OpenCASCADE 麻花钻头造型实例分析
eryar@163.com
Abstract. OpenCASCADE provides a simple twist drill bit modeling example in Tcl script. The blog will give a details of some key points, when you understand the key points, you can modeling the shape like that.
Key Words. OpenCASCADE, Twist Drill Bit Modeling, Tcl
1.Introduction
OpenCASCADE的Draw Test Harness中提供了一个麻花钻头造型的实例，如下图所示：
Figure 1. A Simple Twist Drill Bit by OpenCASCADE
钻头造型主要涉及到旋转形状的造型方法，放样造型及布尔操作。本文结合钻头的Tcl脚本来详细说明造型过程。对于有同样造型需求的，可以在理解造型方法的基础上实现自己需要的参数化形状。
2.Modeling Tcl Script
通过Draw Test Harness的菜单Samples->View samples可以找到Drill例子，如下图所示：
Figure 2. Drill bit modeling sample
对应的Tcl脚本位于samples/tcl文件夹中，列出如下：
# Sample: creation of simple twist drill bit
#Category: Modeling
#Title: Drill
pload MODELING VISUALIZATION
# drill parameters (some terms taken from http://www.drill-bits.cn/drill-bits-quality.asp)
dset R 4. ;# outer radius
dset D 2*R ;# diameter
dset Rr 3.5 ;# chisel radius (outer radius minus body clearance)
dset b 1. ;# web thickness (approximate)
dset d b/2
dset H 80. ;# height of the spiral part
dset a 3.*pi ;# total angle of spiral rotation
dset sigma 118 ;# point angle, in degrees
# Create section profile by sequence of Boolean operations
# on simple planar objects
puts "Creating the drill section profile..."
polyline rectangle1 d -R 0 R -R 0 -d R 0 -R R 0 d -R 0
circle circle1 0 0 0 0 0 1 R
mkedge circle1 circle1
wire circle1 circle1
circle circle2 0 0 0 0 0 1 Rr
mkedge circle2 circle2
wire circle2 circle2
plane p0
mkface rectangle1 p0 rectangle1
mkface circle1 p0 circle1
mkface circle2 p0 circle2
bcommon sec rectangle1 circle1
# note use of 'fuse' instead of 'bfuse' -- we need to get single face
fuse sec sec circle2
# Construct flute profile so as to have cutting lip straight after sharpening.
# Here we need to take into account spiral shift of the flute edge
# along the point length -- the way to do that is to make spiral
# from the desired cutting lip edge and then intersect it by plane
polyline lip d -d/2 0 d -R -R/tan(sigma/2*pi/180)
polyline sp 0 0 0 0 0 H
cylinder cc 0 0 0 0 0 1 0 -4 0 4
line ll 0 0 a 80
trim ll ll 0 sqrt(a*a+H*H)
vertex v1 0 -R 0
vertex v2 0 -R H
trotate v2 0 0 0 0 0 1 180.*a/pi
mkedge ee ll cc v1 v2
wire gg ee
mksweep sp
setsweep -G gg 0 0
addsweep lip
buildsweep spiral -S
mkface f0 p0 -R R -R R
bsection sflute spiral f0
# here we rely on that section curve is parameterized from 0 to 1
# and directed as cutting lip edge;
# note that this can change if intersection algorithm is modified
explode sflute e
mkcurve cflute sflute_1
cvalue cflute 0. x0 y0 z0
cvalue cflute 1. x1 y1 z1
vertex vf0 x0 y0 z0
vertex vf1 x1 y1 z1
# -- variant: replace curve by arc with start at x0,y0,z0 and end at x1,y1,z1,
# -- such that tanget at start point is along Y
#dset Rflute ((x1-x0)*(x1-x0)+(y1-y0)*(y1-y0))/(2*(x1-x0))
#circle aflute x0+Rflute y0 0 0 0 1 Rflute
#mkedge sflute_1 aflute vf0 vf1
# make rounding in the flute; use circle with radius Rr/2
circle cround x0+Rr/2 y0 0 0 0 1 Rr/2
vertex vf3 x0+Rr y0 0
mkedge sflute_2 cround vf3 vf0
vertex vf2 R -R 0
edge sflute_3 vf3 vf2
edge sflute_4 vf2 vf1
wire w2 sflute_1 sflute_2 sflute_3 sflute_4
mkface flute p0 w2
# cut flute from profile
bcut sec sec flute
trotate flute 0 0 0 0 0 1 180.
bcut sec sec flute
donly sec
# sweep profile to get a drill body
puts "Sweeping the profile..."
mksweep sp
setsweep -G gg 0 0
explode sec w
addsweep sec_1
buildsweep base -S
# sharpen the drill (see http://tool-land.ru/zatochka-sverla.php)
puts "Sharpening..."
dset theta a*R/H*sin((90-sigma/2)*pi/180)
plane ax1 d 1.9*D "H+1.9*D/tan(pi/180.*sigma/2.)" 0 -1 -1
pcone sh1 ax1 0 100*sin((sigma-90)/2*pi/180.) 100
trotate sh1 0 0 0 0 0 1 -theta*180/pi
tcopy sh1 sh2
trotate sh2 0 0 0 0 0 1 180
box sh -D/2 -D/2 72 D D 20
bcommon qq sh1 sh2
bcut sharpener sh qq
bcut body base sharpener
# make a shank
puts "Making a shank..."
plane pl2 0 0 -40 0 0 1
pcylinder shank pl2 4 40
pcone transit R 0 R
plane pl3 0 0 -40 0 0 -0.5
pcone tail pl3 R 0 0.5
bfuse shank shank tail
bfuse shank shank transit
bfuse drill body shank
# check result
checkshape drill
# show result
puts "Displaying result..."
incmesh drill 0.01
vdisplay drill
vsetdispmode drill 1
vrenderparams -msaa 8
vfit
# show section and sweep path
ttranslate sec_1 0 0 H; trotate sec_1 0 0 0 0 0 1 a*180/pi; incmesh gg 0.01; vdisplay gg sec_1
下面结合Tcl脚本来对钻头的造型过程进行详细说明。
3.Twist Drill Bit Parameters
脚本首先加载相应的模块：造型模块和显示模块，分别为MODEING和VISUALIZATION。然后设置钻头相关参数变量：
Figure 3. Drill Bit
变量名及其说明：
| 变量名variable |
说明 detail |
| R |
外半径 outer radius |
| D |
直径D |
| Rr |
Chisel Radius |
| b |
Web thickness(approximate) |
| d |
|
| H |
螺旋部分的高度Height of the spiral height |
| a |
螺旋角度Total angle of spiral rotation |
| sigma |
钻尖角度Point angle in degrees. |
先定义这些变量的思想也是参数化的思想，将这些变量值改变就会得到不同的钻头模型。参数化的思想在CAD软件中比较常见，如AutoCAD中的动态块，PDMS中的设备模板Template等，都属于参数化的方法。通过修改参数让一个或多个几何体形状发生变化。
4.Create Drill Section Profile
创建钻头截面的Tcl脚本如下所示：
# Create section profile by sequence of Boolean operations
# on simple planar objects
puts "Creating the drill section profile..."
polyline rectangle1 d -R 0 R -R 0 -d R 0 -R R 0 d -R 0
circle circle1 0 0 0 0 0 1 R
mkedge circle1 circle1
wire circle1 circle1
circle circle2 0 0 0 0 0 1 Rr
mkedge circle2 circle2
wire circle2 circle2
plane p0
mkface rectangle1 p0 rectangle1
mkface circle1 p0 circle1
mkface circle2 p0 circle2
bcommon sec rectangle1 circle1
# note use of 'fuse' instead of 'bfuse' -- we need to get single face
fuse sec sec circle2
生成的图形如下图所示：
Figure 4 Profile construction curves
其中变量Rectangle1是图中绿色表示的四边形，Circle1是黄色表示的外圆，Circle2是红色表示的内圆。通过布尔操作求取Rectangle1和Circle1的公共部分得到如下图所示的截面：
Figure 5 Common of Rectangle1 and Circle1
将公共部分与内圆Circle2合并得到如图所示截面：
Figure 6. Fuse of Common part and Circle2
下面在这个截面的基础上生成钻头的螺旋凹槽，这里是相对关键的地方，生成凹槽的时候还要考虑螺旋。
# Construct flute profile so as to have cutting lip straight after sharpening.
# Here we need to take into account spiral shift of the flute edge
# along the point length -- the way to do that is to make spiral
# from the desired cutting lip edge and then intersect it by plane
polyline lip d -d/2 0 d -R -R/tan(sigma/2*pi/180)
polyline sp 0 0 0 0 0 H
cylinder cc 0 0 0 0 0 1 0 -4 0 4
line ll 0 0 a 80
trim ll ll 0 sqrt(a*a+H*H)
vertex v1 0 -R 0
vertex v2 0 -R H
trotate v2 0 0 0 0 0 1 180.*a/pi
mkedge ee ll cc v1 v2
wire gg ee
mksweep sp
setsweep -G gg 0 0
addsweep lip
buildsweep spiral -S
mkface f0 p0 -R R -R R
bsection sflute spiral f0
代码生成的图形如下图所示：
Figure 7. Flute of the drill bit
通过将线lip沿着路径直线sp和螺旋线gg放样得到螺旋截面spiral，再将螺旋截面spiral和底平面f0求交得到sflute。
# here we rely on that section curve is parameterized from 0 to 1
# and directed as cutting lip edge;
# note that this can change if intersection algorithm is modified
explode sflute e
mkcurve cflute sflute_1
cvalue cflute 0. x0 y0 z0
cvalue cflute 1. x1 y1 z1
vertex vf0 x0 y0 z0
vertex vf1 x1 y1 z1
# -- variant: replace curve by arc with start at x0,y0,z0 and end at x1,y1,z1,
# -- such that tanget at start point is along Y
#dset Rflute ((x1-x0)*(x1-x0)+(y1-y0)*(y1-y0))/(2*(x1-x0))
#circle aflute x0+Rflute y0 0 0 0 1 Rflute
#mkedge sflute_1 aflute vf0 vf1
# make rounding in the flute; use circle with radius Rr/2
circle cround x0+Rr/2 y0 0 0 0 1 Rr/2
vertex vf3 x0+Rr y0 0
mkedge sflute_2 cround vf3 vf0
vertex vf2 R -R 0
edge sflute_3 vf3 vf2
edge sflute_4 vf2 vf1
wire w2 sflute_1 sflute_2 sflute_3 sflute_4
mkface flute p0 w2
通过求得的交线再构造圆弧和线段来构造出钻头凹槽的截面，如下图所示：
Figure 8. Make flute section
将凹槽截面flute从前面构造的截面中去除：
# cut flute from profile
bcut sec sec flute
Figure 9. Make drill bit profile
将凹槽截面绕Z轴旋转180度，再去布尔减原来的截面：
trotate flute 0 0 0 0 0 1 180.
bcut sec sec flute
Figure 10. Make drill bit profile
这样钻头的截面就生成了。
5.Sweeping the Profile
将上面钻头截面沿着直线sp和螺旋线gg放样：
# sweep profile to get a drill body
puts "Sweeping the profile..."
mksweep sp
setsweep -G gg 0 0
explode sec w
addsweep sec_1
buildsweep base -S
生成图形如下图所示：
Figure 11. Sweep the profile
通过放样得到了钻头的螺杆base。
6.Sharpening
得到钻头的螺杆后需要生成钻头的钻尖，根据钻尖角point angle变量sigma来造型。
# sharpen the drill (see http://tool-land.ru/zatochka-sverla.php)
puts "Sharpening..."
dset theta a*R/H*sin((90-sigma/2)*pi/180)
plane ax1 d 1.9*D "H+1.9*D/tan(pi/180.*sigma/2.)" 0 -1 -1
pcone sh1 ax1 0 100*sin((sigma-90)/2*pi/180.) 100
trotate sh1 0 0 0 0 0 1 -theta*180/pi
tcopy sh1 sh2
trotate sh2 0 0 0 0 0 1 180
box sh -D/2 -D/2 72 D D 20
bcommon qq sh1 sh2
bcut sharpener sh qq
bcut body base sharpener
先构造两个圆锥体(sh1和sh2)和一个长方体sh，再通过布尔运算得到两个圆锥体sh1，sh2的公共部分qq。如下图所示：
Figure 12. Modeling point angle
再从长方体sh中减去两个圆锥的公共部分得到如下图所示：
Figure 13. Modeling point angle
将螺杆减去上面得到的sharpener就完成钻尖的造型，如下图所示：
Figure 14. Modeling point angle
7.Make Shank
完成螺杆和钻尖造型后，就来实现麻花钻头的最后部分，钻头柄shank。
# make a shank
puts "Making a shank..."
plane pl2 0 0 -40 0 0 1
pcylinder shank pl2 4 40
pcone transit R 0 R
plane pl3 0 0 -40 0 0 -0.5
pcone tail pl3 R 0 0.5
bfuse shank shank tail
bfuse shank shank transit
bfuse drill body shank
钻头柄部由两个圆锥（transit和tail）和一个圆柱shank组成。最后将这三个形状合并得到完成的钻头的柄部shank。
Figure 15. Modeling Shank
8.Conclusion
通过对OpenCASCADE中麻花钻头的造型实例分析，可见这个钻头造型用到的算法主要是放样及布尔操作。在生成螺旋线时主要是使用pcurve的方式。最后还要理解参数化的造型思想，通过定义变量来生成参数化的模型。