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<title>Structural dynamics</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Structural dynamics</span></span>

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<p><b>Structural dynamics</b> is a branch of <a href="Structural_analysis" title="Structural analysis">structural analysis</a> which covers the behavior of a <a href="Structure" title="Structure">structure</a> subjected to <a href="Dynamics_(physics)" class="mw-redirect" title="Dynamics (physics)">dynamic</a> loading. Dynamic loading is any time-varying loading which changes quickly enough that the response of the structure differs from the response to the same loading applied <a href="Statics" title="Statics">statically</a>. Causes of dynamic loading include people, wind, waves, traffic, <a href="Earthquake" title="Earthquake">earthquakes</a>, and blasts. Dynamic analysis can be used to find dynamic <a href="Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacements</a>, time history, and <a href="Modal_analysis" title="Modal analysis">natural frequencies and mode shapes</a>.

</p><p>Whether a given load should be treated as static or dynamic depends on how quickly the load varies in comparison to the structure's natural frequency. If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis.

</p><p>Dynamic analysis for simple structures can be carried out analytically, but for complex structures <a href="Finite_element_analysis" class="mw-redirect" title="Finite element analysis">finite element analysis</a> is more often used to calculate the mode shapes and frequencies.

</p>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>Structural dynamics is applied in a number of engineering fields, including

</p>

<ul><li><a href="Earthquake_engineering" title="Earthquake engineering">Earthquake engineering</a></li>

<li><a href="Wind_engineering" title="Wind engineering">Wind engineering</a></li>

<li><a href="Coastal_engineering" title="Coastal engineering">Coastal engineering</a></li>

<li>Human-structure interaction in <a href="Structural_engineering" title="Structural engineering">structural engineering</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Dynamic_loading">Dynamic loading</h2></div>

<p>Structural analysis is mainly concerned with finding out the behavior (forces or displacements) of a physical structure when subjected to force. This action can be in the form of <a href="Structural_load" title="Structural load">load</a> due to the weight of things such as people, furniture, wind, snow, etc. or some other kind of excitation such as an earthquake, shaking of the ground due to a blast nearby, etc. All loads are dynamic in the literal sense, because at some point in time they were not present. The distinction is made between the dynamic and the static analysis on the basis of whether the applied action has enough acceleration in comparison to the structure's natural frequency. If a load is applied sufficiently slowly, the inertia forces (<a href="Newton's_first_law_of_motion" class="mw-redirect" title="Newton's first law of motion">Newton's first law of motion</a>) can be ignored and the analysis can be simplified as static analysis.

</p><p>Dynamic loads on a structure can be categorized as periodic or non-periodic. Periodic loads may be <a href="Simple_harmonic_motion" title="Simple harmonic motion">simple harmonic</a>, as in the case of a rotating machine with an unbalanced flywheel (a familiar example is a washing machine operating at a steady speed), or they may be more complex but representable by a <a href="Fourier_series" title="Fourier series">Fourier series</a>. Non-periodic loads include <a href="Impulse_(physics)" title="Impulse (physics)">impulsive</a> (very short duration) loading caused by blasts or <a href="Impact_(physics)" class="mw-redirect" title="Impact (physics)">impacts</a>, and longer duration loads including earthquakes and wind. In the case of <a href="Random_vibration" title="Random vibration">random excitation</a>, the amplitude-time history of the load and the structural response are defined in terms of statistical distributions. <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Displacements">Displacements</h2></div>

<p>A dynamic load can have a significantly larger effect than a static load of the same magnitude due to the structure's inability to respond quickly to the loading (by deflecting). The increase in the effect of a dynamic load is given by the <a href="Dynamic_amplification_factor" title="Dynamic amplification factor">dynamic amplification factor</a> (DAF) or dynamic load factor (DLF):

</p>

</mrow>

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<mfrac>

<msub>

</mrow>

</msub>

<msub>

<mtext>static</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\text{DAF}}={\text{DLF}}={\frac {u_{\max }}{u_{\text{static}}}}}</annotation>

</semantics>

</math></span><img src="./a12ea83fd6325ec1d455f5469273fd1f1fcad24e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.314ex; height:5.009ex;" alt="{\displaystyle {\text{DAF}}={\text{DLF}}={\frac {u_{\max }}{u_{\text{static}}}}}" loading="lazy"></span></dd></dl>

<p>where <i>u</i> is the deflection of the structure due to the applied load.

</p><p>Graphs of dynamic amplification factors vs non-dimensional <a href="Rise_time" title="Rise time">rise time</a> (<i>t</i><sub><i>r</i></sub>/<i>T</i>) exist for standard loading functions (for an explanation of rise time, see <b>time history analysis</b> below). Hence the DAF for a given loading can be read from the graph, the static deflection can be easily calculated for simple structures and the dynamic deflection found.

</p>

<div class="mw-heading mw-heading2"><h2 id="Time_history_analysis">Time history analysis</h2></div>

<p>A full time history will give the response of a structure over time during and after the application of a load. To find the full time history of a structure's response, you must solve the structure's <a href="Equation_of_motion" class="mw-redirect" title="Equation of motion">equation of motion</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

<p>A simple single <a href="Degrees_of_freedom_(mechanics)" title="Degrees of freedom (mechanics)">degree of freedom</a> <a href="System" title="System">system</a> (a <a href="Mass" title="Mass">mass</a>, <i>M</i>, on a <a href="Spring_(device)" title="Spring (device)">spring</a> of <a href="Stiffness" title="Stiffness">stiffness</a> <i>k</i>, for example) has the following equation of motion:

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<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M{\ddot {x}}+kx=F(t)}</annotation>

</semantics>

</math></span><img src="./077c55dd79b796392692653891c8b00b27f28c35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.641ex; height:2.843ex;" alt="{\displaystyle M{\ddot {x}}+kx=F(t)}" loading="lazy"></span></dd>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\ddot {x}}}</annotation>

</semantics>

</math></span><img src="./06e0e705ddda28c6cd06cdc6e18be9abf88bb395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\ddot {x}}}" loading="lazy"></span> is the acceleration (the double <a href="Derivative" title="Derivative">derivative</a> of the displacement) and x is the displacement.

</p><p>If the loading <i>F</i>(<i>t</i>) is a <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> (the sudden application of a constant load), the solution to the equation of motion is:

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<mfrac>

<msub>

</mrow>

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</mfrac>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle x={\frac {F_{0}}{k}}[1-\cos(\omega t)]}</annotation>

</semantics>

</math></span><img src="./b0ef0b18fd3d516d6714dd0706a17fae51ced0f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.315ex; height:5.509ex;" alt="{\displaystyle x={\frac {F_{0}}{k}}[1-\cos(\omega t)]}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\sqrt {\frac {k}{M}}}}">

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\frac {k}{M}}}}</annotation>

</semantics>

</math></span><img src="./5a99259b9950f385afe42285ba606528a4a7b68f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.146ex; height:6.176ex;" alt="{\displaystyle \omega ={\sqrt {\frac {k}{M}}}}" loading="lazy"></span> and the fundamental natural frequency, <span class="mwe-math-element mwe-math-element-inline"><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={\frac {\omega }{2\pi }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f={\frac {\omega }{2\pi }}}</annotation>

</semantics>

</math></span><img src="./79577771d4c01f51150d7c5c89ca1048019ea838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.708ex; height:4.676ex;" alt="{\displaystyle f={\frac {\omega }{2\pi }}}" loading="lazy"></span>.

</p><p>The static deflection of a single degree of freedom system is:

</p>

<msub>

<mtext>static</mtext>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{\text{static}}={\frac {F_{0}}{k}}}</annotation>

</semantics>

</math></span><img src="./62d973f5de52a291efff1687c47c14e1369efb1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.982ex; height:5.509ex;" alt="{\displaystyle x_{\text{static}}={\frac {F_{0}}{k}}}" loading="lazy"></span></dd></dl>

<p>so we can write, by combining the above formulae:

</p>

<msub>

<mtext>static</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=x_{\text{static}}[1-\cos(\omega t)]}</annotation>

</semantics>

</math></span><img src="./d5c97a8a732b6fc2ed30ab2059a336587e777a81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.429ex; height:2.843ex;" alt="{\displaystyle x=x_{\text{static}}[1-\cos(\omega t)]}" loading="lazy"></span></dd></dl>

<p>This gives the (theoretical) time history of the structure due to a load F(t), where the false assumption is made that there is no <a href="Damping_ratio" class="mw-redirect" title="Damping ratio">damping</a>.

</p><p>Although this is too simplistic to apply to a real structure, the Heaviside step function is a reasonable model for the application of many real loads, such as the sudden addition of a piece of furniture, or the removal of a prop to a newly cast concrete floor. However, in reality loads are never applied instantaneously – they build up over a period of time (this may be very short indeed). This time is called the <a href="Rise_time" title="Rise time">rise time</a>.

</p><p>As the number of degrees of freedom of a structure increases it very quickly becomes too difficult to calculate the time history manually – real structures are analysed using <a href="Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> <a href="Finite_element_analysis" class="mw-redirect" title="Finite element analysis">finite element analysis</a> software.

</p>

<div class="mw-heading mw-heading2"><h2 id="Damping">Damping</h2></div>

<p>Any real structure will dissipate energy (mainly through friction). This can be modelled by modifying the DAF

</p>

</mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\text{DAF}}=1+e^{-c\pi }}</annotation>

</semantics>

</math></span><img src="./e512e3cd9034b90944d39647c2b6d93e4fc5b85b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.386ex; height:2.676ex;" alt="{\displaystyle {\text{DAF}}=1+e^{-c\pi }}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {\text{damping coefficient}}{\text{critical damping coefficient}}}}">

<mfrac>

<mtext>damping coefficient</mtext>

<mtext>critical damping coefficient</mtext>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c={\frac {\text{damping coefficient}}{\text{critical damping coefficient}}}}</annotation>

</semantics>

</math></span><img src="./b8d78326b40b955a7d800eb57b84a93a8f21ec82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.079ex; height:5.843ex;" alt="{\displaystyle c={\frac {\text{damping coefficient}}{\text{critical damping coefficient}}}}" loading="lazy"></span> and is typically 2–10% depending on the type of construction:

</p>

<ul><li>Bolted steel ~6%</li>

<li><a href="Reinforced_concrete" title="Reinforced concrete">Reinforced concrete</a> ~5%</li>

<li>Welded steel ~2%</li>

<li>Brick masonry ~10%</li></ul>

<p><b>Methods to increase damping</b>

</p><p>One of the widely used methods to increase damping is to attach a layer of material with a high Damping Coefficient, for example rubber, to a vibrating structure.

</p>

<div class="mw-heading mw-heading2"><h2 id="Modal_analysis">Modal analysis</h2></div>

<p>A <a href="Modal_analysis" title="Modal analysis">modal analysis</a> calculates the frequency <a href="Normal_mode" title="Normal mode">modes</a> or natural frequencies of a given system, but not necessarily its full-time history response to a given input. The natural frequency of a system is dependent only on the <a href="Stiffness" title="Stiffness">stiffness</a> of the structure and the <a href="Mass" title="Mass">mass</a> which participates with the structure (including self-weight). It is not dependent on the load function.

</p><p>It is useful to know the modal frequencies of a structure as it allows you to ensure that the frequency of any applied periodic loading will not coincide with a modal frequency and hence cause <a href="Resonance" title="Resonance">resonance</a>, which leads to large <a href="Oscillations" class="mw-redirect" title="Oscillations">oscillations</a>.

</p><p>The method is:

</p>

<ol><li>Find the natural modes (the shape adopted by a structure) and natural frequencies</li>

<li>Calculate the response of each mode</li>

<li>Optionally superpose the response of each mode to find the full modal response to a given loading</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Energy_method">Energy method</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Rayleigh's_quotient_in_vibrations_analysis" title="Rayleigh's quotient in vibrations analysis">Rayleigh's quotient in vibrations analysis</a></div>

<p>It is possible to calculate the frequency of different mode shape of system manually by the <a href="Energy_principles_in_structural_mechanics" title="Energy principles in structural mechanics">energy method</a>. For a given mode shape of a multiple degree of freedom system you can find an "equivalent" mass, stiffness and applied force for a single degree of freedom system. For simple structures the basic mode shapes can be found by inspection, but it is not a conservative method. Rayleigh's principle states:

</p><p>"The frequency ω of an arbitrary mode of vibration, calculated by the energy method, is always greater than – or equal to – the fundamental frequency <i>ω</i><sub><i>n</i></sub>."

</p><p>For an assumed mode shape <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {u}}(x)}">

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {u}}(x)}</annotation>

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</math></span><img src="./ab48ae082b952a53fb29bdbe36f8df15674c17e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.469ex; height:2.843ex;" alt="{\displaystyle {\bar {u}}(x)}" loading="lazy"></span>, of a <a href="Structural_system" title="Structural system">structural system</a> with mass M; bending stiffness, EI (<a href="Young's_modulus" title="Young's modulus">Young's modulus</a>, <i>E</i>, multiplied by the <a href="Second_moment_of_area" title="Second moment of area">second moment of area</a>, <i>I</i>); and applied force, <i>F</i>(<i>x</i>):

</p>

<mtext>Equivalent mass, </mtext>

</mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Equivalent mass, }}M_{\text{eq}}=\int M{\bar {u}}^{2}\,du}</annotation>

</semantics>

</math></span><img src="./07a9292f26b330b56d01c3880c2a9b28937f3bb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.277ex; height:5.676ex;" alt="{\displaystyle {\text{Equivalent mass, }}M_{\text{eq}}=\int M{\bar {u}}^{2}\,du}" loading="lazy"></span></dd></dl>

<mtext>Equivalent stiffness, </mtext>

</mrow>

<msub>

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</msub>

<msup>

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<mfrac>

<mrow>

<msup>

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<mover>

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<msup>

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</msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Equivalent stiffness, }}k_{\text{eq}}=\int EI\left({\frac {d^{2}{\bar {u}}}{dx^{2}}}\right)^{2}\,dx}</annotation>

</semantics>

</math></span><img src="./cafcf2cfe1ad6deae32ad08d5c500189eb863650.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.387ex; height:6.676ex;" alt="{\displaystyle {\text{Equivalent stiffness, }}k_{\text{eq}}=\int EI\left({\frac {d^{2}{\bar {u}}}{dx^{2}}}\right)^{2}\,dx}" loading="lazy"></span></dd></dl>

<mtext>Equivalent force, </mtext>

</mrow>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\text{Equivalent force, }}F_{\text{eq}}=\int F{\bar {u}}\,dx}</annotation>

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</math></span><img src="./47eeab7380d5575570e2963141d396874b5d2882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.682ex; height:5.676ex;" alt="{\displaystyle {\text{Equivalent force, }}F_{\text{eq}}=\int F{\bar {u}}\,dx}" loading="lazy"></span></dd></dl>

<p>then, as above:

</p>

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\frac {k_{\text{eq}}}{M_{\text{eq}}}}}}</annotation>

</semantics>

</math></span><img src="./e4bdd5325f65fc1f288614942e0929b6c2f70165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.788ex; height:7.509ex;" alt="{\displaystyle \omega ={\sqrt {\frac {k_{\text{eq}}}{M_{\text{eq}}}}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Modal_response">Modal response</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Modal_analysis" title="Modal analysis">Modal analysis</a></div>

<p>The complete modal response to a given load <i>F</i>(<i>x</i>,<i>t</i>) is <span class="mwe-math-element mwe-math-element-inline"><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(x,t)=\sum u_{n}(x,t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(x,t)=\sum u_{n}(x,t)}</annotation>

</semantics>

</math></span><img src="./257aae6ed106f7f33371fede4d1c7015a7efcd91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.541ex; height:3.843ex;" alt="{\displaystyle v(x,t)=\sum u_{n}(x,t)}" loading="lazy"></span>. The summation can be carried out by one of three common methods:

</p>

<ul><li>Superpose complete time histories of each mode (time consuming, but exact)</li>

<li>Superpose the maximum amplitudes of each mode (quick but conservative)</li>

<li>Superpose the square root of the sum of squares (good estimate for well-separated frequencies, but unsafe for closely spaced frequencies)</li></ul>

<p>To superpose the individual modal responses manually, having calculated them by the energy method:

</p><p>Assuming that the rise time t<sub>r</sub> is known (<i>T</i> = 2<span class="texhtml mvar" style="font-style:italic;">π</span>/<i>ω</i>), it is possible to read the DAF from a standard graph. The static displacement can be calculated with <span class="mwe-math-element mwe-math-element-inline"><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_{\text{static}}={\frac {F_{1,{\text{eq}}}}{k_{1,{\text{eq}}}}}}">

<msub>

<mtext>static</mtext>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

<msub>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{\text{static}}={\frac {F_{1,{\text{eq}}}}{k_{1,{\text{eq}}}}}}</annotation>

</semantics>

</math></span><img src="./f3fb69615595dcce5bd6105bebe895c1f058cd1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.037ex; height:6.176ex;" alt="{\displaystyle u_{\text{static}}={\frac {F_{1,{\text{eq}}}}{k_{1,{\text{eq}}}}}}" loading="lazy"></span>. The dynamic displacement for the chosen mode and applied force can then be found from:

</p>

<msub>

</mrow>

</msub>

<msub>

<mtext>static</mtext>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{\max }=u_{\text{static}}{\text{DAF}}}</annotation>

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</math></span><img src="./84570fb25a23a2ed9faeb8a159afd6bce499af2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.254ex; height:2.509ex;" alt="{\displaystyle u_{\max }=u_{\text{static}}{\text{DAF}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Modal_participation_factor">Modal participation factor</h2></div>

<p>For real systems there is often mass participating in the <a href="Forcing_function_(differential_equations)" title="Forcing function (differential equations)">forcing function</a> (such as the mass of ground in an <a href="Earthquake" title="Earthquake">earthquake</a>) and mass participating in <a href="Inertia" title="Inertia">inertia</a> effects (the mass of the structure itself, <i>M</i><sub>eq</sub>). The modal participation factor Γ is a comparison of these two masses. For a single degree of freedom system Γ = 1.

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<annotation encoding="application/x-tex">{\displaystyle \Gamma ={\frac {\sum M_{n}{\bar {u}}_{n}}{\sum M_{n}{\bar {u}}_{n}^{2}}}}</annotation>

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</math></span><img src="./df9805d510aa6c08ee58e87a04566012c4486ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.249ex; height:6.509ex;" alt="{\displaystyle \Gamma ={\frac {\sum M_{n}{\bar {u}}_{n}}{\sum M_{n}{\bar {u}}_{n}^{2}}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://structdynviblab.mcgill.ca/">Structural Dynamics and Vibration Laboratory of McGill University</a></li>

<li><a rel="nofollow" class="external text" href="http://www.noisestructure.com/products/FRF.php">Frequency response function from modal parameters</a></li>

<li><a rel="nofollow" class="external text" href="http://vibrationdata.wordpress.com/category/structural-dynamics/">Structural Dynamics Tutorials & Matlab scripts</a></li>

<li><a rel="nofollow" class="external text" href="http://www.exploringstructuraldynamics.org/">AIAA Exploring Structural Dynamics</a> (<a rel="nofollow" class="external free" href="http://www.exploringstructuraldynamics.org/">http://www.exploringstructuraldynamics.org/</a> ) – Structural Dynamics in Aerospace Engineering: Interactive Demos, Videos & Interviews with Practicing Engineers</li></ul>

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</style><cite id="CITEREFCloughPenzien1975" class="citation book cs1">Clough, Ray W.; Penzien, Joseph (1975). <i>Dynamics of Structures</i>. <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. pp. <span class="nowrap">2–</span>3. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-070-11392-0</bdi>.</cite></span>

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