OpenCASCADE 麻花鉆頭造型實例分析
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中提供了一個麻花鉆頭造型的實例，如下圖所示：
Figure 1. A Simple Twist Drill Bit by OpenCASCADE
鉆頭造型主要涉及到旋轉形狀的造型方法，放樣造型及布爾操作。本文結合鉆頭的Tcl腳本來詳細說明造型過程。對于有同樣造型需求的，可以在理解造型方法的基礎上實現自己需要的參數化形狀。
2.Modeling Tcl Script
通過Draw Test Harness的菜單Samples->View samples可以找到Drill例子，如下圖所示：
Figure 2. Drill bit modeling sample
對應的Tcl腳本位于samples/tcl文件夾中，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
下面結合Tcl腳本來對鉆頭的造型過程進行詳細說明。
3.Twist Drill Bit Parameters
腳本首先加載相應的模塊：造型模塊和顯示模塊，分別為MODEING和VISUALIZATION。然后設置鉆頭相關參數變量：
Figure 3. Drill Bit
變量名及其說明：
| 變量名variable |
說明 detail |
| R |
外半徑 outer radius |
| D |
直徑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 |
鉆尖角度Point angle in degrees. |
先定義這些變量的思想也是參數化的思想，將這些變量值改變就會得到不同的鉆頭模型。參數化的思想在CAD軟件中比較常見，如AutoCAD中的動態塊，PDMS中的設備模板Template等，都屬于參數化的方法。通過修改參數讓一個或多個幾何體形狀發生變化。
4.Create Drill Section Profile
創建鉆頭截面的Tcl腳本如下所示：
# 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
生成的圖形如下圖所示：
Figure 4 Profile construction curves
其中變量Rectangle1是圖中綠色表示的四邊形，Circle1是黃色表示的外圓，Circle2是紅色表示的內圓。通過布爾操作求取Rectangle1和Circle1的公共部分得到如下圖所示的截面：
Figure 5 Common of Rectangle1 and Circle1
將公共部分與內圓Circle2合并得到如圖所示截面：
Figure 6. Fuse of Common part and Circle2
下面在這個截面的基礎上生成鉆頭的螺旋凹槽，這里是相對關鍵的地方，生成凹槽的時候還要考慮螺旋。
# 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
代碼生成的圖形如下圖所示：
Figure 7. Flute of the drill bit
通過將線lip沿著路徑直線sp和螺旋線gg放樣得到螺旋截面spiral，再將螺旋截面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
通過求得的交線再構造圓弧和線段來構造出鉆頭凹槽的截面，如下圖所示：
Figure 8. Make flute section
將凹槽截面flute從前面構造的截面中去除：
# cut flute from profile
bcut sec sec flute
Figure 9. Make drill bit profile
將凹槽截面繞Z軸旋轉180度，再去布爾減原來的截面：
trotate flute 0 0 0 0 0 1 180.
bcut sec sec flute
Figure 10. Make drill bit profile
這樣鉆頭的截面就生成了。
5.Sweeping the Profile
將上面鉆頭截面沿著直線sp和螺旋線gg放樣：
# 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
生成圖形如下圖所示：
Figure 11. Sweep the profile
通過放樣得到了鉆頭的螺桿base。
6.Sharpening
得到鉆頭的螺桿后需要生成鉆頭的鉆尖，根據鉆尖角point angle變量sigma來造型。
# 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
先構造兩個圓錐體(sh1和sh2)和一個長方體sh，再通過布爾運算得到兩個圓錐體sh1，sh2的公共部分qq。如下圖所示：
Figure 12. Modeling point angle
再從長方體sh中減去兩個圓錐的公共部分得到如下圖所示：
Figure 13. Modeling point angle
將螺桿減去上面得到的sharpener就完成鉆尖的造型，如下圖所示：
Figure 14. Modeling point angle
7.Make Shank
完成螺桿和鉆尖造型后，就來實現麻花鉆頭的最后部分，鉆頭柄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
鉆頭柄部由兩個圓錐（transit和tail）和一個圓柱shank組成。最后將這三個形狀合并得到完成的鉆頭的柄部shank。
Figure 15. Modeling Shank
8.Conclusion
通過對OpenCASCADE中麻花鉆頭的造型實例分析，可見這個鉆頭造型用到的算法主要是放樣及布爾操作。在生成螺旋線時主要是使用pcurve的方式。最后還要理解參數化的造型思想，通過定義變量來生成參數化的模型。