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Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
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Reçu: 06 Feb 2022
Accepté: 13 Apr 2022
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Abstract

This article establishes the actual movement mathematical model of CNC machine tools for the precision processing of jewelry. Through analyzing the general geometric error analysis model of CNC machine tools with less than five axes and the method of solving precision CNC instructions, the operating principle of the CNC machine tools is studied. At the same time, we use a transformation matrix to express the relationship between the various moving bodies. The article abstracts the complex motion relationship between entities as the relationship between mathematical matrices. The experimental results show that the theoretical method proposed in this paper can increase the machining accuracy of the machine tool by more than 50%.

Keywords

MSC 2010

Introduction

At present, there are two main methods for modeling spiral bevel gears: The first method is to obtain the discrete points on the tooth surface to fit the spiral bevel gear surface through the tooth surface equation in the three-dimensional software. This method is cumbersome and has poor modeling reusability. The second method simulates the principle of machine tool processing and the actual processing process through simulation processing [1]. The raw material on the tooth blank is removed by the interference of the machining cutter head and the tooth blank and Boolean operation. In the end, the discrete “trajectory” of the cutter head is enveloped into the tooth surface of the spiral bevel gear. Simulating the machining process of spiral bevel gears based on the machining principle of spiral bevel gears has become the main method to establish accurate three-dimensional models. Virtual machining has also become an important idea of spiral bevel gear modeling. A manufacturing simulation system was developed for the spiral bevel gear cutter tilt method based on a 5-axis linkage CNC bevel gear milling machine. The system completely simulates the five-axis machining process of CNC machine tools [2].

The principle of motion conversion between traditional mechanical machine tools and CNC machine tools
The equivalent motion conversion principle of two kinds of machine tools

The CNC machine tool replaces the complex processing adjustment parameters of the transmission machine tool through flexible six-axis five-axis linkage. This makes the CNC machine tool equivalently realize any processing method of the traditional machine tool in the processing process (Figure 1). We need to ensure that the following two formulas are established [3]. The purpose is to reproduce the joint movement relationship between the tool and the workpiece on the traditional machine tool on the CNC machine tool. In this way, the processing of traditional machine tools can be equivalently realized. Lpt(G)(μ,ϕ,ψ)=Lpt(C)(μ,ϕ,ψ) L_{pt}^{\left(G \right)}\left({\mu,\,\phi,\,\psi} \right) = L_{pt}^{\left(C \right)}\left({\mu,\,\phi,\,\psi} \right) (OpOt)p(G)=(OpOt)p(C) \left({{O_p}{O_t}} \right)_p^{\left(G \right)} = \left({{O_p}{O_t}} \right)_p^{\left(C \right)}

Figure 1

Conversion principle of two different types of machine tools

Lpt(μ, ϕ, ψ) represents the rotation transformation matrix from the tool frame to the blank tooth frame in the machine tool coordinate system. (OpOt)p {\left({{O_p}{O_t}} \right)_p} represents the vector length from the origin of the tool frame to the origin of the gear blank in the gear blank coordinate system. The superscript C, G in the vector represents CNC machine tools and traditional mechanical machine tools, respectively [4]. The subscript p, t in the vector represents the gear blank coordinate system and the coordinate tool system, respectively. Among them, formula (1) ensures that the two tools have the same motion posture relative to the tooth blank, and formula (2) ensures that the relative positions of the two are the same.

The coordinate system and mathematical model of traditional mechanical machine tools

A set of coordinate systems, as shown in Figure 2 is established according to traditional mechanical machine tools’ machine setting and processing principles. Sm(xm, ym, zm) is the fixed machine tool coordinate system. Sc(xc, yc, zc) is firmly connected to the cradle, and its origin is the center of the cradle. St(xt, yt, zt) is firmly connected to the cutter head. Sp(xp, yp, zp) is firmly connected to the workpiece. Sn(xn, yn, zn)) is fixedly connected to the saddle, and its origin On is represented by the coordinate (0, −Em, Xb) in Sm. Sq(xq, yq, zq) is the transition coordinate system and parallel to Sn, and its origin Oq is represented by coordinate (X1, 0, 0) in Sn. Sc has an angle around zc and q, Sp has an angle around xp ϕp.

Figure 2

Machine tool processing coordinate system

Descartes coordinate transformation can obtain the transformation matrix Lpt(G)(ζk) L_{pt}^{\left(G \right)}\left({{\zeta _k}} \right) and (OpOt)p(G) \left({\overrightarrow {{O_p}{O_t}}} \right)_p^{\left(G \right)} , Lpt(G)(ζk) L_{pt}^{\left(G \right)}\left({{\zeta _k}} \right) from the cutter head coordinate system to the wheel blank coordinate system as the third-order rotation transformation matrix [5]. Its elements are {a11=cosicosγmsin(qj)sinγmsinia12=cos(qj)cosγma13=cosicosγmsin(qj)+cosisinγm \left\{{\matrix{{{a_{11}} = \cos \,i\,\cos \,{\gamma _m}\,\sin \left({q - j} \right) - \sin \,{\gamma _m}\,\,\sin \,i} \hfill \cr {{a_{12}} = - \cos \left({q - j} \right) - \,\cos \,\,{\gamma _m}} \hfill \cr {{a_{13}} = \cos \,i\,\cos \,{\gamma _m}\,\sin \left({q - j} \right) + \cos \,i\,\,\sin \,{\gamma _m}} \hfill \cr}} \right. (OpOt)p(G)=[SdcosqcosγmXbsinγmX1Sd(sinqcosϕp+cosqsinγmsinϕp)+EmcosϕpXbcosγmsinϕpSd(sinqsinϕp+cosqsinγmsinϕpEmsinϕpXbcosγmcosϕp] \left({{O_p}{O_t}} \right)_p^{\left(G \right)} = \left[ {\matrix{{{S_d}\,\cos \,q\,\cos \,{\gamma _m} - {X_b}\,\,\sin \,{\gamma _m}\, - {X_1}} \hfill \cr {- {S_d}\,\left({\sin \,q\,\,\cos \,\,{\phi _p} + \,\cos \,q\,\sin \,{\gamma _m}\,\,\sin {\phi _p}} \right)} \hfill \cr {+ {E_m}\,\cos \,\,{\phi _p} - \,{X_b}\,\,\cos \,{\gamma _m}\,\,\sin \,{\phi _p}} \hfill \cr \matrix{- {S_d}\,\left({- \sin \,\,q\,\sin \,{\phi _p} + \,\cos \,\,q\,\sin \,{\gamma _m}\,\,\sin \,{\phi _p}} \right. \hfill \cr - \,\,{E_m}\,\sin \,\,{\phi _p} - \,{X_b}\,\,\cos \,{\gamma _m}\,\,\cos \,{\phi _p} \hfill \cr} \hfill \cr}} \right] q = θ0 + ωct, ϕp = Rap ωct in formulas (3) and (4). Where ϕp is the cradle angle. θ0 is the initial cradle angle. ωc is the angular velocity of the cradle. Rap is the rolling ratio. γm is the installation root cone angle of the machine tool. i is the knife inclination angle. j is the corner of the knife. The adjustment angle of the eccentric drum is called the deflection angle and is denoted as β. The adjustment angle of the cradle is called the angle of the cradle, which is recorded as q.

The coordinate system and mathematical model of CNC machine tools

The establishment of a set of coordinate systems of CNC machine tools is shown in Figure 3. Sf(xf, yf, zf) is the coordinate system fixed with the bed. St(xt, yt, zt) and Sp(xp, yp, zp) are respectively fixedly connected with the cutter head and workpiece (small wheel). Sh and Sm are parallel to Sf, and we are connected to the y direction slide table I and z direction slide table III, respectively [6]. The position of Sh origin Oh in Sf is represented by coordinate (x, y, 0), and its function is to describe x, y degrees of freedom. The position of the Sm origin Om in Sf is represented by the coordinate (0, 0, z), and its function is to describe the z degrees of freedom. St rotates the corner β around the coordinate axis zh (that is C, the degree of freedom of rotation). Because Se is fixedly connected to the turntable IV, there is a rotation angle ϕ around the ym axis (that is, B rotation degree of freedom); Sd is the transition coordinate system, parallel to the Se coordinate axis, and the position of the origin Od is determined by L (corresponding to the horizontal wheel position in the mechanical type). Sp rotates around the xp axis (that is, the A rotation degree of freedom), and the rotation angle is ψ. From the Descartes coordinate transformation, Lpt(C)(μ,ϕ,ψ) L_{pt}^{\left(C \right)}\left({\mu,\,\phi,\,\psi} \right) and (OpOt)p(C) \left({{O_p}{O_t}} \right)_p^{\left(C \right)} can be obtained as Lpt(C)(μ,ϕ,ψ)=[cosμcosϕsinμcosϕsinϕsinψcosμsinϕ+cosψsinμsinψsinϕsinμ+cosψcosμsinψcosϕcosψcosμsinϕ+sinψsinμcosψsinϕsinμ+sinψcosμcosψcosϕ] L_{pt}^{\left(C \right)}\left({\mu,\phi,\psi} \right) = \left[ {\matrix{{\cos \,\mu \cos \phi} & {- \sin \,\mu \cos \phi} & {\sin \phi} \cr {\sin \,\psi \cos \mu \sin \phi + \cos \,\psi \sin \mu} & {- \sin \,\psi \sin \phi \sin \mu + \cos \,\psi \cos \mu} & {- \sin \,\psi \cos \phi} \cr {- \cos \,\psi \cos \mu \sin \phi + \sin \,\psi \sin \mu} & {\cos \,\psi \sin \phi \sin \mu + \sin \psi \cos \mu} & {\cos \,\psi \cos \phi} \cr}} \right] (OpOt)p(C)=[xcosϕzsinϕLxsinϕsinψ+ycosψ+zsinψcosϕxsinϕcosψ+ysinψ+zcosψcosϕ] \left({{O_p}{O_t}} \right)_p^{\left(C \right)} = \left[ {\matrix{{x\cos \phi \, - \,z\,\,\sin \,\phi \, - L} \hfill \cr {x\sin \phi \,\,\sin \,\psi \, + y\,\cos \psi \, + \,z\,\,\sin \,\psi \,\cos \phi \,} \hfill \cr {- x\sin \phi \,\,\cos \,\psi \, + y\,\sin \psi \, + \,z\,\,\cos \,\psi \,\cos \phi \,} \hfill \cr}} \right]

Figure 3

CNC machine tool processing coordinate system

Solving the motion parameters of CNC machine tools with different processing methods

From formula (1), we can see that formulas (5) and (6) correspond to the identical elements of the matrix. From equation (2), we can see that equations (4) and (7) correspond to the elements of the matrix are equal. In this way, we can find that [x, y, z, A, B]T is a function of t.

1) When machining spiral bevel gears with a no-tool tilting angle (that is, the generative method i = 0, j = 0), the motion parameter of Phoenix I is [xyzAB]=[Sdcosq(X1L)cosγmSdsinq+Em(X1L)sinγm+Xbϕpγm] \left[ {\matrix{x \cr y \cr z \cr A \cr B \cr}} \right] = \left[ {\matrix{{{S_d}\cos \,q - \left({{X_1} - L} \right)\cos \,{\gamma _m}} \cr {- {S_d}\sin \,q\, + {E_m}} \cr {\left({{X_1} - L} \right)\,\sin \,{\gamma _m} + {X_b}} \cr {{\phi _p}} \cr {{\gamma _m}} \cr}} \right]

2) When processing with the denaturation method, only the proportional relationship between the two quantities is changed. By modifying the roll ratio, the transmission error of the spiral bevel gear meets the pre-designed high-order geometric transmission error curve [7]. In this case, only the proportional relationship between ϕp and t is changed. We can substitute ϕp into equation (7).

3) When the spiral bevel gear is machined with a knife tilting knife angle, the analytical solutions of the motion equations of each axis are obtained by coupling two kinds of machine tool coordinate systems. The formula is as follows {Lx=(a14+L)cosϕ(a34cosψa24sinψ)sinϕLy=a4cosψ+a34sinψLz=(a14+L)sinϕ(a34cosψa24sinψ)cosϕA=arctan(a23/a33)B=arcsin(a13) \left\{{\matrix{{{L_x} = \left({{a_{14}} + L} \right)\,\cos \phi - \left({{a_{34\,}}\,\cos \,\psi \, - \,{a_{24}}\,\sin \,\psi} \right)\sin \,\phi} \hfill \cr {{L_y}\, = \,{a_4}\,\cos \,\psi \, + \,{a_{34}}\,\sin \,\psi} \hfill \cr {{L_z} = - \left({{a_{14}} + L} \right)\sin \,\phi \, - \,\left({{a_{34}}\,\cos \,\psi \, - {a_{24}}\,\sin \,\psi} \right)\,\cos \,\phi} \hfill \cr {A\, = \, - \,\arctan \,\left({{a_{23}}/{a_{33}}} \right)} \hfill \cr {B\, = \,\arcsin \,\left({{a_{13}}} \right)} \hfill \cr}} \right.

The approximate polynomial expression form can be obtained by simplifying the above expression through a set of McLaughlin's formulas: {x=Lx(t)=Lx(0)+Lx(0)t1!+Lx(0)t22!++Lx(n)(0)tnn!+Rn1y=Ly(t)=Ly(0)+Ly(0)t1!+Ly(0)t22!++Ly(n)(0)tnn!+Rn2z=Lz(t)=Lz(0)+Lz(0)t1!+Lz(0)t22!++Lz(n)(0)tnn!+Rn3A=La(t)=La(0)+La(0)t1!+La(0)t22!++La(n)(0)tnn!+Rn4B=Lb(t)=Lb(0)+Lb(0)t1!+Lb(0)t22!++Lb(n)(0)tnn!+Rn5 \left\{{\matrix{{x = {L_x}\left(t \right)\, = {L_x}\left(0 \right) + {{L_x^{'}\left(0 \right) \cdot t} \over {1!}} + {{L_x^{''}\left(0 \right) \cdot {t^2}} \over {2!}} + \cdots + {{L_x^{\left(n \right)}\left(0 \right) \cdot {t^n}} \over {n!}} + {R_{n\,1}}} \hfill \cr {y = {L_y}\left(t \right)\, = {L_y}\left(0 \right) + {{L_y^{'}\left(0 \right) \cdot t} \over {1!}} + {{L_y^{''}\left(0 \right) \cdot {t^2}} \over {2!}} + \cdots + {{L_y^{\left(n \right)}\left(0 \right) \cdot {t^n}} \over {n!}} + {R_{n\,2}}} \hfill \cr {z = {L_z}\left(t \right)\, = {L_z}\left(0 \right) + {{L_z^{'}\left(0 \right) \cdot t} \over {1!}} + {{L_z^{''}\left(0 \right) \cdot {t^2}} \over {2!}} + \cdots + {{L_z^{\left(n \right)}\left(0 \right) \cdot {t^n}} \over {n!}} + {R_{n\,3}}} \hfill \cr {A = {L_a}\left(t \right)\, = {L_a}\left(0 \right) + {{L_a^{'}\left(0 \right) \cdot t} \over {1!}} + {{L_a^{''}\left(0 \right) \cdot {t^2}} \over {2!}} + \cdots + {{L_a^{\left(n \right)}\left(0 \right) \cdot {t^n}} \over {n!}} + {R_{n\,4}}} \hfill \cr {B = {L_b}\left(t \right)\, = {L_b}\left(0 \right) + {{L_b^{'}\left(0 \right) \cdot t} \over {1!}} + {{L_b^{''}\left(0 \right) \cdot {t^2}} \over {2!}} + \cdots + {{L_b^{\left(n \right)}\left(0 \right) \cdot {t^n}} \over {n!}} + {R_{n\,5}}} \hfill \cr}} \right.

The selection of the series order n in the formula is determined according to the required accuracy. Moment t = 0 indicates that the cutter head is at the initial cradle angle position. This paper considers all the processing conditions of face-milling and face-hobbing, and the analytical solution expressions obtained are also applied to face hobbing. Therefore, the cutter head angle participates in the linkage in this process. But in this example, face-hobbing is used, so it is not necessary to consider the angle μ of the cutter head.

Spiral bevel gear simulation processing modeling ideas

The virtual machining modeling thought adopted in this paper has two key points: 1) Adjust the position of the cutter head and the workpiece at the initial time of machining, and simulate the toolset of the CNC machine tool at the beginning of machining. Call the scroll bar in the Visual Basic control, and drag the scroll bar to make the cutter head and workpiece rotate and translate in real-time [8]. In this way, the purpose of adjusting the initial position of the machining can be achieved, and the toolset can be realized. 2) Ensure that the relative positional relationship between the cutter head and the gear blank of the CNC machine tool is correct at each time of the machining process. We use the five-axis motion parameter x, y, z, A, B during the machining of the machine tool as the translation and rotation transformation parameters of the gear blank and the cutter head. At the same time, we use the synchronous change of the relative position of the cutter head and the gear blank to realize the five-axis linkage of the CNC machine tool.

Virtual manufacturing of spiral bevel gears based on five-axis CNC linkage

The movement conversion of the above machine tools provides a theoretical basis for NC machining simulation [9]. This article is based on the phoenix spiral bevel gear processing machine tool. The small wheel adopts the knife tilt method, and the large wheel adopts the forming method.

The composition structure and process of the simulation system

The numerical control machine tool simulation manufacturing system is based on the Visual Basic Application programming language embedded in AutoCAD. We use Active X Automation technology to prepare. The flow structure of the simulation system is shown in Figure 4. The main steps of simulation manufacturing are processing parameter conversion, five-axis linkage equation and related modeling data input, solid modeling of the cutter head and gear blank, simulation processing of tooth cutting, and output of spiral bevel gear model.

Figure 4

Simulation system flow structure

The processing steps of the small wheel cutter tilting method

Start AutoCAD to carry out the parametric modeling of the small wheel “tooth blank” and the machining “cutter head” entities. The models are shown in Figures 5(a) and 5(b), respectively.

Concave surface processing of small wheels. Import the coefficients of the five-axis linkage control equation for concave machining and drag the scroll bar to adjust the initial machining position of the cutter head and the gear blank. In the simulation processing, the gear blank and the cutter head perform translation and rotation transformations according to the changes of the five-axis motion parameters of the machine tool [10]. When the two interfere, the cutter head envelops the concave surface of the bevel gear on the tooth blank. The processing process of the concave surface of the small wheel is shown in Figure 5(c).

Convex surface processing of small wheels. Import the governing equation coefficients for convex machining and make the tooth blank rotate one angle. Then follow step 2) the process of processing the concave surface to complete the processing of the convex surface of the bevel gear.

We copy the processed tooth shape entity in a circumferential direction and then perform the Boolean operation with the tooth blank to obtain the small wheel three-dimensional model. The purpose is to shorten the simulation processing time.

Figure 5

Simulation results

Big wheel forming method processing

The large wheel forming method uses a double-sided cutter to cut out the two sides of the gear simultaneously. The processing process is simpler than that of the small wheel. After adjusting the machining parameters of the machine tool, the two tooth surfaces of the bevel gear can be cut out through a one-time Boolean operation [11]. The solid modeling of the tooth blank and the cutter head and the adjustment of the relative position of the two can refer to the small wheel processing process.

Processing simulation example

We use the simulation manufacturing system to process a pair of gear pairs, the small wheel adopts the knife inclination method, and the large wheel adopts the forming method. The basic parameters of this pair of gears are shown in Tables 1 and 2. The processed model is shown in Figure 6.

Adjustment parameters of small wheel processing machine tools.

Tooth surface Tool nose radius/mm Tooth profile angle/(°)
Concave 285.52 20
Convex 328.94 25
Tooth surface Knife tilt angle/mm Total knife angle/(°)
Concave 46°33′ 11.73848
Convex 53°31′ 13.38259
Tooth surface Machine tool installation root cone angle Horizontal wheel position/mm
Concave −2.00° −7.98
Convex −3°59′ 9.76
Tooth surface Basic knife inclination angle/(°) Eccentric angle
Concave 322.13402 81°54′
Convex 319.0779 102°08′
Tooth surface Bed/mm Vertical wheel position/mm
Concave 17.08 29.12
Convex 22.61 42.62
Tooth surface Rocking angle Radial tool position/mm
Concave 112°12′ 145.6695
Convex 104°56′ 172.8904
Tooth surface Knife corner Roll ratio
Concave −115°39′ 6.33912
Convex −126°07′ 7.26721

Gear blank geometric parameters.

Gear Number of teeth Tooth surface width/mm pressure angle Outer cone distance/mm
Big wheel 41 62 211.77
Small wheel 6 67.46 23°11′/–21°49′ 208.4
gear Cone angle Root cone angle Pitch angle Helix angle
Big wheel 80°11′ 76°01′ 79°44′ 33°50′
Small wheel 13°43′ 9°38′ 10°05′ 45°00′

Figure 6

Machining simulation example model

According to the virtual manufacturing method of gear proposed in this paper, the geometric model of the small wheel is established [12]. According to Gleason's method of measuring total tooth surface error, 45 points were selected, and the actual tooth surface was established (Figure 7).

Figure 7

Normal error analysis of the full tooth surface

Conclusion

This article adopts the movement conversion principle of mechanical machine tools and CNC machine tool processing. The article deduces the six-axis five-linkage control equation of the spiral bevel gear CNC milling machine from the perspective of kinematic equivalent. The calculation process is simple and convenient, and the conversion accuracy is high. This provides a theoretical basis for numerical control simulation processing. This paper's CNC simulation processing system fully realizes the machining simulation of the 5-axis linkage CNC gear milling machine tool inclination method. The system can easily perform tool settings, has a high degree of simulation, and is easy and convenient to operate. This provides an accurate three-dimensional model for some subsequent processing and analysis of spiral bevel gears.

Figure 1

Conversion principle of two different types of machine tools
Conversion principle of two different types of machine tools

Figure 2

Machine tool processing coordinate system
Machine tool processing coordinate system

Figure 3

CNC machine tool processing coordinate system
CNC machine tool processing coordinate system

Figure 4

Simulation system flow structure
Simulation system flow structure

Figure 5

Simulation results
Simulation results

Figure 6

Machining simulation example model
Machining simulation example model

Figure 7

Normal error analysis of the full tooth surface
Normal error analysis of the full tooth surface

Gear blank geometric parameters.

Gear Number of teeth Tooth surface width/mm pressure angle Outer cone distance/mm
Big wheel 41 62 211.77
Small wheel 6 67.46 23°11′/–21°49′ 208.4
gear Cone angle Root cone angle Pitch angle Helix angle
Big wheel 80°11′ 76°01′ 79°44′ 33°50′
Small wheel 13°43′ 9°38′ 10°05′ 45°00′

Adjustment parameters of small wheel processing machine tools.

Tooth surface Tool nose radius/mm Tooth profile angle/(°)
Concave 285.52 20
Convex 328.94 25
Tooth surface Knife tilt angle/mm Total knife angle/(°)
Concave 46°33′ 11.73848
Convex 53°31′ 13.38259
Tooth surface Machine tool installation root cone angle Horizontal wheel position/mm
Concave −2.00° −7.98
Convex −3°59′ 9.76
Tooth surface Basic knife inclination angle/(°) Eccentric angle
Concave 322.13402 81°54′
Convex 319.0779 102°08′
Tooth surface Bed/mm Vertical wheel position/mm
Concave 17.08 29.12
Convex 22.61 42.62
Tooth surface Rocking angle Radial tool position/mm
Concave 112°12′ 145.6695
Convex 104°56′ 172.8904
Tooth surface Knife corner Roll ratio
Concave −115°39′ 6.33912
Convex −126°07′ 7.26721

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