E574656

Lei Peng

Gu Lizhi

College of Mechanical Engineering and Automation

Huaqiao University Quanzhou, P.R. China leipeng900@163.com

Abstract—According to the forming principle of helical surface and space analytic geometry, the mathematical model of helical surface was established. Based on the principle of coordinate conversion, the space envelope theory of single parameter surface, and theorem that the contact line between the cutter revolution surface and the helical surface is a space curve, the theoretical models of the helical surface and the contact line was derived. Based on the thought of numerical operation with MATLAB, 3D mesh model of helical surface and axial section of cutter have been obtained accurately by discretion and digitalization under the control of the number of discrete points. The current modeling, compared with traditional methods of modeling, provides rapidly and accurately the design of products of helical surface and the design of corresponding cutter with axial section. Keywords- helical surface; digital modeling; contact line; axial section; discrete points

I. INTRODUCTION

Helical surface is widely used in mechanical transmission and cutter manufacturing. Because the working surface of many mechanisms, such as screw, worm, gear, drill, cutter etc., is helical surface, the research of machining for helical surface is the important issue in this field. The quality of the helical surface in modeling and the axial section of cutter in design has a significant impact on the accuracy of helical surface in machining. We were driving in detail at the theoretical calculation, including the digital modeling of helical surface, cutter revolution surface, and the method of obtaining axial section of cutter.

Helical surface of cylinder is usually formed with disc cutter in machining as shown 砂轮in Fig.1. Most of the relevant references studied z'on some specific cutters lacking of commonality, while there are also some references talked about the theoretical study for the forming principle of helical surface and axial section ￡of cutter but y'rarely refered to the

corresponding application example. Combining with the numerical analysis method and MATLAB, this paper proposes

ox'a new method to model helical surface and design its corresponding cutter. With this method 3D mesh model of the helical surface and surface of revolution cutter were established by means of programming with numerical method. This modeling method may enhance the design accuracy efficiency greatly and provide theoretical basis and experience for developing CAD simulation system of helical surface machining and corresponding axial section of cutter design based on MATLAB platform.

College of Mechanical Engineering and Automation

Huaqiao University Quanzhou, P.R. China gulizhi888@163.com

II.

ESTABLISHMENT OF MATHEMATICAL MODEL FOR

HELICAL SURFACE

Precise expression for mathematical model of helical surface influences directly the precision of the modeling for the helical surface itself and profile of cutter machining the helical surface. According to the principle of curve generation, there is a fixed 3D coordinate system, （Oxyz）, with its three coordinates in terms of unit vectorsi,j,k, respectively. Let the vector

equation of a space curve ￡ be

r0r0(u) （1）

Or it may be expressed in the form of coordinate values:

x0x0(u) yy(u) （002）

z0z0(u)As shown in Figure 1, suppose that coordinate x coincides

with the axis of helical surface, and ￡ is a generatrix of the cylindrical helical surface. Then ￡ rotates about coordinate x meanwhile moves along coordinate x. This means that ￡ makes spiral motion. The trace of movement of ￡ in space generates helical surface.

xx'cutterz'￡ozyy'

Figure 1 cylindrical helical surface

Equation of cylindrical helical surface can be obtained according to the forming principle of cylindrical helical surface as mentioned above

pv 00x0(u)2X10　　  Y0cosvsinvy0(u)+　　Z0sinvcosvz(u)　　00pvx0(u)2y0(u)cosvz0(u)sinvy(u)sinvz(u)cosv00generatrix. Using coordinate transformation thought change

equation (3) into equation (4) from the workpiece coordinate system O'x'y'z' into the cutter coordinate system as follows

(3)

X'cossin0XY'sincos0YZ'001ZApv (4) where u and v are the parameter of helical surface cross section and rotation angle around x-axis, respectively. Parameter p is helical pitch of the helical surface. With regard to sinistrality helical surface, the sign of p should be changed to minus.

There are diverse generatrices on helical surface through the analysis of the geometric properties of constant curvature and constant torsion. In other words, each curve on helical surface could be its generatrix. The enveloping curve or surface is formed by a curve or surface touching with a family of curves or surfaces. So one of them could be the special contact line which satisfies the tangent conditions between the cutter's cutting circle and the helical surface. It is means that envelope surface is formed during the special contact line’s movement according the theory mentioned above.

III. DESIGN OF PROFILE OF CUTTER FOR MACHINING

HELICAL SURFACE Making sure the installation position relationship between the cutter and the workpiece should be the first job before designing axial section of cutter As shown in figure 2, the angle between the axis of workpiece and the axis of cutter is , which is the workpiece’s helix angle. Thus, disc cutter was able to process along the spiral line tangent direction. The distance between the axes of workpiece and the axes of cutter is A. The workpiece coordinate system Oxyz and the cutter coordinate system O'x'y'z' are established based on the conditions above-mentioned.

cutterworkpiece

Figure 2 Relationship of the cutter and workpiece in installation

Let the normal section (x(u),y(u),z(u)) of helical surface normal to the helical line with helical angle be the

(x0(u)2)cos+(y0(u)cosvz0(u)sinv)sin　　　(xpv0(u))sin(y0(u)cosvz0(u)sinv)cos2yz0(u)sinv0(u)cosvAEquation (5) which was used to describe each point’s normal

vector on helical surface can be expressed as follows

i'j'k'n'X'Y'Z'nx'vvvny'X'Y'Z'nz'uuu[p(y(u)sinvz(u)cosvx(u)(y(u)cosv (5)

z(u)sinv))sin(y(u)y(u)z(u)z(u))cos]　　[p(y(u)sinvz(u)cosvx(u)(y(u)cosvz(u)sinv))cos(y(u)y(u)z(u)z(u))sin]p(x(u)cosvz(u)sinvx(u)(y(u)sinvz(u)cosv)where x(u) is the derivation of x(u) on u. According to the principle of \"envelope forming\on each point of the contact line between workpiece and cutter. Each normal to cutter’s revolution surface had conditions for intersecting with cutter axis. Once they intersected, the normal line turned to be the common normal line. In cutter coordinate system the condition which satisfied tangency and contact is common normal line n', cutter axis and vector radius in the same plane as the formulation of equation (6).

nx'ny'nz'f(u,v)(n',r',i')x'y'z' （6） 100 ny'z'nz'y'0

Put the parameter ny'，nz'，y'and z' into equation (6), and program with numerical method in MATLAB, the corresponding values of uv are worked out through by a series of are given. The relationship of u and v is described in equation (7).

vv(u) (7)

the contact line S which based on the principle of ″envelope forming″ is obtained.

To eliminate v equation (7) is substituted in equation (4),

V. APPLICATION

The radial profile of workpiece as shown in figure 4, its radius R25mm, the radius of circular arc r10mm, the

S(x'(u,v(u)),y'(u,v(u)),z'(u,v(u))) (8) 45, the direction of turning is right. helical angle So equation of helical surface of the workpiece is derived

by the known conditions and equation (3) as follows.

S'(x'(u,v(u)),y'(u,v(u))2z'(u,v(u))2,0) (9)

So the cutter’s axial section can be obtained by equation (8),

IV. THREE-D MESH MODELING BASED ON MATLAB AND SOLUTION TO THE CUTTER PROFILE Although molding helical surface and designing axial

section of cutter need lots of equations which include the general solution of equation, equation of differential etc to be derived and calculated, the thought of numerical operation with MATLAB based on the mathematical model above-established make the process easier.

The helical surface with complex shapes of workpiece is modeled by which the cross-sectional shape of helical surface moving long a spiral lead, so when modeling the helical surface’s 3D mesh model, coordinate value for its surface’s discrete points is enough in MATLAB. There is no need to derive the parametric expression of helical surface before discrete operation. Based on that thought the step of modeling 3D mesh model of helical surface and designing the axial section of cutter as follows:

StartEatablish helicoidal surfaceaccuracy mathematical modelBuild 3D mesh model of helicoidal surface based on numerical operation methodsDerive the contact line of equation accordingto the known parameter βpA

Put u ,v into the contact line of equation (9)to obtain its discrete points' coordinate value Build 3D meshding model of cutterbased on discrete points mentioned aboveExtract discrete points from cutter's 3Dmeshding model by equation (9) and usepolynomial fit technique with MATLABEnd Figure 3 Flow chart

xy　　　　　pv/2（104cosu）cosv4sinusinv (10) z（104cosu）sinv4sinucosvzy0R0 zuroAyFigure 4 Radial profile of workpiece where is the lead of helical surface and it satisfies the relational expression p=2Rcot. Selecting 80mm as the reference distance of cutter axis and workpiece axis based on cutter’s overall dimensions, then cutter disk outside diameter can be certain curtained and its value is 55mm. Because of symmetry of the cutter disk, only need to calculate half of the axial section of cutter.

Table 1 The date of discrete points

Ordinal u /(°) Z /(mm) R /(mm) 1 0 0 -55 2 10 1.692 -54.9 3 20 3.073 -53.841 4 30 4.578 -52.633 5 40 5.832 -50.877 6 50 6.543 -48.842 7 60 7.392-46.595 8 70 8.265-44.091 9 748.498 -43.112 10 79 8.541-42.472

Based on the relational expression umaxcos1(r/2R), the value of u scope is curtained u（0，78.46). Combining with this scope, select out a series of u to obtain axial section of cutter’s discrete points coordinate value. The number of discrete points can be specified by the accuracy requirement. R is the cutter radius, and Z is the coordinate value of cutter axial direction,and the concrete date of R, Z as the Table 1 shows. Finally 3D mesh model of helical surface as shown in Figure 5 and axial section of cutter as shown in Figure 6 are obtained by numerical operation based on MATLAB platform.

Figure 5 Three -D meshding model

Figure 6 Axial section of cutter

VI. CONCLUSION

Modeling helical surface precisely and designing its corresponding ideal axial section of cutter by traditional method is a difficult subject in the field of machining operations at present. In this paper, establishment of mathematical model for helical surface, development of relative program and cross-sectional shape simulation, etc. have been carried out. The modeling of 3D digital helical surface and designing axial section of cutter with MATLAB provides us a new method to study this subject.

Precisely digital modeling for helical surface and the corresponding axial section of cutter have been done through programming the MATLAB program to pick the coordinates of contract line for helical surface and revise the value interval of u to achieve denseness of 3D mesh surface. It has been shown from the current study that the key to the design for the axial section of cutter used in machining helical surface is the proper and precise establishment of the contract line equation. The discrete points of cutter’s axial section were operated by using polynomial fit technique with MATLAB, which guaranteed the accuracy requirement and met the demand of product design. It has been demonstrated that the method proposed and developed to build 3D digital model of helical surface and design axial section of cutter is feasible and it has provided theoretical basis and evidence for developing products of helical surface with CAD and for designing corresponding axial section of cutter.

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