渐开线方程推导

渐开线方程推导(直角坐标系)
鱼板主在前面对渐开线的极坐标方程进行了推导，使大家受益匪浅。

直角坐标方程在本论坛上也出现了很多次，但一些朋友对其中的参数理解上还有一定的偏差。

本贴通过对渐开线直角坐标方程（参数方程）推导，使朋友们对其中的参数更加深入的了解，以便在工作中能很好的使用它。

如果大家觉得没有什么意义的话，本贴就当是灌水。

如图：在渐开线上有一点P（X，Y），X=OB+BC，Y=AB-AN
由渐开线特点可知，弧长AD=AP=r.β
OB=rcosβBC=AP.sinβ=r.β.sinβ
所以X=r.cosβ+r.β.sinβ
同理Y=r.sinrβ-r.β.cosβ
因此，渐开线的直角坐标参数方程就是：
X=r.cosβ+r.β.sinβ
Y=r.sinrβ-r.β.cosβ其中r为基圆半径
在这里大家可以和渐开线的极坐标方程推导进行比较，直角坐标方程中的β就是压力角和展角的和，β=α+θ
图片附件:渐开线.jpg(2005-7-622:40,26.33K)
使用autocadvba绘制渐开线齿轮
Dim mAsDouble'齿轮模数
Dim zAsInteger'齿数
Dim rAsDouble'分度圆半径
Dim raAsDouble'齿顶圆半径
Dim rbAsDouble'基圆半径
Dim rfAsDouble'齿根圆半径
Dim PIAsDouble'定义常数π
PrivateSubCommand1_Click()
PI=4*Atn(1)
m=Val(TextBox1.text)
z=Val(TextBox2.text)
r=m*z/2
ra=r+m
rb=r*Cos(20*PI/180)
rf=r-1.25*m
Dim cobrAsAcadCircle'分度圆
Dim cobraAsAcadCircle'齿顶圆
Dim cobrbAsAcadCircle'基圆
Dim cobrfAsAcadCircle'齿根圆
Dim cp1(0To2)AsDouble
cp1(0)=0:cp1(1)=0:cp1(2)=0
Setcobr=ThisDrawing.ModelSpace.AddCircle(cp1,r)
Setcobra=ThisDrawing.ModelSpace.AddCircle(cp1,ra)
Setcodrb=ThisDrawing.ModelSpace.AddCircle(cp1,rb)
Setcodrf=ThisDrawing.ModelSpace.AddCircle(cp1,rf)
Dim colorAsAcadAcCmColor
Setcolor=AcadApplication.GetInterfaceObject("AutoCAD.AcCmColor.17")
Callcolor.SetRGB(80,100,244)
cobr.TrueColor=color'创建spline
Dim theta0 As Double'定义渐开线展角与压力角之和
Dim InvPoint(0To32) As Double'定义拟合点坐标
Dim SPtan(0To2) As Double'定义起点切线方向
Dim EPtan(0To2) As Double'定义终点切线方向
theta0=Sqr(ra^2-rb^2)/rb'将展角与压力角之和角度转换为弧度
theta1=theta0-Atn(theta0)'展角
delta_theta=theta0/10'单位角
For j=0 To 10
theta=j*delta_theta
InvPoint(j*3)=rb*(Sin(theta)-theta*Cos(theta))
InvPoint(j*3+1)=rb*(Cos(theta)+theta*Sin(theta))
InvPoint(j*3+2)=0
Next j
EPtan(0)=1:EPtan(1)=1/Tan(theta0):EPtan(2)=0
Setinvobj=ThisDrawing.ModelSpace.AddSpline(InvPoint,SPtan,EPtan)'创建半个齿顶圆弧
Dim center1(0To2) As Double
Dim radius1 As Double
Dim startangle As Double,endangle As Double
Dim arc1 As AcadArc
center1(0)=0:center1(1)=0:center1(2)=0
radius1=ra
startangle=PI/2-(Tan(PI/9)-PI/9+PI/2/z)
endangle=PI/2-(Atn((Sin(theta)-theta*Cos(theta))/(Cos(theta)+theta*Sin(theta)))) Setarc1=ThisDrawing.ModelSpace.AddArc(center1,radius1,startangle,endangle)'齿根圆
Dim myplineAsAcadLWPolyline
Dim vpoint(0To5)AsDouble
vpoint(0)=0:vpoint(1)=rb
vpoint(2)=-(rb-rf)*Tan(PI/2/z+Tan(PI/9)-PI/9)/2:vpoint(3)=(rb+rf)/2
vpoint(4)=-rf*Sin(PI/2/z-Tan(PI/9)+PI/9):vpoint(5)=rf*Cos(PI/2/z-Tan(PI/9)+PI/9) Setmypline=ThisDrawing.ModelSpace.AddLightWeightPolyline(vpoint)
mypline.SetBulge1,-1/3
mypline.Update'镜像spline、齿根曲线和齿顶圆弧，形成一个齿廓
Dim mirror_point1(0 To 2) As Double
Dim mirror_point2(0 To 2) As Double
mirror_point1(0)=0:mirror_point1(1)=0:mirror_point1(2)=0
mirror_point2(0)=1:mirror_point2(1)=1/Tan(Tan(PI/9)-PI/9+PI/2/z):mirror_point2(2 )=0
Dim mirrorinvobj As AcadSpline
Dim mirrorarc1 As AcadArc
Dim mirror_mypline As AcadLWPolyline
Setmirrorarc1=arc1.Mirror(mirror_point1,mirror_point2)
086
Setmirrorinvobj=invobj.Mirror(mirror_point1,mirror_point2)
087
Setmirror_mypline=mypline.Mirror(mirror_point1,mirror_point2)
088
'环形阵列齿轮轮齿各部分线段
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Dim noOfObjectsAsInteger
090
Dim angleToFillAsDouble
091
Dim basePnt(0To2)AsDouble
092
noOfObjects=z
093
angleToFill=2*PI*(z-1)/z
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basePnt(0)=0#:basePnt(1)=0#:basePnt(2)=0#
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Dim retobjAsVariant
097
Dim retobj1AsVariant
098
Dim retobjarc1AsVariant
099
Dim retobjarcAsVariant
100
Dim retobj_myplineAsVariant
101
Dim retobj_mirror_myplineAsVariant
102
retobj=invobj.ArrayPolar(noOfObjects,angleToFill,basePnt)
103
retobj1=mirrorinvobj.ArrayPolar(noOfObjects,angleToFill,basePnt)
104
retobjarc1=mirrorarc1.ArrayPolar(noOfObjects,angleToFill,basePnt)
105
retobjarc=arc1.ArrayPolar(noOfObjects,angleToFill,basePnt)
106
retob_mypline=mypline.ArrayPolar(noOfObjects,angleToFill,basePnt)
107
retobj_mirror_mypline=mirror_mypline.ArrayPolar(noOfObjects,angleToFill,basePnt) 108
ZoomAll
109
110
111
Command1.Enabled=False
112
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114
EndSub。