Step 4: The discussion above enables us to verify simultaneous generation of profile-crowned pinion tooth surface σ and worm thread surface w by rack-cutter surface c.. 17.7.8 provides
Trang 117.7 Generation of Double-Crowned Pinion by a Worm 493
Figure 17.7.3: For illustration of axodes of worm, pinion, and rack-cutter.
the rack-cutter corresponds to rotation of the pinion with angular velocity ω ( p) The
relation between v1andω ( p)is defined as
where r pis the radius of the pinion pitch cylinder
Step 3: An additional motion of surface cwith velocity vauxalong direction t−t of
skew rack-cutter teeth (Fig 17.7.3) is performed and this motion does not affect thegeneration of surface σ Vector equation v2= v1+ vaux allows us to obtain velocity
v2of rack-cutter cin a direction that is perpendicular to the axis of the worm Then,
we may represent the generation of worm surface w by rack-cutter c considering
that the rack-cutter performs translational motion v2while the worm is rotated withangular velocityω (w) The relation between v2andω (w)is defined as
where r is the radius of the worm pitch cylinder
Trang 2Figure 17.7.4: Contact lines L cσ and L cwcorresponding to meshing of rack-cutter cwith pinion and worm surfaces σ and w, respectively.
Worm surface w is generated as the envelope to the family of rack-cutter surfaces
c
Step 4: The discussion above enables us to verify simultaneous generation of
profile-crowned pinion tooth surface σ and worm thread surface w by rack-cutter surface
c Each of the two generated surfaces σ and ware in line contact with rack-cuttersurface c However, the contact lines L c σ and L cwdo not coincide but rather intersect
each other as shown in Fig 17.7.4 Here, L c σ and L cwrepresent the lines of contactbetween c and σ, and between c and w , respectively Lines L c σ and L cw areobtained for a chosen value of related parameters of motion between c, σ, and w
Point N of intersection of lines L cw and L c σ (Fig 17.7.4) is the common point oftangency of surfaces c, σ, and w
Profile Crowning of Pinion
Profile-crowned pinion tooth surface σ has been previously obtained by using cutter surface c Direct derivation of generation of σ by the worm w may be ac-complished as follows:
rack-(a) Consider that worm surface w and pinion tooth surface σ perform rotationbetween their crossed axes with angular velocities ω (w) andω ( p) It follows fromprevious discussions that w and σ are in point contact and N is one of the
instantaneous points of contact of wand σ (Fig 17.7.4)
(b) The concept of direct derivation of σ by w is based on the two-parameterenveloping process (see Section 6.10) The process of such enveloping is based
on application of two independent sets of parameters of motion [Litvin & Seol,1996]:
(i) One set of parameters relates the angles of rotation of the worm and the pinionas
Trang 317.7 Generation of Double-Crowned Pinion by a Worm 495
Figure 17.7.5: Schematic of generation: (a) without worm plunging; (b) with worm plunging.
collinear to the axis of the pinion [Fig 17.7.5(a)], and (2) small rotationalmotion of the pinion about the pinion axis determined as
ψ p= s w
where p is the screw parameter of the pinion.
Analytical determination of a surface generated as the envelope to a two-parameterenveloping process is presented in Section 6.10
The schematic generation of σby wis shown in Fig 17.7.5(a) wherein the shortestcenter distance is shown as an extended one for the purpose of better illustration In theprocess of meshing of wand σ, the worm surface wand the profile-crowned pinionsurface perform rotation about crossed axes The shortest distance is executed as
Trang 4crossed axes; (3) s w and ψ p are the components of the screw motion of the feed
motion; and (4) r w and r pare the radii of pitch cylinders
Double Crowning of Pinion
We have presented above the generation by a worm of a profile-crowned surface σ ofthe pinion However, our final goal is the generation by a worm of a double-crownedsurface1of the pinion Two approaches are proposed for this purpose:
WORM PLUNGING. Additional pinion crowning (longitudinal crowning) is provided
by plunging of the worm with respect to the pinion which is shown schematically inFig 17.7.5(b) Plunging of the worm in the process of pinion generation is performed asvariation of the shortest distance between the axes of the grinding worm and the pinion
The instantaneous shortest center distance E wp(s w) between the grinding worm andthe pinion is executed as [Fig 17.7.5(b)]:
E wp(s w)= E(0)
wp − a pl(s w)2. (17.7.8)Here, s w is measured along the pinion axis from the middle of the pinion; a pl is
the parabola coefficient of the function a pl(s w)2; and E(0)wp is the nominal value ofthe shortest distance defined by Eq (17.7.7) Plunging of the worm with observation of
Eq (17.7.8) provides a parabolic function of transmission errors in the process of ing of the pinion and the gear of the proposed new version of the Novikov–Wildhaberhelical gear drive
mesh-MODIFIED ROLL OF FEED MOTION. Conventionally, the feed motion of the worm is vided by observation of relation (17.7.6) between componentss wandψ p For thepurpose of pinion longitudinal crowning, the following functionψ p(s w) is observed:
pro-ψ p(s w)= s w
p + a mr(s w)2 (17.7.9)
where a mris the parabola coefficient of the parabolic function in Eq (17.7.9) Modifiedroll motion is provided to the worm instead of worm plunging Application of func-tionψ p(s w) enables us to modify the pinion tooth surface and provide a parabolicfunction of transmission errors of the proposed gear drive
The derivation of double-crowned surface 1 of the pinion by application of bothpreviously mentioned approaches is based on determination of1as a two-parameterenveloping process:
Step 1: We consider that surface wis determined as the envelope to the rack-cuttersurface c The determination of wis a one-parameter enveloping process
Step 2: Double-crowned surface1of the pinion is determined as an envelope of atwo-parameter enveloping process by application of the following equations:
r1(u w , θ w , ψ w , s w)= M1w(ψ w , s w)rw (u w , θ w) (17.7.10)
Nw· v(w1 ,ψ w)
Nw· v(w1 ,s w) = 0. (17.7.12)
Trang 517.8 TCA of a Gear Drive with a Double-Crowned Pinion 497
Here, (u w , θ w) are the worm surface parameters, and (ψ w , s w) are the generalized rameters of motion of the two-parameter enveloping process Vector equation (17.7.10)represents the family of surfaces w of the worm in coordinate system S1of the pin-
pa-ion Equations (17.7.11) and (17.7.12) represent two equations of meshing Vector Nw
is the normal to the worm tooth surface w and is represented in system S w Vector
v(w1 ,ψ w)
w represents the relative velocity between the worm and pinion determined der the condition that parameter ψ w of motion is varied and the other parameter s w
un-is held at rest Vector v(w1 ,s w)
w is determined under the condition that parameter s w isvaried and the other parameter of motionψ w is held at rest Both vectors of relative
velocity are represented in coordinate system S w Vector equations (17.7.10), (17.7.11),and (17.7.12) (considered simultaneously) determine a double-crowned pinion toothsurface obtained by a two-parameter enveloping process (see Section 6.10)
17.8 TCA OF A GEAR DRIVE WITH A DOUBLE-CROWNED PINION
Simulation of meshing of a gear drive with a double-crowned pinion is investigated
by application of the same algorithm discussed in Section 17.5 for a gear drive with aprofile-crowned pinion and gear tooth surfaces The TCA has been performed for thefollowing cases:
(1) The new version of the Novikov–Wildhaber helical gear drive
(2) The modified involute helical gear drive, whose design is based on the followingideas: (i) a pinion rack-cutter with a parabolic profile and a conventional gear rack-cutter with a straight profile are applied for the generation of the pinion and thegear, respectively; and (ii) the pinion of the gear drive is double-crowned
The applied design parameters are shown in Table 17.8.1 for both the new version
of the Novikov–Wildhaber gear drive (case 1) and the modified involute helical gear
Table 17.8.1: Design parameters
Parabolic coefficient of pinion rack-cutter(a) , a c 0.016739 mm−1
Parabolic coefficient of pinion rack-cutter(b) , a c 0.016739 mm−1
(a)Novikov–Wildhaber helical gear drive
(b)Modified involute helical gear drive
Trang 6cutter a chas been used for both the new version of the Novikov–Wildhaber gear driveand the modified involute helical gear drive The parabolic coefficient of longitudinal
crowning a pl used in each case provides a limited error of 8 arcsec of the predesignedfunction of transmission errors for a gear drive without errors of alignment Figures17.8.1(a) and 17.8.1(b) show the path of contact for cases (1) and (2), respectively.Figure 17.8.1(c) shows the function of transmission errors for case (1) The function oftransmission errors for case (2) is similar and also provides a maximum transmissionerror of 8 arcsec The TCA output shows that a parabolic function of transmissionerrors is indeed obtained in the meshing of the pinion and the gear due to application
of a double-crowned pinion
The approaches chosen for TCA cover application of (i) a disk-shaped tool tion 17.6), (ii) a plunging worm (Section 17.7), and (iii) modified roll of feed motion
Trang 7(Sec-17.8 TCA of a Gear Drive with a Double-Crowned Pinion 499
Figure 17.8.2: Influence of errors of alignment in the shift of the path of contact for a Novikov– Wildhaber helical gear drive wherein the pinion is generated by plunging of the generating worm: (a) with error E [70 µm]; (b) with error γ [3 arcmin]; (c) with error λ [3 arcmin]; (d) with
γ + λ1 = 0 arcmin.
(Section 17.7) These approaches yield almost the same output of TCA The simulation
of meshing is performed for the following errors of alignment: (i) change of centerdistance E = 70 µm, (ii) change of shaft angle γ = 3 arcmin, (iii) error λ = 3
arcmin, and (iv) combination of errorsγ and λ as γ + λ = 0.
The results of TCA accomplished for the design parameters represented inTable 17.8.1 are as follows:
(1) Figures 17.8.1(a) and 17.8.1(b) show that the paths of contact of aligned geardrives are oriented longitudinally in both cases of design Novikov–Wildhabergears and modified helical gears Deviation from the longitudinal direction is lessfor modified involute helical gear drives in comparison with the new version ofthe Novikov–Wildhaber helical gear drive However, the advantage of the newNovikov–Wildhaber gear drive is the reduction of stresses (see Section 17.10).(2) Figures 17.8.2(a), 17.8.2(b), and 17.8.2(c) show the shift of the paths of contactcaused by errors of alignmentE, γ , and λ, respectively The shift of paths of
contact caused byγ may be compensated by correction λ1of the pinion (orλ2
of the gear) Figure 17.8.2(d) shows that the location of the path of contact can berestored by correction ofλ1of the pinion by takingγ + λ1= 0 This meansthat correction ofλ1can be used for the restoration of the location of the path
of contact Correction ofλ1orλ2may be applied in the process of generation
of the pinion or the gear, respectively
Trang 8It was previously mentioned (see Section 17.5) that double crowning of the pinionprovides a predesigned parabolic function of transmission errors Therefore, linear func-tions of transmission errors caused byγ , λ, and other errors are absorbed by the
predesigned parabolic function of transmission errors φ2(φ1) The final function oftransmission errorsφ2(φ1) remains a parabolic one However, increase of the magni-tude of errorsγ and λ may result in the final function of transmission errors φ2(φ1)
becoming a discontinued one In such a case, the predesigned parabolic function φ2(φ1)has to be of larger magnitude or it becomes necessary to limit the range ofγ , λ, and
other errors
17.9 UNDERCUTTING AND POINTING
The pinion of the drive is more sensitive to undercutting than the gear because the pinionhas a smaller number of teeth
F (u c , θ c , ψ σ)= 0 (17.9.3)
that relates parameters u c , θ c , and ψ σat a point of singularity of surface σ.The limitation of generating surface c for avoidance of singularities of generatedsurface σis based on the following procedure:
(1) Using equation of meshing f σc (u c , θ c , ψ σ)= 0 between the rack-cutter and the
pinion, we may obtain in plane of parameters (u c , θ c) the family of contact lines ofthe rack-cutter and the pinion Each contact line is determined for a fixed parameter
of motionψ σ
(2) The sought-for limiting line L [Fig 17.9.1(a)] that limits the rack-cutter surface
is determined in the space of parameters (u c , θ c) by simultaneous consideration
of equations f σ c = 0 and F = 0 [Fig 17.9.1(a)] Then we can obtain the limiting
Trang 917.9 Undercutting and Pointing 501
(mm)
θ
(mm)
Figure 17.9.1: Contact lines L σc and limiting line L: (a) in plane (u c,θ c), and (b) on surface c.
line L on the surface of the rack-cutter [Fig 17.9.1(b)] The limiting line L on the rack-cutter surface is formed by regular points of the rack-cutter, but these points will generate singular points on the pinion tooth surface.
Limitations of the rack-cutter surface by L enables us to avoid singular points on the
pinion tooth surface Singular points on the pinion tooth surface can be obtained by
coordinate transformation of line L on rack-cutter surface c to surface σ
Pointing
Pointing of the pinion means that the width of the topland becomes equal to zero.Figure 17.9.2(a) shows cross sections of the pinion and the pinion rack-cutter Point
A c of the rack-cutter generates the point A σ that is the limiting point of the cross
section of the pinion tooth surface which is still free of singularities Point B c of the
rack-cutter generates point B σ of the pinion profile Parameter s a indicates the chosenwidth of the pinion topland Parameter α t indicates the pressure angle at point Q Parameters h1and h2indicate the limitation of location of limiting points A c and B c
of the rack-cutter profiles Figure 17.9.2(b) shows functions h1(N1) and h2(N1) (N1isthe pinion tooth number) obtained for the following data:α d= 25◦,β = 20◦, parabola
Trang 10Figure 17.9.2: Permissible dimensions
h1and h2of rack-cutter: (a) cross tions of pinion and rack-cutter; (b) func-
[Litvin et al., 2001c] and is formed by a double-crowned helical pinion and a
conven-tional involute helical gear The second type of gear drive has been predesigned with aparabolic function of transmission errors, similar to the function of transmission errors
of the proposed version of Novikov–Wildhaber gear drives (see Section 17.8)
The difference between the two types of gear drives that have been investigated is thatthe Novikov–Wildhaber gear drives are generated by two parabolic rack-cutters that
Trang 11Development of Finite Element Models
The approach followed for finite element models is summarized in Section 9.5 and hasthe following characteristics:
(i) Finite element models of the gear drive are automatically obtained for any position
of pinion and gear obtained from tooth contact analysis (TCA) Stress convergence
is assured because there is at least one point of contact between contacting surfaces.(ii) Assumption of load distribution in the contact area is not required because the al-gorithm of contact of the general computer program [Hibbit, Karlsson & Sirensen,Inc., 1998] will determine it by application of torque to the pinion, whereas thegear is considered at rest
(iii) Finite element models of any number of teeth can be obtained As an example,Figure 17.10.1 shows a whole gear drive finite element model However, three- or
Figure 17.10.1: Whole gear drive finite
element model.
Trang 12Figure 17.10.2: Finite element model with three pairs of teeth.
Figure 17.10.3: Contact and bending stresses at the middle point of the path of contact on the pinion tooth surface for a Novikov–Wildhaber gear drive wherein the generation is performed by plunging of the grinding worm.
Trang 1317.10 Stress Analysis 505
Figure 17.10.4: Contact and bending stresses at the middle point of the path of contact on the pinion tooth surface for a modified involute helical gear drive wherein the generation is performed by plunging
of the grinding worm.
five-tooth models are more adequate for consideration of a more refined meshthat will allow the contact ellipses to be determined accurately The use of sev-eral pairs of contacting teeth in the finite element models has the followingadvantages:
(a) Boundary conditions are far enough from the loaded areas of the teeth.(b) Simultaneous meshing of two pairs of teeth can occur due to the elasticity ofsurfaces Therefore, the load transition at the beginning and at the end of thepath of contact can be studied
Numerical Example
Finite element analyses have been performed for the following cases:
(1) the new version of the Novikov–Wildhaber helical gear drive
(2) a modified involute helical gear drive
The applied design parameters are shown in Table 17.8.1 (see Section 17.8) The output
of TCA [see Figs 17.8.1(a) and 17.8.1(b)] allows the designer to design the finite elementmodel at any point of contact
Trang 14Contact Stresses (MPa)
Contact Stresses (MPa)
the properties of Young’s Modulus E = 2.068 × 105MPa and Poisson’s ratio of 0.29.
A torque of 500 Nm is applied to the pinion in both cases Figure 17.10.2 shows thefinite element mesh for case (1) at the mean contact point Figures 17.10.3 and 17.10.4show the maximum contact and bending stresses obtained at the mean contact pointfor cases (1) and (2), respectively
Trang 1517.10 Stress Analysis 507
Bending Stresses (MPa)
Bending Stresses (MPa)
in comparison with the modified involute helical gear drive