348 lines
14 KiB
Python
348 lines
14 KiB
Python
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#!/usr/bin/env python3
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# Based on gearUtils-03.js by Dr A.R.Collins
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# Latest version: <www.arc.id.au/gearDrawing.html>
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# Calculation of Bezier coefficients for
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# Higuchi et al. approximation to an involute.
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# ref: YNU Digital Eng Lab Memorandum 05-1
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from math import *
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def rotate(p, t):
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return (p[0] * cos(t) - p[1] * sin(t), p[0] * sin(t) + p[1] * cos(t))
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def SVG_move(p, t):
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pp = rotate(p, t)
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return 'M ' + str(pp[0]) + ',' + str(pp[1]) + '\n'
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def SVG_line(p, t):
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pp = rotate(p, t)
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return 'L ' + str(pp[0]) + ',' + str(pp[1]) + '\n'
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def SVG_circle(p, r, sweep, t):
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pp = rotate(p, t)
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return 'A ' + str(r) + ',' + str(r) + ' 0 0,' + str(sweep) + ' ' + str(pp[0]) + ',' + str(pp[1]) + '\n'
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def SVG_curve(p, c1, c2, t):
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pp = rotate(p, t)
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c1p = rotate(c1, t)
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c2p = rotate(c2, t)
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return 'C ' + str(pp[0]) + ',' + str(pp[1]) + ' ' + str(c1p[0]) + ',' + str(c1p[1]) + ' ' + str(c2p[0]) + ',' + str(c2p[1]) + '\n'
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def SVG_curve2(p1, c11, c12, p2, c21, c22, t):
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p1p = rotate(p1, t)
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c11p = rotate(c11, t)
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c12p = rotate(c12, t)
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p2p = rotate(p2, t)
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c21p = rotate(c21, t)
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c22p = rotate(c22, t)
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return 'C ' + str(p1p[0]) + ',' + str(p1p[1]) + ' ' + str(c11p[0]) + ',' + str(c11p[1]) + ' ' + str(c12p[0]) + ',' + str(c12p[1]) + ' ' + str(p2p[0]) + ',' + str(p2p[1]) + ' ' + str(c21p[0]) + ',' + str(c21p[1]) + ' ' + str(c22p[0]) + ',' + str(c22p[1]) + '\n'
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def SVG_close():
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return 'Z\n'
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def genInvolutePolar(Rb, R): # Rb = base circle radius
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# returns the involute angle as function of radius R.
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return (sqrt(R*R - Rb*Rb) / Rb) - acos(Rb / R)
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def rotate(pt, rads): # rotate pt by rads radians about origin
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sinA = sin(rads)
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cosA = cos(rads)
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return [pt[0] * cosA - pt[1] * sinA,
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pt[0] * sinA + pt[1] * cosA]
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def toCartesian(radius, angle): # convert polar coords to cartesian
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return [radius * cos(angle), radius * sin(angle)]
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def CreateExternalGear(m, Z, phi):
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# ****** external gear specifications
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addendum = m # distance from pitch circle to tip circle
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dedendum = 1.25 * m # pitch circle to root, sets clearance
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clearance = dedendum - addendum
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# Calculate radii
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Rpitch = Z * m / 2 # pitch circle radius
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Rb = Rpitch*cos(phi * pi / 180) # base circle radius
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Ra = Rpitch + addendum # tip (addendum) circle radius
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Rroot = Rpitch - dedendum # root circle radius
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fRad = 1.5 * clearance # fillet radius, max 1.5*clearance
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Rf = sqrt((Rroot + fRad) * (Rroot + fRad) - (fRad * fRad)) # radius at top of fillet
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if (Rb < Rf):
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Rf = Rroot + clearance
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# ****** calculate angles (all in radians)
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pitchAngle = 2 * pi / Z # angle subtended by whole tooth (rads)
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baseToPitchAngle = genInvolutePolar(Rb, Rpitch)
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pitchToFilletAngle = baseToPitchAngle # profile starts at base circle
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if (Rf > Rb): # start profile at top of fillet (if its greater)
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pitchToFilletAngle -= genInvolutePolar(Rb, Rf)
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filletAngle = atan(fRad / (fRad + Rroot)) # radians
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# ****** generate Higuchi involute approximation
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fe = 1 # fraction of profile length at end of approx
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fs = 0.01 # fraction of length offset from base to avoid singularity
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if (Rf > Rb):
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fs = (Rf * Rf - Rb * Rb) / (Ra * Ra - Rb * Rb) # offset start to top of fillet
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# approximate in 2 sections, split 25% along the involute
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fm = fs + (fe - fs) / 4 # fraction of length at junction (25% along profile)
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dedBez = BezCoeffs(m, Z, phi, 3, fs, fm)
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addBez = BezCoeffs(m, Z, phi, 3, fm, fe)
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dedInv = dedBez.involuteBezCoeffs()
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addInv = addBez.involuteBezCoeffs()
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# join the 2 sets of coeffs (skip duplicate mid point)
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inv = dedInv + addInv[1:]
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# create the back profile of tooth (mirror image)
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invR = [0 for i in range(0, len(inv))] # involute profile along back of tooth
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for i in range(0, len(inv)):
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# rotate all points to put pitch point at y = 0
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pt = rotate(inv[i], -baseToPitchAngle - pitchAngle / 4)
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inv[i] = pt
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# generate the back of tooth profile nodes, mirror coords in X axis
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invR[i] = [pt[0], -pt[1]]
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# ****** calculate section junction points R=back of tooth, Next=front of next tooth)
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fillet = toCartesian(Rf, -pitchAngle / 4 - pitchToFilletAngle) # top of fillet
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filletR = [fillet[0], -fillet[1]] # flip to make same point on back of tooth
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rootR = toCartesian(Rroot, pitchAngle / 4 + pitchToFilletAngle + filletAngle)
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rootNext = toCartesian(Rroot, 3 * pitchAngle / 4 - pitchToFilletAngle - filletAngle)
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filletNext = rotate(fillet, pitchAngle) # top of fillet, front of next tooth
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# Draw the shapes in SVG
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t_inc = 2.0 * pi / float(Z)
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thetas = [(x * t_inc) for x in range(Z)]
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svg = SVG_move(fillet, 0) # start at top of fillet
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for theta in thetas:
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if (Rf < Rb):
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svg += SVG_line(inv[0], theta) # line from fillet up to base circle
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svg += SVG_curve2(inv[1], inv[2], inv[3],
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inv[4], inv[5], inv[6], theta)
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svg += SVG_circle(invR[6], Ra, 1, theta) # arc across addendum circle
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# svg = SVG_move(invR[6]) # TEMP
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svg += SVG_curve2(invR[5], invR[4], invR[3],
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invR[2], invR[1], invR[0], theta)
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if (Rf < Rb):
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svg += SVG_line(filletR, theta) # line down to topof fillet
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if (rootNext[1] > rootR[1]): # is there a section of root circle between fillets?
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svg += SVG_circle(rootR, fRad, 0, theta) # back fillet
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svg += SVG_circle(rootNext, Rroot, 1, theta) # root circle arc
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svg += SVG_circle(filletNext, fRad, 0, theta)
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svg += SVG_close()
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return svg
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def CreateInternalGear(m, Z, phi):
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addendum = 0.6 * m # pitch circle to tip circle (ref G.M.Maitra)
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dedendum = 1.25 * m # pitch circle to root radius, sets clearance
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# Calculate radii
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Rpitch = Z * m / 2 # pitch radius
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Rb = Rpitch * cos(phi * pi / 180) # base radius
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Ra = Rpitch - addendum # addendum radius
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Rroot = Rpitch + dedendum# root radius
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clearance = 0.25 * m # gear dedendum - pinion addendum
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Rf = Rroot - clearance # radius of top of fillet (end of profile)
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fRad = 1.5 * clearance # fillet radius, 1 .. 1.5*clearance
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# ****** calculate subtended angles
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pitchAngle = 2 * pi / Z # angle between teeth (rads)
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baseToPitchAngle = genInvolutePolar(Rb, Rpitch)
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tipToPitchAngle = baseToPitchAngle # profile starts from base circle
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if (Ra > Rb):
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tipToPitchAngle -= genInvolutePolar(Rb, Ra) # start profile from addendum
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pitchToFilletAngle = genInvolutePolar(Rb, Rf) - baseToPitchAngle
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filletAngle = 1.414 * clearance / Rf # to make fillet tangential to root
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# ****** generate Higuchi involute approximation
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fe = 1 # fraction of involute length at end of approx (fillet circle)
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fs = 0.01 # fraction of length offset from base to avoid singularity
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if (Ra > Rb):
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fs = (Ra*Ra - Rb*Rb) / (Rf*Rf - Rb*Rb) # start profile from addendum (tip circle)
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# approximate in 2 sections, split 25% along the profile
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fm = fs + (fe - fs) / 4
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addBez = BezCoeffs(m, Z, phi, 3, fs, fm)
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dedBez = BezCoeffs(m, Z, phi, 3, fm, fe)
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addInv = addBez.involuteBezCoeffs()
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dedInv = dedBez.involuteBezCoeffs()
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# join the 2 sets of coeffs (skip duplicate mid point)
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invR = addInv + dedInv[1:]
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# create the front profile of tooth (mirror image)
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inv = [0 for i in range(0, len(invR))] # back involute profile
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for i in range(0, len(inv)):
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# rotate involute to put center of tooth at y = 0
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pt = rotate(invR[i], pitchAngle / 4 - baseToPitchAngle)
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invR[i] = pt
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# generate the back of tooth profile, flip Y coords
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inv[i] = [pt[0], -pt[1]]
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# ****** calculate coords of section junctions
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fillet = [inv[6][0], inv[6][1]] # top of fillet, front of tooth
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tip = toCartesian(Ra, -pitchAngle / 4 + tipToPitchAngle) # tip, front of tooth
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tipR = [tip[0], -tip[1]] # addendum, back of tooth
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rootR = toCartesian(Rroot, pitchAngle / 4 + pitchToFilletAngle + filletAngle)
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rootNext = toCartesian(Rroot, 3 * pitchAngle / 4 - pitchToFilletAngle - filletAngle)
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filletNext = rotate(fillet, pitchAngle) # top of fillet, front of next tooth
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# Draw the shapes in SVG
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t_inc = 2.0 * pi / float(Z)
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thetas = [(x * t_inc) for x in range(Z)]
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svg = SVG_move(fillet, 0) # start at top of fillet
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for theta in thetas:
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svg += SVG_curve2(inv[5], inv[4], inv[3],
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inv[2], inv[1], inv[0], theta)
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if (Ra < Rb):
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svg += SVG_line(tip, theta) # line from end of involute to addendum (tip)
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svg += SVG_circle(tipR, Ra, 1, theta) # arc across tip circle
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if (Ra < Rb):
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svg += SVG_line(invR[0], theta) # line from addendum to start of involute
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svg += SVG_curve2(invR[1], invR[2], invR[3],
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invR[4], invR[5], invR[6], theta)
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if (rootR[1] < rootNext[1]): # there is a section of root circle between fillets
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svg += SVG_circle(rootR, fRad, 1, theta) # fillet on back of tooth
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svg += SVG_circle(rootNext, Rroot, 1, theta) # root circle arc
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svg += SVG_circle(filletNext, fRad, 1, theta) # fillet on next
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return svg
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class BezCoeffs:
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def chebyExpnCoeffs(self, j, func):
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N = 50 # a suitably large number N>>p
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c = 0
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for k in range(1, N + 1):
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c += func(cos(pi * (k - 0.5) / N)) * cos(pi * j * (k - 0.5) / N)
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return 2 *c / N
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def chebyPolyCoeffs(self, p, func):
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coeffs = [0, 0, 0, 0]
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fnCoeff = [0, 0, 0, 0]
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T = [[1, 0, 0, 0, 0],
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[0, 1, 0, 0, 0],
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[0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0],
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[0, 0, 0, 0, 0]
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]
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# now generate the Chebyshev polynomial coefficient using
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# formula T(k+1) = 2xT(k) - T(k-1) which yields
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# T = [ [ 1, 0, 0, 0, 0, 0], # T0(x) = +1
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# [ 0, 1, 0, 0, 0, 0], # T1(x) = 0 +x
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# [-1, 0, 2, 0, 0, 0], # T2(x) = -1 0 +2xx
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# [ 0, -3, 0, 4, 0, 0], # T3(x) = 0 -3x 0 +4xxx
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# [ 1, 0, -8, 0, 8, 0], # T4(x) = +1 0 -8xx 0 +8xxxx
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# [ 0, 5, 0,-20, 0, 16], # T5(x) = 0 5x 0 -20xxx 0 +16xxxxx
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# ... ]
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for k in range(1, p + 1):
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for j in range(0, len(T[k]) - 1):
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T[k + 1][j + 1] = 2 * T[k][j]
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for j in range(0, len(T[k - 1])):
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T[k + 1][j] -= T[k - 1][j]
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# convert the chebyshev function series into a simple polynomial
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# and collect like terms, out T polynomial coefficients
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for k in range(0, p + 1):
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fnCoeff[k] = self.chebyExpnCoeffs(k, func)
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coeffs[k] = 0
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for k in range(0, p + 1):
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for pwr in range(0, p + 1):
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coeffs[pwr] += fnCoeff[k] * T[k][pwr]
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coeffs[0] -= self.chebyExpnCoeffs(0, func) / 2 # fix the 0th coeff
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return coeffs
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# Equation of involute using the Bezier parameter t as variable
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def involuteXbez(self, t):
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# map t (0 <= t <= 1) onto x (where -1 <= x <= 1)
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x = t * 2 - 1
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# map theta (where ts <= theta <= te) from x (-1 <=x <= 1)
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theta = x * (self.te - self.ts) / 2 + (self.ts + self.te) / 2
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return self.Rb * (cos(theta) + theta * sin(theta))
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def involuteYbez(self, t):
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# map t (0 <= t <= 1) onto x (where -1 <= x <= 1)
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x = t * 2 - 1
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# map theta (where ts <= theta <= te) from x (-1 <=x <= 1)
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theta = x * (self.te - self.ts) / 2 + (self.ts + self.te) / 2
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return self.Rb * (sin(theta) - theta * cos(theta))
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def binom(self, n, k):
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coeff = 1
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for i in range(n - k + 1, n + 1):
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coeff *= i
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for i in range(1, k + 1):
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coeff /= i
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return coeff
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def bezCoeff(self, i, func):
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# generate the polynomial coeffs in one go
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polyCoeffs = self.chebyPolyCoeffs(self.p, func)
|
||
|
|
|
||
|
|
bc = 0
|
||
|
|
for j in range(0, i + 1):
|
||
|
|
bc += self.binom(i, j) * polyCoeffs[j] / self.binom(self.p, j)
|
||
|
|
|
||
|
|
return bc
|
||
|
|
|
||
|
|
def involuteBezCoeffs(self):
|
||
|
|
# calc Bezier coeffs
|
||
|
|
bzCoeffs = []
|
||
|
|
for i in range(0, self.p + 1):
|
||
|
|
bcoeff = [0, 0]
|
||
|
|
bcoeff[0] = self.bezCoeff(i, self.involuteXbez)
|
||
|
|
bcoeff[1] = self.bezCoeff(i, self.involuteYbez)
|
||
|
|
bzCoeffs.append(bcoeff)
|
||
|
|
|
||
|
|
return bzCoeffs
|
||
|
|
|
||
|
|
# Parameters:
|
||
|
|
# module - sets the size of teeth (see gear design texts)
|
||
|
|
# numTeeth - number of teeth on the gear
|
||
|
|
# pressure angle - angle in degrees, usually 14.5 or 20
|
||
|
|
# order - the order of the Bezier curve to be fitted [3, 4, 5, ..]
|
||
|
|
# fstart - fraction of distance along tooth profile to start
|
||
|
|
# fstop - fraction of distance along profile to stop
|
||
|
|
def __init__(self, module, numTeeth, pressureAngle, order, fstart, fstop):
|
||
|
|
self.Rpitch = module * numTeeth / 2 # pitch circle radius
|
||
|
|
self.phi = pressureAngle # pressure angle
|
||
|
|
self.Rb = self.Rpitch * cos(self.phi * pi / 180) # base circle radius
|
||
|
|
self.Ra = self.Rpitch + module # addendum radius (outer radius)
|
||
|
|
self.ta = sqrt(self.Ra * self.Ra - self.Rb * self.Rb) / self.Rb # involute angle at addendum
|
||
|
|
self.stop = fstop
|
||
|
|
|
||
|
|
if (fstart < self.stop):
|
||
|
|
self.start = fstart
|
||
|
|
|
||
|
|
self.te = sqrt(self.stop) * self.ta # involute angle, theta, at end of approx
|
||
|
|
self.ts = sqrt(self.start) * self.ta # involute angle, theta, at start of approx
|
||
|
|
self.p = order # order of Bezier approximation
|