Optimization of Grinding Parameters for Twist Drill in Biglide Parallel Machine

Abstract

In view of the Biglide parallel machine, this paper presented the theoretical and experimental basis for grinding the designated geometrical parameters of twist drill based on optimizing of the grinding parameters. The kinematics structure of the Biglide parallel machine was firstly introduced for grinding of the twist drill, and then the mathematical model of the twist drill flank was derived. And the relations between the geometric parameters of the drill point and grinding parameters of the grinding machine were analyzed, through which the optimal grinding parameters were obtained by using genetic algorithm. It was verified that the optimized grinding parameters improved the grinding precision,depending on the customers’ demands.

1 Introduction

Within the recent years, the parallel mechanisms have attracted many researchers’ and machine manufacturers’ attractions as machine tools because of their conceptual potentials in high motion dynamics and accuracy combined with high structural rigidity [1, 2].

Due to its high precision and versatility for various machining, PKM became increasingly attractive for grinding of twist drill-complex geometry of cutting tool, using the grinding system based on the Stewart platform [3]. But because of its disadvantages such as small workspace and complex foreword kinematics, many researchers paid many attentions to PKM with limited degrees of freedom [4]. Using the fluted drill for grinding work piece in many cases, its grinding movement is relatively simple, its process does not need a big workspace and its grinding force is relatively small, therefore, the Biglide parallel machine can be reasonable to grind twist drill in high precision by only using the grinding parameters [5, 6].

The geometry of twist drill is more complex than the other cutting tool, so many modeling methods and numeric approximation approaches have been reported [7, 8]. However, it is still remained as a difficult problem to grind out the required geometry of twist drill along with the demand of users for improving its cutting performance [5, 9, 10].

In this paper, the grinding system of twist drill is constructed using parallel kinematics machine and the modeling of twist drill flank is made based on grinding parameters. For the practical grinding of twist drill, those key geometric parameters are analyzed and optimized based on the grinding parameters.

Optimizing the grinding parameters based on genetic algorithm, the theoretical basis is obtained for grinding of the high-precision twist drill flank according to the user’s needs of the twist drill geometry.

2 Kinematics of Biglide Parallel Machine

The structure of the Biglide parallel machine, as shown Fig. 1, has three translations, which is comprised of a certain length of biglides, coupling with the moving platform and the slide units by revolute joint. The work table moves up and down by the unit of worm and worm wheel. The linear movement of each slider is provided with the ball screw and actuated by a stepping motor.

Fig. 1

Structure chart of Biglide parallel machine: 1 sliding carriage, 2 ball screw, 3 slider, 4 rod, 5 drill, 6 moving platform, 7 grinding wheel, 8 work table, 9 worm gearing

The velocity of the drill point can be decomposed into $v_{x}$ and $v_{z}$, according to the relationship as shown in Fig. 2; they are given as:

Fig. 2

Schematic movement Biglide parallel machine

$$v_{x} = \frac{v}{2},\;v_{z} = \frac{{v(L_{1} - vt - 2r_{0} )}}{{2\sqrt {4L_{2}^{2} - (L_{1} - vt - 2r_{0} )^{2} } }}$$

(1)

The motion of the drill point in z-axis generates acceleration, however, the value of the acceleration is very small (about 0.008 mm/s²) and the grinding time is very short (2 s), thus its motion can be linearized.

In order to guarantee the whole surface of the drill flank to be ground, the angle between the drill axis and the moving direction of the drill point in grinding should not be larger than the angle between the grinding wheel axis and the drill axis.

$$\tan^{ - 1} (\frac{{v_{x} }}{{v_{z} }}) \le \theta$$

(2)

3 Modeling of Twist Drill Flank

The drill point geometry is uniquely configured with the drill flank and of the flute. The flute feature being designed by the manufacturer, the drill flank one becomes the principal factor for developing new drill point geometry, as well as improving the grinding accuracy and efficiency for the drill point concerned by its user and manufacturer.

As shown in Fig. 3, the moving coordinate system $S_{2} (x_{2} ,y_{2} ,z_{2} )$ performs a screw motion with respect to global coordinate system $S(x,y,z)$ and together with a translation along negative $X$-direction. The angle of rotation and axial displacements in motion are described with $\varphi$, $k_{z} \varphi$ and $k_{x} \varphi$, respectively.

Fig. 3

Coordinate system

Here $k_{x}$ and $k_{z}$ are given by:

$$k_{z} = \frac{{v_{z} }}{\omega },\;k_{x} = \frac{{v_{x} }}{\omega }$$

(3)

When angle of drill revolution is $\varphi$, an arbitrary point on the intersection curve of the drill flank and the grinding wheel is expressed as $P(x_{p} ,y_{p} ,z_{p} )$ in coordinate system $S_{2}$. Then, in coordinate system $S$, the ground flank surface can be represented as following.

$$\left[ {\begin{array}{*{20}c} x \\ y \\ z \\ 1 \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos \varphi } & { - \sin \varphi } & 0 & { - k_{x} \varphi } \\ {\sin \varphi } & {\cos \varphi } & 0 & 0 \\ 0 & 0 & 1 & {k_{z} \varphi } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x_{p} } \\ {y_{p} } \\ {z_{p} } \\ 1 \\ \end{array} } \right]$$

(4)

$$\left\{ \begin{aligned} x &= x_{p} \cos \varphi - y_{p} \sin \varphi - k_{x} \varphi \hfill \\ y &= x_{p} \sin \varphi + y_{p} \cos \varphi \hfill \\ z &= z_{p} \cot \varphi + k_{z} \varphi \hfill \\ \end{aligned} \right.$$

(5)

4 Parametric Model of Twist Drill

In this section, three important geometric parameters for cutting force, cutting temperature and durability are represented in terms of grinding ones such as relief angle, semi-point angle and chisel angle, one by one.

As shown Fig. 4, the relief angle in an arbitrary point $A$ on cutting edge is defined as the angle between the drill flank surface and the drill point end plane, measured on the section tangent to cylinder with radius $r_{a}$ crossing the point $A$.

Fig. 4

Schematic figure of relief angle

Generally, the relief angle means the one of the outside corner $C$.

Setting the coordinate system $O_{a} (x_{a} ,y_{a} ,z_{a} )$ with $x_{a}$ axis passing the point $A$, the relief angle of its location is expressed as following.

$$\tan \alpha_{fa} = \left[ {\frac{{dz_{a} }}{{dy_{a} }}} \right]_{a}$$

(6)

The relationship equation of the coordinate system $O_{a} (x_{a} ,y_{a} ,z_{a} )$ and $O(x,y,z)$ is expressed as

$$\left[ {\begin{array}{*{20}c} {x_{a} } \\ {y_{a} } \\ {z_{a} } \\ \end{array} } \right] = \left[ \begin{array}{*{20}c} {\cos \delta_{a} } & {\sin \delta_{a} } & 0 \\ - \sin \delta_{a} & {\cos \delta_{a} } & 0 \\ 0 & 0 & 1 \\ \end{array} \right] \left[ {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right]$$

(7)

According to Eqs. (5) and (7), the relief angle of the drill point is given as

$$\tan \alpha_{fc} = \frac{{\frac{v}{2}(\frac{{L_{1} }}{2} - r_{0} )}}{{(\omega r_{c} \cos \delta_{c} - \frac{v}{2}\sin \delta_{c} )\sqrt {L_{2}^{2} - (\frac{{L_{1} }}{2} - r_{0} )^{2} } }}$$

(8)

The semi-point angle is defined as the one on the outer corner point $C$, is expressed as follow.

$$\tan \rho = \left[ \frac{dz}{dx} \right]_{C}$$

(9)

According to Eq. (5), the semi-point angle of the drill point is given as

$$\rho = \theta$$

(10)

According to Fig. 5, the chisel edge angle ψ can be expressed as follows.

Fig. 5

Schematic figure of chisel edge angle

$$\tan \psi = \left[ \frac{dy}{dx} \right]_{O}$$

(11)

Considering the point $M$ forms the profile of the chisel edge and the projected curve of the one to $X$ axis is very small, the chisel edge angle can be simply expressed as the bevel on the point M instead of the one on the point $O.$

$$\tan \psi = \frac{{2\omega r_{m} }}{v}$$

(12)

5 Optimizing of Grinding Parameters

Compared with the other traditional optimization methods, it is known that genetic algorithm has the obvious advantages in solving nonlinear and multi-objective optimization problems.

The objective function and constraint are discussed below:

Objective function:

The objective is to maximize the fitness function as given by Eq. (13).

$$F(L_{1} ,v,\omega ) = \frac{1}{D}$$

(13)

$$D = \eta_{\alpha } \left| {\alpha_{fc} - \alpha_{0} } \right| + \eta_{\psi } \left| {\psi - \psi_{0} } \right|$$

(14)

$$\eta_{\alpha } + \eta_{\psi } = 1$$

(15)

$\alpha_{fc}$ and $\psi$ are get in Eqs. (8) and (9), respectively.

$\alpha_{0}$ and $\psi_{0}$ are specified by users.

Constraint:

• (a) Structural constraint

  400 mm < L ₁ < 1100 mm, 0 < v < 20 mm/s, 0 < ω < 15 rad/s

• (b) Grinding constraint

  $$\frac{{\sqrt {4L_{2}^{2} - (L_{1} - vt - 2r_{0} )^{2} } }}{{(L_{1} - vt - 2r_{0} )}} \le { \tan }\,\theta$$

  (16)

6 Application of the Proposed Method

For the comparison of grinding results before and after optimization, two drill blanks are selected with the same material and dimension.

Drill parameters: radius 3 mm, material k20 (tungsten carbide), helix angle 30°, initial semi-point angle 59°, initial chisel edge angle 50°, δ _c  = 1.66°.

Grinding wheel parameters: radius 125 mm, width 16 mm (cylindrical grinder), power 180 w, normal revolutions 2800 r/min.

Machine parameters: $L_{2} = 500$ mm, θ = 60°, $r_{0} = 65$ mm

Grinding parameters before optimization: $L_{1} = 8 30$ mm, $v = 2. 5$ mm/s, $\omega = 1. 5 7$ rad/s

Grinding parameters after optimization: $L_{1} = 9 6 5$ mm, $v = 2. 8 6$ mm/s, $\omega = 2. 5 2$ rad/s

For optimization of the grinding parameters, the following parameters are used.

Computational parameters: population size 100, maximum evolution 200, selection probability 0.5, crossover probability 0.8, mutation probability 0.05.

Target values: relief angle α ₀ = 16°, chisel edge angle ${\psi}_{0}$ = 53°, semi-point angle ρ = 60°,

Weight coefficient: $\eta_{\alpha } = 0. 8$, $\eta_{\psi } = 0. 2$

After grinding of drill point on the Biglide parallel machine as shown Fig. 1, the relief angle is measured with dial indicator installed on it.

Its measurement principle is shown in Fig. 6, $r$ means the distance from the drill axis to contacting point of the probe and the drill flank, $\delta$ is the rotational angle of drill point, and $p$ means the changed reading value of dial indicator while the drill point rotates up to $\delta$.

Fig. 6

Cylindrical surface measurement schematic. 1 Drill, 2 probe of dial indicator

$$\tan \alpha_{fc} = \frac{p}{r\delta }$$

(17)

The measurement is repeated ten times for each ground drill and their average value is get as measurement value. The measurement of chisel edge angel is conducted by processing image of ground drill.

The measurement values are below for drill ground before/after optimizing.

Before optimizing: α _mfc  = 18.5° (error 15.6 %), ${\psi}_{m}$ = 46.5° (error 12.3 %)

After optimizing: α _mfc  = 16.3° (error 1.9 %), ${\psi}_{m}$ =  53.5° (error 1 %)

The original drill, ground drill before/after optimizing are shown in Fig. 7, it is relatively clear that the optimized grinding parameters improve the grinding performance for twist drill on Biglide parallel machine.

Fig. 7

Compartment of drill points. a Original. b Ground before optimizing. c Ground after optimizing

7 Conclusions

(1) The grinding of twist drill flank is easily achieved by using Biglide parallel machine, meeting the requirements of the grinding precision and the users.

The only two movements-translational motion of the slider and rotational motion of the grinding wheel can form the needed ones for grinding of the twist drill in the Biglide parallel machine.

(2) It is verified optimized grinding parameters based on generic algorithm help the users with improving the grinding precision of the twist drill.

This work can be further extended to the analysis and design of the reasonable geometry of the twist drill under the different cutting conditions.