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class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> AHL Damped and driven oscillations <a href="#" class="mark-page-favorite pull-right" data-pid="1248" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../physics.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../849/engineering.html">Engineering</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">AHL Damped and driven oscillations</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../ahl-engineering/shatter.jpg" style="float: left; width: 250px; height: 141px;">In the <a href="../1117/simple-harmonic-motion.html" title="Simple harmonic motion">Simple harmonic motion</a> topic we considered natural oscillations of pendula and masses on springs. But so far we have ignored the effect of the fluid in which the oscillations are taking place and therefore the effects of damping. And what if these oscillations are forced? Resonance may ensue!</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div class="panel panel-turquoise panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Natural frequency</p> </div> </div> <div class="panel-body"> <div> <p>Pendula and masses on springs oscillate at a natural frequency when released in the absence of a driving or damping force. This is determined using the equations for time period previously studied (<a href="../1117/simple-harmonic-motion.html" title="Simple harmonic motion">Simple harmonic motion</a>) and then finding the reciprocal:</p> <p style="text-align: center;"><span class="math-tex">\(T_m=2\pi \sqrt{m\over k}\)</span></p> <ul> <li><span class="math-tex">\(T\)</span> is time period for a mass on a spring (s)</li> <li><span class="math-tex">\(m\)</span> is mass (kg)</li> <li><span class="math-tex">\(k\)</span> is spring constant (N m<sup>-1</sup>)</li> </ul> <p style="text-align: center;"><span class="math-tex">\(T_p=2\pi\sqrt{l\over g}\)</span></p> <ul> <li><span class="math-tex">\(T\)</span> is time period for a pendulum (s)</li> <li><span class="math-tex">\(l\)</span> is the length of the string</li> <li><span class="math-tex">\(g\)</span> is gravitational field strength (N kg<sup>-1</sup>)</li> </ul> <p style="text-align: center;"><span class="math-tex">\(f={1\over T}\)</span></p> <ul> <li><span class="math-tex">\(f\)</span> is frequency (Hz)</li> <li><span class="math-tex">\(T\)</span> is time period (s)</li> </ul> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/388883325"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Damping</p> </div> </div> <div class="panel-body"> <div> <p>Damping opposes oscillations. It is caused by the resistance of the fluid in which the oscillations take place. There are three types of damping:</p> <ul> <li>Under-damping - in which the body continues to oscillate with decreasing amplitude (may be light or heavy)</li> <li>Critical damping - in which the body returns to its equilibrium position in the shortest possible time without over-shooting</li> <li>Over-damping - in which the body returns to its equilibrium position slowly without over-shooting</li> </ul> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/388879880"></iframe></div> <p>In under-damped systems, the amplitude decreases according to an exponential decay relationship.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/388879892"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-turquoise panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Q factor</p> </div> </div> <div class="panel-body"> <div> <p>The Q (quality) factor for oscillations is a dimensionless quantity that indicates the extent of damping. It is calculated as follows:</p> <p style="text-align: center;"><span class="math-tex">\(Q=2\pi {\text{energy stored}\over \text{energy dissipated per cycle}}\)</span></p> <ul> <li><span class="math-tex">\(Q\)</span> is the Q factor (dimensionless)</li> <li>The energy stored is the energy at the beginning of a cycle (J)</li> <li>The energy dissipated per cycle is the energy lost in the system as a result of that same cycle (J)</li> </ul> <p>Critical damping occurs when <span class="math-tex">\(Q={1\over 2}\)</span>. A system with <span class="math-tex">\(Q>{1\over 2}\)</span> is under-damped and a system with <span class="math-tex">\(Q<{1\over 2}\)</span> is over-damped.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/388879973"></iframe></div> <p>Since the energy dissipated in one cycle is equal to the power loss multiplied by the time for that cycle:</p> <p style="text-align: center;"><span class="math-tex">\(Q=2\pi \times {\text{energy stored}\over \text{power loss}\times T}\)</span></p> <p style="text-align: center;"><span class="math-tex">\(\Rightarrow Q=2\pi \times \text{resonant frequency}\times {\text{energy stored}\over \text{power loss}}\)</span></p> <ul> <li><span class="math-tex">\(Q\)</span> is the Q factor (dimensionless)</li> <li>The resonant frequency is the frequency of the oscillations (Hz)</li> <li>The energy stored is the energy at the beginning of a cycle (J)</li> <li>The power loss is the energy lost per unit time during that same cycle (W)</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Driving frequency</p> </div> </div> <div class="panel-body"> <div> <p>Oscillations may be driven by a continuous force. Pushing a child on a swing is a close example, but not quite, as the pusher is not in constant contact with the swing. A better example is to hold the top of a spring on which a mass is hanging and to move your hand up and down.</p> <p>The driving frequency has significant effects on the amplitude of the oscillations:</p> <ul> <li>For low frequency movements, the whole system of driver and oscillating body moves as one. The driver and the oscillating body are in phase. The amplitude of the oscillations is equal to that of the driver.</li> <li>As the driving frequency approaches the natural frequency, the amplitude rises to a maximum. The driver is one quarter of a cycle ahead of the oscillating body (phase difference of <span class="math-tex">\(\pi\over 2\)</span>).</li> <li>For high frequency movements, the driver and the oscillating body move in antiphase. The amplitude decreases with increased frequency to approach zero.</li> </ul> <p style="text-align: center;"><img alt="" src="../../ahl-engineering/524px-resonant_frequency_amplitude.svg.png" style="width: 250px; height: 177px;"></p> <p>When the driver frequency matches the natural frequency of the oscillating body, it is at the resonant frequency. Resonance occurs.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/388883502"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Resonance</p> </div> </div> <div class="panel-body"> <div> <p>Forced oscillations and resonance are considered in electrical and civil engineering designs. In the latter, safety is the leading priority; there must be no risk of a driving frequency approximating the resonant frequency. In situations where the driving frequency is fixed, there are two ways to make civil structures safer:</p> <ol> <li>Alter the natural frequency. Ever seen someone stick chewing gum to their rear-view mirror to stop it shaking? The addition of the gum changes the mass of the mirror and thus the time period of its oscillations. The driving frequency due to the motion of the car is unaffected and therefore there is a larger difference between the driving frequency and natural frequency.</li> <li>Increase damping. Damping comes from the interaction of the oscillating body and the fluid in which it is oscillating, so a more viscous fluid or a larger surface area for the body will increase damping. This reduces the resonant frequency.</li> </ol> <p>Resonance has useful effects including in watch mechanisms, musical instruments and selection of frequencies in a radio. However, when destructive, bridges and buildings can be damaged due to the large amplitudes in a fixed quantity of material.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/388879987"></iframe></div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p><em>Use flashcards to practise your recall.</em></p> <div class="tib-flashcard"><a class="show-flashcards btn btn-success btn-xs-block btn-block " data-levels="3" data-mode="Normal" data-topics="1045" data-subject-id="6" data-n-flashcards="16" style="text-align:center">Show flashcards</a></div><hr> <p><em>Use quizzes to practise application of theory.</em></p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#606da684"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="606da684"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> AHL Damping and resonance <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="6-398-1248" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Under-damping causes an exponential decay in amplitude.</p><p>The equation for amplitude as a function of time is therefore of the form:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(A = A_0\text{e}^{kt}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(A = A_0\text{e}^{-{k\over t}}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(A = A_0\text{e}^{k\over t}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(A = A_0\text{e}^{-kt}\)</span></span></label></p></div><div class="q-explanation"><p>As <span class="math-tex">\(t\)</span> increases, <span class="math-tex">\(\text{e}^{-kt}\)</span> decreases.</p><p>Note: <span class="math-tex">\(\text{e}^{k\over t}\)</span> also causes a decay but is infinite when <span class="math-tex">\(t=0\)</span>.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A pendulum is made from a small steel ball and 30 cm of thin string.</p><p>The damping due to air resistance would be:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>over</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>under (light)</span></label> </p><p><label class="radio"> <input type="radio"> <span>critical</span></label> </p><p><label class="radio"> <input type="radio"> <span>under (heavy)</span></label> </p></div><div class="q-explanation"><p>Think of the real world! This pendulum is likely to swing for dozens of cycles.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A pendulum is made from a 20 cm diameter balloon and 30 cm of thin string.</p><p>The damping due to air resistance would be:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>critical</span></label> </p><p><label class="radio"> <input type="radio"> <span>over</span></label> </p><p><label class="radio"> <input type="radio"> <span>under (light)</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>under (heavy)</span></label> </p></div><div class="q-explanation"><p>The pendulum would probably swing for a couple of cycles.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The dampers in a car suspension system damp the oscillations of the springs.</p><p>The optimum level of damping is:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>over</span></label> </p><p><label class="radio"> <input type="radio"> <span>heavy</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>critical</span></label> </p><p><label class="radio"> <input type="radio"> <span>light</span></label> </p></div><div class="q-explanation"><p>The springs should return to their equilibrium position in the minimum amount of time, thus reducing the likelihood of damage to the car and discomfort for the passengers.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The simulation represents a damped pendulum.</p><p style="text-align: center;"><img alt="" height="242" class="gifffer" data-gifffer="/media/physics/qfac1.gif" width="343"></p><p>The Q factor is approximately:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>24</span></label> </p><p><label class="radio"> <input type="radio"> <span>100</span></label> </p><p><label class="radio"> <input type="radio"> <span>2</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>12</span></label> </p></div><div class="q-explanation"><p>Original <span class="math-tex">\(E_p = mg\times 0.2\)</span></p><p>After one cycle, <span class="math-tex">\(E_p = mg\times 0.1\)</span></p><p><span class="math-tex">\(Q = 2π \times {\text{energy stored}\over \text{energy lost per cycle}} = 4π\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>This PhET simulation represents a forced harmonic motion:</p><p style="text-align: center;"><img alt="" height="251" class="gifffer" data-gifffer="/media/physics/reso1.gif" width="355"></p><p>The driving frequency...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>is equal to resonant frequency</span></label> </p><p><label class="radio"> <input type="radio"> <span>could be less than or greater than the resonant frequency</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>is less than resonant frequency</span></label> </p><p><label class="radio"> <input type="radio"> <span>is greater than resonant frequency</span></label> </p></div><div class="q-explanation"><p>The driver and oscillator are in phase.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>This PhET simulation represents a forced harmonic motion.</p><p style="text-align: center;"><img alt="" height="247" class="gifffer" data-gifffer="/media/physics/reso2.gif" width="350"></p><p>The driving frequency...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>could be less than or greater than the resonant frequency</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>is greater than resonant frequency</span></label> </p><p><label class="radio"> <input type="radio"> <span>is equal to resonant frequency</span></label> </p><p><label class="radio"> <input type="radio"> <span>is less than resonant frequency</span></label> </p></div><div class="q-explanation"><p>The driver and oscillator are out of phase.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Two equal masses on springs are driven by an oscillator.</p><p style="text-align: center;"><img alt="" height="241" class="gifffer" data-gifffer="/media/physics/reso4.gif" width="184"></p><p>Mass 2 is resonating. What change could make mass 1 resonate?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>increase the frequency</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>reduce the mass</span></label> </p><p><label class="radio"> <input type="radio"> <span>increase the amplitude</span></label> </p><p><label class="radio"> <input type="radio"> <span>reduce the spring constant</span></label> </p></div><div class="q-explanation"><p>Mass 1 is <span class="math-tex">\(\pi\)</span> out of phase with the driver so the driving frequency is exceeding the natural frequency. We must therefore find a way of increasing the natural frequency. Since time period is linked to <span class="math-tex">\(m\over k\)</span>, reducing <span class="math-tex">\(m\)</span> will reduce time period and increase natural frequency.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>A mass on a spring stops resonating when the damping is increased.</p><p style="text-align: center;"><img alt="" height="238" class="gifffer" data-gifffer="/media/physics/-reo5.gif" width="182"></p><p>Resonance could be acheived again by:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>decreasing the amplitude</span></label> </p><p><label class="radio"> <input type="radio"> <span>increasing the driving frequency</span></label> </p><p><label class="radio"> <input type="radio"> <span>increasing the amplitude</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>reducing the driving frequency</span></label> </p></div><div class="q-explanation"><p>Damping reduces the resonant frequency; we must therefore reduce the driving frequency to approach this.</p><p>Note: amplitude of the driver does not affect whether or not the oscillator is at resonance.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is not an example of resonance?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>A note is produced when air is blown across the top of a bottle.</span></label> </p><p><label class="radio"> <input type="radio"> <span>Plates rattle when a truck drives by.</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>A tuning fork sounds louder when the end is held on a table.</span></label> </p><p><label class="radio"> <input type="radio"> <span>An opera singer shatters a glass with their voice.</span></label> </p></div><div class="q-explanation"><p>The tuning fork sounds louder because of the increased vibrating area, not resonance.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button 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