chapter6.page12

chapter6.page12 - 00 kx = f t models the movement of a mass...

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12 1. LAPLACE TRANSFORM METHODS In modelling the movement of a baseball after being hit by bat. In spring-mass system, where a object with given mass it at- tached to one end of a spring whose other end is attached to a steady object (such as wall), modelling the movement when hits the object with a hammer. Modelling the current in a closed circuit when the switch is turn on and off instantly. It turns out that we can stretch ourself a little and compute the Laplace transform of δ a ( t ) . Let δ ( t ) = δ 0 ( t ) then δ a ( t ) = δ ( t - a ) . Theorem 4.1 . b δ ( s ) = 1 and b δ a ( s ) = e - as . Example 4.1 . A mass m = 1 is attached to a spring with constant k = 4 ; there is not dashpot. The mass is released from the rest with x (0) = 3 . At the instant t = π the mass is struck with a hammer, providing an impulse p = 8 . Determine the motion of the mass. Solution In general mx
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Unformatted text preview: 00 + kx = f ( t ) models the movement of a mass attached to the end of spring with no dashpot. So we need to solve x 00 + 4 x = 8 δ π ( t ) , x (0) = 3 ,x (0) = 0 . Applying Laplace transform to both sides of the equation, b x 00 ( s ) + 4 b x ( s ) = 8 δ π ( s ) s 2 b x-sx (0)-x (0) + 4 b x ( s ) = 8 e-πs ( s 2 + 4) b x ( s )-3 s = 8 e-πs Solve for b x ( s ), we get b x ( s ) = 3 s s 2 + 4 + 8 s 2 + 4 e-πs Using Mathcad , we can find inverse Laplace transform, x ( t ) = 2 cos(2 t ) + 4 u π ( t )sin(2 t ) = ‰ 2cos(2 t ) if 0 ≤ t < π 2cos(2 t ) + 4sin(2 t ) if t ≥ π So clearly the impact of the impulse is felt by the spring and change the its movement immediately. The solution curve is shown below, a...
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