P1: JTH CB672-11 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:24 11.3 Conditions of Nonundercutting 313 and N c (N 2 is given): r ac N c [ −N c sin(φ c + ) cos φ 2 + N 2 cos(φ c + ) sin φ 2 ] =−r a2 sin  w a2 2r a2  r ac N c [ N c sin(φ c + ) sin φ 2 + N 2 cos(φ c + ) cos φ 2 ] = r a2 cos  w a2 2r a2  . (11.3.11) The first guess for the solution of system (11.3.11) is based on considerations similar to those previously discussed: Step 1: Transforming equation system (11.3.11), we obtain cos 2 (φ c + ) = r 2 a2 −r 2 ac r 2 ac  N 2 N c  2 − 1  . (11.3.12) We take for the first guess N c = 0.8N 2 and obtain (φ c + ) from Eq. (11.3.12). Param- eter φ 2 is determined from Eq. (11.3.10). Step 2: Knowing N c , φ c , and φ 2 for the first guess, and using the subroutine for the solutions of equation system (11.3.11), we can determine the exact solution for Axial Generation Radial Generation 0 0 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200 No. of Gear Teeth No. of Shaper Teeth Figure 11.3.5: Design chart for pressure angle α c = 30 ◦ . P1: JTH CB672-11 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:24 314 Internal Involute Gears Table 11.3.1: Maximal number of shaper teeth Pressure angle Generation method Gear teeth Shaper teeth Axial 25 ≤ N 2 ≤ 31 N c ≤ 0.82N 2 − 3.20 α c = 20 ◦ Axial 32 ≤ N 2 ≤ 200 N c ≤ 1.004N 2 − 9.162 Two-parameter 36 ≤ N 2 ≤ 200 N c ≤ N 2 − 17.6 Axial 17 ≤ N 2 ≤ 31 N c ≤ 0.97N 2 − 5.40 α c = 25 ◦ Axial 32 ≤ N 2 ≤ 200 N c ≤ N 2 − 6.00 Two-parameter 23 ≤ N 2 ≤ 200 N c ≤ N 2 − 11.86 α c = 30 ◦ Axial 15 ≤ N 2 ≤ 200 N c ≤ N 2 − 4.42 Two-parameter 17 ≤ N 2 ≤ 200 N c ≤ N 2 − 8.78 N c . Computations based on the above algorithms allow us to develop charts for de- termination of the maximal number of shaper teeth, N c , as a function of N 2 and the pressure angle α c . An example of such a chart developed for axial generation and two- parameter generation is shown in Fig. 11.3.5. Table 11.3.1 (developed by Litvin et al. [1994]) allows us to determine the maximal number of shaper teeth for various pressure angles. 11.4 INTERFERENCE BY ASSEMBLY We consider that the internal gear with the tooth number N 2 was generated by the shaper with tooth number N c and the condition of nonundercutting was observed. Then, we consider that the internal gear is assembled with the pinion with the tooth number N 1 > N c . The question is what is the limiting tooth number N 1 that allows us to avoid interference by assembly. Henceforth, we consider two possible cases of assembly – axial and radial. Axial Assembly Axial assembly is performed when the final center distance E (2) = (N 2 − N 1 )/2P is initially installed and the pinion is put into mesh with the internal gear by the axial displacement of the pinion. Radial assembly means that the pinion is put into mesh with the internal gear by translational displacement along the center distance. The center distance E by the radial displacement of the pinion is changed from (N 2 − N 1 − 4)/2P to (N 2 − N 1 )/2P . Interference in the axially assembled drive occurs if the tip of the pinion tooth gener- ates in relative motion a trajectory that intersects the gear involute profile. The trajectory is an extended hypocycloid. The solution is based on the same approach that was applied for axial generation. The limiting number N 1 of pinion teeth is a little larger than N c due to the lessened dimension of the pinion addendum in comparison with the shaper addendum. P1: JTH CB672-11 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:24 11.4 Interference by Assembly 315 Radial Assembly The investigation of interference in the radial assembly is based on the following con- siderations: (i) We represent in fixed coordinate system S f equations of profiles of several gear spaces and pinion teeth by the following vector functions: r (2) f (θ 2 , j δ 2 , N 2 ), r (1) f (θ 1 , j δ 1 , E, N 1 ). (11.4.1) Here, θ i is the parameter of the involute profile (i = 1, 2); δ i = (2π )/N i is the angular pitch; j is the space (tooth) number; and E is the variable center distance that is installed by the assembly. The superscripts “1” and “2” indicate the pinion and the gear, respectively; N 2 is considered as given. (ii) Interference of pinion and gear involute profiles occurs if r (2) f (θ 2 , j δ 2 , N 2 ) − r (1) f (θ 1 , j δ 1 , E, N 1 ) = 0. (11.4.2) (iii) Equations (11.4.2) provide a system of two scalar equations f j (θ 2 ,θ 1 , j δ 2 , j δ 1 , E, N 1 , N 2 ) = 0(j = 0, 1, 2, ,m). (11.4.3) We consider the most unfavorable case when the point of interference lies on the addendum circles of the pinion and the gear (Fig. 11.4.1), and therefore parameters θ 1 and θ 2 are known. We will determine (N 1 , E) if the solution of equation system (11.4.3) exists. The solution for N 1 = N (r ) 1 determines the maximal number N (r ) 1 of the pinion that is allowed by radial assembly. Figure 11.4.1: Interference by radial assembly. P1: JTH CB672-11 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:24 316 Internal Involute Gears Figure 11.4.2: Radial assembly. After the gears are radially assembled and the final center distance E (2) is installed, the tip of the pinion generates an extended hypocycloid while the pinion and gear perform rotational motions. Interference of the hypocycloid with the gear involute profile is avoided by making the number of pinion teeth N 1 ≤ N (a) 1 , where N (a) 1 is the number of pinion teeth allowed by axial assembly. The designed number of pinion teeth should not exceed N (a) 1 and N (r ) 1 . Figure 11.4.2 illustrates the computerized simulation of radial assembly of the pinion and gear. The computations were performed for a gear drive with N 1 = 25, N 2 = 40, diametral pitch P = 8, and pressure angle α c = 20 ◦ . We can avoid the investigation of interference by radial assembly if the pinion tooth number N (r ) 1 satisfies the inequality N (r ) 1 ≤ N (r ) c where N (r ) c is the shaper tooth number allowed by radial–axial generation (see Table 11.3.1). Nomenclature E c distance between gear and cutter axes (Fig. 11.2.1) N 1 pinion teeth number N 2 gear teeth number N c shaper teeth number P diametral pitch j tooth number m ij transmission ratio of gear i to gear j r a1 radius of pinion addendum circle r a2 radius of gear addendum circle (Fig. 11.3.2) r ac radius of cutter addendum circle (Fig. 11.2.2) r b1 radius of pinion base circle r b2 radius of gear base circle (Fig. 11.3.2) r bc radius of cutter base circle (Fig. 11.2.2) r p2 radius of gear pitch circle (Fig. 11.3.2) P1: JTH CB672-11 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:24 11.4 Interference by Assembly 317 r pc radius of cutter pitch circle (Fig. 11.2.2) s ac tooth thickness of the cutter on the addendum circle (Fig. 11.2.2) s pc tooth thickness of the cutter on the pitch circle (Fig. 11.2.2) w a2 space width on the gear addendum circle (Fig. 11.3.2) w p2 space width on the gear pitch circle (Fig. 11.3.2) 2 angle of tooth thickness on the cutter addendum circle (Fig. 11.2.2) α c pressure angle of cutter θ i parameter of gear involute profile (i = 1, 2) (Fig. 11.3.1) φ 2 angle of gear rotation (Fig. 11.2.1) φ c angle of cutter rotation (Fig. 11.2.1) P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 12 Noncircular Gears 12.1 INTRODUCTION Noncircular gears transform rotation between parallel axes with the prescribed gear ratio function m 12 = ω (1) ω (2) = f (φ 1 ) where φ 1 is the angle of rotation of the driving gear. The center distance between the axes of rotation is constant. The most typical examples of application of noncircular gears are (i) as the driving mechanism for a linkage to modify the displacement function or the velocity function, and (ii) for the generation of a prescribed function. Figure 12.1.1 shows the Geneva mechanism that is driven by elliptical gears. The application of elliptical gears enables it to change the angular velocity of the crank of the mechanism during the crank revolution. A crank–slider linkage that is driven by elliptical gears is shown in Fig. 12.1.2. A kinematical sketch of the mechanism is shown in Fig. 12.1.3(a). Application of elliptical gears enables it to modify the velocity function v(φ) of the slider [Fig. 12.1.3(b)]. Oval gears (Fig. 12.1.4) are applied in the Bopp and Reuter meters for the measurement of the discharge of liquid; the oval gears are shown in the figure in three positions. Figure 12.1.5 shows noncircular gears with unclosed centrodes that are applied in instruments for the generation of functions. Figure 12.1.6 shows a noncircular gear of a drive that is able to transform rotation between parallel axes for a cycle that exceeds one gear revolution. During the cycle the gears perform axial translational motions in addition to rotational motions. Noncircular gears have not yet found a broad application although modern man- ufacturing methods enable their makers to provide conjugate profiles using the same tools as are applied for spur circular gears. The following sections are based on work by Litvin [1956]. 12.2 CENTRODES OF NONCIRCULAR GEARS We consider two cases, assuming as given either (i) the gear ratio function m 12 (φ 1 ), or (ii) the function y(x) to be generated. 318 P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 12.2 Centrodes of Noncircular Gears 319 Chain conveyor Elliptical gears Figure 12.1.1: Conveyor driven by the Geneva mechanism and elliptical gears. Case 1: The gear ratio function m 12 (φ 1 ) ∈ C 1 , 0 ≤ φ 1 ≤ φ ∗ 1 (12.2.1) where φ 1 is the angle of rotation of the driving gear 1 is given. Here, m 12 (φ 1 ) = ω (1) ω (2) = dφ 1 dφ 2 where ω (i ) (i = 1, 2) is the gear angular velocity. Figure 12.1.2: Conveyor based on application of the crank–slider linkage and elliptical gears. P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 Figure 12.1.3: Combination of elliptical gears with a crank–slider linkage. I II III Figure 12.1.4: Oval gears of a liquid meter. 320 P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 12.2 Centrodes of Noncircular Gears 321 Figure 12.1.5: Noncircular gears applied in in- struments. Figure 12.1.6: Twisted noncircular gear. P1: GDZ/SPH P2: GDZ CB672-12 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:32 322 Noncircular Gears The centrode of gear 1 is represented in polar form by the equation r 1 (φ 1 ) = E 1 m 12 (φ 1 ) ± 1 (12.2.2) where E is the center distance. The centrode of the driven gear 2 is determined with the equations r 2 (φ 2 ) = E m 12 (φ 1 ) m 12 (φ 1 ) ± 1 ,φ 2 =  φ 1 0 dφ 1 m 12 (φ 1 ) . (12.2.3) Function φ 2 (φ 1 ) ∈ C 2 is a monotonic increasing function, and the gear ratio function m 12 (φ 1 ) ∈ C 1 must be positive. The difference between m 12 max and m 12 min is to be limited to avoid undesirable pressure angles (see Section 12.12). We have to differentiate the angle of rotation φ i of gear i from the polar angle θ i that determines the position vector of the centrode (i = 1, 2). Angles φ i and θ i are equal, but they are measured in opposite directions. The orientation of the tangent with respect to the current position vector of the centrode is designated by angle µ, where tan µ i = r i (φ i ) dr i dφ i . (12.2.4) Equations (12.2.1), (12.2.3), and (12.2.4) yield tan µ 1 =− m 12 (φ 1 ) ± 1 m  12 (φ 1 ) (12.2.5) tan µ 2 =± m 12 (φ 1 ) ± 1 m  12 (φ 1 ) . (12.2.6) Here, m  12 = (∂/∂φ 1 )[m 12 (φ 1 )]. Function µ i (φ 1 )(i = 1, 2) is used for determination of variations of the pressure angle in the process of meshing (see Section 12.12). The upper (lower) sign in the above expressions with double signs corresponds to the case of external (internal) gears. Angle µ i is measured in the same direction as θ i . The following discussion is limited to the case of external noncircular gears. The subscripts “1” and “2” in expressions for µ 1 and µ 2 indicate gears 1 and 2, respectively. Case 2: Function y(x) to be generated is given y(x) ∈ C 2 , x 2 ≥ x ≥ x 1 . Rotation angles of the gears are determined as φ 1 = k 1 (x − x 1 ),φ 2 = k 2 [y(x) − y(x 1 )] (12.2.7) where k 1 and k 2 are the scale coefficients of constant values. Equations (12.2.7) represent in parametric form the displacement function of the gears. The gear ratio function is m 12 = dφ 1 dφ 2 = k 1 k 2 y x (12.2.8) [...]... φ 2 π −π 2 − b2 )0.5 The derivations above confirm Eq ( 12. 3.6) Using Eq ( 12. 3.6), we obtain the following expression for E: E= 1 p 1 + [1 + (n2 − 1)(1 − e 2 )] 2 2 1−e For the case when n = 1, we obtain E= 2p = 2a, 1 − e2 and the gear centrode is an ellipse as well ( 12. 3.7) P1: GDZ/SPH CB6 72- 12 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0: 32 326 Noncircular Gears 12. 4 ELLIPTICAL AND. .. α and δ the angles of rotation of gears 1 and 4 (Fig 12. 8.1) and introduce the equations α = k1 (x − x1 ), δ = k 4 (y − y1 ) ( 12. 8.1) where k1 = αmax , x2 − x1 k4 = δmax y2 − y1 ( 12. 8 .2) P1: GDZ/SPH CB6 72- 12 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0: 32 334 Noncircular Gears Figure 12. 8.1: Combined gear mechanism with identical centrodes for (i) gears 1 and 3, and (ii) gears 2 and. .. ( 12. 2 .2) and ( 12. 2.3), we can represent the condition of centrode convexity in terms of function m 12 (φ1 ) and its derivatives: (i) For the driving gear we have 1 + m 12 (φ1 ) + m 12 (φ1 ) ≥ 0 ( 12. 5.3) (ii) For the driven gear we obtain 2 1 + m 12 (φ1 ) + (m 12 (φ1 )) − m 12 (φ1 )m 12 (φ1 ) ≥ 0 ( 12. 5.4) Here, m 12 = (d/dφ1 )(m 12 (φ1 )), m 12 = (d 2 /d 2 φ1 )(m 12 (φ1 )) When the inequalities ( 12. 5.3) and ( 12. 5.4)... The centrode of gear 1 is represented in polar form by the equation r 1 (φ1 ) = p 1 + e cos φ Here, p = a(1 − e 2 ), e = c/a (Fig 12. 3.1) Figure 12. 3.1: Elliptical centrode ( 12. 3.5) P1: GDZ/SPH CB6 72- 12 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0: 32 12. 3 Closed Centrodes 325 Equation ( 12. 3.4) yields 2 = n 2 0 p dφ1 = E − p + Ee cos φ1 2 p 1 [(E − p )2 − E 2 e 2 ] 2 ( 12. 3.6) The derivation... as [see Eq ( 12. 4.3)] r1 = a(1 − e 2 ) 1 − e cos 2 1 Determine the condition of centrode convexity Solution The gear ratio function and its derivatives are m 12 (φ1 ) = 1 − 2e cos 2 1 + e 2 E − r 1 (φ1 ) = r 1 (φ1 ) 1 − e2 ( 12. 5.7) because E = 2a m 12 (φ1 ) = 4e sin 2 1 , 1 − e2 m 12 (φ1 ) = 8e cos 2 1 1 − e2 ( 12. 5.8) Equations ( 12. 5.3), ( 12. 5.8), and ( 12. 5.7) yield 1 + 3e cos 2 1 ≥ 0, ( 12. 5.9) which... − 12) 2 4n ( 12. 6.4) The centrode of gear 2 is determined with the following equations: r 2 = E − r 1 = a[c − (1 − ε 2 sin2 φ1 )1 /2 + ε cos φ1 ] φ1 2 = 0 (1 − ε 2 sin2 φ1 )1 /2 − ε cos φ1 c − (1 − 2 sin2 φ1 )1 /2 + ε cos φ1 dφ1 ( 12. 6.5) ( 12. 6.6) The curvature radius of the gear 2 centrode is 2 = a r 1 (φ1 )[E − r 1 (φ1 )] [r 1 (φ1 ) ]2 + Ee cos φ1 ( 12. 6.7) The condition of convexity of the gear 2. .. GDZ/SPH CB6 72- 12 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0: 32 12. 3 Closed Centrodes 323 where yx = dy/dx; yx (x) ∈ C 1 , and x1 ≤ x ≤ x2 The gear centrodes are represented by the equations φ1 = k1 (x − x1 ), r1 = E 2 = k 2 [y(x) − y(x1 )], k 2 yx k1 + k 2 yx r2 = E k1 k1 + k 2 yx ( 12. 2.9) ( 12. 2.10) In the case when the derivative yx changes its sign in the area x1 ≤ x ≤ x2 , the direct... CB6 72- 12 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0: 32 330 Noncircular Gears In the case of generation of given function f (x), we obtain the following conditions of convexity for the driving and driven gears, respectively: (i) 2 k1 k 2 [ f (x)]3 + k1 [ f (x) ]2 + 2[ f (x) ]2 − f (x) f (x) ≥ 0 ( 12. 5.5) k 2 [ f (x)]3 [k1 + k 2 f (x)] + f (x) f (x) − [ f (x) ]2 ≥ 0 ( 12. 5.6) (ii) Problem 12. 5.1... = φ1 max , x2 − x1 2 = k 2 k2 = 1 1 − x1 x 2 max 1 1 − x1 x ( 12. 7.5) ( 12. 7.6) where φ1 max = 2 max The displacement function of the gears is 2 = F (φ1 ) = a 2 φ1 a 3 + a 4 φ1 ( 12. 7.7) 2 where a 2 = k 2 , a 3 = k1 x1 , a 4 = x1 Coefficients a 2 , a 3 , and a 4 are related as 2 max = φ1 max = a 2 φ1 max , a 3 + a 4 φ1 max ( 12. 7.8) which yields a2 − a3 = φ1 max a4 ( 12. 7.9) Using Eq ( 12. 7.7), we... of gear 1 is represented as a closed curve by the periodic function r 1 (φ1 ) ∈ C 2 and r 1 (2 ) = r 1 (2 /n1 ) = r 1 (0) (ii) The angle of rotation of gear 2, 2 = 2 /n2 , must be performed while gear 1 performs rotation of the angle φ1 = 2 /n1 (iii) Taking into account that 2 = n2 2 n1 dφ1 m 12 (φ1 ) ( 12. 3 .2) E − r 1 (φ1 ) r 2 (φ1 ) = r 1 (φ1 ) r 1 (φ1 ) ( 12. 3.3) 0 and m 12 (φ1 ) = we obtain 2 . ( 12. 2.4) Equations ( 12. 2.1), ( 12. 2.3), and ( 12. 2.4) yield tan µ 1 =− m 12 (φ 1 ) ± 1 m  12 (φ 1 ) ( 12. 2.5) tan µ 2 =± m 12 (φ 1 ) ± 1 m  12 (φ 1 ) . ( 12. 2.6) Here, m  12 = (∂/∂φ 1 )[m 12 (φ 1 )]. Function. Oval gears of a liquid meter. 320 P1: GDZ/SPH P2: GDZ CB6 72- 12 CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0: 32 12. 2 Centrodes of Noncircular Gears 321 Figure 12. 1.5: Noncircular gears applied. N 2 ≤ 31 N c ≤ 0.82N 2 − 3 .20 α c = 20 ◦ Axial 32 ≤ N 2 ≤ 20 0 N c ≤ 1.004N 2 − 9.1 62 Two-parameter 36 ≤ N 2 ≤ 20 0 N c ≤ N 2 − 17.6 Axial 17 ≤ N 2 ≤ 31 N c ≤ 0.97N 2 − 5.40 α c = 25 ◦ Axial 32