P1: GDZ/SPH P2: GDZ References CB672/Litvin CB672/Litvin-v1.cls February 27, 2004 2:52 References 793 Niemann, G., & Heyer, E. 1953. Investigations of Worm Gears. VDI 95, 141–157. Niemann, G. 1961. Novikov Gear System and Other Special Gear Systems for High Load Car- rying Capacity. VDI Berichte, 47. Novikov, M. L. 1956. USSR Patent No. 109,750. Nutbourne, A. W., & Martin, R. R. 1988. Differential Geometry Applied to Curve and Surface Design. Vol. 1. Chichester, West Sussex, England: Ellis Horwood Limited. Polder, J. W. 1991. Interference of Internal Gears. In Dudley’s Gear Handbook, P. Townsend, ed. New York: McGraw-Hill. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992. Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. New York: Cambridge University Press. Rashevski, P. K. 1956. Course of Differential Geometry, 4th ed. State Publishing House of Technical and Theoretical Literature, Moscow (in Russian). Saada, A., & Velex, P. 1995. An Extended Model for the Analysis of the Dynamic Behavior of Planetary Trains. ASME Journal of Mechanical Design, 117(June), 241–247. Schreier, G. 1961. Stirnrad-Verzahnung. Berlin: VEB Verlag “Technik.” Seol, I. H., & Litvin, F. L. 1996. Computerized Design, Generation and Simulation of Meshing and Contact of Worm-Gear Drives with Improved Geometry. Computer Methods in Applied Mechanics and Engineering, 138(1-4), 73–103. Shabana, A. A. 1989. Dynamics of Multibody Systems. New York: Wiley. Sheveleva, G. I. 1999. Theory of Formation and Contact of Moving Bodies. Moscow State Technological University Stankin, Moscow (in Russian). Smith, W. C. 1995. The Math of Noncircular Gearing. Gear Technology, 1(January/February). Stadtfeld, H. J. 1993. Handbook of Bevel and Hypoid Gears: Calculation, Manufacturing, and Optimization. Rochester, New York: Rochester Institute of Technology. Stadtfeld, H. J. 1995. Gleason Bevel Gear Technology – Manufacturing, Inspection and Op- timization, Collected Publications. Rochester, New York: The Gleason Works. Stosic, N. 1998. On Gearing of Helical Screw Compressor Rotors. Proceedings of IMechE, Journal of Mechanical Engineering Science, 212, 587–594. Strang, G. 1988. Linear Algebra and Its Application 3rd. ed. San Diego: Harcourt Brace Jovanovich. Suh, C. H., & Radcliffe, E. 1978. Kinematics and Mechanism Design. New York: Wiley. Townsend, D. P. 1991. Dudley’s Gear Handbook, 2nd ed. New York: McGraw-Hill. Vecchiato, D., Demenego, A., Argyris, J., & Litvin, F. L. 2001. Geometry of a Cycloidal Pump. Computer Methods in Applied Mechanics and Engineering, 190(18–19), 2309–2330. Velex, P., & Flamand, L. 1996. Dynamic Response of Planetary Trains to Mesh Parametric Excitations. ASME Journal of Mechanical Design, 118(March), 7–14. Vigodsky, M. Y. 1949. Differential Geometry. Leningrad: Gos Tech Isdat. Visual Numerics, Inc. 1998. IMSL Fortran 90 MP Library, v. 3.0. info@boulder.vni.com. Wells, C. F., & Shotter, B. A. 1962. The Development of ‘Circarc’ Gearing. London: AEI Engineering. Wildhaber, E. 1926. Helical Gearing. United States Patent Office, Patent No. 1,601,750. Wildhaber, E. 1956. Surface Curvature. Production Engineering, 27, 184–191. Willis, R. I. 1841. Principles of Mechanisms. London: Longman, Green and Co. Winter, H., & Jooman, J. 1961. Tools for Making Helical Circular Arc Spur Gears. VDI Berichte, 47. Zalgaller, V. A. 1975. Theory of Envelopes. Moscow: Publishing House Nauka (in Russian). Zalgaller, V. A., & Litvin, F. L. 1977. Sufficient Condition of Existence of Envelope to Contact Lines and Edge of Regression on the Surface of the Envelope to the Parametric Family of Sur- faces Represented in Parametric Form. Proceedings of Universities: Mathematics (in Russian), 178(3), 20–23. Zhang, Y., Litvin, F. L., & Handschuh, R. F. 1995. Computerized Design of Low-Noise Face- Milled Spiral Bevel Gears. Mechanism and Machine Theory, 30(8), 1171–1178. Zienkiewicz, O. C., & Taylor, R. L. 2000. The Finite Element Method, 5th ed. New York: Wiley. P1: GDZ/SPH P2: GDZ References CB672/Litvin CB672/Litvin-v1.cls February 27, 2004 2:52 794 P1: JsY cb672index CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:5 Index Addendum, 49, 276, 279, 296, 298, 300, 302, 392 circle, 279, 291, 292, 301, 309 cyclinder, 461, 463 cycloidal gear tooth profile, of, 356, 357, 370 Archimedes screw surface, 82, 93, 94, 96 Archimedes spiral, 27, 96, 272 Area of meshing, 606, 607, 609, 610 Axes of meshing, 128–135, 597, 598 Axial diametral pitch, 618 Axodes, 51, 52, 55, 56, 375, 383, 390, 394, 407, 408, 425, 426, 449, 456, 473, 478, 479, 492, 493, 510, 511, 513, 548, 679, 688 Backlash regulation (planetary gear trains), 697, 711–714, 716 Base circle, cylinder, 268, 273–275, 278, 281, 283, 287–289, 377, 379, 443, 445, 450, 461, 576 Bearing contact, 241, 260, 404, 435, 508, 528, 531, 633, 661, 665 localization, 406, 508, 512, 513 Bevel gear differential, 701, 702 Binormal, 154, 155, 164 Blade installation (for worm generation) ZA worms, of, 557 ZI worms, of, 578 ZK worms, of, 581 ZN worms, of, 561, 562, 567 Blade profile, 557, 558, 563, 581 Bobilier construction, 352, 353 Boundary conditions (for FEA), 257, 258, 260 Canonical design, 441, 446, 459–461, 467, 473 Carrier, 697 Center distance, 288, 289, 294, 296–298, 302 Centrodes circular gears, of, 38, 45, 49, 50, 115, 137, 211, 274, 280, 288, 289, 296, 298, 352 noncircular gears, of, 318, 322–334, 337, 338, 341, 343–346, 348, 349, 355, 356, 359, 360 Circular pitch, 390 Clearance, 609 Computer controlled machine (CNC), 746 Phoenix, 747 Star, 748, 749 Cone generation, of, 31 surface, 31 Contact (of surfaces) characteristic, 110, 202, 226 line (contact), instantaneous, 97, 110, 111, 128, 202, 226, 242, 249, 262, 404, 405, 408, 427, 440, 441, 444, 445, 450, 456, 508, 538, 656 point (contact), instantaneous, 202, 225, 230, 242, 252, 260, 261, 264, 405, 408, 414, 427, 440, 441, 444, 508, 525, 533, 538, 656 Contact ellipse, 234–240, 247–249 Contact lines, 387–389, 395, 553, 586, 590, 593, 599, 604, 605, 621, 622, 624–626, 721–723, 731 Contact path, 252, 263–265 Contact ratio, 292–294, 398 Coordinate measurement, 782–785 Coordinate transformation, 1, 5 Coordinate transformation, inverse, 4 Coordinates Cartesian, 78 curvilinear (Gaussian, for surfaces), 78 homogeneous, 1, 2, 14 Cradle, 631, 634 Cradle angle, 635, 653, 692 Crossing angle, 446, 448, 449, 452, 453, 458–464 Crowning double, 408, 417, 419, 429, 431, 485, 487, 496, 500, 611, 711, 716 longitudinal, 405, 406, 419, 429, 430, 477, 487, 496, 498, 611, 711 profile, 405, 406, 414, 419, 427, 477, 482, 487, 494, 498, 611, 711 795 P1: JsY cb672index CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:5 796 Index Curvature Gaussian, 189, 190, 193, 194, 529 geodesic, 166, 167, 172–174, 195 normal, 166–173, 177, 179, 180, 188, 190, 193, 198 planar curves, of, 68, 72–75, 77 principal (curvatures), 175, 180, 182, 184, 185, 188, 190, 191, 193, 198–200 principal directions, 175, 180–189, 192, 199, 200 radius of, 68, 72, 74 spatial curves, of, 156, 158, 159, 161, 163, 164, 170–173 Curvature matrix, 218, 231, 232 diagonalization, 231–234 Curvature relations normal curvatures of surfaces, of, 226–230 planar gearing, for, 204–218 principal curvatures, of, 218–225 Cutter mean radius, 637 Cutter point radius, 637, 639, 645, 693 Cutter point width, 637 Cutting ratio, 617, 618, 621 Cycloid extended, 27 generation, of, 27 ordinary, 27 shortened, 27 Cycloidal gearing, 350 Cycloidal gearing pump, 115, 116 Decomposition of motion, 202, 226 Dedendum, 49, 276, 279, 283, 296, 297, 302, 392 circle, 279, 283, 285, 300, 301 cyclinder, 461, 463 cycloidal gear tooth profile, of, 356, 357, 359, 370 Deviations (of real tooth surface), 787 determination, 762, 763 minimization, 763, 764, 782, 787 Diametral pitch, 49, 50 Direction cosines, 4 Directions (on surface) principal, 175, 180–189, 192, 199, 200 Disk-shaped milling cutter, 719, 720, 726, 727, 729, 730, 732 Displacement functions (for generation of non-circular gears) rack-cutter, by, 345–347 shaper, by, 348, 349 Double-enveloping worm gear drive, 614 modified gearing, 617, 620, 621, 625 unmodified gearing, 617–625 Eccentric gears, 289, 330 Edge contact, 260–266, 404, 408, 439–441, 443, 446, 448, 452, 459, 460 Edge of regression, 80 Efficiency (of a planetary gear train), 709–711 Elastic deformation, 234, 235, 238 Elliptical gears, 318, 319, 326–328 Envelope to family of contact lines, 112, 113, 118, 119 formation of branches of envelope, 114, 115, 117 necessary and sufficient conditions of existence, 112, 113 Envelope to family of surfaces, 97–100, 103, 105–110, 202, 203 necessary conditions of existence, 97, 98 sufficient conditions of existence, 97, 107–110 Envelope to two-parametric family of surfaces, 125, 126, 427, 428 necessary conditions of existence, 126 Epicycloid extended, 24, 25, 59, 63, 64, 66, 67, 77, 350, 352, 354 generation, of, 24, 351, 352 ordinary, 25, 67, 77, 350, 353 shortened, 25, 350 Equation of meshing, 98–101, 104, 106, 361–363, 366, 372, 383, 392, 583, 584, 586, 588, 593, 595, 596, 599, 603, 604, 620, 621, 641, 648, 650, 652, 653, 657, 695, 737, 740, 742, 754, 757–759, 785 Evolute, 268, 274, 341–343 Fillet, 119–124, 144 face gears, of, 508, 510, 524 internal gears, of, 305–307 Finger-shaped milling cutter, 718–720, 722–725 Finite element analysis, 257–259 Flyblade, 734 Force transmission, 399, 401 Forms (fundamental) first and second, 175–177, 180 Frenet equations, 71 Frenet trihedron, 69, 70, 73 Gaussian coordinates (surface parameters), 376 Gear ratio, 44, 45, 50–52, 58, 256, 287–289, 318, 322, 323, 327, 330, 407, 418, 549, 550, 552, 555, 604, 617, 697, 698, 701–703, 706, 709 Generating lines (for worm generation), 557, 563–567, 569, 570, 572, 614, 779 Geneva mechanism, 318 Geodesic line, 167, 169, 194–198 Grinding, 404 Grinding wheel, 579–581, 591–594, 597–600, 603, 641, 718, 729 Helical gears, 375 nonstandard, 375 parallel axes, with, 375, 376 profile angles, 385, 386, 402 standard, 375 P1: JsY cb672index CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:5 Index 797 Helicoid, 90–95, 376, 395, 547, 597, 603, 611, 731, 773, 774, 781 generalized, 601, 602, 604, 606 generation, of, 28, 90, 718, 721 Helix (on a surface), 163, 175, 376, 377, 380, 388, 396, 547–550, 553, 555, 569, 573, 574, 578, 680 Helix angle, 375, 390, 554 Herringbone teeth (helical gears), 401 Hobbing, 404 Honing, 441 Hyperboloids of revolution, 55–57, 679 Hypocycloid extended, 304, 307, 308, 311, 312, 314, 316, 352, 353, 355 ordinary, 353, 354 pseudo, 304, 307, 308 Hypoid gears, 679 face-hobbed, 685, 687, 689 face-milled, 685, 686, 690–692 formate-cut, 687, 690, 691 Instantaneous axis of rotation, 51, 99, 375, 383, 392, 395 Instantaneous center of rotation, 37, 44, 45, 48, 101, 120, 121, 137, 138, 144, 271, 272, 274, 275, 287, 290, 302, 350 Interference, 290 internal gears, of, 314–316 spur gears, of, 291 Internal gears, generation of axial, 305, 309 axial and step-by-step radial, 305 axial–radial (two-parametric), 305, 309 Involute conventional, 268–270, 272, 309, 310 extended, 270–272 function, 270 shortened, 270, 271 Involute curve extended, 25, 59, 76, 95, 572 generation, of, 25, 26 ordinary, 25, 63, 65, 76 shortened, 25 Jacobian, 252, 418, 452 Knots of meshing, 134–137 Lead angle, 552, 553, 734, 768 Limit contact normal (Wildhaber’s concept), 118, 119 Limiting position of rays (for a planar curve), 60, 61 at a regular point, 60 at a singular point, 60 Limiting position of rays (for a surface) at a regular point, 80 set of rays, 80 Line of action, 33, 126, 129, 252, 255–257, 287, 441, 445–448, 450, 459, 460 cycloidal gears, of, 357, 363, 366, 372 Local synthesis, 245–249, 628, 629, 631, 633, 641, 649, 650, 652, 656 Localized bearing contact, 416 Machine center to back, 642, 644, 692 Machine offset, 692 Machine root angle, 634, 642, 644, 692 Machine-tool settings (of spiral bevel gears), 635, 642–645, 650, 655 Matrix column, 1, 4 determinant, of, 4 direct, 5 identity, 5, 9 inverse, 4, 5, 15, 17, 19 rotational, 14, 15 row, 1 skew-symmetric, 9, 156, 166 symmetric, 222, 229, 231 translational, 14, 16, 18 transpose, 1 Misalignment, 242, 243, 249, 250, 262, 402, 403 Modified roll, 406, 429, 430, 478, 496, 498, 631–633, 644, 648, 649 Module, 279, 408, 430, 435, 454, 457, 461, 479 Motion generalized parameter, 39, 98, 100, 106, 109, 110, 115, 122 planar, 44 relative, 47, 102, 103, 129, 161, 202 rotational, 44 transfer, 102, 119, 161, 202 translational, 44 Noise, 404, 406, 416, 417, 419, 475, 477, 478, 485, 487, 627–631, 649 Noncircular gear applications combined, as, 320 crank-slider linkage, with, 319 Geneva mechanism, 318, 319 instruments, for, 321 liquid meter, for, 320 twisted, as, 321 Noncircular gear generation enveloping method by rack-cutter, 337–339 enveloping method by shaper, 341 master-gears, by application of, 335, 336 worm-gear master mechanism, by application of, 336 Nonstandard gears, 280, 284, 294, 295, 300, 302 general system, 295, 298 long–short addendum system, 295 P1: JsY cb672index CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:5 798 Index Normal, unit normal planar curve, to, 62, 63, 69–71, 74 principal, 154, 155 spatial curve, to, 154 surface, to, 81, 82, 165 Osculating circle, 68–70 plane, 153–157, 161–164, 168–173, 197 Oval gears, 318, 328 Overcentrode cycloidal gearing, 367–370 Overwire (ball) measurement, 769 Parabola vertex location parameter, 638 Path of contact, 252, 263–265, 514 Phase angle (of planet gears), 707–709 Pin gearing external, 359–364 internal, 365–367 Pitch angular, 292 axial, 553 base, 280, 293 circular, 278, 526 diametral, 274, 278, 279, 526 normal, 553 transverse, 553 Pitch circle, 49, 274–276, 278, 379 operating, 50 Pitch cones, 51, 511, 514, 679–681 design, 685–690 operating, 679, 680 Pitch cylinder, 390, 404, 420, 424, 425, 428, 449, 459–461, 488, 491, 495, 496 operating, 375, 418, 449, 459 Pitch diameter hob, of, 611 worm, of, 552, 553 Pitch line, 511, 512, 514 Pitch plane, 681–684, 686–689 Pitch point, 510, 511, 513, 514, 517, 518, 522, 681–684, 686 Pitch surfaces, 510, 511, 548 operating, 548 ordinary, 548 Planar curve parametric representation, 59, 63, 72 regular, 59, 60 representation by implicit function, 60 simple, 59, 60 Plane normal to surface, 154, 164, 169–172, 178, 180, 197, 198 osculating, 153–157, 161–164, 168–173, 197 rectifying, 154 surface parameters, of, 78 Plane of action, 389, 396, 443–445 Plunging (of a tool), 406, 419, 429–432, 438 Plunging motion, 422, 423, 487, 489, 490 Pointing face gears, of, 522–524 modified helical gears, of, 434, 435 Novikov-Wildhaber helical gears, of, 501 spur gears, of, 283 Points of planar curve regression, 60, 61, 66, 67 singular, 60, 61, 67 Points of surface elliptic, 190, 193, 194, 529 hyperbolic, 190–195, 529 parabolic, 191, 192, 194, 529 rectification, 153, 155 Poisson’s ratio, 440, 506, 542, 665 Predesigned parabolic function of transmission errors, 242–245, 477, 478, 482, 498, 500 Pressure angle, 288, 291, 298, 344, 409, 415, 418, 430, 434, 523 Profile angles, of worm axial, 555, 556, 586 normal, 555, 556, 581 transverse, 555, 556, 576, 581 Rack, rack-cutter circular arc profile, with, 145, 146 external pin gearing, for, 361, 364 internal pin gearing, for, 367 parabolic profile, with, 408, 410, 477, 479, 481, 482, 487, 502, 515, 525, 526 straight line profile, with, 105, 119, 273–276, 280–283, 408, 411, 525, 526 watch gearing, for, 359 Radial distance, 635 Radial setting, 692 Ratio of roll, 644, 648 Root’s blower, 369, 371 Rotation between crossed axes, 33, 52, 441 between intersected axes, 51, 52 between parallel axes, 35, 44, 441 Screw involute surface, 405, 443–445, 456 Screw motion, 28, 54, 414, 415, 421, 423, 428, 443, 444, 456, 488–490, 495, 496 instantaneous axis, of, 55 parameter, 28, 32, 55, 90, 94 Screw rotors compressors, of, 351 pump, of, 351 Screw surface generation, of, 32 Settings (or rack-cutter) conventional, 280–282 limiting, 280, 282 non-conventional, 280, 282, 283, 295 P1: JsY cb672index CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:5 Index 799 Shaper, 508, 512, 513, 527 edged top, with, 541–545 rounded top, with, 524, 541 Shaping, 404 Shaving, 441 Singularities (see undercutting), 103 Sliding base, 642, 644, 692 Spiral bevel gears, 627 face-milled generated, 627, 650, 666 formate-cut, 627, 633–635, 641–644, 650, 656 Spur involute gears, 267, 268, 273 generation by a hob, 276, 277 generation by a rack-cutter, 273–276 generation by a shaper, 278 Standard center distance, 50 Stress analysis crossed helical gears, of, 465–467 face gears, of, 541–546 modified helical gears, of, 435–440 Novikov-Wildhaber helical gears, of, 502–507 spiral bevel gears, of, 670–676 Surface normal, 6 point, 6 revolution, of, 29 spherical, 30 Surface of action, 112, 134, 389, 392, 395, 606 Surface points pseudosingular, 86, 88 regular, 80 singular, 80, 81 Surface representation by implicit function, 82 in parametric form, 78 Surface types cone, of, 82, 83, 88, 183, 185, 186 helicoid, 90–95 involute screw, 82, 83, 93, 95, 185, 547, 570, 573, 575, 576, 579, 588, 712, 724 regular, 82, 97, 107, 108, 110 revolution, of, 83–85, 185, 195, 196 ruled, 82, 83, 91–93, 191, 192, 195, 196 screw, 82, 149, 150 simple, 78, 82 spherical, 85–88, 171, 172, 182, 195, 196 Swivel angle, 692 Tangent plane “half” tangent plane, 80 surface regular point, at, 79–82, 153, 176, 179 Tangent, unit tangent “half” tangent (to planar curve), 60, 61, 66, 67 planar curve regular point, at, 60, 61, 63, 69, 70, 77 spatial curve, to, 153–155, 161, 164, 165, 167, 168, 170–172, 176, 180 Theorem Bonnet, 167, 196 Camus, 355, 356 Clariaut, 195 Dupin’s (for indicatrix), 193–195 Euler, 188, 189 Euler–Rodrigues, 8, 9, 14 Frenet, 69, 70, 73 Frenet-Serret, 156, 159 Gauss, 175, 189 Implicit function system existence, 106, 108, 251, 413, 418, 452, 484, 518 Lewis, 101 Meusnier, 171, 172 Rodrigues, 8, 182 Wildhaber, 118, 119 Zalgaller, 60, 80, 107, 110 Thickness of tooth, 279, 283, 285–287, 294, 295, 298, 299, 380, 392 Tilt angle, 692 Tooth contact analysis (TCA), 249–256, 531 Tooth thickness, 455, 457, 461–463, 473 Torsion spatial curve, of, 156–159, 162–164 surface, of, 166, 167, 194, 196–198, 227 Transfer of meshing, 289, 293 Transition point, 242 Transmission errors, function of, 404, 408, 415, 417–419, 430, 431, 498, 500, 533, 548, 606, 611, 612, 661, 665, 714, 716 integrated one, 716, 717 linear one, 242–244, 264, 404, 406, 416, 419, 431, 475, 487, 500, 548, 611, 629, 712 parabolic one, 496, 498, 500, 535, 611, 613, 628–631, 633, 667, 711, 712 predesigned parabolic one, 242–245, 406, 416, 429–431, 526, 535 Transmission function, 242–244 Trihedron planar curve, of, 69, 70 spatial curve, of, 153–155, 158, 162, 163 surface spatial curve, 164–166, 172 Undercutting, 103–107, 109, 110, 118, 126, 127 face gears, of, 519–522, 531, 539 helical gears, of, 396, 397 internal involute gears, of, 304, 307, 309, 311, 312 modified helical gears, of, 432–434 Novikov-Wildhaber helical gears, of, 500, 501 spur involute gears, of, 280, 281, 285 Unitless coefficient of face gears, 529 Unitless stress parameter, 545, 546 P1: JsY cb672index CB672/Litvin CB672/Litvin-v2.cls April 15, 2004 16:5 800 Index Vector free, 6 free, components of, 6 moment, 33, 54 sliding, 6, 33, 38, 54 unit, 1, 3, 4, 6, 8–11 Velocity relative, 33, 34, 39, 102, 128, 584, 603, 652, 657, 658, 660, 679, 683, 709, 755, 757, 785 relative angular, 304 sliding, 34–36, 38, 41, 42, 99, 102, 106, 203, 213, 701 sliding, matrix representation, 39 transfer, 102 Velocity ratio, 644, 648 Vibration, 404, 406, 416, 417, 419, 475, 485, 487, 606, 611, 629, 649 Watch gearing, 358 Width (of space), 279, 283, 295, 297–299, 377, 379, 392, 455, 473 Willis’ equation, 698 Worm surface types ZA (Archimedes) worm, 547, 557, 561, 573, 604, 605, 609, 610, 738 ZF (Flender) F-I worm, 590, 591, 594–597, 599, 743 ZF (Flender) F-II worm, 597–600, 744 ZI (Involute) worm, 547, 574–579 ZK (Klingelnberg) worm, 547, 581, 740 ZN (Convolute) worm, 547, 561–573 Young’s Modulus, 440, 506, 542, 665 . 30 2 Centrodes circular gears, of, 38 , 45, 49, 50, 115, 137 , 211, 274, 280 , 288 , 289 , 296, 2 98, 35 2 noncircular gears, of, 31 8, 32 2 33 4, 33 7, 33 8, 34 1, 34 3 34 6, 34 8, 34 9, 35 5, 35 6, 35 9, 36 0 Circular pitch, 39 0 Clearance,. 77, 35 0, 35 2, 35 4 generation, of, 24, 35 1, 35 2 ordinary, 25, 67, 77, 35 0, 35 3 shortened, 25, 35 0 Equation of meshing, 98 101, 104, 106, 36 1 36 3, 36 6, 37 2, 38 3, 39 2, 5 83 , 584 , 586 , 588 , 5 93, 595,. 679 Hypocycloid extended, 30 4, 30 7, 30 8, 31 1, 31 2, 31 4, 31 6, 35 2, 35 3, 35 5 ordinary, 35 3, 35 4 pseudo, 30 4, 30 7, 30 8 Hypoid gears, 679 face-hobbed, 685 , 687 , 689 face-milled, 685 , 686 , 690–692 formate-cut, 687 , 690,