{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "[](https://drivetrainhub.com)\n", "\n", "
Stresses on gear tooth.
\n", "Image credit: G. M. Maitra, Handbook of Gear Design " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Root Bending Stress Calculation\n", "\n", "The bending stress is organized as:\n", "\n", "$$\\sigma_b = \\frac{F_t}{b m_n} \\frac{6m_n h_q \\cos\\alpha_a \\prime}{S_q^2 \\cos\\alpha}$$\n", "\n", "The second fraction is defined as the tooth form factor according to: \n", "\n", "$$q_k = \\frac{6m_n h_q \\cos\\alpha_a \\prime}{S_q^2 \\cos\\alpha}$$\n", "\n", "where values for $q_k$ are obtained from the figure below.\n", "\n", "\n", "Tooth form factor for bending stress.
\n", "Image credit: G. M. Maitra, Handbook of Gear Design " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The final form of the tooth bending stress is,\n", "\n", "$$ \\sigma_{b} = \\frac{F_t}{b_1 m_n} q_e q_k$$\n", "\n", "where $F_t$ is the tangential gear mesh force, $b_1$ is the facewidth of the pinion, $m_n$ is the normal module, $q_e = \\frac{1}{CR}$ is a factor from contact ratio, and $q_{k,1}$ is the tooth form factor of the pinion obtained from the figure above. \n", "\n", "Tangential force acting on the gear tooth is found from the torque on the pinion, using\n", "\n", "$$ F_t = \\frac{T_1}{\\frac{1}{2}d_{p,1}}$$\n", "\n", "The permissible root bending stress is denoted by $\\sigma_{bp}$ for the material. For example, the permissible bending stress for induction hardened steel 40 Cr 4 is given as 200 MPa. The stress at the root must be less than the permissible bending stress, $\\sigma_b < \\sigma_{bp}$. Substituting this and the tangential force expression, and using milimeters for units for facewidth and module gives the minimum module as,\n", "\n", "$$ m > \\sqrt[3]{\\frac{2000 T_1 q_e q_k }{\\frac{b_1}{d_{p,1}}z_{1}^2 \\sigma_{bp}} }$$\n", " \n", "Idential expression with the parameters for the wheel gives the root stress on the wheel. Generally, root stress on pinion is higher than the wheel. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "