Timing Belt Calculator (console)¶
This command-line Python script is a utility to help calculate parameters for timing belt drive systems. A more general discussion of designing with timing belts can be found on the Timing Belt Guide pages.
Installation Requirements¶
The code requires a working installation of Python 3 with scipy. For suggestions on setting up your system please see Python 3 Installation.
Examples¶
The script accepts parameters on the command line and chooses a calculation based on the specific combination of supplied values.
To see all the command-line options:
python3 timing_belt_calculator.py --help
To calculate the timing belt length for a 20 tooth GT2 pulley driving a 60 tooth GT2 pulley separated by 100 mm:
python3 timing_belt_calculator.py -N1 20 -N2 60 -C 100
Timing belt has 140.811669871759 teeth.
Note that this solution isn’t feasible, since endless belts must have an integer number of teeth selected from specific available sizes. To choose the correct separation for the available 140 tooth size:
python3 timing_belt_calculator.py -N1 20 -N2 60 -T 140
Pulley center distance is 99.181641 mm.
Full Code¶
Direct download: timing_belt_calculator.py
1#!/usr/bin/env python3
2
3import math, argparse
4
5# Use the scipy library for numerical solutions.
6# https://docs.scipy.org/doc/scipy/reference/optimize.html
7import scipy.optimize
8import numpy as np
9
10#================================================================
11
12# Each belt drive system is described by a set of properties. This allows the
13# same code to work with different systems.
14
15GT2 = { 'units' : 'mm',
16 'pitch' : 2.00,
17 'tooth_height' : 0.76,
18 'pitch_line_offset' : 0.25,
19 'belt_thickness' : 1.52,
20 }
21
22MXL = { 'units' : 'inch',
23 'pitch' : 0.080,
24 'tooth_height' : 0.020,
25 'pitch_line_offset' : 0.010,
26 'belt_thickness' : 0.045,
27 }
28
29XL = { 'units' : 'inch',
30 'pitch' : 0.200,
31 'tooth_height' : 0.050,
32 'pitch_line_offset' : 0.010,
33 'belt_thickness' : 0.090,
34 }
35
36L = { 'units' : 'inch',
37 'pitch' : 0.375,
38 'tooth_height' : 0.075,
39 'pitch_line_offset' : 0.015,
40 'belt_thickness' : 0.140,
41 }
42
43#================================================================
44# Timing pulley center distance formulas.
45# See section 22 in https://www.sdp-si.com/PDFS/Technical-Section-Timing.pdf
46
47# Given:
48# C center distance
49# L belt length
50# R1 larger pulley pitch radius
51# R2 smaller pulley pitch radius
52# phi One half angle of wrap on smaller pulley (radians)
53
54# C * cos(phi) = R1 - R2
55# 2 * C * sin(phi) = L - pi*(R1+R2) - (pi - 2*phi)*(R1-R2)
56
57# Rewriting:
58# phi = acos((R1 - R2) / C)
59# L = 2 * C * sin(phi) + pi * (R1 + R2) + (pi - 2 * phi) * (R1 - R2)
60
61def timing_belt_length(N1, N2, C, system=GT2):
62 """Calculate the belt length in teeth for a pair of pulleys with N1 and N2 teeth
63 separated by center distance C. The result may be fractional and thus
64 unrealizable; endless belts have an integer number of teeth chosen from a
65 specific set of available sizes.
66 """
67
68 # Make sure N1 >= N2:
69 if N2 > N1:
70 N1, N2 = N2, N1
71
72 # Look up belt properties.
73 pitch = system['pitch']
74
75 # Calculate the radius of each pulley in the system units. The circumference
76 # at the pitch diameter is the number of teeth multiplied by the belt tooth pitch.
77 R1 = N1 * pitch / (2 * math.pi)
78 R2 = N2 * pitch / (2 * math.pi)
79
80 # Check the pulley separation:
81 if R1 + R2 >= C:
82 print("Warning: pulleys collide, solution not feasible.")
83
84 # The essential calculation does not depend on units:
85 phi = math.acos((R1 - R2) / C)
86 L = 2 * C * math.sin(phi) + math.pi * (R1 + R2) + (math.pi - 2 * phi) * (R1 - R2)
87
88 # Convert the length to a tooth count (possibly non-integer).
89 belt_teeth = L / pitch
90 return belt_teeth
91
92#================================================================
93def timing_belt_center_distance(N1, N2, T, system=GT2):
94 """Calculate the center to center distance for a pair of pulleys with N1 and N2
95 teeth driving a belt with T teeth.
96 """
97
98 # Look up belt properties.
99 pitch = system['pitch']
100
101 # Use the scipy fmin algorithm to calculate an inverse solution using the
102 # length function. The initial guess is based on the pulley sizes.
103 x0 = np.array(((N1+N2) * pitch))
104
105 result = scipy.optimize.fmin(lambda x: abs(timing_belt_length(N1, N2, x[0]) - T), x0, disp=False)
106 return result[0]
107
108#================================================================
109def timing_pulley_dimensions(N, system=GT2):
110 """Calculate properties for a timing pulley of a given size and system.
111
112 :param N: integer number of teeth
113 :param system: dictionary of belt system properties.
114 :return: dictionary of pulley properties
115 """
116
117 # Look up belt properties.
118 pitch = system['pitch']
119 tooth_height = system['tooth_height']
120
121 # Calculate pulley properties.
122
123 # The pitch diameter is the effective diameter of the pulley acting as a
124 # wheel; it falls inside the belt, so outside the actual pulley.
125 pitch_diameter = N * pitch / math.pi
126
127 # The stock diameter is the maximum diameter of the pulley teeth,
128 # corresponding to the stock circular diameter before cutting grooves.
129 stock_diameter = pitch_diameter - tooth_height
130
131 return { 'teeth' : N,
132 'pitch_diameter' : pitch_diameter,
133 'stock_diameter' : stock_diameter
134 }
135
136
137#================================================================
138# Main script follows. This sequence is executed when the script is initiated from the command line.
139
140if __name__ == "__main__":
141 parser = argparse.ArgumentParser(description="Timing belt calculator.")
142 parser.add_argument('--debug', action='store_true', help='Enable debugging logging to console.')
143 parser.add_argument('-N1', type=int, help="Number of teeth on first pulley.")
144 parser.add_argument('-N2', type=int, help="Number of teeth on second pulley.")
145 parser.add_argument('-T', type=int, help="Number of teeth on timing belt.")
146 parser.add_argument('-C', type=float,help="Distance between pulley centers in default units.")
147 parser.add_argument('-sys', type=str, default='GT2', help="Belt system: GT2, MXL, XL, or L (default %(default)s).")
148
149 args = parser.parse_args()
150
151 system = {'GT2' : GT2, 'MXL' : MXL, 'XL' : XL, 'L' : L}.get(args.sys)
152 if system is None:
153 print("Warning: unrecognized belting system type, defaulting to GT2.")
154 system = GT2
155
156 # Choose a calculation depending on the combination of arguments.
157 if args.N1 is not None and args.N2 is not None and args.C is not None and args.T is None:
158 belt_teeth = timing_belt_length(args.N1, args.N2, args.C, system=system)
159 print(f"Timing belt has {belt_teeth} teeth.")
160
161 elif args.N1 is not None and args.N2 is not None and args.C is None and args.T is not None:
162 center_distance = timing_belt_center_distance(args.N1, args.N2, args.T, system=system)
163 print(f"Pulley center distance is %f %s." % (center_distance, system['units']))
164
165 else:
166 print("No calculation defined for this combination of arguments.")