Gear Geometry and Applied Theory Episode 3 Part 1 potx

P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.7 Geometry and Generation of K Worms 583 Figure 19.7.2: Coordinate systems applied for generation of K worms is represented as the family of lines of contact of surfaces equations: c and r1 (u c , θc , ψ) = M1o Moc rc (u c , θc ) Nc (θc ) · (c1) vc (u c , θc ) = f (u c , θc ) = by the following (19.7.1) (19.7.2) Equation (19.7.1) represents the family of tool surfaces; (u c , θc ) are the Gaussian coordinates of the tool surface, and ψ is the angle of rotation in the screw motion (c1) Equation (19.7.2) is the equation of meshing Vectors Nc and vc are represented in S c and indicate the normal to c and the relative (sliding) velocity, respectively It is proven below [see Eq (19.7.8)] that Eq (19.7.2) does not contain parameter ψ Equations (19.7.1) and (19.7.2) considered simultaneously represent the surface of the worm in terms of three related parameters (u c , θc , ψ) For further derivations we will consider that the surface side I of a right-hand worm is generated The cone surface is represented by the equations (Fig 19.7.3) rc = u c cos αc (cos θc ic + sin θc jc ) + (u c sin αc − a) kc (19.7.3) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 584 1:28 Worm-Gear Drives with Cylindrical Worms Figure 19.7.3: Generating cone surface Here, u c determines the location of a current point on the cone generatrix; “a” determines the location of the cone apex The unit normal to the cone surface is determined as Nc ∂rc ∂rc , Nc = × , (19.7.4) nc = |Nc | ∂u c ∂θc which yields nc = [− sin αc cos θc − sin αc sin θc cos αc ]T (19.7.5) The relative velocity is represented as the velocity in screw motion (Fig 19.7.4) v(c1) = −ω c × rc − Rc × ω c − p ω c c (19.7.6) where Rc = −E c ic is the position vector of point O1 of the line of action of ω Equation (19.7.6) yields   − sin γc z c + cos γc yc     (c1) (19.7.7) vc = ω − cos γc (xc + E c ) − p sin γc    sin γc (xc + E c ) − p cos γc The equation of meshing of the grinding surface with the worm surface after elimination of (−ω sin γc cos θc ) is represented as nc · v(c1) = f (u c , θc ) = a sin αc − (E c sin αc cot γc + p sin αc ) tan θc c − (E c − p cot γc ) cos αc − uc = cos θc (19.7.8) where u c > Equation (19.7.8) with the given value of u c provides two solutions for θc and determines two curves, I and II in the plane (u c , θc ) (Fig 19.7.5) Only curve I is the real contact line in the space of parameters (u c , θc ) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 19.7 Geometry and Generation of K Worms 1:28 585 Figure 19.7.4: Installment of grinding cone: (a) illustration of installment parameter E c ; (b) illustration of installment parameter γc Figure 19.7.5: Line of contact between generating cone and K worm surface: representation in plane of parameters P1: JsY CB672-19 CB672/Litvin 586 CB672/Litvin-v2.cls February 27, 2004 1:28 Worm-Gear Drives with Cylindrical Worms Figure 19.7.6: Contact lines between generating cone and worm on worm surface Equations (19.7.3) and (19.7.8) considered simultaneously represent in S c the line of contact between c and The line of contact is not changed in the screw motion of the worm because equation of meshing (19.7.8) does not contain parameter of motion ψ The worm surface is represented by Eqs (19.7.1) and (19.7.8) considered simultaneously Figure 19.7.6 shows the contact lines on between and c The design parameters of the worm surface are related with the equations tan αc = tan αa x cos λp (19.7.9) where αa x is the profile angle of the worm in its axial section, and λp is the lead angle on the worm pitch cylinder, and s c ≈ wa x cos λp (19.7.10) where wa x is the width of worm space in the axial section, and wa x is measured on the pitch cylinder The exact value of required s c can be determined using the equations of the axial section of the generated worm The design parameters r c and a are represented as r c = Ec − rp a = r c tan αc + (19.7.11) sc (19.7.12) The derivation of Eqs (19.7.11) and (19.7.12) is based on Figs (19.7.1) and (19.7.2) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.7 Geometry and Generation of K Worms 587 The final expressions for both sides of the right-hand and left-hand worms and the surface unit normals are represented by the following equations: (i) Surface side I, right-hand worm: x1 = u c (cos αc cos θc cos ψ + cos αc cos γc sin θc sin ψ − sin αc sin γc sin ψ) + a sin γc sin ψ + E c cos ψ y1 = u c (− cos αc cos θc sin ψ + cos αc cos γc sin θc cos ψ (19.7.13) − sin αc sin γc cos ψ) + a sin γc cos ψ − E c sin ψ z = u c (sin αc cos γc + cos αc sin γc sin θc ) − pψ − a cos γc nx1 = cos ψ sin αc cos θc + sin ψ(cos γc sin αc sin θc + sin γc cos αc ) ny1 = − sin ψ sin αc cos θc + cos ψ(cos γc sin αc sin θc + sin γc cos αc ) (19.7.14) nz1 = sin γc sin αc sin θc − cos γc cos αc where u c = a sin αc − (E c sin αc cot γc + p sin αc ) tan θc − (E c − p cot γc ) cos αc cos θc (19.7.15) (ii) Surface side II, right-hand worm: x1 = u c (cos αc cos θc cos ψ + cos αc cos γc sin θc sin ψ + sin αc sin γc sin ψ) − a sin γc sin ψ + E c cos ψ y1 = u c (− cos αc cos θc sin ψ + cos αc cos γc sin θc cos ψ (19.7.16) + sin αc sin γc cos ψ) − a sin γc cos ψ − E c sin ψ z = u c (− sin αc cos γc + cos αc sin γc sin θc ) − pψ + a cos γc nx1 = cos ψ sin αc cos θc + sin ψ(cos γc sin αc sin θc − sin γc cos αc ) ny1 = − sin ψ sin αc cos θc + cos ψ(cos γc sin αc sin θc − sin γc cos αc ) (19.7.17) nz1 = sin γc sin αc sin θc + cos γc cos αc where u c = a sin αc + (E c sin αc cot γc + p sin αc ) tan θc − (E c − p cot γc ) cos αc cos θc (19.7.18) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 588 1:28 Worm-Gear Drives with Cylindrical Worms (iii) Surface side I, left-hand worm: x1 = u c (cos αc cos θc cos ψ + cos αc cos γc sin θc sin ψ + sin αc sin γc sin ψ) − a sin γc sin ψ + E c cos ψ y1 = u c (− cos αc cos θc sin ψ + cos αc cos γc sin θc cos ψ (19.7.19) + sin αc sin γc cos ψ) − a sin γc cos ψ − E c sin ψ z = u c (sin αc cos γc − cos αc sin γc sin θc ) + pψ − a cos γc nx1 = cos ψ sin αc cos θc + sin ψ(cos γc sin αc sin θc − sin γc cos αc ) ny1 = − sin ψ sin αc cos θc + cos ψ(cos γc sin αc sin θc − sin γc cos αc ) (19.7.20) nz1 = − sin γc sin αc sin θc − cos γc cos αc where u c = a sin αc + (E c sin αc cot γc + p sin αc ) tan θc − (E c − p cot γc ) cos αc cos θc (19.7.21) (iv) Surface side II, left-hand worm: x1 = u c (cos αc cos θc cos ψ + cos αc cos γc sin θc sin ψ − sin αc sin γc sin ψ) + a sin γc sin ψ + E c cos ψ y1 = u c (− cos αc cos θc sin ψ + cos αc cos γc sin θc cos ψ (19.7.22) − sin αc sin γc cos ψ) + a sin γc cos ψ − E c sin ψ z = u c (− sin αc cos γc − cos αc sin γc sin θc ) + pψ + a cos γc nx1 = cos ψ sin αc cos θc + sin ψ(cos γc sin αc sin θc + sin γc cos αc ) ny1 = − sin ψ sin αc cos θc + cos ψ(cos γc sin αc sin θc + sin γc cos αc ) (19.7.23) nz1 = − sin γc sin αc sin θc + cos γc cos αc where u c = a sin αc − (E c sin αc cot γc + p sin αc ) tan θc − (E c − p cot γc ) cos αc cos θc (19.7.24) Particular Case It can be proven that for the case when γc = the generated worm surface is a screw involute surface This statement is correct for all four types of worm surfaces represented by Eqs (19.7.13), (19.7.16), (19.7.19), and (19.7.22), respectively The proof is based on the following considerations: (i) The equation of meshing (19.7.15) provides that sin θc = p cot αc Ec (19.7.25) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.7 Geometry and Generation of K Worms 589 This means that θc is constant and c contacts along a straight line, the generatrix of the cone (ii) The worm surface is generated by a straight line, that is, it is a ruled surface It is a developed surface as well because the surface normal does not depend on surface coordinate u c Recall that u c determines the location of a current point on the generating line (iii) Considering the equations of the worm surface and the unit normal to the surface, we may represent a current point of the surface normal by the equation R1 (u c , ψ, m) = r1 (u c , ψ) + mn1 (ψ) (19.7.26) where the variable parameter m determines the location of the current point on the surface normal Function R1 (u c , ψ, m) represents the one-parameter family of curves that are traced out in S by a current point of the surface normal (iv) The envelope to the family of curves is determined with Eq (19.7.26) and the equation (see Section 6.1) ∂R1 ∂R1 × ∂u c ∂ψ · ∂R1 = ∂m (19.7.27) (v) Equations (19.7.26) and (19.7.27) yield that the normals to the worm surface are tangents to the cylinder of radius rb and form the angle of (90◦ − λb ) with the worm axis Here, rb = E c sin θc = p cot αc , λb = αc (19.7.28) Problem 19.7.1 Consider that the worm surface represented by Eqs (19.7.13) is cut by the plane y1 = Axis x1 is the axis of symmetry of the space in axial section The point of intersection of the axial profile with the pitch cylinder is determined with the coordinates x1 = r p , y1 = 0, z1 = − pa x π wa x =− =− 4Pa x Here, wa x is the nominal value of the space width in axial section that is measured along the generatrix of the pitch cylinder; pa x is the distance between two neighboring threads along the generatrix of the pitch cylinder, and Pa x = π/ pa x is the worm diametral pitch in axial section Derive the system of equations to be applied to determine s c (Fig 19.7.1) considering r p , r c , E c , αc , p, and wa x as given P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 590 1:28 Worm-Gear Drives with Cylindrical Worms Solution u c = a sin αc − (E c sin αc cot γc + p sin αc ) tan θc − tan ψ = (E c − p cot γc ) cos αc cos θc u c (cos αc sin θc cos γc − sin αc sin γc ) + a sin γc u c cos αc cos θc + E c u c cos αc cos θc + E c − rp = cos ψ u c (sin αc cos γc + cos αc sin γc sin θc ) − pψ − a cos γc + wa x =0 where a = r c tan αc + sc The derived equation system contains four equations in four unknowns: θc , ψ, u c , and a The solution of the system for the unknowns provides the sought-for value of s c Problem 19.7.2 Consider the particular case of the installment of the tool when γc = Derive (i) the equation of meshing (19.7.27), and (ii) the equations of the envelope to the family of normals to the worm surface (19.7.13) Recall that the envelope is represented by Eqs (19.7.26) and (19.7.27) which have to be considered simultaneously Solution (i) u c cos αc + m sin αc + E c cos θc = (ii) X1 = E c sin θc sin(θc − ψ) Y1 = −E c sin θc cos(θc − ψ) Z1 = uc + E c cot αc cos θc − pψ − a sin αc 19.8 GEOMETRY AND GENERATION OF F-I WORMS (VERSION I) F worms with concave–convex surfaces have been proposed by Niemann and Heyer (1953) and applied in practice by the Flender Co., Germany The great advantage of the F worm-gear drives is the improvement of conditions of lubrication that is achieved due to the favorable shape of contact lines between the worm and the worm-gear surfaces We consider two versions of F worms: (i) the original one, F-I, and (ii) the modified one, F-II, proposed by Litvin (1968) Both versions of worm-gear drives are designed (o) as nonstandard ones: the radius r p of the worm operating pitch cylinder differs from (o) the radius r p of the worm pitch cylinder, and r p − r p ≈ 1.3/Pa x To avoid pointing of P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.8 Geometry and Generation of F-I Worms (Version I) 591 Figure 19.8.1: Installation of grinding wheel generating worm F-I: (a) illustration of installation parameter γc ; (b) illustration of installation parameter E c teeth of worm-gears, the tooth thickness of the worm on the pitch cylinder is designed as t p = 0.4 pa x = 0.4π/Pa x Installment of the Grinding Wheel for F-I The surface of the grinding wheel is a torus The axial section of the grinding wheel is the arc α–α of radius ρ [Fig 19.8.1(b)] In the following discussion we consider the generation of the surface side II of the right-hand worm The radius ρ is chosen as approximately equal to the radius r p of the worm pitch cylinder The installation of the grinding wheel with respect to the worm is shown in Fig 19.8.1(a) The axes of the grinding wheel and the worm form the angle γc = λp , where λp is the lead angle on the worm pitch cylinder, and the shortest distance between these axes is E c Figure 19.8.2(a) shows the section of the grinding wheel and the worm by a plane that is drawn through the z c axis, which is the axis of rotation of the grinding wheel, and the shortest distance Oc O1 [Fig 19.8.1(b)] It is assumed that the line of shortest distance passes through the mean point M of the worm profile; a and b determine the location of center Ob of the circular arc α–α with respect to Oc Here, b = ρ cos αn where ρ is the radius of arc α–α (19.8.1) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 592 1:28 Worm-Gear Drives with Cylindrical Worms (a) (b) Figure 19.8.2: Generation of grinding wheel with torus surface: (a) section of the grinding wheel and (b) applied coordinate systems Equations of Generating Surface Σc We set up coordinate systems S c and S p that are rigidly connected to the grinding wheel; coordinate systems S b and S a are rigidly connected to the circular arc of radius ρ (Fig 19.8.2) The circular arc α–α is represented in S b by the equation rb = ρ[− sin θ 1]T cos θ (19.8.2) Figure 19.8.2(a) shows coordinate systems S a and S b in the initial position The surface of the grinding wheel is generated in S c while the circular arc with coordinate systems S a and S b is rotated about the z p axis [Fig 19.8.2(b)] The coordinate transformation is based on the following matrix equation: rc (θ, ν) = Mc p M pa Mab rb = Mcb rb (19.8.3) Here,  Mc p  Mab 1  0  = 0   0  = 0  0  0    , −b   0 0 −d  0 0    ,   0  M pa  Mcb cos ν   − sin ν  =   sin ν cos ν   − sin ν  =   0  0   0  0 cos ν  0 sin ν cos ν 0 0 −d cos ν  (19.8.4)  d sin ν    −b   We use the following designations [Fig 19.8.1(b)]: a = r p + ρ sin αn (19.8.5) d = E c − a = E c − (r p + ρ sin αn) (19.8.6) P1: JsY CB672-19 CB672/Litvin 598 CB672/Litvin-v2.cls February 27, 2004 1:28 Worm-Gear Drives with Cylindrical Worms Figure 19.9.1: Axes of meshing in the case of grinding of worm F-II meshing, II–II, parameters a and δ, respectively, are determined with the equations a = p cot γc (19.9.1) where p is the screw parameter and γc is the angle formed by the axes of the grinding wheel and the worm, and δ = arctan p Ec (19.9.2) where E c is the shortest distance between the previously mentioned axes The installation of the grinding wheel is based on observation of the following requirements: (a) Center Ob of the circular arc α–α (Fig 19.9.2) is located on the xc axis which is the line of shortest distance between the axes of the grinding wheel and the worm (b) The distance a from the worm axis (Fig 19.9.1) and the crossing angle γc must be related by the equation γc = arctan p a (19.9.3) where p is the screw parameter of the screw motion of the worm in the process of grinding The normal to c already intersects the axis of the grinding wheel, that is the axis of meshing, I –I , as well The normal to c also intersects the other axis of meshing, II–II, because the Ob center of the circular arc α–α is located on II–II P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.9 Geometry and Generation of F-II Worms (Version II) 599 Figure 19.9.2: Grinding wheel for worm F-II Equation (19.9.3) requires only the relation between a and γc , but a can be chosen arbitrarily However, the shape of lines of contact between the worm and the worm-gear surfaces, and , depends on a Based on preliminary investigation, the recommended choice is a = r p + p sin αn (19.9.4) Summarizing, we may formulate the difference in the installations of the grinding wheel for generation of worms F-I and F-II as follows: Version 1: b = 0; the line of shortest distance passes through the middle point M of circular arc α–α; γc = λp ; a = r p + ρ sin αn (Figs 19.8.1 and 19.8.2) Version 2: b = 0; the line of shortest distance passes through Ob ; γc = λp , but γc and a are related with Eq (19.9.1) (Fig 19.9.1) Equation of Meshing We may derive for the F-II worm the equation of meshing between surfaces c and considering the previously derived Eq (19.8.10) but taking b = 0, d = E c − a, and a tan γc = p After the derivations, we obtain the following equation of meshing for the F-II worm: sin θ (E c cot γc + p) sin ν − (E c − a)(1 − cos ν) cos θ = (19.9.5) There are two solutions to Eq (19.9.5): (i) with ν = and any value of θ, and (ii) with the relation between θ and ν determined as tan (E c cot γc + p) tan θ ν − = Ec − a (19.9.6) The meaning of the first solution is that the line of contact between c and is the circular arc α–α, the axial section of the grinding wheel The second solution provides a contact line on c that is out of the working part of the grinding wheel Both contact lines in the space of parameters θ and ν are shown in Fig 19.9.3 P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 600 1:28 Worm-Gear Drives with Cylindrical Worms Figure 19.9.3: Line of contact between grinding wheel and worm F-II surfaces Following an approach similar to that applied for F-I worms, we have derived the following equations for the surface of F-II worms and the surface unit normal: (i) Surface side I, right-hand worm: x1 = −ρ(sin θc cos ψ − cos θc sin ψ sin γc ) + a cos ψ y1 = ρ(sin θc sin ψ + cos θc cos ψ sin γc ) − a sin ψ (19.9.7) z = −ρ cos θc cos γc − pψ + a o where ao = − wa x + ρ cos θc cos γc + pψ (19.9.8) nx1 = − sin θc cos ψ + sin γc cos θc sin ψ ny1 = sin θc sin ψ + sin γc cos θc cos ψ (19.9.9) nz1 = − cos γc cos θc (ii) Surface side II, right-hand worm: x1 = −ρ(sin θc cos ψ + cos θc sin ψ sin γc ) + a cos ψ y1 = ρ(sin θc sin ψ − cos θc cos ψ sin γc ) − a sin ψ (19.9.10) z = ρ cos θc cos γc − pψ + a o where ao = wa x − ρ cos θc cos γc + pψ (19.9.11) nx1 = − sin θc cos ψ − sin γc cos θc sin ψ ny1 = sin θc sin ψ − sin γc cos θc cos ψ nz1 = cos γc cos θc (19.9.12) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.10 Generalized Helicoid Equations 601 (iii) Surface side I, left-hand worm: x1 = −ρ(sin θc cos ψ + cos θc sin ψ sin γc ) + a cos ψ y1 = ρ(sin θc sin ψ − cos θc cos ψ sin γc ) − a sin ψ (19.9.13) z = −ρ cos θc cos γc + pψ + a o where ao = − wa x + ρ cos θc cos γc − pψ (19.9.14) nx1 = − sin θc cos ψ − sin γc cos θc sin ψ ny1 = sin θc sin ψ − sin γc cos θc cos ψ (19.9.15) nz1 = − cos γc cos θc (iv) Surface side II, left-hand worm: x1 = −ρ(sin θc cos ψ − cos θc sin ψ sin γc ) + a cos ψ y1 = ρ(sin θc sin ψ + cos θc cos ψ sin γc ) − a sin ψ (19.9.16) z = ρ cos θc cos γc + pψ + a o where ao = wa x − ρ cos θc cos γc − pψ (19.9.17) nx1 = − sin θc cos ψ + sin γc cos θc sin ψ ny1 = sin θc sin ψ + sin γc cos θc cos ψ (19.9.18) nz1 = cos γc cos θc Axis x1 in all four cases is the axis of symmetry of the axial section of the worm space (see Fig 19.8.4) 19.10 GENERALIZED HELICOID EQUATIONS Consider that the cross section of the worm is represented in parametric form in the auxiliary coordinate system S a as [Fig 19.10.1(a)] (θ ) = r (θ ) cos θ ia + r (θ) sin θ ja (19.10.1) where r (θ) is the polar equation of the cross section The worm surface now can be represented as the surface that is generated by the curve (θ ) that is performing the screw motion about the worm z axis [Fig 19.10.1(b)] The worm surface can be determined by the matrix equation r1 (θ, ζ ) = M1a (ζ )ra (θ ) (19.10.2) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 602 1:28 Worm-Gear Drives with Cylindrical Worms Figure 19.10.1: For derivation of generalized helicoid where [Fig 19.10.1(b)]  M1a cos ζ  cos ζ 0     pζ     sin ζ  =   − sin ζ 0 (19.10.3) Using Eqs (19.10.1) to (19.10.3), we represent the worm surface as follows: r1 (θ, ζ ) = r cos(θ + ζ ) i1 + r sin(θ + ζ ) j1 + pζ k1 (19.10.4) For the following derivations we need angle µ that is formed between the position vector (θ) and the tangent to this curve [Fig 19.10.1(a)] It is known that µ = arctan r (θ ) rθ rθ = dr dθ (19.10.5) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.11 Equation of Meshing of Worm and Worm-Gear Surfaces 603 An alternative equation for determination of µ is based on the expression [Fig 19.10.1(a)] µ = 90◦ − θ + δ = 90◦ − θ + arctan N ya Nxa (19.10.6) where Na is the normal to the planar curve (θ) The unit normal to surface (19.10.4) is determined with the equations n1 = N1 , |N1 | N1 = ∂r1 ∂r1 × , ∂θ ∂ζ (19.10.7) which yield n1 = ( p2 + r2 [ p sin(θ + ζ + µ) i1 − p cos(θ + ζ + µ) j1 + r cos µ k1 ] cos2 µ)0.5 (19.10.8) We recall that because the worm surface is a helicoid, the coordinates of the worm surface and the surface unit normal are related by the following equation (see Section 5.5): y1 nx1 − x1 ny1 − pnz1 = (19.10.9) The screw parameter p is positive for a right-hand worm The advantage of Eqs (19.10.4) and (19.10.8) is that the worm surface and its normal are represented in two-parameter form However, this approach requires the analytical or numerical determination of the worm cross section The discussed approach is especially effective in the case when the worm is generated by the surface of a grinding wheel and the worm surface is represented by three parameters 19.11 EQUATION OF MESHING OF WORM AND WORM-GEAR SURFACES The equation of meshing determines the relation between the worm surface parameters and the angle of rotation φ1 of the worm that is in mesh with the worm-gear surface Surfaces and are in contact along a line (L) at every instant The determination of L is based on the requirement that at any point of L the following equations must hold: (12) Ni · vi =0 (i = 1, 2, f ) (19.11.1) Here, the subscripts (1, 2, f ) designate coordinate systems S , S , and S f that are rigidly connected to the worm, the gear, and the frame (housing); Ni is the normal to the worm (12) surface; vi is the relative velocity of with respect to (see Chapter 2) We can simplify the equation of meshing, taking into account that the worm is a helicoid For simplification of the equation of meshing, we can use Eq (19.10.9) or the equation y f nx f − x f ny f − pnz f = (19.11.2) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 604 1:28 Worm-Gear Drives with Cylindrical Worms Using Eq (19.11.1) with i = 1, and Eq (19.10.9), we represent the equation of meshing in S as follows: (z cos φ1 + E cot γ sin φ1 )Nx1 + (−z sin φ1 + E cot γ cos φ1 )N y1 − (x1 cos φ1 − y1 sin φ1 + E) − p − m21 cos γ m21 sin γ Nz1 = (19.11.3) Here, m21 = N1 /N2 is the gear ratio; (x1 , y1 , z ) are the coordinates of the worm surface; (Nx1 , N y1 , Nz1 ) are the projections of the normal N1 to the worm surface; γ is the twist angle The equation of meshing can be represented in the coordinate system S f as z f Nx f + E cot γ N y f − x f + E − p − m21 cos γ m21 sin γ Nz f = (19.11.4) Consider now the case when the worm surface is represented as a generalized helicoid (see Section 19.10) The equation of meshing is represented in this case as r r cos(θ + ζ + φ1 ) + E − p − m21 cos γ cos µ + E p cot γ cos τ m21 sin γ = pz f sin τ = p2 ζ sin τ (19.11.5) Here, r = r (θ ) is the magnitude of the position vector of the current point of the worm cross section [Fig 19.10.1(a)]; τ = θ + ζ + φ1 + µ The coordinates of a current contact point can be expressed in S f by the equations x f = r cos(θ + ζ + φ1 ), y f = r sin(θ + ζ + φ1 ), z f = pζ (19.11.6) Any of equations (19.11.3), (19.11.4), and (19.11.5) yields the relation between the worm surface parameters (u, θ ) and the angle of worm rotation, that is, f (u, θ, φ1 ) = (19.11.7) Equations r1 = r1 (u, θ), f (u, θ, φ1 ) = (19.11.8) where r1 = r1 (u, θ ) is the worm surface , represent in S the family of contact lines on surface ; φ1 is a fixed-in parameter of motion, the parameter of the family of contact lines Contact lines on the worm-gear surface are represented by equations r2 (u, θ, φ1 ) = M21 r1 (u, θ), f (u, θ, φ1 ) = (19.11.9) where M21 is the matrix that describes the coordinate transformation from coordinate system S to coordinate system S Here, S and S are rigidly connected to the worm and the worm-gear, respectively Figure 19.11.1 shows the contact lines on the surface of an Archimedes worm Figure 19.11.2 shows the contact lines on the worm-gear surface It was mentioned in Section 6.6 that the contact lines on the generating surface may have an envelope In the case of a worm-gear drive, the generating surface is the worm P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.11 Equation of Meshing of Worm and Worm-Gear Surfaces 605 Envelope Contact Lines Figure 19.11.1: Contact lines on worm surface surface The envelope to contact lines on an Archimedes worm surface is shown in Fig 19.11.1 Figures 19.11.1 and 19.11.2 correspond to a worm-gear drive with the following parameters: the number of worm threads and gear teeth are N1 = and N2 = 30, respectively; the axial worm module is ma x = mm; the twist angle is γ = 90◦ ; the shortest distance between the axes of the worm and the worm-gear is E = 176 mm Figure 19.11.2: Contact lines on worm-gear surface P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 606 1:28 Worm-Gear Drives with Cylindrical Worms The instantaneous line contact exists only for an ideal worm-gear drive, without misalignment and errors of manufacturing In reality, the contact of surfaces and is an instantaneous point contact, which might be accompanied with the shift of the bearing contact to the edge and an undesirable shape of the function of transmission errors Such transmission errors cause vibration during the meshing To minimize the influence of misalignment and errors of manufacturing, it is necessary to localize the bearing contact between and using the proper mismatch between the theoretical and real worm surfaces 19.12 AREA OF MESHING The area of meshing is the active part of the surface of action The surface of action is the set of lines of contact between the worm and worm-gear surfaces that are represented in the fixed coordinate system S f Knowing the area of meshing, we are able to determine the working axial length of the worm and the working axial width of the worm-gear (see below) The following derivations are based on representation of the worm surface as a generalized helicoid (see Section 19.11) Figure 19.12.1(b) shows the area of meshing of an orthogonal worm-gear drive that is limited in plane (z f , y f ) with curves a–a and b–b The area of meshing is represented Figure 19.12.1: For derivation of area of meshing P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.12 Area of Meshing 607 in the fixed coordinate system S f Curve a–a corresponds to the entry into meshing of those points of the worm surface that belong to the worm addendum cylinder of radius r a [Fig 19.12.1(a)] Curve b–b corresponds to the entry into meshing of those points of the worm-gear surface that belong to the gear addendum cylinder Current point M of curve a–a is determined by the following equations: sin(θa + ζ + φ1 ) = yf p cos µa m21 p sin[µa + (θa + ζ + φ1 )] (19.12.1) r a r a cos(θa + ζ + φ1 ) + E − zf = x f = r a cos(θa + ζ + φ1 ) (19.12.2) (19.12.3) The input for the solution of the system of Eqs (19.12.1) to (19.12.3) is the current value of y f ; r a , θa , and µa are considered as known Equations of the system above are represented in echelon form Varying y f , we can determine the corresponding values of z f and x f of curve a–a Equation (19.12.1) provides two solutions for the angle (θa + ζ + φ1 ), but only the solution that corresponds to x f < must be used This consideration is based on the specific location of the area of meshing (Figs 19.12.1 and 19.12.2) Figure 19.12.2: Intersection of worm and wormgear surfaces by plane m−n P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 608 1:28 Worm-Gear Drives with Cylindrical Worms Figure 19.12.3: For derivation of curve b–b shown in Fig 19.12.1 Let us now consider the determination of current point N of curve b–b [Fig 19.12.1(b)] We limit the discussion to the shape of the addendum surface of the wormgear that is shown in Fig 19.2.2 Surface AB (or CD) of the worm-gear addendum ∗ surface is a cylinder of radius R a ; the axis of the cylinder coincides with the worm-gear axis Surface BC of the worm-gear addendum surface is a cylinder of radius r i ; the axis of this cylinder coincides with the axis of the worm The intersection of surface BC by plane mn is the arc of the circle of radius R a (Figs 19.12.2 and 19.12.3) Point N of curve b–b can be determined as the point of intersection of curve z f (x f ) and the circle of radius R a (Fig 19.12.3) Curve z f (x f ) is obtained as the result of intersection of the surface of action by plane y f = const The determination of current point N of curve b–b for the BC gear addendum surface is based on the following equations: y f − r (θ ) sin(θ + ζ + φ1 ) = f (θ, (ζ + φ1 )) = (19.12.4) p cos µ(θ) m21 = f (θ, (ζ + φ1 )) = p sin[µ(θ) + (θ + (ζ + φ1 ))] r (θ) r (θ) cos(θ + (ζ + φ1 )) + E − zf − (19.12.5) x f − r (θ ) cos(θ + (ζ + φ1 )) = f (θ, (ζ + φ1 )) = [E + x f (θ, (ζ + φ1 ))]2 + z (θ, (ζ + φ1 )) f = f (θ, (ζ + φ1 )) = 0.5 (19.12.6) − E + r i2 − y (θ, (ζ + φ1 )) f 0.5 (19.12.7) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1:28 19.13 Prospects of New Developments 609 Here, r i = r d + c, where r d is the radius of the worm dedendum cylinder and c is the clearance; usually, c = 0.25/P Equations (19.12.5) and (19.12.6) are designated for determination of x f (θ, (ζ + φ1 )) and z f (θ, (ζ + φ1 )) used in Eq (19.12.7); coordinate y f is considered as the input data; tan µ = r (θ )/r θ , where r θ = dr /dθ Equation system (19.12.4) to (19.12.7) may be considered as a system of two nonlinear equations in two unknowns, θ and (ζ + φ1 ) The two-equation system is formed by Eqs (19.12.4) and (19.12.7) and can be solved by using a numerical subroutine [More et al., 1980; Visual Numerics, Inc., 1998] An iterative process for the solution based on the following procedure can be applied as well: Step 1: We use Eq (19.12.4) considering y f as given and choosing a value of θ Then, we can determine sin(θ + ζ + φ1 ) from Eq (19.12.4) This equation provides two solutions for (θ + ζ + φ1 ), but only the solution with x f (θ + ζ + φ1 ) < should be selected (see the location of the area of meshing in Figs 19.12.1 and 19.12.2) Step 2: We determine the values of z f (θ + ζ + φ1 ) and x f (θ + ζ + φ1 ) using Eqs (19.12.5) and (19.12.6), respectively Step 3: We check if Eq (19.12.7) is satisfied with the chosen value of θ and the respective value of (ζ + φ1 ) determined from Eq (19.12.4) If not, it is necessary to start a new iteration with the new value of θ The determination of current point N of curve b–b (Fig 19.12.1) for the AB (and CD) gear addendum surface (Fig 19.12.2) is based on the equation system that contains Eqs (19.12.4), (19.12.5), (19.12.6), and the equation ∗ R a − (E + x f )2 + z f 0.5 = f (θ, (ζ + φ1 )) = (19.12.8) ∗ Equation (19.12.8) is used instead of Eq (19.12.7) Parameter R a is shown in Fig 19.12.2 Equations (19.12.4) and (19.12.8) considered simultaneously represent a system of two nonlinear equations in the unknowns θ and (ζ + φ1 ) Equations (19.12.5) and (19.12.6) are used in this case for determination of coordinates x f and z f for current point N of curve b–b The determination of the area of meshing enables us to determine the length L of the working part of the worm and the width B of the working part of the gear The area of meshing for a worm-gear drive with the ZA worm (Archimedes worm) is represented in Figs 19.12.4 and 19.12.5 The input data for computation is as follows: N1 = 2, (o) N2 = 30, r p = 46 mm, ma x = mm The operating pitch radius is r p = r p + ζ ma x , where ζ = (Fig 19.12.4) and ζ = (Fig 19.12.5) The center distance is E = r p + N2 ma x /2 + ζ ma x 19.13 PROSPECTS OF NEW DEVELOPMENTS Introductory Remarks Worm-gear drives with cylindrical worms are still an example of gear drives for which a satisfactory bearing contact is obtained by lapping under a load in the gear drive house However, such lapping is expensive in terms of time and is not sufficiently effective P1: JsY CB672-19 CB672/Litvin 610 CB672/Litvin-v2.cls February 27, 2004 1:28 Worm-Gear Drives with Cylindrical Worms Figure 19.12.4: Area of meshing for a standard worm-gear drive with the ZA (Archimedes) worm (ζ = 0) The quality of gear drives of existing design depends substantially on the matching of the hob to the worm of the drive The instantaneous contact of the worm and the worm-gear is a line contact New trends toward localization of bearing contact have still neglected the area of worm-gear drives Modification of the geometry of wormgear drives is inevitable The previous sections of this chapter cover the geometry of Figure 19.12.5: Area of meshing for a nonstandard worm-gear drive with the ZA (Archimedes) worm (ζ = 1) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 19.13 Prospects of New Developments 1:28 611 cylindrical worm-gear drives with instantaneous line contact of worms and worm-gears The purpose of this section is to briefly describe perspectives on a new geometry Double Crowning of the Worm The approach toward modification of the geometry of worm-gear drives with brighter prospects should be based on double crowning of the worm with respect to the hob This means that the surfaces of the worm will be properly deviated from the surfaces of the hob The basic principle of existing design is based on application of worms and hobs that are identical to each other The proposed modification of geometry achieved by double crowning of the worm is an extension of the approach that has been already developed for spiral bevel gears, hypoid gears, helical gears, and spur gears Double crowning of the worm means that its surfaces are deviated in the profile and longitudinal directions, respectively, from the hob surface Profile crowning of the worm with respect to the hob is equivalent to application of two mismatched helicoids where one helicoid represents the worm of the drive and the other one is the hob that generates the gear The surfaces of mismatched helicoids are in tangency along a common helix The mismatch of helicoids is the precondition of localization of contact between the surfaces of the worm of the drive and the worm-gear It was mentioned above that longitudinal crowning of the worm has to be applied in addition to profile crowning The purpose of longitudinal crowning is to reduce the shift of the bearing contact, avoid edge contact, and reduce transmission errors All of these defects are caused by misalignments Longitudinal crowning of the worm provides a parabolic function of transmission errors of the worm-gear drive in the process of meshing Such a function is able to absorb discontinuous linear functions of transmission errors caused by misalignments Double crowning of the worm as a combination of profile and longitudinal crowning is especially effective for worm-gear drives with multi-thread worms Gear drives with multi-thread worms are more sensitive to misalignment that cause larger transmission errors and vibrations These defects are reduced due to the effect of application of a parabolic function of transmission errors (see Sections 17.4, 17.6, and 17.7) Application of Oversized Hob Modification of the geometry of worm-gear drives has been based in the past on application of oversized hobs [Colbourne, 1989; Seol & Litvin, 1996] The main idea of design of an oversized hob is based on the increase of the number of threads of the hob with respect to the worm of the worm-gear drive This approach requires an increase in the pitch diameter of the hob We may illustrate the idea of application of an oversized hob considering the hob and the worm of the drive to be in internal meshing and their axes crossed (Fig 19.13.1) The main features of meshing of the oversized hob with the worm are as follows: (i) The pitch cylinder of the hob is larger than the one of the worm, and λ and r are the crossing angle and the shortest center distance between the axes (Fig 19.13.1) P1: JsY CB672-19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 612 1:28 Worm-Gear Drives with Cylindrical Worms Figure 19.13.1: Tangency of pitch cylinders of worm and hob λ is the crossing angle of axes; r is the shortest center distance (ii) Point P of tangency of the hob and the worm pitch cylinders belongs to the shortest distance between the hob and the worm-gear, and to axes II–II of meshing (see Section 6.11) It is easy to verify that the normals to the surfaces of the hob, worm, and worm-gear pass through point “P” and that these surfaces are in simultaneous tangency in the beginning of meshing (iii) The hob is provided with the same type of thread surface as that of the worm (iv) It is obvious that the surfaces of the hob and the worm-gear being generated are in line contact at every instant, but surfaces of the worm and the worm-gear are in point contact at every instant (v) The chosen oversized r affects the magnitude of the major axis of the instantaneous contact ellipses and the level of transmission errors (vi) The generation of the worm-gear by an oversized hob must be accomplished with the following installation parameters E hg = E wg + r, γhg = 90◦ − γ Figure 19.13.2: Example of TCA for localization of contact obtained by an oversized hob: (a) path of contact; (b) function of transmission errors of parabolic type ... + (19 .7 .11 ) sc (19 .7 .12 ) The derivation of Eqs (19 .7 .11 ) and (19 .7 .12 ) is based on Figs (19 .7 .1) and (19 .7.2) P1: JsY CB672 -19 CB672/Litvin CB672/Litvin-v2.cls February 27, 2004 1: 28 19 .7 Geometry. .. Eq (19 .11 .1) with i = 1, and Eq (19 .10 .9), we represent the equation of meshing in S as follows: (z cos ? ?1 + E cot γ sin ? ?1 )Nx1 + (−z sin ? ?1 + E cot γ cos ? ?1 )N y1 − (x1 cos ? ?1 − y1 sin ? ?1 +... equations (19 .11 .3) , (19 .11 .4), and (19 .11 .5) yields the relation between the worm surface parameters (u, θ ) and the angle of worm rotation, that is, f (u, θ, ? ?1 ) = (19 .11 .7) Equations r1 = r1 (u,