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involute.py
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involute.py
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#!/usr/bin/env python3
# Based on gearUtils-03.js by Dr A.R.Collins
# Latest version: <www.arc.id.au/gearDrawing.html>
# Calculation of Bezier coefficients for
# Higuchi et al. approximation to an involute.
# ref: YNU Digital Eng Lab Memorandum 05-1
from math import *
from svg import *
def genInvolutePolar(Rb, R): # Rb = base circle radius
# returns the involute angle as function of radius R.
return (sqrt(R*R - Rb*Rb) / Rb) - acos(Rb / R)
def rotate(pt, rads): # rotate pt by rads radians about origin
sinA = sin(rads)
cosA = cos(rads)
return [pt[0] * cosA - pt[1] * sinA,
pt[0] * sinA + pt[1] * cosA]
def toCartesian(radius, angle): # convert polar coords to cartesian
return [radius * cos(angle), radius * sin(angle)]
def CreateExternalGear(m, Z, phi):
# ****** external gear specifications
addendum = m # distance from pitch circle to tip circle
dedendum = 1.25 * m # pitch circle to root, sets clearance
clearance = dedendum - addendum
# Calculate radii
Rpitch = Z * m / 2 # pitch circle radius
Rb = Rpitch*cos(phi * pi / 180) # base circle radius
Ra = Rpitch + addendum # tip (addendum) circle radius
Rroot = Rpitch - dedendum # root circle radius
fRad = 1.5 * clearance # fillet radius, max 1.5*clearance
Rf = sqrt((Rroot + fRad) * (Rroot + fRad) - (fRad * fRad)) # radius at top of fillet
if (Rb < Rf):
Rf = Rroot + clearance
# ****** calculate angles (all in radians)
pitchAngle = 2 * pi / Z # angle subtended by whole tooth (rads)
baseToPitchAngle = genInvolutePolar(Rb, Rpitch)
pitchToFilletAngle = baseToPitchAngle # profile starts at base circle
if (Rf > Rb): # start profile at top of fillet (if its greater)
pitchToFilletAngle -= genInvolutePolar(Rb, Rf)
filletAngle = atan(fRad / (fRad + Rroot)) # radians
# ****** generate Higuchi involute approximation
fe = 1 # fraction of profile length at end of approx
fs = 0.01 # fraction of length offset from base to avoid singularity
if (Rf > Rb):
fs = (Rf * Rf - Rb * Rb) / (Ra * Ra - Rb * Rb) # offset start to top of fillet
# approximate in 2 sections, split 25% along the involute
fm = fs + (fe - fs) / 4 # fraction of length at junction (25% along profile)
dedBez = BezCoeffs(m, Z, phi, 3, fs, fm)
addBez = BezCoeffs(m, Z, phi, 3, fm, fe)
dedInv = dedBez.involuteBezCoeffs()
addInv = addBez.involuteBezCoeffs()
# join the 2 sets of coeffs (skip duplicate mid point)
inv = dedInv + addInv[1:]
# create the back profile of tooth (mirror image)
invR = [0 for i in range(0, len(inv))] # involute profile along back of tooth
for i in range(0, len(inv)):
# rotate all points to put pitch point at y = 0
pt = rotate(inv[i], -baseToPitchAngle - pitchAngle / 4)
inv[i] = pt
# generate the back of tooth profile nodes, mirror coords in X axis
invR[i] = [pt[0], -pt[1]]
# ****** calculate section junction points R=back of tooth, Next=front of next tooth)
fillet = toCartesian(Rf, -pitchAngle / 4 - pitchToFilletAngle) # top of fillet
filletR = [fillet[0], -fillet[1]] # flip to make same point on back of tooth
rootR = toCartesian(Rroot, pitchAngle / 4 + pitchToFilletAngle + filletAngle)
rootNext = toCartesian(Rroot, 3 * pitchAngle / 4 - pitchToFilletAngle - filletAngle)
filletNext = rotate(fillet, pitchAngle) # top of fillet, front of next tooth
# Draw the shapes in SVG
t_inc = 2.0 * pi / float(Z)
thetas = [(x * t_inc) for x in range(Z)]
svg = SVG_move(fillet, 0) # start at top of fillet
for theta in thetas:
if (Rf < Rb):
svg += SVG_line(inv[0], theta) # line from fillet up to base circle
svg += SVG_curve2(inv[1], inv[2], inv[3],
inv[4], inv[5], inv[6], theta)
svg += SVG_circle(invR[6], Ra, 1, theta) # arc across addendum circle
# svg = SVG_move(invR[6]) # TEMP
svg += SVG_curve2(invR[5], invR[4], invR[3],
invR[2], invR[1], invR[0], theta)
if (Rf < Rb):
svg += SVG_line(filletR, theta) # line down to topof fillet
if (rootNext[1] > rootR[1]): # is there a section of root circle between fillets?
svg += SVG_circle(rootR, fRad, 0, theta) # back fillet
svg += SVG_circle(rootNext, Rroot, 1, theta) # root circle arc
svg += SVG_circle(filletNext, fRad, 0, theta)
svg += SVG_close()
return svg
def CreateInternalGear(m, Z, phi):
addendum = 0.6 * m # pitch circle to tip circle (ref G.M.Maitra)
dedendum = 1.25 * m # pitch circle to root radius, sets clearance
# Calculate radii
Rpitch = Z * m / 2 # pitch radius
Rb = Rpitch * cos(phi * pi / 180) # base radius
Ra = Rpitch - addendum # addendum radius
Rroot = Rpitch + dedendum# root radius
clearance = 0.25 * m # gear dedendum - pinion addendum
Rf = Rroot - clearance # radius of top of fillet (end of profile)
fRad = 1.5 * clearance # fillet radius, 1 .. 1.5*clearance
# ****** calculate subtended angles
pitchAngle = 2 * pi / Z # angle between teeth (rads)
baseToPitchAngle = genInvolutePolar(Rb, Rpitch)
tipToPitchAngle = baseToPitchAngle # profile starts from base circle
if (Ra > Rb):
tipToPitchAngle -= genInvolutePolar(Rb, Ra) # start profile from addendum
pitchToFilletAngle = genInvolutePolar(Rb, Rf) - baseToPitchAngle
filletAngle = 1.414 * clearance / Rf # to make fillet tangential to root
# ****** generate Higuchi involute approximation
fe = 1 # fraction of involute length at end of approx (fillet circle)
fs = 0.01 # fraction of length offset from base to avoid singularity
if (Ra > Rb):
fs = (Ra*Ra - Rb*Rb) / (Rf*Rf - Rb*Rb) # start profile from addendum (tip circle)
# approximate in 2 sections, split 25% along the profile
fm = fs + (fe - fs) / 4
addBez = BezCoeffs(m, Z, phi, 3, fs, fm)
dedBez = BezCoeffs(m, Z, phi, 3, fm, fe)
addInv = addBez.involuteBezCoeffs()
dedInv = dedBez.involuteBezCoeffs()
# join the 2 sets of coeffs (skip duplicate mid point)
invR = addInv + dedInv[1:]
# create the front profile of tooth (mirror image)
inv = [0 for i in range(0, len(invR))] # back involute profile
for i in range(0, len(inv)):
# rotate involute to put center of tooth at y = 0
pt = rotate(invR[i], pitchAngle / 4 - baseToPitchAngle)
invR[i] = pt
# generate the back of tooth profile, flip Y coords
inv[i] = [pt[0], -pt[1]]
# ****** calculate coords of section junctions
fillet = [inv[6][0], inv[6][1]] # top of fillet, front of tooth
tip = toCartesian(Ra, -pitchAngle / 4 + tipToPitchAngle) # tip, front of tooth
tipR = [tip[0], -tip[1]] # addendum, back of tooth
rootR = toCartesian(Rroot, pitchAngle / 4 + pitchToFilletAngle + filletAngle)
rootNext = toCartesian(Rroot, 3 * pitchAngle / 4 - pitchToFilletAngle - filletAngle)
filletNext = rotate(fillet, pitchAngle) # top of fillet, front of next tooth
# Draw the shapes in SVG
t_inc = 2.0 * pi / float(Z)
thetas = [(x * t_inc) for x in range(Z)]
svg = SVG_move(fillet, 0) # start at top of fillet
for theta in thetas:
svg += SVG_curve2(inv[5], inv[4], inv[3],
inv[2], inv[1], inv[0], theta)
if (Ra < Rb):
svg += SVG_line(tip, theta) # line from end of involute to addendum (tip)
svg += SVG_circle(tipR, Ra, 1, theta) # arc across tip circle
if (Ra < Rb):
svg += SVG_line(invR[0], theta) # line from addendum to start of involute
svg += SVG_curve2(invR[1], invR[2], invR[3],
invR[4], invR[5], invR[6], theta)
if (rootR[1] < rootNext[1]): # there is a section of root circle between fillets
svg += SVG_circle(rootR, fRad, 1, theta) # fillet on back of tooth
svg += SVG_circle(rootNext, Rroot, 1, theta) # root circle arc
svg += SVG_circle(filletNext, fRad, 1, theta) # fillet on next
return svg
class BezCoeffs:
def chebyExpnCoeffs(self, j, func):
N = 50 # a suitably large number N>>p
c = 0
for k in range(1, N + 1):
c += func(cos(pi * (k - 0.5) / N)) * cos(pi * j * (k - 0.5) / N)
return 2 *c / N
def chebyPolyCoeffs(self, p, func):
coeffs = [0, 0, 0, 0]
fnCoeff = [0, 0, 0, 0]
T = [[1, 0, 0, 0, 0],
[0, 1, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]
]
# now generate the Chebyshev polynomial coefficient using
# formula T(k+1) = 2xT(k) - T(k-1) which yields
# T = [ [ 1, 0, 0, 0, 0, 0], # T0(x) = +1
# [ 0, 1, 0, 0, 0, 0], # T1(x) = 0 +x
# [-1, 0, 2, 0, 0, 0], # T2(x) = -1 0 +2xx
# [ 0, -3, 0, 4, 0, 0], # T3(x) = 0 -3x 0 +4xxx
# [ 1, 0, -8, 0, 8, 0], # T4(x) = +1 0 -8xx 0 +8xxxx
# [ 0, 5, 0,-20, 0, 16], # T5(x) = 0 5x 0 -20xxx 0 +16xxxxx
# ... ]
for k in range(1, p + 1):
for j in range(0, len(T[k]) - 1):
T[k + 1][j + 1] = 2 * T[k][j]
for j in range(0, len(T[k - 1])):
T[k + 1][j] -= T[k - 1][j]
# convert the chebyshev function series into a simple polynomial
# and collect like terms, out T polynomial coefficients
for k in range(0, p + 1):
fnCoeff[k] = self.chebyExpnCoeffs(k, func)
coeffs[k] = 0
for k in range(0, p + 1):
for pwr in range(0, p + 1):
coeffs[pwr] += fnCoeff[k] * T[k][pwr]
coeffs[0] -= self.chebyExpnCoeffs(0, func) / 2 # fix the 0th coeff
return coeffs
# Equation of involute using the Bezier parameter t as variable
def involuteXbez(self, t):
# map t (0 <= t <= 1) onto x (where -1 <= x <= 1)
x = t * 2 - 1
# map theta (where ts <= theta <= te) from x (-1 <=x <= 1)
theta = x * (self.te - self.ts) / 2 + (self.ts + self.te) / 2
return self.Rb * (cos(theta) + theta * sin(theta))
def involuteYbez(self, t):
# map t (0 <= t <= 1) onto x (where -1 <= x <= 1)
x = t * 2 - 1
# map theta (where ts <= theta <= te) from x (-1 <=x <= 1)
theta = x * (self.te - self.ts) / 2 + (self.ts + self.te) / 2
return self.Rb * (sin(theta) - theta * cos(theta))
def binom(self, n, k):
coeff = 1
for i in range(n - k + 1, n + 1):
coeff *= i
for i in range(1, k + 1):
coeff /= i
return coeff
def bezCoeff(self, i, func):
# generate the polynomial coeffs in one go
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