#!/usr/bin/env python3 import math, argparse # Use the scipy library for numerical solutions. # https://docs.scipy.org/doc/scipy/reference/optimize.html import scipy.optimize import numpy as np #================================================================ # Each belt drive system is described by a set of properties. This allows the # same code to work with different systems. GT2 = { 'units' : 'mm', 'pitch' : 2.00, 'tooth_height' : 0.76, 'pitch_line_offset' : 0.25, 'belt_thickness' : 1.52, } MXL = { 'units' : 'inch', 'pitch' : 0.080, 'tooth_height' : 0.020, 'pitch_line_offset' : 0.010, 'belt_thickness' : 0.045, } XL = { 'units' : 'inch', 'pitch' : 0.200, 'tooth_height' : 0.050, 'pitch_line_offset' : 0.010, 'belt_thickness' : 0.090, } L = { 'units' : 'inch', 'pitch' : 0.375, 'tooth_height' : 0.075, 'pitch_line_offset' : 0.015, 'belt_thickness' : 0.140, } #================================================================ # Timing pulley center distance formulas. # See section 22 in https://www.sdp-si.com/PDFS/Technical-Section-Timing.pdf # Given: # C center distance # L belt length # R1 larger pulley pitch radius # R2 smaller pulley pitch radius # phi One half angle of wrap on smaller pulley (radians) # C * cos(phi) = R1 - R2 # 2 * C * sin(phi) = L - pi*(R1+R2) - (pi - 2*phi)*(R1-R2) # Rewriting: # phi = acos((R1 - R2) / C) # L = 2 * C * sin(phi) + pi * (R1 + R2) + (pi - 2 * phi) * (R1 - R2) def timing_belt_length(N1, N2, C, system=GT2): """Calculate the belt length in teeth for a pair of pulleys with N1 and N2 teeth separated by center distance C. The result may be fractional and thus unrealizable; endless belts have an integer number of teeth chosen from a specific set of available sizes. """ # Make sure N1 >= N2: if N2 > N1: N1, N2 = N2, N1 # Look up belt properties. pitch = system['pitch'] # Calculate the radius of each pulley in the system units. The circumference # at the pitch diameter is the number of teeth multiplied by the belt tooth pitch. R1 = N1 * pitch / (2 * math.pi) R2 = N2 * pitch / (2 * math.pi) # Check the pulley separation: if R1 + R2 >= C: print("Warning: pulleys collide, solution not feasible.") # The essential calculation does not depend on units: phi = math.acos((R1 - R2) / C) L = 2 * C * math.sin(phi) + math.pi * (R1 + R2) + (math.pi - 2 * phi) * (R1 - R2) # Convert the length to a tooth count (possibly non-integer). belt_teeth = L / pitch return belt_teeth #================================================================ def timing_belt_center_distance(N1, N2, T, system=GT2): """Calculate the center to center distance for a pair of pulleys with N1 and N2 teeth driving a belt with T teeth. """ # Look up belt properties. pitch = system['pitch'] # Use the scipy fmin algorithm to calculate an inverse solution using the # length function. The initial guess is based on the pulley sizes. x0 = np.array(((N1+N2) * pitch)) result = scipy.optimize.fmin(lambda x: abs(timing_belt_length(N1, N2, x[0]) - T), x0, disp=False) return result[0] #================================================================ def timing_pulley_dimensions(N, system=GT2): """Calculate properties for a timing pulley of a given size and system. :param N: integer number of teeth :param system: dictionary of belt system properties. :return: dictionary of pulley properties """ # Look up belt properties. pitch = system['pitch'] tooth_height = system['tooth_height'] # Calculate pulley properties. # The pitch diameter is the effective diameter of the pulley acting as a # wheel; it falls inside the belt, so outside the actual pulley. pitch_diameter = N * pitch / math.pi # The stock diameter is the maximum diameter of the pulley teeth, # corresponding to the stock circular diameter before cutting grooves. stock_diameter = pitch_diameter - tooth_height return { 'teeth' : N, 'pitch_diameter' : pitch_diameter, 'stock_diameter' : stock_diameter } #================================================================ # Main script follows. This sequence is executed when the script is initiated from the command line. if __name__ == "__main__": parser = argparse.ArgumentParser(description="Timing belt calculator.") parser.add_argument('--debug', action='store_true', help='Enable debugging logging to console.') parser.add_argument('-N1', type=int, help="Number of teeth on first pulley.") parser.add_argument('-N2', type=int, help="Number of teeth on second pulley.") parser.add_argument('-T', type=int, help="Number of teeth on timing belt.") parser.add_argument('-C', type=float,help="Distance between pulley centers in default units.") parser.add_argument('-sys', type=str, default='GT2', help="Belt system: GT2, MXL, XL, or L (default %(default)s).") args = parser.parse_args() system = {'GT2' : GT2, 'MXL' : MXL, 'XL' : XL, 'L' : L}.get(args.sys) if system is None: print("Warning: unrecognized belting system type, defaulting to GT2.") system = GT2 # Choose a calculation depending on the combination of arguments. if args.N1 is not None and args.N2 is not None and args.C is not None and args.T is None: belt_teeth = timing_belt_length(args.N1, args.N2, args.C, system=system) print(f"Timing belt has {belt_teeth} teeth.") elif args.N1 is not None and args.N2 is not None and args.C is None and args.T is not None: center_distance = timing_belt_center_distance(args.N1, args.N2, args.T, system=system) print(f"Pulley center distance is %f %s." % (center_distance, system['units'])) else: print("No calculation defined for this combination of arguments.")