Sage Journals: Discover world-class research

Abstract

The logarithmic spiral bevel gear is a new type of spiral bevel gear and has received great attention in the industry for its excellent engineering characteristics. The objective of this study is to develop an accurate geometric model for the logarithmic spiral bevel gear. Based on the gear tooth flank formation mechanism, a conical logarithmic spiral curve with a constant spiral angle was constructed as the tooth trace curve. The profiles for the exterior and interior transverse of the tooth were built with an accurate involutes curve, with transition being circular arcs and straight lines. The first tooth model was established by sweeping the tooth profile accurately along the tooth trace curve, and the rest of the teeth were created by an array operation. The accurate three-dimensional model of logarithmic spiral bevel gear was finally obtained by means of a Boolean addition operation among all of the gear teeth and the root cone. An experimental study was carried out for such a gear whose number of teeth was 37, with modules being 4.5 mm, normal pressure angle being 20° and spiral angle being 35°. A DMU 40 monoBLOCK five-axis computer numerical control milling machine tool was used to produce the prototype, and a Zeiss CONTURA G3 three-dimensional coordinate instrument was employed to measure the exterior transverse addendum diameter. The linear error between the theoretical model value and the measured average value was 0.0027 mm, indicating the effectiveness and practicability of this modeling method, which provides an accurate geometric model for design and for subsequent tasks like computer-aided engineering analysis and manufacturing.

Keywords

Logarithmic spiral bevel gear accurate modeling conical logarithmic spiral curve Boolean addition operation exterior transverse addendum diameter

Introduction

Spiral bevel gear (SBG) with the premium advantages of stable transmission, low noise characteristics and high coincidence ratio is widely used in aerospace, machine tool, shipbuilding and automobile industries.¹ The varying spiral angle in conventional SBG directly affects the stability of transmission, the contact state of tooth surface and the service life. One of the basic requirements for SBG meshing is that the spiral angle of meshing contact point at a point needs to be equal. A large number of parameters of the cutters and machine tools need to be adjusted for the manufacture of SBG because of the different spiral angles at a point on the tooth trace curve, and even so, it cannot ensure the spiral angle to remain a constant from point to point. A new type of SBG, known as logarithmic spiral bevel gear (LSBG), has been proposed in the literature,^2,3 with characteristic of constant spiral angle because the tooth trace curve is a conical logarithmic curve. Although the new type of SBG is advanced in theory, the actual effects in the process of design, manufacturing, assembly and computer-aided engineering (CAE) analysis still need to be researched and verified.⁴ Therefore, the primary problem for the next series research is to establish the accurate three-dimensional (3D) geometric model of LSBG.

The modeling technology for conventional SBG is becoming more and more matured^5–9 and the technology is already built into commercial software packages, such as UG NX after version 7.5 coming with a GC Toolkit and Solidworks 2011 with Geartrax toolbar. Generally, there are mainly two methods for the modeling of conventional SBG.^10–13 The first method is fitting modeling based on tooth surface equation differentiation.^14–16 The tooth surface model is generated by fitting the discrete points on the tooth surface which is solved by the data processing software tools (such as MATLAB, Mathematica, Maple, Excel and Access), and the model of SBG is generated from the tooth surface model in the end.^17–21 This modeling method needs high mathematical requirements, the operation switches among multiple software tools is relatively tedious and the modeling accuracy is limited because of the interpolation error during discretization and fitting.^22–26 The second method for conventional SBG modeling is the virtual simulation process modeling based on virtual processing technology.^27–30 The SBG model is generated by making use of the virtual machine tools for 3D virtual simulation processing on semi-finished product of SBG model.^31–34 This modeling method with the good advantage of consistency between the model and the actual processed prototype reflects the effect of processing. The apparent drawback of this modeling method is the system error between the 3D model and the mathematical model which influences the modeling precision.

There are few reports on the modeling of LSBG at present. Li and Yan¹ built a LSBG model with two deficiencies, one is that the transverse tooth profiles just are the projections of the normal surface involutes tooth profiles, which lead the size of face width b is difficult to ensure, and the other is that the top clearance coefficient C* = 0.7 and addendum coefficient $h_{a}^{*} = 0.25$ , which did not meet the requirements of the Gleason SBG system. The accurate model of LSBG whose top clearance coefficient C* = 0.85 and addendum coefficient $h_{a}^{*} = 0.188$ was built based on the formation mechanism of LSBG tooth flank in this article. The transverse tooth profiles are accurate involutes. The error of this modeling method is theoretically zero, while the actual linear error is less than 0.1 µm because of the influence of the truncation error and rounded error of computer numerical calculation during the actual modeling.

The derivation of conical logarithmic spiral curve equation

The constant spiral angle spiral curve formed on a cone (the ratio between curvature and torsion is a constant) is called conical logarithmic spiral curve as shown in Figure 1. To form such a curve, let a point M move on the cone surface. When the point moves from M₀ to M₁ with a turning angle τ on the cone surface, the angle will have a change from τ to τ + dτ when it approaches M₂ from M₁, and the vector r defined from the coordinate origin point O, which is the top point of the cone, is changed with an incremental dr at the same time. The spiral angle β defined at point M₂ at the tangent direction of the curve M₁M₂ and the cone generatrix line OM₂ direction will remain constant. The vector equation of the conical logarithmic spiral curve can be derived as follows, assuming that the spiral angle β is constant

r = a e^{φ \sin α \cot β}

(1)

where a is the distance from the start point of conical logarithmic spiral curve to the cone top point O, φ is an independent variable (a projection angle on the bottom surface of turning angle of point M), α is a half cone angle and β is the spiral angle.

Figure 1.

The conical logarithmic spiral curve.

Equation (1) can be rewritten in the form of parametric format as follows

{\begin{cases} x = O M_{x} = a e^{φ \sin α \cot β} \sin α \cos φ \\ y = O M_{y} = a e^{φ \sin α \cot β} \sin α \sin φ \\ z = O M_{z} = a e^{φ \sin α \cot β} \cos α \end{cases}

(2)

The LSBG tooth trace curve is the conical logarithmic spiral curve. One of the important characteristics of conical logarithmic spiral curve is the spiral angle β at a point defined between curve tangent direction and the cone generatrix line direction is a constant. The spiral angle β is a constant as shown in Figure 1.

The accurate modeling of LSBG based on Boolean addition operation

Some parameters of LSBG

There are a pair of conventional Gleason SBG which are installed on a mini bus main reducer. Replace the conventional Gleason SBG with LSBG.

Creation of the basic curves of LSBG

The basic curves of LSBG include the axis line of the wheel, the generatrix line of the face cone, the generatrix line of the pitch cone, the generatrix line of the root cone, the generatrix line of the base cone, the generatrix line of the back cone and the generatrix line of the front cone. The radius of the addendum circle is the distance from the generatrix line of face cone on exterior transverse plane to the axis line of the wheel. The radius of the pitch circle is the distance from the generatrix line of the pitch cone on exterior transverse plane to the axis line of the wheel. The radius of the dedendum circle is the distance from the generatrix line of the root cone on exterior transverse plane to the axis line of the LSBG. The radius of the base circle is the distance from the generatrix line of the base cone on exterior transverse plane to the axis line of the wheel. The basic curve relationships of the LSBG of exterior transverse can be deduced as follows according to the geometric relations

{\begin{cases} r_{2} = m \times z_{2} / 2 \\ r_{2 a} = m \times z_{2} / 2 + m \times h_{a}^{*} \times \cos (δ_{2}) \\ r_{2 b} = d_{2} \times \cos (α_{n}) / 2 \\ r_{2 f} = m \times z_{2} / 2 - (h_{a}^{*} + c^{*}) \times m \times \cos (δ_{2}) \end{cases}

(3)

where r₂ is the radius of the exterior transverse pitch circle of the LSBG, r_2a is the radius of the exterior transverse addendum circle of the LSBG, r_2b is the radius of the exterior transverse base circle of the LSBG, r_2f is the radius of the exterior transverse dedendum circle of the LSBG, δ₂ is the pitch cone angle of LSBG, Z₂ is the number of teeth of the LSBG, $h_{a}^{*}$ is the addendum coefficient, c* is the top clearance coefficient and α_n is the normal pressure angle. The symbols are shown in Figure 2.

Figure 2.

The basic curves.

According to equation (3) and the values of parameters in Table 1, we can obtain the specific value of a parameter in equation (3) and thus can draw the basic sketch curves of the LSBG.

Table 1.

Some parameters of LSBG.

Parameters	Symbol (unit)	Wheel
Number of teeth	Z ₂	37
Exterior transverse modulus	m (mm)	4.5
Shaft angle	Σ (°)	90
Pitch cone angle	δ ₂ (°)	76.3287
Face width	b (mm)	28
Rotation direction		Right
Diameter of pitch circle	d (mm)	166.5
Normal pressure angle	α_n (°)	20
Top clearance coefficient	C ^*	0.188
Addendum coefficient	$h_{a}^{*}$	0.85
Spiral angle	β (°)	35
Face width coefficient	Φ_R	0.3268
Addendum	h_a (mm)	3.825
Dedendum	h_f (mm)	4.671

The basic curve relationship equations of the LSBG of interior transverse are similar with the exterior transverse.

Creation of the basis circles of equivalent LSBG

The exterior transverse basic circles of equivalent LSBG include the exterior transverse pitch circle of equivalent LSBG, the exterior transverse addendum circle of equivalent LSBG, the exterior transverse base circle of equivalent LSBG and the exterior transverse dedendum circle of equivalent LSBG. The numerical calculation equations are shown in equation (4)

{\begin{cases} r_{22} = m \times z_{2} / (2 \times \cos (δ_{2})) \\ r_{2 a 2} = (m \times z_{2} + 2 \times m \times h_{a}^{*} \times \cos (δ_{2})) / (2 \times \cos (δ_{2})) \\ r_{2 b 2} = (d_{2} \times \cos (α_{n})) / (2 \times \cos (δ_{2})) \\ r_{2 f 2} = (m \times z 1 - 2 \times (h_{a}^{*} + c^{*}) \times m \times \cos (δ_{2})) / (2 \times \cos (δ_{2})) \end{cases}

(4)

where r₂₂ is the radius of the exterior transverse pitch circle of equivalent LSBG, r_2a2 is the radius of the exterior transverse addendum circle of equivalent LSBG, r_2b2 is the radius of the exterior transverse base circle of equivalent LSBG and r_2f2 is the radius of the exterior transverse dedendum circle of equivalent LSBG. The symbols are shown in Figure 2.

The creation of the basic circles of the exterior transverse equivalent LSBG can be done according to equation (4) and the parameters in Table 1.

The numerical calculation equations for interior transverse basic circles of equivalent LSBG are similar with the exterior transverse.

The modeling principle based on Boolean addition operation

Suppose A and B represent two models, respectively. The Boolean addition operation computation equation between model A and model B can be expressed by the following equation according to the knowledge of computational geometry and computer graphics

A + B = A out B + B out A

(5)

where A out B means the part of model A out of model B, and B out A means the part of model B out of model A. The modeling principal chart for LSBG based on Boolean addition operation computation is demonstrated in Figure 3.

Figure 3.

The modeling principle of Boolean addition operation.

Creation of the conical logarithmic spiral curve

A conical logarithmic spiral curve, expressed in equation (2), can be generated using UG NX software package which offers a curve creation dialog box based on parametric equation as shown in Figure 4.

Figure 4.

The creation dialog box of conical logarithmic spiral curve.

The use of the expression function to create a conical logarithmic spiral curve on the pitch cone is shown in Figure 5 by selecting t, logx, logy, and logz for the internal variables, x coordinate, y coordinate and z coordinate, respectively, where t is an internal variable in UG NX software package ranging from 0 to 1. The conical logarithmic spiral curve has the characteristic that the start point r₀ = 1, half cone angle α = δ₂, spiral angle β = 35° and the bottom surface projection angle φ of M is 190°. The conical logarithmic spiral curve on the pitch cone is the tooth trace curve, and it is also the sweep trace curve for the gear tooth accurate modeling of LSBG.

Figure 5.

The logarithmic spiral curve on the pitch cone.

The descriptions and visual expressions of the exterior transverse and interior transverse involutes

The exterior transverse and interior transverse tooth profiles of LSBG are both involutes. The involute equations of exterior transverse can be written in the form of

{\begin{cases} x_{t 2} = r_{2 b 2} \times \cos (θ) + r_{2 b 2} \times \sin (θ) \times t \times π / 2 \\ y_{t 2} = r_{2 b 2} \times \sin (θ) - r_{2 b 2} \times \cos (θ) \times t \times π / 2 \\ z_{t 2} = 0 \end{cases}

(6)

where x_t₂ is the exterior transverse involutes’x coordinate of LSBG, y_t₂ is the exterior transverse involutes’y coordinate of LSBG, z_t₂ is the exterior transverse involutes’z coordinate of LSBG, r_2b2 is the exterior transverse equivalent base circle radius of LSBG, θ is the involutes’ spread angle and t is an internal variable in UG NX software package ranging from 0 to 1. The symbols are shown in Figure 6.

Figure 6.

The exterior transverse involutes.

One side involutes curve of LSBG exterior transverse was constructed by making use of the curve function of UG NX software package, and the other side involutes curve as shown in Figure 7 was generated by mirroring the constructed involutes curve along the symmetry center axis. The transition curve between the involutes curve and the dedendum circle consists of two segments. The first segment is a transition straight line which starts from the start point (intersection point of involutes and base circle) and is tangent to the involutes curve. The second section is a transition circular arc which connects the first section transition straight line and the dedendum circle. The radius of the transition circular arc r_t is defined by the following equation

r_{t} = {\begin{cases} 0.38 \times m & 1 ⩽ h_{a}^{*} \\ 0.46 \times m & 0 < h_{a}^{*} < 1 \end{cases}

(7)

The symbol r_t is shown in Figure 7.

Figure 7.

The equivalent gear involutes tooth of exterior transverse.

The interior transverse involutes curve construction of LSBG is similar to the exterior transverse LSBG.

The solid line shown in Figure 8 is the tooth trace curve (swept path) for the gear tooth modeling of the LSBG. The tooth trace curve was generated by selecting the inner surface and the outer surface of the pitch cone for truncating the conical logarithmic spiral curve in Figure 8.

Figure 8.

The modeling flow of LSBG.

The sweeping operation of the model can be done by making use of the sweep function of UG NX software package. The user may select the involutes gear tooth profiles from both sides, as shown in Figure 8, as the sweeping section, and choose the tooth trace curve (conical logarithmic spiral curve) with the right curve direction^35,36 as the sweeping path. A resulting model of one gear tooth is shown in Figure 8.

The construction of the completed model of LSBG

One tooth in model is created as shown in Figure 8. Now we produce the entire gear wheel by performing an “array” operation. The first tooth is chosen as the source for the operation, the number for array is set to Z₂ − 1 and the array center line is set to the root cone axis. The result is shown in Figure 8. Next, the root cone model can be established according to the root cone parameters. Finally, a Boolean addition operation can be performed between the root cone and all gear teeth. The final 3D model of LSBG is generated by a Boolean addition operation between the root cone and all gear teeth.

The machining of the prototype

The prototype processing on five-axis computer numerical control milling machine tool

A medium carbon steel 45 was used to produce the prototype on a DMU 40 MonoBLOCK five-axis computer numerical control (CNC) milling using ball end milling^37–40 as shown in Figure 9.

Figure 9.

Processing the LSBG on a DMU 40 monoBLOCK five-axis CNC milling machine tool.

The NC programs were generated automatically using the software package of UG CAD system.^41–43 UG CAD system has the capability of automatic generation of NC codes for manufacturing of parts as long as a model is input.

The derivations of circle center coordinates and diameter of exterior transverse addendum circle

When the coordinates of three points R(x₁, y₁, z₁), S(x₂, y₂, z₂) and T(x₃, y₃, z₃) in space are measured on exterior transverse addendum circle of the LSBG, then the center point O₁(x, y, z) and the diameter of the LSBG can be calculated using the following equation

[\begin{matrix} x \\ y \\ z \end{matrix}] = - {[\begin{matrix} A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3} \end{matrix}]}^{- 1} [\begin{matrix} D_{1} \\ D_{2} \\ D_{3} \end{matrix}]

(8)

where

\begin{array}{l} A_{1} = y_{1} z_{2} - y_{1} z_{3} - z_{1} y_{2} + z_{1} y_{3} + y_{2} z_{3} - y_{3} z_{2} \\ B_{1} = x_{1} z_{3} - x_{1} z_{2} + z_{1} x_{2} - z_{1} x_{3} - x_{3} z_{3} + x_{3} z_{2} \\ C_{1} = x_{1} y_{2} - x_{1} y_{3} - y_{1} x_{2} + y_{1} x_{3} + x_{2} y_{3} - x_{3} y_{2} \\ D_{1} = x_{1} y_{3} z_{2} - x_{1} y_{2} z_{3} + x_{2} y_{1} z_{3} - x_{3} y_{1} z_{2} - x_{2} y_{3} z_{1} + x_{3} y_{2} z_{1} \end{array}

\begin{array}{l} A_{2} = 2 (x_{2} - x_{1}) \\ B_{2} = 2 (y_{2} - y_{1}) \\ C_{2} = 2 (z_{2} - z_{1}) \\ D_{2} = x_{1}^{2} + y_{1}^{2} + z_{1}^{2} - x_{2}^{2} - y_{2}^{2} - z_{2}^{2} \end{array}

\begin{matrix} A_{3} = 2 (x_{3} - x_{1}) \\ B_{3} = 2 (y_{3} - y_{1}) \\ C_{3} = 2 (z_{3} - z_{1}) \\ D_{3} = x_{1}^{2} + y_{1}^{2} + z_{1}^{2} - x_{3}^{2} - y_{3}^{2} - z_{3}^{2} \end{matrix}

The diameter D is determined by the three points in the form of following expression

D = 2 \sqrt{{(x_{1} - x)}^{2} + {(y_{1} - y)}^{2} + {(z_{1} - z)}^{2}}

(9)

D = 2 \sqrt{{(x_{2} - x)}^{2} + {(y_{2} - y)}^{2} + {(z_{3} - z)}^{2}}

(10)

D = 2 \sqrt{{(x_{3} - x)}^{2} + {(y_{3} - y)}^{2} + {(z_{3} - z)}^{2}}

(11)

Measuring the exterior transverse addendum circle diameter of LSBG prototype by 3D coordinate measuring instrument

The coordinates of three teeth on LSBG exterior transverse addendum circle were measured using a Zeiss CONTURA G3 3D coordinate measuring instrument. The measured coordinate values are shown in Table 2.

Table 2.

The data of Zeiss CONTURA G3 3D coordinate measuring experiment (mm).

Group	Order of tooth	x coordinate	y coordinate	z coordinate	The theoretical value	The measured value	Error
Group 1	1	−57.430	51.995	−2.572	168.308089	168.3054	0.002689
	13	85.758	58.077	−2.573
	26	12.465	−73.123	−2.570
Group 2	5	−15.340	89.038	−2.573	168.308089	168.3050	0.003089
	17	99.886	3.817	−2.573
	30	−39.561	−52.217	−2.570
Group 3	9	40.678	91.418	−2.574	168.308089	168.3058	0.002289
	21	76.791	−47.275	−2.571
	34	−66.906	−3.268	−2.571
Average value					168.308089	168.3054	0.002689

The axial linear error in an axis of DMU 40 monoBLOCK five-axis CNC milling machine tool was measured and got with the value of 0.001 mm. The axial linear error in an axis of Zeiss CONTURA G3 3D coordinate measuring instrument was measured and with the value of 0.0018 mm. The axial truncation error and rounded error of numerical calculation using UG NX software package for modeling are 0.0001 mm. So, the cumulative linear errors in an axis are the sum of those three components

0.001 + 0.0018 + 0.0001 = 0.0029 mm

The total linear error in space can be calculated as follows

0.0029 \times \sqrt{1^{2} + 1^{2} + 1^{2}} = 0.0051 mm

The x, y, z coordinates of exterior transverse addendum circle of LSBG prototype were measured on Zeiss CONTURA G3 3D coordinate measuring instrument. The exterior transverse addendum circle diameter of each group was calculated according to equations (8) and (9) with the measured data, and the average value of exterior transverse addendum circle diameter of 3 groups was calculated. The measured average value of the exterior transverse addendum circle diameter of LSBG prototype was 168.3054 mm as shown in Table 2.

The calculated theoretical values of exterior transverse addendum circle diameter of LSBG can be written as follows according to equation (3)

\begin{matrix} d_{2 a} = m \times z_{2} + 2 \times m \times h_{a}^{*} \times \cos δ_{2} \\ = 4.5 \times 37 + 2 \times 4.5 \times 0.85 \\ \times \cos (76 . 3287^{\circ}) = 168.308089 mm \end{matrix}

The average error for exterior transverse addendum circle diameter of the LSBG prototype between theoretical calculations and measured data is

168.308089 - 168.3054 = 0.002689 mm

It is about 0.0027 mm after date are rounded.

Discussion and conclusion

The method based on Boolean addition operation between the tooth models and the root cone of LSBG is used for LSBG accurate modeling, whose number of teeth was 37, with modules being 4.5 mm, normal pressure angle being 20° and spiral angle being 35°. The theoretical error of this modeling method is zero, and the actual model had a linear error less than 0.1 µm.

The prototype of LSBG was successfully processed on a DMU 40 monoBLOCK five-axis CNC milling machine, and the material of the LSBG was medium carbon steel 45.

The Zeiss CONTURA G3 3D coordinate measuring instrument was used to measure x, y, z coordinate values of three points on exterior transverse addendum circle of the LSBG prototype with three groups. According to the space geometry, the average diameter of the exterior transverse addendum circle can be calculated by the x, y, z coordinate values of three points. The greatest linear error between the theoretical value and the measured average value is 0.0027 mm.

An accurate model of LSBG was created based on the gear tooth flank formation mechanism and the Boolean addition operation. The fabrication of the LSBG prototype was completely carried out on the above model with rather high precision. Both may pave the way for LSBG design and manufacture for the applications in industry.

The parametric modeling of the LSBG as well as the machining efficiency for the product is still in expertise of the future work.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was sponsored by the National Natural Science Foundation of China named “Research on standing wave mechanism and experiment for the failure of high-speed tire blow-out” under grant no. 51475399 and the Science and Technology Projects named “Research on meshing dynamics of logarithmic spiral bevel gear” under grant no. JA14239 of Education Department of Fujian Province, China.

References

Yan

HB.

The logarithmic spiral bevel gear meshing theory. Beijing, China: Metallurgical Industry Press, 2012.

Weng

Wang

GP.

Gearing theory and tooth area’s formation of logarithmic spiral bevel gear(I). J Liaoning Tech Univ 2008; 27(1): 107–109.

Tan

Chen

Peng

. Study on spatial curve meshing and its application for logarithmic spiral bevel gears. Mech Mach Theory 2015; 86(4): 172–190.

Ani

Gerd

Paul

. Three-dimensional, parallel, finite element simulation of fatigue crack growth in a spiral bevel pinion gear. Eng Fract Mech 2015; 72(8): 1148–1170.

Deng

Hua

Han

XH.

Numerical and experimental investigation of cold rotary forging of a 20CrMnTi alloy spur bevel gear. Mater Des 2011; 32(3): 1376–1389.

Fang

Optimization of face cone element for spiral bevel and hypoid gears. J Mech Des: T ASME 2011; 133(9): 091002 (7 pp.).

Suh

Jih

Hong

. Sculptured surface machining of spiral bevel gears with CNC milling. Int J Mach Tool Manu 2001; 41(6): 833–850.

Yang

Huang

GX.

The OPC technology research about spiral bevel gear machine tools for machining simulation problems. Procedia Eng 2011; 15(8): 1266–1270.

Jin

Zhang

Yang

HL.

Research on the 3D solid modeling of high efficient gear tooth based on ProE. Procedia Eng 2012; 29(2): 2990–2994.

10.

Ding

Adayi

XZD

. Methods of and outlook for the modeling of spiral bevel gear. Mach Des Manuf 2014; 51(1): 257–260.

11.

Tsai

Hsu

WY.

The study on the design of spiral bevel gear sets with circular-arc contact paths and tooth profiles. Mech Mach Theory 2008; 43(9): 1158–1174.

12.

Jerzy

Albert

Roman

. 3D modeling of induction hardening of gear wheels. J Comput Appl Math 2014; 270(11): 231–240.

13.

Chen

Fong

ZH.

Study on the cutting time of the hypoid gear tooth flank. Mech Mach Theory 2015; 84(2): 113–124.

14.

Vilmos

VS.

Influence of tooth modifications on tooth contact in face-hobbed spiral bevel gears. Mech Mach Theory 2011; 46(12): 1980–1998.

15.

Vilmos

VS.

Optimization of face-hobbed hypoid gears. Mech Mach Theory 2014; 77(7): 164–181.

16.

Lin

Tsay

Fong

ZH.

Computer-aided manufacturing of spiral bevel and hypoid gears by applying optimization techniques. J Mater Process Tech 2001; 114(1): 22–35.

17.

Wei

Liu

. The kinematic synthesis of involute spiral bevel gears and their tooth contact analysis. Mech Mach Theory 2014; 79(9): 141–157.

18.

Cui

Fang

. Precise modeling of arc tooth face-gear with transition curve. Chinese J Aeronaut 2013; 26(5): 1346–1351.

19.

Vilmos

VS.

Advanced manufacture of spiral bevel gears on CNC hypoid generating machine. J Mech Des: T ASME 2010; 132(3): 031001 (8 pp.).

20.

Shih

Fong

ZH.

Flank correction for spiral bevel and hypoid gears on a six-axis CNC hypoid generator. J Mech Des: T ASME 2008; 130(6): 062604 (11 pp.).

21.

Tang

Dai

Accurate modeling method of a SGM-manufactured spiral bevel gear. J Mech Transm 2008; 32(1): 43–46, 83.

22.

Lin

Tsay

Fong

ZH.

Mathematical model of spiral bevel and hypoid gears manufactured by the modified roll method. Mech Mach Theory 1997; 32(2): 121–136.

23.

Sun

Liu

Research on the method of accurate NURBS surface fitting to scattered points. Chin J Mech Eng 2005; 40(3): 10–14.

24.

Jehng

WH.

Computer solid modeling technologies applied to develop and form mathematical parametric tooth profiles of bevel gear and skew gear sets. J Mater Process Tech 2002; 122(2): 160–172.

25.

Wei

Zhang

LH.

Solid modeling of spiral bevel gear considering cutter fillet and undercutting. J Mach Des 2011; 28(3): 9–12.

26.

Fong

ZH.

Mathematical model of universal hypoid generator with supplemental kinematic flank correction motions. J Mech Des: T ASME 2000; 122(1): 136–142.

27.

Zhang

Wei

Virtual machining on monolithic structure spiral bevel gear milling machine. Trans CSAE 2008; 24(12): 71–75.

28.

Ferenc

Zoltan

Mixed CAD method to develop gear surfaces using the relative cutting movements and NURBS surfaces. Procedia Technol 2015; 19(2): 20–27.

29.

Fan

Computerized modeling and simulation of spiral bevel and hypoid gears manufactured by Gleason face hobbing process. J Mech Design 2006; 128(11): 1315–1327.

30.

Litvin

Wang

Handschuh

RF.

Computerized generation and simulation of meshing and contact of spiral bevel gears with improved geometry. Comput Method Appl M 1998; 158(3): 35–64.

31.

Litvin

Alfonso

Fan

. An invariant approach for gear generation with supplemental motions. Mech Mach Theory 2007; 42(5): 275–295.

32.

Chiang

Fong

Chang

KL.

Computerized gear cutting simulation using a psuedo-planar method. Proc IMechE, Part B: J Engineering Manufacture 2009; 223(12): 1541–1551.

33.

Wang

. 3D solid accurate modeling of spiral bevel gear based on manufacture method and meshing theory. J Jilin Univ 2008; 38(6): 1315–1319.

34.

Suh

Lee

Kim

. Geometric error measurement of spiral bevel gears using a virtual gear model for STEP-NC. Int J Mach Tool Manu 2002; 42(3): 335–342.

35.

Wang

FY.

Application of shape comparing technology in CAD. Die Mould Ind 2011; 37(3): 21–23.

36.

Wang

Niu

. Algorithms for determining the direction and inclusivity of 2D curves. Die Mould Ind 2013; 39(11): 16–20.

37.

Chen

Zhao

Dong

. Research on the machined surface integrity under combination of various inclination angles in multi-axis ball end milling. Proc IMechE, Part B: J Engineering Manufacture 2014; 228(1): 31–50.

38.

Chen

Zhao

. Numerical modelling of high-speed ball end milling with cutter inclination angle. Proc IMechE, Part B: J Engineering Manufacture 2012; 226(4): 606–616.

39.

Liu

Loftus

Prediction of surface quality from ball-nose milling in high-speed machining applications. Proc IMechE, Part B: J Engineering Manufacture 2006; 220(4): 571–578.

40.

Jung

Lee

. Swept volume with self-intersection for five-axis ball-end milling. Proc IMechE, Part B: J Engineering Manufacture 2003; 217(8): 1173–1178.

41.

Prieto

Wright

Qin

. A novel desktop computer-aided design system for early form design developments. Proc IMechE, Part B: J Engineering Manufacture 2007; 221(2): 277–288.

42.

Williams

Torrens

Hodgson

AR.

Integration of anthropometric data within a computer aided design model. Proc IMechE, Part B: J Engineering Manufacture 2004; 218(10): 1417–1421.

43.

Huang

Zhang

Bai

. An effective numerical control machining process reuse approach by merging feature similarity assessment and data mining for computer-aided manufacturing models. Proc IMechE, Part B: J Engineering Manufacture 2015; 229(7): 1229–1242.

Accurate modeling of logarithmic spiral bevel gear based on the tooth flank formation and Boolean addition operation

Abstract

Keywords

Introduction

The derivation of conical logarithmic spiral curve equation

The accurate modeling of LSBG based on Boolean addition operation

Some parameters of LSBG

Creation of the basic curves of LSBG

Creation of the basis circles of equivalent LSBG

The modeling principle based on Boolean addition operation

Creation of the conical logarithmic spiral curve

The descriptions and visual expressions of the exterior transverse and interior transverse involutes

The construction of the completed model of LSBG

The machining of the prototype

The prototype processing on five-axis computer numerical control milling machine tool

The derivations of circle center coordinates and diameter of exterior transverse addendum circle

Measuring the exterior transverse addendum circle diameter of LSBG prototype by 3D coordinate measuring instrument

Discussion and conclusion

Footnotes

Appendix 1

Declaration of Conflicting Interests

Funding

References