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MassSpring

# MassSpring - The Mass-Spring System Math 308 This Maple...

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The Mass-Spring System Math 308 This Maple session uses the mass-spring system to demonstrate the phase plane, direction fields, solution curves (``trajectories''), the extended phase space, and more. The differential equation is = + + m ° ± ² ² ³ ´ µ µ 2 t 2 ( ) y t b ° ± ² ² ³ ´ µ µ t ( ) y t k ( ) y t 0 (I've used b instead of the usual γ for the coefficient of friction.) We convert this to a system by defining = ( ) v t t ( ) y t Then = t ( ) v t - - k ( ) y t m b ° ± ² ² ³ ´ µ µ t ( ) y t m or = t ( ) v t - - k ( ) y t m b ( ) v t m This equation along with the equation that defines v(t) make up a system of two equations. > with(DEtools): We first define the two equations that make up the system: > eq1 := diff(y(t),t) = v(t); := eq1 = t ( ) y t ( ) v t > eq2 := diff(v(t),t) = -(k/m)*y(t)-(b/m)*v(t); := eq2 = t ( ) v t - - k ( ) y t m b ( ) v t m Now define a variable that holds the system: > sys := [eq1,eq2];

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:= sys · ¸ ¸ ¹ º » » , = t ( ) y t ( ) v t = t ( ) v t - - k ( ) y t m b ( ) v t m We substitute in some numerical values for the parameters, so we can plot some solutions.
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MassSpring - The Mass-Spring System Math 308 This Maple...

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