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Research Papers: Design of Direct Contact Systems

Revisiting Generation and Meshing Properties of Beveloid Gears OPEN ACCESS

[+] Author and Article Information
Alessio Artoni

Dipartimento di Ingegneria Civile e Industriale,
Università di Pisa,
Largo Lucio Lazzarino 2,
Pisa 56122, Italy
e-mail: alessio.artoni@ing.unipi.it

Massimo Guiggiani

Dipartimento di Ingegneria Civile e Industriale,
Università di Pisa,
Largo Lucio Lazzarino 2,
Pisa 56122, Italy
e-mail: massimo.guiggiani@ing.unipi.it

1Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 6, 2017; final manuscript received June 30, 2017; published online July 26, 2017. Assoc. Editor: Hai Xu.

J. Mech. Des 139(9), 093301 (Jul 26, 2017) (9 pages) Paper No: MD-17-1251; doi: 10.1115/1.4037345 History: Received April 06, 2017; Revised June 30, 2017

The teeth of ordinary spur and helical gears are generated by a (virtual) rack provided with planar generating surfaces. The resulting tooth surface shapes are a circle-involute cylinder in the case of spur gears, and a circle-involute helicoid for helical gears. Advantages associated with involute geometry are well known. Beveloid gears are often regarded as a generalization of involute cylindrical gears involving one additional degree-of-freedom, in that the midplane of their (virtual) generating rack is inclined with respect to the axis of the gear being generated. A peculiarity of their generation process is that the motion of the generating planar surface, seen from the fixed space, is a rectilinear translation (while the gear blank is rotated about a fixed axis); the component of such translation that is orthogonal to the generating plane is the one that ultimately dictates the shape of the generated, envelope surface. Starting from this basic fact, we set out to revisit this type of generation-by-envelope process and to profitably use it to explore peculiar design layouts, in particular for the case of motion transmission between skew axes (and intersecting axes as a special case). Analytical derivations demonstrate the possibility of involute helicoid profiles (beveloids) transmitting motion between skew axes through line contact and, perhaps more importantly, they lead to the derivation of designs featuring insensitivity of the transmission ratio to all misalignments within relatively large limits. The theoretical developments are confirmed by various numerical examples.

Involute helicoid gears are generally known as conical involute gears or taper gears or beveloid gears. Fundamental contributions to this field were made by Merritt [1], Beam [2], Smith [3], and Mitome [46], among others. Subsequently, studies concerned with analytical derivations of tooth geometry and tooth contact analysis (TCA) were presented (e.g., see Refs. [79]). Beveloid gears have been often defined, from a somewhat narrow perspective, as gears obtained by a suitable (usually small) inclination of the generating rack commonly used with spur/helical gears with respect to the axis of the gear being generated, and indeed they are also defined as gears with variable addendum correction across the face width.

This paper, also inspired by the enthralling work by Phillips [10] but pursuing a completely different approach, proposes a novel geometric framework to favor understanding and design of some fundamental concepts that underlie beveloid gears (and spur and helical gears as special instances of the general theory). The case of conjugate involute helicoid gears transmitting motion between skew axes through exact line contact is also obtained, which, to the best of our knowledge, has not been envisioned with beveloid gears (however, the recent works [11] and [12] do so approximately).

The geometric characteristics of the resulting involute helicoid tooth surfaces possess advantageous properties during meshing: (i) their transmission ratio is constant and insensitive to misalignments and (ii) regardless of misalignments, they possess a straight line-of-action, and therefore (1) reduced dynamic excitation and noise emission (due to the absence of built-in transmission errors), (2) larger tolerances allowed at assembly, and (3) the moment arms of the normal contact forces with respect to the axes of the two gear wheels do not change, which improves meshing. Also, fresh concepts are proposed to provide designers with practical geometric insights.

This section is devoted to the main subject of this work: gears whose tooth surfaces are generated (by envelope) by a tool provided with planar surfaces, hence a rack-cutter or any of its kinematically equivalent implementations, such as a hob or a shaper. This part is focused on the generation of tooth surfaces of a single gear. The most relevant relative motions between the gear and its generating plane are examined. We start out by briefly reviewing the well-known arrangement of spur (cylindrical) gears, which is necessary to contextualize the problem and to establish the notation; as we shall see, the very same concepts apply to beveloid gears with straight teeth. Subsequently, the more general framework pertaining to helical and beveloid gears is presented, and the resulting tooth surfaces are investigated.

The Planar Case: Spur Gears and Straight Beveloid Gears.

In order to mathematically describe the generation process on which spur gears are based, let us refer to the geometric arrangement in Fig. 1. A generating half-plane Π (hereafter generating plane) is displaced (translated) by d along the unit vector j, while the gear blank (whose shape is unimportant here) is rotated by θ about the z-axis, marked by the unit vector k. The reference frame F = (x, y, z) is fixed as well as its unit vectors (i, j, k). Note that the displacement of the plane is parallel to its unit normal nΠ (=j). Parameters s and t are the plane's Gaussian coordinates, and h is the distance of its lower edge from the (y, z) plane.

The displacement d and the rotation θ are not independent; they are related by a pre-assigned generation ratio r = d/θ, fixed and positive, dictated in practice by the axodes (centrodes in the planar case) of the tool and the gears being generated [13, chap. 3]. The rotation θ can be used as motion parameter; as θ varies, one obtains an enveloping family of planes whose envelope surface seen from the gear blank is the involute surface that characterizes spur gear teeth. In the following framework, the invariant approach [14] will be used, which treats vectors as such and does not need any chain of reference frames. It is based on the well-known rotation operator R: given a rotation axis a marked by a unit vector a and a vector p whose initial point belongs to a, R compactly expresses the rigid rotation of p about a by an angle ϕ as Display Formula

(1)R(p,a,ϕ)=(1cosϕ)(pa)a+cos(ϕ)p+sin(ϕ)(a×p)

With reference to Fig. 1, when θ = 0 (hence d = 0), a generic point of the plane Π is represented by the position vector pΠ(s, t) = (s + h) i + tk (with origin in point O), and the enveloping family of planes can be obtained by activating the rotation θ and the ensuing plane translation d (with d = ) Display Formula

(2)pe(s,t,θ)=(s+h)i+tk+d(θ)j=(s+h)i+tk+rθj

where h is constant. As we are interested in the envelope surface seen from the rotating blank, we need to superimpose a counter-rotation −θk to the whole system to make the blank stationary. By applying the counter-rotation −θk to pe about the z-axis, one obtains the enveloping family of planes p̂e(s,t,θ)=R(pe,k,θ) (now seen from the stationary blank), whose components in the fixed frame F read Display Formula

(3)p̂e(s,t,θ)=((h+s)cosθ+rθsinθ,(h+s)sinθ+rθcosθ,t)

The plane's normal nΠ must undergo the same counter-rotation, which results in Display Formula

(4)n̂Π(θ)=R(j,k,θ)=(sinθ,cosθ,0)

In order to obtain the envelope surface, the equation of meshing now requires that the normal n̂Π be orthogonal to the relative (geometric) velocity of the generating plane with respect to the (stationary) blank. This condition can be expressed by requiring Display Formula

(5)n̂Π(θ)θ(p̂e(s,t,θ))=0,whichresultsins+h=r

The necessary condition expressed by Eq. (5) does not involve θ nor t. For any value of θ, the unique line on Π characterized by s = r − h (and free t), hence fixed, is the generating contact line. Substituting Eq. (5) into the expression of the enveloping family of planes, Eq. (3), one obtains the following expression for the envelope surface (gear tooth surface): Display Formula

(6)pg(t,θ)=(r(cosθ+θsinθ),r(sinθ+θcosθ),t)

which is the parametric representation of a circle-involute cylinder in Cartesian coordinates. For any given value of t, which is equivalent here to cross sectioning the gear with a plane orthogonal to the z-axis at z = t, the resulting curve pg(t, θ) is a circle involute, whose base circle radius is just rb = r (Fig. 2). Also, for any given value of θ, the curve pg(t, θ) represents a contact line “printed” by the plane Π during generation.

The surface of action is the family of contact lines between the plane Π and the gear blank during generation as seen from the fixed space. In our framework, we just need to “undo” the counter-rotation that had been superimposed to the system, and in particular we only need now to rotate the envelope surface pg about the z-axis by θDisplay Formula

(7)pa(t,θ)=R(pg(t,θ),k,θ)=(r,rθ,t)

It is straightforward to see that this simple expression is that of a plane, and it is thus called plane of action (Fig. 2). It is parallel to the (y, z) plane, and its distance from it is just the base cylinder radius rb = r.

As is well-known, envelope surfaces may be affected by undercut during generation. The regularity of a surface is guaranteed wherever its normal vector does not vanish. Points where the surface is not regular are said to form lines of singular points. The normal vector of the envelope surface pg is Display Formula

(8)ng(θ)=pgt×pgθ=(rθsinθ,rθcosθ,0)

therefore it vanishes for θ = 0. The corresponding line of singular points, also known as edge of regression, is expressed by pg(t, 0) = (r, 0, t), which, not surprisingly, coincides with the start of the involute surface at the base cylinder. Singular points represent a warning about potential undercut during generation. Indeed, all the points of the plane Π having s < (r − h) would penetrate the base cylinder at θ = 0, resulting in undercut. The plane Π should therefore be bound below by the line having s = (r − h), which also coincides with the generating line. Such a limited active surface (just a line) is one of the reasons why racks with inclined (planar) surfaces have been devised (cf. next paragraph).

It is worth highlighting that, as a consequence of the equation of meshing, any displacement of the generating plane parallel to itself is irrelevant to the enveloping process. Figure 3 shows a situation where Π is displaced by d, inclined by an angle α with respect to the horizontal y-axis. Displacement d is not orthogonal to Π; the envelope tooth surface only depends on its component d and not on d. In positions (I) and (II), Π generates lines aI and aII, respectively, of the plane of action, with a* being the last line that can be generated. This is the way an ordinary generating rack operates; angle α in Fig. 3 is just its pressure angle. It is important to note that, provided r = d/θ is unaltered, α has no effect on the tooth surface shape.

If one now reviews the steps taken with spur gears, it is easy to realize that the very same concepts apply to beveloid gears with straight teeth (Fig. 4). With a given generation ratio, straight beveloids and spur gears have the same tooth surface shape, i.e., the same involute cylinder: practical differences between the two gear types are typically introduced by a rotation of Π about an axis parallel to its normal vector (needed with beveloids to accommodate the required root angle), and by the shape of their blanks, respectively, conical and cylindrical. This amounts to selecting two different portions of the same involute cylinder as final tooth surfaces. In this scenario, straight beveloids can be regarded as spur gears having a linearly varying profile shift along the face-width direction.

Generalization Through Helicality: Helical Gears and Beveloid Gears.

While it is now clear that displacements of the generating plane parallel to itself do not affect the final tooth shape, what can be said in terms of (absolute) rotations? It is plain to see that any rotation of the plane about its normal vector (parallel to j) is irrelevant, as the plane would just “slide on itself.” Also, rotation about an axis parallel to k would just lead to the situation in Fig. 3, with no implications if d = d. The remaining rotation, namely, that about an axis parallel to i, is indeed the only additional rotation of Π required to add full generality to the enveloping process. With reference to Fig. 5, and for reasons that will be clearer later, we call such rotation helicality and parameterize it by the angle δ. We assume again a predefined generation ratio r.

By following the same reasoning as in Sec. 2.1, analytical derivations result in the following necessary condition from the equation of meshing (cf. Eq. (5)): Display Formula

(9)s+h=rcosδ

which identifies the unique generating line on Π, and in the following expression for the envelope tooth surface (cf. Eq. (6)): Display Formula

(10)pg(t,θ)=[rcosδcosθ+(rθcosδtsinδ)sinθrcosδsinθ+(rθcosδtsinδ)cosθrθsinδ+tcosδ]

By introducing z=rθsinδ+tcosδ, the latter can be rewritten as Display Formula

(11)pg(z,θ)=[rcosδ(cosθ+θsinθ)ztanδsinθrcosδ(sinθ+θcosθ)ztanδcosθz]

At the cross section z = 0, the resulting curve is clearly a circle involute with base radius rb=r/cosδ. It can be easily shown that the tooth surface expressed by Eq. (11) is a generalized helicoid, namely, an involute helicoid, that can be obtained by rotating the circle involute about the z-axis by an angle ψ and, at the same time, displacing it parallel to k by z according to the constant ratio z/ψ=r/sinδ. Its base cylinder has radius rb=r/cosδ, which becomes rb = r in the planar case (all the relations of the planar case can be recovered by setting δ = 0).

In the presence of helicality, the plane of action reads Display Formula

(12)pa(z,θ)=(rcosδ,rcosδθztanδ,z)

and a comparative view with that of the planar case (δ = 0) is given in Fig. 6. Regarding undercut, the edge of regression now coincides with the curve, namely, a helix, where the involute helicoid meets its base cylinder and observations analogous to those made for the planar case apply here. Figure 7 shows involute tooth surfaces extending out of their base cylinders in the two cases δ = 0 and δ > 0.

It should now be evident that the fundamental shape of a tooth surface generated in the presence of helicality is just the same as that of ordinary helical gears, namely, an involute helicoid. In order to characterize helical and (nonstraight) beveloid gears, comments in perfect analogy with those in the last paragraph of Sec. 2.1 can be made.

Other aspects are also worth remarking. The fact that general beveloid gears are conical in shape is not a consequence of the fundamental geometric aspects described thus far; they are conical because their spatial arrangement generally requires that their blanks be conically cut for proper macrogeometry and tooth proportions. However, this is not a rule: as a matter of fact, a beveloid gear can correctly mesh with a cylindrical spur or helical gear. The tooth shapes of both helical and beveloid gears are the described involute helicoids, and their generation axodes are the described base cylinders. During meshing, their operating axodes (also called operating pitch surfaces) are cones in the case of intersecting axes, hyperboloids of revolution in the case of skew axes, and cylinders in the case of parallel axes. The terms involute helicoid gears and beveloid gears will be used interchangeably in the rest of this paper.

Once an involute helicoid gear has been generated, it can be regarded as intrinsically bound with its plane of action. Taking this into account, let us consider mating of involute helicoid gears and set out a principled reasoning to explore relevant geometric configurations. We start out with the well-known case of spur gears to facilitate presentation of the underlying concepts.

Generation/Meshing of Spur Gears.

For a pair of mating spur gears, installed at their nominal center distance c, one can imagine (Fig. 8) that the two gears first undergo suitable initial rotations αg and αp (synchronization) that make their planes of action coincident, then meshing proceeds (parameterized, e.g., by θg(θp)) line by line on the common plane of action. This is equivalent to considering a simultaneous generation of the two gears by a virtual, common generating plane Π (which, for the reasons discussed about undercut, should coalesce into a single line); at each instant, the mating tooth flanks and the plane are in contact at one common contact line.

Contact Analysis of Spur Gears.

Based on these considerations, one can perform TCA using directly the planes of action rather than the corresponding tooth surfaces (a one-to-one correspondence exists between the two), which makes the analytical expressions simple. To analyze contact at any meshing position, one needs to analytically determine which points of the rotated (synchronized) pinion and gear planes of action have

  1. (1)same position vector (with origin in a fixed point Of, taken on the gear axis) and
  2. (2)parallel contact normals (orthogonal to the common generating plane).

This can be mathematically required by the set of simultaneous equations Display Formula

(13a)ĝa(tg,θg;αg)=p̂a(tp,θp;αp)
Display Formula
(13b)n̂ag(αg)×n̂ap(αp)=0

where ĝa and p̂a are the position vectors of the planes of action of gear and pinion, respectively, rotated about their respective axes by the synchronization angles αg and αp, while n̂ag and n̂ap are their rotated normals. System (13) is a set of four independent scalar equations in six unknowns. Using the pinion rotation angle θp and its surface parameter tp as input, solution of Eq. (13) provides Display Formula

(14a)αp=±arccos(rp+rgc)
Display Formula
(14b)αg=αp+π
Display Formula
(14c)θg=(rprg)θp1rgc2(rp+rg)2
Display Formula
(14d)tg=tp

Equation (14a) shows that the pinion synchronization angle αp depends on the pinion and gear generation ratios (rp, rg), i.e., on the radii of their base cylinders as well as on the center distance c. The upper/lower sign selects drive/coast side operation. It should be noted that (the absolute value of) αp is the operating pressure angle. Equation (14c) expresses the dependence between gear rotation θg and pinion rotation θp (plus an offset): with a given transmission ratio τ=|ωg/ωp|, it poses the following condition on (rp, rg): Display Formula

(15)τ=|dθg/dθp|rg=rp/τ

It is also evident from Eq. (14c) that τ=|dθg/dθp|does not depend on c, which proves that the transmission ratio is insensitive to center distance variations (as is well known). Finally, Eq. (14d) specifies that any tg = tp satisfies the TCA conditions, thus revealing line contact for any input θp.

Generation/Meshing of Beveloid Gears.

This section describes the key ideas on which this work is based. Let us consider Fig. 9, which introduces rotations δg for the gear blank (positive if concordant with the unit vector λg) and δp for the pinion blank (positive if concordant with λp). If the orientation of the generating plane is unchanged, these rotation angles coincide with the gear and pinion helicality angles defined earlier. When such rotations are applied, we can distinguish between two cases:

  1. (1)δg = −δp: The pinion and gear rotation axes are still parallel. All other conditions being equal, the base cylinders undergo the same increase in radius, and the corresponding planes of action are still coincident. Or else, the initial base cylinders can be retained by properly reducing rp and rg = rp/τ, namely, by multiplying them by cosδg. In this scenario, gear and pinion are in line contact, and this involute helicoid gear pair is just a fully conjugate helical gear pair of ordinary practice.
  2. (2)δg ≠ −δp: In this circumstance, the pinion and gear axes are skew. Due to such nonparallelism, the base cylinders undergo different enlargements, and coincidence of the two planes of action cannot be recovered by a trivial synchronization. However, a special geometric layout can be designed, which allows to obtain an involute helicoid gear pair for motion transmission between skew axes with line contact, hence fully conjugate. The details are presented in Sec. 3.4.

The geometric arrangements described thus far enable motion transmission between parallel and skew axes, and they do so with line contact. The important case of intersecting axes seems to be excluded. However, to recover this case, only one additional rotation needs to be introduced, namely, a relative rotation between the pinion and the gear about an axis belonging to the common plane of action and directed as the generating plane's normal nΠ. With reference to Fig. 9, we apply such rotation to the pinion blank and call it spin. It is parameterized by the angle σ, positive if concordant with the unit vector nΠ. After skewing the pinion and gear axes through δp and δg, intersecting axes can be obtained by spinning the pinion axis through a suitable angle σ. This operation has an important implication, namely, that the two planes of action are not coincident any more. They now intersect at a line, parallel to nΠ, and along which contact evolves: the line-of-action. As a result, line contact is changed into point contact, but conjugality is not affected. This contact localization should not cause too much concern; indeed, microgeometry corrections (e.g., profile and lead crowning) change line contact into point contact. Obviously, this localization effect should not be too severe in order to avoid large contact stresses which penalize load-carrying capacity. In Sec. 3.4, the claims made here will be demonstrated analytically using the TCA tools already employed for spur gears.

Contact Analysis of Beveloid Gears.

With the help of Fig. 10, let us first define a geometric setup for the skew axes case with zero spin. If Σ is the nominal shaft angle, the pinion and gear helicality angles must be such that

Display Formula

(16)Σ=δg+δp

We erect one fixed reference system Sf = (Of; xf, yf, zf) as indicated in Fig. 10, which shows the overall setup (compare with Fig. 9). The fixed point Of is taken on the gear axis. Symbol c denotes here the shortest distance between the pinion and gear axes. During generation/meshing, the common generating plane is always parallel to the fixed (xf, zf) plane, while the two planes of action, orthogonal to it and tangent to the base cylinders, need to be coincident for meshing.

When spin σ is applied to the pinion, the situation changes as shown in Fig. 11. Now contact evolves point by point along the line-of-action, such line coincides with the spin axis, whose coordinates (xs, zs) can be selected at will provided the axis belongs to the gear's plane of action and is parallel to the yf-axis (i.e., orthogonal to the common generating plane). As such, it will be tangent to the base cylinders. It is interesting to note that with larger zs values, one can obtain increased distance of the meshing zone from the base cylinders, which results in favorable tooth curvatures. However, a tradeoff with the increased overall dimensions of the gear pair has to be made.

Comparing the proportions of the various geometric entities in Fig. 11 with those in Fig. 10, they appear quite different. As a matter of fact, introduction of spin causes the actual shaft angle and shortest distance values to deviate from their nominal specifications (see, e.g., the effective c in Fig. 11). In order to restore their nominal values, some geometric constraints (omitted here for brevity) need to be satisfied, particularly in terms of pinion or gear helicality angle and pinion vertical displacement c0 (tantamount to c in Fig. 10). Eventually, one has the following set of free design parameters (other choices are possible): shaft angle Σ, transmission ratio τ, shortest distance c, spin angle σ, gear helicality angle δg, and z-coordinate of the spin axis zs.

We are now ready to analyze meshing and contact for general involute helicoid gears with both helicality and spin by imposing the TCA conditions Display Formula

(17a)ĝa(tg,θg;αg)=p̂a(tp,θp;αp)
Display Formula
(17b)n̂ag(αg)=μn̂ap(αp)

which constitute a system of six scalar equations in seven unknowns. Its solution is (with θp as input parameter) Display Formula

(18a)αp=0
Display Formula
(18b)αg=π
Display Formula
(18c)θg=(rprg)θp
Display Formula
(18d)tp=zs+(c0+xsrpcosδp)cotσ(xs+rgcosδg)cscσ
Display Formula
(18e)tg=zs+(c0+xsrpcosδp)cscσ(xs+rgcosδg)cotσ

Equations (18a) and (18b) specify the required initial rotations for synchronization, while Eq. (18c) requires rg = rp/τ to obtain the nominal transmission ratio. For any input θp, Eqs. (18d) and (18e) provide two distinct values for tp and tg, implying point contact.

Solution of the special case with zero spin can be obtained from Eq. (18) by setting σ = 0. Equations (18d) and (18e) become Display Formula

(19a)tp=tg
Display Formula
(19b)c=c0=rpcosδp+rgcosδg=rbp+rbg

As anticipated while discussing this special geometric arrangement for motion transmission between skew axes, Eq. (19a) is a confirmation of line contact. On the other hand, Eq. (19b) expresses a stringent geometric constraint, i.e., the base cylinder radii must add up to the nominal shortest distance c: any shortest distance variation would not be tolerated (or better, that would be accommodated, but it would cause line contact to change into point contact).

The fact that involute helicoid gears are generated by a translating plane has remarkable consequences in terms of sensitivity of contact properties to misalignments.

Let us consider the general case of motion transmission between skew/intersecting axes, in the presence of both helicality and spin. The same misalignments (or assembly errors) as in the case of hypoid gears (Fig. 12) apply here, namely, offset (or shortest distance) error E, pinion axial error P, gear axial error G, and shaft angle error β (usually called α, but designated β here to avoid confusion with the synchronization angles αg and αp).

Taking Fig. 11 into account, let us first examine linear errors E, P, and G. As they do not change the orientations of the planes of action of pinion and gear in mesh, collinearity of contact normals is unaltered. As a consequence of such displacements, the line-of-action will change its location in the fixed space, but not its orientation, and the transmission function will include a constant offset, but with no repercussions on transmission ratio.

Regarding the angular misalignment β, its effect is a bit more involved. The TCA condition n̂ag(αg)=μn̂ap(αp) requires parallelism between the in-plane normals of the pinion and gear planes of action (each normal is orthogonal to the contact lines forming the respective plane of action). As already discussed, proper rotations αg and αp of the two blanks are required to obtain such parallelism. In particular, as we are transmitting motion between external gears, our geometric setup requires μ = −1, i.e., that the two normals be opposite. With some geometric manipulation, it turns out that the previously mentioned TCA condition can be satisfied only if the shaft angle error β ranges within

βmin=δg+δpΣandβmax=32π2δg+δpΣ

While its upper bound is wide enough with ordinary design architectures, its lower bound is rather restrictive, and it is zero with no spin (which δp depends on). However, as the spin angle is increased, βmin quickly drops by a few degrees, which is satisfactory for practical applications. When βmin < β < βmax, the line-of-action will generally change both its location and orientation, and the transmission function will include again a constant offset without producing any transmission errors.

Such a low sensitivity to misalignments is a remarkable feature of beveloid gears. Absence of built-in transmission errors and fixed orientation of the line-of-action are certainly advantageous in containing noise and vibration levels, and they grant wider tolerances at assembly. More quantitative details will be given in a numerical example.

Visualization of relevant results discussed in this paper and quantitative assessment of sensitivity to misalignments are the topic of this section. The figures presented in this section have been generated by numerically evaluating the theoretical results described thus far. As the primary application of the beveloid gears under study is motion transmission between skew and intersecting axes, visual examples for these two cases will be provided; then, the effects of misalignments on transmission ratio and line-of-action will be numerically assessed.

Skew Axes (With Spin).

Figure 13 shows the pair of involute helicoid tooth surfaces synthesized from the following design data:

τ=1:3Σ=45degc=20mmδg=25degσ=10degzs=60mm

In the shown meshing position, it can be noticed that the two instantaneous contact lines form an angle equal to the spin angle σ, and they intersect at the line-of-action (the latter being tangent to the base cylinders). Which portions of the involute helicoid flanks should be selected for the actual teeth is an important matter to be dealt with. As already mentioned, the more the flanks depart from their base cylinders, the more tooth curvatures decrease, which reduces contact pressures. Also, reduction of sliding velocity is crucial in terms of mechanical efficiency; therefore, it would be beneficial to keep the meshing zone in close proximity to the instantaneous screw axis (ISA) of the relative motion.

Skew Axes (Zero Spin).

Using the same data as in the previous example, but setting the spin angle to zero, the situation represented in Fig. 14 was obtained. Line contact is now clear to see. It should be recalled that this case is an “overconstrained” geometric scenario, not accommodating shortest distance variations without altering the nature of contact. Any misalignment would change line contact into point contact, but conjugate action would be preserved (allowing for the discussed limitations on the shaft angle error β). On this subject, it should be realized that spur and helical gears are overconstrained too; as an example, they are not tolerant to certain angular misalignments, in particular to the so-called line-of-action misalignment, which causes the contact pattern to abruptly shift to a tooth end, originating edge-loading. As a matter of fact, flank modifications (or microgeometry corrections, see Refs. [16,17]) are always introduced to allow proper operation, which could also be the case for the special involute helicoids at hand, and some slight spin would entail the same effect (but small enough to remain in the neighborhood of line contact conditions, which is obviously advantageous in restraining contact pressures).

Intersecting Axes.

Figure 15 shows the pair of involute helicoid tooth surfaces synthesized from the following design data:

τ=1:2Σ=90degc=0mmδg=40degσ=40degzs=30mm

This case of intersecting axes obviously requires the presence of spin. In the shown meshing position, the two instantaneous contact lines meet at a point belonging to the ISA: the relative velocity (sliding) between the two (extended) tooth surfaces at their contact point is zero. In order to maximize mechanical efficiency, portions of the two involute helicoid surfaces in the neighborhood of such point should be selected as the actual tooth surfaces.

Sensitivity to Misalignments.

The results presented in this section were obtained by numerically solving the TCA equations including the presence of all the four misalignments (E, P, G, β). This example was created from the following basic design data:

τ=1:2Σ=49degc=6.625mmδg=25degσ=20degzs=50mm

The effects of the following two sets of misalignments were assessed:

m1=(E1,P1,G1,β1)=(1.5mm,2.5mm,1.5mm,0.0deg)m2=(E2,P2,G2,β2)=(0.0mm,0.0mm,0.0mm,1.0deg)

In set m1, only linear misalignments are nonzero, while set m2 only includes a nonzero angular misalignment. In both cases, the misalignment values are very large.

The results in terms of transmission ratio sensitivity confirm the analysis done in Sec. 4; in all cases, the transmission ratio remains unchanged, that is, τ=|dθg/dθp|=1/2. Regarding the effects on the line-of-action, Fig. 16 shows the results corresponding to the three cases (nominal, m1, and m2). We still have lines of action: the one pertaining to case m1 is parallel to the nominal one, while the one corresponding to case m2 is not. Again, these results provide a numerical confirmation of the theoretical analysis.

In this paper, we have investigated a novel geometric framework to foster understanding of involute helicoid/beveloid gears. The newly defined concepts of helicality and spin should be taken into account when dealing with macrogeometry design of such gears. Theoretical derivations and numerical examples in support of the proposed framework have been presented, and the remarkable characteristics of involute helicoid surfaces regarding sensitivity of their meshing properties to generic misalignments have been demonstrated. Current work is devoted to delving further into practical macrogeometry design in terms of blank dimensions, tooth size and proportions, contact ratio, etc. Also, mechanical performance of involute helicoid gears will be assessed, particularly in comparison with spiral bevel and hypoid gears of ordinary practice.

The support of Avio Aero under contract SPINS is gratefully acknowledged.

Merritt, H. E. , 1954, “ Conical Involute Gears,” Gears, Issac Pitman and Sons, London, pp. 165–170.
Beam, A. S. , 1954, “ Beveloid Gearing,” Mach. Des., 26, pp. 220–238.
Smith, L. J. , 1989, “ The Involute Helicoid and the Universal Gear,” American Gear Manufacturers Association, Alexandria, VA, AGMA Technical Paper No. 89FTM10. https://www.geartechnology.com/articles/1190/The_Involute_Helicoid_and_The_Universal_Gear/
Mitome, K. , 1983, “ Conical Involute Gear—Part 1: Design and Production System,” Bull. JSME, 26(212), pp. 299–305. [CrossRef]
Mitome, K. , 1983, “ Conical Involute Gear—Part 2: Design and Production System of Involute Pinion-Type Cutter,” Bull. JSME, 26(212), pp. 306–312. [CrossRef]
Mitome, K. , 1985, “ Conical Involute Gear—Part 3: Tooth Action of a Pair of Gears,” Bull. JSME, 28(245), pp. 2757–2764. [CrossRef]
Brauer, J. , 2002, “ Analytical Geometry of Straight Conical Involute Gears,” Mech. Mach. Theory, 37(1), pp. 127–141. [CrossRef]
Innocenti, C. , 1997, “ Analysis of Meshing of Beveloid Gears,” Mech. Mach. Theory, 32(3), pp. 363–373. [CrossRef]
Liu, C.-C. , and Tsay, C.-B. , 2002, “ Contact Characteristics of Beveloid Gears,” Mech. Mach. Theory, 37(4), pp. 333–350. [CrossRef]
Phillips, J. , 2003, General Spatial Involute Gearing, Springer-Verlag, Berlin. [CrossRef]
Wu, S. , and Tsai, S. , 2009, “ Geometrical Design of Skew Conical Involute Gear Drives in Approximate Line Contact,” Proc. Inst. Mech. Eng. C, 223(9), pp. 2201–2211. [CrossRef]
Zhu, C. , Song, C. , Lim, T. C. , and Vijayakar, S. , 2012, “ Geometry Design and Tooth Contact Analysis of Crossed Beveloid Gears for Marine Transmissions,” Chin. J. Mech. Eng., 25(2), pp. 328–337. [CrossRef]
Litvin, F. L. , and Fuentes, A. , 2004, Gear Geometry and Applied Theory, Cambridge University Press, Cambridge, UK. [CrossRef]
Di Puccio , F., Gabiccini , M., and Guiggiani , M. , 2005, “ Alternative Formulation of the Theory of Gearing,” Mech. Mach. Theory, 40(5), pp. 613–637. [CrossRef]
Artoni, A. , Gabiccini, M. , and Kolivand, M. , 2013, “ Ease-Off Based Compensation of Tooth Surface Deviations for Spiral Bevel and Hypoid Gears: Only the Pinion Needs Corrections,” Mech. Mach. Theory, 61, pp. 84–101. [CrossRef]
Artoni, A. , Guiggiani, M. , Kahraman, A. , and Harianto, J. , 2013, “ Robust Optimization of Cylindrical Gear Tooth Surface Modifications Within Ranges of Torque and Misalignments,” ASME J. Mech. Des., 135(12), p. 121005. [CrossRef]
Artoni, A. , Gabiccini, M. , Guiggiani, M. , and Kahraman, A. , 2011, “ Multi-Objective Ease-Off Optimization of Hypoid Gears for Their Efficiency, Noise, and Durability Performances,” ASME J. Mech. Des., 133(12), p. 121007. [CrossRef]
Copyright © 2017 by ASME
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References

Merritt, H. E. , 1954, “ Conical Involute Gears,” Gears, Issac Pitman and Sons, London, pp. 165–170.
Beam, A. S. , 1954, “ Beveloid Gearing,” Mach. Des., 26, pp. 220–238.
Smith, L. J. , 1989, “ The Involute Helicoid and the Universal Gear,” American Gear Manufacturers Association, Alexandria, VA, AGMA Technical Paper No. 89FTM10. https://www.geartechnology.com/articles/1190/The_Involute_Helicoid_and_The_Universal_Gear/
Mitome, K. , 1983, “ Conical Involute Gear—Part 1: Design and Production System,” Bull. JSME, 26(212), pp. 299–305. [CrossRef]
Mitome, K. , 1983, “ Conical Involute Gear—Part 2: Design and Production System of Involute Pinion-Type Cutter,” Bull. JSME, 26(212), pp. 306–312. [CrossRef]
Mitome, K. , 1985, “ Conical Involute Gear—Part 3: Tooth Action of a Pair of Gears,” Bull. JSME, 28(245), pp. 2757–2764. [CrossRef]
Brauer, J. , 2002, “ Analytical Geometry of Straight Conical Involute Gears,” Mech. Mach. Theory, 37(1), pp. 127–141. [CrossRef]
Innocenti, C. , 1997, “ Analysis of Meshing of Beveloid Gears,” Mech. Mach. Theory, 32(3), pp. 363–373. [CrossRef]
Liu, C.-C. , and Tsay, C.-B. , 2002, “ Contact Characteristics of Beveloid Gears,” Mech. Mach. Theory, 37(4), pp. 333–350. [CrossRef]
Phillips, J. , 2003, General Spatial Involute Gearing, Springer-Verlag, Berlin. [CrossRef]
Wu, S. , and Tsai, S. , 2009, “ Geometrical Design of Skew Conical Involute Gear Drives in Approximate Line Contact,” Proc. Inst. Mech. Eng. C, 223(9), pp. 2201–2211. [CrossRef]
Zhu, C. , Song, C. , Lim, T. C. , and Vijayakar, S. , 2012, “ Geometry Design and Tooth Contact Analysis of Crossed Beveloid Gears for Marine Transmissions,” Chin. J. Mech. Eng., 25(2), pp. 328–337. [CrossRef]
Litvin, F. L. , and Fuentes, A. , 2004, Gear Geometry and Applied Theory, Cambridge University Press, Cambridge, UK. [CrossRef]
Di Puccio , F., Gabiccini , M., and Guiggiani , M. , 2005, “ Alternative Formulation of the Theory of Gearing,” Mech. Mach. Theory, 40(5), pp. 613–637. [CrossRef]
Artoni, A. , Gabiccini, M. , and Kolivand, M. , 2013, “ Ease-Off Based Compensation of Tooth Surface Deviations for Spiral Bevel and Hypoid Gears: Only the Pinion Needs Corrections,” Mech. Mach. Theory, 61, pp. 84–101. [CrossRef]
Artoni, A. , Guiggiani, M. , Kahraman, A. , and Harianto, J. , 2013, “ Robust Optimization of Cylindrical Gear Tooth Surface Modifications Within Ranges of Torque and Misalignments,” ASME J. Mech. Des., 135(12), p. 121005. [CrossRef]
Artoni, A. , Gabiccini, M. , Guiggiani, M. , and Kahraman, A. , 2011, “ Multi-Objective Ease-Off Optimization of Hypoid Gears for Their Efficiency, Noise, and Durability Performances,” ASME J. Mech. Des., 133(12), p. 121007. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Basic geometric layout for generation of spur gear teeth

Grahic Jump Location
Fig. 2

Spur gear scenario: front view of involute tooth surfaces, their base cylinder, and the plane of action (obtained with θ ranging between θ = −70 deg and θ+ = 70 deg)

Grahic Jump Location
Fig. 3

Generation with the generating plane displaced not orthogonally to itself (negative θ only)

Grahic Jump Location
Fig. 4

A straight beveloid gear pair for motion transmission between intersecting axes

Grahic Jump Location
Fig. 5

Basic geometric layout for generation with helicality

Grahic Jump Location
Fig. 6

Planes of action and base cylinders for the cases δ = 0 (planar, spur gear case) and δ = 20 deg, the generation ratio r being equal

Grahic Jump Location
Fig. 7

Involute helicoid tooth surfaces extending out of their base cylinders for the cases δ = 0 (planar, spur gear case) and δ = 20 deg

Grahic Jump Location
Fig. 8

A pair of spur gears in mesh: (a) gears to be synchronized and (b) synchronized gears in mesh

Grahic Jump Location
Fig. 9

Helicality rotations δp and δg of pinion and gear blanks and pinion spin σ

Grahic Jump Location
Fig. 10

Three-dimensional setup for involute helicoid gears transmitting motion between skew axes (no spin)

Grahic Jump Location
Fig. 11

Three-dimensional setup for involute helicoid gears transmitting motion between skew axes in the presence of spin (σ)

Grahic Jump Location
Fig. 12

Pitch cones expressing the general relative position between hypoid pinion and gear, in the presence of misalignments E, P, G, and β. Here, deviations from the common 90 deg shaft angle layout are depicted. (Other symbols: e, shaft offset or nominal shortest distance; Σ, nominal shaft angle; and C.P., crossing point. Adapted from Ref. [15].)

Grahic Jump Location
Fig. 13

Involute helicoid tooth surfaces for motion transmission between skew axes, with spin

Grahic Jump Location
Fig. 14

Involute helicoid tooth surfaces for motion transmission between skew axes, zero spin

Grahic Jump Location
Fig. 15

Involute helicoid tooth surfaces for motion transmission between intersecting axes (with spin)

Grahic Jump Location
Fig. 16

Influence of misalignments on the line-of-action (all units in millimeters)

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