lecture02 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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T s (t) ϖ (t) Inertia,J Bearings,T 0 ,b MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.004 Dynamics and Control II Fall 2007 Lecture 2 Solving the Equation of Motion Goals for today Modeling of the 2.004 Lab’s rotational system Analytical solution of the equation of motion for a 1 st –order system using the time domain Next lecture (Monday): Solution of the equations of motion in the Laplace domain ( s –domain). As we saw in lecture 1, the Equation of Motion of a mechanical system is, in general, an Ordinary Differential Equation (ODE). In this lecture, we will remind ourselves how to solve ODEs analytically in the time domain and in Matlab ( i.e. , numerically.) We will consider the following rotational system (plant) of a motor attached to a shaft with viscous and/or Coulomb friction: The motor applies torque T s ( t ) which is zero for t < 0 and increases to a step T 0 at t = 0. The shaft inertia is J while we assume that the motor inertia is negligible. We will also neglect the system compliance. (Justify these assumptions.) As input to the system we consider the torque, while the output is the angular velocity. We will consider three cases for the friction applied by the bearings: (i) viscous friction of coefficient f v b (units N m sec); (ii) Coulomb friction of magnitude f c · · (units N m sec); and (iii) both viscous and Coulomb friction. · · 1
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1. Viscous friction The system equation of motion is d ω J 1 J + = T s ( t ) ω ˙ + ω = T s ( t ) . (1) d t b b This is a linear 1 st –order ODE with constant coefficients. We are interested in the evolution of the system’s output (angular velocity) after application of the input (motor
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