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Pitch cone design and influence of misalignments on tooth contact behaviors of crossed beveloid gears
Mechanism and Machine Theory 59 (2013) 48–64
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Pitch cone design and influence of misalignments on tooth contact behaviors of crossed beveloid gears Caichao Zhu a, Chaosheng Song a, b,⁎, Teik C. Lim b, Tao Peng c a b c
The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China School of Dynamic Systems, Mechanical Engineering, 598 Rhodes Hall, P.O. Box 210072, University of Cincinnati, Cincinnati, OH 45221,USA ArvinMeritor Inc., Troy, MI 48084, USA
a r t i c l e
i n f o
Article history: Received 8 September 2011 Received in revised form 23 February 2012 Accepted 7 August 2012 Available online 27 September 2012 Keywords: Beveloid gears Pitch cone design Tooth contact analysis Gear misalignments
a b s t r a c t A pitch cone design theory assuming line contact is formulated to analyze the tooth engagement behavior of crossed beveloid gears. The analysis applies the spatial gearing model and beveloid gear manufacturing technique to determine the cutter parameters and geometry. Using the proposed procedure, a crossed beveloid gear pair with small shaft angle is studied. Also, the proposed formulation is capable of quantifying the influence of misalignments including offset and angular position errors on the contour and position of contact path and lines over a complete mesh cycle. The results show that both offset and angular position errors of the mating members worsen the conjugate action of the mating tooth surfaces. Furthermore, both the contact path and the transmission error are more sensitive to the angular error than the offset error. The loaded tooth contact characteristics are subsequently modeled using the finite element method for comparison purpose. The proposed model is partially validated by comparing the numerical simulation results. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction In modern gearing application of power transmission between two crossed shafts at large angles (close to 90°), hypoid or spiral bevel gears, face gears and worm gears are superior to involute helical gears. However, it is not practical to use them when the crossed shaft angle is less than 45°, due to design and manufacturing difficulties [1]. Instead, beveloid gears that are also known as involute conical gears are better suited to accommodate applications having a small shaft angle with tapered tooth thickness. Furthermore, these types of gears can be manufactured like cylindrical gears using a hobbing process or a threaded-wheel grinding process. One of the advantages of beveloid gears is the allowance to backlash. The main challenge in crossed axis application is the fact that the contact path is theoretically pointwise with high sliding velocity that can affect tooth surface durability. In the last couple of decades, a number of studies have been performed on the beveloid gearing theory, design and manufacturing [2–9]. For instance, Mitome and Yamazaki [2,3] proposed a new type of concave-conical gearing with better curvature relation at the contact points for 45-degree crossed shafts. The work also includes an experimental verification of the predicted contact pattern. Brauer [4] derived a mathematical model of spur beveloid gear with parallel axes and applied the model for backlash analysis. Liu and Tsay [5] derived the equations to simulate the generating process, and studied the theoretical meshing and bearing contact of the beveloid gears. Tsai and Wu [6,7] proposed two methods to design the geometry of beveloid gear with crossed axes. Also the generation process and contact characteristics for conventional beveloid gears were studied. Ohmachi et al. [8] conducted the tooth surface contact analysis using a conjugate theory for the practical design of conical involute gears. Li et al. [9] proposed a new type of non-involute beveloid gear and analyzed the meshing characteristics. In spite of the ⁎ Corresponding author at: The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China. Tel.: +86 13883402720. E-mail address: [email protected] (C. Song). 0094-114X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2012.08.008
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Nomenclature di mounting distance Di heights of pitch cones dwi working mounting distances E offset ki tooth contact lines mn normal module of the beveloid gear mnw normal module of the common rack j j j nxc , nyc , nzc cutter surface normal vectors ⇀n′ i normal vectors of pitch cone surface nfi normal vectors at pitch point in the global fixed coordinate system Ni tooth numbers ni normal vector in the coordinate system fixed to pinion and gear Pn gear diametral pitch pc circular pitch Rfi position vectors at pitch point in the fixed global coordinate system Ri position vectors of pinion and gear rj reference pitch circle radius Sf fixed global coordinate system S′i pinion or gear coordinates used in pitch cone design Sb fixed coordinate system to define assembly errors Sc plane axode coordinate system fixed to the imaginary rack cutter Sn, Sp auxiliary rack cutter coordinate systems Si pinion or gear coordinate system ui generatrix coordinates of pitch cone θi rotational coordinates of pitch cone x j, y j, z j beveloid gear surface coordinates xcj, ycj, zcj frame coordinates fixed to cutter x′i, y′i, z′i surface coordinates of pitch cones αn normal pressure angle αnw common rack normal pressure angle αPL(R)j transverse plane pressure angles αPLw(R)j transverse plane working pressure angle β helix angle βwi working spiral angle γj angle relative to the plane axode coordinate system γwi working pitch angle δ shaft angle η angle between the generatrices ϕj gear rotation angle during generating process ϕ′i rotational angles τi generatrices of the pitch cones ξ crossing angle between the first principal directions of the tooth surface curvatures ξLj, ξRj intersection angle between the contact line k and the pitch plane of the common rack ξn normal angular factor ξtj transverse angular factor Σi gear and pinion tooth surfaces Subscripts i refers to the pinion or gear: i = 1 for pinion, i = 2 for gear j refers to the cutter surfaces: j = P for pinion, j = G for gear L refers to the left tooth surface R refers to the right tooth surface f refers to fixed coordinate system
above-mentioned studies focusing on geometry design and manufacturing of beveloid gears, none applied the pitch cone design with line contact and none studied the influence of misalignments on tooth contact behavior in crossed beveloid gears with line contact condition. These gaps will be the emphasis of our current paper.
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This paper presents an analytical derivation of working pitch cone design with line contact. The influence of misalignments including offset and angular position errors on the profile and position of contact path and transmission error is studied through a theoretical tooth contact model based on the idealized geometry. Also, loaded tooth contact analysis using the finite element theory is also performed to demonstrate the salient features of the proposed approach. The analysis results can be applied to guide the design of crossed beveloid gear geometry. 2. Beveloid gear tooth profile According to the generation concept for beveloid gear proposed by Mitome [10], the envelope surface of the hob during the manufacturing process can be considered as a rack cutter. As shown in Fig. 1, the normal section of the rack cutter consists mainly of two parts including the straight edges and the fillet curves that generate the working tooth surfaces and fillet surfaces of the gear, respectively. Fig. 2 illustrates a schematic of the generation mechanism and the coordinate relationship between the normal section of the rack cutter and the generated beveloid gear. Based on the theory of gearing, the cutting process can be expressed by using the a series of transformation matrices given by Sn → Sp → Sc → Sb → S1. Then, the rack cutter surfaces can be represented in coordinate system Sc as follows: h i j j j j j j j j j j j x c ¼ l cosα n −a cosγ þ ∓ l sinα n −a tanα n −b sinβ þ u cosβ sinγ
ð1Þ
j j j j j j j yc ¼ l sinα n −a tanα n −b cosβ þ u sinβ
ð2Þ
h i j j j j j j j j j j j zc ¼ − l cosα n −a sinγ þ ∓ l sinα n −a tanα n −b sinβ þ u cosβ cosγ
ð3Þ
where superscript j refers to the cutter surfaces (j = P for pinion, j = G for gear). The upper sign of Eqs. (1)–(3) indicates the left-side straight edge, while the lower sign represents the right-side straight edge. The notations xcj,ycj,zcj represent the position vector on the cutter surface, αn denotes the normal pressure angle, symbols Pn and pc represent the gear diametral pitch and
Fig. 1. Normal cross section of a rack cutter.
xP
γj
β
xb
xn
xc j
zP zc
o p on = u j
zn on
I
yn
v oc , oP
x1 ϕ1 I O1 Ob
yc , yP
ϕ1 y1
zb , z1 yb
rP1
ω1 Fig. 2. Coordinate relationship between the normal cross section of rack cutter and the beveloid gear.
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circular pitch, respectively, β j represents the helix angle on the pitch plane of imaginary rack cutter, and γ j is the angle with respect to the plane axode coordinate system. Furthermore, l = |M0M1| represents the distance measured from the initial point M0, moving along the straight line M0M2, to point M1. Hence, as proposed by Tsay [7], the mathematical model of the beveloid gear tooth surface in coordinate S1 can be represented follows: j j j x ¼ x c cosϕ j −y c sinϕ j þ r j cosϕ j þ ϕ j sinϕ j
ð4Þ
j j j y ¼ x c sinϕ j þ y c cosϕ j þ r j sinϕ j −ϕ j cosϕ j
ð5Þ
j
j
z ¼ zc
ð6Þ
where the meshing equation is given by j j j j j ϕj ¼ nxc yc −nyc xc =nxc r j j
j
ð7Þ
j
and where x ,y ,z represent the position vector on the tooth surface on the beveloid gear, rj denotes the reference pitch circle j j radius, ϕj represents the gear rotation angle in the generating process, and nxc ,nyc represent the components of a normal vector of the point on the cutter surface. 3. Working pitch cone design 3.1. Pitch cone parameters According to Litvin's gearing theory [11], the design of pitch cones is used to express the relationship between parameters γw1,γw2,βw1,βw2, where γwi is the working pitch angle and βwi is the working spiral angle (i = 1 for the pinion, and i = 2 for the gear). Fig. 3 shows a specific layout for a beveloid gear pair with three distinct coordinate systems to be used in the mathematical derivation. The first two coordinate systems S′i are attached to the pinion and gear bodies. The third one is the global coordinate system Sf that is fixed to a reference frame supporting the gear pair. The two cones are tangent at point P. Unit vectors τ 1 and τ 2 represent the generatrices of the pitch cones that lie in the pitch plane and intersect each other at the pitch point P. A pitch cone surface can be represented analytically in the local coordinate system S′i, where M is one position point on the surface of the cone and (ui, θi) are the surface coordinates (i.e. the Gaussian coordinates), as shown in Fig. 4. The mathematical model of the pitch cone can be represented by ′
′
′
x i ¼ ui sinγ wi cosθi ; y i ¼ ui sinγwi sinθi ; z i ¼ ui cosγwi :
ð8Þ
The surface unit normal is represented by ′
⇀n i ¼ ½ cosθi cosγwi
sinθi cosγ wi
T
− sinγ wi :
ð9Þ
According to Fig. 3, the pitch cones are in tangency at the pitch point P, and the equations of tangency are given by, Rf 1 ðθ1 ; u1 Þ ¼ Rf 2 ðθ2 ; u2 Þ
Fig. 3. Relationship between the local coordinate system S′i where i = 1,2, and the fixed coordinate system Sf.
ð10Þ
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Fig. 4. Operating pitch cone and its parameters.
nf 1 ðθ1 ; u1 Þ ¼ nf 2 ðθ2 ; u2 Þ:
ð11Þ
Substituting (sin θi) 2 + (cos θi) 2 = 1, and the working pitch circle radius rwi = ui sin γwi into the above equations, it yields the following set of formulas: E cosγw1 sinδ cosγ w1 −r w2 r w1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosγ w2 2 2 2 sin δ− sin γw1 − sin γw2 −2 sinγw1 sinγ w2 cosδ r w2 þ d1 ¼ − sinγ w1 cosγ w2
d2 ¼
E sin2 δ− sin2 γ w1 − sinγ w1 sinγw2 cosδ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinγ w1 sinδ sin2 δ− sin2 γ w1 − sin2 γw2 −2 sinγ w1 sinγw2 cosδ
rw2 Eð sinγw1 cosδ þ sinγ w2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − sinγw2 cosγ w2 sinδ sin2 δ− sin2 γ w1 − sin2 γ w2 −2 sinγ w1 sinγ w2 cosδ
ð12Þ
ð13Þ
ð14Þ
where Rfi and nfi(i = 1, 2) are the surface coordinates and normal vectors of the points on the pitch cone, δ is the shaft angle, E is the offset, and di(i = 1, 2) is the mounting distance of the pinion and gear. Fig. 5 shows the longitudinal shapes that are in tangency at P. The longitudinal shape of the tooth in the pitch plane is the intersecting curve of the tooth surface with the pitch plane. The generatrices of the pitch cones form the angle η that is represented by the equation 1 2
cosη ¼ τ τ :
ð15Þ
Then, taking into account that η = βw1 ± βw2 as shown in Fig. 5, the following equation can be obtained cosðβw1 βw2 Þ ¼ tanγ w1 tanγ w2 þ
cosδ : cosγ w1 cosγw2
ð16Þ
Using Eqs. (12) and (16), the offset can be derived as E¼
ðr Pw1 cosγ w2 þ r Pw2 cosγ w1 Þ sinðβw1 βw2 Þ : sinδ
ð17Þ
3.2. Line contact condition In theory, a beveloid gear pair with non-parallel axes engages in point contact. However, that may not be the case in practice. Therefore, our aim here is to determine the relations of pitch cone parameters to realize the approximate line contact. As shown in Fig. 6(a), a beveloid gear pair engages with the common rack in line contact. Here, k1 and k2 are the contact lines with the direction of observation from the toe to the heel. The angle ξ is defined as the crossing angle between the first principal directions of the tooth surface curvatures (FPD-angle).
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(a) Same direction of helices
(b) Opposite direction of helices
Fig. 5. Orientation of relative velocity at the pitch point.
According to Fig. 6(a), the cross angle between k1 and k2 can be represented as follows ξRðLÞ ¼ ξRðLÞ1 þ ξRðLÞ2 :
ð18Þ
Based on spatial geometry relationship between the beveloid gear and the common rack, the following equations can be obtained tanξLi ¼ tanβwi sinα nw þ
tanγwi cosα nw cosβwi
ð19Þ
tanξRi ¼ tanβwi sinα nw −
tanγ wi cosα nw : cosβwi
ð20Þ
It follows that the approximate line contact condition can be represented as ξRðLÞ ¼ ξRðLÞ1 þ ξRðLÞ2 ≈0:
ð21Þ
ξR(L) = 0 is an ideal line contact condition. Since the instantaneous line contact of gear tooth surfaces may only exist in perfect geared drives without misalignment and manufacturing errors. In practical application, due to the elasticity of gear tooth surfaces, the theoretical point contact of crossed beveloid gear pair is spread over an elliptic area, and the center of the contact ellipse is the theoretical point of contact. However, as the FPD-angle decreases, the ellipse contact becomes an approximate line contact type as shown in Fig. 6(b). From the equations and analysis, we use a small value of ξR(L) to obtain the transition from the surface ellipse contact to the approximate line contact. 3.3. Reference and working gearing relationships Similar to the formulation used by Tsai [6], the transverse angular factor ξt of a beveloid gear is defined as follows: ξti ¼
cosα PwLðRÞi ri ¼ r wi cosα PLðRÞi
ð22Þ
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C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Contact lines on the tooth surface Pitch Plane of The Common rack
k1
R
k2
ξR1
ξR2 The direction of observation
ξR R
b) Contact ellipse on the tooth surface Right tooth surface
Contact ellipses Imaginary contact lines
FPD -angles
Fig. 6. Approximate line contact condition.
where αPL(R)i is the pressure angle on the transverse plane, and αPwL(R)i is the working pressure angle on the transverse plane. According to the involute gearing theory [11], the following equations can be obtained
ξn ¼
mn cosα nw cosβi ¼ ¼ ξti mnw cosα n cosβwi
ð23Þ
2 tanα n sinγi secβi 1 tanα PwRi þ tanα PwLi ξti
ð24Þ
tanγwi ¼
sinγ wi ¼ sinγ i
tanα n sinα n ¼ sinγi sinα nw ξn tanα nw
ð25Þ
sinβwi ¼ sinβi
cosγ i 1 cosγwi ξn
ð26Þ
where mn is the normal module of the beveloid gear, mnw is the normal module of the common rack, and ξn is the normal angular factor. The equations above defined the relations between the reference gearing data that contains mn, αn, γi and βi, and the working gearing data that contains mnw, αnw, γwi and βwi.
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A backlash-free condition is still necessary for the design of a beveloid gear pair having crossed axis. According to He and Wu [12], the backlash-free equation can be represented as follows: cosγ 1 cosγ2 N 2 tanα n xt1 ¼ 1 ðinvα PwL1 þ invα PwR1 −invα PL1 −invα PR1 Þ þ xt2 cosβ1 cosβ2 2 N2 ð invα PwL2 þ invα PwR2 −invα PL2 −invα PR2 Þ þ 2
ð27Þ
where xt1 and xt2 are the profile-shifted coefficients in the pinion and gear transverse sections. 3.4. Pitch cone design procedure Based on the geometry design procedure by Tsai [6], two pitch cone design methods considering the approximate line contact condition are developed including the direct and in-direct pitch cone formulations. The main difference between these two methods is that they have different input parameters and cyclical variables. Both methods consider the same approximate line contact condition. The flowchart in Fig. 7 illustrates the pitch cone design calculation procedure.
a) Direct design
b) In-direct design
Fig. 7. Pitch cone design chart.
Fig. 8. Meshing model for crossed beveloid gears.
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C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
4. Meshing model The meshing model is formulated next as shown in Fig. 8. Here, five coordinate systems denoted by S1, S2, Sf, Sg and Sq are used. The coordinate systems Sf, S1 and S2 are fixed to the frame, pinion and gear, respectively. The beveloid pinion Σ1 and gear Σ2 tooth surfaces can be considered as two imaginary cones with a pair of respective working cone angles γw1 and γw2. The coordinate systems Sg and Sq are the reference coordinate systems for the gear coordinate system S2 used to characterize the severity of misalignments. The parameter bc1,2 indicates the distance between the working pitch circle and the reference pitch circle. Also, rw1,2 and rP1,2 denote the working and reference pitch radii respectively. To investigate the meshing of beveloid gear pairs with misalignments, five types of assembly errors related to gear offset and crossing angle errors, are studied here. As shown in Fig. 9, Δxq, Δyq and Δzq are the offset errors of the gear along the x, y and z axes, respectively, and Δγv and Δγh are the misaligned angle errors of the gear. To perform tooth contact analysis (TCA), the homogeneous surface coordinates and the normal vector of the pinion and gear have to be transferred to the fixed coordinate system Sf. This transformation is given by Rfi ¼ Mfi Ri
ð28Þ
nfi ¼ Lfi ni
ð29Þ
where Rfi (i = 1,2) is the position vector of the pinion and gear in coordinate system Sf, and nfi (i = 1,2) is the normal vector in coordinate system Sf. The transformation matrix Mf1 and Lf1 of the pinion is represented as follows. 2
Mf 1
cosφ′ 1 6 sinφ′ ¼6 4 0 1 0 2
Lf 1
′
cosφ 1 ¼ 4 sinφ′ 1 0
− sinφ′ 1 cosφ′ 1 0 0 ′
− sinφ 1 cosφ′ 1 0
3 0 0 7 0 0 7 1 dw1 −bc1 5 0 1
ð30Þ
3 0 0 5: 1
ð31Þ
Also, assuming misalignments are present, the transformation matrices Mf2 and Lf2 of the gear is represented by Mf 2 ¼ M fq M q2 2
1 0 60 cosδ Mfq ¼ 6 4 0 − sinδ 0 0 2
Mq2
ð32Þ 0 sinδ cosδ 0
cosΔγv 6 − sinΔγ v sinΔγ h ¼6 4 − sinΔγ cosΔγ v h 0
3
E
sinδðdw2 −bc2 Þ 7 7 cosδðdw2 −bc2 Þ 5 1 0 cosΔγ h − sinΔγ h 0
ð33Þ
sinΔγv sinΔγ h cosΔγ v cosΔγh cosΔγv 0
32 Δxq cosφ′ 2 6 sinφ′ Δyq 7 76 2 Δzq 54 0 1 0
− sinφ′ 2 cosφ′ 2 0 0
0 0 1 0
3 0 07 7 05 1
Zg X g, X f Δ
v
Δzq Δxq Δ
Δyq
h
Yg
Og Of Fig. 9. Misalignments of crossed beveloid gears.
Zf
ð34Þ
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2
Lf 2
1 0 ¼ 40 cosδ 0 − sinδ
32 cosΔγ v 0 sinδ 54 − sinΔγv sinΔγ h cosδ − sinΔγ v cosΔγ h
0 cosΔγh − sinΔγh
57
32 sinΔγ v cosφ′ 2 4 5 sinΔγh cosΔγv sinφ′ 2 cosΔγ h cosΔγ v 0
− sinφ′ 2 cosφ′ 2 0
3 0 0 5: 1
ð35Þ
Throughout the gear meshing process, the tooth surfaces Σ1 and Σ2 should be in continuous tangency, and therefore they must have a common contact and normal vector. Therefore, the following equations can then be obtained: ′ ′ Rf 1 l1 ; u1 ; φ 1 ¼ Rf 2 l2 ; u2 ; φ 2
ð36Þ
′ ′ nf 1 l1 ; u1 ; φ 1 ¼ nf 2 l2 ; u2 ; φ 2 :
ð37Þ
It should be noted that |nf1| = |nf2| = 1, which reduces the above set of equations to a system of 5 independent scalar algebraic equations given by ′ ′ f i l1 ; u1 ; φ 1 ; l2 ; u2 ; φ 2 ¼ 0; i1e5:
ð38Þ
The solution to the above algebraic problem yields the relationships between the roll angles of the pinion and gear, path of contact, transmission error as well as meshing position and normal vectors for any angular position, which is demonstrated in the next section. 5. Numerical examples A pair of computational studies applying the proposed model presented above is shown next. The first study employs the analytical tooth contact analysis (TCA) formulation expressed by Eqs. (36)–(38). In this numerical TCA solution, the influence of the misalignments in the mating gear members on the contact path and transmission error (TE) is investigated for the unloaded condition. The second computational study employs a hybrid finite element-analytical formulation available in a specialized gear tooth contact analysis program that is available commercially [13]. This loaded TCA is used to verify the analytical formulation presented in this paper. These TCA studies are applied to two pairs of beveloid gears defined in Table 1. Prior to the tooth contact analyses, the positional coordinates of the tooth surface are determined from the beveloid gear generation Eqs. (1)–(7). Results of the TCA are discussed in detail next for both approaches. 5.1. Misalignment sensitivity analysis: case 1 5.1.1. Contact path The contact path results with gear offset errors of Δq = ± 0.5 mm, + 1.0 mm (where Δq = Δxq = Δyq = Δzq) are shown in Fig. 10. It can be observed that the position of the contact path moves from the middle of the tooth flank to the heel or toe with the increase in the offset error. When the gear offset of Δq is equal to 1.0 mm, theoretical edge contact begins to happen. Fig. 11 shows the distribution of the contact path when the angular misalignment of gear axes in the horizontal plane occurs. The basic contact paths shown in blue thick lines are located in the middle region of the right tooth flank when no misalignments Table 1 Gearing parameters for a pair of beveloid gears. Symbol
Normal module (mm) Normal pressure angle (°) Number of teeth Shaft angle (°) Offset (mm) Cone angle (°) Helix angle (°) Profile shifting factor Total face-width (mm) Transverse angular factor Mounting distance Working cone angle (°) Working helix angle (°) FPD-angle (°) LH: left hand; RH: right hand.
Case 1
mn αn N δ E γ β xt W ξt d γw βw ξL ξR
Case 2
Pinion (beveloid)
Gear (beveloid)
Pinion (beveloid)
Gear (beveloid)
6 20 29
6 20 45
6 20 29
6 20 45
13 229 2.0 23 (RH) 0.15 50 0.9904 2687.7 1.8873 23.19
13 229 0.8 −10.38 (LH) 0.1927 60 0.9915 10297.0 0.7549 −10.47
7.5702 2.4081
1.0 20 (RH) 0.16 50 0.9895 5589.1 0.9370 20.19
0.8 −7.24 (LH) 0.2191 60 0.9906 10,368 7496 −7.3036 6.5815 3.3511
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C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Pinion
b) Gear
Fig. 10. Influence of the gear offset errors on contact path.
a) Pinion
b) Gear
Fig. 11. Influence of the angular misalignment in the horizontal plane on contact path.
exist. The positive error which is equal to 0.03° moves the contact path towards the heel and a negative error moves the contact path towards to the toe. In either situation when the errors are large enough, the mating pinion and gear pair cannot engage well with each other. Fig. 12 shows the distribution of the contact path with angular misalignment along the vertical direction. The contact path moves out of the gear blank when the positive error is equal to ±0.2° similar to the trend obtained in Fig. 11. Also, from these two sets of results, it is obvious that the contact path is more sensitive to the angular misalignment of gear axes in the horizontal plane than the angular misalignment of gear axes in the vertical plane. 5.1.2. Transmission error The influences of misalignments in the mating members on predicted transmission error (TE) are presented in Figs. 13–15. As the contact paths move to the heel, the theoretical angular TE is in dog-leg shape and positive offset error caused a lower peak–
a) Pinion
b) Gear
Fig. 12. Influence of the angular misalignment in the vertical plane on contact path.
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Transmission error
Baseline with no error
59
b) Peak-Peak value of TE
Δq=0.5mm
Δq=1.0mm
Δq=-0.5mm
Fig. 13. Influence of the gear offset errors on predicted TE.
a) Transmission error
Baseline with no error
b) Peak-Peak value of TE
h
=0.03deg
h
= -0.03deg
Fig. 14. Influence of the angular misalignment of gear axes in the horizontal plane on predicted TE.
a) Transmission error
Baseline with no error
b) Peak-Peak value of TE
Δ γ v =0.20deg
Δ γ v = -0.20deg
Fig. 15. Influence of the angular misalignment of gear axes in the vertical plane on predicted TE.
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peak value of the angular TE as shown in Fig. 13. However, the peak–peak value of angular TE for the negative offset error has the opposite trend where it becomes higher due to contact path shifting towards the toe. Figs. 14 and 15 show the effect of the angular misalignments of gear axes in the horizontal plane and vertical plane on the predicted TE. For both the horizontal and vertical angular misalignments defined in Fig. 9, the negative angular error caused higher peak-to-peak value of the angular TE and the positive angular error has the opposite trend of lowering peak-to-peak angular TE. Also, the angular misalignment in the horizontal plane causes more change than the angular misalignment of gear axes in the vertical plane. These results are similar to previous study for hypoid and spiral bevel gears [14].
5.2. Misalignment sensitivity analysis: case 2 5.2.1. Contact path The influences of misalignments of the mating gear members on contact path for example case 2 are presented in Figs. 16–18. In this case, the FPD-angle of the beveloid gear pair example is equal to 3.35° as compared to 2.41° for case 1. For the two cases, the same teeth numbers, normal module, pressure angle, tooth width, shaft angle and offset are used. The main difference in the design between cases 1 and 2 is that they have different cone angles, helical angles and profile shifting factors which caused a different FPD-angle. As observed in Fig. 16, the position of the contact path moves towards to the heel or toe when the offset error occurs. The effect of offset error on contact path for case 2 under consideration here is more prevalent than that of case 1. Fig. 17 shows the distribution of the contact path with angular misalignment along the horizontal direction. Note that the contact path actually moves out of the gear blank when the positive error is equal to ± 0.04°. Fig. 18 shows the distribution of the contact path with angular misalignment along the vertical direction. The contact path moves out of the gear blank when the positive error is set to either 0.25 or –0.25°. Overall, the effect on contact path for case 2 is less sensitive than that of case 1.
a) Pinion
b) Gear
Fig. 16. Influence of the gear offset errors on contact path.
a) Pinion
b) Gear
Fig. 17. Influence of the angular misalignment in the horizontal plane on contact path.
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Pinion
61
b) Gear
Fig. 18. Influence of the angular misalignment in the vertical plane on contact path.
a) Transmission error
Baseline with no error
b) Peak-Peak value of TE
Δq=0.5mm
Δq=1.0mm
Δq=-0.5mm
Δq=-1.0mm
Fig. 19. Influence of the gear offset errors on predicted TE.
5.2.2. Transmission error The influences of misalignments of the mating gear members on TE for case 2 are presented in Figs. 19–21. As shown in these results for both the offset error and angular errors, the positive errors slightly lower the maximum peak–peak value of the TE, while the negative errors increase the peak–peak value of TE. Both the shape of the TE and the above trends are consistent with those results shown in case 1.
a) Transmission error
Baseline with no error
b) Peak-Peak value of TE
Δγ h =0.04deg
Δ γ h =-0.04deg
Fig. 20. Influence of the angular misalignment of gear axes in the horizontal plane on predicted TE.
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C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Transmission error
b) Peak-Peak value of TE
Baseline with no error
Δ γ v =0.25deg
Δγ v =-0.25deg
Fig. 21. Influence of the angular misalignment of gear axes in the vertical plane on predicted TE.
Fig. 22 shows the comparison results of TE without any assembly error for cases 1 and 2. Here, the maximum TE value for case 1 when the FPD-angle is equal to 2.4° is lower than that for case 2 when the FPD-angle is equal to 3.35°. The peak-to-peak value of TE for case 1 is 7.5 micro-radians, which is lower than that for case 2 given by 14.3 micro-radians.
ξ R1 + ξ R 2 = 2.40 deg
ξ R1 + ξ R 2 = 3.35 deg
Fig. 22. Transmission error of beveloid gear pairs with different line contact design angle.
a) Case 1
b) Case 2
Fig. 23. Solid model for beveloid gear pairs.
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) T = 5 Nm
b) Gear T = 200 Nm
c) Pinion T= 400 Nm
d) Gear T = 600Nm
63
Fig. 24. Contact patterns for different torque loads for beveloid gear design case 1 with 2.40° FPD-angle.
5.3. Loaded tooth contact analysis To setup the loaded tooth contact analysis problem, firstly, the solid models of the beveloid gears defined by the geometrical parameters given in Table 1 are created as shown in Fig. 23. Secondly, the solid models are used to create a finite element representation that is subsequently analyzed in a commercially available, specialized tooth contact analysis (TCA) program [15]. The specialized TCA program employs a hybrid finite element-analytical mechanics approach to simulate the physics of tooth engagement. Additional information on the theoretical basis can be found in the work published by Vijayakar [16]. Figs. 24–25 show that the contact patterns of the gears resulted from the different torque loads ranging from 5 to 600 Nm for the two beveloid gear designs of interest. The results for the design defined by case 1 with 2.4° of line contact design angle are given by Fig. 24, while the results of the design given by case 2 with 3.35° of line contact design angle are illustrated in Fig. 25. Under very light torque load (5 Nm), the contact pattern compares well with the contact path resulting from the theoretical tooth contact analysis for the unloaded condition as computed earlier. However, for larger torque loads of more than 200 Nm, the beveloid pinion and gear engages with each other as line contact. With the increase of the torque loads from 200 Nm to 600 Nm, the area of the contact grows larger from approximately 50% to 70% of the total tooth surface. For the comparison between cases 1 and 2, the area of contacts is smaller for the larger FPD-angle. This result agrees with the practical application of the FPD-angle. With the increase of the FPD-angle from zero to a very large value, the contact of crossed beveloid gear pair changes from the ideal line contact to point contact. However, due to the elasticity, the gear tooth contact is spread over a finite area around the contact point, which is shown in Figs. 24 and 25. 6. Conclusions A numerical approach based on the pitch cone design equations is applied to perform tooth contact analysis with line contact and to simulate the influence of misalignments on contact path and transmission error. The computational results are validated by comparing them to loaded tooth contact calculations performed using a hybrid finite element-analytical mechanics model. The model is further used to perform a series of parametric studies that yield the following conclusions: 1. All assembly errors considered in this study tend to worsen the contact path, but only have modest effect on TE. Both the contact path and theoretical TE are more sensitive to the angular misalignment of gear axes in the horizontal plane than the offset error and the angular misalignment of gear axes in the vertical plane. For the gear design with lower FPD-angle, the influence of misalignments on tooth contact including the contact path and TE is more obvious. 2. Under light torque load, the contact pattern predicted through the loaded tooth contact analysis based on the hybrid finite element-analytical mechanics model correlates well with the contact path predicted by the theoretical tooth contact analysis
a) T = 5 Nm
b) Gear T = 200 Nm
c) Pinion T= 400 Nm
d) Gear T = 600Nm
Fig. 25. Contact patterns for different torque loads for beveloid gear design case 2 with 3.35° FPD-angle.
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employing the pitch cone equations. For a larger torque load, the beveloid gear pairs depict line contact. The results show that the contact pattern area can reach 75% of the tooth surface. Acknowledgments The authors would like to acknowledge the support from the China Scholarship Council (CSC), Fundamental Research Funds for Central Universities (project no. CDJXS11111138), Program for Key Projects in the National Science and Technology Pillar Program in the Twelfth Five-year Plan Period of China (grant no. 2011BAF09B07), the National Natural Science Foundation of China (grant no. 51175523) and Mechanical Engineering in the School of Dynamic Systems at the University of Cincinnati. References [1] Y. Zhang, Z. Fang, Analysis of tooth contact and load distribution of helical gears with crossed axes, Mechanism and Machine Theory 34 (1999) 41–57. [2] K. Mitome, Design of miter conical involute gears based on tooth bearing, JSME International, Part C 38 (2) (1995) 307–311. [3] K. Mitome, T. Yamazaki, Design of conical involute gear engaged with profile spur gear on intersecting shafts (In Japanese), Transactions of the Hapan Society of Mechanical Engineers, Part C 62 (598) (1996) 2436–2441. [4] J. Brauer, Transmission error in anti-backlash conical involute gear transmissions: a global–local FE approach, Finite Elements in Analysis and Design 41 (2005) 431–457. [5] C.-C. Liu, C.-B. Tsay, Contact characteristics of beveloid gears, Mechanism and Machine Theory 37 (2002) 333–350. [6] S.J. Tsai, S.-H. Wu, Geometrical design of conical gear drives with profile-shifted transmission, in: 12th IFToMM World Congress, Besançon (France), June, 18–21 2007. [7] S.-H. Wu, S.J. Tsai, Contact stress analysis of skew conical involute gear drives in approximate line contact, Mechanism and Machine Theory 223 (9) (2009) 2201–2211. [8] T. Ohmachi, A. Uchino, H. Komatsubara, Analysis of the tooth surface contact of the conical involute gear with the conjugate surface, in: Proc. of MPT2009-Sendai, 2009, pp. 52–57. [9] L. Guixian, W. Jianmin, Z. Xin, L. Yu, Meshing theory and simulation of noninvolute beveloid gears, Mechanism and Machine Theory 39 (8) (2004) 883–892. [10] K. Mitome, Inclining Work-arbor taper hobbing of conical gear using cylindrical hob, Journal of Mechanisms, Transmissions and Automation in Design 108 (1986) 135–140. [11] F.L. Litvin, Gear Geometry and Applied Theory, Cambridge University Press, Cambridge, U. K., New York, 2004. [12] J. He, X. Wu, Study on geometry of conical involute gear pair with nonintersecting–nonparallel-axes, Journal of Xi'an Jiatong University 37 (5) (May, 2003) 471–474. [13] Y. Cheng, T.C. Lim, Vibration analysis of hypoid transmissions applying an exact geometry-based gear mesh theory, Journal of Sound and Vibration 240 (3) (2001) 519–543. [14] T. Peng, T.C. Lim, Effects of assembly errors on spiral bevel gear mesh characteristics and dynamic response, in: Proceedings of MPT2009-Sendai, JSME International Conference on Motion and Power Transmissions, Matsushima Isles Resort, Japan, May 2009, pp. 13–15. [15] S.M. Vijayakar, Contact Analysis Program Package: HypoidK, Advanced Numerical Solutions, Hilliard, OH, 2009. [16] S.M. Vijayakar, Finite element methods for quasi-prismatic bodies with application to gears. Ph.D Dissertation, The Ohio State University, Columbus, Ohio, 1987.
Contents lists available at SciVerse ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Pitch cone design and influence of misalignments on tooth contact behaviors of crossed beveloid gears Caichao Zhu a, Chaosheng Song a, b,⁎, Teik C. Lim b, Tao Peng c a b c
The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China School of Dynamic Systems, Mechanical Engineering, 598 Rhodes Hall, P.O. Box 210072, University of Cincinnati, Cincinnati, OH 45221,USA ArvinMeritor Inc., Troy, MI 48084, USA
a r t i c l e
i n f o
Article history: Received 8 September 2011 Received in revised form 23 February 2012 Accepted 7 August 2012 Available online 27 September 2012 Keywords: Beveloid gears Pitch cone design Tooth contact analysis Gear misalignments
a b s t r a c t A pitch cone design theory assuming line contact is formulated to analyze the tooth engagement behavior of crossed beveloid gears. The analysis applies the spatial gearing model and beveloid gear manufacturing technique to determine the cutter parameters and geometry. Using the proposed procedure, a crossed beveloid gear pair with small shaft angle is studied. Also, the proposed formulation is capable of quantifying the influence of misalignments including offset and angular position errors on the contour and position of contact path and lines over a complete mesh cycle. The results show that both offset and angular position errors of the mating members worsen the conjugate action of the mating tooth surfaces. Furthermore, both the contact path and the transmission error are more sensitive to the angular error than the offset error. The loaded tooth contact characteristics are subsequently modeled using the finite element method for comparison purpose. The proposed model is partially validated by comparing the numerical simulation results. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction In modern gearing application of power transmission between two crossed shafts at large angles (close to 90°), hypoid or spiral bevel gears, face gears and worm gears are superior to involute helical gears. However, it is not practical to use them when the crossed shaft angle is less than 45°, due to design and manufacturing difficulties [1]. Instead, beveloid gears that are also known as involute conical gears are better suited to accommodate applications having a small shaft angle with tapered tooth thickness. Furthermore, these types of gears can be manufactured like cylindrical gears using a hobbing process or a threaded-wheel grinding process. One of the advantages of beveloid gears is the allowance to backlash. The main challenge in crossed axis application is the fact that the contact path is theoretically pointwise with high sliding velocity that can affect tooth surface durability. In the last couple of decades, a number of studies have been performed on the beveloid gearing theory, design and manufacturing [2–9]. For instance, Mitome and Yamazaki [2,3] proposed a new type of concave-conical gearing with better curvature relation at the contact points for 45-degree crossed shafts. The work also includes an experimental verification of the predicted contact pattern. Brauer [4] derived a mathematical model of spur beveloid gear with parallel axes and applied the model for backlash analysis. Liu and Tsay [5] derived the equations to simulate the generating process, and studied the theoretical meshing and bearing contact of the beveloid gears. Tsai and Wu [6,7] proposed two methods to design the geometry of beveloid gear with crossed axes. Also the generation process and contact characteristics for conventional beveloid gears were studied. Ohmachi et al. [8] conducted the tooth surface contact analysis using a conjugate theory for the practical design of conical involute gears. Li et al. [9] proposed a new type of non-involute beveloid gear and analyzed the meshing characteristics. In spite of the ⁎ Corresponding author at: The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China. Tel.: +86 13883402720. E-mail address: [email protected] (C. Song). 0094-114X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2012.08.008
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
49
Nomenclature di mounting distance Di heights of pitch cones dwi working mounting distances E offset ki tooth contact lines mn normal module of the beveloid gear mnw normal module of the common rack j j j nxc , nyc , nzc cutter surface normal vectors ⇀n′ i normal vectors of pitch cone surface nfi normal vectors at pitch point in the global fixed coordinate system Ni tooth numbers ni normal vector in the coordinate system fixed to pinion and gear Pn gear diametral pitch pc circular pitch Rfi position vectors at pitch point in the fixed global coordinate system Ri position vectors of pinion and gear rj reference pitch circle radius Sf fixed global coordinate system S′i pinion or gear coordinates used in pitch cone design Sb fixed coordinate system to define assembly errors Sc plane axode coordinate system fixed to the imaginary rack cutter Sn, Sp auxiliary rack cutter coordinate systems Si pinion or gear coordinate system ui generatrix coordinates of pitch cone θi rotational coordinates of pitch cone x j, y j, z j beveloid gear surface coordinates xcj, ycj, zcj frame coordinates fixed to cutter x′i, y′i, z′i surface coordinates of pitch cones αn normal pressure angle αnw common rack normal pressure angle αPL(R)j transverse plane pressure angles αPLw(R)j transverse plane working pressure angle β helix angle βwi working spiral angle γj angle relative to the plane axode coordinate system γwi working pitch angle δ shaft angle η angle between the generatrices ϕj gear rotation angle during generating process ϕ′i rotational angles τi generatrices of the pitch cones ξ crossing angle between the first principal directions of the tooth surface curvatures ξLj, ξRj intersection angle between the contact line k and the pitch plane of the common rack ξn normal angular factor ξtj transverse angular factor Σi gear and pinion tooth surfaces Subscripts i refers to the pinion or gear: i = 1 for pinion, i = 2 for gear j refers to the cutter surfaces: j = P for pinion, j = G for gear L refers to the left tooth surface R refers to the right tooth surface f refers to fixed coordinate system
above-mentioned studies focusing on geometry design and manufacturing of beveloid gears, none applied the pitch cone design with line contact and none studied the influence of misalignments on tooth contact behavior in crossed beveloid gears with line contact condition. These gaps will be the emphasis of our current paper.
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This paper presents an analytical derivation of working pitch cone design with line contact. The influence of misalignments including offset and angular position errors on the profile and position of contact path and transmission error is studied through a theoretical tooth contact model based on the idealized geometry. Also, loaded tooth contact analysis using the finite element theory is also performed to demonstrate the salient features of the proposed approach. The analysis results can be applied to guide the design of crossed beveloid gear geometry. 2. Beveloid gear tooth profile According to the generation concept for beveloid gear proposed by Mitome [10], the envelope surface of the hob during the manufacturing process can be considered as a rack cutter. As shown in Fig. 1, the normal section of the rack cutter consists mainly of two parts including the straight edges and the fillet curves that generate the working tooth surfaces and fillet surfaces of the gear, respectively. Fig. 2 illustrates a schematic of the generation mechanism and the coordinate relationship between the normal section of the rack cutter and the generated beveloid gear. Based on the theory of gearing, the cutting process can be expressed by using the a series of transformation matrices given by Sn → Sp → Sc → Sb → S1. Then, the rack cutter surfaces can be represented in coordinate system Sc as follows: h i j j j j j j j j j j j x c ¼ l cosα n −a cosγ þ ∓ l sinα n −a tanα n −b sinβ þ u cosβ sinγ
ð1Þ
j j j j j j j yc ¼ l sinα n −a tanα n −b cosβ þ u sinβ
ð2Þ
h i j j j j j j j j j j j zc ¼ − l cosα n −a sinγ þ ∓ l sinα n −a tanα n −b sinβ þ u cosβ cosγ
ð3Þ
where superscript j refers to the cutter surfaces (j = P for pinion, j = G for gear). The upper sign of Eqs. (1)–(3) indicates the left-side straight edge, while the lower sign represents the right-side straight edge. The notations xcj,ycj,zcj represent the position vector on the cutter surface, αn denotes the normal pressure angle, symbols Pn and pc represent the gear diametral pitch and
Fig. 1. Normal cross section of a rack cutter.
xP
γj
β
xb
xn
xc j
zP zc
o p on = u j
zn on
I
yn
v oc , oP
x1 ϕ1 I O1 Ob
yc , yP
ϕ1 y1
zb , z1 yb
rP1
ω1 Fig. 2. Coordinate relationship between the normal cross section of rack cutter and the beveloid gear.
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
51
circular pitch, respectively, β j represents the helix angle on the pitch plane of imaginary rack cutter, and γ j is the angle with respect to the plane axode coordinate system. Furthermore, l = |M0M1| represents the distance measured from the initial point M0, moving along the straight line M0M2, to point M1. Hence, as proposed by Tsay [7], the mathematical model of the beveloid gear tooth surface in coordinate S1 can be represented follows: j j j x ¼ x c cosϕ j −y c sinϕ j þ r j cosϕ j þ ϕ j sinϕ j
ð4Þ
j j j y ¼ x c sinϕ j þ y c cosϕ j þ r j sinϕ j −ϕ j cosϕ j
ð5Þ
j
j
z ¼ zc
ð6Þ
where the meshing equation is given by j j j j j ϕj ¼ nxc yc −nyc xc =nxc r j j
j
ð7Þ
j
and where x ,y ,z represent the position vector on the tooth surface on the beveloid gear, rj denotes the reference pitch circle j j radius, ϕj represents the gear rotation angle in the generating process, and nxc ,nyc represent the components of a normal vector of the point on the cutter surface. 3. Working pitch cone design 3.1. Pitch cone parameters According to Litvin's gearing theory [11], the design of pitch cones is used to express the relationship between parameters γw1,γw2,βw1,βw2, where γwi is the working pitch angle and βwi is the working spiral angle (i = 1 for the pinion, and i = 2 for the gear). Fig. 3 shows a specific layout for a beveloid gear pair with three distinct coordinate systems to be used in the mathematical derivation. The first two coordinate systems S′i are attached to the pinion and gear bodies. The third one is the global coordinate system Sf that is fixed to a reference frame supporting the gear pair. The two cones are tangent at point P. Unit vectors τ 1 and τ 2 represent the generatrices of the pitch cones that lie in the pitch plane and intersect each other at the pitch point P. A pitch cone surface can be represented analytically in the local coordinate system S′i, where M is one position point on the surface of the cone and (ui, θi) are the surface coordinates (i.e. the Gaussian coordinates), as shown in Fig. 4. The mathematical model of the pitch cone can be represented by ′
′
′
x i ¼ ui sinγ wi cosθi ; y i ¼ ui sinγwi sinθi ; z i ¼ ui cosγwi :
ð8Þ
The surface unit normal is represented by ′
⇀n i ¼ ½ cosθi cosγwi
sinθi cosγ wi
T
− sinγ wi :
ð9Þ
According to Fig. 3, the pitch cones are in tangency at the pitch point P, and the equations of tangency are given by, Rf 1 ðθ1 ; u1 Þ ¼ Rf 2 ðθ2 ; u2 Þ
Fig. 3. Relationship between the local coordinate system S′i where i = 1,2, and the fixed coordinate system Sf.
ð10Þ
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Fig. 4. Operating pitch cone and its parameters.
nf 1 ðθ1 ; u1 Þ ¼ nf 2 ðθ2 ; u2 Þ:
ð11Þ
Substituting (sin θi) 2 + (cos θi) 2 = 1, and the working pitch circle radius rwi = ui sin γwi into the above equations, it yields the following set of formulas: E cosγw1 sinδ cosγ w1 −r w2 r w1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosγ w2 2 2 2 sin δ− sin γw1 − sin γw2 −2 sinγw1 sinγ w2 cosδ r w2 þ d1 ¼ − sinγ w1 cosγ w2
d2 ¼
E sin2 δ− sin2 γ w1 − sinγ w1 sinγw2 cosδ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinγ w1 sinδ sin2 δ− sin2 γ w1 − sin2 γw2 −2 sinγ w1 sinγw2 cosδ
rw2 Eð sinγw1 cosδ þ sinγ w2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − sinγw2 cosγ w2 sinδ sin2 δ− sin2 γ w1 − sin2 γ w2 −2 sinγ w1 sinγ w2 cosδ
ð12Þ
ð13Þ
ð14Þ
where Rfi and nfi(i = 1, 2) are the surface coordinates and normal vectors of the points on the pitch cone, δ is the shaft angle, E is the offset, and di(i = 1, 2) is the mounting distance of the pinion and gear. Fig. 5 shows the longitudinal shapes that are in tangency at P. The longitudinal shape of the tooth in the pitch plane is the intersecting curve of the tooth surface with the pitch plane. The generatrices of the pitch cones form the angle η that is represented by the equation 1 2
cosη ¼ τ τ :
ð15Þ
Then, taking into account that η = βw1 ± βw2 as shown in Fig. 5, the following equation can be obtained cosðβw1 βw2 Þ ¼ tanγ w1 tanγ w2 þ
cosδ : cosγ w1 cosγw2
ð16Þ
Using Eqs. (12) and (16), the offset can be derived as E¼
ðr Pw1 cosγ w2 þ r Pw2 cosγ w1 Þ sinðβw1 βw2 Þ : sinδ
ð17Þ
3.2. Line contact condition In theory, a beveloid gear pair with non-parallel axes engages in point contact. However, that may not be the case in practice. Therefore, our aim here is to determine the relations of pitch cone parameters to realize the approximate line contact. As shown in Fig. 6(a), a beveloid gear pair engages with the common rack in line contact. Here, k1 and k2 are the contact lines with the direction of observation from the toe to the heel. The angle ξ is defined as the crossing angle between the first principal directions of the tooth surface curvatures (FPD-angle).
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(a) Same direction of helices
(b) Opposite direction of helices
Fig. 5. Orientation of relative velocity at the pitch point.
According to Fig. 6(a), the cross angle between k1 and k2 can be represented as follows ξRðLÞ ¼ ξRðLÞ1 þ ξRðLÞ2 :
ð18Þ
Based on spatial geometry relationship between the beveloid gear and the common rack, the following equations can be obtained tanξLi ¼ tanβwi sinα nw þ
tanγwi cosα nw cosβwi
ð19Þ
tanξRi ¼ tanβwi sinα nw −
tanγ wi cosα nw : cosβwi
ð20Þ
It follows that the approximate line contact condition can be represented as ξRðLÞ ¼ ξRðLÞ1 þ ξRðLÞ2 ≈0:
ð21Þ
ξR(L) = 0 is an ideal line contact condition. Since the instantaneous line contact of gear tooth surfaces may only exist in perfect geared drives without misalignment and manufacturing errors. In practical application, due to the elasticity of gear tooth surfaces, the theoretical point contact of crossed beveloid gear pair is spread over an elliptic area, and the center of the contact ellipse is the theoretical point of contact. However, as the FPD-angle decreases, the ellipse contact becomes an approximate line contact type as shown in Fig. 6(b). From the equations and analysis, we use a small value of ξR(L) to obtain the transition from the surface ellipse contact to the approximate line contact. 3.3. Reference and working gearing relationships Similar to the formulation used by Tsai [6], the transverse angular factor ξt of a beveloid gear is defined as follows: ξti ¼
cosα PwLðRÞi ri ¼ r wi cosα PLðRÞi
ð22Þ
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C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Contact lines on the tooth surface Pitch Plane of The Common rack
k1
R
k2
ξR1
ξR2 The direction of observation
ξR R
b) Contact ellipse on the tooth surface Right tooth surface
Contact ellipses Imaginary contact lines
FPD -angles
Fig. 6. Approximate line contact condition.
where αPL(R)i is the pressure angle on the transverse plane, and αPwL(R)i is the working pressure angle on the transverse plane. According to the involute gearing theory [11], the following equations can be obtained
ξn ¼
mn cosα nw cosβi ¼ ¼ ξti mnw cosα n cosβwi
ð23Þ
2 tanα n sinγi secβi 1 tanα PwRi þ tanα PwLi ξti
ð24Þ
tanγwi ¼
sinγ wi ¼ sinγ i
tanα n sinα n ¼ sinγi sinα nw ξn tanα nw
ð25Þ
sinβwi ¼ sinβi
cosγ i 1 cosγwi ξn
ð26Þ
where mn is the normal module of the beveloid gear, mnw is the normal module of the common rack, and ξn is the normal angular factor. The equations above defined the relations between the reference gearing data that contains mn, αn, γi and βi, and the working gearing data that contains mnw, αnw, γwi and βwi.
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55
A backlash-free condition is still necessary for the design of a beveloid gear pair having crossed axis. According to He and Wu [12], the backlash-free equation can be represented as follows: cosγ 1 cosγ2 N 2 tanα n xt1 ¼ 1 ðinvα PwL1 þ invα PwR1 −invα PL1 −invα PR1 Þ þ xt2 cosβ1 cosβ2 2 N2 ð invα PwL2 þ invα PwR2 −invα PL2 −invα PR2 Þ þ 2
ð27Þ
where xt1 and xt2 are the profile-shifted coefficients in the pinion and gear transverse sections. 3.4. Pitch cone design procedure Based on the geometry design procedure by Tsai [6], two pitch cone design methods considering the approximate line contact condition are developed including the direct and in-direct pitch cone formulations. The main difference between these two methods is that they have different input parameters and cyclical variables. Both methods consider the same approximate line contact condition. The flowchart in Fig. 7 illustrates the pitch cone design calculation procedure.
a) Direct design
b) In-direct design
Fig. 7. Pitch cone design chart.
Fig. 8. Meshing model for crossed beveloid gears.
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C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
4. Meshing model The meshing model is formulated next as shown in Fig. 8. Here, five coordinate systems denoted by S1, S2, Sf, Sg and Sq are used. The coordinate systems Sf, S1 and S2 are fixed to the frame, pinion and gear, respectively. The beveloid pinion Σ1 and gear Σ2 tooth surfaces can be considered as two imaginary cones with a pair of respective working cone angles γw1 and γw2. The coordinate systems Sg and Sq are the reference coordinate systems for the gear coordinate system S2 used to characterize the severity of misalignments. The parameter bc1,2 indicates the distance between the working pitch circle and the reference pitch circle. Also, rw1,2 and rP1,2 denote the working and reference pitch radii respectively. To investigate the meshing of beveloid gear pairs with misalignments, five types of assembly errors related to gear offset and crossing angle errors, are studied here. As shown in Fig. 9, Δxq, Δyq and Δzq are the offset errors of the gear along the x, y and z axes, respectively, and Δγv and Δγh are the misaligned angle errors of the gear. To perform tooth contact analysis (TCA), the homogeneous surface coordinates and the normal vector of the pinion and gear have to be transferred to the fixed coordinate system Sf. This transformation is given by Rfi ¼ Mfi Ri
ð28Þ
nfi ¼ Lfi ni
ð29Þ
where Rfi (i = 1,2) is the position vector of the pinion and gear in coordinate system Sf, and nfi (i = 1,2) is the normal vector in coordinate system Sf. The transformation matrix Mf1 and Lf1 of the pinion is represented as follows. 2
Mf 1
cosφ′ 1 6 sinφ′ ¼6 4 0 1 0 2
Lf 1
′
cosφ 1 ¼ 4 sinφ′ 1 0
− sinφ′ 1 cosφ′ 1 0 0 ′
− sinφ 1 cosφ′ 1 0
3 0 0 7 0 0 7 1 dw1 −bc1 5 0 1
ð30Þ
3 0 0 5: 1
ð31Þ
Also, assuming misalignments are present, the transformation matrices Mf2 and Lf2 of the gear is represented by Mf 2 ¼ M fq M q2 2
1 0 60 cosδ Mfq ¼ 6 4 0 − sinδ 0 0 2
Mq2
ð32Þ 0 sinδ cosδ 0
cosΔγv 6 − sinΔγ v sinΔγ h ¼6 4 − sinΔγ cosΔγ v h 0
3
E
sinδðdw2 −bc2 Þ 7 7 cosδðdw2 −bc2 Þ 5 1 0 cosΔγ h − sinΔγ h 0
ð33Þ
sinΔγv sinΔγ h cosΔγ v cosΔγh cosΔγv 0
32 Δxq cosφ′ 2 6 sinφ′ Δyq 7 76 2 Δzq 54 0 1 0
− sinφ′ 2 cosφ′ 2 0 0
0 0 1 0
3 0 07 7 05 1
Zg X g, X f Δ
v
Δzq Δxq Δ
Δyq
h
Yg
Og Of Fig. 9. Misalignments of crossed beveloid gears.
Zf
ð34Þ
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
2
Lf 2
1 0 ¼ 40 cosδ 0 − sinδ
32 cosΔγ v 0 sinδ 54 − sinΔγv sinΔγ h cosδ − sinΔγ v cosΔγ h
0 cosΔγh − sinΔγh
57
32 sinΔγ v cosφ′ 2 4 5 sinΔγh cosΔγv sinφ′ 2 cosΔγ h cosΔγ v 0
− sinφ′ 2 cosφ′ 2 0
3 0 0 5: 1
ð35Þ
Throughout the gear meshing process, the tooth surfaces Σ1 and Σ2 should be in continuous tangency, and therefore they must have a common contact and normal vector. Therefore, the following equations can then be obtained: ′ ′ Rf 1 l1 ; u1 ; φ 1 ¼ Rf 2 l2 ; u2 ; φ 2
ð36Þ
′ ′ nf 1 l1 ; u1 ; φ 1 ¼ nf 2 l2 ; u2 ; φ 2 :
ð37Þ
It should be noted that |nf1| = |nf2| = 1, which reduces the above set of equations to a system of 5 independent scalar algebraic equations given by ′ ′ f i l1 ; u1 ; φ 1 ; l2 ; u2 ; φ 2 ¼ 0; i1e5:
ð38Þ
The solution to the above algebraic problem yields the relationships between the roll angles of the pinion and gear, path of contact, transmission error as well as meshing position and normal vectors for any angular position, which is demonstrated in the next section. 5. Numerical examples A pair of computational studies applying the proposed model presented above is shown next. The first study employs the analytical tooth contact analysis (TCA) formulation expressed by Eqs. (36)–(38). In this numerical TCA solution, the influence of the misalignments in the mating gear members on the contact path and transmission error (TE) is investigated for the unloaded condition. The second computational study employs a hybrid finite element-analytical formulation available in a specialized gear tooth contact analysis program that is available commercially [13]. This loaded TCA is used to verify the analytical formulation presented in this paper. These TCA studies are applied to two pairs of beveloid gears defined in Table 1. Prior to the tooth contact analyses, the positional coordinates of the tooth surface are determined from the beveloid gear generation Eqs. (1)–(7). Results of the TCA are discussed in detail next for both approaches. 5.1. Misalignment sensitivity analysis: case 1 5.1.1. Contact path The contact path results with gear offset errors of Δq = ± 0.5 mm, + 1.0 mm (where Δq = Δxq = Δyq = Δzq) are shown in Fig. 10. It can be observed that the position of the contact path moves from the middle of the tooth flank to the heel or toe with the increase in the offset error. When the gear offset of Δq is equal to 1.0 mm, theoretical edge contact begins to happen. Fig. 11 shows the distribution of the contact path when the angular misalignment of gear axes in the horizontal plane occurs. The basic contact paths shown in blue thick lines are located in the middle region of the right tooth flank when no misalignments Table 1 Gearing parameters for a pair of beveloid gears. Symbol
Normal module (mm) Normal pressure angle (°) Number of teeth Shaft angle (°) Offset (mm) Cone angle (°) Helix angle (°) Profile shifting factor Total face-width (mm) Transverse angular factor Mounting distance Working cone angle (°) Working helix angle (°) FPD-angle (°) LH: left hand; RH: right hand.
Case 1
mn αn N δ E γ β xt W ξt d γw βw ξL ξR
Case 2
Pinion (beveloid)
Gear (beveloid)
Pinion (beveloid)
Gear (beveloid)
6 20 29
6 20 45
6 20 29
6 20 45
13 229 2.0 23 (RH) 0.15 50 0.9904 2687.7 1.8873 23.19
13 229 0.8 −10.38 (LH) 0.1927 60 0.9915 10297.0 0.7549 −10.47
7.5702 2.4081
1.0 20 (RH) 0.16 50 0.9895 5589.1 0.9370 20.19
0.8 −7.24 (LH) 0.2191 60 0.9906 10,368 7496 −7.3036 6.5815 3.3511
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C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Pinion
b) Gear
Fig. 10. Influence of the gear offset errors on contact path.
a) Pinion
b) Gear
Fig. 11. Influence of the angular misalignment in the horizontal plane on contact path.
exist. The positive error which is equal to 0.03° moves the contact path towards the heel and a negative error moves the contact path towards to the toe. In either situation when the errors are large enough, the mating pinion and gear pair cannot engage well with each other. Fig. 12 shows the distribution of the contact path with angular misalignment along the vertical direction. The contact path moves out of the gear blank when the positive error is equal to ±0.2° similar to the trend obtained in Fig. 11. Also, from these two sets of results, it is obvious that the contact path is more sensitive to the angular misalignment of gear axes in the horizontal plane than the angular misalignment of gear axes in the vertical plane. 5.1.2. Transmission error The influences of misalignments in the mating members on predicted transmission error (TE) are presented in Figs. 13–15. As the contact paths move to the heel, the theoretical angular TE is in dog-leg shape and positive offset error caused a lower peak–
a) Pinion
b) Gear
Fig. 12. Influence of the angular misalignment in the vertical plane on contact path.
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Transmission error
Baseline with no error
59
b) Peak-Peak value of TE
Δq=0.5mm
Δq=1.0mm
Δq=-0.5mm
Fig. 13. Influence of the gear offset errors on predicted TE.
a) Transmission error
Baseline with no error
b) Peak-Peak value of TE
h
=0.03deg
h
= -0.03deg
Fig. 14. Influence of the angular misalignment of gear axes in the horizontal plane on predicted TE.
a) Transmission error
Baseline with no error
b) Peak-Peak value of TE
Δ γ v =0.20deg
Δ γ v = -0.20deg
Fig. 15. Influence of the angular misalignment of gear axes in the vertical plane on predicted TE.
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peak value of the angular TE as shown in Fig. 13. However, the peak–peak value of angular TE for the negative offset error has the opposite trend where it becomes higher due to contact path shifting towards the toe. Figs. 14 and 15 show the effect of the angular misalignments of gear axes in the horizontal plane and vertical plane on the predicted TE. For both the horizontal and vertical angular misalignments defined in Fig. 9, the negative angular error caused higher peak-to-peak value of the angular TE and the positive angular error has the opposite trend of lowering peak-to-peak angular TE. Also, the angular misalignment in the horizontal plane causes more change than the angular misalignment of gear axes in the vertical plane. These results are similar to previous study for hypoid and spiral bevel gears [14].
5.2. Misalignment sensitivity analysis: case 2 5.2.1. Contact path The influences of misalignments of the mating gear members on contact path for example case 2 are presented in Figs. 16–18. In this case, the FPD-angle of the beveloid gear pair example is equal to 3.35° as compared to 2.41° for case 1. For the two cases, the same teeth numbers, normal module, pressure angle, tooth width, shaft angle and offset are used. The main difference in the design between cases 1 and 2 is that they have different cone angles, helical angles and profile shifting factors which caused a different FPD-angle. As observed in Fig. 16, the position of the contact path moves towards to the heel or toe when the offset error occurs. The effect of offset error on contact path for case 2 under consideration here is more prevalent than that of case 1. Fig. 17 shows the distribution of the contact path with angular misalignment along the horizontal direction. Note that the contact path actually moves out of the gear blank when the positive error is equal to ± 0.04°. Fig. 18 shows the distribution of the contact path with angular misalignment along the vertical direction. The contact path moves out of the gear blank when the positive error is set to either 0.25 or –0.25°. Overall, the effect on contact path for case 2 is less sensitive than that of case 1.
a) Pinion
b) Gear
Fig. 16. Influence of the gear offset errors on contact path.
a) Pinion
b) Gear
Fig. 17. Influence of the angular misalignment in the horizontal plane on contact path.
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Pinion
61
b) Gear
Fig. 18. Influence of the angular misalignment in the vertical plane on contact path.
a) Transmission error
Baseline with no error
b) Peak-Peak value of TE
Δq=0.5mm
Δq=1.0mm
Δq=-0.5mm
Δq=-1.0mm
Fig. 19. Influence of the gear offset errors on predicted TE.
5.2.2. Transmission error The influences of misalignments of the mating gear members on TE for case 2 are presented in Figs. 19–21. As shown in these results for both the offset error and angular errors, the positive errors slightly lower the maximum peak–peak value of the TE, while the negative errors increase the peak–peak value of TE. Both the shape of the TE and the above trends are consistent with those results shown in case 1.
a) Transmission error
Baseline with no error
b) Peak-Peak value of TE
Δγ h =0.04deg
Δ γ h =-0.04deg
Fig. 20. Influence of the angular misalignment of gear axes in the horizontal plane on predicted TE.
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C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) Transmission error
b) Peak-Peak value of TE
Baseline with no error
Δ γ v =0.25deg
Δγ v =-0.25deg
Fig. 21. Influence of the angular misalignment of gear axes in the vertical plane on predicted TE.
Fig. 22 shows the comparison results of TE without any assembly error for cases 1 and 2. Here, the maximum TE value for case 1 when the FPD-angle is equal to 2.4° is lower than that for case 2 when the FPD-angle is equal to 3.35°. The peak-to-peak value of TE for case 1 is 7.5 micro-radians, which is lower than that for case 2 given by 14.3 micro-radians.
ξ R1 + ξ R 2 = 2.40 deg
ξ R1 + ξ R 2 = 3.35 deg
Fig. 22. Transmission error of beveloid gear pairs with different line contact design angle.
a) Case 1
b) Case 2
Fig. 23. Solid model for beveloid gear pairs.
C. Zhu et al. / Mechanism and Machine Theory 59 (2013) 48–64
a) T = 5 Nm
b) Gear T = 200 Nm
c) Pinion T= 400 Nm
d) Gear T = 600Nm
63
Fig. 24. Contact patterns for different torque loads for beveloid gear design case 1 with 2.40° FPD-angle.
5.3. Loaded tooth contact analysis To setup the loaded tooth contact analysis problem, firstly, the solid models of the beveloid gears defined by the geometrical parameters given in Table 1 are created as shown in Fig. 23. Secondly, the solid models are used to create a finite element representation that is subsequently analyzed in a commercially available, specialized tooth contact analysis (TCA) program [15]. The specialized TCA program employs a hybrid finite element-analytical mechanics approach to simulate the physics of tooth engagement. Additional information on the theoretical basis can be found in the work published by Vijayakar [16]. Figs. 24–25 show that the contact patterns of the gears resulted from the different torque loads ranging from 5 to 600 Nm for the two beveloid gear designs of interest. The results for the design defined by case 1 with 2.4° of line contact design angle are given by Fig. 24, while the results of the design given by case 2 with 3.35° of line contact design angle are illustrated in Fig. 25. Under very light torque load (5 Nm), the contact pattern compares well with the contact path resulting from the theoretical tooth contact analysis for the unloaded condition as computed earlier. However, for larger torque loads of more than 200 Nm, the beveloid pinion and gear engages with each other as line contact. With the increase of the torque loads from 200 Nm to 600 Nm, the area of the contact grows larger from approximately 50% to 70% of the total tooth surface. For the comparison between cases 1 and 2, the area of contacts is smaller for the larger FPD-angle. This result agrees with the practical application of the FPD-angle. With the increase of the FPD-angle from zero to a very large value, the contact of crossed beveloid gear pair changes from the ideal line contact to point contact. However, due to the elasticity, the gear tooth contact is spread over a finite area around the contact point, which is shown in Figs. 24 and 25. 6. Conclusions A numerical approach based on the pitch cone design equations is applied to perform tooth contact analysis with line contact and to simulate the influence of misalignments on contact path and transmission error. The computational results are validated by comparing them to loaded tooth contact calculations performed using a hybrid finite element-analytical mechanics model. The model is further used to perform a series of parametric studies that yield the following conclusions: 1. All assembly errors considered in this study tend to worsen the contact path, but only have modest effect on TE. Both the contact path and theoretical TE are more sensitive to the angular misalignment of gear axes in the horizontal plane than the offset error and the angular misalignment of gear axes in the vertical plane. For the gear design with lower FPD-angle, the influence of misalignments on tooth contact including the contact path and TE is more obvious. 2. Under light torque load, the contact pattern predicted through the loaded tooth contact analysis based on the hybrid finite element-analytical mechanics model correlates well with the contact path predicted by the theoretical tooth contact analysis
a) T = 5 Nm
b) Gear T = 200 Nm
c) Pinion T= 400 Nm
d) Gear T = 600Nm
Fig. 25. Contact patterns for different torque loads for beveloid gear design case 2 with 3.35° FPD-angle.
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employing the pitch cone equations. For a larger torque load, the beveloid gear pairs depict line contact. The results show that the contact pattern area can reach 75% of the tooth surface. Acknowledgments The authors would like to acknowledge the support from the China Scholarship Council (CSC), Fundamental Research Funds for Central Universities (project no. CDJXS11111138), Program for Key Projects in the National Science and Technology Pillar Program in the Twelfth Five-year Plan Period of China (grant no. 2011BAF09B07), the National Natural Science Foundation of China (grant no. 51175523) and Mechanical Engineering in the School of Dynamic Systems at the University of Cincinnati. References [1] Y. Zhang, Z. Fang, Analysis of tooth contact and load distribution of helical gears with crossed axes, Mechanism and Machine Theory 34 (1999) 41–57. [2] K. Mitome, Design of miter conical involute gears based on tooth bearing, JSME International, Part C 38 (2) (1995) 307–311. [3] K. Mitome, T. Yamazaki, Design of conical involute gear engaged with profile spur gear on intersecting shafts (In Japanese), Transactions of the Hapan Society of Mechanical Engineers, Part C 62 (598) (1996) 2436–2441. [4] J. Brauer, Transmission error in anti-backlash conical involute gear transmissions: a global–local FE approach, Finite Elements in Analysis and Design 41 (2005) 431–457. [5] C.-C. Liu, C.-B. Tsay, Contact characteristics of beveloid gears, Mechanism and Machine Theory 37 (2002) 333–350. [6] S.J. Tsai, S.-H. Wu, Geometrical design of conical gear drives with profile-shifted transmission, in: 12th IFToMM World Congress, Besançon (France), June, 18–21 2007. [7] S.-H. Wu, S.J. Tsai, Contact stress analysis of skew conical involute gear drives in approximate line contact, Mechanism and Machine Theory 223 (9) (2009) 2201–2211. [8] T. Ohmachi, A. Uchino, H. Komatsubara, Analysis of the tooth surface contact of the conical involute gear with the conjugate surface, in: Proc. of MPT2009-Sendai, 2009, pp. 52–57. [9] L. Guixian, W. Jianmin, Z. Xin, L. Yu, Meshing theory and simulation of noninvolute beveloid gears, Mechanism and Machine Theory 39 (8) (2004) 883–892. [10] K. Mitome, Inclining Work-arbor taper hobbing of conical gear using cylindrical hob, Journal of Mechanisms, Transmissions and Automation in Design 108 (1986) 135–140. [11] F.L. Litvin, Gear Geometry and Applied Theory, Cambridge University Press, Cambridge, U. K., New York, 2004. [12] J. He, X. Wu, Study on geometry of conical involute gear pair with nonintersecting–nonparallel-axes, Journal of Xi'an Jiatong University 37 (5) (May, 2003) 471–474. [13] Y. Cheng, T.C. Lim, Vibration analysis of hypoid transmissions applying an exact geometry-based gear mesh theory, Journal of Sound and Vibration 240 (3) (2001) 519–543. [14] T. Peng, T.C. Lim, Effects of assembly errors on spiral bevel gear mesh characteristics and dynamic response, in: Proceedings of MPT2009-Sendai, JSME International Conference on Motion and Power Transmissions, Matsushima Isles Resort, Japan, May 2009, pp. 13–15. [15] S.M. Vijayakar, Contact Analysis Program Package: HypoidK, Advanced Numerical Solutions, Hilliard, OH, 2009. [16] S.M. Vijayakar, Finite element methods for quasi-prismatic bodies with application to gears. Ph.D Dissertation, The Ohio State University, Columbus, Ohio, 1987.