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chapter6.page10

chapter6.page10 - f t models a spring system that with one...

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10 1. LAPLACE TRANSFORM METHODS 3.2. Solve ODE With Piecewise Input Function. Example 3.3 . Find solution to x 00 + 4 x = f ( t ) , x (0) = 0 , x 0 (0) = - 1 , where f ( t ) = cos(2 t ) if 0 t < π 0 if t π Solution First we express f ( t ) as linear combination of step function, f ( t ) = cos(2 t )( u ( t ) - u 2 π ( t )) Apply Laplace transform to both sides of the equation and using Mathcad we get ( s 2 + 4) b x ( s ) = - 1 + (1 - e - πs ) s s 2 + 4 b x ( s ) = - 1 s 2 + 4 + (1 - e - πs ) s ( s 2 + 4) 2 Apply inverse Laplace transform, we get, x ( t ) = - 1 2 sin(2 t ) + 1 4 t sin(2 t ) if 0 t < π 0 if π - 2 4 sin(2 t ) π a The equation x 00
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Unformatted text preview: f ( t ) models a spring system that with one spring, one end of the spring is fixed and a object with 1 unit mass attached at one end. The Hook’s constant k = 4 . f ( t ) is external force applied to the system. In the Example 3.3, the force sin(2 t ) only applied at first π unit of time (second). The following diagram compares the solution of this equation with different input f ( t ) ....
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