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Click here for more information."><img alt="This is a good article. Click here for more information." src="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/19px-Symbol_support_vote.svg.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/29px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/94/Symbol_support_vote.svg/39px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Analogy representing a mechanical system by an electrical one</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Analogy_(disambiguation)" class="mw-disambig" title="Analogy (disambiguation)">Analogy (disambiguation)</a>.</div> <p> The <b>mobility analogy, </b>also called <b>admittance analogy</b> or <b>Firestone analogy</b>, is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of <a href="/wiki/Electronic_filter" title="Electronic filter">filters</a>.<sup id="cite_ref-Talbot186_1-0" class="reference"><a href="#cite_note-Talbot186-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in <a href="/wiki/Electromechanics" title="Electromechanics">electromechanical systems</a>: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole. </p><p>The mathematical behaviour of the simulated electrical system is identical to the mathematical behaviour of the represented mechanical system. Each <a href="/wiki/Electrical_element" title="Electrical element">element</a> in the electrical domain has a corresponding element in the mechanical domain with an analogous <a href="/wiki/Constitutive_equation" title="Constitutive equation">constitutive equation</a>. All laws of <a href="/wiki/Circuit_analysis" class="mw-redirect" title="Circuit analysis">circuit analysis</a>, such as <a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's laws</a>, that apply in the electrical domain also apply to the mechanical mobility analogy. </p><p>The mobility analogy is one of the two main <a href="/wiki/Mechanical%E2%80%93electrical_analogies" title="Mechanical–electrical analogies">mechanical–electrical analogies</a> used for representing mechanical systems in the electrical domain, the other being the <a href="/wiki/Impedance_analogy" title="Impedance analogy">impedance analogy</a>. The roles of voltage and current are reversed in these two methods, and the electrical representations produced are the <a href="/wiki/Dual_impedance" title="Dual impedance">dual circuits</a> of each other. The mobility analogy preserves the topology of the mechanical system when transferred to the electrical domain whereas the impedance analogy does not. On the other hand, the impedance analogy preserves the analogy between <a href="/wiki/Electrical_impedance" title="Electrical impedance">electrical impedance</a> and <a href="/wiki/Mechanical_impedance" title="Mechanical impedance">mechanical impedance</a> whereas the mobility analogy does not. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=1" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mobility analogy is widely used to model the behaviour of <a href="/wiki/Mechanical_filter" title="Mechanical filter">mechanical filters</a>. These are filters that are intended for use in an electronic circuit, but work entirely by mechanical vibrational waves. <a href="/wiki/Transducer" title="Transducer">Transducers</a> are provided at the input and output of the filter to convert between the electrical and mechanical domains.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Another very common use is in the field of audio equipment, such as loudspeakers. Loudspeakers consist of a transducer and mechanical moving parts. Acoustic waves themselves are waves of mechanical motion: of air molecules or some other fluid medium.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Elements">Elements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=2" title="Edit section: Elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Before an electrical analogy can be developed for a mechanical system, it must first be described as an abstract <a href="/wiki/Mechanical_network" title="Mechanical network">mechanical network</a>. The mechanical system is broken down into a number of ideal elements each of which can then be paired with an electrical analogue.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The symbols used for these mechanical elements on network diagrams are shown in the following sections on each individual element. </p><p>The mechanical analogies of lumped <a href="/wiki/Electrical_element" title="Electrical element">electrical elements</a> are also <a href="/wiki/Lumped_elements" class="mw-redirect" title="Lumped elements">lumped elements</a>, that is, it is assumed that the mechanical component possessing the element is small enough that the time taken by <a href="/wiki/Mechanical_wave" title="Mechanical wave">mechanical waves</a> to propagate from one end of the component to the other can be neglected. Analogies can also be developed for <a href="/wiki/Distributed_elements" class="mw-redirect" title="Distributed elements">distributed elements</a> such as <a href="/wiki/Transmission_line" title="Transmission line">transmission lines</a> but the greatest benefits are with lumped-element circuits. Mechanical analogies are required for the three passive electrical elements, namely, <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a>, <a href="/wiki/Inductance" title="Inductance">inductance</a> and <a href="/wiki/Capacitance" title="Capacitance">capacitance</a>. What these analogies are is determined by what mechanical property is chosen to represent <a href="/wiki/Voltage" title="Voltage">voltage</a>, and what property is chosen to represent <a href="/wiki/Electric_current" title="Electric current">current</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In the mobility analogy the analogue of voltage is <a href="/wiki/Velocity" title="Velocity">velocity</a> and the analogue of current is <a href="/wiki/Force" title="Force">force</a>.<sup id="cite_ref-Eargle,_p._5_6-0" class="reference"><a href="#cite_note-Eargle,_p._5-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Mechanical_impedance" title="Mechanical impedance">Mechanical impedance</a> is defined as the ratio of force to velocity, thus it is not analogous to <a href="/wiki/Electrical_impedance" title="Electrical impedance">electrical impedance</a>. Rather, it is the analogue of <a href="/wiki/Admittance" title="Admittance">electrical admittance</a>, the inverse of impedance. Mechanical admittance is more commonly called mobility,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> hence the name of the analogy.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Resistance">Resistance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=3" title="Edit section: Resistance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mobility_analogy_resistor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Mobility_analogy_resistor.svg/170px-Mobility_analogy_resistor.svg.png" decoding="async" width="170" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Mobility_analogy_resistor.svg/255px-Mobility_analogy_resistor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Mobility_analogy_resistor.svg/340px-Mobility_analogy_resistor.svg.png 2x" data-file-width="287" data-file-height="304" /></a><figcaption>The mechanical symbol for a damper (left) and its electrical analogy (right).<sup id="cite_ref-Eargle4_9-0" class="reference"><a href="#cite_note-Eargle4-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The symbol is meant to be evocative of a <a href="/wiki/Dashpot" title="Dashpot">dashpot</a>.<sup id="cite_ref-Kleiner71_10-0" class="reference"><a href="#cite_note-Kleiner71-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The mechanical analogy of electrical resistance is the loss of energy of a moving system through such processes as <a href="/wiki/Friction" title="Friction">friction</a>. A mechanical component analogous to a <a href="/wiki/Resistor" title="Resistor">resistor</a> is a <a href="/wiki/Shock_absorber" title="Shock absorber">shock absorber</a> and the property analogous to inverse resistance (conductance) is <a href="/wiki/Damping" title="Damping">damping</a> (inverse, because electrical impedance is the analogy of the inverse of mechanical impedance). A resistor is governed by the constitutive equation of <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=vG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mi>v</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=vG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08f6a7f277f853e13d7a462c6320593a2e0db31c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.855ex; height:2.176ex;" alt="{\displaystyle i=vG}"></span></dd></dl> <p>The analogous equation in the mechanical domain is, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=uR_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>u</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=uR_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/913e29b4bda6dc6a4b0933257a1081c225128f03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.534ex; height:2.509ex;" alt="{\displaystyle F=uR_{\mathrm {m} }}"></span></dd></dl> <dl><dd>where,</dd> <dd><i>G</i> = 1/<i>R</i> is conductance</dd> <dd><i>R</i> is resistance</dd> <dd><i>v</i> is voltage</dd> <dd><i>i</i> is current</dd> <dd><i>R</i><sub>m</sub> is mechanical resistance, or damping</dd> <dd><i>F</i> is force</dd> <dd><i>u</i> is velocity induced by the force.<sup id="cite_ref-Kleiner71_10-1" class="reference"><a href="#cite_note-Kleiner71-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Electrical conductance represents the <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> of <a href="/wiki/Admittance" title="Admittance">electrical admittance</a>. Likewise, mechanical resistance is the real part of <a href="/wiki/Mechanical_impedance" title="Mechanical impedance">mechanical impedance</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Inductance">Inductance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=4" title="Edit section: Inductance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mobility_analogy_inductor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Mobility_analogy_inductor.svg/170px-Mobility_analogy_inductor.svg.png" decoding="async" width="170" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Mobility_analogy_inductor.svg/255px-Mobility_analogy_inductor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Mobility_analogy_inductor.svg/340px-Mobility_analogy_inductor.svg.png 2x" data-file-width="287" data-file-height="304" /></a><figcaption>The mechanical symbol for a compliance element (left) and its electrical analogy (right).<sup id="cite_ref-Eargle,_p._5_6-1" class="reference"><a href="#cite_note-Eargle,_p._5-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The symbol is meant to be evocative of a spring.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The mechanical analogy of inductance in the mobility analogy is compliance. It is more common in mechanics to discuss <a href="/wiki/Stiffness" title="Stiffness">stiffness</a>, the inverse of compliance. A mechanical component analogous to an <a href="/wiki/Inductor" title="Inductor">inductor</a> is a <a href="/wiki/Spring_(device)" title="Spring (device)">spring</a>. An inductor is governed by the constitutive equation, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=L{\frac {di}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>i</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=L{\frac {di}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6eee1371ecb78ac96b11af80775099c8484b36a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.701ex; height:5.509ex;" alt="{\displaystyle v=L{\frac {di}{dt}}}"></span></dd></dl> <p>The analogous equation in the mechanical domain is a form of <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=C_{\mathrm {m} }{\frac {dF}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>F</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=C_{\mathrm {m} }{\frac {dF}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90feaef76319add49a6091b216a9f8dce7dff090" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.484ex; height:5.509ex;" alt="{\displaystyle u=C_{\mathrm {m} }{\frac {dF}{dt}}}"></span></dd></dl> <dl><dd>where,</dd> <dd><i>L</i> is inductance</dd> <dd><i>t</i> is time</dd> <dd><i>C</i><sub>m</sub> = 1/<i>S</i> is mechanical compliance</dd> <dd><i>S</i> is stiffness<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The impedance of an inductor is purely <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary</a> and is given by, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=j\omega L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>j</mi> <mi>ω</mi> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=j\omega L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da026a80b88331acf5708dfe34bb6b9b419772b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.766ex; height:2.509ex;" alt="{\displaystyle Z=j\omega L}"></span></dd></dl> <p>The analogous mechanical admittance is given by, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\mathrm {m} }=j\omega C_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>j</mi> <mi>ω</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{\mathrm {m} }=j\omega C_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/536bf3b4e905c2e17449a7987c21c3aeb8d6393f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.717ex; height:2.509ex;" alt="{\displaystyle Y_{\mathrm {m} }=j\omega C_{\mathrm {m} }}"></span></dd></dl> <dl><dd>where,</dd> <dd><i>Z</i> is electrical impedance</dd> <dd><i>j</i> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a></dd> <dd><i>ω</i> is <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a></dd> <dd><i>Y</i><sub>m</sub> is mechanical admittance.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Capacitance">Capacitance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=5" title="Edit section: Capacitance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mobility_analogy_capacitor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Mobility_analogy_capacitor.svg/170px-Mobility_analogy_capacitor.svg.png" decoding="async" width="170" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Mobility_analogy_capacitor.svg/255px-Mobility_analogy_capacitor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Mobility_analogy_capacitor.svg/340px-Mobility_analogy_capacitor.svg.png 2x" data-file-width="313" data-file-height="305" /></a><figcaption>The mechanical symbol for a mass (left) and its electrical analogy (right).<sup id="cite_ref-Eargle,_p._5_6-2" class="reference"><a href="#cite_note-Eargle,_p._5-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The square angle below the mass is meant to indicate that movement of the mass is relative to a frame of reference.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The mechanical analogy of capacitance in the mobility analogy is <a href="/wiki/Mass" title="Mass">mass</a>. A mechanical component analogous to a <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> is a large, rigid weight or a mechanical <a href="/wiki/Inerter_(mechanical_networks)" title="Inerter (mechanical networks)">Inerter</a>. </p><p>A capacitor is governed by the constitutive equation, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=C{\frac {dv}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>v</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=C{\frac {dv}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba97f795e58f5009bad0a5491bb6c3987da14d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.847ex; height:5.509ex;" alt="{\displaystyle i=C{\frac {dv}{dt}}}"></span></dd></dl> <p>The analogous equation in the mechanical domain is <a href="/wiki/Newton%27s_second_law_of_motion" class="mw-redirect" title="Newton's second law of motion">Newton's second law of motion</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=M{\frac {du}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=M{\frac {du}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc430a1b528a5df8353645d558ea055ed2e8379e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.663ex; height:5.509ex;" alt="{\displaystyle F=M{\frac {du}{dt}}}"></span></dd></dl> <dl><dd>where,</dd> <dd><i>C</i> is capacitance</dd> <dd><i>M</i> is mass</dd></dl> <p>The impedance of a capacitor is purely imaginary and is given by, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z={1 \over j\omega C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>j</mi> <mi>ω</mi> <mi>C</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z={1 \over j\omega C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87c26d09bc2b50cbb1a34b96590bc00a563ebe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.785ex; height:5.676ex;" alt="{\displaystyle Z={1 \over j\omega C}}"></span></dd></dl> <p>The analogous mechanical admittance is given by, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{\mathrm {m} }={1 \over j\omega M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>j</mi> <mi>ω</mi> <mi>M</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{\mathrm {m} }={1 \over j\omega M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5e193f2358ae6d0f6c4a8208123e5afb3d4b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.732ex; height:5.676ex;" alt="{\displaystyle Y_{\mathrm {m} }={1 \over j\omega M}}"></span>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Inertance">Inertance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=6" title="Edit section: Inertance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A curious difficulty arises with mass as the analogy of an electrical element. It is connected with the fact that in mechanical systems the velocity of the mass (and more importantly, its acceleration) is always measured against some fixed reference frame, usually the earth. Considered as a two-terminal system element, the mass has one terminal at velocity ''u'', analogous to electric potential. The other terminal is at zero velocity and is analogous to electric ground potential. Thus, mass cannot be used as the analogue of an ungrounded capacitor.<sup id="cite_ref-Busch-Vishniac,_p._20_17-0" class="reference"><a href="#cite_note-Busch-Vishniac,_p._20-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>This led <a href="/wiki/Malcolm_C._Smith" title="Malcolm C. Smith">Malcolm C. Smith</a> of the <a href="/wiki/University_of_Cambridge" title="University of Cambridge">University of Cambridge</a> in 2002 to define a new energy storing element for mechanical networks called <i>inertance</i>. A component that possesses inertance is called an <a href="/wiki/Inerter_(mechanical_networks)" title="Inerter (mechanical networks)">inerter</a>. The two terminals of an inerter, unlike a mass, are allowed to have two different, arbitrary velocities and accelerations. The constitutive equation of an inerter is given by,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=B\left({\frac {du_{\mathrm {2} }}{dt}}-{\frac {du_{\mathrm {1} }}{dt}}\right)=B{\frac {d\Delta u}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>B</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="normal">Δ</mi> <mi>u</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=B\left({\frac {du_{\mathrm {2} }}{dt}}-{\frac {du_{\mathrm {1} }}{dt}}\right)=B{\frac {d\Delta u}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f8f96aee3295c156029e87f9ca66e199be05b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.304ex; height:6.176ex;" alt="{\displaystyle F=B\left({\frac {du_{\mathrm {2} }}{dt}}-{\frac {du_{\mathrm {1} }}{dt}}\right)=B{\frac {d\Delta u}{dt}}}"></span></dd></dl> <dl><dd>where,</dd> <dd><i>F</i> is an equal and opposite force applied to the two terminals</dd> <dd><i>B</i> is the inertance</dd> <dd><i>u</i><sub>1</sub> and <i>u</i><sub>2</sub> are the velocities at terminals 1 and 2 respectively</dd> <dd>Δ<i>u</i> = <i>u</i><sub>2</sub> − <i>u</i><sub>1</sub></dd></dl> <p>Inertance has the same units as mass (kilograms in the <a href="/wiki/SI_system" class="mw-redirect" title="SI system">SI system</a>) and the name indicates its relationship to <a href="/wiki/Inertia" title="Inertia">inertia</a>. Smith did not just define a network theoretic element, he also suggested a construction for a real mechanical component and made a small prototype. Smith's inerter consists of a plunger able to slide in or out of a cylinder. The plunger is connected to a <a href="/wiki/Rack_and_pinion" title="Rack and pinion">rack and pinion</a> gear which drives a flywheel inside the cylinder. There can be two counter-rotating flywheels in order to prevent a <a href="/wiki/Torque" title="Torque">torque</a> developing. Energy provided in pushing the plunger in will be returned when the plunger moves in the opposite direction, hence the device stores energy rather than dissipates it just like a block of mass. However, the actual mass of the inerter can be very small, an ideal inerter has no mass. Two points on the inerter, the plunger and the cylinder case, can be independently connected to other parts of the mechanical system with neither of them necessarily connected to ground.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>Smith's inerter has found an application in <a href="/wiki/Formula_One" title="Formula One">Formula One</a> racing where it is known as the J-damper. It is used as an alternative to the now banned <a href="/wiki/Tuned_mass_damper" title="Tuned mass damper">tuned mass damper</a> and forms part of the vehicle suspension. It may have been first used secretly by <a href="/wiki/McLaren" title="McLaren">McLaren</a> in 2005 following a collaboration with Smith. Other teams are now believed to be using it. The inerter is much smaller than the tuned mass damper and smoothes out <a href="/wiki/Contact_patch" title="Contact patch">contact patch</a> load variations on the tyres.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Smith also suggests using the inerter to reduce machine vibration.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>The difficulty with mass in mechanical analogies is not limited to the mobility analogy. A corresponding problem also occurs in the impedance analogy, but in that case it is ungrounded inductors, rather than capacitors, that cannot be represented with the standard elements.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Resonator">Resonator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=7" title="Edit section: Resonator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A mechanical resonator consists of both a mass element and a compliance element. Mechanical resonators are analogous to electrical <a href="/wiki/LC_circuit" title="LC circuit">LC circuits</a> consisting of inductance and capacitance. Real mechanical components unavoidably have both mass and compliance so it is a practical proposition to make resonators as a single component. In fact, it is more difficult to make a pure mass or pure compliance as a single component. A spring can be made with a certain compliance and mass minimised, or a mass can be made with compliance minimised, but neither can be eliminated altogether. Mechanical resonators are a key component of mechanical filters.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Generators">Generators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=8" title="Edit section: Generators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mobility_analogy_voltage.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Mobility_analogy_voltage.svg/170px-Mobility_analogy_voltage.svg.png" decoding="async" width="170" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Mobility_analogy_voltage.svg/255px-Mobility_analogy_voltage.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Mobility_analogy_voltage.svg/340px-Mobility_analogy_voltage.svg.png 2x" data-file-width="293" data-file-height="305" /></a><figcaption>The mechanical symbol for a constant velocity generator (left) and its electrical analogy (right)<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mobility_analogy_current.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Mobility_analogy_current.svg/170px-Mobility_analogy_current.svg.png" decoding="async" width="170" height="177" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Mobility_analogy_current.svg/255px-Mobility_analogy_current.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Mobility_analogy_current.svg/340px-Mobility_analogy_current.svg.png 2x" data-file-width="293" data-file-height="305" /></a><figcaption>The mechanical symbol for a constant force generator (left) and its electrical analogy (right)<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Analogues exist for the active electrical elements of the <a href="/wiki/Voltage_source" title="Voltage source">voltage source</a> and the <a href="/wiki/Current_source" title="Current source">current source</a> (generators). The mechanical analogue in the mobility analogy of the constant current generator is the constant force generator. The mechanical analogue of the constant voltage generator is the constant velocity generator.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>An example of a constant force generator is the <a href="/wiki/Constant-force_spring" title="Constant-force spring">constant-force spring</a>. An example of a practical constant velocity generator is a lightly loaded powerful machine, such as a <a href="/wiki/Electric_motor" title="Electric motor">motor</a>, driving a <a href="/wiki/Belt_(mechanical)" title="Belt (mechanical)">belt</a>. This is analogous to a real voltage source, such as a battery, which remains near constant-voltage with load provided that the load resistance is much higher than the battery internal resistance.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Transducers">Transducers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=9" title="Edit section: Transducers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Electromechanics" title="Electromechanics">Electromechanical systems</a> require <a href="/wiki/Transducer" title="Transducer">transducers</a> to convert between the electrical and mechanical domains. They are analogous to <a href="/wiki/Two-port_network" title="Two-port network">two-port networks</a> and like those can be described by a pair of simultaneous equations and four arbitrary parameters. There are numerous possible representations, but the form most applicable to the mobility analogy has the arbitrary parameters in units of admittance. In matrix form (with the electrical side taken as port 1) this representation is, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}i\\u\end{bmatrix}}={\begin{bmatrix}y_{11}&y_{12}\\y_{21}&y_{22}\end{bmatrix}}{\begin{bmatrix}v\\F\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>v</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}i\\u\end{bmatrix}}={\begin{bmatrix}y_{11}&y_{12}\\y_{21}&y_{22}\end{bmatrix}}{\begin{bmatrix}v\\F\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48426dc55f987ff859802a7dbf445d6c6a991a86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.142ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}i\\u\end{bmatrix}}={\begin{bmatrix}y_{11}&y_{12}\\y_{21}&y_{22}\end{bmatrix}}{\begin{bmatrix}v\\F\end{bmatrix}}}"></span></dd></dl> <p>The element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{22}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{22}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d63a19b3737fdae896d66a5afda794660c56d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.403ex; height:2.009ex;" alt="{\displaystyle y_{22}\,}"></span> is the short circuit mechanical admittance, that is, the admittance presented by the mechanical side of the transducer when zero voltage (short circuit) is applied to the electrical side. The element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{11}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{11}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae1fbb75c823ff4255faaf60b273e0b1718306f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.403ex; height:2.009ex;" alt="{\displaystyle y_{11}\,}"></span>, conversely, is the unloaded electrical admittance, that is, the admittance presented to the electrical side when the mechanical side is not driving a load (zero force). The remaining two elements, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{21}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{21}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b4ca2f53580fe4b6032e97d3819b6c7e3ce8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.403ex; height:2.009ex;" alt="{\displaystyle y_{21}\,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{12}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{12}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d8581c05272d197f39cb3214f86b8687e6fed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.403ex; height:2.009ex;" alt="{\displaystyle y_{12}\,}"></span>, describe the transducer forward and reverse transfer functions respectively. They are both analogous to <a href="/wiki/Transconductance" title="Transconductance">transfer admittances</a> and are hybrid ratios of an electrical and mechanical quantity.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Transformers">Transformers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=10" title="Edit section: Transformers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mechanical analogy of a <a href="/wiki/Transformer" title="Transformer">transformer</a> is a <a href="/wiki/Simple_machine" title="Simple machine">simple machine</a> such as a <a href="/wiki/Pulley" title="Pulley">pulley</a> or a <a href="/wiki/Lever" title="Lever">lever</a>. The force applied to the load can be greater or less than the input force depending on whether the <a href="/wiki/Mechanical_advantage" title="Mechanical advantage">mechanical advantage</a> of the machine is greater or less than unity respectively. Mechanical advantage is analogous to the inverse of transformer turns ratio in the mobility analogy. A mechanical advantage less than unity is analogous to a step-up transformer and greater than unity is analogous to a step-down transformer.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Power_and_energy_equations">Power and energy equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=11" title="Edit section: Power and energy equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <caption>Table of analogous power and energy equations in the mobility analogy </caption> <tbody><tr> <th>Electrical quantity</th> <th>Electrical expression</th> <th>Mechanical analogy</th> <th>Mechanical expression </th></tr> <tr> <td>Energy supplied</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\int vi\ dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo>∫</mo> <mi>v</mi> <mi>i</mi> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\int vi\ dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97780bc99394116c23c27bf5719618316e4ac6db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.021ex; height:5.676ex;" alt="{\displaystyle E=\int vi\ dt}"></span></td> <td>Energy supplied</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\int Fu\ dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo>∫</mo> <mi>F</mi> <mi>u</mi> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\int Fu\ dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4527d86d6c8209f3f8218967b90e96cfdad389d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.161ex; height:5.676ex;" alt="{\displaystyle E=\int Fu\ dt}"></span> </td></tr> <tr> <td>Power supplied</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=vi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>v</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=vi}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a78e727d0b6dcbf4953b558fa1e8fe61e6bb34e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.774ex; height:2.176ex;" alt="{\displaystyle P=vi}"></span></td> <td>Power supplied</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=Fu}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>F</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=Fu}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/012ad408ac99e89bfe7c1816f783034f0811f20a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.914ex; height:2.176ex;" alt="{\displaystyle P=Fu}"></span> </td></tr> <tr> <td>Power dissipation in a resistor</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=i^{2}R={v^{2} \over R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=i^{2}R={v^{2} \over R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5110585feb8efa4e222c15240e7fdce91d3481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.581ex; height:5.843ex;" alt="{\displaystyle P=i^{2}R={v^{2} \over R}}"></span></td> <td>Power dissipation in a damper<sup id="cite_ref-Eargle4_9-1" class="reference"><a href="#cite_note-Eargle4-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=u^{2}R_{\mathrm {m} }={F^{2} \over R_{\mathrm {m} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=u^{2}R_{\mathrm {m} }={F^{2} \over R_{\mathrm {m} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c917e44fce5b4b5d808bd376bf2592a85a83a77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.893ex; height:6.009ex;" alt="{\displaystyle P=u^{2}R_{\mathrm {m} }={F^{2} \over R_{\mathrm {m} }}}"></span> </td></tr> <tr> <td>Energy stored in an inductor magnetic field</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\tfrac {1}{2}}Li^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>L</mi> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\tfrac {1}{2}}Li^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf918b376888df6148696548f475ac67964a1a5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.972ex; height:3.509ex;" alt="{\displaystyle E={\tfrac {1}{2}}Li^{2}}"></span></td> <td>Potential energy stored in a spring<sup id="cite_ref-Talbot186_1-1" class="reference"><a href="#cite_note-Talbot186-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\tfrac {1}{2}}C_{\mathrm {m} }F^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\tfrac {1}{2}}C_{\mathrm {m} }F^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835c0bf8a3f372fd275f54928b86b44738a47a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.664ex; height:3.509ex;" alt="{\displaystyle E={\tfrac {1}{2}}C_{\mathrm {m} }F^{2}}"></span> </td></tr> <tr> <td>Energy stored in a capacitor electric field</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\tfrac {1}{2}}Cv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>C</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\tfrac {1}{2}}Cv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db18ed95514ee737ef68e28b20ad9aa8cca13d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.48ex; height:3.509ex;" alt="{\displaystyle E={\tfrac {1}{2}}Cv^{2}}"></span></td> <td>Kinetic energy of a moving mass<sup id="cite_ref-Talbot186_1-2" class="reference"><a href="#cite_note-Talbot186-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\tfrac {1}{2}}Mu^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>M</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\tfrac {1}{2}}Mu^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2471d80aca3cc4bc40876260dd7dc4ba8e3a98f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.358ex; height:3.509ex;" alt="{\displaystyle E={\tfrac {1}{2}}Mu^{2}}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=12" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Simple_resonant_circuit">Simple resonant circuit</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=13" title="Edit section: Simple resonant circuit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Mobility_analogy_resonator.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Mobility_analogy_resonator.svg/660px-Mobility_analogy_resonator.svg.png" decoding="async" width="660" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Mobility_analogy_resonator.svg/990px-Mobility_analogy_resonator.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Mobility_analogy_resonator.svg/1320px-Mobility_analogy_resonator.svg.png 2x" data-file-width="1320" data-file-height="425" /></a><figcaption>Simple mechanical resonator (left) and its mobility analogy equivalent circuit (right)</figcaption></figure> <p>The figure shows a mechanical arrangement of a platform of mass <i>M</i> that is suspended above the substrate by a spring of stiffness <i>S</i> and a damper of resistance <i>R</i><sub>m</sub>. The mobility analogy equivalent circuit is shown to the right of this arrangement and consists of a <a href="/wiki/RLC_circuit#Parallel_RLC_circuit" title="RLC circuit">parallel resonant circuit</a>. This system has a <a href="/wiki/Resonant_frequency" class="mw-redirect" title="Resonant frequency">resonant frequency</a>, and may have a <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequency</a> of oscillation if not too heavily damped.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Advantages_and_disadvantages">Advantages and disadvantages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=14" title="Edit section: Advantages and disadvantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The principal advantage of the mobility analogy over its alternative, the <a href="/wiki/Impedance_analogy" title="Impedance analogy">impedance analogy</a>, is that it preserves the topology of the mechanical system. Elements that are in series in the mechanical system are in series in the electrical equivalent circuit and elements in parallel in the mechanical system remain in parallel in the electrical equivalent.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>The principal disadvantage of the mobility analogy is that it does not maintain the analogy between electrical and mechanical impedance. Mechanical impedance is represented as an electrical admittance and a mechanical resistance is represented as an electrical conductance in the electrical equivalent circuit. Force is not analogous to voltage (<a href="/wiki/Generator_(circuit_theory)" title="Generator (circuit theory)">generator</a> voltages are often called <a href="/wiki/Electromotive_force" title="Electromotive force">electromotive force</a>), but rather, it is analogous to current.<sup id="cite_ref-Busch-Vishniac,_p._20_17-1" class="reference"><a href="#cite_note-Busch-Vishniac,_p._20-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=15" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Historically, the impedance analogy was in use long before the mobility analogy. Mechanical admittance and the associated mobility analogy were introduced by F. A. Firestone in 1932 to overcome the issue of preserving topologies.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> W. Hähnle independently had the same idea in Germany. <a href="/wiki/Horace_M._Trent" title="Horace M. Trent">Horace M. Trent</a> developed a treatment for analogies in general from a mathematical <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> perspective and introduced a new analogy of his own.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 23em;"> <ol class="references"> <li id="cite_note-Talbot186-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Talbot186_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Talbot186_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Talbot186_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Talbot-Smith, p. 1.86</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Carr, pp. 170–171</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Eargle, pp. 5–8</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Kleiner, pp. 69–70</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Busch-Vishniac, pp. 18–20</span> </li> <li id="cite_note-Eargle,_p._5-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-Eargle,_p._5_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Eargle,_p._5_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Eargle,_p._5_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Eargle, p. 5</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Fahy & Gardonio, p. 71</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Busch-Vishniac, p. 19</span> </li> <li id="cite_note-Eargle4-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Eargle4_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Eargle4_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Eargle, p. 4</span> </li> <li id="cite_note-Kleiner71-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kleiner71_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kleiner71_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Kleiner, p. 71</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Atkins & Escudier, p. 216</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Kleiner, p. 73</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Smith, p. 1651</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Kleiner, pp. 73–74</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Kleiner, p. 74</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Kleiner, pp. 72–73</span> </li> <li id="cite_note-Busch-Vishniac,_p._20-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Busch-Vishniac,_p._20_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Busch-Vishniac,_p._20_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Busch-Vishniac, p. 20</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Smith, pp. 1649–1650</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Smith, pp. 1650–1651</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">De Groote</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Smith, p. 1661</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Smith, p. 1649</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Taylor & Huang, pp. 377–383</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li>Kleiner, p. 77</li><li>Beranek & Mellow, p. 70</li></ul></div></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li>Kleiner, p. 76</li><li>Beranek & Mellow, p. 70</li></ul></div></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Kleiner, pp. 76–77</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Kleiner, p. 77</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Debnath & Roy, pp. 566–567</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li>Kleiner, pp. 74–76</li><li>Beranek & Mellow, pp. 76–77</li></ul></div></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Eargle, pp. 4–5</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li>Busch-Vishniac, pp. 20–21</li><li>Eargle, pp. 4–5</li></ul></div></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li>Pierce, p. 321</li><li>Firestone</li><li>Pusey, p. 547</li></ul></div></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="display:inline-flex;--size:100%; max-width:max(15em, calc(var(--size) - 3.2em));"><ul style="display:inline-block"><li>Findeisen, p. 26</li><li>Busch-Vishniac, pp. 19–20</li><li>Hähnle</li><li>Trent</li></ul></div></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mobility_analogy&action=edit&section=17" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Atkins, Tony; Escudier, Marcel, <i>A Dictionary of Mechanical Engineering</i>, Oxford University Press, 2013 <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0199587434" title="Special:BookSources/0199587434">0199587434</a>.</li> <li>Beranek, Leo Leroy; Mellow, Tim J., <i>Acoustics: Sound Fields and Transducers</i>, Academic Press, 2012 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0123914213" title="Special:BookSources/0123914213">0123914213</a>.</li> <li>Busch-Vishniac, Ilene J., <i>Electromechanical Sensors and Actuators</i>, Springer Science & Business Media, 1999 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/038798495X" title="Special:BookSources/038798495X">038798495X</a>.</li> <li>Carr, Joseph J., <i>RF Components and Circuits</i>, Newnes, 2002 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7506-4844-9" title="Special:BookSources/0-7506-4844-9">0-7506-4844-9</a>.</li> <li>Debnath, M. C.; Roy, T., <a rel="nofollow" class="external text" href="http://www.hindawi.com/journals/ijmms/1987/575903/abs/">"Transfer scattering matrix of non-uniform surface acoustic wave transducers"</a>, <i>International Journal of Mathematics and Mathematical Sciences,</i> vol. 10, iss. 3, pp. 563–581, 1987.</li> <li>De Groote, Steven, <a rel="nofollow" class="external text" href="http://www.f1technical.net/features/10586">"J-dampers in Formula One"</a>, F1 Technical, 27 September 2008.</li> <li>Eargle, John, <i>Loudspeaker Handbook</i>, Kluwer Academic Publishers, 2003 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1402075847" title="Special:BookSources/1402075847">1402075847</a>.</li> <li>Fahy, Frank J.; Gardonio, Paolo, <i>Sound and Structural Vibration: Radiation, Transmission and Response</i>, Academic Press, 2007 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0080471102" title="Special:BookSources/0080471102">0080471102</a>.</li> <li>Findeisen, Dietmar, <i>System Dynamics and Mechanical Vibrations</i>, Springer, 2000 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3540671447" title="Special:BookSources/3540671447">3540671447</a>.</li> <li>Firestone, Floyd A., "A new analogy between mechanical and electrical systems", <i>Journal of the Acoustical Society of America</i>, vol. 4, pp. 249–267 (1932–1933).</li> <li>Hähnle, W., <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007%2F978-3-642-99668-9_1">"Die Darstellung elektromechanischer Gebilde durch rein elektrische Schaltbilder"</a>, <i>Wissenschaftliche Veröffentlichungen aus dem Siemens-Konzern</i>, vol. 1, iss. 11, pp. 1–23, 1932.</li> <li>Kleiner, Mendel, <i>Electroacoustics</i>, CRC Press, 2013 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1439836183" title="Special:BookSources/1439836183">1439836183</a>.</li> <li>Pierce, Allan D., <i>Acoustics: an Introduction to its Physical Principles and Applications</i>, Acoustical Society of America 1989 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0883186128" title="Special:BookSources/0883186128">0883186128</a>.</li> <li>Pusey, Henry C. (ed), <i>50 years of shock and vibration technology</i>, Shock and Vibration Information Analysis Center, Booz-Allen & Hamilton, Inc., 1996 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0964694026" title="Special:BookSources/0964694026">0964694026</a>.</li> <li>Smith, Malcolm C., <a rel="nofollow" class="external text" href="https://doi.org/10.1109/TAC.2002.803532">"Synthesis of mechanical networks: the inerter"</a>, <i>IEEE Transactions on Automatic Control</i>, vol. 47, iss. 10, pp. 1648–1662, October 2002.</li> <li>Talbot-Smith, Michael, <i>Audio Engineer's Reference Book</i>, Taylor & Francis, 2013 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1136119736" title="Special:BookSources/1136119736">1136119736</a>.</li> <li>Taylor, John; Huang, Qiuting, <i>CRC Handbook of Electrical Filters</i>, CRC Press, 1997 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0849389518" title="Special:BookSources/0849389518">0849389518</a>.</li> <li>Trent, Horace M., <a rel="nofollow" class="external text" href="http://scitation.aip.org/content/asa/journal/jasa/27/3/10.1121/1.1907949">"Isomorphisms between oriented linear graphs and lumped physical systems"</a>, <i>The Journal of the Acoustical Society of America</i>, vol. 27, pp. 500–527, 1955.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Mobility_analogy&oldid=1195666399">https://en.wikipedia.org/w/index.php?title=Mobility_analogy&oldid=1195666399</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Electrical_analogies" title="Category:Electrical analogies">Electrical analogies</a></li><li><a href="/wiki/Category:Electromechanical_engineering" title="Category:Electromechanical engineering">Electromechanical engineering</a></li><li><a href="/wiki/Category:Electronic_design" title="Category:Electronic design">Electronic design</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Good_articles" title="Category:Good articles">Good articles</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 January 2024, at 20:35<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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