Question $212$ marks

Figure $3$ shows a single mass that can only move horizontally, in one plane. The mass does not have any elastic or damping element that ties it to the ground. The mass is acted on by a spring with stiffness $k,$ the far end of which has a prescribed displacement $y(t).$ The system has properties $m=40kg,k=4000\frac{N}{m}.$ The forcing parameters are $Y_o=0\mathrm{.}025m$ and $=4ra\frac{d}{s}.$

Figure $3$: Single degree of freedom system

$($a$)$ For the case where $y(t)=Y_{o}sin(t),$ draw the Free Body Diagram for the mass

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and hence find the equation of motion. Write the differential equation of motion symbolically $($in terms of $k,m,Y_o$ etc...$).$

$($b$)$ If the mass has zero initial displacement but an initial velocity of $x_0^{}=0\mathrm{.}5\frac{m}{s},$ find

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the Laplace Transform of the displacement response $x(s).$ Write $x(s)$ symbolically in terms of $k,m,x_0^{}$ etc. Do not perform the Partial Fraction Decomposition at this stage.

$($c$)$ The system has properties: $m=40kg$ and $k=4000\frac{N}{m}.$ The forcing parameters

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are $Y_o=0\mathrm{.}025m$ and $=4ra\frac{d}{s}.$ Using the tables of Partial Fraction Decompositions, re$-$write $x(s)$ as fully decomposed partial fractions. Use the Inverse Laplace Transform to find $x(t).$