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:

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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.

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Question:

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

Inverse Laplace: