{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "[](https://drivetrainhub.com)\n", "\n", "
Symbol | \n", "Description | \n", "
---|---|
$m_n$ | \n", "Module | \n", "
$\\alpha_n$ | \n", "Pressure angle | \n", "
$p$ | \n", "Pitch | \n", "
$p_b$ | \n", "Base pitch | \n", "
$h_a$ | \n", "Addendum | \n", "
$h_f$ | \n", "Dedendum | \n", "
$c$ | \n", "Bottom clearance | \n", "
$\\rho_F$ | \n", "Root radius | \n", "
$\\alpha_{FP}$ | \n", "Undercut angle | \n", "
$U_{FP}$ | \n", "Undercut size | \n", "
$p_{bP}$ | \n", "Undercut base pitch | \n", "
Basic rack tooth profile
\n", "\n", "Basic rack with mating rack and bottom clearance
\n", "\n", "As observed in the figure above, the mathematical expression for bottom clearance can be written as:\n", "\n", "$$c = h_f - h_a$$\n", "\n", "Additionally, the root radius height is equal to the bottom clearance, thus the root radius can be expressed as:\n", "\n", "$$\\rho_F = \\frac{c}{1 - \\sin\\alpha_n}$$\n", "\n", "Full root radius of basic rack
\n", "\n", "If root radius is not limited by bottom clearance, the maximum root radius corresponds to a full root radius. The solution of a full root radius can be obtained by formulating a system of equations. First, an equation can be written for the horizontal coordinate:\n", "\n", "$$\\rho_F = \\frac{s_\\text{fillet}}{2 \\cos\\alpha_n}$$\n", "\n", "where $s_\\rm{fillet}$ is the spacewidth at the start of the root fillet, an unknown parameter. Next, an equation can be written for the vertical coordinate:\n", "\n", "$$\\rho_F = \\frac{h_\\text{fillet}}{1 - \\sin\\alpha_n}$$\n", "\n", "where $h_\\text{fillet}$ is the height of the root fillet, an unknown parameter. This equation was applied in the case of a root radius limited by bottom clearance, where $h_\\rm{fillet} = c$. We currently have two equations with three unknowns, so we need to introduce additional expressions to create a solvable system of equations. By observing that $s_\\text{fillet}$ and $h_\\text{fillet}$ can be expressed as functions of a vertical coordinate, $f\\{y\\}$, we can create a solvable system.\n", "\n", "$$s_\\text{fillet} = \\frac{p}{2} + 2 y_\\text{fillet} \\tan\\alpha_n$$\n", "\n", "$$h_\\text{fillet} = h_f + y_\\text{fillet}$$\n", "\n", "where $y_\\text{fillet}$ is a negative value of the vertical coordinate from the basic rack datum line to the start of the full root fillet. By solving the system of equations, we obtain an expression for $y_\\text{fillet}$, which can then be used to find the full root radius.\n", "\n", "$$y_\\text{fillet} = \\frac{e_p \\tan\\alpha_n - e_p \\sec\\alpha_n + 2 h_f}{2 (\\tan\\alpha_n \\sec\\alpha_n -\\tan^2\\alpha_n - 1)}$$\n", "\n", "where $e_p = \\pi m_n / 2$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Basic Rack Plot\n", "\n", "By parametrically defining a basic rack, we can produce a piece-wise plot of the resulting geometry. Below is a such a plot, drawn in the Cartesian xy-plane per the table of input parameters used to define the basic rack." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "tags": [ "hide_input" ] }, "outputs": [ { "data": { "text/html": [ "$m_n$ | $\\alpha_n$ | $h_a^*$ | $h_f^*$ | $\\rho_F^*$ | $\\rho_{F,\\rm{max}}^*$ | $p$ | $p_b$ | $c^*$ |
---|---|---|---|---|---|---|---|---|
$1\\text{mm}$ | $20^\\circ$ | $1$ | $1.25$ | $0.38$ | $0.38$ | $3.14\\text{mm}$ | $2.95\\text{mm}$ | $0.25$ |
Undercut basic rack tooth profile
\n", "\n", "Notice the base pitch of the undercut profile, $p_{bP}$, differs from the tooth profile base pitch, $p_b$, despite each profile having the same pitch, $p$. As we will see in the chapters on spur and helical gear geometry, this difference explains why only gears of the same module and reference pressure angle can mesh. However, there are some exceptions for the generating tool geometry that will be discussed in the relevant notebooks. \n", "\n", "$m_n$ | $\\alpha_n$ | $h_a^*$ | $h_f^*$ | $\\rho_F^*$ | $\\alpha_{FP}$ | $U_{FP}$ |
---|---|---|---|---|---|---|
$1\\text{mm}$ | $20^\\circ$ | $1$ | $1.25$ | $0.38$ | $10^\\circ$ | $0.15\\text{mm}$ |
\n", " | A | \n", "B | \n", "C | \n", "D | \n", "
---|---|---|---|---|
Pressure angle, $\\alpha_n$ | \n", "$20^\\circ$ | \n", "$20^\\circ$ | \n", "$20^\\circ$ | \n", "$20^\\circ$ | \n", "
Addendum coefficient, $h_a^*$ | \n", "$1.00$ | \n", "$1.00$ | \n", "$1.00$ | \n", "$1.00$ | \n", "
Dedendum coefficient, $h_f^*$ | \n", "$1.25$ | \n", "$1.25$ | \n", "$1.25$ | \n", "$1.40$ | \n", "
Root radius coefficient, $\\rho_F^*$ | \n", "$0.38$ | \n", "$0.30$ | \n", "$0.25$ | \n", "$0.39$ | \n", "
\n", " | I | \n", "II | \n", "III | \n", "IV | \n", "
---|---|---|---|---|
Pressure angle, $\\alpha_n$ | \n", "$25^\\circ$ | \n", "$15^\\circ$ | \n", "$20^\\circ$ | \n", "$20^\\circ$ | \n", "
Addendum coefficient, $h_a^*$ | \n", "$0.90$ | \n", "$1.10$ | \n", "$1.00$ | \n", "$1.00$ | \n", "
Dedendum coefficient, $h_f^*$ | \n", "$1.15$ | \n", "$1.35$ | \n", "$1.25$ | \n", "$1.25$ | \n", "
Root radius coefficient, $\\rho_F^*$ | \n", "$0.30$ | \n", "$0.30$ | \n", "$0.30$ | \n", "$0.30$ | \n", "
Undercut angle, $\\alpha_{FP}$ | \n", "- | \n", "- | \n", "$10^\\circ$ | \n", "$5^\\circ$ | \n", "
Undercut size, $U_{FP}$ | \n", "- | \n", "- | \n", "$0.15$ | \n", "$0.15$ | \n", "