5 YES To be investigated YES NO Limited Moderate Sensitive NO (opposite hand) CV6(a) YES To be investigated YES NO Limited Excellent Sensitive NO (opposite hand) NO (female cutter CV6 YES To be investigated NO Limited Moderate Sensitive NO (opposite hand) in one piece)

NO (variable rake CC1 NO Possible NO Limited Good Insensitive YES angle) NO (single point CC2(b) YES To be investigated NO Unlimited Moderate Insensitive YES cutting) Only substantial profile Good; better CC2 YES YES YES Limited Insensitive YES shifts are feasible with crowning Compulsory Only substantial profile CC3 YES YES Limited Excellent Insensitive YES crowning shifts are feasible Limited to a small Good; better CC4 YES Possible YES YES Insensitive YES number with crowning CC5 NO Possible YES NO Limited Moderate Insensitive NO (opposite hand)

3.3 Description of the added C-gear types was pointed out along with the inferior surface quality The C-gears types that are added to the eleven types (roughness) of these gears [20]. Another problem with this gear previously gathered by Arafa [4] are briefly described. cutting process is the variation of tooth depth across the face width. The described cutting process dictates that the tool digs CV6(a)-gears. These gears combine features of CV2- and deeper in the middle of the tooth [16]. Also, the tooth becomes CV5-gears; being generated by one male cutter in continuous asymmetric in the side planes, which is attributed to the indexing. The most significant advantage of this new type, in variation in rack inclination resulting from the conical shaped addition to the higher productivity of its manufacture, is the racks. Accordingly, line contact is not achieved and edge localization of gear tooth-surface contact (point contact) that loading (due to interference) may develop if longitudinal teeth leads to a better capability of accommodating misalignments crowning is not imparted. and avoiding edge loading. This type is suggested here to fill a void in the assortment of CV-gears. 3.4 Discussion Line contact is achieved if the meshing gear teeth profiles CC2(b)-gears. Andrei et al. [16] recently proposed CC2(b)- are conjugate in all transverse planes. In this case, the tooth gears, which are based on the principle of tool-tip cutting. traces of the driving tooth flank and the driven tooth flank are However, the authors refer to their cutting process as having a identical. On the other hand, point contact is achieved if there straight edge cutting. In fact, the bulk material removal may be exists a difference in curvature between the tooth traces in the done by the cutter edge, but the finishing is done by only one developed pitch plane of the driving and the driven tooth point on that edge (point cutting). Consequently, the straight flanks. Although point contact lowers the load carrying cutting edge leads, in this case, to deviations from the involute capacity of the tooth, it also greatly enhances the tolerance to towards the side planes, as a result of the conical rack shape misalignment. In addition to these two types of contact, the so- formed by the cutter rotation. This deviation from the involute called “point-line contact” can be identified when the radii of

8 Copyright © 2012 by ASME curvature of the two “conformal” flanks are slightly different, all issues of concern. Following are some prospective so that the teeth “theoretically” have point contact, which application areas of C-gears in which their dexterous nature can becomes a line contact upon the slightest loading. CC2-gears lead to improved performance, extended life, and reduced furnish a good example for this type of contact, with the slight weight. difference in the curved tooth radii being inherent to the errors dictated by using a rounded nose cutter. 4.1 Wind turbine gearboxes The limitation on the maximum number of teeth to cut With the present state-of-the-art of horizontal axis wind arises from two different sources. In some cases, the turbines (HAWT) of multi-megawatt power ratings and rotor relationship between the cutter head and gear blank diameters speeds in the vicinity of 10 rpm, the torque input to the gear would cause the cutters, on their way round, to interfere with box assumes values of multi MN.m; torques that could only be the gear blank near either extreme position of the cutter head at encountered in the propeller shafts of super tankers and aircraft the start/end of the involute generating process as such carriers. Since a wind turbine gear box cannot nearly be as (depending on whether an outside or inside cutter is used). heavy as marine gearing, it has now become known that elastic Added to this, in the case of some CC-gears, there will be a deformations of the gear box casing, shafts, planet carriers, and constraint on the base circle to be larger than the root circle in the gear and pinion bodies themselves lead to unprecedented order to be able to complete the generation. amounts of misalignment in the gear meshes. If not properly CC-gears have favorable characteristics over CV-gears due designed, this misalignment causes tooth-edge loading, which to their constant pressure angle across the gear face width. The would ultimately lead to catastrophic failure. It has been variation in pressure angle makes CV-gears sensitive to center proposed that the inherent self-aligning capability of C-gears distance variations and impose tight tolerances on gear can be utilized to enhance the reliability of the step-up mounting. On the contrary, all CC-gears possess the merit of transmissions employed in HAWTs [28]. In the midst of today’s insensitivity to center distance variation, which is a genuine quest for reliable sources of renewable energy, any attribute of involute gearing. Moreover, profile shifting can be improvements in the design of such equipment would have a imparted to all CC-gears—except for CC2 and CC3-gears, in significant economic payoff. which only a substantial amount of profile shift is practical. Due the complexity of the tooth flank geometry of CV-gears, 4.2 Rotorcraft transmissions the possibility of profile shifting needs to be investigated. This The use of C-gears has also been suggested as a issue is considered beyond the scope of the present work. replacement for double-helical gears incorporated in the final As shown from the above Tables, a CC4-gear has the stage bull-gear of split-path transmissions, which are used in simplest rack surface geometry, which is a patch of a the main drive of helicopters [31]. This would lead to a cylindrical surface; it is the only type of C-gears with a substantial weight saving due to both the closer-to-unity load- describable, relatively simple, tooth surface geometry, which is distribution factor, and the geometric continuity of the tooth as called an involute tube. This tube is formed by the motion of all opposed to double-helical gears with a central recess. In points on a circle to unfold off the base cylinder [27]. Although addition, the self-aligning qualities of C-gears can lead to line contact is typically achieved between CC4-gears, localized improved operating performance and lower noise. (point-line) contact can easily be obtained by crowning. Owing to the simple geometry of their rack flanks, and to the fact that 4.3 Other power transmission applications their operation is insensitive to center distance variations, CC4- In much the same way, turboprop aircraft may furnish a gears, as well as CC1-gears (previously called CCA-gears and good example in which C-gears would also result in CCB-gears [28]) are the most promising of all C-gear types. considerable weight savings. Huge split-path marine transmissions would benefit from using C-gears as well. 4. APPLICATIONS OF C-GEARS It was only in 1943 that Wittmann [29] reported on a new 4.4 Gear pumps so-called Forster-toothing (CV5-gears according to the present In all the applications listed hitherto, C-gears were used as coding) and reproduced a photograph of a continuous-indexing, power transmission elements to step-up or step-down the twin-cutter machine built by a Swiss manufacturer for its rotational speed. However, C-gears, can be used in gear pumps generation. The photograph matches the drawings in a patent as well, where a fluid is being pumped as a result of the by Forster [30], which was assigned to that same company. rotation of a pair of meshing gears. Gear pumps are also used Since then, nothing could be seen on actual manufacturing of in pumping polymer melts due to their intrinsic preciseness in C-gears. Thus, despite the advantages that C-gears can promise, flow-rate control, and their potential for pumping highly they have not found widespread applications. This can be viscous fluids with satisfactory efficiency. Spur, helical, and attributed to the absence of a persisting need for exploiting herringbone gears are used in state-of-the-art melt pumps. these advantages spurred by an application that justifies their Helical and herringbone gears typically offer a smoother flow complexity. Nevertheless, C-gears hold a great potential for than spur gears, but the use of herringbone gears results in a being used in several applications where their merits outweigh defect in the extruded polymer foils known as a “center strip”.

9 Copyright © 2012 by ASME This defect manifests itself in the form of a longitudinally Journal of Mechanical Engineering Science, Vol. 219, pp. 709- running middle trace generated by the apex between the right- 726, 2005. hand and the left-hand halves of the herringbone gear. In 2003, [5] Böttcher, Paul, “Curved Teeth for Gear Wheels” GB C-gears were proposed for use in melt pumps by Witte [32], Patent 15,278, March 5, 1914. replacing herringbone gears to eliminate the above-mentioned [6] Stepanov, J. S., et al. “Method of Machining Spur defect due to the continuous geometry of the curved tooth Wheels with Circular Teeth”, RU Patent 2,147,976, April 27 , trace. It is worth mentioning that the gear-meshing 2000. characteristic that is indispensable to gear pumps is the line [7] Boor, Francis H., “Curved Tooth Gear and Pinion contact between meshing gear teeth. Hence, types of C-gears Wheels”, US Patent 2,248,158, July 8, 1941. with point contact such as CV2-gears, or point-line contact [8] Cantrell, Dan R., “Apparatus and Method for Cutting a such as CC2-gears are not suitable for incorporation in gear Radius Toothed Gear”, US Patent 3,492,916, February 3, 1970. pumps. [9] Tseng, Rui-Tang; and Tsay, Chung-Biau, “Mathematical Model and Undercutting of Cylindrical Gears with Curvilinear 5. CONCLUSIONS Shaped Teeth”, Mechanism and Machine Theory, Vol. 36, pp. Several types of longitudinally curved gears (C-gears) 1189-1202, 2001. have been proposed in the literature in the past 160 years, some [10] Sidorenko, Aleksandr K., et al. “Method of Cutting of which were reinvented several times during the course of Convex and Concave Sides of Arched Teeth of Cylindrical this period. Some of these disclosures presented detailed Toothed Wheels”, SU Patent 1,722,719, March 30, 1992. information pertaining to the gear tooth geometry, and some [11] Ishibashi, Akira, “The Characteristics of Circular-Arc- even delved in studying their meshing characteristics. Others Toothed Cylindrical Gears”, Bulletin of JSME, Vol. 9, No. 33, envisaged manufacturing processes and invented machine tools pp. 200-208, 1966. for generating their proposed curved teeth. Nevertheless, the [12] Dai, Yutang; Ariga, Yukinori; and Nagata, Shigeyoshi, absence of an application that would justify designing and “Study on a Cylindrical Gear with Curved Tooth Traces”, manufacturing C-gears led to discarding the idea immediately Proceedings of the Tenth World Congress on the Theory of after its proposal. Sometimes, the idea of C-gears would die for Machines and Mechanisms. Vol. 6, pp. 2337-2342, Oulu, some years—or even decades—only to be reinvented again. In Finland, 1999. the past few years, the subject of C-gears is clearly revived as [13] Koga, Tamotsu, “Method for Cutting Paired Gears evidenced by the increasing number of recent research papers Having Arcuate Tooth Traces”, US Patent 3,915,060, October and patents. 28, 1975. Despite this history, many engineers are not even aware of [14] Waguri, Akira, “Grinding Method and Grinding Head for the existence of C-gears. It is aimed that this work would better Grinding Tooth Surfaces of Gears with Circular Tooth Trace”, inform the engineering community of the potential applications US Patent 3,127,709, April 7, 1964. of C-gears, which would justify further R & D. It is deemed [15] Lee, C-K; and Chen, C-K, “Mathematical Models, that the development of the first commercial C-gear will be for Meshing Analysis and Transmission Design for a Robust rotorcraft transmissions or wind turbine gearboxes. It will not Cylindrical Gear Set Generated by Two Blade-Discs with be until the technology of designing and manufacturing C-gears Parabolic Cutting Edges”, Proceedings of the Institution of is fully developed that they can be encountered in other Mechanical Engineers, Part C: Journal of Mechanical applications to replace double-helical gears. Engineering Science, Vol. 218, pp. 1539-1553, 2004. The most promising types of the assortment of C-gear [16] Andrei, L., et al., “Numerical Simulation and Generation types are CC4-gears and CC1-gears that have constant pressure of Curved Face Width Gears., International Journal of Machine angle across their face. Further work needs to be done for Tools & Manufacture, Vol. 42, pp. 1-6, 2002. studying the manufacturing methods proposed so far. [17] Yamada, Silvio M.; Lee, Hong-Tao; and Vickers, Doug, “Gear Tooth of Circoid Shape,” JP Patent 2002,070,989, March REFERENCES 8, 2002. [1] Arafa, Hani A., “Mechanical Design Pitfalls”, [18] Zablonskij, K. I.; Chekin, B. M.; and Matsej, R. A., Proceedings of the Institution of Mechanical Engineers, Part C: “Cylindrical Involute Toothed Gearing with Arched Teeth and Journal of Mechanical Engineering Science, Vol. 220, pp. 887- Method of their Production”, SU Patent 987,232, January 7, 899, 2006. 1983. [2] Semple, Amzi C., “Rack and Pinion”, US Patent 5,647, [19] Litvin, Faydor L., et al. “Modified Involute Helical June 27, 1848. Gears: Computerized Design, Simulation of Meshing, and [3] Wanyan, Xueming; Wanyan, Zhihan; and Wanyan, Yan, Stress Analysis,” NASA CR 2003-212229, 2003. “Arc Helix Cylindrical Gear and Arc Rack”, CN Patent [20] Andrei, L, et al., “Experimental Assessment of Plastic 101,149,104 (B), September 7, 2011. 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10 Copyright © 2012 by ASME [21] Shurr, Charles H., “Method of Generating Gear-Teeth”, US Patent 1,355,919, October 19, 1920. [22] Farnum, William C., “Gear-Cutting Machine”, US Patent 1,373,956, April 5, 1921. [23] Wingqvist, Erik H., “Improvements in Gear Wheels and Method of Manufacturing Same”, GB Patent 113,966, July 4, 1918. [24] Sidorenko, Aleksandr K.; Naletov, Sergej P.; and Korotkov, Vyacheslav D., “Method of Machining Wheels with Curvilinear Shape of Involute Teeth”, SU Patent 1,526,935, December 7, 1989. [25] Lewis, Frank M., “Improvements in Cutting Toothed Gears”, GB Patent 155,181, December 6, 1920. [26] Mammano, B., “Improvements in or Relating to the Cutting of Gear Teeth”, GB Patent 462,709, March 15, 1937. [27] Inoue, Jin, “Improvements Relating to the Manufacture of Curved Tooth Involute Gears”, GB Patent 846,275, August 31, 1960. [28] Arafa, H. A.; and Bedewy, M., “Quasi-Exact-Constraint Design of Wind Turbine Gearing,” Proceedings of the ASME 2010 Power Conference, pp.607-616, Chicago, IL, USA, 2010. [29] Wittmann, H., “Leistungssteigerung im Getriebebau”, Maschinenbau/Der Betrieb, Vol. 22, No. 1, pp. 9-13, 1943. [30] Forster, Albert, “Machine for Cutting Curved Self- Conjugate Indentations”, US Patent 2,406,009, August 20, 1946. [31] Arafa, H. A.; and Bedewy, M., “C-Gears: a Novel Design Paradigm for Rotorcraft Transmissions,” Proceedings of the AHS/AIAA/SAE/RAeS 2010 International Powered Lift Conference (IPLC), Philadelphia, PA, USA, 2010; submitted to the Journal of the American Helicopter Society (AHS). [32] Witte, Reinhard, “Gearwheel pump has two intermeshing gear wheels each with curved toothed spline for smoother engagement to avoid centre strip”, DE Patent 10,148,476, April 30, 2003.