Engineering: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=164
en-us2016 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 07 Feb 2016 05:59:37 GMTSun, 07 Feb 2016 05:59:37 GMTNew applications in the Engineering categoryhttp://www.mapleprimes.com/images/mapleapps.gifEngineering: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=164
Statically Indeterminate Structure
http://www.maplesoft.com/applications/view.aspx?SID=153940&ref=Feed
The application allows you to determine the constraint reactions, build diagrams of the normal forces N, shear forces Q and bending moments M for beams and frames with any number of sections and degree of static indefinability.
The application calculates the deformation (displacements) of the structure in millimeters and displays the displacements of nodes in the horizontal and vertical. It is also possible to calculate the displacement of any point of the structure.<img src="/view.aspx?si=153940/391e24e981ea8d11454375def604a185.gif" alt="Statically Indeterminate Structure" align="left"/>The application allows you to determine the constraint reactions, build diagrams of the normal forces N, shear forces Q and bending moments M for beams and frames with any number of sections and degree of static indefinability.
The application calculates the deformation (displacements) of the structure in millimeters and displays the displacements of nodes in the horizontal and vertical. It is also possible to calculate the displacement of any point of the structure.153940Mon, 25 Jan 2016 05:00:00 ZDr. Aleksey ShirkoDr. Aleksey ShirkoDemo Worksheet for Numerical Delay Differential Equation Solution
http://www.maplesoft.com/applications/view.aspx?SID=153939&ref=Feed
<P>This application shows several examples of modeling using delay differential equations in Maple. These examples are from the webinar <A HREF="http://www.maplesoft.com/products/maple/demo/player/2015/solvingdelaydiffeq.aspx">Solving Delay Differential Equations</A>.</P>
<P>Note: Requires Maple 2015.2 or later.</P><img src="/view.aspx?si=153939/dde.PNG" alt="Demo Worksheet for Numerical Delay Differential Equation Solution" align="left"/><P>This application shows several examples of modeling using delay differential equations in Maple. These examples are from the webinar <A HREF="http://www.maplesoft.com/products/maple/demo/player/2015/solvingdelaydiffeq.aspx">Solving Delay Differential Equations</A>.</P>
<P>Note: Requires Maple 2015.2 or later.</P>153939Wed, 16 Dec 2015 05:00:00 ZAllan WittkopfAllan WittkopfPV Diode Parameter Estimation
http://www.maplesoft.com/applications/view.aspx?SID=153908&ref=Feed
This application fits experimental I-V data to an equation that describes a photovoltaic diode.<img src="/view.aspx?si=153908/pvdiode.png" alt="PV Diode Parameter Estimation" align="left"/>This application fits experimental I-V data to an equation that describes a photovoltaic diode.153908Fri, 30 Oct 2015 04:00:00 ZSamir KhanSamir KhanGain of an Ideal and Non-Ideal Amplifier
http://www.maplesoft.com/applications/view.aspx?SID=153907&ref=Feed
This application models the ideal and non-ideal behavior of an amplifier.<img src="/view.aspx?si=153907/amplifiergain.png" alt="Gain of an Ideal and Non-Ideal Amplifier" align="left"/>This application models the ideal and non-ideal behavior of an amplifier.153907Fri, 30 Oct 2015 04:00:00 ZSamir KhanSamir KhanEconomic Pipe Sizer for Process Plants
http://www.maplesoft.com/applications/view.aspx?SID=153659&ref=Feed
<p>Pipework is a large part of the cost of a process plant. Plant designers need to minimize the total cost of this pipework across the lifetime of the plant. The total overall cost is a combination of individual costs related to the:</p>
<ul>
<li>pipe material,</li>
<li>installation, </li>
<li>maintenance, </li>
<li>depreciation, </li>
<li>energy costs for pumping, </li>
<li>liquid parameters, </li>
<li>required flowrate,</li>
<li>pumping efficiencies,</li>
<li>taxes,</li>
<li>and more.</li>
</ul>
<p>The total cost is not a simple linear sum of the individual costs; a more complex relationship is needed.</p>
<p>This application uses the approach described in [1] to find the pipe diameter that minimizes the total lifetime cost. The method involves the iterative solution of an empirical equation using <a href="/support/help/Maple/view.aspx?path=fsolve">Maple’s fsolve function</a> (the code for the application is in the Startup code region).</p>
<p>Users can choose the pipe material (carbon steel, stainless steel, aluminum or brass), and specify the desired fluid flowrate, fluid viscosity and density. The application then solves the empirical equation (using Maple’s fsolve() function) and returns the economically optimal pipe diameter.</p>
<p>Bear in mind that the empirical parameters used in the application vary as economic conditions change. Those used in this application are correct for 1998 and 2008.</p>
<p><em>[1]: "Updating the Rules for Pipe Sizing", Durand et al., Chemical Engineering, January 2010</em></p><img src="/applications/images/app_image_blank_lg.jpg" alt="Economic Pipe Sizer for Process Plants" align="left"/><p>Pipework is a large part of the cost of a process plant. Plant designers need to minimize the total cost of this pipework across the lifetime of the plant. The total overall cost is a combination of individual costs related to the:</p>
<ul>
<li>pipe material,</li>
<li>installation, </li>
<li>maintenance, </li>
<li>depreciation, </li>
<li>energy costs for pumping, </li>
<li>liquid parameters, </li>
<li>required flowrate,</li>
<li>pumping efficiencies,</li>
<li>taxes,</li>
<li>and more.</li>
</ul>
<p>The total cost is not a simple linear sum of the individual costs; a more complex relationship is needed.</p>
<p>This application uses the approach described in [1] to find the pipe diameter that minimizes the total lifetime cost. The method involves the iterative solution of an empirical equation using <a href="/support/help/Maple/view.aspx?path=fsolve">Maple’s fsolve function</a> (the code for the application is in the Startup code region).</p>
<p>Users can choose the pipe material (carbon steel, stainless steel, aluminum or brass), and specify the desired fluid flowrate, fluid viscosity and density. The application then solves the empirical equation (using Maple’s fsolve() function) and returns the economically optimal pipe diameter.</p>
<p>Bear in mind that the empirical parameters used in the application vary as economic conditions change. Those used in this application are correct for 1998 and 2008.</p>
<p><em>[1]: "Updating the Rules for Pipe Sizing", Durand et al., Chemical Engineering, January 2010</em></p>153659Fri, 15 Aug 2014 04:00:00 ZSamir KhanSamir KhanOptimizing the Design of a Coil Spring
http://www.maplesoft.com/applications/view.aspx?SID=153608&ref=Feed
<p>The design optimization of helical springs is of considerable engineering interest, and demands strong solvers. While the number of constraints is small, the coil and wire diameters are raised to higher powers; this makes the optimization difficult for gradient-based solvers working in standard floating-point precision; a larger number of working digits is needed.</p>
<p>Maple lets you increase the number of digits used in calculations; hence numerically difficult problems, like this, can be solved.</p>
<p>This application minimizes the mass of a helical spring. The constraints include the minimum deflection, the minimum surge wave frequency and the maximum stress, and a loading condition.</p>
<ul>
<li>the minimum deflection, </li>
<li>the minimum surge wave frequency, </li>
<li>the maximum stress, </li>
<li>and a loading condition.</li>
</ul>
<p>The design variables are the</p>
<ul>
<li>diameter of the wire, </li>
<li>the outside diameter of the spring,</li>
<li>and the number of coils</li>
</ul>
<p> Reference: "Introduction to Optimum Design", Jasbir S. Arora, 3<sup>rd</sup> Edition 2012.</p><img src="/view.aspx?si=153608/695d991fff8fb4975d1e1dcd90bb771d.gif" alt="Optimizing the Design of a Coil Spring" align="left"/><p>The design optimization of helical springs is of considerable engineering interest, and demands strong solvers. While the number of constraints is small, the coil and wire diameters are raised to higher powers; this makes the optimization difficult for gradient-based solvers working in standard floating-point precision; a larger number of working digits is needed.</p>
<p>Maple lets you increase the number of digits used in calculations; hence numerically difficult problems, like this, can be solved.</p>
<p>This application minimizes the mass of a helical spring. The constraints include the minimum deflection, the minimum surge wave frequency and the maximum stress, and a loading condition.</p>
<ul>
<li>the minimum deflection, </li>
<li>the minimum surge wave frequency, </li>
<li>the maximum stress, </li>
<li>and a loading condition.</li>
</ul>
<p>The design variables are the</p>
<ul>
<li>diameter of the wire, </li>
<li>the outside diameter of the spring,</li>
<li>and the number of coils</li>
</ul>
<p> Reference: "Introduction to Optimum Design", Jasbir S. Arora, 3<sup>rd</sup> Edition 2012.</p>153608Tue, 17 Jun 2014 04:00:00 ZSamir KhanSamir KhanWelded Beam Design Optimization
http://www.maplesoft.com/applications/view.aspx?SID=153592&ref=Feed
<p>A rigid member is welded onto a beam, with a load applied to the end of the member. The total cost of production is equal to the labor costs (a function of the weld dimensions) plus the cost of the weld and beam material.</p>
<p>The design of the beam is optimized to minimize the production costs by varying the weld and member dimensions.</p>
<p>The constraints include limits on the shear stress, bending stress, buckling load and end deflection, and several size constraints.</p>
<p>The application uses Maple’s non-linear optimizers</p><img src="/view.aspx?si=153592/0621a9aba622112f66506495e21f68d9.gif" alt="Welded Beam Design Optimization" align="left"/><p>A rigid member is welded onto a beam, with a load applied to the end of the member. The total cost of production is equal to the labor costs (a function of the weld dimensions) plus the cost of the weld and beam material.</p>
<p>The design of the beam is optimized to minimize the production costs by varying the weld and member dimensions.</p>
<p>The constraints include limits on the shear stress, bending stress, buckling load and end deflection, and several size constraints.</p>
<p>The application uses Maple’s non-linear optimizers</p>153592Fri, 30 May 2014 04:00:00 ZSamir KhanSamir KhanTuned Mass-Spring-Damper Design
http://www.maplesoft.com/applications/view.aspx?SID=153572&ref=Feed
<p>A mass-spring-damper is disturbed by a force that resonates at the natural frequency of the system.</p>
<p>This application calculates the optimum spring and damping constant of a parasitic tuned-mass damper that the minimizes the vibration of the system.</p>
<p>The vibration of system with and without the tuned mass-spring-damper is viewed as a frequency response, time-domain simulation and power spectrum.</p><img src="/view.aspx?si=153572/cdf00085048c6b59e75db56bb6c0210b.gif" alt="Tuned Mass-Spring-Damper Design" align="left"/><p>A mass-spring-damper is disturbed by a force that resonates at the natural frequency of the system.</p>
<p>This application calculates the optimum spring and damping constant of a parasitic tuned-mass damper that the minimizes the vibration of the system.</p>
<p>The vibration of system with and without the tuned mass-spring-damper is viewed as a frequency response, time-domain simulation and power spectrum.</p>153572Wed, 07 May 2014 04:00:00 ZSamir KhanSamir KhanOptimizing the Design of a Fuel Pod with NX and Maple
http://www.maplesoft.com/applications/view.aspx?SID=153573&ref=Feed
<p>A manufacturer has designed a fuel pod in NX. The fuel pod has a hemispherical and conical end, and a cylindrical mid-section. To minimize material costs, the manufacturer wants to minimize the surface area of the fuel pod while maintaining the existing volume.</p>
<p>This application:</p>
<ul>
<li>pulls the current dimensions of the fuel pod (radius of the hemispherical end, length of the cylindrical midsection, and height of the conical end) from the NX CAD model, </li>
<li>calculates the current volume of the fuel pod,</li>
<li>optimizes the dimensions to minimize the surface area while maintaining the existing volume,</li>
<li>and pushes the optimized dimensions back into the NX CAD model.</li>
</ul>
<p>NOTE: To use this application, you must</p>
<ul>
<li>have a supported version of NX installed, </li>
<li>load canisterOptimization.prt in NX (this is the CAD model of the fuel pod),</li>
<li>ensure the NX-Maple link works correctly.</li>
</ul><img src="/view.aspx?si=153573/fuelpod.jpg" alt="Optimizing the Design of a Fuel Pod with NX and Maple" align="left"/><p>A manufacturer has designed a fuel pod in NX. The fuel pod has a hemispherical and conical end, and a cylindrical mid-section. To minimize material costs, the manufacturer wants to minimize the surface area of the fuel pod while maintaining the existing volume.</p>
<p>This application:</p>
<ul>
<li>pulls the current dimensions of the fuel pod (radius of the hemispherical end, length of the cylindrical midsection, and height of the conical end) from the NX CAD model, </li>
<li>calculates the current volume of the fuel pod,</li>
<li>optimizes the dimensions to minimize the surface area while maintaining the existing volume,</li>
<li>and pushes the optimized dimensions back into the NX CAD model.</li>
</ul>
<p>NOTE: To use this application, you must</p>
<ul>
<li>have a supported version of NX installed, </li>
<li>load canisterOptimization.prt in NX (this is the CAD model of the fuel pod),</li>
<li>ensure the NX-Maple link works correctly.</li>
</ul>153573Wed, 07 May 2014 04:00:00 ZSamir KhanSamir KhanCollision detection between toolholder and workpiece on ball nut grinding
http://www.maplesoft.com/applications/view.aspx?SID=153477&ref=Feed
<p>In this worksheet a collision detection performed to determine the minimum safety distance between a tool holder and ball nut on grinding manufacturing. A nonlinear quartic equation system have to be solved by <em>Newton's</em> and <em>Broyden's</em> methods and results are compared with <em>Maple fsolve()</em> command. Users can check the different results by embedded components and animated 3D surface plot.</p><img src="/view.aspx?si=153477/Collision_Detection_image1.jpg" alt="Collision detection between toolholder and workpiece on ball nut grinding" align="left"/><p>In this worksheet a collision detection performed to determine the minimum safety distance between a tool holder and ball nut on grinding manufacturing. A nonlinear quartic equation system have to be solved by <em>Newton's</em> and <em>Broyden's</em> methods and results are compared with <em>Maple fsolve()</em> command. Users can check the different results by embedded components and animated 3D surface plot.</p>153477Mon, 23 Dec 2013 05:00:00 ZGyörgy HegedûsGyörgy HegedûsClassroom Tips and Techniques: Mathematical Thoughts on the Root Locus
http://www.maplesoft.com/applications/view.aspx?SID=153452&ref=Feed
Under suitable assumptions, the roots of the equation <em>f</em>(<em>z, c</em>) = 0, namely, <em>z</em> = <em>z</em>(<em>c</em>), trace a curve in the complex plane. In engineering feedback-control, such curves are called a <em>root locus</em>. This article examines the parameter-dependence of roots of polynomial and transcendental equations.<img src="/view.aspx?si=153452/thumb.jpg" alt="Classroom Tips and Techniques: Mathematical Thoughts on the Root Locus" align="left"/>Under suitable assumptions, the roots of the equation <em>f</em>(<em>z, c</em>) = 0, namely, <em>z</em> = <em>z</em>(<em>c</em>), trace a curve in the complex plane. In engineering feedback-control, such curves are called a <em>root locus</em>. This article examines the parameter-dependence of roots of polynomial and transcendental equations.153452Tue, 29 Oct 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezSymmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE
http://www.maplesoft.com/applications/view.aspx?SID=151383&ref=Feed
<p>Hybrid structures were made by assembling monolayers (MLs) of closely packed colloidal microspheres on a metal-coated glass substrate . In fact, this architecture is one of several realizations of hybrid plasmonic-photonic crystals (PHs), which differ in photonic crystals dimensionality and metal ﬁlm corrugation [1,2].</p>
<p>The main challenge to us were exploring of those properties of structures which are caused by their space symmetry. In particular, it was necessary to establish the so-called "rules of selection", i.e. the list of the allowed transitions between electronic states of different symmetry and energy that can be induced by light of varying polarization. Additional interest for us was to demonstrate the possibilities of MAPLE within this specific field.</p><img src="/view.aspx?si=151383/440fb9a2994e797b26c18564d860131b.gif" alt="Symmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE" align="left"/><p>Hybrid structures were made by assembling monolayers (MLs) of closely packed colloidal microspheres on a metal-coated glass substrate . In fact, this architecture is one of several realizations of hybrid plasmonic-photonic crystals (PHs), which differ in photonic crystals dimensionality and metal ﬁlm corrugation [1,2].</p>
<p>The main challenge to us were exploring of those properties of structures which are caused by their space symmetry. In particular, it was necessary to establish the so-called "rules of selection", i.e. the list of the allowed transitions between electronic states of different symmetry and energy that can be induced by light of varying polarization. Additional interest for us was to demonstrate the possibilities of MAPLE within this specific field.</p>151383Thu, 05 Sep 2013 04:00:00 ZOlga V. DvornikOlga V. DvornikWavelet analysis of the blood pressure and pulse frequency measurements with Maple
http://www.maplesoft.com/applications/view.aspx?SID=149420&ref=Feed
<p>A significant part of medical signals, or observations, is non-stationary, discrete time sequences. Thus, the computer methods analysis, as well as refinement and compression, are very helpful as for the problems of recognition and detection of their key diagnostic features. We are going to illustrate here this statement with examples of very common, and even routine medical measurements of blood pressure as well as pulse rate and with possibilities of Maple.<br />The package of Discrete Wavelet transforms (DWT) within Maple 16 [1] was recently added as new research software just for such tasks. The practical testing of this package was additional goal of present study.</p><img src="/view.aspx?si=149420/4b9024ee653d2c7be8febb717b1df52a.gif" alt="Wavelet analysis of the blood pressure and pulse frequency measurements with Maple" align="left"/><p>A significant part of medical signals, or observations, is non-stationary, discrete time sequences. Thus, the computer methods analysis, as well as refinement and compression, are very helpful as for the problems of recognition and detection of their key diagnostic features. We are going to illustrate here this statement with examples of very common, and even routine medical measurements of blood pressure as well as pulse rate and with possibilities of Maple.<br />The package of Discrete Wavelet transforms (DWT) within Maple 16 [1] was recently added as new research software just for such tasks. The practical testing of this package was additional goal of present study.</p>149420Sun, 14 Jul 2013 04:00:00 ZIrina A. DanishewskaIrina A. DanishewskaDispersion of arterial pulse waves
http://www.maplesoft.com/applications/view.aspx?SID=145362&ref=Feed
<p>In this paper, we are primarily focusing on the arterial pulse waves, more exactly even their dispersion. It is important because presence of this together with the non-linearity generates the conditions of existence of localized waveforms. Thus, we would like to obtain here the dispersion law for pulse waves with Maple.</p><img src="/view.aspx?si=145362/arterialpulsewaves_thumb.png" alt="Dispersion of arterial pulse waves" align="left"/><p>In this paper, we are primarily focusing on the arterial pulse waves, more exactly even their dispersion. It is important because presence of this together with the non-linearity generates the conditions of existence of localized waveforms. Thus, we would like to obtain here the dispersion law for pulse waves with Maple.</p>145362Tue, 02 Apr 2013 04:00:00 ZShiyan S.I.Shiyan S.I.Fuzzy Sets in Examples
http://www.maplesoft.com/applications/view.aspx?SID=141714&ref=Feed
<p>This worksheet has been created first as a practical part of short course on the pattern recognition theory for my students. It had intended to their introduce, including visually impressions, with fuzzy sets and basic rules of simple operations with them. MAPLE tools were extremely comfortable for such a task and this experience may be useful for community colleagues.</p><img src="/view.aspx?si=141714/fuzzy-sets.jpg" alt="Fuzzy Sets in Examples" align="left"/><p>This worksheet has been created first as a practical part of short course on the pattern recognition theory for my students. It had intended to their introduce, including visually impressions, with fuzzy sets and basic rules of simple operations with them. MAPLE tools were extremely comfortable for such a task and this experience may be useful for community colleagues.</p>141714Sat, 22 Dec 2012 05:00:00 ZProf. Gennady P. ChuikoProf. Gennady P. ChuikoGeneration and Interaction of Solitons
http://www.maplesoft.com/applications/view.aspx?SID=141102&ref=Feed
<p>Classic computer experiments demonstrating the generation of solitons first time, has been published by N. J. Zabusky and M. D. Kruskal in 1965. Considered that was an earlier idea of Enrico Fermi. In 2006, Frank Wang has created a demonstration on the same subject with Maple tools . We would like to show both the origin and the interaction of Korteweg de Vries solitons as a development of approach of above cited publications.</p><img src="/view.aspx?si=141102/fig.jpg" alt="Generation and Interaction of Solitons" align="left"/><p>Classic computer experiments demonstrating the generation of solitons first time, has been published by N. J. Zabusky and M. D. Kruskal in 1965. Considered that was an earlier idea of Enrico Fermi. In 2006, Frank Wang has created a demonstration on the same subject with Maple tools . We would like to show both the origin and the interaction of Korteweg de Vries solitons as a development of approach of above cited publications.</p>141102Tue, 04 Dec 2012 05:00:00 ZS.I. ShyanS.I. ShyanClassroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package
http://www.maplesoft.com/applications/view.aspx?SID=134198&ref=Feed
The OrthogonalExpansions package contributed to the Maple Application Center by Dr. Sergey Moiseev is considered as a tool for generating a Fourier series and its partial sums. This package provides commands for expansions in 17 other bases of orthogonal functions. In addition to looking at the Fourier series option, this article also considers the Bessel series expansion.<img src="/view.aspx?si=134198/thumb.jpg" alt="Classroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package" align="left"/>The OrthogonalExpansions package contributed to the Maple Application Center by Dr. Sergey Moiseev is considered as a tool for generating a Fourier series and its partial sums. This package provides commands for expansions in 17 other bases of orthogonal functions. In addition to looking at the Fourier series option, this article also considers the Bessel series expansion.134198Mon, 14 May 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezVan der Waals equation of state (I)
http://www.maplesoft.com/applications/view.aspx?SID=134131&ref=Feed
<p>This is the first of a series of Maple worksheets developed for teaching chemical thermodynamics.</p>
<p>Given the van der Waals' constants, this worksheet plots: a) the PV van der Waals' isotherms as-given by the equation, b) the PV isotherms based on Maxwell's construction, c) the compressibility factor isotherms based on the low pressure series expansion, and d) the compressibility factor isotherms based on Maxwell's construction. All procedures are independent and were developed to work with the classic interface.</p><img src="/view.aspx?si=134131/436827\41b8ef0a763a603085145e0cf8cd9b47.gif" alt="Van der Waals equation of state (I)" align="left"/><p>This is the first of a series of Maple worksheets developed for teaching chemical thermodynamics.</p>
<p>Given the van der Waals' constants, this worksheet plots: a) the PV van der Waals' isotherms as-given by the equation, b) the PV isotherms based on Maxwell's construction, c) the compressibility factor isotherms based on the low pressure series expansion, and d) the compressibility factor isotherms based on Maxwell's construction. All procedures are independent and were developed to work with the classic interface.</p>134131Sat, 12 May 2012 04:00:00 ZChristian Viales MonteroChristian Viales MonteroAnalysis of basic equations of state (II)
http://www.maplesoft.com/applications/view.aspx?SID=134136&ref=Feed
<p>This is the second of a series of Maple worksheets developed for teaching chemical thermodynamics.</p>
<p>Given a two-constant equation of state (the worksheet includes the most common), this worksheet calculates its critical point, reduced form, volumetric coefficients, Boyle's temperature, virial expansion and internal pressure.</p><img src="/view.aspx?si=134136/436839\ae00ea34ed62fd64822a9ee2652b3c1c.gif" alt="Analysis of basic equations of state (II)" align="left"/><p>This is the second of a series of Maple worksheets developed for teaching chemical thermodynamics.</p>
<p>Given a two-constant equation of state (the worksheet includes the most common), this worksheet calculates its critical point, reduced form, volumetric coefficients, Boyle's temperature, virial expansion and internal pressure.</p>134136Sat, 12 May 2012 04:00:00 ZChristian Viales MonteroChristian Viales MonteroClassroom Tips and Techniques: An Undamped Coupled Oscillator
http://www.maplesoft.com/applications/view.aspx?SID=129521&ref=Feed
<p>Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system <em>M</em> ü + <em>K</em> u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize <em>M</em> and <em>K</em>, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.</p><img src="/view.aspx?si=129521/thumb.jpg" alt="Classroom Tips and Techniques: An Undamped Coupled Oscillator" align="left"/><p>Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system <em>M</em> ü + <em>K</em> u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize <em>M</em> and <em>K</em>, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.</p>129521Tue, 10 Jan 2012 05:00:00 ZDr. Robert LopezDr. Robert Lopez