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Analysis of loaded characteristics of helical curve face gear
Mechanism and Machine Theory 115 (2017) 267–282
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Research paper
Analysis of loaded characteristics of helical curve face gear Chunjiang He, Chao Lin∗ State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China
a r t i c l e
i n f o
Article history: Received 29 January 2017 Revised 14 April 2017 Accepted 14 May 2017
Keywords: Variable transmission ratio LTCA Contact stress Helical curve face gear Complex tooth surface
a b s t r a c t As a new gear pair, the helical curve face gear pair have not been widely applied as normal face gear. It can be used to substitute crank-slider mechanism to achieve variable transmission ratio and improve the structure and characteristics to some extent, such as grinders, planers and so on. The mathematical model of the helical curve face gear pair and the tooth contact analysis are necessary for the theoretical analysis of this gear pair. Based on the theory of gear meshing and coordinate transmission, the meshing equations for noncircular gear contacting with helical curve face gear without misalignment are obtained. On the basis of Hertz’s theory, the principal curvatures at contact point are represented. And the contact area on every tooth and the half cycle of helical curve face gear have been established separately. As well as the contact stress on every tooth and half cycle of helical curve face gear. However, there may be misalignment during the meshing of this gear pair, this paper also covers the contact path and the functions of transmission errors with misalignment, includes the comparison of helical curve face gear and straight curve face gear. The developed theories for complex tooth surface are verified by experiment. These analysis show the contact characteristics of this gear pair during the meshing process and provide the theoretical support for further analysis and application of helical curve face gear pair. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Face gear drive is a gear pair with bevel gear and cylinder gear meshing in intersect or cross axes [1]. Comparing with bevel gear drive, face gear drive has many advantages such as much compact structure, less axial force and better bearing capacity; also, it has much simpler structure, large transmission ratio with better contact situation and better flow of power than planetary gear set. With the good characteristics of face gear pair, it can be used in helicopters, truck with overloaded and so on. For the better understanding of face gear drive, a lot of researches have be done by the scholars at home and abroad. The computerized generation and simulation, the contact ellipse and functions of transmission errors [2] have been proposed for the analysis of helical gear. The base curves of non-circular gears with a constant pressure angle was developed by Figliolini [3]. Based on the curves of non-circular gear, the meshing of non-circular gear pair can be classified into fixed center distance and variable center distance. The mathematical model, driving principle and mechanical properties of these two gear type [4,5] was presented by Zheng et al. Two of the major cutters for the generation of non-circular gear are spread-blade cutter and fixed-setting cutters, the advantages and disadvantages of different cutters was proposed by Alfonso
∗
Corresponding author. E-mail address: [email protected] (C. Lin).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.05.014 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. The derivation of non-circular gear.
Fuentes with numerical examples [6]. Non-circular gear can be utilized with linkage model, such as five-bar linkages model [7], to obtain the exact path and dynamic qualities of different gear type [8]. According to Gear Geometry and Applied Theory [9], the mathematical model [10] and meshing analysis [11] for localized contact on hypoid gear, localized contact on the surface of tooth is proved to be good for the contact situation of gear drive. So the finite analysis [12], meshing characteristics of orthogonal face gear drive [13] were based on the localized contact method. Litvin introduced the contact path and stress analysis of helical face gear [14]. And the tooth contact analysis for helical face gear with variable tooth thickness was discussed by Hsu [15], so, it is feasible to get some parameters (such as tooth thickness, transmission ratio and so on) of face gear pair changed to get the needed transmission characteristics. Helical curve face gear is a new gear for complex tooth surface, which includes the advantages of face gear, helical gear and non-circular gear, it can mesh with non-circular gear to achieve continuously variable transmission with changeable transmission ratio. So, this face gear can be applied for electric percussion drill and alleviate some of the shortages of normal face gear. The bending stress of curve face gear has been explored roughly by Lin [16,17]. The theoretical analysis of tooth contact and the ANSYS simulation of curve face gear pair were investigated by Lin [18].The theory of processing non-circular gear and helical curve face gear with the same shaper cutter was putted by Lin [19]. This paper establishes the mathematical equations for the surface of helical curve face gear pair in Section 2 and the formula of point contact, UTCA in Section 3. Sections 4 and 5 discuss the contact area and the transmission errors of helical curve face gear. And covers the comparison of contact path and functions of transmission errors between helical curve face gear and straight curve face gear [20] in Section 6.Also, this paper presents the effect with the assembly deflection errors of helical curve face gear. The analysis of contact stress on every tooth and the whole half cycle of helical curve face gear has been covered in this paper. These results provide the theoretical support for further theoretical analysis and practical application of complex tooth surface. 2. Derivation of contact point 2.1. Contact line of non-circular gear with shaper cutter With the use of face gear pair getting more and more widely, the method of manufacturing of face gear has been proposed recently and it can be classified into shaping processing, cutting processing and generating processing. Generating processing is based on the meshing trajectory of gear pair and it is the most widely used method for the theoretical analysis of gear pair. According to the method of gear meshing with point contact on the surface [9], the tooth numbers of helical shaper cutter NS should be more than the tooth numbers of helical non-circular gear N1 and the difference between NS and N1 is 2 or 3 to getting good contact situation. According to the generating method, the tooth surface of non-circular gear is generated by the tooth surface of shaper cutter. As shown in Fig. 1(a), the meshing process of non-circular gear with shaper cutter can be considered as shaper cutter meshing with rack-cutter o and the non-circular gear meshing with the same rack-cutter simultaneously, Fig. 1(b) shows the relationship of the three pitch surfaces. In Fig. 1(a), So (Xo Yo Zo ) is the fixed coordinate system of rack-cutter,
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Ss (Xs Ys Zs ) is the fixed coordinate system of shaper cutter and S1 (X1 Y1 Z1 ) is the fixed coordinate system of non-circular gear. So (Xo Yo Zo ) is the movable coordinate system of rack-cutter, Ss (Xs Ys Zs ) is the movable coordinate system of shaper cutter and S1 (X1 Y1 Z1 ) is the movable coordinate system of non-circular gear. Any one of the three pitch surfaces, which were shown in Fig. 1(b), is doing pure rolling with the other two surfaces at the same time. So, the displacement Loo can be replaced as rs · ϕ s and r1 · ϕ 1 . The mathematical equation of ϕ 1 , r1 and Loo can be defined as:
⎧ ⎨Loo = rs · ϕs ϕ1 = Lroo = rs 0ϕs r11 dϕ ⎩r = 1a(1−k2 ) 1
(1)
1−k cos(n1 ϕ1 )
Where, r1 is the pitch radius of non-circular gear; rs is the pitch radius of shaper cutter; ϕ 1 is the rotation angle of noncircular gear; ϕ s is the rotation angle of shaper cutter; a is the semi-major axis of non-circular gear; k is the eccentricity ratio of non-circular gear; n1 is the order of non-circular gear. With the method of coordinate transformation, the coordinate transformation matrix M1 s are presented as follows:
M1 s = M1 1 · M1s · Ms s
⎡
cos ϕ1 cos ϕs − sin ϕ1 sin ϕs ⎢cos ϕ1 sin ϕs + sin ϕ1 cos ϕs =⎣ 0 0
− cos ϕ1 sin ϕs − sin ϕ1 cos ϕs cos ϕ1 cos ϕs − sin ϕ1 sin ϕs 0 0
0 0 1 0
⎤
− sin ϕ1 (r1 + rs ) cos ϕ1 (r1 + rs ) ⎥ ⎦ 0 1
(2)
The shaper cutter, which is shown in Fig. 1, can be deemed as cylindrical gear and the mathematical equation of shaper cutter can be shown as follows:
⎡
⎤
rbs [cos(ϕos + ϕs − ξs ) − ϕs sin(ϕos + ϕs − ξs )] ⎢−rbs [sin(ϕos + ϕs − ξs ) + ϕs cos(ϕos + ϕs − ξs )]⎥ −→(s ) r s ( ϕ s , ξs ) = ⎣ ⎦ pξs 1
(3)
Where, p is spiral parameter; rbs is the pitch radius of shaper cutter; ϕ os is the helix angle of base circle in shaper cutter;
ϕ s is the rotational angle and ξ s is torsion angle. And the unit normal of shaper cutter:
−→(s ) ns (βs , ϕs ) =
− cos βs sin(ϕos + ϕs − ξs ) − cos βs cos(ϕos + ϕs − ξs ) − sin βs
(4)
Where, β s is the base spiral angle of shaper cutter. The shaper cutter is pure rolling with non-circular gear during the meshing process, so the contact line of non-circular gear and shaper cutter can be derived as follows:
→ − → − fs1 (ϕ1 , βs , ϕs , ξs ) = ns · vs1 = 0
(5)
According to the coordinate transformation matrix (2) of the shaper cutter and non-circular gear, the meshing equation of non-circular gear with shaper cutter can be represented as follows:
− → − → − − → → − → fs1 (ϕ1 , βs , ϕs , ξs ) = ns · (i1 s ωs · (M1 s · ks ) × (M1 s · rs ) − ωs · ks × rs )
(6)
So, the tooth surface equation of non-circular gear can be derived by the meshing Eq. (6) and coordinate transformation matrix (2), shown as follows:
−→
−→ r 1 ( 1 ) ( ϕ 1 , ξ1 ) = M 1 s · r s ( s ) ( ϕ s , ξs ) fs1 (βs , ϕs , ξs ) = 0
(7)
2.2. Contact line of helical curve face gear with variable transmission ratio According to the principle of gear meshing [21], the generating of face gear and shaper cutter can be seen as a composite process that is the face gear rotating and shaper cutter rotating with moving simultaneously. Fig. 2 shows the relationship of the displacement and rotating angles of shaper cutter and curve face gear. The coordinate system Ss (Xs Ys Zs ), shown in Fig. 2, is the fixed coordinate system of shaper cutter, and the coordinate system Ss (Xs Ys Zs ) is the movable coordinate system of shaper cutter. The coordinate system S2 (X2 Y2 Z2 ) is the fixed coordinate system of helical curve face gear, and the coordinate system S2 (X2 Y2 Z2 ) is the movable coordinate system of helical curve face gear. L1 is the radius of helical curve face gear and equal R. L2 is the displacement caused be the moving of shaper cutter and ϕ s , ϕ 2 are the angles caused by the rotating of shaper cutter and helical curve face gear respectively. Based on
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Fig. 2. The meshing of helical curve face gear with shaper cutter.
the coordinate transformation theory, the coordinate transformation matrix M2 s from S2 (X2 Y2 Z2 ) to Ss (Xs Ys Zs )in the meshing process, can be established as follows:
M2 s = M2 2 · M2s
⎡
cos ϕ2 cos ϕs + sin ϕ2 sin ϕs ⎢− cos ϕs sin ϕ2 + + cos ϕ2 sin ϕs =⎣ sin ϕs 0
− sin ϕs cos ϕ2 + sin ϕ2 cos ϕs sin ϕ2 sin ϕs + + cos ϕ2 cos ϕs cos ϕs 0
− sin ϕ2 − cos ϕ2 1 0
⎤
−L1 sin ϕ2 −L1 cos ϕ2 ⎥ ⎦ −L2 1
(8)
The shaper cutter and helical curve face gear are doing pure rolling on the pitch surface, so the contact line of helical curve face gear and shaper cutter during the meshing process, can be derived as follows:
→ − → − fs2 (ϕ2 , βs , ϕs , ξs ) = ns · vs2 = 0
(9)
According to the coordinate transformation matrix (8) of the shaper cutter and face gear, the meshing equation of helical curve face gear and shaper cutter can be proposed as follows:
− → − → − − → − → → fs2 (ϕ2 , βs , ϕs , ξs ) = ns · (i2 s · ωs · (M2 s · ks ) × (M2 s · rs ) − ωs · ks × rs )
(10)
So, the tooth surface equation of helical curve face gear can be derived by the meshing Eq. (10) and coordinate transformation matrix (8), shown as follows:
−→
−→ r 2 ( 2 ) ( ϕ 2 , ξ2 ) = M 2 s · r s ( s ) ( ϕ s , ξs ) fs2 (βs , ϕs , ξs ) = 0
(11)
2.3. The fillet surface and range of radius During the meshing process of curve face gear, there will be fillet surface at the bottom of tooth surface, and the tooth undercut and tooth pointing will limit the width of tooth. The fillet surface is formed by the rounded corner of shaper. In Fig. 2 the rounded corner of shaper cutter can be presented as follows:
⎡
⎤
rbs [cos(ϕos + ϕsc − ξsc ) − ϕsc sin(ϕos + ϕsc − ξsc )] + ρ cos ϕcc −rbs [sin(ϕos + ϕsc − ξsc ) + ϕsc cos(ϕos + ϕsc − ξsc )] − ρ sin ϕcc ⎥ ⎢ −−→(s ) r c (ϕsc , ξsc , ξcc ) = ⎣ ⎦ p(ξsc + ξcc ) 1
(12)
Where, ϕ sc is the rotational angle of top point c of tooth surface, ξ sc is the torsion angle of point c and ξcc is torsion angle between point c and c on the rounded corner. And the unit normal of rounded corner:
−−→(s ) nc (βs , ϕcc ) =
cos ϕcc − sin ϕcc − sin βs
(13)
So the fillet surface can be established with the same method as tooth surface:
−−→
−−→ r2c (2 ) (ϕ2c , ξ2c ) = M2 s · r c (s ) (ϕsc , ξsc , ξcc ) −→ −→ fs2c (βs , ϕsc , ξsc ) = nsc · vsc = 0
(14)
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Fig. 3. The point contact of helical curve face gear pair.
The torsion angle and the width of tooth are small, so the torsion angle ϕ sc can be omitted. The approximation algorithm of the range of radius can be recognized as straight curve face gear. During the undercut process, the minimum radius R1 can be derived by the limit line on the tooth surface of shaper cutter and presented as follows:
⎧ −−−→ −→(2 ) ⎪ r 2 ( ϕ 2 , ξ2 ) = M s s · r s ( s ) ( ϕ s , ξs ) ⎪ ⎪ ⎨ f (βs , ϕs , ξs ) = 0 −−→(2 )
( 2 )
∂ r−−→2 − → dϕs = −v2s (s) dt ∂ ϕs ∂ f dϕs ∂ f dϕ1 ∂ us dt + ∂ ϕs dt = − ∂ ϕ1 dt
∂ r2 ⎪ ⎪ ⎪ ⎩ ∂ f∂ udsus
dus dt
+
(15)
So the minimum radius R1 is:
R1 =
y ∗2k s +(R + zk∗ s )
2
1
(16)
1
Where, y∗ k1 s and z∗ k1 s is the critical coordinate value on the surface of shaper with undercut happened. And the maximum radius without tooth pointing R2 can be established as the similar method with R1 :
R2 =R+
rbs R rbs − rs cos α2 cos α2
(17)
Where, α 2 is the critical pressure angle and α2 is the pressure angle at radius R with tooth pointing happened. 3. Analysis of tooth contact Based on the formation of conjugate tooth surface and the space meshing theory, Fig. 3 shows the contact point on surface 2 , 2 is tooth surface of helical curve face gear and 1 is tooth surface of non-circular gear. Part (a) is the simulating meshing of helical curve face gear pair, Part (b) is the local magnification of part (a) and the theoretical analysis of contact point is shown in part (c). Ls1 is the line of Eq. (7), it shows the contact situation of shaper cutter and non-circular gear and Ls2 is the line of Eq. (11), it shows the situation of shaper cutter and curve helical face gear. During the meshing process, the shaper cutter is meshing with non-circular gear and helical curve face gear simultaneously. So the contact line Ls1 and Ls2 appeared at the same time and the intersection point p, shown in part (c). So, the contact point without misalignment in the meshing process of helical curve face gear pair can be discussed as follows:
fs2 (βs , ϕs , ξs ) = fs1 (βs , ϕs , ξs ) fs1 (βs , ϕs , ξs ) = 0 fs2 (βs , ϕs , ξs ) = 0
(18)
And the parameters (β s , ϕ s , ξ s ) for the contact point should be limited on the surfaces of the gear pair with Eq. (19).
r 2 ( 2 ) ( ϕ 2 , ξ2 ) = M 2 s · r s ( s ) ( ϕ s , ξs ) r 1 ( 1 ) ( ϕ 1 , ξ1 ) = M 1 s · r s ( s ) ( ϕ s , ξs )
(19)
And as the structure of helical curve face gear changes periodically, so any adjacent five teeth of curve face gear in half a cycle can show the characteristics in detail. The tooth number is shown in Fig. 4.
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Fig. 4. The tooth number of helical curve-face gear. Table 1 Parameters of helical curve face gear and shaper cutter. Parameter
Value
Number of tooth of shaper cutter zs Number of tooth of non-circular gear z1 Number of tooth of helical curve face gear z2 Modulus of shaper cutter m (mm) Pressure angle of shaper cutter α s (°) Torsion angle ξ s (°) Inner radius of helical curve face gearR1 (mm) Outer radius of helical curve face gearR2 (mm)
21 18 36 4 20 10 70 83
Fig. 5. The contact path and transmission errors with e = 1 mm.
The helical curve face gear and non-circular gear are both built by the surface of the same shaper cutter. So most parameters of the curve face gear pair are decided by the shaper cutter. Some parameters of the helical curve face gear pair are shown in Table 1. According to Eqs. (18) and (19) and with the help of MATLAB (Matrix Laboratory), the contact points can be obtained. And the function of transmission errors is shown in Eq. (20).
ϕ2 = 0ϕ1 1/i12 dϕ = R1 0ϕ1 r (ϕ1 )dϕ
ϕ 2 = ϕ 2 − ϕ 2
(20)
Where, ϕ2 is the rotating angle with misalignment. ϕ 2 is the difference between theoretical angle ϕ 2 and actual angle
ϕ2 .
As shown in Figs. 5-7, the principle of tooth contact is regular and near the inner part of helical curve face gear in the half cycle.
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Fig. 6. The contact path and transmission errors with q = 1 mm.
Fig. 7. The contact path and transmission errors with γ = 1°.
In the three figures, e is the error of alignment in radial direction of tooth, q is the error of alignment in the height direction of tooth and γ is the amount of angular displacement to the right alignment. The errors e and q are position deviation amount to the right alignment and generating a new contact point to the right and bottom of the one, which is generated by correct alignment, at the same rotating angle. So, the new contact path on every tooth is below the correct one, shown as (a) parts In Figs. 5 and 6. The error γ generating a new contact point to the left and top of the one, which is generated by correct alignment, at the same rotating angle. So, the new contact path on every tooth is on top of the correct one, shown as (a) part In Fig. 7. Part (b) in every figure shows the transmission errors of the five tooth in half a cycle. The transmission errors increase from tooth 1 to tooth 3 and decrease from tooth 3 to tooth 5. The contact path and transmission errors of every alignment condition can represent the generating situation and taking the three errors of alignment into comparison, error e doing the minimal effect to the generating process, while error γ doing the maximal effect.
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Fig. 8. Force of helical curve face gear.
4. Loaded tooth contact analysis The face gear pair always cover a certain load in practical application, so the tooth contact analysis (TCA) is not enough, and further loaded tooth contact analysis (LTCA) is needed to evaluate the characteristics of helical curve face gear. According to Hertz’s theory on the contact between two surface elastomers, the contact ellipse, maximum contact stresses and surface stress distribution on the surface of curve face gear are obtained. For those results, the force on the surface of tooth should be available at first. During the meshing process of helical curve face gear pair, the Force on meshing point P have been shown in Fig. 8. So the force on the surface could be calculated as follows:
⎧ ⎨Ft (θ1 ) = TR1 Ft (θ1 ) Fn (θ1 ) = cos (αn ) ⎩ Fa (θ1 ) = Ft (θ1 ) tan(αn )
(21)
Where, α n is the changeable pressure angle of curve face gear and αn = α0 +π /2 − arctan(r (θ1 )/r (θ1 ) ). In the meshing process, the contact ratio ɛ of this helical curve face gear pair changes from 1.3 to 2.5, so the curve face gear will have 2 or 3 teeth meshing with non-circular gear simultaneously at some rotating angle in the half cycle. The force on the surface of contact point will change with the number of instantaneous meshing teeth. 4.1. The principal curvatures at the center of contact area According to the mathematical equation of non-circular gear, the principal curvatures can be derived as
K1 (1) = 0 K2 (1) = − θ
(22)
1 o1 rb1
Where θ o1 and rb1 are the abduction angle of tooth profile and base radius on non-circular gear. The Eq. (22) shows that the principal curvature K1 (1) always equal 0 and K2 (1) always less than 0 with rb1 increase from root to top on the tooth surface. With the same theory of principal curvatures of non-circular gear, the K1(2 ) and K2(2 ) of helical curve face gear can be derived and shown as below:
−→ −→ −−−→ r θ1 × r θ2 n2 ( f ) = −→ −→ r θ1 × r θ2
(23)
−−−−→ −−−−→ −→ −→ Where, r θ1 is the partial derivative of r2 (2 ) (θ1 , θ2 ) about θ 1 , r θ2 is the partial derivative of r2 (2 ) (θ1 , θ2 )about θ 2 . The surface of curve face gear is irregular surface, so the principal curvatures of curve face gear can be derived by The Second Basic Homogeneous Form of Arbitrary Surface, shown as follows: (2 )
K1,2
2MF − LG − NE =− ± 2(EG − F 2 )
(
LN − M2 2MF − LG − NE 2 ) − 2 2(EG − F ) (EG − F 2 )
(24)
−→ −→ −→ −→ −→ −→ − → −−−→ − → −−−→ − → −−−→ Where, E = r θ1 · r θ1 , F = r θ1 · r θ2 , G = r θ2 · r θ2 , L = n1 · r θ1 θ1 , M = n1 · r θ1 θ2 and N = n1 · r θ2 θ2 .θ 1 is the rotating angle of non-circular gear and θ 2 is the rotating angle of curve face gear.
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Fig. 9. Principal curvatures of curve-face gear.
The principal curvatures K1(2 ) , K2(2 ) of the contact point on curve face gear can be established as follows
As shown in Fig. 9, the principal curvatures K1(2 ) at meshing point are changing with K1(2 ) > 0, in the half cycle with five
teeth. As the meshing goes from the root of tooth to the top, K1(2 ) keeps decreasing at every tooth. For the same height on
tooth surface, K1(2 ) increases from tooth 1 to 3 and decreases from tooth 3 to 5, so the maximum K1(2 ) are obtained at the tooth 3 in half cycle of the curve face gear. The principal curvatures K2(2 ) at meshing point are changing with K2(2 ) < 0, in the half cycle with five teeth. And with the meshing goes from the root of tooth to the top, K2(2 ) keeps increasing at every tooth. For the same height on tooth surface,
the maximum K2(2 ) of every tooth increases from tooth 1 to 3 and decreases from tooth3 to 5, so the maximum K2(2 ) of
every tooth are obtained at the tooth 3 and the minimum K2(2 ) of every tooth increases from tooth 1 to 5 in half cycle of the curve face gear. 4.2. The derivation of contact area One advantage of gear pair is the transmission goes smoothly with force, the contact point will extend to be an area with load. And the contact area is an important representation of the characteristics of curve face gear. According to the Hertz theory, the contact area is an ellipse with its curvature radius related to the principal curvatures K1(2 ) , K2(2 ) of helical curve face gear. The curvature radius at major axis ρ x and curvature radius at minor axis ρ y can be established as follows:
⎧ 3(τ1 +τ2 ) ⎪ ⎪ ⎨ρx (θ1 ) = μ0 3 8 K1(2) +K2(2) +K1(1) +K2(1) Fn (θ1 ) 3(τ1 +τ2 ) ⎪ Fn (θ1 ) ρ ( θ ) = v ⎪ 3 y 1 0 ⎩ 8 K1(2) +K2(2) +K1(1) +K2(1)
(25)
−1 ) Where, τ 1 , τ 2 is the parameters of the material and τ = 4(M . E M2 As the meshing process goes, some curvature radius ρ x , ρ y with no alignment errors can be obtained and shown in Table 2. Show the values of coordinate values at every axis, curvature radius at major axis ρ x and curvature radius at minor axis ρ y on the surface of curve face gear by MATLAB. Fig. 10 shows the trajectory and distribution of contact ellipses on the surface of curve face gear in the half cycle with five teeth during the meshing process. And Figs. 11-15 show the change of contact ellipses in detail, which are caused by different tooth height. As shown in Figs. 11-15: 1, The change of ρ x and ρ y basically have the same trend with load at the surface of every surface, which also meet the Eq. (25). 2, The contact area decreases from tooth 1 to tooth 3 and increases from tooth 3 to tooth 5, the ratio ρ x /ρ y increases with the meshing goes. 3, The meshing point on the tooth surface of helical curve face gear is always in the vicinity of pitch surface. The contact ellipse is formed at meshing point with major axis ρ x near X axis and minor axis ρ y vertical. The contact area can’t show 2
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C. He, C. Lin / Mechanism and Machine Theory 115 (2017) 267–282 Table 2 Some parameters of curve face gear pair. Tooth number
Coordinate values at every axis
1
2.98 6.75 4.21 8.19 7.67 11.93 9.47 15.24 8.56 12.85
2 3 4 5
67.78 67.82 66.24 66.09 62.53 62.2 56.77 56.1 49.58 48.71
−2.37 −3.39 −14.56 −15.57 −26.27 −27.23 −37.12 −38.3 −46.22 −47.27
ρx
ρy
2.49 5.32 2.75 2.77 5.92 3.07 3.65 5.24 1.84 3.39
2.36 4.63 1.51 1.43 3.73 1.88 1.97 3.24 1.43 2.31
Fig. 10. Contact ellipses at different contact points of curve face gear.
on the tooth surface completely when the load is too large. Therefore, when the inner radius of curve face gear R0 = 70 mm and outer radius R2 = 83 mm, the contact ellipse on the surface of helical curve face gear is too close to the inner side of tooth. And with the increase of load, there will be contact with the edge of tooth, making loaded deviation, affecting the meshing of helical curve face gear pair. 4.3. The derivation of contact stress According to the Hertz theory, the contact area on the surface of curve face gear will extend to be ellipse with the increase of load. The contact stress on contact area can be shown as ellipse, which are changing with different load and height on the surface of every tooth. The maximum contact stress σ Hmax can be presented as follows:
σH max (θ1 ) =
0.92
μv
3
K1(2) − K2(2) + K1(1) − K2(1)
( τ1 + τ2 ) 2
2 Tn (θ1 )
(26)
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Fig. 11. Contact ellipses on the surface of tooth 1.
Fig. 12. Contact ellipses on the surface of tooth 2.
Fig. 13. Contact ellipses on the surface of tooth 3.
Fig. 14. Contact ellipses on the surface of tooth 4.
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Fig. 15. Contact ellipses on the surface of tooth 5.
Fig. 16. Contact stress on the surface of tooth 1.
Fig. 17. Contact stress on the surface of tooth 2.
Figs. 16-20 show the change of contact stress with different height on the surface of every tooth, which are caused by the load on the surface of helical face gear. As shown in Figs. 16-20, the contact stress can’t be presented on the tooth surface completely with the increase of height. The stress increase at first and then decrease with the increase of height. The maximum contact stress of every tooth is ranged from the middle to the top, because of the different instantaneous contact ratio and transmission ratio. And the maximum stress appeared at different height for different tooth. In the half cycle, the contact stress increase with the increase of tooth number at first, getting the maximum value at tooth 3, and then decrease from tooth 3 to tooth 5.Taking the conclusion shown in Figs. 11-15 into consideration, the contact ellipse on tooth 3 is the worst one, so the contact situation on tooth 3 is the worst and it determines the transmission characteristics of curve face gear.
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Fig. 18. Contact stress on the surface of tooth 3.
Fig. 19. Contact stress on the surface of tooth 4.
5. FEM analysis During the meshing process, the contact ratio of helical curve face gear pair is ranging from 1.3 to 2.5, so the FEM analysis of helical curve face gear pair will be divided into single meshing and double meshing. The FEM analysis in this paper was obtained with ϕ1 = −0.01π , Fig. 21(a) shows the analysis with single meshing and Fig. 21(b) shows the analysis with double meshing. With the help of ANSYS, the contact stress can be shown on the surface of helical curve face gear with single meshing and double meshing. As Fig. 22(a) and (b) shown, the contact area on the surface of helical curve face gear are appeared at the inner part of tooth. The area is ellipse at every moment and the errors for the theoretical contact stress compared with FEM analysis are shown in Fig. 23. From Fig. 23, the theoretical contact stress is different from FEM analysis during the meshing process. But the trend for the change of theoretical analysis is the same with FEM analysis and the maximum error is 6.54%. During the FEM analysis, there will be some errors, which are caused by 3D model and assemble of gear pair. And taking those errors into consideration, the maximum error 6.54% can be accepted.
6. Experiment The experiment is operated at the State Key Laboratory of Mechanical Transmission of Chongqing University. The noncircular gear was mounted on the drive shaft with helical curve face gear mounted on the driven shaft, shown in Fig. 24. The errors of alignment e, q and γ are represented by the adjustment of the relative position of drive shaft and driven shaft.
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Fig. 20. Contact stress on the surface of tooth 5.
Fig. 21. FEM models with different meshing tooth.
Fig. 22. Contact stress nephograms of helical curve face gear pair.
Fig. 23. The comparison of contact stress.
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Fig. 24. The experimental meshing.
Fig. 25. The comparison between theoretical area and experimental area.
Because of the variable transmission ratio of this helical face gear pair, the surface of every tooth is not the same to either adjacent one in half a cycle. And a half cycle with five teeth of the helical curve face gear should be taken into consideration to get the principle of tooth contact completely. The experimental area are not the same with theoretical result completely, because of the misalignment during experimental meshing. But, it still presented the result for contact area and contact stress, which are obtained by theoretical analysis in part 4. The contact path is irregular in the half cycle. And the contact situation on the 3rd tooth is the worst in the experimental meshing, but the edge contact appeared on the 5th tooth firstly. This experimental result can be considered as the same to the theoretical analysis. 7. Conclusion According to the basic theorem of gear meshing and coordinate transmission, the mathematical equations for non-circular gear contacting with helical curve face gear without misalignment are represented. The contact path and functions of transmission errors on the surface of helical curve face gear are obtained with different errors e= 1 mm, q= 1 mm, γ = 1°. And the further conclusions of helical curve face gear are shown as follows: (1) The tooth contact area on every tooth is irregular and different with either adjacent one in half a cycle and the contact area on tooth 3 is the worst.
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(2) The transmission errors increase from tooth 1 to tooth 3 and decrease from tooth 3 to tooth 5. The errors e and
q generate a new contact area in the right and bottom position of the one without misalignment and error γ generate a new contact area in the left and top position. Error γ does the minimum effect to the meshing process with the biggest difference of transmission errors getting about 15 , while error e does the maximum effect to the meshing process with the biggest difference of transmission errors getting about 20 . (3) The change of curvature radius ρ x and ρ y basically have the same trend with the change of load and the ratio ρ x /ρ y increases with the meshing goes. The contact point and contact area on the tooth surface of helical curve face gear are always in the vicinity of pitch surface. The contact area can’t be shown on the tooth surface completely when the load is too large. Therefore, with the increase of load, there will be contact with the edge of tooth, making load deviation, affecting the meshing of helical curve face gear pair. (4) The contact stress increase at first and then decrease with the increase of height on every tooth. And the trend for the change of contact stress with different height are the same from tooth 1 to toot h 5. The maximum contact stress, which is ranged from the middle to the top, is obtained with different height on different tooth. In the half cycle, the maximum contact stress is obtained at tooth 3. And taking the contact area into consideration, the contact situation on tooth 3 determines the transmission characteristics of curve face gear. (5) Comparing the helical curve face gear with straight curve face gear, the contact ratio of helical curve face gear is larger than straight curve face gear and it ranges from 1.3 to 2.5 with the change of order n1 , modulus m, eccentricity k and other parameters. In half a cycle, the helical curve face gear has less single-tooth meshing and more 2-3 teeth meshing than straight one with the same parameters. And the meshing with 2-3 teeth simultaneously is better than single teeth. So the contact situation of helical face gear is much better than straight face gear. Acknowledgments The author would like to thank the National Natural Science Foundation of China (No. 51675060) and the Chongqing University Postgraduates’ innovation project (CYB15019) for financially supporting this work. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2017.05.014. References [1] C. Lin, H. Gong, N. Nie, Geometry design, three-dimensional modeling and kinematic analysis of orthogonal fluctuating gear ratio face gear drive, Proc. Inst. Mech.Eng. C J. Mech. Eng. Sci. 227 (2013) 779–793. [2] F.L. Litvin, Q. Fan, D. Vecchiato, et al., Computerized generation and simulation of meshing of modified spur and helical gears manufactured by shaving, Comput. Methods Appl. Mech. Eng. 190 (2001) 5037–5055. [3] G. Figliolini, H. Stachel, J. Angeles, Synthesis of the base curves of non-circular gears via the return circle, in: International Gear Conference 2014 26-28, 2014, pp. 540–550. [4] F.Y. Zheng, L. Hua, X.H. Han, The mathematical model and mechanical properties of variable center distance gears based on screw theory, Mech. Mach. Theory 101 (2016) 116–139. [5] F.Y. Zheng, L. Hua, X.H. Han, et al., Linkage model and manufacturing process of shaping non-circular gears, Mech. Mach. Theory 96 (2016) 192–212. [6] A. Fuentes, R. Ruiz-Orzaez, I. Gonzalez-Perez, Comput. Methods Appl, Mech. Eng. 272 (2014) 321–339. [7] D. Mundo, G. Gatti, D.B. Dooner, Optimized five-bar linkages with non-circular gears, Mech. Mach. Theory 44 (2009) 751–760. [8] L. Xia, Y. Liu, D. Li, J. Han, A linkage model and applications of hobbing non-circular helical gears with axial shift of hob, Mech. Mach. Theory 70 (2013) 32–44. [9] F.L. Litvin, Gear Geometry and Applied Theory, 2008. [10] C.B. Tsay, J.-Y. Lin, A mathematical model for the tooth geometry of hypoid gears, Math. Comput. Modell. 18 (2) (1993) 23–34. [11] X.T. Wu, N. Hu, Z.H. Chen, Theoretical formulae for the meshing analysis of point-contact tooth surfaces of hypoid gears, Chin. J. Mech. Eng. 41 (6) (2005) 81–85 (in Chinese). [12] J.Y. Tang, Y.P. Liu, Loaded meshing simulation of face-gear drive with spur involute pinion based on finite element analysis, J. Mech. Eng. 48 (5) (2012) 124–131. [13] Z.Q.M. Li, R.P. Zhu, Impact of assembly deflection error on contact characteristics of orthogonal face gear drive, Acta Aeronaut. Astronaut. Sinica 30 (7) (2009) 1353–1360 (in Chinese). [14] F.L. Litvin, I. Gonzalez-Pererz, A. Fuentes, et al., Design, generation and stress analysis of face-gear drive with helical pinon, Comput. Methods Appl. Mech. Eng. 194 (2005) 3870–3901. [15] R.H. Hsu, H.H. Su, Tooth contact analysis for helical gear pairs generated by a modified hob with variable tooth thickness, Mech. Mach. Theory 71 (2014) 40–51. [16] Y. Wang, Meshing Characteristics and Load Capacity Analysis of Orthogonal Curve-Face Gear, Chongqing University, 2015 in Chinese. [17] S.J. Gu, Analysis on Characteristics of Tooth Load Contact of Curve-face Gear, Chongqing University, 2016 in Chinese. [18] C. Lin, Y. Hou, H. Gong, Design and analysis of transmission mode for high-order deformed elliptic bevel gears, China J. Mech. Eng. 47 (2011) 131–139. [19] C. Lin, D Zeng, Design, generation and tooth width analysis of helical curve-face gear, J. Adv. Mech. Des. Syst. Manuf. 9 (5) (2015) 1–13. [20] C. Lin, C.J. He, Analysis of tooth contact and transmission errors of curve face gear, Proc. Inst. Mech. Eng. Part C (2016) 1–11 0(0). [21] X.T. Wu, Principle of Gearing, 2009 in Chinese.
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Research paper
Analysis of loaded characteristics of helical curve face gear Chunjiang He, Chao Lin∗ State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China
a r t i c l e
i n f o
Article history: Received 29 January 2017 Revised 14 April 2017 Accepted 14 May 2017
Keywords: Variable transmission ratio LTCA Contact stress Helical curve face gear Complex tooth surface
a b s t r a c t As a new gear pair, the helical curve face gear pair have not been widely applied as normal face gear. It can be used to substitute crank-slider mechanism to achieve variable transmission ratio and improve the structure and characteristics to some extent, such as grinders, planers and so on. The mathematical model of the helical curve face gear pair and the tooth contact analysis are necessary for the theoretical analysis of this gear pair. Based on the theory of gear meshing and coordinate transmission, the meshing equations for noncircular gear contacting with helical curve face gear without misalignment are obtained. On the basis of Hertz’s theory, the principal curvatures at contact point are represented. And the contact area on every tooth and the half cycle of helical curve face gear have been established separately. As well as the contact stress on every tooth and half cycle of helical curve face gear. However, there may be misalignment during the meshing of this gear pair, this paper also covers the contact path and the functions of transmission errors with misalignment, includes the comparison of helical curve face gear and straight curve face gear. The developed theories for complex tooth surface are verified by experiment. These analysis show the contact characteristics of this gear pair during the meshing process and provide the theoretical support for further analysis and application of helical curve face gear pair. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Face gear drive is a gear pair with bevel gear and cylinder gear meshing in intersect or cross axes [1]. Comparing with bevel gear drive, face gear drive has many advantages such as much compact structure, less axial force and better bearing capacity; also, it has much simpler structure, large transmission ratio with better contact situation and better flow of power than planetary gear set. With the good characteristics of face gear pair, it can be used in helicopters, truck with overloaded and so on. For the better understanding of face gear drive, a lot of researches have be done by the scholars at home and abroad. The computerized generation and simulation, the contact ellipse and functions of transmission errors [2] have been proposed for the analysis of helical gear. The base curves of non-circular gears with a constant pressure angle was developed by Figliolini [3]. Based on the curves of non-circular gear, the meshing of non-circular gear pair can be classified into fixed center distance and variable center distance. The mathematical model, driving principle and mechanical properties of these two gear type [4,5] was presented by Zheng et al. Two of the major cutters for the generation of non-circular gear are spread-blade cutter and fixed-setting cutters, the advantages and disadvantages of different cutters was proposed by Alfonso
∗
Corresponding author. E-mail address: [email protected] (C. Lin).
http://dx.doi.org/10.1016/j.mechmachtheory.2017.05.014 0094-114X/© 2017 Elsevier Ltd. All rights reserved.
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Fig. 1. The derivation of non-circular gear.
Fuentes with numerical examples [6]. Non-circular gear can be utilized with linkage model, such as five-bar linkages model [7], to obtain the exact path and dynamic qualities of different gear type [8]. According to Gear Geometry and Applied Theory [9], the mathematical model [10] and meshing analysis [11] for localized contact on hypoid gear, localized contact on the surface of tooth is proved to be good for the contact situation of gear drive. So the finite analysis [12], meshing characteristics of orthogonal face gear drive [13] were based on the localized contact method. Litvin introduced the contact path and stress analysis of helical face gear [14]. And the tooth contact analysis for helical face gear with variable tooth thickness was discussed by Hsu [15], so, it is feasible to get some parameters (such as tooth thickness, transmission ratio and so on) of face gear pair changed to get the needed transmission characteristics. Helical curve face gear is a new gear for complex tooth surface, which includes the advantages of face gear, helical gear and non-circular gear, it can mesh with non-circular gear to achieve continuously variable transmission with changeable transmission ratio. So, this face gear can be applied for electric percussion drill and alleviate some of the shortages of normal face gear. The bending stress of curve face gear has been explored roughly by Lin [16,17]. The theoretical analysis of tooth contact and the ANSYS simulation of curve face gear pair were investigated by Lin [18].The theory of processing non-circular gear and helical curve face gear with the same shaper cutter was putted by Lin [19]. This paper establishes the mathematical equations for the surface of helical curve face gear pair in Section 2 and the formula of point contact, UTCA in Section 3. Sections 4 and 5 discuss the contact area and the transmission errors of helical curve face gear. And covers the comparison of contact path and functions of transmission errors between helical curve face gear and straight curve face gear [20] in Section 6.Also, this paper presents the effect with the assembly deflection errors of helical curve face gear. The analysis of contact stress on every tooth and the whole half cycle of helical curve face gear has been covered in this paper. These results provide the theoretical support for further theoretical analysis and practical application of complex tooth surface. 2. Derivation of contact point 2.1. Contact line of non-circular gear with shaper cutter With the use of face gear pair getting more and more widely, the method of manufacturing of face gear has been proposed recently and it can be classified into shaping processing, cutting processing and generating processing. Generating processing is based on the meshing trajectory of gear pair and it is the most widely used method for the theoretical analysis of gear pair. According to the method of gear meshing with point contact on the surface [9], the tooth numbers of helical shaper cutter NS should be more than the tooth numbers of helical non-circular gear N1 and the difference between NS and N1 is 2 or 3 to getting good contact situation. According to the generating method, the tooth surface of non-circular gear is generated by the tooth surface of shaper cutter. As shown in Fig. 1(a), the meshing process of non-circular gear with shaper cutter can be considered as shaper cutter meshing with rack-cutter o and the non-circular gear meshing with the same rack-cutter simultaneously, Fig. 1(b) shows the relationship of the three pitch surfaces. In Fig. 1(a), So (Xo Yo Zo ) is the fixed coordinate system of rack-cutter,
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Ss (Xs Ys Zs ) is the fixed coordinate system of shaper cutter and S1 (X1 Y1 Z1 ) is the fixed coordinate system of non-circular gear. So (Xo Yo Zo ) is the movable coordinate system of rack-cutter, Ss (Xs Ys Zs ) is the movable coordinate system of shaper cutter and S1 (X1 Y1 Z1 ) is the movable coordinate system of non-circular gear. Any one of the three pitch surfaces, which were shown in Fig. 1(b), is doing pure rolling with the other two surfaces at the same time. So, the displacement Loo can be replaced as rs · ϕ s and r1 · ϕ 1 . The mathematical equation of ϕ 1 , r1 and Loo can be defined as:
⎧ ⎨Loo = rs · ϕs ϕ1 = Lroo = rs 0ϕs r11 dϕ ⎩r = 1a(1−k2 ) 1
(1)
1−k cos(n1 ϕ1 )
Where, r1 is the pitch radius of non-circular gear; rs is the pitch radius of shaper cutter; ϕ 1 is the rotation angle of noncircular gear; ϕ s is the rotation angle of shaper cutter; a is the semi-major axis of non-circular gear; k is the eccentricity ratio of non-circular gear; n1 is the order of non-circular gear. With the method of coordinate transformation, the coordinate transformation matrix M1 s are presented as follows:
M1 s = M1 1 · M1s · Ms s
⎡
cos ϕ1 cos ϕs − sin ϕ1 sin ϕs ⎢cos ϕ1 sin ϕs + sin ϕ1 cos ϕs =⎣ 0 0
− cos ϕ1 sin ϕs − sin ϕ1 cos ϕs cos ϕ1 cos ϕs − sin ϕ1 sin ϕs 0 0
0 0 1 0
⎤
− sin ϕ1 (r1 + rs ) cos ϕ1 (r1 + rs ) ⎥ ⎦ 0 1
(2)
The shaper cutter, which is shown in Fig. 1, can be deemed as cylindrical gear and the mathematical equation of shaper cutter can be shown as follows:
⎡
⎤
rbs [cos(ϕos + ϕs − ξs ) − ϕs sin(ϕos + ϕs − ξs )] ⎢−rbs [sin(ϕos + ϕs − ξs ) + ϕs cos(ϕos + ϕs − ξs )]⎥ −→(s ) r s ( ϕ s , ξs ) = ⎣ ⎦ pξs 1
(3)
Where, p is spiral parameter; rbs is the pitch radius of shaper cutter; ϕ os is the helix angle of base circle in shaper cutter;
ϕ s is the rotational angle and ξ s is torsion angle. And the unit normal of shaper cutter:
−→(s ) ns (βs , ϕs ) =
− cos βs sin(ϕos + ϕs − ξs ) − cos βs cos(ϕos + ϕs − ξs ) − sin βs
(4)
Where, β s is the base spiral angle of shaper cutter. The shaper cutter is pure rolling with non-circular gear during the meshing process, so the contact line of non-circular gear and shaper cutter can be derived as follows:
→ − → − fs1 (ϕ1 , βs , ϕs , ξs ) = ns · vs1 = 0
(5)
According to the coordinate transformation matrix (2) of the shaper cutter and non-circular gear, the meshing equation of non-circular gear with shaper cutter can be represented as follows:
− → − → − − → → − → fs1 (ϕ1 , βs , ϕs , ξs ) = ns · (i1 s ωs · (M1 s · ks ) × (M1 s · rs ) − ωs · ks × rs )
(6)
So, the tooth surface equation of non-circular gear can be derived by the meshing Eq. (6) and coordinate transformation matrix (2), shown as follows:
−→
−→ r 1 ( 1 ) ( ϕ 1 , ξ1 ) = M 1 s · r s ( s ) ( ϕ s , ξs ) fs1 (βs , ϕs , ξs ) = 0
(7)
2.2. Contact line of helical curve face gear with variable transmission ratio According to the principle of gear meshing [21], the generating of face gear and shaper cutter can be seen as a composite process that is the face gear rotating and shaper cutter rotating with moving simultaneously. Fig. 2 shows the relationship of the displacement and rotating angles of shaper cutter and curve face gear. The coordinate system Ss (Xs Ys Zs ), shown in Fig. 2, is the fixed coordinate system of shaper cutter, and the coordinate system Ss (Xs Ys Zs ) is the movable coordinate system of shaper cutter. The coordinate system S2 (X2 Y2 Z2 ) is the fixed coordinate system of helical curve face gear, and the coordinate system S2 (X2 Y2 Z2 ) is the movable coordinate system of helical curve face gear. L1 is the radius of helical curve face gear and equal R. L2 is the displacement caused be the moving of shaper cutter and ϕ s , ϕ 2 are the angles caused by the rotating of shaper cutter and helical curve face gear respectively. Based on
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Fig. 2. The meshing of helical curve face gear with shaper cutter.
the coordinate transformation theory, the coordinate transformation matrix M2 s from S2 (X2 Y2 Z2 ) to Ss (Xs Ys Zs )in the meshing process, can be established as follows:
M2 s = M2 2 · M2s
⎡
cos ϕ2 cos ϕs + sin ϕ2 sin ϕs ⎢− cos ϕs sin ϕ2 + + cos ϕ2 sin ϕs =⎣ sin ϕs 0
− sin ϕs cos ϕ2 + sin ϕ2 cos ϕs sin ϕ2 sin ϕs + + cos ϕ2 cos ϕs cos ϕs 0
− sin ϕ2 − cos ϕ2 1 0
⎤
−L1 sin ϕ2 −L1 cos ϕ2 ⎥ ⎦ −L2 1
(8)
The shaper cutter and helical curve face gear are doing pure rolling on the pitch surface, so the contact line of helical curve face gear and shaper cutter during the meshing process, can be derived as follows:
→ − → − fs2 (ϕ2 , βs , ϕs , ξs ) = ns · vs2 = 0
(9)
According to the coordinate transformation matrix (8) of the shaper cutter and face gear, the meshing equation of helical curve face gear and shaper cutter can be proposed as follows:
− → − → − − → − → → fs2 (ϕ2 , βs , ϕs , ξs ) = ns · (i2 s · ωs · (M2 s · ks ) × (M2 s · rs ) − ωs · ks × rs )
(10)
So, the tooth surface equation of helical curve face gear can be derived by the meshing Eq. (10) and coordinate transformation matrix (8), shown as follows:
−→
−→ r 2 ( 2 ) ( ϕ 2 , ξ2 ) = M 2 s · r s ( s ) ( ϕ s , ξs ) fs2 (βs , ϕs , ξs ) = 0
(11)
2.3. The fillet surface and range of radius During the meshing process of curve face gear, there will be fillet surface at the bottom of tooth surface, and the tooth undercut and tooth pointing will limit the width of tooth. The fillet surface is formed by the rounded corner of shaper. In Fig. 2 the rounded corner of shaper cutter can be presented as follows:
⎡
⎤
rbs [cos(ϕos + ϕsc − ξsc ) − ϕsc sin(ϕos + ϕsc − ξsc )] + ρ cos ϕcc −rbs [sin(ϕos + ϕsc − ξsc ) + ϕsc cos(ϕos + ϕsc − ξsc )] − ρ sin ϕcc ⎥ ⎢ −−→(s ) r c (ϕsc , ξsc , ξcc ) = ⎣ ⎦ p(ξsc + ξcc ) 1
(12)
Where, ϕ sc is the rotational angle of top point c of tooth surface, ξ sc is the torsion angle of point c and ξcc is torsion angle between point c and c on the rounded corner. And the unit normal of rounded corner:
−−→(s ) nc (βs , ϕcc ) =
cos ϕcc − sin ϕcc − sin βs
(13)
So the fillet surface can be established with the same method as tooth surface:
−−→
−−→ r2c (2 ) (ϕ2c , ξ2c ) = M2 s · r c (s ) (ϕsc , ξsc , ξcc ) −→ −→ fs2c (βs , ϕsc , ξsc ) = nsc · vsc = 0
(14)
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Fig. 3. The point contact of helical curve face gear pair.
The torsion angle and the width of tooth are small, so the torsion angle ϕ sc can be omitted. The approximation algorithm of the range of radius can be recognized as straight curve face gear. During the undercut process, the minimum radius R1 can be derived by the limit line on the tooth surface of shaper cutter and presented as follows:
⎧ −−−→ −→(2 ) ⎪ r 2 ( ϕ 2 , ξ2 ) = M s s · r s ( s ) ( ϕ s , ξs ) ⎪ ⎪ ⎨ f (βs , ϕs , ξs ) = 0 −−→(2 )
( 2 )
∂ r−−→2 − → dϕs = −v2s (s) dt ∂ ϕs ∂ f dϕs ∂ f dϕ1 ∂ us dt + ∂ ϕs dt = − ∂ ϕ1 dt
∂ r2 ⎪ ⎪ ⎪ ⎩ ∂ f∂ udsus
dus dt
+
(15)
So the minimum radius R1 is:
R1 =
y ∗2k s +(R + zk∗ s )
2
1
(16)
1
Where, y∗ k1 s and z∗ k1 s is the critical coordinate value on the surface of shaper with undercut happened. And the maximum radius without tooth pointing R2 can be established as the similar method with R1 :
R2 =R+
rbs R rbs − rs cos α2 cos α2
(17)
Where, α 2 is the critical pressure angle and α2 is the pressure angle at radius R with tooth pointing happened. 3. Analysis of tooth contact Based on the formation of conjugate tooth surface and the space meshing theory, Fig. 3 shows the contact point on surface 2 , 2 is tooth surface of helical curve face gear and 1 is tooth surface of non-circular gear. Part (a) is the simulating meshing of helical curve face gear pair, Part (b) is the local magnification of part (a) and the theoretical analysis of contact point is shown in part (c). Ls1 is the line of Eq. (7), it shows the contact situation of shaper cutter and non-circular gear and Ls2 is the line of Eq. (11), it shows the situation of shaper cutter and curve helical face gear. During the meshing process, the shaper cutter is meshing with non-circular gear and helical curve face gear simultaneously. So the contact line Ls1 and Ls2 appeared at the same time and the intersection point p, shown in part (c). So, the contact point without misalignment in the meshing process of helical curve face gear pair can be discussed as follows:
fs2 (βs , ϕs , ξs ) = fs1 (βs , ϕs , ξs ) fs1 (βs , ϕs , ξs ) = 0 fs2 (βs , ϕs , ξs ) = 0
(18)
And the parameters (β s , ϕ s , ξ s ) for the contact point should be limited on the surfaces of the gear pair with Eq. (19).
r 2 ( 2 ) ( ϕ 2 , ξ2 ) = M 2 s · r s ( s ) ( ϕ s , ξs ) r 1 ( 1 ) ( ϕ 1 , ξ1 ) = M 1 s · r s ( s ) ( ϕ s , ξs )
(19)
And as the structure of helical curve face gear changes periodically, so any adjacent five teeth of curve face gear in half a cycle can show the characteristics in detail. The tooth number is shown in Fig. 4.
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Fig. 4. The tooth number of helical curve-face gear. Table 1 Parameters of helical curve face gear and shaper cutter. Parameter
Value
Number of tooth of shaper cutter zs Number of tooth of non-circular gear z1 Number of tooth of helical curve face gear z2 Modulus of shaper cutter m (mm) Pressure angle of shaper cutter α s (°) Torsion angle ξ s (°) Inner radius of helical curve face gearR1 (mm) Outer radius of helical curve face gearR2 (mm)
21 18 36 4 20 10 70 83
Fig. 5. The contact path and transmission errors with e = 1 mm.
The helical curve face gear and non-circular gear are both built by the surface of the same shaper cutter. So most parameters of the curve face gear pair are decided by the shaper cutter. Some parameters of the helical curve face gear pair are shown in Table 1. According to Eqs. (18) and (19) and with the help of MATLAB (Matrix Laboratory), the contact points can be obtained. And the function of transmission errors is shown in Eq. (20).
ϕ2 = 0ϕ1 1/i12 dϕ = R1 0ϕ1 r (ϕ1 )dϕ
ϕ 2 = ϕ 2 − ϕ 2
(20)
Where, ϕ2 is the rotating angle with misalignment. ϕ 2 is the difference between theoretical angle ϕ 2 and actual angle
ϕ2 .
As shown in Figs. 5-7, the principle of tooth contact is regular and near the inner part of helical curve face gear in the half cycle.
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Fig. 6. The contact path and transmission errors with q = 1 mm.
Fig. 7. The contact path and transmission errors with γ = 1°.
In the three figures, e is the error of alignment in radial direction of tooth, q is the error of alignment in the height direction of tooth and γ is the amount of angular displacement to the right alignment. The errors e and q are position deviation amount to the right alignment and generating a new contact point to the right and bottom of the one, which is generated by correct alignment, at the same rotating angle. So, the new contact path on every tooth is below the correct one, shown as (a) parts In Figs. 5 and 6. The error γ generating a new contact point to the left and top of the one, which is generated by correct alignment, at the same rotating angle. So, the new contact path on every tooth is on top of the correct one, shown as (a) part In Fig. 7. Part (b) in every figure shows the transmission errors of the five tooth in half a cycle. The transmission errors increase from tooth 1 to tooth 3 and decrease from tooth 3 to tooth 5. The contact path and transmission errors of every alignment condition can represent the generating situation and taking the three errors of alignment into comparison, error e doing the minimal effect to the generating process, while error γ doing the maximal effect.
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Fig. 8. Force of helical curve face gear.
4. Loaded tooth contact analysis The face gear pair always cover a certain load in practical application, so the tooth contact analysis (TCA) is not enough, and further loaded tooth contact analysis (LTCA) is needed to evaluate the characteristics of helical curve face gear. According to Hertz’s theory on the contact between two surface elastomers, the contact ellipse, maximum contact stresses and surface stress distribution on the surface of curve face gear are obtained. For those results, the force on the surface of tooth should be available at first. During the meshing process of helical curve face gear pair, the Force on meshing point P have been shown in Fig. 8. So the force on the surface could be calculated as follows:
⎧ ⎨Ft (θ1 ) = TR1 Ft (θ1 ) Fn (θ1 ) = cos (αn ) ⎩ Fa (θ1 ) = Ft (θ1 ) tan(αn )
(21)
Where, α n is the changeable pressure angle of curve face gear and αn = α0 +π /2 − arctan(r (θ1 )/r (θ1 ) ). In the meshing process, the contact ratio ɛ of this helical curve face gear pair changes from 1.3 to 2.5, so the curve face gear will have 2 or 3 teeth meshing with non-circular gear simultaneously at some rotating angle in the half cycle. The force on the surface of contact point will change with the number of instantaneous meshing teeth. 4.1. The principal curvatures at the center of contact area According to the mathematical equation of non-circular gear, the principal curvatures can be derived as
K1 (1) = 0 K2 (1) = − θ
(22)
1 o1 rb1
Where θ o1 and rb1 are the abduction angle of tooth profile and base radius on non-circular gear. The Eq. (22) shows that the principal curvature K1 (1) always equal 0 and K2 (1) always less than 0 with rb1 increase from root to top on the tooth surface. With the same theory of principal curvatures of non-circular gear, the K1(2 ) and K2(2 ) of helical curve face gear can be derived and shown as below:
−→ −→ −−−→ r θ1 × r θ2 n2 ( f ) = −→ −→ r θ1 × r θ2
(23)
−−−−→ −−−−→ −→ −→ Where, r θ1 is the partial derivative of r2 (2 ) (θ1 , θ2 ) about θ 1 , r θ2 is the partial derivative of r2 (2 ) (θ1 , θ2 )about θ 2 . The surface of curve face gear is irregular surface, so the principal curvatures of curve face gear can be derived by The Second Basic Homogeneous Form of Arbitrary Surface, shown as follows: (2 )
K1,2
2MF − LG − NE =− ± 2(EG − F 2 )
(
LN − M2 2MF − LG − NE 2 ) − 2 2(EG − F ) (EG − F 2 )
(24)
−→ −→ −→ −→ −→ −→ − → −−−→ − → −−−→ − → −−−→ Where, E = r θ1 · r θ1 , F = r θ1 · r θ2 , G = r θ2 · r θ2 , L = n1 · r θ1 θ1 , M = n1 · r θ1 θ2 and N = n1 · r θ2 θ2 .θ 1 is the rotating angle of non-circular gear and θ 2 is the rotating angle of curve face gear.
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Fig. 9. Principal curvatures of curve-face gear.
The principal curvatures K1(2 ) , K2(2 ) of the contact point on curve face gear can be established as follows
As shown in Fig. 9, the principal curvatures K1(2 ) at meshing point are changing with K1(2 ) > 0, in the half cycle with five
teeth. As the meshing goes from the root of tooth to the top, K1(2 ) keeps decreasing at every tooth. For the same height on
tooth surface, K1(2 ) increases from tooth 1 to 3 and decreases from tooth 3 to 5, so the maximum K1(2 ) are obtained at the tooth 3 in half cycle of the curve face gear. The principal curvatures K2(2 ) at meshing point are changing with K2(2 ) < 0, in the half cycle with five teeth. And with the meshing goes from the root of tooth to the top, K2(2 ) keeps increasing at every tooth. For the same height on tooth surface,
the maximum K2(2 ) of every tooth increases from tooth 1 to 3 and decreases from tooth3 to 5, so the maximum K2(2 ) of
every tooth are obtained at the tooth 3 and the minimum K2(2 ) of every tooth increases from tooth 1 to 5 in half cycle of the curve face gear. 4.2. The derivation of contact area One advantage of gear pair is the transmission goes smoothly with force, the contact point will extend to be an area with load. And the contact area is an important representation of the characteristics of curve face gear. According to the Hertz theory, the contact area is an ellipse with its curvature radius related to the principal curvatures K1(2 ) , K2(2 ) of helical curve face gear. The curvature radius at major axis ρ x and curvature radius at minor axis ρ y can be established as follows:
⎧ 3(τ1 +τ2 ) ⎪ ⎪ ⎨ρx (θ1 ) = μ0 3 8 K1(2) +K2(2) +K1(1) +K2(1) Fn (θ1 ) 3(τ1 +τ2 ) ⎪ Fn (θ1 ) ρ ( θ ) = v ⎪ 3 y 1 0 ⎩ 8 K1(2) +K2(2) +K1(1) +K2(1)
(25)
−1 ) Where, τ 1 , τ 2 is the parameters of the material and τ = 4(M . E M2 As the meshing process goes, some curvature radius ρ x , ρ y with no alignment errors can be obtained and shown in Table 2. Show the values of coordinate values at every axis, curvature radius at major axis ρ x and curvature radius at minor axis ρ y on the surface of curve face gear by MATLAB. Fig. 10 shows the trajectory and distribution of contact ellipses on the surface of curve face gear in the half cycle with five teeth during the meshing process. And Figs. 11-15 show the change of contact ellipses in detail, which are caused by different tooth height. As shown in Figs. 11-15: 1, The change of ρ x and ρ y basically have the same trend with load at the surface of every surface, which also meet the Eq. (25). 2, The contact area decreases from tooth 1 to tooth 3 and increases from tooth 3 to tooth 5, the ratio ρ x /ρ y increases with the meshing goes. 3, The meshing point on the tooth surface of helical curve face gear is always in the vicinity of pitch surface. The contact ellipse is formed at meshing point with major axis ρ x near X axis and minor axis ρ y vertical. The contact area can’t show 2
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C. He, C. Lin / Mechanism and Machine Theory 115 (2017) 267–282 Table 2 Some parameters of curve face gear pair. Tooth number
Coordinate values at every axis
1
2.98 6.75 4.21 8.19 7.67 11.93 9.47 15.24 8.56 12.85
2 3 4 5
67.78 67.82 66.24 66.09 62.53 62.2 56.77 56.1 49.58 48.71
−2.37 −3.39 −14.56 −15.57 −26.27 −27.23 −37.12 −38.3 −46.22 −47.27
ρx
ρy
2.49 5.32 2.75 2.77 5.92 3.07 3.65 5.24 1.84 3.39
2.36 4.63 1.51 1.43 3.73 1.88 1.97 3.24 1.43 2.31
Fig. 10. Contact ellipses at different contact points of curve face gear.
on the tooth surface completely when the load is too large. Therefore, when the inner radius of curve face gear R0 = 70 mm and outer radius R2 = 83 mm, the contact ellipse on the surface of helical curve face gear is too close to the inner side of tooth. And with the increase of load, there will be contact with the edge of tooth, making loaded deviation, affecting the meshing of helical curve face gear pair. 4.3. The derivation of contact stress According to the Hertz theory, the contact area on the surface of curve face gear will extend to be ellipse with the increase of load. The contact stress on contact area can be shown as ellipse, which are changing with different load and height on the surface of every tooth. The maximum contact stress σ Hmax can be presented as follows:
σH max (θ1 ) =
0.92
μv
3
K1(2) − K2(2) + K1(1) − K2(1)
( τ1 + τ2 ) 2
2 Tn (θ1 )
(26)
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Fig. 11. Contact ellipses on the surface of tooth 1.
Fig. 12. Contact ellipses on the surface of tooth 2.
Fig. 13. Contact ellipses on the surface of tooth 3.
Fig. 14. Contact ellipses on the surface of tooth 4.
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Fig. 15. Contact ellipses on the surface of tooth 5.
Fig. 16. Contact stress on the surface of tooth 1.
Fig. 17. Contact stress on the surface of tooth 2.
Figs. 16-20 show the change of contact stress with different height on the surface of every tooth, which are caused by the load on the surface of helical face gear. As shown in Figs. 16-20, the contact stress can’t be presented on the tooth surface completely with the increase of height. The stress increase at first and then decrease with the increase of height. The maximum contact stress of every tooth is ranged from the middle to the top, because of the different instantaneous contact ratio and transmission ratio. And the maximum stress appeared at different height for different tooth. In the half cycle, the contact stress increase with the increase of tooth number at first, getting the maximum value at tooth 3, and then decrease from tooth 3 to tooth 5.Taking the conclusion shown in Figs. 11-15 into consideration, the contact ellipse on tooth 3 is the worst one, so the contact situation on tooth 3 is the worst and it determines the transmission characteristics of curve face gear.
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Fig. 18. Contact stress on the surface of tooth 3.
Fig. 19. Contact stress on the surface of tooth 4.
5. FEM analysis During the meshing process, the contact ratio of helical curve face gear pair is ranging from 1.3 to 2.5, so the FEM analysis of helical curve face gear pair will be divided into single meshing and double meshing. The FEM analysis in this paper was obtained with ϕ1 = −0.01π , Fig. 21(a) shows the analysis with single meshing and Fig. 21(b) shows the analysis with double meshing. With the help of ANSYS, the contact stress can be shown on the surface of helical curve face gear with single meshing and double meshing. As Fig. 22(a) and (b) shown, the contact area on the surface of helical curve face gear are appeared at the inner part of tooth. The area is ellipse at every moment and the errors for the theoretical contact stress compared with FEM analysis are shown in Fig. 23. From Fig. 23, the theoretical contact stress is different from FEM analysis during the meshing process. But the trend for the change of theoretical analysis is the same with FEM analysis and the maximum error is 6.54%. During the FEM analysis, there will be some errors, which are caused by 3D model and assemble of gear pair. And taking those errors into consideration, the maximum error 6.54% can be accepted.
6. Experiment The experiment is operated at the State Key Laboratory of Mechanical Transmission of Chongqing University. The noncircular gear was mounted on the drive shaft with helical curve face gear mounted on the driven shaft, shown in Fig. 24. The errors of alignment e, q and γ are represented by the adjustment of the relative position of drive shaft and driven shaft.
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Fig. 20. Contact stress on the surface of tooth 5.
Fig. 21. FEM models with different meshing tooth.
Fig. 22. Contact stress nephograms of helical curve face gear pair.
Fig. 23. The comparison of contact stress.
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Fig. 24. The experimental meshing.
Fig. 25. The comparison between theoretical area and experimental area.
Because of the variable transmission ratio of this helical face gear pair, the surface of every tooth is not the same to either adjacent one in half a cycle. And a half cycle with five teeth of the helical curve face gear should be taken into consideration to get the principle of tooth contact completely. The experimental area are not the same with theoretical result completely, because of the misalignment during experimental meshing. But, it still presented the result for contact area and contact stress, which are obtained by theoretical analysis in part 4. The contact path is irregular in the half cycle. And the contact situation on the 3rd tooth is the worst in the experimental meshing, but the edge contact appeared on the 5th tooth firstly. This experimental result can be considered as the same to the theoretical analysis. 7. Conclusion According to the basic theorem of gear meshing and coordinate transmission, the mathematical equations for non-circular gear contacting with helical curve face gear without misalignment are represented. The contact path and functions of transmission errors on the surface of helical curve face gear are obtained with different errors e= 1 mm, q= 1 mm, γ = 1°. And the further conclusions of helical curve face gear are shown as follows: (1) The tooth contact area on every tooth is irregular and different with either adjacent one in half a cycle and the contact area on tooth 3 is the worst.
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(2) The transmission errors increase from tooth 1 to tooth 3 and decrease from tooth 3 to tooth 5. The errors e and
q generate a new contact area in the right and bottom position of the one without misalignment and error γ generate a new contact area in the left and top position. Error γ does the minimum effect to the meshing process with the biggest difference of transmission errors getting about 15 , while error e does the maximum effect to the meshing process with the biggest difference of transmission errors getting about 20 . (3) The change of curvature radius ρ x and ρ y basically have the same trend with the change of load and the ratio ρ x /ρ y increases with the meshing goes. The contact point and contact area on the tooth surface of helical curve face gear are always in the vicinity of pitch surface. The contact area can’t be shown on the tooth surface completely when the load is too large. Therefore, with the increase of load, there will be contact with the edge of tooth, making load deviation, affecting the meshing of helical curve face gear pair. (4) The contact stress increase at first and then decrease with the increase of height on every tooth. And the trend for the change of contact stress with different height are the same from tooth 1 to toot h 5. The maximum contact stress, which is ranged from the middle to the top, is obtained with different height on different tooth. In the half cycle, the maximum contact stress is obtained at tooth 3. And taking the contact area into consideration, the contact situation on tooth 3 determines the transmission characteristics of curve face gear. (5) Comparing the helical curve face gear with straight curve face gear, the contact ratio of helical curve face gear is larger than straight curve face gear and it ranges from 1.3 to 2.5 with the change of order n1 , modulus m, eccentricity k and other parameters. In half a cycle, the helical curve face gear has less single-tooth meshing and more 2-3 teeth meshing than straight one with the same parameters. 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