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Laplace Transforms: Mass on springs

Question:


A particle of mass m is attached between two horizontal springs of stiffness 2k and 5k, each of un-stretched length a (see figure below). The mass is held stationary at a displacement from its resting position of x = 1.03m.


Show that the system satisfies the differential equation given by:




and use Laplace transforms of derivatives to find L{x}.


Answer:


Resolving horizontally (+ve direction from left to right):



Take Laplace transforms of both sides:



Sub in Laplace transform of derivatives:



Sub in initial conditions x0 and x'0:



 

Question:

Determine the poles and zeros of the system and plot them on a pole-zero diagram. Use the initial value theorem to check your Laplace Transform.


Answer:

Find the Poles:


Find the Zeros:



Draw the Pole-Zero diagram:


Determine initial values:



We know this is correct as the displacement at time = 0 was known.


 

Question:

By using tables to find the Inverse Laplace Transform, determine the equation for the displacement x(t)


Inverse Laplace:


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