c11 - 18.03 Class 11, March 3, 2006 Second order equations:...

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_____________ _____ 18.03 Class 11, March 3, 2006 Second order equations: Physical model, characteristic polynomial, real roots, structure of solutions, initial conditions [1] F = ma is the basic example. Take a spring attached to a wall, spring mass dashpot || || || || | |-------> F_ext | ___|___ | | | || || || || ||---VVVVVVV---| |------|_____| |-------|| || |_______| _____________| || || O | O || || | || |-------> | x Set up the coordinate system so that at x = 0 the spring is relaxed. The cart is influenced by three forces: the spring, the "dashpot" (which is a way to make friction explicit), and an external force: mx" = F_spr + F_dash + F_ext The spring force is characterized by depending only on position: write F_spr(x). 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 a linearizing approxmimation. The dashpot force is frictional. This means that it depends only on the velocity. Write F_dash(x'). It acts against the velocity: If x' > 0 , F_dash(x') < 0 If x' = 0 , F_dash(x') = 0 If x' < 0 , F_dash(x') > 0 The simplest way to model this behavior (and one which is valid in
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c11 - 18.03 Class 11, March 3, 2006 Second order equations:...

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