P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 12.11 Evolute of Tooth Profiles 343 Figure 12.11.3: Gear centrode and evolutes of left- and right-hand sides of tooth profile. of the centrode; K is the curvature center at M; L and N are the current points of both profile evolutes. An approximate representation of tooth profiles of a noncircular gear is based on the local substitution of the gear centrode by the pitch circle of a circular involute gear. Figure 12.11.4 shows that the profiles of tooth number 1 can be approximately represented Figure 12.11.4: Local representation of a noncircular gear by the respective circular gear. P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 344 Noncircular Gears as tooth profiles of the circular involute gear with the pitch circle ρ A , where ρ A is the curvature radius of the noncircular gear centrode at point A. Similarly, tooth profiles of tooth number 10 are represented approximately as tooth profiles of the circular gear with the pitch circle of radius ρ B . The number of teeth of the substituting circular gear is determined as N i = 2Pρ i (12.11.2) where ρ i is the curvature radius of the centrode of the noncircular gear, and P is the diametral pitch of the tool applied for generation. 12.12 PRESSURE ANGLE Consider Fig. 12.12.1 which shows conjugate tooth profiles of the noncircular gears. The rotation centers of the gears are O 1 and O 2 ; the driving and resisting torques are m 1 and m 2 . The tooth profiles are in tangency at the instantaneous center of rotation I. The common normal to the tooth profiles is directed along n–n. The pressure angle α 12 is formed by the velocity v I 2 of driven point I and the reaction R (12) n that is transmitted from the driving gear 1 to driven gear 2. (Friction of profiles is neglected.) The pressure angle is determined with the following equations: (i) When the driving profile is the left-side one, as shown in Fig. 12.12.1, we have α 12 (θ 1 ) = µ 1 (θ 1 ) +α c − π 2 . (12.12.1) Figure 12.12.1: Pressure angle of noncircular gears. P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 Appendix 12.A Displacement Functions for Generation by Rack-Cutter 345 (ii) When the driving profile is the right-side one, we have α 12 (θ 1 ) = µ 1 (θ 1 ) −α c − π 2 . (12.12.2) The pressure angle α 12 (θ 1 ) is varied in the process of motion because µ 1 is not con- stant. The negative sign of α 12 indicates that the profile normal passes through another quadrant in comparison with the case shown in Fig. 12.12.1. APPENDIX 12.A: DISPLACEMENT FUNCTIONS FOR GENERATION BY RACK-CUTTER Gear Centrode The gear centrode is represented in polar form by angle θ 1 and function ρ 1 (θ 1 ) =|ρ 1 (θ 1 )|, where ρ 1 (θ 1 ) = O 1 M is the position vector of current point M (see Fig. 12.A.1). The tangent τ 1 to the centrode and ρ 1 (θ 1 ) form angle µ where tan µ(θ 1 ) = ρ 1 (θ 1 ) ρ θ ,  ρ θ = dρ 1 dθ 1  . (12.A.1) Point M 0 and angle µ 0 are determined with θ 1 = 0 and µ 0 = µ(0). The coordinate axis x 1 is parallel to τ 0 . The Cartesian coordinates of the centrode, the unit tangent τ 1 , and the unit normal n 1 are represented by the equations x 1 = ρ 1 (θ 1 ) cos(θ 1 − µ 0 ), y 1 =−ρ 1 (θ 1 ) sin(θ 1 − µ 0 ) (12.A.2) Figure 12.A.1: Representation of gear centrode, its unit tangent, and its unit normal. P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 346 Noncircular Gears where µ 0 = 90 ◦ − ψ. τ 1 = [cos(θ 1 + µ −µ 0 ) − sin(θ 1 + µ −µ 0 )0] T (12.A.3) n 1 = τ 1 × k 1 = [−sin(θ 1 + µ −µ 0 ) − cos(θ 1 + µ −µ 0 )0] T . (12.A.4) Rack-Cutter Centrode The centrode of the rack-cutter coincides with the x 2 axis [Fig. 12.10.2(b)]. The rack- cutter centrode and its unit normal are represented in S 2 by the equations ρ 2 = [u 001] T (12.A.5) n 2 = [0 − 10] T . (12.A.6) Coordinate Transformation Our next goal is to represent the gear and rack-cutter centrodes in fixed coordinate sys- tem S f . The coordinate transformation from S i (i = 1, 2) to S f is based on the following matrix equations [Fig. 12.10.2(b)]: r (i ) f = M fi ρ i , n (i ) f = L fi n i . (12.A.7) After transformations we obtain r (1) f = [ρ 1 (θ 1 ) cos q − ρ 1 (θ 1 ) sin q + y (O 1 ) f 01] T . (12.A.8) Here, q = θ 1 − µ 0 − φ 1 , where φ 1 is the angle of gear rotation; y (O 1 ) f = (O f O 1 ) · j f ; n (1) f = [−sin δ − cos δ 0] T (12.A.9) where δ = θ 1 + µ −µ 0 − φ 1 . r (2) f = [u + x (O 2 ) f 001] T (12.A.10) where x (O 2 ) f = (O f O 2 ) ·i 2 . n (2) f = [0 − 10] T . (12.A.11) Equations of Centrode Tangency The gear and rack-cutter centrodes are in tangency at any instant. Thus r (1) f − r (2) f = 0 (12.A.12) n (1) f − n (2) f = 0. (12.A.13) Equations (12.A.12) and (12.A.13) yield a system of only three independent scalar equations because |n (1) f |=|n (2) f |=1. These equations are ρ 1 (θ 1 ) cos q −u − x (O 2 ) f = 0 (12.A.14) −ρ 1 (θ 1 ) sin q + y (O 1 ) f = 0 (12.A.15) −sin δ = 0, − cos δ =−1. (12.A.16) P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 Appendix 12.A Displacement Functions for Generation by Rack-Cutter 347 Equations (12.A.16) yield φ 1 (θ 1 ) = θ 1 + µ −µ 0 (12.A.17) where µ = arctan(ρ 1 (θ 1 )/ρ θ ), µ 0 = arctan(ρ 1 (0)/ρ θ ). Transformations of Eqs. (12.A.14) are based on the following considerations: (i) At the start of motion we have φ 1 = 0, θ 1 = 0, origins O f and O 2 coincide with each other, and x (O 2 ) f (0) = 0. Then, we obtain ρ 1 (0) cos µ 0 = ρ 0 cos µ 0 = u 0 (12.A.18) where u 0 determines the initial position of the point of tangency of the centrodes on the x 2 axis. (ii) The displacement of the point of tangency of the centrodes along the rack-cutter centrode is s 2 = u − u 0 . (12.A.19) (iii) The displacement of the point of tangency along the gear centrode is determined as s 1 =  θ 1 0 ds 1 (θ 1 ) =  θ 1 0  dx 2 1 + dy 2 1  1/2 =  θ 1 0 ρ 1 (θ 1 ) sin µ dθ 1 . (12.A.20) Due to pure rolling of the centrodes, we have that s 1 = s 2 and u − u 0 = u − ρ 0 cos µ 0 =  θ 1 0 ρ 1 (θ 1 ) sin µ dθ 1 . (12.A.21) (iv) Equation (12.A.17) yields q = θ 1 − µ 0 − φ 1 =−µ. (12.A.22) (v) Equations (12.A.14), (12.A.21), and (12.A.22) yield the following final expression for x (O 2 ) f : x (O 2 ) f (θ 1 ) = ρ 1 (θ 1 ) cos µ −ρ 0 cos µ 0 − s 1 . (12.A.23) Similar transformations of Eq. (12.A.15) yield y (O 1 ) f (θ 1 ) =−ρ 1 (θ 1 ) sin µ. (12.A.24) Computational Procedure The final system of displacement equations is φ 1 (θ 1 ) = θ 1 + µ − µ 0 x (O 2 ) f = ρ 1 (θ 1 ) cos µ − ρ 0 cos µ 0 − s 1 (θ 1 ) y (O 1 ) f =−ρ 1 (θ 1 ) sin µ. (12.A.25) Here, µ = arctan  ρ 1 (θ 1 ) ρ θ  ,ρ θ = dρ 1 dθ 1 , s 1 (θ 1 ) =  θ 1 0 ρ 1 (θ 1 ) sin µ dθ 1 . P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 348 Noncircular Gears Generally, s 1 (θ 1 ) can be determined by numerical integration. Equation system (12.A.25) is used for the numerical control of motions of the cutting machine (or for the design of cams if a mechanical cutting machine is used). APPENDIX 12.B: DISPLACEMENT FUNCTIONS FOR GENERATION BY SHAPER The applied coordinate systems S 1 , S 2 , and S f are rigidly connected to the gear being generated, the shaper, and the frame of the cutting machine, respectively. The orienta- tion of these coordinate systems at the initial position is as shown in Fig. 12.B.1. The gear centrode and its unit normal are represented in S 1 by Eqs. (12.A.2) and (12.A.4), respectively. The shaper centrode is a circle of radius ρ 2 , and this centrode and its normal are represented in S 2 by the equations ρ 2 (θ 2 ) = ρ 2 [sin θ 2 − cos θ 2 01] T (12.B.1) n 2 = [sin θ 2 − cos θ 2 0] T . (12.B.2) The coordinate transformations from S 1 and S 2 to S f (Fig. 12.B.2) are based on the following equations: ρ (i ) f (θ i ,φ i ) = M fi ρ i (θ i ,φ i )(i = 1, 2) (12.B.3) n (i ) f (θ i ,φ i ) = L fi n i (θ i ,φ i )(i = 1, 2). (12.B.4) Figure 12.B.1: Coordinate systems applied for generation by a shaper. P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 Appendix 12.B Displacement Functions for Generation by Shaper 349 Figure 12.B.2: Derivation of displacement functions for generation by a shaper. The centrodes of the gear and the shaper are in tangency with each other at point I, represented as [ 0 − ρ 2 01] T . Equations of centrode tangency provide the following displacement functions: φ 1 (θ 1 ) = θ 1 + µ − µ 0 (12.B.5) x (O 1 ) f (θ 1 ) =−ρ 1 (θ 1 ) cos µ (12.B.6) y (O 1 ) f (θ 1 ) =−ρ 2 − ρ 1 (θ 1 ) sin µ (12.B.7) φ 2 (θ 1 ) = 1 ρ 2  θ 1 0 ρ 1 (θ 1 ) sin µ dθ 1 . (12.B.8) Displacement functions (12.B.5)–(12.B.8) enable us to execute the motions of the shaper and gear being generated in the process for generation. P1: GDZ/SPH P2: GDZ CB672-13 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:36 13 Cycloidal Gearing 13.1 INTRODUCTION The predecessor of involute gearing is the cycloidal gearing that has been broadly used in watch mechanisms. Involute gearing has replaced cycloidal gearing in many areas but not in the watch industry. There are several examples of the application of cycloidal gearing not only in instruments but also in machines that show the strength of positions that are still kept by cycloidal gearing: Root’s blower (see Section 13.8), rotors of screw compressors (Fig. 13.1.1), and pumps (Fig. 13.1.2). This chapter covers (1) generation and geometry of cycloidal curves, (2) Camus’ theorem and its application for conjugation of tooth profiles, (3) the geometry and design of pin gearing for external and internal tangency, (4) overcentrode cycloidal gearing with a small difference of numbers of teeth, and (5) the geometry of Root’s blower. 13.2 GENERATION OF CYCLOIDAL CURVES A cycloidal curve is generated as the trajectory of a point rigidly connected to the circle that rolls over another circle (over a straight line in a particular case). Henceforth, we differentiate ordinary, extended, and shortened cycloidal curves. Figure 13.2.1 shows the generation of an extended epicycloid as the trajectory of point M that is rigidly connected to the rolling circle of radius r . In the case when generating point M is a point of the rolling circle, it will generate an ordinary epicycloid, but when M is inside of circle r it will generate a shortened epicycloid. Point P of tangency of circles r and r 1 is the instantaneous center of rotation. The velocity v of point M of the rolling circle is determined as v = ω × PM. (13.2.1) Vector v is directed along the tangent to the cycloidal curve  being generated, and PM is directed along the normal to  at point M. There is an alternative approach for generation of the same curve . The generation is performed by the rolling of circle r  over the circle r  1 . The same tracing point M is rigidly connected now to circle r  . The new instantaneous center of rotation is P  which is determined as the point of intersection of two straight lines: (i) the extended straight 350 P1: GDZ/SPH P2: GDZ CB672-13 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:36 Figure 13.1.1: Screw rotors of a compressor. Figure 13.1.2: Screw rotors of a pump. Figure 13.2.1: Generation of extended epicycloid. 351 P1: GDZ/SPH P2: GDZ CB672-13 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:36 352 Cycloidal Gearing line PM – the normal to the generated curve, and (ii) line O  P  that passes through point O 1 and is drawn parallel to OM. Point O 1 is the common center of circles r 1 and r  1 . Point O  is a vertex of parallelogram OMO  O 1 and is also the center of circle r  . The radii r  and r  1 of the circles used in the alternative approach are determined with the equations r  = a r 1 +r r (13.2.2) r  1 = a r 1 r (13.2.3) where a =| OM|. The segments |PM| and |P  M| are related with the equation PM P  M = r r 1 +r . (13.2.4) The velocity v of generating point M is the same in both approaches if the angular velocities ω P and ω P  are related as ω P ω P  = r 1 +r r . (13.2.5) The Bobilier construction [Hall, 1966] enables us to determine the curvature center C of the generated curve. The theorem states: Consider as known the centers of curvature of two centrodes and the centers of curvature of two conjugate profiles that are rigidly connected to the respective centrodes. Draw two straight lines such as those that interconnect the curvature center of the respective centrode and the curvature center of the profile rigidly connected to the centrode. These two straight lines intersect at point K of line PK that passes through the instantaneous center of rotation P and is perpendicular to the common normal to the conjugate profiles. In our case, one of the mating profiles is the tracing point M, and the other mating profile is the generated extended epicycloid. The determination of curvature center C of the extended epicycloid in accordance with the Bobilier construction is based on the following procedure (Fig. 13.2.1): Step 1: Identify as given (a) the centers of curvature O and O 1 of the centrodes r and r 1 , (b) point M is one of the mating profiles that is rigidly connected to centrode r , (c) point P is the point of tangency of the centrodes r and r 1 , and (d) the normal PM to the generated curve. Step 2: Draw straight line MO that interconnects points M and O. Step 3: Draw through point P line PK that is perpendicular to the normal PM. Step 4: It is evident that the sought-for center of curvature of the extended epicycloid is point C. The two straight lines OM and O 1 C intersect each other at point K . Step 5: The Bobilier construction can be similarly applied for the alternative method of generation of the extended epicycloid where the rolling centrodes are the circles of radii r  1 and r  (Fig. 13.2.1). The two straight lines O  M and O 1 C intersect each other at point K  of the straight line P  K  ; line P  K  passes through point P  and is perpendicular to the curve normal P  M. Figure 13.2.2 shows the generation of an extended hypocycloid by two alternative approaches. In this case it is necessary to take (r 1 −r ) instead of (r 1 +r ) in equations [...]... CB6 72- 13 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0 :36 13 .2 Generation of Cycloidal Curves 35 3 Figure 13 .2. 2: Generation of extended hypocycloid that are similar to ( 13 .2. 2), ( 13 .2. 4), and ( 13 .2. 5) The Bobilier construction has been applied in this case as well to illustrate the geometric way of determining the curvature center C for the extended hypocycloid Figures 13 .2. 3 and 13 .2. 4... X1 = −r 1 sin φ1 , Y1 = −r 1 cos φ1 ( 13. 5 .3) P1: GDZ/SPH CB6 72- 13 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 3 62 Figure 13. 5 .3: Applied coordinate systems Figure 13. 5.4: For derivation of equation of meshing 0 :36 Cycloidal Gearing P1: GDZ/SPH CB6 72- 13 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0 :36 13. 5 External Pin Gearing 36 3 Thus, −r 1 cos φ1 + r 1 + ρ cos θ −r 1... the equations x1 = 2 sin 2 , y1 = r 1 − 2 cos 2 (90◦ ≥ 2 ≥ 0) ( 13. 6 .3) (ii) Curve II is represented in S 1 by the equations 2 ≥ 90◦ , 2 = 2( 2 − 90◦ ), φ1 = r2 2 r1 x1 = −r 2 sin(φ1 − 2 ) + (r 2 − r 1 ) sin φ1 + 2 sin [ 2 + (φ1 − 2 )] ( 13. 6.4) y1 = r 2 cos(φ1 − 2 ) − (r 2 − r 1 ) cos φ1 − 2 cos [ 2 + (φ1 − 2 )] Equations ( 13. 6.4) represent the right-side profile of gear 1 teeth Similarly,... equation rt ( 2 , 2 ) = Mt2 ( 2 )r2 ( 2 ), ( 13. 6.5) which yields 2 ≥ 90◦ , 2 = 2( 2 − 90◦ ) xt = −r 2 ( 2 − sin 2 ) + 2 sin( 2 − 2 ) ( 13. 6.6) yt = −r 2 (1 − cos 2 ) − 2 cos( 2 − 2 ) Axis yt is the axis of symmetry of the tooth profile of the rack-cutter 13. 7 OVERCENTRODE CYCLOIDAL GEARING Basic Idea The term “overcentrode” means that the gear teeth are displaced with respect to the gear centrodes... Figure 13. 5.5: Gear 2 tooth profile (ii) Curve II is represented by the equations θ > 90◦ , φ1 = 2( θ − 90◦ ), φ 2 = φ1 r1 r2 r2 (θ, φ1 ) = M21 (φ1 , 2 )r1 (θ ) ( 13. 5.8) Axis y2 is the axis of symmetry of gear 2 space (Fig 13. 5.5) Using Eqs ( 13. 5.8), we obtain after transformations θ > 90◦ , φ1 = 2( θ − 90◦ ), φ 2 = φ1 r1 r2 x2 = r 1 sin(φ1 + 2 ) − (r 1 + r 2 ) sin 2 − ρ sin [θ − (φ1 + 2 )] ( 13. 5.9) y2... 2 ) y f = r 2 cos 2 − 2 cos( 2 − 2 ) ( 13. 6 .2) It is easy to verify that if 2 = 0, the line of action is a circular arc of radius r 2 P1: GDZ/SPH CB6 72- 13 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0 :36 13. 7 Overcentrode Cycloidal Gearing 36 7 Pinion Tooth Profile The pinion tooth profile is formed by two curves, I and II : (i) Curve I in S 1 is an arc of the circle of radius 2 and. .. rigidly connected to gear 2 epicycloids P α and Pβ Figure 13. 5.1: Pin gearing P1: GDZ/SPH CB6 72- 13 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0 :36 13. 5 External Pin Gearing 36 1 Figure 13. 5 .2: For conjugation of tooth profiles for pin gearing Step 2: We can imagine now that the conjugate pinion -gear tooth profiles are (i) point P of the pinion, and (ii) the two branches P α and Pβ of the epicycloid... centrode 3 rolls over the gear centrodes 1 and 2 Centrodes 3 and 1 are in external tangency and centrodes 3 and 2 are in internal tangency Point P of auxiliary centrode 3 generates in coordinate system S 1 rigidly connected to gear 1 the epicycloid P α as the profile of the addendum of gear 1 P1: GDZ/SPH CB6 72- 13 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0 :36 13. 4 Camus’ Theorem and Its... 13. 6 .2) : (i) Solution 1 provides any value of 2 at the position 2 = 0 The gear 2 pin of circle of radius 2 copies the same circle in coordinate system S 1 (ii) Solution 2 provides the relation 2 = 2( 2 − 90◦ ), 2 > 90◦ ( 13. 6.1) Line of Action The current point of the line of action is M (Fig 13. 6 .2) , determined in S f by the equations 2 ≥ 90◦ , 2 = 2( 2 − 90◦ ) x f = r 2 sin 2 + 2 sin( 2. .. 13 .2. 4: Generation of ordinary hypocycloid 13. 3 EQUATIONS OF CYCLOIDAL CURVES Extended Epicycloid The position vector O1M (Fig 13. 3.1) is represented as O1M = O1O + O M ( 13. 3.1) ψr = φr 1 ( 13. 3 .2) Due to pure rolling, we have Figure 13. 3.1: For derivation of equations of extended epicycloid P1: GDZ/SPH CB6 72- 13 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0 :36 13. 4 Camus’ Theorem and . P1: GDZ/SPH P2: GDZ CB6 72- 12 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0: 32 12. 11 Evolute of Tooth Profiles 34 3 Figure 12. 11 .3: Gear centrode and evolutes of left- and right-hand sides of. GDZ CB6 72- 13 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0 :36 13 .2 Generation of Cycloidal Curves 35 3 Figure 13 .2. 2: Generation of extended hypocy- cloid. that are similar to ( 13 .2. 2), ( 13 .2. 4),. to gear 2 epicycloids P α and Pβ. Figure 13. 5.1: Pin gearing. P1: GDZ/SPH P2: GDZ CB6 72- 13 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0 :36 13. 5 External Pin Gearing 36 1 Figure 13. 5 .2: