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      <text>19.4.2.1  Modelling free mechanical oscillations</text>
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      <text><![CDATA[<h3>Modelling free mechanical oscillations<br></h3>

<p>Consider the following schematic diagram of a shock absorber:</p>

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    <img src="@@PLUGINFILE@@/fig3.svg" alt="Image of a shock absorber. A spring is attached to a dashpot and a mass is on top." width="150" height="213" class="atto_image_button_text-bottom">
</figure>

<p>The equation of motion can be described in terms of the vertical displacement \(x\) of the mass.</p>
<p>Let \(m\) be the mass, \(k \dfrac{dx}{dt}\) the damping force resulting from the dashpot and \(nx\) the restoring force resulting from the spring. Here, \(k\) and \(n\) are constants.</p>
<p>Then the equation of motion is</p>
<p>\(\qquad m\dfrac{d^2x}{dt^2} = -k\dfrac{dx}{dt} -nx.\)</p>

<p>Suppose that the mass is displaced a distance \(x_0\) initially and released from rest. Then at \(t=0\), \(x=x_0\) and \(\dfrac{dx}{dt}=0.\) Writing the differential equation in standard form gives</p>
<p>\(\qquad m\dfrac{d^2x}{dt^2} + k\dfrac{dx}{dt}+nx = 0.\)</p>
<p>We shall see that the nature of the oscillations described by this differential equation depends crucially upon the relative values of the mechanical constants \(m,k\) and \(n.\) This will be explored in subsequent Tasks.</p>

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<h4 class="HELM_example">Example</h4>
<p>Find and solve the auxiliary equation of the differential equation \[m{d^2x\over dt^2}+k{dx\over dt}+nx=0.\]</p>

<h4 class="HELM_solution">Solution</h4>
<p>Putting \(x=\text{e}^{\lambda t}\), the auxiliary equation is</p>
<p>\(\quad m\,\lambda^2+k\,\lambda+n=0.\)</p>
<p>Hence \(\lambda=\dfrac{-k\pm\sqrt{k^2-4m\,n}}{2m}.\)</p>

<p><br></p>
<p>The value of \(k\) controls the amount of damping in the system. We explore the solution for various values of \(k.\)</p>]]></text>
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