MIT18_03S10_c11

MIT18_03S10_c11 - 18.03 Class 11, Feb 26, 2010 Second order...

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18.03 Class 11 , Feb 26, 2010 Second order linear equations: Physical model, solutions in homogeneous case. Characteristic polynomial, distinct real roots. [1] Springs and masses [2] Dashpots [3] Second order linear equations [4] Solutions in homogeneous case: Superposition I [5] Exponential solutions: characteristic polynomial [1] Second order equations are the basis of analysis of mechanical and electrical systems. We'll build this up slowly. A spring is attached to a wall and a cart: spring mass | || |-------> F_ext || | || ___|___ || | | ||---VVVVVVV---| | || |_______| || O | O || | |-------> | x Set up the coordinate system so that at x = 0 the spring is relaxed, which means that it is exerting no force. In addition to the spring, suppose that there is another force acting on the cart -- an "external force," maybe wind blowing on a sail attached to it, maybe gravity, or some other force. Then mx" = F_spr + F_ext The spring force is characterized by the fact that it depends only on position. In fact: If x > 0 , F_spr(x) < 0 If x = 0 , F_spr(x) = 0 If x < 0 , F_spr(x) > 0 I sketched a graph of F_spr(x) as a function of x . The simplest way to model this behavior (and one which is valid in general for small x , by the tangent line approximation) is F_spr(x) = -kx k > 0 the "spring constant." "Hooke's Law" This is another example of the linear approximation that Linn was
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discussing on Monday. So we get mx" + kx = F_ext . I displayed a weight on a rubber band. This is not a spring, as you usually think of one, but it behaves like one, at least in a range. Lay a rubber band laid out on a table. Fix the right end of it and set x = 0 where the left end is in a relaxed state, then the graph of the force exerted by the rubber band looks something like this . .. | | || .------- fastened here || | || linear spring | || | | || | | .---- end of unstretched band | \ | | | | \ /__/ | | | \ \ V V | \ | \ *=========* | \ ______________| \__________.__________ ________________ \ | \ | \ | linear spring -----> \ | \ | \ | \ | || || || || || | | | | ^ | | the rubber band breaks! ----- [2] Any real mechanical system has friction. Friction takes many forms.
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This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c11 - 18.03 Class 11, Feb 26, 2010 Second order...

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