Mechanical: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=199
en-us2016 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemFri, 05 Feb 2016 22:13:41 GMTFri, 05 Feb 2016 22:13:41 GMTNew applications in the Mechanical categoryhttp://www.mapleprimes.com/images/mapleapps.gifMechanical: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=199
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 ShirkoGas Orifice Flow Meter Calculator
http://www.maplesoft.com/applications/view.aspx?SID=153739&ref=Feed
<p>This application calculates the flowrate through a large-diameter orifice using the approach outlined in ISO 5167 2:2003</p>
<p>Orifice meters use the pressure loss across a constriction (that is, the orifice plate) in a pipe to determine the flowrate. </p>
<p>The formulas are valid for</p>
<ul>
<li>pipes diameters between 50mm and 1000 mm, </li>
<li>and pressure ratios greater than 0.75.</li>
</ul>
<p>Reference: <span ><a href="http://en.wikipedia.org/wiki/Orifice_plate">http://en.wikipedia.org/wiki/Orifice_plate</a></span></p><img src="/view.aspx?si=153739/gasorifice.png" alt="Gas Orifice Flow Meter Calculator" align="left"/><p>This application calculates the flowrate through a large-diameter orifice using the approach outlined in ISO 5167 2:2003</p>
<p>Orifice meters use the pressure loss across a constriction (that is, the orifice plate) in a pipe to determine the flowrate. </p>
<p>The formulas are valid for</p>
<ul>
<li>pipes diameters between 50mm and 1000 mm, </li>
<li>and pressure ratios greater than 0.75.</li>
</ul>
<p>Reference: <span ><a href="http://en.wikipedia.org/wiki/Orifice_plate">http://en.wikipedia.org/wiki/Orifice_plate</a></span></p>153739Tue, 20 Jan 2015 05:00:00 ZSamir KhanSamir KhanThe Comet 67P/Churyumov-Gerasimenko, Rosetta & Philae
http://www.maplesoft.com/applications/view.aspx?SID=153706&ref=Feed
<p> Abstract<br /><br />The Rosetta space probe launched 10 years ago by the European Space Agency (ESA) arrived recently (November 12, 2014) at the site of the comet known as 67P/Churyumov-Gerasimenco after a trip of 4 billions miles from Earth. After circling the comet, Rosetta released its precious load : the lander Philae packed with 21 different scientific instruments for the study of the comet with the main purpose : the origin of our solar system and possibly the origin of life on our planet.<br /><br />Our plan is rather a modest one since all we want is to get , by calculations, specific data concerning the comet and its lander.<br />We shall take a simplified model and consider the comet as a perfect solid sphere to which we can apply Newton's laws.<br /><br />We want to find:<br /><br />I- the acceleration on the comet surface ,<br />II- its radius,<br />III- its density,<br />IV- the velocity of Philae just after the 1st bounce off the comet (it has bounced twice),<br />V- the time for Philae to reach altitude of 1000 m above the comet .<br /><br />We shall compare our findings with the already known data to see how close our simplified mathematical model findings are to the duck-shaped comet already known results.<br />It turned out that our calculations for a sphere shaped comet are very close to the already known data.<br /><br />Conclusion<br /><br />Even with a shape that defies the application of any mechanical laws we can always get very close to reality by adopting a simplified mathematical model in any preliminary study of a complicated problem.<br /><br /></p><img src="/applications/images/app_image_blank_lg.jpg" alt="The Comet 67P/Churyumov-Gerasimenko, Rosetta & Philae" align="left"/><p> Abstract<br /><br />The Rosetta space probe launched 10 years ago by the European Space Agency (ESA) arrived recently (November 12, 2014) at the site of the comet known as 67P/Churyumov-Gerasimenco after a trip of 4 billions miles from Earth. After circling the comet, Rosetta released its precious load : the lander Philae packed with 21 different scientific instruments for the study of the comet with the main purpose : the origin of our solar system and possibly the origin of life on our planet.<br /><br />Our plan is rather a modest one since all we want is to get , by calculations, specific data concerning the comet and its lander.<br />We shall take a simplified model and consider the comet as a perfect solid sphere to which we can apply Newton's laws.<br /><br />We want to find:<br /><br />I- the acceleration on the comet surface ,<br />II- its radius,<br />III- its density,<br />IV- the velocity of Philae just after the 1st bounce off the comet (it has bounced twice),<br />V- the time for Philae to reach altitude of 1000 m above the comet .<br /><br />We shall compare our findings with the already known data to see how close our simplified mathematical model findings are to the duck-shaped comet already known results.<br />It turned out that our calculations for a sphere shaped comet are very close to the already known data.<br /><br />Conclusion<br /><br />Even with a shape that defies the application of any mechanical laws we can always get very close to reality by adopting a simplified mathematical model in any preliminary study of a complicated problem.<br /><br /></p>153706Mon, 17 Nov 2014 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyOptimizing 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 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 KhanDescartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies
http://www.maplesoft.com/applications/view.aspx?SID=153515&ref=Feed
<p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p><img src="/view.aspx?si=153515/Elastic_Collision_image1.jpg" alt="Descartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies" align="left"/><p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p>153515Fri, 07 Mar 2014 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyHohmann Elliptic Transfer Orbit with Animation
http://www.maplesoft.com/applications/view.aspx?SID=151351&ref=Feed
<p>Abstract<br /><br />The main purpose of this article is to show how to use Hohmann elliptic transfer in two situations:<br />a- When one manned spaceship is trying to catch up with an other one <br />on the same circular orbit around Earth.<br />b- When delivering a payload from Earth to a space station on a circular <br />orbit around Earth using 2-stage rocket .<br /><br />The way we set up the problem is as follows:<br />Consider two manned spaceships with astronauts Sally & Igor , the latter<br />lagging behind Sally by a given angle = 4.5 degrees while both are on the same<br />circular orbit C2 about Earth. A 2d lower circular orbit C1 is given. <br />Find the Hohmann elliptic orbit that is tangent to both orbits which allows<br />Sally to maneuver on C1 then to get back to the circular orbit C2 alongside Igor.<br /><br />Though the math was correct , however the final result we found was not !! <br />It was somehow tricky to find the culprit!<br />We have to restate the problem to get the correct answer. <br />The animation was then set up using the correct data. <br />The animation is a good teaching help for two reasons:<br />1- it gives a 'hand on' experience for anyone who wants to fully understand it,<br />2- it is a good lesson in Maple programming with many loops of the type 'if..then'.<br /><br />Warning<br /><br />This particular animation is a hog for the CPU memory since data accumulated <br />for plotting reached 20 MB! This is the size of this article when animation is <br />executed. For this reason and to be able to upload it I left the animation <br />procedure non executed which drops the size of the article to 300KB.<br /><br />Conclusion<br /><br />If I can get someone interested in the subject of this article in such away that he or <br />she would seek further information for learning from other sources, my efforts<br />would be well rewarded.</p><img src="/view.aspx?si=151351/Elliptic_image1.jpg" alt="Hohmann Elliptic Transfer Orbit with Animation" align="left"/><p>Abstract<br /><br />The main purpose of this article is to show how to use Hohmann elliptic transfer in two situations:<br />a- When one manned spaceship is trying to catch up with an other one <br />on the same circular orbit around Earth.<br />b- When delivering a payload from Earth to a space station on a circular <br />orbit around Earth using 2-stage rocket .<br /><br />The way we set up the problem is as follows:<br />Consider two manned spaceships with astronauts Sally & Igor , the latter<br />lagging behind Sally by a given angle = 4.5 degrees while both are on the same<br />circular orbit C2 about Earth. A 2d lower circular orbit C1 is given. <br />Find the Hohmann elliptic orbit that is tangent to both orbits which allows<br />Sally to maneuver on C1 then to get back to the circular orbit C2 alongside Igor.<br /><br />Though the math was correct , however the final result we found was not !! <br />It was somehow tricky to find the culprit!<br />We have to restate the problem to get the correct answer. <br />The animation was then set up using the correct data. <br />The animation is a good teaching help for two reasons:<br />1- it gives a 'hand on' experience for anyone who wants to fully understand it,<br />2- it is a good lesson in Maple programming with many loops of the type 'if..then'.<br /><br />Warning<br /><br />This particular animation is a hog for the CPU memory since data accumulated <br />for plotting reached 20 MB! This is the size of this article when animation is <br />executed. For this reason and to be able to upload it I left the animation <br />procedure non executed which drops the size of the article to 300KB.<br /><br />Conclusion<br /><br />If I can get someone interested in the subject of this article in such away that he or <br />she would seek further information for learning from other sources, my efforts<br />would be well rewarded.</p>151351Wed, 04 Sep 2013 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyEquilibrium Configurations of Cantilever Beam under Terminal Load
http://www.maplesoft.com/applications/view.aspx?SID=142643&ref=Feed
<p>This worksheet implements the calculation of the equilibrium shapes of<br /> initially straight inextensible and unshearable elastic cantilever beam.Three types of load conditions are implemented:<br />1) Follower load problem.<br />2) Unknown load parameter problem<br />3) Conservative load problem</p>
<p>For each of these problems animation of beam deformation are given.</p><img src="/view.aspx?si=142643/5a35294a40cc0022a2745d42d103edbb.gif" alt="Equilibrium Configurations of Cantilever Beam under Terminal Load" align="left"/><p>This worksheet implements the calculation of the equilibrium shapes of<br /> initially straight inextensible and unshearable elastic cantilever beam.Three types of load conditions are implemented:<br />1) Follower load problem.<br />2) Unknown load parameter problem<br />3) Conservative load problem</p>
<p>For each of these problems animation of beam deformation are given.</p>142643Sat, 26 Jan 2013 05:00:00 ZDr. milan batistaDr. milan batistaHardening 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 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 LopezSpherical Pendulum with Animation
http://www.maplesoft.com/applications/view.aspx?SID=132143&ref=Feed
<p>Some years ago I have written a Maple document ( already on Maple's online) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch.<br />Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one.<br />The idea is:<br />1- to get the differential equations of motion for the Spherical Pendulum (SP),<br />2- to solve them,<br />3- to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time,<br />4- to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting,<br />5- finally to work out the necessary steps for the purpose of animation.<br />It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports:<br /> " the length of the output exceeds 1 million".<br />Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. <br />Note that with only 60 positions of the bob, the present article length is equal to 11.3 MB! To be able to upload it, I have to save it without running the last command related to the animation. Doing so I reduced it to a mere 570 KB.<br /><br />It was tiring to get through a jumble of formulas, calculations and programming so I wonder why I have to go through all this trouble to get this animation and yet one can get the same thing with much better animation from the internet. I think the reason is the challenge to be able to do things that others have done before and secondly the idea of creating something form nothing then to see it working as expected, gives (at least to me) a great deal of pleasure and satisfaction.<br />This is beside the fact that, to my knowledge, no such animation for (SP) has been published on Maple online with detailed calculations & programming as I did.<br /><br /></p><img src="/view.aspx?si=132143/433082\Spherical_Pendulum_p.jpg" alt="Spherical Pendulum with Animation" align="left"/><p>Some years ago I have written a Maple document ( already on Maple's online) on the subject of animating a simple pendulum for large angles of oscillation. This gave me the chance to test Maple command JacobiSN(time, k). I was very much pleased to see Maple do a wonderful job in getting these Jacobi's elliptic functions without a glitch.<br />Today I am back to these same functions for a similar purpose though much more sophisticated than the previous one.<br />The idea is:<br />1- to get the differential equations of motion for the Spherical Pendulum (SP),<br />2- to solve them,<br />3- to use Maple for finding the inverse of these Elliptic Integrals i.e. finding the displacement z as function of time,<br />4- to get a set of coordinates [x, y, z] for the positions of the bob at different times for plotting,<br />5- finally to work out the necessary steps for the purpose of animation.<br />It turns out that even with only 3 oscillations where each is defined with only 20 positions of the bob for a total of 60 points on the graph, the animation is so overwhelming that Maple reports:<br /> " the length of the output exceeds 1 million".<br />Not withstanding this warning, Maple did a perfect job by getting the animation to my satisfaction. <br />Note that with only 60 positions of the bob, the present article length is equal to 11.3 MB! To be able to upload it, I have to save it without running the last command related to the animation. Doing so I reduced it to a mere 570 KB.<br /><br />It was tiring to get through a jumble of formulas, calculations and programming so I wonder why I have to go through all this trouble to get this animation and yet one can get the same thing with much better animation from the internet. I think the reason is the challenge to be able to do things that others have done before and secondly the idea of creating something form nothing then to see it working as expected, gives (at least to me) a great deal of pleasure and satisfaction.<br />This is beside the fact that, to my knowledge, no such animation for (SP) has been published on Maple online with detailed calculations & programming as I did.<br /><br /></p>132143Mon, 26 Mar 2012 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyClassroom 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 LopezDamage Effective Stress Concepts
http://www.maplesoft.com/applications/view.aspx?SID=129456&ref=Feed
<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 BettenClassroom 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 LopezRelaxation von Brain Tissue
http://www.maplesoft.com/applications/view.aspx?SID=125272&ref=Feed
<p>In seiner Dissertation hat <em>AIMEDIEU </em>(2004) nichtlineare Relaxation von <em>Brain Tissue </em>untersucht. Er fand gute Übereinstimmungen zwischen dem Wurzel-t-Gesetz (<em>BETTEN, </em>2008) und seinen eigenen Experimenten.</p>
<p> Im Folgenden wird gezeigt, dass <em>PRONY</em>-Reihen weniger geeignet sind, den experimentellen Befund zu beschreiben.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Relaxation von Brain Tissue" align="left"/><p>In seiner Dissertation hat <em>AIMEDIEU </em>(2004) nichtlineare Relaxation von <em>Brain Tissue </em>untersucht. Er fand gute Übereinstimmungen zwischen dem Wurzel-t-Gesetz (<em>BETTEN, </em>2008) und seinen eigenen Experimenten.</p>
<p> Im Folgenden wird gezeigt, dass <em>PRONY</em>-Reihen weniger geeignet sind, den experimentellen Befund zu beschreiben.</p>125272Tue, 30 Aug 2011 04:00:00 ZProf. Josef BettenProf. Josef BettenPlastic method of structural analysis
http://www.maplesoft.com/applications/view.aspx?SID=102534&ref=Feed
<p>This worksheet contains a step-by-step method for the analysis of 2D frames with all kind of boundary conditions or joints between elements.</p>
<p>In each step, determined by the creation of a new plastic hinge, numeric information (displacements, reactions and stresses) and graphic information (beam diagrams and deformed shape) are obtained.</p>
<p>Finally, accumulated beam diagrams and accumulated deformed shapes are also obtained.<br /><br /></p><img src="/view.aspx?si=102534/Image1.PNG" alt="Plastic method of structural analysis" align="left"/><p>This worksheet contains a step-by-step method for the analysis of 2D frames with all kind of boundary conditions or joints between elements.</p>
<p>In each step, determined by the creation of a new plastic hinge, numeric information (displacements, reactions and stresses) and graphic information (beam diagrams and deformed shape) are obtained.</p>
<p>Finally, accumulated beam diagrams and accumulated deformed shapes are also obtained.<br /><br /></p>102534Mon, 14 Mar 2011 04:00:00 ZAntolín Lorenzana IbánAntolín Lorenzana IbánRelaxation due to the Sqrt(t)-Law in Comparison with the MAXWELL-Fluid
http://www.maplesoft.com/applications/view.aspx?SID=100763&ref=Feed
<p>This document is concerned with the relaxation functions (11.72a,b) taken</p>
<p>from BETTEN's book: Creep Mechanics, Third Edition, Springer-Verlag 2008,</p>
<p>in more detail. The parameters in the MAXWELL-model (11.72b) and the</p>
<p>sqrt(t)-law (11.72a) are denoted by b and B, respectively. A relation between</p>
<p>b and B has been found, so that the simple MAXWELL-model can be regarded</p>
<p>as the best approximation to the sqrt(t)-law.</p><img src="/view.aspx?si=100763/maple_icon.jpg" alt="Relaxation due to the Sqrt(t)-Law in Comparison with the MAXWELL-Fluid" align="left"/><p>This document is concerned with the relaxation functions (11.72a,b) taken</p>
<p>from BETTEN's book: Creep Mechanics, Third Edition, Springer-Verlag 2008,</p>
<p>in more detail. The parameters in the MAXWELL-model (11.72b) and the</p>
<p>sqrt(t)-law (11.72a) are denoted by b and B, respectively. A relation between</p>
<p>b and B has been found, so that the simple MAXWELL-model can be regarded</p>
<p>as the best approximation to the sqrt(t)-law.</p>100763Sun, 09 Jan 2011 05:00:00 ZProf. Josef BettenProf. Josef BettenElastic-Plastic Transition # Appendix.mws
http://www.maplesoft.com/applications/view.aspx?SID=99879&ref=Feed
<p><strong>Abstract </strong></p>
<p><strong><br /></strong>In addition to the recently submitted worksheet, dated by November 21, 2010,</p>
<p>uni-axial stress-strain relations have been examined, if they are suitable to</p>
<p>describe the elastic-plastic transition.</p>
<p><em>Keywords: </em>Envelope; modified tanh approach; modified <em>GAUSS </em>error function;</p>
<p> modified rational function;</p><img src="/view.aspx?si=99879/maple_icon.jpg" alt="Elastic-Plastic Transition # Appendix.mws" align="left"/><p><strong>Abstract </strong></p>
<p><strong><br /></strong>In addition to the recently submitted worksheet, dated by November 21, 2010,</p>
<p>uni-axial stress-strain relations have been examined, if they are suitable to</p>
<p>describe the elastic-plastic transition.</p>
<p><em>Keywords: </em>Envelope; modified tanh approach; modified <em>GAUSS </em>error function;</p>
<p> modified rational function;</p>99879Sun, 05 Dec 2010 05:00:00 ZProf. Josef BettenProf. Josef BettenTwo Bodies Revolving Around Their Center of Mass with ANIMATION
http://www.maplesoft.com/applications/view.aspx?SID=99587&ref=Feed
<p>For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see. <br />For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.<br />For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference. <br />The reason is that:<br /> the mass of the satellite being insignificant compared to Earth's <br /> mass, the revolving satellite doesn't affect Earth at all so<br /> that the CM of Earth-satellite system is still the center of the Earth.<br /> Hence we use the center of the Earth as the origin of a rectangular<br /> coordinates system.<br /> <br />In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing: <br />1- their combined motion as seen from their common Center of Mass,<br />2- their relative motion as if one of them is fixed and the other one is moving. <br />In this last instance the frame of reference is attached to the the body that is supposed to be at rest.<br /><br /></p><img src="/view.aspx?si=99587/thumb.jpg" alt="Two Bodies Revolving Around Their Center of Mass with ANIMATION" align="left"/><p>For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see. <br />For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.<br />For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference. <br />The reason is that:<br /> the mass of the satellite being insignificant compared to Earth's <br /> mass, the revolving satellite doesn't affect Earth at all so<br /> that the CM of Earth-satellite system is still the center of the Earth.<br /> Hence we use the center of the Earth as the origin of a rectangular<br /> coordinates system.<br /> <br />In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing: <br />1- their combined motion as seen from their common Center of Mass,<br />2- their relative motion as if one of them is fixed and the other one is moving. <br />In this last instance the frame of reference is attached to the the body that is supposed to be at rest.<br /><br /></p>99587Mon, 29 Nov 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyElastic-Plastic Transition
http://www.maplesoft.com/applications/view.aspx?SID=99226&ref=Feed
<p>This worksheet is concerned with the generalization of nonlinear material laws found in experiments to multi-axial states of stress. For engineering applications it is very important to generalize uni-axial relations to tensorial constitutive equations valid for multi-axial states of stress. This can be achieved by utilizing interpolation methods for tensor functions developed by the Author. It can be shown that the scalar coefficients in tensorial constitutive equations are functions of the set of irreducible invariants (<em>integrity basis) </em>and experimental data. In the following uni-axial relations describing the elastic-plastic transition in uni-axial tests have been generalized to multi-axial states of stress.</p>
<p><em>Keywords: </em>Elastic-plastic transitions under uni-axial load; tensorial generalization; irreducible invariants; tensorial interpolation; example;</p><img src="/view.aspx?si=99226/maple_icon.jpg" alt="Elastic-Plastic Transition" align="left"/><p>This worksheet is concerned with the generalization of nonlinear material laws found in experiments to multi-axial states of stress. For engineering applications it is very important to generalize uni-axial relations to tensorial constitutive equations valid for multi-axial states of stress. This can be achieved by utilizing interpolation methods for tensor functions developed by the Author. It can be shown that the scalar coefficients in tensorial constitutive equations are functions of the set of irreducible invariants (<em>integrity basis) </em>and experimental data. In the following uni-axial relations describing the elastic-plastic transition in uni-axial tests have been generalized to multi-axial states of stress.</p>
<p><em>Keywords: </em>Elastic-plastic transitions under uni-axial load; tensorial generalization; irreducible invariants; tensorial interpolation; example;</p>99226Sun, 21 Nov 2010 05:00:00 ZProf. Josef BettenProf. Josef Betten