Engineering Mathematics: New Applications
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en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 16 Apr 2014 16:50:33 GMTWed, 16 Apr 2014 16:50:33 GMTNew applications in the Engineering Mathematics categoryhttp://www.mapleprimes.com/images/mapleapps.gifEngineering Mathematics: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=138
The House Warming Model
http://www.maplesoft.com/applications/view.aspx?SID=153491&ref=Feed
In this guest article in the Tips and Techniques series, Dr. Michael Monagan discusses a model of heat-flow in a house, and shows how he uses this model in his class.<img src="/view.aspx?si=153491/thumb.jpg" alt="The House Warming Model" align="left"/>In this guest article in the Tips and Techniques series, Dr. Michael Monagan discusses a model of heat-flow in a house, and shows how he uses this model in his class.153491Wed, 22 Jan 2014 05:00:00 ZProf. Michael MonaganProf. Michael MonaganCollision 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 LopezClassroom Tips and Techniques: Gems 31-35 from the Red Book of Maple Magic
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Five additional "gems" from the Red Book of Maple Magic are detailed. Gem 31 shows how the updated Explore command can be applied to the numeric solution of an initial-value problem containing parameters. Gem 32 shows some list manipulations. Gem 33 clarifies some issues with the contourplot command, while Gem 34 clarifies some issues with the sample option in the plot command. Finally, Gem 36 examines the Equate command, and its alternatives.<img src="/view.aspx?si=147092/thumb.jpg" alt="Classroom Tips and Techniques: Gems 31-35 from the Red Book of Maple Magic" align="left"/>Five additional "gems" from the Red Book of Maple Magic are detailed. Gem 31 shows how the updated Explore command can be applied to the numeric solution of an initial-value problem containing parameters. Gem 32 shows some list manipulations. Gem 33 clarifies some issues with the contourplot command, while Gem 34 clarifies some issues with the sample option in the plot command. Finally, Gem 36 examines the Equate command, and its alternatives.147092Fri, 10 May 2013 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Least-Squares Fits
http://www.maplesoft.com/applications/view.aspx?SID=140942&ref=Feed
<p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p><img src="/view.aspx?si=140942/image.jpg" alt="Classroom Tips and Techniques: Least-Squares Fits" align="left"/><p><span id="ctl00_mainContent__documentViewer" ><span ><span class="body summary">The least-squares fitting of functions to data can be done in Maple with eleven different commands from four different packages. The <em>CurveFitting</em> and LinearAlgebra packages each have a LeastSquares command; the Optimization package has the LSSolve and NLPSolve commands; and the Statistics package has the seven commands Fit, LinearFit, PolynomialFit, ExponentialFit, LogarithmicFit, PowerFit, and NonlinearFit, which can return some measure of regression analysis.</span></span></span></p>140942Wed, 28 Nov 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezHardening of Aluminium Alloy AA 7075 T 7351
http://www.maplesoft.com/applications/view.aspx?SID=140361&ref=Feed
<p><span id="ctl00_mainContent__documentViewer"><span><span class="body summary">This worksheet is concerned with the hardening of aluminium alloy, the behaviour of which can be expressed by a simple power law with two hardening parameters. Based upon experimental data these parameters have been determined by both a linear regrssion and the nonlinear <em>MARQUARDT-LEVENBERG algorithm.</em></span></span></span></p><img src="/applications/images/app_image_blank_lg.jpg" alt="Hardening of Aluminium Alloy AA 7075 T 7351" align="left"/><p><span id="ctl00_mainContent__documentViewer"><span><span class="body summary">This worksheet is concerned with the hardening of aluminium alloy, the behaviour of which can be expressed by a simple power law with two hardening parameters. Based upon experimental data these parameters have been determined by both a linear regrssion and the nonlinear <em>MARQUARDT-LEVENBERG algorithm.</em></span></span></span></p>140361Wed, 14 Nov 2012 05:00:00 ZJosef BettenJosef BettenLAGRANGE Interpolation
http://www.maplesoft.com/applications/view.aspx?SID=139076&ref=Feed
<p>Zur Aufstellung von Interpolationspolynomen n - ten Grades bei n + 1 gegebenen Messpunkten wird die LAGRANGEsche Interpolationsmethode ausführlich diskutiert und mit anderen Verfahren verglichen. Als numerisches Beispiel wird ein Polynom fünften Grades mut sechs Stützpunkten konstruiert.</p>
<p><em>Keywords: </em>LAGRANGEsche Grundfunktionen, Determinantenmethode, lineare, kubische und Splinefunktionen fünften und höheren Grades, Glättung von Interpolationspolynomen hohen Grades, Formfunktionen vom LAGRANGEschen Typ für finite Elemente</p><img src="/view.aspx?si=139076/b14d4036a98db9e3dc88c1915625b142.gif" alt="LAGRANGE Interpolation" align="left"/><p>Zur Aufstellung von Interpolationspolynomen n - ten Grades bei n + 1 gegebenen Messpunkten wird die LAGRANGEsche Interpolationsmethode ausführlich diskutiert und mit anderen Verfahren verglichen. Als numerisches Beispiel wird ein Polynom fünften Grades mut sechs Stützpunkten konstruiert.</p>
<p><em>Keywords: </em>LAGRANGEsche Grundfunktionen, Determinantenmethode, lineare, kubische und Splinefunktionen fünften und höheren Grades, Glättung von Interpolationspolynomen hohen Grades, Formfunktionen vom LAGRANGEschen Typ für finite Elemente</p>139076Thu, 01 Nov 2012 04:00:00 ZJosef BettenJosef BettenClassroom 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 LopezParameterizing Motion along a Curve
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<p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Parameterizing Motion along a Curve" align="left"/><p>We use the Euler-Lagrange equation to parameterize the motion of a bead on a parabola, a helix, and a piecewise defined combination of the two.</p>130465Wed, 08 Feb 2012 05:00:00 ZShawn HedmanShawn HedmanClassroom Tips and Techniques: An Undamped Coupled Oscillator
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<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 LopezDamage Effective Stress Concepts
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<p>During the last two or three decades many scientists have devoted much effort to the stress analysis in a damaged material, and the notation <em>damage effective stress </em>has been introduced. </p>
<p>In the following some various <em>damage effective stress concepts </em>should be reviewed.</p><img src="/view.aspx?si=129456/428320\12aff582b7c2a4ef667abfccba992187.gif" alt="Damage Effective Stress Concepts" align="left"/><p>During the last two or three decades many scientists have devoted much effort to the stress analysis in a damaged material, and the notation <em>damage effective stress </em>has been introduced. </p>
<p>In the following some various <em>damage effective stress concepts </em>should be reviewed.</p>129456Sun, 08 Jan 2012 05:00:00 ZProf. Josef BettenProf. Josef BettenSimple Harmonic Motion
http://www.maplesoft.com/applications/view.aspx?SID=87640&ref=Feed
<p>The aim of this topic is to visualize the motion of a simple pendulum which consist of a small mass m
suspended by a light inextensible cord of length L from a fixed support.
</p><img src="/view.aspx?si=87640/harmonic_sm.png" alt="Simple Harmonic Motion" align="left"/><p>The aim of this topic is to visualize the motion of a simple pendulum which consist of a small mass m
suspended by a light inextensible cord of length L from a fixed support.
</p>87640Fri, 09 Dec 2011 05:00:00 ZAli Abu OamAli Abu OamClassroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem
http://www.maplesoft.com/applications/view.aspx?SID=128444&ref=Feed
<p>This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of <em>n × n</em> matrices.</p>
<p>Given the <em>n × n</em> matrices <em>A</em> and <em>B</em>, the <em>generalized eigenvalue problem</em> seeks the eigenpairs <em>(lambda<sub>k</sub>, x<sub>k</sub>)</em>, solutions of the equation <em>Ax = lambda Bx</em>, or <em>(A - lambda B) x = 0</em>. If <em>B</em> is nonsingular, the eigenpairs of <em>B<sup>-1</sup> A</em> are solutions. If a matrix <em>S</em> exists for which<em> S<sup>T</sup> A S = Lambda</em>, and <em>S<sup>T</sup> B S = I</em>, where <em>Lambda</em> is a diagonal matrix and <em>I</em> is the <em>n × n</em> identity, then <em>A</em> and <em>B</em> are said to be <em>diagonalized simultaneously</em>, in which case the diagonal entries of <em>Lambda</em> are the generalized eigenvalues for <em>A</em> and <em>B</em>. Such a matrix <em>S</em> exists if <em>A</em> is symmetric and <em>B</em> is positive definite. (Our definition of positive definite includes symmetry.)</p><img src="/view.aspx?si=128444/thumb.jpg" alt="Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem" align="left"/><p>This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of <em>n × n</em> matrices.</p>
<p>Given the <em>n × n</em> matrices <em>A</em> and <em>B</em>, the <em>generalized eigenvalue problem</em> seeks the eigenpairs <em>(lambda<sub>k</sub>, x<sub>k</sub>)</em>, solutions of the equation <em>Ax = lambda Bx</em>, or <em>(A - lambda B) x = 0</em>. If <em>B</em> is nonsingular, the eigenpairs of <em>B<sup>-1</sup> A</em> are solutions. If a matrix <em>S</em> exists for which<em> S<sup>T</sup> A S = Lambda</em>, and <em>S<sup>T</sup> B S = I</em>, where <em>Lambda</em> is a diagonal matrix and <em>I</em> is the <em>n × n</em> identity, then <em>A</em> and <em>B</em> are said to be <em>diagonalized simultaneously</em>, in which case the diagonal entries of <em>Lambda</em> are the generalized eigenvalues for <em>A</em> and <em>B</em>. Such a matrix <em>S</em> exists if <em>A</em> is symmetric and <em>B</em> is positive definite. (Our definition of positive definite includes symmetry.)</p>128444Tue, 06 Dec 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezThe Origin of Complex Numbers
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The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.<img src="/applications/images/app_image_blank_lg.jpg" alt="The Origin of Complex Numbers" align="left"/>The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.126618Fri, 14 Oct 2011 04:00:00 ZDr. John MathewsDr. John MathewsA new algorithm for computing the multivariate Faà di Bruno’s formula
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<p>We provide a new algorithm for computing the multivariate Faà di Bruno's formula. We follow a symbolic approach based on the classical umbral calculus that leads back the computation of the multivariate Faà di Bruno's formula to a suitable multinomial expansion. The resulting computational times are faster compared with procedures existing in the literature.</p><img src="/view.aspx?si=101396/319207\UMFB.JPG" alt="A new algorithm for computing the multivariate Faà di Bruno’s formula" align="left"/><p>We provide a new algorithm for computing the multivariate Faà di Bruno's formula. We follow a symbolic approach based on the classical umbral calculus that leads back the computation of the multivariate Faà di Bruno's formula to a suitable multinomial expansion. The resulting computational times are faster compared with procedures existing in the literature.</p>101396Thu, 03 Feb 2011 05:00:00 ZDr. Giuseppe GuarinoDr. Giuseppe GuarinoClassroom Tips and Techniques: Partial Derivatives by Subscripting
http://www.maplesoft.com/applications/view.aspx?SID=100266&ref=Feed
As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.<img src="/view.aspx?si=100266/thumb.jpg" alt="Classroom Tips and Techniques: Partial Derivatives by Subscripting" align="left"/>As output, Maple can display the partial derivative ∂/∂<em>x f</em>(<em>x,y</em>) as <em>f</em><sub>x</sub>; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.100266Wed, 15 Dec 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezLead and Lag Root Locus Design
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<p>Root locus plots can provide a great deal of information about a system. Maple's DynamicSystems package provides the RootContourPlot and the RootLocusPlot commands for visualizing the behavior of a system when a control parameter is varied. This worksheet shows how systems with multiple free parameters can be analyzed.</p>
<p>This application is part of the <A HREF="/contact/webforms/ControlTheory/">Classroom Content: Control Theory</A> collection.</p><img src="/view.aspx?si=87682/thumb.jpg" alt="Lead and Lag Root Locus Design" align="left"/><p>Root locus plots can provide a great deal of information about a system. Maple's DynamicSystems package provides the RootContourPlot and the RootLocusPlot commands for visualizing the behavior of a system when a control parameter is varied. This worksheet shows how systems with multiple free parameters can be analyzed.</p>
<p>This application is part of the <A HREF="/contact/webforms/ControlTheory/">Classroom Content: Control Theory</A> collection.</p>87682Sun, 14 Nov 2010 05:00:00 ZMaplesoftMaplesoftClassroom Tips and Techniques: Diffusion with a Generalized Robin Condition
http://www.maplesoft.com/applications/view.aspx?SID=96958&ref=Feed
<p>The one-dimensonal heat equation with a generalized Robin condition is solved on [0, 1] by a finite-difference scheme and by the Laplace transform, with the inversion implemented numerically. The left end is insulated and the initial temperature is zero. The Robin condition at the right end is driven by a function governed by an ODE, that is in turn, driven by the endpoint temperature.</p><img src="/view.aspx?si=96958/thumb.jpg" alt="Classroom Tips and Techniques: Diffusion with a Generalized Robin Condition" align="left"/><p>The one-dimensonal heat equation with a generalized Robin condition is solved on [0, 1] by a finite-difference scheme and by the Laplace transform, with the inversion implemented numerically. The left end is insulated and the initial temperature is zero. The Robin condition at the right end is driven by a function governed by an ODE, that is in turn, driven by the endpoint temperature.</p>96958Fri, 17 Sep 2010 04:00:00 ZRobert LopezRobert LopezMotion in one, two and three dimention
http://www.maplesoft.com/applications/view.aspx?SID=88359&ref=Feed
<p>The purpose of this worksheet is to show the powerfull of maple in modeling and visualizing the motion in different dimentions</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Motion in one, two and three dimention" align="left"/><p>The purpose of this worksheet is to show the powerfull of maple in modeling and visualizing the motion in different dimentions</p>88359Mon, 24 May 2010 04:00:00 ZAli Abu OamAli Abu OamThe Relationship between Pole Locations and Time-Domain Performance for a Second Order System
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<P>An interactive worksheet that goes through the effect of pole locations on a second order system. The worksheet visually shows how changing the poles in the S-plane effects the step response in the time domain.
<p>This application is part of the <A HREF="/contact/webforms/ControlTheory/">Classroom Content: Control Theory</A> collection.</p><img src="/view.aspx?si=87681/thumb.jpg" alt="The Relationship between Pole Locations and Time-Domain Performance for a Second Order System" align="left"/><P>An interactive worksheet that goes through the effect of pole locations on a second order system. The worksheet visually shows how changing the poles in the S-plane effects the step response in the time domain.
<p>This application is part of the <A HREF="/contact/webforms/ControlTheory/">Classroom Content: Control Theory</A> collection.</p>87681Fri, 21 May 2010 04:00:00 ZMaplesoftMaplesoft