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

Transcription

1 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 DETC MANUFACTURABILITY AND VIABILITY OF DIFFERENT C-GEAR TYPES: A COMPARATIVE STUDY Hani A. Arafa Professor Department of Mechanical Design and Production Engineering, Cairo University, Cairo, Egypt hani_arafa41@yahoo.com Mostafa Bedewy Assistant Lecturer Department of Mechanical Design and Production Engineering, Cairo University Currently at the University of Michigan, Ann Arbor, MI, USA mbedewy@umich.edu ABSTRACT The family of cylindrical parallel-axis involute gearing currently include spur, helical and double-helical gears, all having straight tooth traces in the developed pitch plane. However, gears with curved tooth traces have also been proposed. One of the obvious merits of this configuration is the insensitivity to shaft misalignment. Although this and other merits of gears with lengthwise curved teeth (C-gears) were highlighted, they have never been mass-manufactured. Many types and shapes of C-gears have been envisioned, a dozen or so, but the particular merits and demerits of each type were 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 inclination between the direction of their tooth trace and the gear blank axis. Helical gears run quieter than spur gears and have a higher load carrying capacity because the axial ratio introduced by the helix angle results in the simultaneous meshing of more teeth (higher overall ratio). Moreover, helical teeth are characterized by uniform wear as opposed to spur gears. This is attributed to their meshing characteristics that dictate an inclination of the progressive lines between the involute helicoidal flanks in mesh. This feature also renders helical gears favorable running-in characteristics. However, due to the helix angle of helical gear teeth, an unwanted axial thrust force component is developed in addition to the tangential and radial force components that normally act on the tooth surface. Fig. 1. C-gears as members of the family of cylindrical parallel-axis involute gears 1 Copyright 2012 by ASME

2 Fig. 2. Types of parallel-axis involute gearing: (a) spur, (b) helical, (c) double-helical gears 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 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 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 doublehelical 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 Fig. 3: Double-helical gears: (a) without central recess; herringbone, (b) with central recess 2. LONGITUDINALLY CURVED TOOTH GEARS Although all the parallel-axis involute gears discussed so far (spur, helical and double-helical gears) have straight tooth traces in the developed pitch plane, gears with curved tooth traces traces have also been suggested several times in the past 160 years, or so. Cylindrical gears with curved tooth traces in the form of circular or other closely similar curves, shown 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 was documented. Because it was treated as a new invention, almost each time it was proposed, many names were given to describe the curvature of the tooth trace. This fact made it even more difficult for researchers to find and cite previous related work. Table 1 summarizes most of these names that are too many to be listed in the list of keywords of any publication. The simplest and most straightforward name is Gears with longitudinally/lengthwise curved teeth (abridged to Cgears ). This short name was recently coined by Arafa [4] for 2 Copyright 2012 by ASME

3 collectively designating all gear types belonging to this category. By closely inspecting the literature on C-gears published so far, one can categorize the available references into three main groups. The first category includes those references (mostly patents) having a more descriptive nature with little or no detailing for the gear generation possibilities or the respective gear tooth geometry. A recent example for this category is the patent granted to Yamada et al. in 2002 [17]. As a result of their vagueness, most of these patents are not even holding any intellectual property rights, as they are just documenting the century-and-a-half-old idea of making gears with lengthwise curved tooth and merely highlighting their merits. The second category comprises references with elaboration on the machining details and kinematics without correlation with the resulting gear tooth geometry and their meshing characteristics such as the patent awarded to Zablonskij et al. [18]. In these documents either new machine tools are designed, or modifications to existing machines are proposed. The third category, which is the scarcest of all, consists of detailed research papers/patents in which geometric features of the proposed gear tooth are discussed in connection with their fabrication methods such as the work of Ishibashi [11]. The early work of Ishibashi [11] published in 1966 can be considered the oldest research paper found on C-gears. Research papers on C-gears became more abundant with the beginning of the third millennium, when several research groups from Japan [12], the Republic of China [15], Romania and the United Kingdom [16], and Egypt [4] conducted research on C-gears and started publishing their work. This indicates that the topic is becoming a rather appealing research topic. Owing to the contemporary dynamicity and advancement of scientific directories and search engines, cross-citing between the above mentioned scholarly publications is a recent trend in the amassed literature on C-gears. In spite of all these references, C-gears have not yet been actually incorporated in commercial products. C-gear is a generic manifestation for any gear geometry involving a curved tooth trace. Several variations were suggested throughout the years, with the only common attribute of having lengthwise curved tooth. The first research paper to bring all these gear forms together and categorize them was published in 2005 by Arafa [4]. In this reference, C-gears were classified, according to the variation of angle across the face width. The name CV-gears was suggested for designating gears with variable angle, and CC-gears for designating gears with constant angle. In that article, eleven gear forms were scrutinized, discussing their machining methods, in connection with their consequential geometric features, and with correlation to their meshing characteristics. Table 1: Previously proposed names for C-gears Name Reference Comments Gears With Teeth Cylindrical Gears With Circular Teeth Gear Radius Gear Tooth Toothed Cylindrical Gears With Curvilinear Shaped Teeth Cylindrical Gear With Arched Teeth Circular-Arc- Toothed Cylindrical Gears Cylindrical Gear With Tooth Traces Gear Arcuate Traces Gear Circular Traces With Tooth With Tooth Cylindrical Gears With Convex- Concave Tooth Traces Face Width Gears Circoid Gear Böttcher [5] Stepanov et al. [6] Boor [7] Cantrell [8] Tseng and Tsay [9] Sidorenko et al. [10] Ishibashi [11] Dai et al. [12] Koga [13] Waguri [14] Lee and Chen [15] Andrei al. [16] et Yamada et al. [17] This name refers to the shape of the teeth as curved; however, it is rather ambiguous, as it does not state whether this curved shape is the tooth profile, or it is in the longitudinal direction. This name is more specific, but it may confuse the reader with circular arc (conformal) gears. This name is ambiguous as well. This name is ambiguous as well. This name is ambiguous as well. This name is ambiguous as well. This name is descriptive enough, but may confuse the reader with circular arc (conformal) gears This name is linguistically accurate for describing this type of gears, but this name refers to the shape of the tooth trace in the developed pitch plane. This name also refers to the shape of the tooth trace in the developed pitch plane. This name is more specific, but it also refers to the shape of the tooth trace in the developed pitch plane. This name refers to the shape of the tooth trace in the developed pitch plane, although a curved line cannot be referred to as being convex or concave. This name is not accurate because, by definition, the face width of a cylindrical gear is a distance measured along the axis of the cylindrical gear blank The word circoid cannot be found as an entry to any dictionary. Most probably, The authors just joined together the prefix circ- with the suffix -oid to create a new word to describe this type of gear. (They published their patent in Japanese and in German, nevertheless, the English title and abstract are found on the website of the European patent office < >) C-Gears Arafa [4] This name was suggested as a short name for all cylindrical gears with longitudinally (or lengthwise) curved teeth 3 Copyright 2012 by ASME

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

5 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 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, thickness, and symmetry of the tooth trace in the developed pitch plane; the cutting rack flank surface, and its profile in the side ; and the tooth profile, its whole depth, and conjugacy. On the other hand, the category of meshing characteristics combined with the category of manufacturing details comprises the base surface, the surface of action, tooth, the complementarity of the generating racks, indexing, number of cutter heads and the number of cutting cycles, the generating rolling surface, and the cutter head inclination to the cutter head axis. The final category highlights the merits such as the possibility of finish-grinding, crowning, and profile shifting; favorable cutting conditions, and the maximum number of teeth to be cut; insensitivity to center distance variation; tolerance to misalignment; and interchangeability. The essential nomenclature needed for this comparison is shown on a C-rack in Fig. 5. Fig. 5. C-rack nomenclature (drawing with a tooth trace inclination at the side of 31 and its radius of curvature to module ratio of 15) 3.2 Comparison results The results of this comparative study are presented herein in a tabulated format. The abbreviations in reference [4] are adopted, viz. CV for gears with a angle that varies across the face, and CC for gears with a constant angle. Table 2 presents a detailed comparison of the different geometric features of each type of the seven CV-gears and six CC-gears. It is noteworthy that the first three columns in this Table refer to the tooth trace geometry in the developed pitch plane. In addition, further gear meshing characteristics are compared in Table 3 alongside some manufacturing details. Finally, all thirteen types are judged in a merit-based comparison, which is given in Table 4. 5 Copyright 2012 by ASME

6 Table 2: Comparing the tooth geometries of C-gears Name Oldest reference Tooth trace symmetry Tooth trace(s) Tooth thickness Rack tooth surface(s) Side-plane rack profile Profile adhering to involute Whole depth Conjugacy CV1 Böttcher [5] Identical circular Constant in all transverse Identical cones Hyperbola midplane Constant CV2 Shurr [21] Circular with Decreases towards side radii that differ by a (constant normal Dissimilar cones half-pitch tooth space) Hyperbola midplane Constant midplane CV3 Farnum [22] Circular with radii that differ by a Decreases towards side half-pitch in one gear; constant Dissimilar cones (interchanged in the other gear between the two meshing gears Hyperbola midplane Constant CV4 Koga [13] Slightly elliptical Decreases towards side in one gear; constant Dissimilar cones in the other gear Hyperbola midplane; deviations Variable at side are larger than in CV3 CV5 Wingqvist [23] NO Prolate trochoidal Constant in all transverse Identical trochoidal cones Only in one plane close to the midplane Constant CV6(a) Suggested NO Prolate trochoidal Decreases towards side Dissimilar trochoidal cones Only in one plane close to the midplane Constant midplane CV6 Dai et al. [12] NO Prolate trochoidal Decreases towards side Dissimilar in one gear; trochoidal cones increases in the other gear Only in one plane close to the midplane Constant CC1 Cantrell [8] Identical circular Constant in all transverse Oblique cylinder Straight Constant CC2(b) Andrei al.[16] et Identical circular Nearly constant in the transverse Oblique cylinder Hyperbola midplane, Variable approximate in all others; pseudo-inv. midplane CC2 Sidorenko et al. [10] Nearly identical Approximately constant in circular all transverse Oblique cylinder Straight midplane, Variable approximate in all others; pseudo-inv. midplane CC3 Sidorenko et al. [24] Circular with different radii Decreases towards side Oblique cylinder midplane, Variable approximate in all others; pseudo-inv. midplane CC4 Lewis [25] Identical slightly elliptical Constant in all transverse Cylinder (simplest of all) Straight Variable CC5 Mammano [26] NO Nearly prolate trochoidal Constant in all transverse Trochoidal cylinder Constant 6 Copyright 2012 by ASME

7 Table 3: Comparing the characteristics and manufacture of C-gears Name Base surface Surface of action CV1 Barrel-shaped surface CV2 CV3 Two barrel-shaped surfaces: the smaller for convex N.A flanks, the other for concave flanks. Two barrel-shaped surfaces: the smaller for convex flanks, the other for concave flanks (opposite for the mating gear). CV4 Two barrel-shaped surfaces Warped CV5 CV6(a) CV6 Oppositely barrel- Unsymmetrical shaped surface Tooth Oppositely warped, symmetrical ruled surface, inflects about the pitch line Oppositely warped, symmetrical ruled surface, inflects about the pitch line Self-complementary only in the midplane (for Singleindexing each of the two nonidentical racks) Two unsymmetrical barrelshaped surfaces: the smaller N.A. for convex flanks, the other for concave flanks. Two unsymmetrical barrelshaped surfaces: the smaller for convex flanks, the other for concave flanks (opposite for the mating gear). CC1 Cylinder CC2(b) Barrel-shaped surface CC2 CC3 CC4 warped, un-symmetrical ruled surface, inflects about the pitch line Oppositely warped, un-symmetrical ruled surface, inflects about the pitch line Plane N.A Cylinder with a diameter appropriately smaller than N.A. the root diameter Cylinder with a diameter appropriately smaller than N.A. the root diameter Cylinder that may be larger than the root diameter for Plane small N CC5 Cylinder Plane Point Point Point-- line Point-- line Point Remarks 1. The generating rack is, by definition, complementary to a rack being cut; hence, if the generating rack is selfcomplementary, then the rack being cut is also selfcomplementary. However, a generating rack is said to be fully self-complementary if the rack profiles in all transverse plane complement themselves. Cutting/ generating racks Generating process (Indexing) Self-complementary only in the midplane Fully self-complementary Singleindexing Singleindexing Self-complementary only in the midplane (for Singleindexing each of the two nonidentical racks) Fully self-complementary Continuous (for each of the two indexing opposite hand racks) Self-complementary only in the midplane (for Continuous each of the two opposite indexing hand racks) Self-complementary only in the midplane (for Continuous each of the two opposite indexing hand racks) Self-complementary only in the midplane Self-complementary only in the midplane Self-complementary only in the midplane Fully self-complementary Singleindexing Singleindexing Singleindexing Singleindexing Fully self-complementary Singleindexing Fully self-complementary Continuous (each of the two opposite indexing hand racks) No. of cutter heads / No. of cycles Two / Two Generating rolling surface Pitch cylinder Pitch cylinder (male cutter) and Pitch cylinder (female cutter) (male cutter) and Pitch cylinder (female cutter) Two / Two or Two / One Pitch cylinder Pitch cylinder (male cutter) and Pitch cylinder (female cutter) Two / Two Cutter edge inclination to cutter head axis angle (φ m ) angle (φ m) angle (φ m ) (φ m -θ) for the outside male and female cutting edges; (φ m +θ) for other two edges angle (φ m ) angle (φ m) angle (φ m) Pitch cylinder Pressure angle (φ) base circle Single point; not edge Two / Two Base cylinder Single point; not edge Base cylinder Single point; not edge Two / Two Base cylinder Parallel Base cylinder Parallel 2. Single indexing (also called face milling) means that the cutting process is discrete and repeated for each tooth after indexing. On the other hand, continuous indexing (also called face hobbing) means that gear cutting is done continuously; no individually repeated indexing. 7 Copyright 2012 by ASME

8 Table 4: Comparing the merits and demerits of C-gears Name Crowning possibility Profile shifting possibility Favorable cutting conditions Can be finish ground Max. no. of teeth to be cut Tolerance to misalignment Insensitivity to center-distance variation Interchangeability CV1 To be investigated Limited Good Sensitive CV2 Compulsory crowning by a substantial amount To be investigated Limited Excellent Sensitive CV3 To be investigated More limited than in CV1 Good Sensitive NO CV4 To be investigated Unlimited Good Sensitive NO CV5 To be investigated NO Limited Moderate Sensitive NO (opposite hand) CV6(a) To be investigated NO Limited Excellent Sensitive NO (opposite hand) CV6 To be investigated CC1 NO Possible CC2(b) To be investigated CC2 CC3 Compulsory crowning Only substantial profile shifts are feasible Only substantial profile shifts are feasible NO (female cutter in one piece) NO (variable rake angle) NO (single point cutting) Limited CC4 Possible NO Limited Moderate Sensitive NO (opposite hand) NO Limited Good Insensitive NO Unlimited Moderate Insensitive Good; better Insensitive with crowning Limited Excellent Insensitive Limited to a small number Good; better Insensitive with crowning CC5 NO Possible NO Limited Moderate Insensitive NO (opposite hand) 3.3 Description of the added C-gear types The C-gears types that are added to the eleven types previously gathered by Arafa [4] are briefly described. CV6(a)-gears. These gears combine features of CV2- and CV5-gears; being generated by one male cutter in continuous indexing. The most significant advantage of this new type, in addition to the higher productivity of its manufacture, is the localization of gear tooth-surface (point ) that leads to a better capability of accommodating misalignments and avoiding edge loading. This type is suggested here to fill a void in the assortment of CV-gears. CC2(b)-gears. Andrei et al. [16] recently proposed CC2(b)- gears, which are based on the principle of tool-tip cutting. However, the authors refer to their cutting process as having a straight edge cutting. In fact, the bulk material removal may be done by the cutter edge, but the finishing is done by only one point on that edge (point cutting). Consequently, the straight cutting edge leads, in this case, to deviations from the involute towards the side, as a result of the conical rack shape formed by the cutter rotation. This deviation from the involute was pointed out along with the inferior surface quality (roughness) of these gears [20]. Another problem with this gear cutting process is the variation of tooth depth across the face width. The described cutting process dictates that the tool digs deeper in the middle of the tooth [16]. Also, the tooth becomes asymmetric in the side, which is attributed to the variation in rack inclination resulting from the conical shaped racks. Accordingly, line is not achieved and edge loading (due to interference) may develop if longitudinal teeth crowning is not imparted. 3.4 Discussion is achieved if the meshing gear teeth profiles are conjugate in all transverse. In this case, the tooth traces of the driving tooth flank and the driven tooth flank are identical. On the other hand, point is achieved if there exists a difference in curvature between the tooth traces in the developed pitch plane of the driving and the driven tooth flanks. Although point lowers the load carrying capacity of the tooth, it also greatly enhances the tolerance to misalignment. In addition to these two types of, the socalled point-line can be identified when the radii of 8 Copyright 2012 by ASME

9 curvature of the two conformal flanks are slightly different, so that the teeth theoretically have point, which becomes a line upon the slightest loading. CC2-gears furnish a good example for this type of, with the slight difference in the curved tooth radii being inherent to the errors dictated by using a rounded nose cutter. The limitation on the maximum number of teeth to cut arises from two different sources. In some cases, the relationship between the cutter head and gear blank diameters would cause the cutters, on their way round, to interfere with the gear blank near either extreme position of the cutter head at the start/end of the involute generating process as such (depending on whether an outside or inside cutter is used). Added to this, in the case of some CC-gears, there will be a constraint on the base circle to be larger than the root circle in order to be able to complete the generation. CC-gears have favorable characteristics over CV-gears due to their constant angle across the gear face width. The variation in angle makes CV-gears sensitive to center distance variations and impose tight tolerances on gear mounting. On the contrary, all CC-gears possess the merit of insensitivity to center distance variation, which is a genuine attribute of involute gearing. Moreover, profile shifting can be imparted to all CC-gears except for CC2 and CC3-gears, in which only a substantial amount of profile shift is practical. Due the complexity of the tooth flank geometry of CV-gears, the possibility of profile shifting needs to be investigated. This issue is considered beyond the scope of the present work. As shown from the above Tables, a CC4-gear has the simplest rack surface geometry, which is a patch of a cylindrical surface; it is the only type of C-gears with a describable, relatively simple, tooth surface geometry, which is called an involute tube. This tube is formed by the motion of all points on a circle to unfold off the base cylinder [27]. Although line is typically achieved between CC4-gears, localized (point-line) can easily be obtained by crowning. Owing to the simple geometry of their rack flanks, and to the fact that their operation is insensitive to center distance variations, CC4- gears, as well as CC1-gears (previously called CCA-gears and CCB-gears [28]) are the most promising of all C-gear types. 4. APPLICATIONS OF C-GEARS It was only in 1943 that Wittmann [29] reported on a new so-called Forster-toothing (CV5-gears according to the present coding) and reproduced a photograph of a continuous-indexing, twin-cutter machine built by a Swiss manufacturer for its generation. The photograph matches the drawings in a patent by Forster [30], which was assigned to that same company. Since then, nothing could be seen on actual manufacturing of C-gears. Thus, despite the advantages that C-gears can promise, they have not found widespread applications. This can be attributed to the absence of a persisting need for exploiting these advantages spurred by an application that justifies their complexity. Nevertheless, C-gears hold a great potential for being used in several applications where their merits outweigh all issues of concern. Following are some prospective application areas of C-gears in which their dexterous nature can lead to improved performance, extended life, and reduced weight. 4.1 Wind turbine gearboxes With the present state-of-the-art of horizontal axis wind turbines (HAWT) of multi-megawatt power ratings and rotor speeds in the vicinity of 10 rpm, the torque input to the gear box assumes values of multi MN.m; torques that could only be encountered in the propeller shafts of super tankers and aircraft carriers. Since a wind turbine gear box cannot nearly be as heavy as marine gearing, it has now become known that elastic deformations of the gear box casing, shafts, planet carriers, and the gear and pinion bodies themselves lead to unprecedented amounts of misalignment in the gear meshes. If not properly designed, this misalignment causes tooth-edge loading, which would ultimately lead to catastrophic failure. It has been proposed that the inherent self-aligning capability of C-gears can be utilized to enhance the reliability of the step-up transmissions employed in HAWTs [28]. In the midst of today s quest for reliable sources of renewable energy, any improvements in the design of such equipment would have a significant economic payoff. 4.2 Rotorcraft transmissions The use of C-gears has also been suggested as a replacement for double-helical gears incorporated in the final stage bull-gear of split-path transmissions, which are used in the main drive of helicopters [31]. This would lead to a substantial weight saving due to both the closer-to-unity loaddistribution factor, and the geometric continuity of the tooth as opposed to double-helical gears with a central recess. In addition, the self-aligning qualities of C-gears can lead to improved operating performance and lower noise. 4.3 Other power transmission applications In much the same way, turboprop aircraft may furnish a good example in which C-gears would also result in considerable weight savings. Huge split-path marine transmissions would benefit from using C-gears as well. 4.4 Gear pumps In all the applications listed hitherto, C-gears were used as power transmission elements to step-up or step-down the rotational speed. However, C-gears, can be used in gear pumps as well, where a fluid is being pumped as a result of the rotation of a pair of meshing gears. Gear pumps are also used in pumping polymer melts due to their intrinsic preciseness in flow-rate control, and their potential for pumping highly viscous fluids with satisfactory efficiency. Spur, helical, and herringbone gears are used in state-of-the-art melt pumps. Helical and herringbone gears typically offer a smoother flow than spur gears, but the use of herringbone gears results in a defect in the extruded polymer foils known as a center strip. 9 Copyright 2012 by ASME

10 This defect manifests itself in the form of a longitudinally running middle trace generated by the apex between the righthand and the left-hand halves of the herringbone gear. In 2003, C-gears were proposed for use in melt pumps by Witte [32], replacing herringbone gears to eliminate the above-mentioned defect due to the continuous geometry of the curved tooth trace. It is worth mentioning that the gear-meshing characteristic that is indispensable to gear pumps is the line between meshing gear teeth. Hence, types of C-gears with point such as CV2-gears, or point-line such as CC2-gears are not suitable for incorporation in gear pumps. 5. CONCLUSIONS Several types of longitudinally curved gears (C-gears) have been proposed in the literature in the past 160 years, some of which were reinvented several times during the course of this period. Some of these disclosures presented detailed information pertaining to the gear tooth geometry, and some even delved in studying their meshing characteristics. Others envisaged manufacturing processes and invented machine tools for generating their proposed curved teeth. Nevertheless, the absence of an application that would justify designing and manufacturing C-gears led to discarding the idea immediately after its proposal. Sometimes, the idea of C-gears would die for some years or even decades only to be reinvented again. In the past few years, the subject of C-gears is clearly revived as evidenced by the increasing number of recent research papers and patents. Despite this history, many engineers are not even aware of the existence of C-gears. It is aimed that this work would better inform the engineering community of the potential applications of C-gears, which would justify further R & D. It is deemed that the development of the first commercial C-gear will be for rotorcraft transmissions or wind turbine gearboxes. It will not be until the technology of designing and manufacturing C-gears is fully developed that they can be encountered in other applications to replace double-helical gears. The most promising types of the assortment of C-gear types are CC4-gears and CC1-gears that have constant angle across their face. Further work needs to be done for studying the manufacturing methods proposed so far. 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11 [21] Shurr, Charles H., Method of Generating Gear-Teeth, US Patent 1,355,919, October 19, [22] Farnum, William C., Gear-Cutting Machine, US Patent 1,373,956, April 5, [23] Wingqvist, Erik H., Improvements in Gear Wheels and Method of Manufacturing Same, GB Patent 113,966, July 4, [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, [25] Lewis, Frank M., Improvements in Cutting Toothed Gears, GB Patent 155,181, December 6, [26] Mammano, B., Improvements in or Relating to the Cutting of Gear Teeth, GB Patent 462,709, March 15, [27] Inoue, Jin, Improvements Relating to the Manufacture of Tooth Involute Gears, GB Patent 846,275, August 31, [28] Arafa, H. A.; and Bedewy, M., Quasi-Exact-Constraint Design of Wind Turbine Gearing, Proceedings of the ASME 2010 Power Conference, pp , Chicago, IL, USA, [29] Wittmann, H., Leistungssteigerung im Getriebebau, Maschinenbau/Der Betrieb, Vol. 22, No. 1, pp. 9-13, [30] Forster, Albert, Machine for Cutting Self- Conjugate Indentations, US Patent 2,406,009, August 20, [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, Copyright 2012 by ASME