P1: GDZ/SPH P2: GDZ CB672-13 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:36 13.8 Root’s Blower 373 Figure 13.8.3: For derivation of relation between the design parameters. Figure 13.8.4: Applied coordinate systems. P1: GDZ/SPH P2: GDZ CB672-13 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:36 374 Cycloidal Gearing Table 13.8.1: Properties of  2 Lobe number Convex Concave–Convex With singularities 20< a r < 0.50.5 < a r < 0.9288 a r > 0.9288 30< a r < 0.50.5 < a r < 0.9670 a r > 0.9670 which yield x f = ρ sin(θ − φ) −a sin φ y f = ρ cos(θ − φ) +a cos φ r sin(θ − φ) −a sin θ = 0. (13.8.5) Equations of Dedendum Curve Σ 2 of Rotor 2 Profile  2 is represented in S 2 by the equations r 2 = M 21 r 1 , f (θ,φ) = 0, (13.8.6) which yield x 2 = ρ sin(θ − 2φ) −a sin 2φ + 2r sin φ y 2 = ρ cos(θ − 2φ) +a cos 2φ − 2r cos φ r sin(θ − φ) −a sin θ = 0. (13.8.7) Depending on the ratio a/r , profile  2 may be represented by (i) a convex curve, (ii) a concave–convex curve, and (iii) a curve with singularities. The third case may be investi- gated by considering the conditions of “nonundercutting” of  2 by  1 (see Section 6.3). The first and second cases may be investigated by considering the relations between the curvatures of conjugate shapes (see Section 8.3). The results of the investigations are presented in Table 13.8.1. P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 14 Involute Helical Gears with Parallel Axes 14.1 INTRODUCTION Cycloidal gears (Chapter 13) and involute gears (Chapters 10, 11, 14, 15, and 16) have different areas of application. This chapter covers involute gears with parallel axes, whose design is based on the assumption that the gear tooth surfaces are in instanta- neous contact along a line (line contact) in the case of aligned gear drives. Although the influence of errors of alignment should be considered in the study of the real meshing (see Chapters 15, 16, and 17), in this chapter we consider a preliminary study limited to the theoretical study of meshing. This allows the reader to focus initially on the theo- retical study of involute gears. However, we have to emphasize that the modern design of helical gear drives is directed at observation of localized bearing contact (obtained by tooth surfaces being in point contact instead of line contact), simulation of mesh- ing of misaligned gear drives, and stress analysis (see Chapters 15, 16, and 17). The nomenclature used in this chapter is presented in Section 14.10. 14.2 GENERAL CONSIDERATIONS Helical gears that transform rotation between parallel axes in opposite directions are in external meshing and are provided with screw tooth surfaces of opposite directions. The axodes of nonstandard gears are two cylinders of radii r o1 and r o2 related as r o2 r o1 = ω (1) ω (2) = m 12 . (14.2.1) These cylinders are called the operating pitch cylinders as well. Henceforth, we differen- tiate standard and nonstandard helical gears. The operating pitch cylinders (the axodes) coincide with the pitch cylinders in the case of standard helical gears, and they differ from the pitch cylinders for nonstandard helical gears (see below). Axodes of standard gears are the gear pitch cylinders. The line of tangency of the axodes is the instanta- neous axis of rotation of the gears in relative motion. The cylinders of radii r o1 and r o2 roll over each other without sliding. The helices on the operating pitch cylinders are of opposite direction but the magnitude of the lead angle (or the helix angle) is the same for both helices. 375 P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 376 Involute Helical Gears with Parallel Axes The tooth surface of a helical gear is a helicoid that is represented by Eq. (1.7.5). It was assumed in the derivation of this equation that the helicoid is generated by the screw motion of a cross profile about the gear axis. The cross profile is represented in a plane that is perpendicular to the gear axis. However, a helicoid may also be generated by the screw motion of the axial profile, which is a curve represented in the plane drawn through the axis of the helical gear. Consider that the tooth surface is represented by the vector equation r 1 = x 1 (u,θ) i 1 + y 1 (u,θ) j 1 + z 1 (u,θ, p) k 1 (14.2.2) where (u,θ) are the surface parameters (Gaussian coordinates); p is the screw parameter in the screw motion about the z 1 axis. The normal to the surface is represented as N 1 = ∂r 1 ∂θ × ∂r 1 ∂u , (14.2.3) and N 1 = 0 is for a regular surface. The requirement for surface (14.2.2) to be a helicoid is expressed by the equation x 1 N y1 − y 1 N x1 + pN z1 = x 1 n y1 − y 1 n x1 + pn z1 = 0. (14.2.4) Here, the screw parameter p in Eq. (14.2.4) is considered as an algebraic value; n 1 = N 1 |N 1 | is the surface unit normal; p > 0 for a right-hand gear. Figure 14.2.1(a) shows a helicoid, an involute screw surface, which is generated by the screw motion of an involute curve; r b is the radius of the base cylinder. The intersection of the helicoid by a cylinder of radius ρ is a helix [Fig. 14.2.1(b)]; H is the lead of the helicoid; λ ρ is the lead angle on the cylinder of radius ρ. Figure 14.2.1(c) shows that the cylinder of radius ρ and the helix have been developed on a plane. It is easy to verify that tan λ ρ = cot β ρ = H 2πρ (14.2.5) where λ ρ and β ρ are the lead angle and the helix angle, respectively. The ratio H/2π = p is the screw parameter, which is the axial displacement in screw motion corresponding to rotation through the angle of one radian. Equation (14.2.5) yields that the product ρ tan λ ρ = p (14.2.6) is an invariant with respect to the radius ρ of the cylinder that intersects the helicoid being considered. The investigation of meshing of helical gears with parallel axes requires solutions to the following problems (see Section 14.5): (i) determination of surface  2 that is conjugate to given surface  1 , (ii) determination of the lines of contact between  1 and  2 , and (iii) determination of the surface of action. P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 14.3 Screw Involute Surface 377 Figure 14.2.1: Development of a cylinder and its helix. 14.3 SCREW INVOLUTE SURFACE An involute helical gear may be considered as a multi-thread involute worm. Equations of tooth surfaces of an involute helical gear are the same as for an involute worm (see Section 19.6). The equations below describe tooth side I and II surfaces for right-hand and left-hand gears. Equations of the gear tooth surfaces are presented separately for the driving gear 1 and driven gear 2, in coordinate systems S 1 and S 2 , respectively. A right-hand helical gear 1 is shown in Fig. 14.3.1. Figure 14.3.2 shows the cross section of tooth surfaces of gear 1 obtained by intersec- tion by plane z 1 = 0. Axis x 1 is the axis of symmetry of the space. Half of the angular width of the space on the base circle is formed by axis x 1 and the position vector O 1 B 1 (k) (k = I, II) and is determined with angle µ 1 . Here (Fig. 14.3.2), µ 1 = w t1 2r p1 − invα t1 (14.3.1) P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 378 Involute Helical Gears with Parallel Axes Figure 14.3.1: Right-hand helical gear. Figure 14.3.2: Cross section of helical gear 1. P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 14.3 Screw Involute Surface 379 Figure 14.3.3: Cross section of helical gear 2. where w t1 is the nominal value of space width on the pitch circle and α t1 is the profile angle in the cross section, at the point of intersection of the profile with the pitch circle. The tooth surface equations are represented in terms of surface parameters (u 1 , θ 1 ) (see Section 19.6). Parameter θ 1 is measured from position vector O 1 B 1 (k) (k = I, II) in the direction shown in Fig. 14.3.2. The direction of measurements of θ 1 and µ 1 is clockwise for surface I and counterclockwise for surface II. It is assumed that the observer is located on the negative axis z 1 . Figure 14.3.3 shows the tooth profiles in the cross section of tooth surfaces of gear 2. The surface equations are represented in S 2 in terms of surface parameters u 2 and θ 2 . The concept of parameters u 2 and θ 2 is based on considerations similar to those used in Section 19.6 for u 1 and θ 1 . Parameter θ 2 is measured from the position vector O 2 B (k) 2 (k = I, II), clockwise for surface I and counterclockwise for surface II, as shown in Fig. 14.3.3. Recall that the observer is located on the negative axis z 2 . Half of the angular tooth thickness on the base circle is represented by angle η 2 where η 2 = s t2 2r p2 + invα t2 . (14.3.2) P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 380 Involute Helical Gears with Parallel Axes Here, s t2 is the nominal value of tooth thickness on the pitch circle, and α t2 is the profile angle in the cross section, at the point of intersection of the profile with the pitch circle. The direction of measurements of η 2 is opposite to the direction of measurements of θ 2 . The gear tooth surfaces are represented in S i by the vector function r i (u i ,θ i )(i = 1, 2). (14.3.3) The unit normals to the tooth surfaces of gear 1 are represented as n 1 =∓ ∂r 1 ∂u 1 × ∂r 1 ∂θ 1     ∂r 1 ∂u 1 × ∂r 1 ∂θ 1     . (14.3.4) The upper and lower signs correspond to surfaces of gear 1 for a right-hand and left- hand orientation, respectively. The direction chosen below of the surface unit normal n 2 enables us to provide the coincidence of n 1 and n 2 when the tangency of tooth surfaces of gears 1 and 2 is considered. The u i coordinate line on gear tooth surface (14.3.3) (θ i is fixed) is a straight line, which generates the gear tooth surface while performing a screw motion (see Section 19.6). The θ i line (u i is fixed) is a helix on the gear tooth surface. The screw parameters p 1 and p 2 in the equations below are always considered as positive values for either right-hand gears or left-hand gears. Here are the derived equations of gear tooth surfaces and the surface unit normals: (i) Right-hand gear 1, side surface I (Fig. 14.3.2): x 1 = r b1 cos(θ 1 + µ 1 ) + u 1 cos λ b1 sin(θ 1 + µ 1 ) y 1 = r b1 sin(θ 1 + µ 1 ) − u 1 cos λ b1 cos(θ 1 + µ 1 ) z 1 =−u 1 sin λ b1 + p 1 θ 1 (14.3.5) n 1 = [ −sin λ b1 sin(θ 1 + µ 1 ) sin λ b1 cos(θ 1 + µ 1 ) −cos λ b1 ] T . (14.3.6) Angle θ 1 and µ 1 are measured clockwise. (ii) Right-hand gear 1, side surface II (Fig. 14.3.2): x 1 = r b1 cos(θ 1 + µ 1 ) + u 1 cos λ b1 sin(θ 1 + µ 1 ) y 1 =−r b1 sin(θ 1 + µ 1 ) + u 1 cos λ b1 cos(θ 1 + µ 1 ) z 1 = u 1 sin λ b1 − p 1 θ 1 (14.3.7) n 1 = [ −sin λ b1 sin(θ 1 + µ 1 ) −sin λ b1 cos(θ 1 + µ 1 ) cos λ b1 ] T . (14.3.8) Angles θ 1 and µ 1 are measured counterclockwise. P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 14.3 Screw Involute Surface 381 (iii) Left-hand gear 1, side surface II (Fig. 14.3.2): x 1 = r b1 cos(θ 1 + µ 1 ) + u 1 cos λ b1 sin(θ 1 + µ 1 ) y 1 =−r b1 sin(θ 1 + µ 1 ) + u 1 cos λ b1 cos(θ 1 + µ 1 ) z 1 =−u 1 sin λ b1 + p 1 θ 1 (14.3.9) n 1 = [ −sin λ b1 sin(θ 1 + µ 1 ) −sin λ b1 cos(θ 1 + µ 1 ) −cos λ b1 ] T . (14.3.10) Angles θ 1 and µ 1 are measured counterclockwise. (iv) Left-hand gear 1, side surface I (Fig. 14.3.2): x 1 = r b1 cos(θ 1 + µ 1 ) + u 1 cos λ b1 sin(θ 1 + µ 1 ) y 1 = r b1 sin(θ 1 + µ 1 ) − u 1 cos λ b1 cos(θ 1 + µ 1 ) z 1 = u 1 sin λ b1 − p 1 θ 1 (14.3.11) n 1 = [ −sin λ b1 sin(θ 1 + µ 1 ) sin λ b1 cos(θ 1 + µ 1 ) cos λ b1 ] T . (14.3.12) Angles θ 1 and µ 1 are measured clockwise. Similarly, we represent equations of tooth surfaces of gear 2. We remind the reader that angle η 2 is measured in a direction opposite to the direction of measurements of θ 2 (Fig. 14.3.3). (i) Right-hand gear 2, side-surface I (Fig. 14.3.3): x 2 = r b2 cos(θ 2 − η 2 ) + u 2 cos λ b2 sin(θ 2 − η 2 ) y 2 = r b2 sin(θ 2 − η 2 ) − u 2 cos λ b2 cos(θ 2 − η 2 ) z 2 =−u 2 sin λ b2 + p 2 θ 2 (14.3.13) n 2 = [sin λ b2 sin(θ 2 − η 2 ) − sin λ b2 cos(θ 2 − η 2 ) cos λ b2 ] T . (14.3.14) Angle θ 2 is measured clockwise. (ii) Right-hand gear 2, side-surface II (Fig. 14.3.3): x 2 = r b2 cos(θ 2 − η 2 ) + u 2 cos λ b2 sin(θ 2 − η 2 ) y 2 =−r b2 sin(θ 2 − η 2 ) + u 2 cos λ b2 cos(θ 2 − η 2 ) z 2 = u 2 sin λ b2 − p 2 θ 2 (14.3.15) n 2 = [sin λ b2 sin(θ 2 − η 2 ) sin λ b2 cos(θ 2 − η 2 ) − cos λ b2 ] T . (14.3.16) Angle θ 2 is measured counterclockwise. (iii) Left-hand gear 2, side-surface II (Fig. 14.3.3): x 2 = r b2 cos(θ 2 − η 2 ) + u 2 cos λ b2 sin(θ 2 − η 2 ) y 2 =−r b2 sin(θ 2 − η 2 ) + u 2 cos λ b2 cos(θ 2 − η 2 ) z 2 =−u 2 sin λ b2 + p 2 θ 2 (14.3.17) n 2 = [sin λ b2 sin(θ 2 − η 2 ) sin λ b2 cos(θ 2 − η 2 ) cos λ b2 ] T . (14.3.18) Angle θ 2 is measured counterclockwise. P1: GDZ/SPH P2: GDZ CB672-14 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 0:39 382 Involute Helical Gears with Parallel Axes (iv) Left-hand gear 2, side-surface I (Fig. 14.3.3): x 2 = r b2 cos(θ 2 − η 2 ) + u 2 cos λ b2 sin(θ 2 − η 2 ) y 2 = r b2 sin(θ 2 − η 2 ) − u 2 cos λ b2 cos(θ 2 − η 2 ) z 2 = u 2 sin λ b2 − p 2 θ 2 (14.3.19) n 2 = [sin λ b2 sin(θ 2 − η 2 ) − sin λ b2 cos(θ 2 − η 2 ) − cos λ b2 ] T . (14.3.20) Angle θ 2 is measured clockwise. 14.4 MESHING OF A HELIC AL GEAR WITH A RACK We may consider the meshing of a screw involute gear, say gear 1, with the respective rack in plane z 1 = 0 that is perpendicular to the z 1 gear axis. The cross section of the gear tooth surface by plane z 1 = 0 is an involute curve. Then, we may consider that a spur gear with an infinitesimally small tooth length is in mesh with a rack whose tooth length is also infinitesimally small. It is known that the profile of such a rack is a straight line (see Chapter 10). The derivation of tooth surface  r of the rack is represented as the determination of the envelope to the family of screw involute surfaces  1 . Consider that movable coordinate systems S 1 and S r are rigidly connected to the gear and the rack; coordinate system S f is the fixed one (Fig. 14.4.1). The gear and the rack perform rotational and translational motions, respectively. The velocities of these motions are related as follows: v ω = ρ. (14.4.1) Figure 14.4.1: For investigation of meshing of a helical gear with a rack. [...]... equations in three unknowns (u, φ, and m): a 11 u + a 13 m = b1 a 21 u + a 22 φ + a 23 m = b2 a 31 u + a 32 φ + a 33 m = b3 ( 14. 4. 14) P1: GDZ/SPH CB6 72- 14 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:39 14. 4 Meshing of a Helical Gear with a Rack 385 Here,  a 11 a 12 a 13 b1   cos λb sin αt     a a a b  =  − cos λ cos α b t  21 22 23 2   a 31 a 32 a 33 b3 − sin λb 0 −nxr r b... cylinder of radius r b [Fig 14. 4 .4( a)] We remind the reader that contact lines on the tooth surface of a spur gear are straight lines that are parallel to the gear axis [Fig 14. 4 .4( b)] Figure 14. 4 .4: Contact lines on tooth surfaces of a helical gear and a spur gear P1: GDZ/SPH CB6 72- 14 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:39 14. 4 Meshing of a Helical Gear with a Rack 389 Surface... pn = ( 14. 4 .41 ) s t = wt = = 2 2 sin λ p 2Pn sin λ p 14. 5 MESHING OF MATING HELICAL GEARS We apply movable coordinate systems S 1 and S 2 that are rigidly connected to gears 1 and 2, and the fixed coordinate system S f (Fig 14. 5.1) The gear ratio m 12 is constant and the rotation angles φ1 and 2 are related as follows: m 12 = ω(1) 2 φ1 = = ω (2) ρ1 2 ( 14. 5.1) The shortest distance between the gear axes... λ p cos β p ( 14. 4.33) Figure 14. 4.6: Design parameters of a rack-cutter in normal and transverse sections (k = n, t) P1: GDZ/SPH CB6 72- 14 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:39 14. 4 Meshing of a Helical Gear with a Rack 391 Figure 14. 4.7: Sections of a rack-cutter for a helical gear Step 2: Determination of pt and Pt : pn pt = , sin λ p Pt = Pn sin λ p ( 14. 4. 34) Step 3: Determination... coordinate system S n by the equation r xnnxn + ynnyn = 0 ( 14. 4 .22 ) P1: GDZ/SPH CB6 72- 14 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:39 14. 4 Meshing of a Helical Gear with a Rack 387 because the origin On lies in plane r and nzn = 0 because the normal to r is perpendicular to z n [Fig 14. 4 .2( c)] Equation ( 14. 4 .22 ) represents a straight line and the profile angle of the rack tooth surface in the... L21 n1 ( 14. 5.13) and the equation of meshing ( 14. 5 .4) Here, L21 is the (3 × 3) submatrix of M21 Surface 2 is a helicoid because the equation y2 nx2 − x2 ny2 − p2 nz2 = 0 ( 14. 5. 14) P1: GDZ/SPH CB6 72- 14 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:39 14. 5 Meshing of Mating Helical Gears 395 is satisfied Here, p2 = − p1 m 12 ( 14. 5.15) This surface is a developed ruled helicoid because the... Projections of F( 12, n) on the coordinate axes of S f are designated as X f , Y f , and P1: GDZ/SPH CB6 72- 14 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 0:39 14. 8 Force Transmission 40 1 Figure 14. 8.3: Components of contact force ( 12) Zf (Fig 14. 8.3) and are shown in Fig 14. 8.3 Here, ( 12) = −F ( 12, n) sin αn ( 12) = F ( 12, n) cos αn cos β p ( 12) = F ( 12, n) cos αn sin β p Xf Yf Zf ( 14. 8.6) It is... 14. 5.1) and is represented as   − cos(φ1 + 2 ) − sin(φ1 + 2 ) 0 E cos 2      sin(φ1 + 2 ) − cos(φ1 + 2 ) 0 −E sin 2    M21 =     0 0 1 0   0 0 0 1 ( 14. 5.6) where φ1 and 2 are related by Eq ( 14. 5.1) Figure 14. 5.1 shows the radii ρ1 and 2 of two axodes of helical gears Taking into account that E = ρ1 + 2 , m 12 = ω(1) 2 = , ω (2) ρ1 ρ1 = r b1 cos αo ( 14. 5.7) and using Eqs ( 14. 5 .4) ,... as n2 = N2 , |N2 | N2 = 2 ( 14. 5.10) in terms of surface parameters ∂r2 ∂r2 × ∂ 2 ∂u 1 ( 14. 5.11) We obtain after derivations that n2 = −[ sin λb1 sin( 2 − αo ) sin λb1 cos( 2 − αo ) (provided that r b1 tan αo (1 + m 12 ) − u 1 cos λb1 = 0) cos λb1 ] T ( 14. 5. 12) An alternative approach for derivation of n2 is based on application of the equation n2 = L21 n1 ( 14. 5.13) and the equation of meshing ( 14. 5 .4) ... 0 ( 14. 4.8) P1: GDZ/SPH CB6 72- 14 P2: GDZ CB6 72/ Litvin CB6 72/ Litvin-v2.cls February 27 , 20 04 3 84 0:39 Involute Helical Gears with Parallel Axes Here,  Mr 1 cos φ  − sin φ =   0 0 sin φ cos φ 0 0  −ρ  ρφ    0  0 0 1 0 ( 14. 4.9) 1 Equations ( 14. 3.5), ( 14. 4.7), and ( 14. 4.9) yield the following equations of r: xr = r b cos αt + u cos λb sin αt − ρ yr = r b sin αt − u cos λb cos αt + ρφ ( 14. 4.10) . (Fig. 14. 3.3): x 2 = r b2 cos(θ 2 − η 2 ) + u 2 cos λ b2 sin(θ 2 − η 2 ) y 2 =−r b2 sin(θ 2 − η 2 ) + u 2 cos λ b2 cos(θ 2 − η 2 ) z 2 = u 2 sin λ b2 − p 2 θ 2 ( 14. 3.15) n 2 = [sin λ b2 sin(θ 2 −. λ b2 sin(θ 2 − η 2 ) y 2 =−r b2 sin(θ 2 − η 2 ) + u 2 cos λ b2 cos(θ 2 − η 2 ) z 2 =−u 2 sin λ b2 + p 2 θ 2 ( 14. 3.17) n 2 = [sin λ b2 sin(θ 2 − η 2 ) sin λ b2 cos(θ 2 − η 2 ) cos λ b2 ] T . ( 14. 3.18) Angle. (Fig. 14. 3.3): x 2 = r b2 cos(θ 2 − η 2 ) + u 2 cos λ b2 sin(θ 2 − η 2 ) y 2 = r b2 sin(θ 2 − η 2 ) − u 2 cos λ b2 cos(θ 2 − η 2 ) z 2 = u 2 sin λ b2 − p 2 θ 2 ( 14. 3.19) n 2 = [sin λ b2 sin(θ 2 −