{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Free Body Diagram for Rigid Bodies\n", "\n", "> Renato Naville Watanabe \n", "> [Laboratory of Biomechanics and Motor Control](http://pesquisa.ufabc.edu.br/bmclab) \n", "> Federal University of ABC, Brazil" ] }, { "cell_type": "markdown", "metadata": { "toc": 1 }, "source": [ "

# Contents

\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Python setup" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "ExecuteTime": { "end_time": "2020-05-08T03:39:04.259704Z", "start_time": "2020-05-08T03:39:04.130050Z" }, "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Equivalent systems\n", "\n", "\n", "A set of forces and moments is considered equivalent if its resultant force and sum of the moments computed relative to a given point are the same. Normally, we want to reduce all the forces and moments being applied to a body into a single force and a single moment.\n", "\n", "We have done this with particles for the resultant force. The resultant force is simply the sum of all the forces being applied to the body.\n", "\n", "\n", "\$$\n", " \\vec{F} = \\sum\\limits_{i=1}^n \\vec{F_i}\n", "\$$\n", "\n", "\n", "where $\\vec{F_i}$ is each force applied to the body.\n", "\n", "Similarly, the total moment applied to the body relative to a point O is:\n", "\n", "\n", "\$$\n", " \\vec{M_O} = \\sum\\limits_{i}\\vec{r_{i/O}} \\times \\vec{F_i}\n", "\$$\n", "\n", "\n", "where $\\vec{r_{i/O}}$ is the vector from the point O to the point where the force $\\vec{\\bf{F_i}}$ is being applied.\n", "\n", "But where the resultant force should be applied in the body? If the resultant force were applied to any point different from the point O, it would produce an additional moment to the body relative to point O. So, the resultant force must be applied to the point O.\n", "\n", "So, any set of forces can be reduced to a moment relative to a chosen point O and a resultant force applied to the point O. \n", "\n", "To compute the resultant force and moment relative to another point O', the new moment is:\n", "\n", "\n", "\$$\n", " \\vec{M_{O'}} = \\vec{M_O} + \\vec{r_{O'/O}} \\times \\vec{F}\n", "\$$\n", "\n", "\n", "And the resultant force is the same.\n", "\n", "It is worth to note that if the resultant force $\\vec{F}$ is zero, than the moment is the same relative to any point.\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Steps to draw a free-body diagram\n", "\n", "The steps to draw the free-body diagram of a body is very similar to the case of particles.\n", "\n", "1 - Draw separately each object considered in the problem. How you separate depends on what questions you want to answer." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "2 - Identify the forces acting on each object. If you are analyzing more than one object, remember the third Newton law (action and reaction), and identify where the reaction of a force is being applied. Whenever a movement of translation of the body is constrained, a force at the direction of the constraint must exist." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "3 - Identify the moments acting on each object. In the case you are analyzing more than one object, you must consider the action and reaction law (third Newton-Euler law). Whenever a movement of rotation of the body is constrained, a moment at the direction of the constraint must exist." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "4 - Draw all the identified forces, representing them as vectors. The vectors should be represented with the origin in the object. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "5 - Draw all the identified moments, representing them as vectors. In planar movements, the moments we be orthogonal to the considered plane. In these cases, normally the moment vectors are represented as curved arrows." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "6 - If necessary, you should represent the reference frame (or references frames in case you use more than one reference frame) in the free-body diagram." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "7 - After this, you can solve the problem using the First and Second Newton-Euler Laws (see, e.g, [Newton-Euler Laws](newton_euler_equations.ipynb)) to find the motion of the body.\n", "\n", "\n", "\$$\n", " \\vec{F} = m\\vec{a_{cm}} = m\\frac{d^2\\vec{r_{cm}}}{dt^2}\n", "\$$\n", "\n", "\n", "\n", "\$$\n", " \\vec{M_O} = I_{zz}^{cm}\\vec{\\alpha} + m \\vec{r_{cm/O}} \\times \\vec{a_{cm}}=I_{zz}^{cm}\\vec{\\frac{d^2\\theta}{dt^2}} + m \\vec{r_{cm/O}} \\times \\frac{d^2\\vec{r_{cm}}}{dt^2}\n", "\$$\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Examples\n", "\n", "Below, we will see some examples of how to draw the free-body diagram and obtain the equation of motion." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Horizontal fixed bar \n", "\n", "The first example is an example of statics. The bar has **no velocity** and **no acceleration**. We can find the **force** and **moment** the wall is applying to the bar.\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Free-body diagram\n", "\n", "The free-body diagram of the bar is depicted below. At the point where the bar is connected to the wall, there is a force $\\vec{F_1}$ constraining the translation movement of the point O and a moment $\\vec{M}$ constraining the rotation of the bar.\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Sum of the forces and moments\n", "\n", "The resultant force being applied to the bar is:\n", "\n", "\n", "\$$\n", "\\vec{F} = -mg\\hat{\\bf{j}} + \\vec{F_1}\n", "\$$\n", "\n", "\n", "And the total moment in the z direction around the point O is:\n", "\n", "\n", "\$$\n", "\\vec{M_O} = \\vec{r_{C/O}}\\times-mg\\hat{\\bf{j}} + \\vec{M}\n", "\$$\n", "\n", "\n", "The vector from the point O to the point C is given by $\\vec{r_{C/O}} =\\frac{l}{2}\\hat{\\bf{i}}$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Newton-Euler laws\n", "As the bar is fixed, all the accelerations are zero. So we can find the forces and the moment at the constraint.\n", "\n", "\n", "\$$\n", "\\vec{F} = \\vec{0}\n", "\$$\n", "\n", "\n", "\n", "\$$\n", "-mg\\hat{\\bf{j}} + \\vec{F_1} = \\vec{0}\n", "\$$\n", "\n", "\n", "\n", "\$$\n", "\\vec{F_1} = mg\\hat{\\bf{j}}\n", "\$$\n", "\n", "\n", "\n", "\$$\n", "\\vec{M_O} = \\vec{0}\n", "\$$\n", "\n", "\n", "\n", "\$$\n", "\\vec{r_{C/O}}\\times-mg\\hat{\\bf{j}} + \\vec{M} = \\vec{0} \n", "\$$\n", "\n", "\n", "\n", "\$$\n", "\\frac{l}{2}\\hat{\\bf{i}}\\times-mg\\hat{\\bf{j}} + \\vec{M} = \\vec{0}\n", "\$$\n", "\n", "\n", "\n", "\$$\n", "-\\frac{mgl}{2}\\hat{\\bf{k}} + \\vec{M} = \\vec{0} \n", "\$$\n", "\n", "\n", "\n", "\$$\n", "\\vec{M} = \\frac{mgl}{2}\\hat{\\bf{k}} \n", "\$$\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Rotating ball with drag force \n", "\n", "\n", "A **basketball** has a mass **$\\bf{m = 0.63}$ kg** and radius equal $\\bf{R = 12}$ **cm**. A basketball player shots the ball from the free-throw line (**4.6 m from the basket**) with a **speed of 9.5 m/s**, **angle of 51 degrees with the court**, **height of 2 m** and **angular velocity of 42 rad/s**. At the ball is acting a **drag force** $\\bf{b_l = 0.5}$ **N.s/m** proportional to the modulus of the ball velocity in the opposite direction of the velocity and a **drag moment** $\\bf{b_r = 0.001}$ **N.m.s**, proportional and in the opposite direction of the angular velocity of the ball. Consider the moment of inertia of the ball as $\\bf{I_{zz}^{cm} = \\frac{2mR^2}{3}}$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Free-body diagram\n", "\n", "Below is depicted the free-body diagram of the ball.\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Sum of forces and moments\n", "\n", "The resultant force being applied at the ball is:\n", "\n", "\n", "\$$\n", "\\vec{F} = -mg\\hat{\\bf{j}} - b_l\\vec{v} = -mg\\hat{\\bf{j}} - b_l\\frac{d\\vec{r}}{dt} \n", "\$$\n", "\n", "\n", "\n", "\$$\n", " \\vec{M_C} = - b_r\\omega\\hat{\\bf{k}}=- b_r\\frac{d\\theta}{dt}\\hat{\\bf{k}}\n", "\$$\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Newton-Euler laws\n", "\n", "So, by the second Newton-Euler law:\n", "\n", "\n", "\$$\n", " I_{zz}^{C}\\frac{d^2\\theta}{dt^2}=\\vec{M_C}\n", "\$$\n", "\n", "\n", " \n", "\$$\n", " I_{zz}^{C}\\frac{d^2\\theta}{dt^2} = - b_r\\frac{d\\theta}{dt}\n", "\$$\n", "\n", "\n", "and by the first Newton-Euler law (for a revision on Newton-Euler laws, [see this notebook](newton_euler_equations.ipynb)):\n", "\n", "\n", "\$$\n", " m\\frac{d^2\\vec{r}}{dt^2}=\\vec{F} \\rightarrow \\frac{d^2\\vec{r}}{dt^2} = -g\\hat{\\bf{j}} - \\frac{b_l}{m}\\frac{d^2\\vec{r}}{dt^2} \n", "\$$\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Equations of motion\n", "\n", "So, we can split the differential equations above in three equations:\n", "\n", "\n", "\$$\n", " \\frac{d^2\\theta}{dt^2} = - \\frac{b_r}{I_{zz}^{C}}\\frac{d\\theta}{dt}\n", "\$$\n", "\n", "\n", "\n", "\$$\n", " \\frac{d^2x}{dt^2} = - \\frac{b_l}{m}\\frac{dx}{dt} \n", "\$$\n", "\n", "\n", "\n", "\$$\n", " \\frac{d^2y}{dt^2} = -g - \\frac{b_l}{m}\\frac{dy}{dt} \n", "\$$\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Numerical solution of the equations\n", "\n", "To solve these equations numerically, we can split each of these equations in first-order equations and the use a numerical method to integrate them. The first-order equations we be written in a matrix form, considering the moment of inertia of the ball as $I_{zz}^{C}=\\frac{2mR^2}{3}$:\n", "\n", "\$$\n", " \\left[\\begin{array}{c}\\frac{d\\omega}{dt}\\\\\\frac{dv_x}{dt}\\\\\\frac{dv_y}{dt}\\\\\\frac{d\\theta}{dt}\\\\\\frac{dx}{dt}\\\\\\frac{dy}{dt} \\end{array}\\right] = \\left[\\begin{array}{c}- \\frac{3b_r}{2mR^2}\\omega\\\\- \\frac{b_l}{m}v_x\\\\-g - \\frac{b_l}{m}v_y\\\\\\omega\\\\v_x\\\\v_y\\end{array}\\right] \n", "\$$\n", "\n", "\n", "Below, the equations were solved numerically by using the Euler method (for a revision on numerical methods to solve ordinary differential equations, [see this notebook](OrdinaryDifferentialEquation.ipynb))." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "ExecuteTime": { "end_time": "2020-05-08T03:39:07.463212Z", "start_time": "2020-05-08T03:39:07.408148Z" }, "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": "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\n", "text/plain": [ "