MaplePrimes Posts

MaplePrimes Posts are for sharing your experiences, techniques and opinions about Maple, MapleSim and related products, as well as general interests in math and computing.

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  • Sometimes, it’s the little things. Those little improvements that make a good tool even better. Sometimes, it’s as simple as an easy shorthand notation that allows you to create and label points on a graph with a single command. Just to pick a totally random example.

     

    A screenshot of a Maple Learn document containing a math cell and a plot. The math cell reads 'A(1,2)'. The plot show a point plotted at (1,2) with the label 'A'.

    Okay, maybe it’s not totally random. Maybe this new point notation is one of our newest features in Maple Learn, and maybe it’s now easy and quick for you to create labeled points to your heart’s content. Maybe you could learn more about all the ins and outs of this new feature by checking out the how-to document.

    But I can’t make any guarantees, of course.

    That said, if this hypothetical scenario were true, you would also be able to see it in action in our new document on the proof of the triangle inequality.

    A screenshot of a Maple Learn document. The left side shows an explanation of how the triangles are constructed for Euclid's proof of the triangle inequality. The right side shows an adjustable graph of said triangles.

    With this document, you can explore a detailed (and interactive!) visualization of the proof using Euclidean geometry. You can adjust the triangles to see for yourself that the sum of the lengths of any two sides must be greater than the third side, read through the explanation to see the mathematical proof, and challenge yourself with the questions it leaves you to answer. And those points on those triangles? Labeled. Smoothly and easily. I wonder how they might have done it?

    We hope you enjoy the new update! Let us know what other features you want to see in Maple Learn, and we’ll do our best to turn those dreams into reality.

    This is a reminder that we are seeking presentation proposals for the Maple Conference.

    Details on how to submit your proposal can be found on the Call for Participation page. Applications are due July 11, 2023.

    We would love to hear about your work and experiences with Maple! Presentations about your work with Maple Learn are also welcome.

    We’re now coming to the end of Pride Month, but that doesn’t mean it’s time to stop celebrating! In keeping with our celebration of queer mathematicians this month, we wanted to take some time to highlight the works of LGBT+ mathematicians throughout history. While it’s impossible to say how some of these individuals would have identified according to our modern labels, it’s still important to recognize that queer people have always existed, and have made and continue to make valuable contributions to the field of mathematics. It’s challenging to find records of LGBT+ people who lived in times when they would have been persecuted for being themselves, and because of that many contributions made by queer individuals have slipped through the cracks of history. So let’s take the time to highlight the works we can find, acknowledge the ones we can’t, and celebrate what the LGBT+ community has brought to the world of mathematics.

    If you ask anyone to name a queer mathematician, chances are—well, chances are they won’t have an answer, because unfortunately the LGBT+ community is largely underrepresented in mathematics. But if they do have an answer, they’ll likely say Alan Turing. Turing (1912-1954) is widely considered the father of theoretical computer science, largely due to his invention of the Turing machine, which is a mathematical model that can implement any computer algorithm. So if you’re looking for an example of his work, look no further than the very device you’re using to read this! He also played a crucial role in decoding the Enigma machine in World War II, which was instrumental in the Allies’ victory. If you want to learn more about cryptography and how the field has evolved since Turing’s vital contributions, check out these Maple MathApps on Vigenère ciphers, password security, and RSA encryption. And as if that wasn’t enough, Turing also made important advances in the field of mathematical biology, and his work on morphogenesis remains a key theory in the field to this day. His mathematical model was confirmed using living vegetation just this year!

    In 1952, Turing’s house was burgled, and in the course of the investigation he acknowledged having a relationship with another man. This led to both men being charged with “gross indecency”, and Turing was forced to undergo chemical castration. He was also barred from continuing his work in cryptography with the British government, and denied entry to the United States. He died in 1954, from what was at the time deemed a suicide by cyanide poisoning, although there is also evidence to suggest his death may have been accidental. Either way, it’s clear that Turing was treated unjustly. It’s an undeniable tragedy that a man whose work had such a significant impact on the modern era was treated as a criminal in his own time just because of who he loved.

    An image of Alan Turing next to a diagram of a Turing machine computing the busy beaver problem.

    Antonia J. Jones (1943-2010) was a mathematician and computer scientist. She worked at a variety of universities, including as a Professor of Evolutionary and Neural Computing at Cardiff University, and lived in a farmhouse with her partner Barbara Quinn. Along with her work with computers and number theory, she also wrote the textbook Game Theory: Mathematical Models of Conflict. If you want to learn more about that field, check out this collection of Maple Learn documents on game theory. As a child, Jones contracted polio and lost the use of both of her legs. This created a barrier to her work with computers, as early computers were inaccessible to individuals with physical disabilities. Luckily, as the technology developed and became more accessible, she was able to make more contributions to the field of computing. And that’s especially lucky for banks who like having their money be secure—she then exposed several significant security flaws at HSBC! That just goes to show you the importance of making mathematics accessible to everyone—who knows how many banks’ security flaws aren’t being exposed because the people who could find them are being stopped by barriers to accessibility?

    An image of Antonia J. Jones next to the cover of the textbook she wrote: Game Theory: Mathematical models of conflict.

    James Stewart (1941-2014) was a gay Canadian mathematician best known for his series of calculus textbooks—yes, those calculus textbooks, the Stewart Calculus series. I’m a 7th edition alumni myself, but I have to admit the 8th edition has the cooler cover. To give you a sense of his work, here’s an example of an optimization problem that could have come straight from the pages of Stewart Calculus. Questions just like this have occupied the evenings of high school and university students for over 25 years. I suspect not all of those students really appreciate that achievement, but nonetheless his works have certainly made an impact! Stewart was also a violinist in the Hamilton Philharmonic Orchestra, and got involved in LGBT+ activism. In the early 1970s, a time where acceptance for LGBT+ people was not particularly widespread (to put it lightly), he brought gay rights activist George Hislop to speak at McMaster University. Stewart is also known for the Integral House, which he commissioned and had built in Toronto. Some may find the interior of the house a little familiar—it was used to film the home of Vulcan ambassador Sarek in Star Trek: Discovery!

    An image of James Stewart next to the cover of the 8th edition Stewart Calculus textbook.

    Agnes E. Wells (1876-1959) was a professor of mathematics and astronomy at Indiana University. She wrote her dissertation on the relative proper motions and radial velocities of stars, which you can learn more about from this document on the speed of orbiting bodies and this document on linear and angular speed conversions. Wells was also a woman’s rights activist, and served as the chair of the National Woman’s Party. In her activism, she argued that the idea of women “belonging in the home” overlooked unmarried women who needed to earn a living—and women like her who lived with another woman as their partner, although she didn’t mention that part. There is a long-standing prejudice against women in mathematics, and it’s the work of women like Wells that has helped our gradual progress towards eliminating that prejudice. To be a queer woman on top of that only added more barriers to Wells’ career, and by overcoming them, she helped pave the way for all queer women in math.

    An image of Agnes E. Wells next to a table from her dissertation on the movement of stars.

    Now, there is a fair amount of debate as to whether or not our next mathematician really was LGBT+, but there is sufficient possibility that it’s worth giving Sir Isaac Newton a mention. Newton (1642-1727) is most known for his formulation of the laws of gravity, his invention of calculus (contended as it is), his work on optics and colour, the binomial theorem, his law of temperature change… I could keep going; the list goes on and on. It’s unquestionable that he had a significant impact on the field of mathematics, and on several other fields of study to boot. While we can’t know how Newton may have identified with any of our modern labels, we do know that he never married, nor “had any commerce with women”[a], leading some to believe he may have been asexual. He also had a close relationship with mathematician Nicolas Fatio de Duillier, which some believe may have been romantic in nature. In the end, we can never say for sure, but it’s worth acknowledging the possibility. After all, now that more and more members of the LGBT+ community are feeling safe enough to tell the world who they are, we’re getting a better sense of just how many people throughout history were forced to hide. Maybe Newton was one of them. Or maybe he wasn’t, but maybe there’s a dozen other mathematicians who were and hid it so well we’ll never find out. In the end, what matters more is that queer mathematicians can see themselves in someone like Newton, and we don’t need historical certainty for that.

    An image of Isaac Newton next to a Maple Learn document depicting how light passing through a prisim becomes a rainbow.

    So there you have it! Of course, this is by no means a comprehensive list, and it’s important to recognize who’s missing from it—for example, this list doesn’t include any people of colour, or any transgender people. Sadly, because of the historical prejudices and modern biases against these groups, they often face greater barriers to entry into the field of mathematics, and their contributions are frequently buried. It’s up to us in the math community to recognize these contributions and, by doing so, ensure that everyone feels like they can be included in the study of mathematics.

    Some texts distinguish between unary and binary negation signs, using short dashes for unary negation and a longer dash for binary subtraction. How important is this distinction to users of Maple?

    Some earlier versions of Maple used to have short dashes for negation (in some places). Maple 2023 has apparently abandoned the short dash for unary negation, and all such signs are now a long dash.

    How about math books? Do all texts make this short-long distinction? The typesetters for my 2001 Advanced Engineering Math book also opted for all long dashes and that book was set from the LaTeX exported from Maple 20+ years ago. But I also have texts in my library that use a short dash for unary negation, on the grounds that -a, the additive inverse of "a" is a complete symbol unto itself, the short dash being part of the symbol for that additive inverse.

    Apparently, this issue bugs me. Am I making a tempest in a teapot?

    What do you think is the acceptable limit to the effort required to answer a question?

    At what point does the question-and-answer game between two contributors become unreasonable?

    How do you, the most highly ranked, deal with situations that last for days?

    We all know that math is beautiful in and of itself—but sometimes students might need a little convincing. What better way to do that then sprucing up your math with a little colour? With Maple Learn, plot colours are fully customizable. We have several colour palettes to choose from—want your document to evoke the delicate tones of springtime? Looking for a palette that’s colourblind friendly? Or maybe you’re just nostalgic for the colours of Maple V? All these options and more are available for making your graphs colourful and coordinated. But maybe you’re the kind of person who wants to go against the grain, and you laugh in the face of predetermined colour coordination. Don’t worry, we’ve got you covered too! With our colour selector, you can also choose your own custom colours. The full colour spectrum is right at your fingertips. To learn more about how to customize the colours on your document, check out this How-To guide.

    And of course, the potential for colour inevitably leads to the potential for art. Our Maple Learn Art Gallery has plenty of fun and colourful works you can admire and contemplate (and maybe even draw inspiration from!). One of our most recent and most colourful additions is this document showcasing the history of the rainbow pride flag, in honour of June being Pride Month. You can use the slider to move through time, letting you see how the colours on the flag evolved and read about the meanings behind them. And, thanks to the colour selector, the colours match the precise shades used for the original flags! That’s the magic of hexadecimal colours for you.


    Hold on—the magic of hexadecimal colours, I hear you ask? What an enticing concept. If only we had some kind of document, perhaps one made in Maple Learn, that explained how hexadecimal colours worked and included an interactive example so that you could easily see how the red, green, and blue colour values blend together to create any given colour… Too bad we don’t!

    Just kidding. Of course we do.

    If all these colours have inspired you, be sure to check out our Call for Creative Works for the upcoming Maple Conference! Maybe your colourful creation could be this year’s winner.

    If you've seen Paulina's announcement then you know that we are once again holding a virtual Maple Conference this year.  As well, we are once again going to have a virtual gallery featuring artwork and creative projects submitted by the Maple community!

    Last year we had a number of great submissions to our Maple Art Gallery and our Maple Learn Creative Showcase.  These were our excellent prize winners.

    From left to right we have A visualization of all the primitive roots of 10037 created by Simon Plouffe, winner of the Judge’s Choice, Mother’s Day Rose created with Maple plots by Greg Wheaton, winner of the People’s Choice, and Mona Lisa in Maple Learn created by Paul DeMarco (with help from Leonardo DaVinci), the winner of the People’s Choice for the Maple Learn Showcase.

    This year we are expanding the Gallery into two collections to encourage more people to submit.  They are

    • The Art Gallery - A small gallery to highlight high effort, mathematically interesting works (with stricter criteria)

    • The Creative Works Showcase - A larger showcase for nearly any interesting visual works created with Maplesoft products like Maple Learn and Maple

    Feel free to submit nearly anything cool for the Creative Works Showcase, if we find it particularly impressive we might even ask you to let us consider it for the gallery.  Also, do not be intimidated by the title "Art Gallery" we're looking for anything that has taken some artistic effort and tells a mathematical story.

    For more information on critera and how to submit, please visit our Call for Creative Works.  The important deadline to know is the September 14th deadline for submission of works with virtual gallery reception and awards ceremony durring the conference October 26-27.

    I look forward to seeing all the submissions for the Maple community again this year!

    We are happy to announce another Maple Conference this year, to be held October 26 and 27, 2023!

    It will be a free virtual event again this year, and it will be an excellent opportunity to meet other members of the Maple community and get the latest news about our products. More importantly, it's a chance for you to share the work work you've been doing with Maple and Maple Learn. There are two ways to do this.

    First, we have just opened the Call for Participation. We are inviting submissions of presentation proposals on a range of topics related to Maple, including Maple in education, algorithms and software, and applications. We also encourage submission of proposals related to Maple Learn. 

    You can find more information about the themes of the conference and how to submit a presentation proposal at the Call for Participation page. Applications are due July 11, 2023.

    Presenters will have the option to submit papers and articles to a special Maple Conference issue of the Maple Transactions journal after the conference.

    The second way in which to share your work is through our Maple Art Gallery and Creative Works Showcase. Details on how to submit your work, due September 14, 2023, are given on the Web page.

    Registration for attending the conference will open later this month. Watch for further announcements in the coming weeks.

    I encourage all of you here in the Maple Primes community to consider joining us for this event, whether as a presenter or attendee!

    Paulina Chin
    Contributed Program Chair

    Happy Pride Month, everyone! June is a month for recognizing and celebrating the LGBT+ community. It was started to mark the anniversary of the Stonewall riots, which were a landmark event in the fight for LGBT+ rights. We celebrate Pride Month to honour those who have fought for their rights, acknowledge the struggles the LGBT+ community continues to face to this day, and celebrate LGBT+ identities and culture.

    This Pride Month, I want to give a special shoutout to those in the math community who also identify as LGBT+. As a member of the LGBT+ community myself, I’ve noticed a fair amount of stigma against being queer in math spaces—and surprisingly often coming from within the community itself. It’s one thing for us to make jokes amongst ourselves about how none of us can sit in chairs properly (I don’t even want to describe how I’m sitting as I write this), but the similar jokes I’ve heard my LGBT+ friends making about being bad at math are a lot more harmful than they might realize. And of course it isn’t just coming from within the community—many people have a notion (whether conscious or unconscious) that all LGBT+ people are artistically inclined, not mathematical or scientific. Obviously, that’s just not true! So I want to spend some time celebrating queerness in mathematics, and I invite you to do the same.

    One of the ways we’re celebrating queerness in math here at Maplesoft is with new Pride-themed Maple Learn documents, created by Miles Simmons. What better way to celebrate Pride than with trigonometry? This document uses sinusoidal transformations to mimic a pride flag waving in the wind. You can adjust the phase shift, vertical shift, horizontal stretch, and vertical stretch to see how that affects the shape of the flag. Then, you can watch the animation bring the flag to life! It’s a great way to learn about and visualize the different ways sinusoidal waves can be transformed, all while letting your colours fly!

     

    A screenshot of a Maple Learn document. The plot window shows a pride flag constructed from sine waves, and the document describes how we will be adding wind in the form of function transformations.

    For more trigonometry, you can also check out this fun paint-by-numbers that can help you practice the sines, cosines, and tangents of special angles. And, as you complete the exercise, you can watch the Pride-themed image come to life! Nothing like adding a little colour to your math practice to make it more engaging.

     

    A screenshot of a Maple Learn document showing a partially coloured paint-by-numbers grid, where the numbers are special trig angles.

    If you’re looking for more you can do to support LGBT+ mathematicians this Pride Month, take a look at Spectra, an association for LGBT+ mathematicians. Their website includes an “Outlist” of openly LGBT+ mathematicians around the world, and contact information if you want to learn more about their experiences. The Fields Institute has also hosted LGBT+Math Days in the past, which showcases the research of LGBT+ mathematicians and their experiences of being queer in the math community. Blog posts like this one by Anthony Bonato, a math professor at Toronto Metropolitan University, and interviews like this one with Autumn Kent, a math professor at the University of Wisconsin-Madison, can also help allies in mathematics to understand the experiences of their queer colleagues and how to best support them. Math is everywhere and for everyone—so let’s make sure that the systems we use to teach and explore math are for everyone too!

    Happy Pride! 🏳️‍🌈

    We've just launched Maple Flow 2023!

    The new release offers many enhancements that help you calculate and write reports faster, resulting in polished technical documents. Let me describe a few of my favorite new features below.

    You can now change the units of results inline in the canvas, without taking your hands off the keyboard. You can still use the Context Panel, but the new method is faster and enhances the fluid workflow that Flow exemplifies.

    You can also enter a partial unit inline; Flow will automatically insert more units to dimensionally balance the system.

    This is useful when results are returned in base dimensions (like time, length and mass) but you want to rescale to higher-level derived units. For an energy analysis, for example, you might guess that the result should contain units of Joules, plus some other units, but you don't know what those other units are; now, you can request that the result contains Joules, and Flow fills the rest in automatically.

    The new Variables Palette lists all the user-defined variables and functions known to Flow at the point of the cursor. If you move your grid cursor up or down, the variables palette intelligently removes or adds entries.

    You can now import an image by simply dragging it from a file explorer into the canvas.

    This is one of those small quality-of-life enhancement that makes Flow a pleasure to use.

    You can now quickly align containers to create ordered, uncluttered groups.

    We've packed a lot more into the new release - head on over here for a complete rundown. And if you're tempted, you can get a trial here.

    We have a lot more in the pipeline - the next 12 months will be very exciting. Let me know what you think!

    On this day 181 years ago, Christian Doppler first presented the effect that would later become known as the Doppler effect. In his paper “On the coloured light of the binary stars and some other stars of the heavens”, he proposed (with a great deal of confidence and remarkably little evidence) that the observed frequency of a wave changes if either the source or observer is moving. Luckily for Doppler, he did turn out to be right! Or at least, right about the effect, not right about supernovas actually being binary stars that are moving really fast. The effect was experimentally confirmed a few years later, and it’s now used in a whole variety of interesting applications.

    To learn more about how the Doppler effect works, take a look at this Maple MathApp. You can adjust the speed of the jet to see how the frequency of the sound changes, and add an observer to see what they perceive the sound as. You can even break the sound barrier, although the poor observer might not like that so much!

     

    A screenshot of a Maple MathApp, showing a visual representation of sound waves coming off a moving jet, with sliders to adjust the speed.

     

    For Maple users, you can also check out the MathApp on the relativistic Doppler effect. You’ll find it in the Natural Sciences section, under Astronomy and Earth Sciences. Settle in to watch those colours come to life!

    A screenshot of a Maple MathApp showing a spectrum of colours, with sliders to control the initial wavelength of the light and a dial showing the current velocity of the viewer

     

    But wait, I mentioned interesting applications, didn’t I? And don’t worry, I’m not just here to talk about sirens moving past you or figuring out the speed of stars (although admittedly, that one is pretty interesting too). No, I’m talking about robots. Some robots make use of the Doppler effect to help monitor their own speed, by bouncing sound waves off the floor and measuring the frequency of the reflected wave. A large change in frequency means that robot is zooming!

    The Doppler effect is also used in the medical field—Doppler ultrasonography uses the Doppler effect to determine and visualize the movement of tissues and body fluids like blood. It works by bouncing sound waves off of moving objects (like red blood cells) and measuring the result. The difference in frequency tells you the speed and direction of the blood flow, in accordance with the Doppler effect! Pretty neat, if you ask me.

    And like any good scientific phenomena, the Doppler effect can be used for both work and pleasure. The Leslie speaker is a type of speaker invented in the 1940s that modifies the sound by rotating a baffle chamber, or drum, in front of the loudspeakers. The change in frequency dictated by the Doppler effect causes the pitch to fluctuate, creating a distinct sound that I can only describe as “woobly”. The speaker can be set to either “chorus” or “tremolo”, depending on how much woobliness the user wants. It was typically used with the Hammond organ, and you can hear it in action here!

    You know who else uses the Doppler effect? Bats. Since they rely on echolocation to get around, they need some way to account for the fact that the returning sound waves won’t be at the same pitch that they were sent out at. This fantastic video explains it far better than I ever could, and involves putting bats on a swing, which I think should be enough of a recommendation all on its own.

    That’s it for our little foray into the Doppler effect, although there’s still a lot more that could be said about it. Try checking out those Maple MathApps for inspiration—who knows, maybe you’ll find a whole new use for this fascinating effect!

    I have been working in the building construction industry for more than 30 years, and one of the big things which has been coming up is parametric design.

    While every major big software company has come up with its own package (Bentley with Generative Components, Autodesk with Dynamo), there is one software now that has won over all them - Grasshopper 3D.

    Nowadays all major software products are trying to write bidirectional links to Grasshopper. Inhouse we do have Tekla on the BIM side and FEM-Design on the calculation side where both are capable to link to Grasshopper.

    Grasshopper itself is also capable of running Python code, though in a reduced kind of way.

    I would very much like to see Maple to have a Grasshopper link, probably achievable through Python.

    At least Maplesoft should have a quick look at it, to see if this is possible or not.

    Ever wonder how to show progress updates from your executing code without printing new lines each time?

    One way to do this is to use a TextArea component and the DocumentTools package. The TextArea could be inserted from the Components Palette in Maple, or programmatically like so:

    restart;

    with(DocumentTools):

    with(DocumentTools:-Components):

    with(DocumentTools:-Layout):

    s := "0": #initial text value

    T := TextArea(s, identity = "TextArea0"):
    xml := Worksheet(Group(Input(Textfield(T)))):

    insertedname:=InsertContent(xml)[1,1]: #find the inserted component name in case changed

    for i to 10 do #start the demonstration procedure
       Threads:-Sleep(1);
       SetProperty(insertedname,value,sprintf("%d",i),refresh=true);
    end do:

    Maplets:-Examples:-Message("Done");


    Download text-area-update-progress.mw

    Probability distributions can be used to predict many things in life: how likely you are to wait more than 15 minutes at a bus stop, the probability that a certain number of credit card transactions are fraudulent, how likely it is for your favorite sports team to win at least three games in a row, and many more. 

    Different situations call for different probability distributions. For instance, probability distributions can be divided into two main categories – those defined by discrete random variables and those defined by continuous random variables. Discrete probability distributions describe random variables that can only take on countable numbers of values, while continuous probability distributions are for random variables that take values from continuums, such as the real number line.

    Maple Learn’s Probability Distributions section provides introductions, examples, and simulations for a variety of discrete and continuous probability distributions and how they can be used in real life. 

    One of the distributions highlighted in Maple Learn’s Example Gallery is the binomial distribution. The binomial distribution is a discrete probability distribution that models the number of n Bernoulli trials that will end in a success.

    This distribution is used in many real-life scenarios, including the fraudulent credit card transactions scenario mentioned earlier. All the information needed to apply this distribution is the number of trials, n, and the probability of success, p. A common usage of the binomial distribution is to find the probability that, for a recently produced batch of products, the number that are defective crosses a certain threshold; if the probability of having too many defective products is high enough, a company may decide to test each product individually rather than spot-checking, or they may decide to toss the entire batch altogether.

    An interesting feature of the binomial distribution is that it can be approximated using a different type of distribution. If the number of trials, n, is large enough and the probability of success, p, is small enough, a Poisson Approximation to the Binomial Distribution can be applied to avoid potentially complex calculations. 

    To see some binomial distribution calculations in action and how accurate the probabilities supplied by the distribution are, try out the Binomial Distribution Simulation document and see how the Law of Large Numbers relates to your results. 

    You can also try your hand at some Binomial Distribution Example Problems to see some realistic examples and calculations.

    Visit the Binomial Distribution: Overview document for a more in-depth explanation of the distribution. The aforementioned Probability Distributions section also contains overviews for the geometric distribution, Poisson distribution, exponential distribution, and several others you may find interesting!

    This post discusses a solution for modeling a traveling load on Maplesim's Flexible Beam component and provides an example of a bouncing load.

    The idea for the above example came from an attempt to reproduce a model of a mass sliding on a beam from MapleSim's model gallery. However, reproducing it using contact components in combination with the Flexible Beam component turned out to be not straightforward, and this will be discussed in the following.

    To simulate a traveling load on the Flexible Beam component, one could apply forces at discrete locations for a certain duration. However, the fidelity of this approach is limited by the number of discrete locations, which must be defined using the Flexible Beam Frame component, as well as the way in which the forces are activated.

    One potential solution to address the issue of temporal activation of forces is to attach contact elements (such as Rectangle components) at distinct locations along the beam, which are defined by Flexible Beam Frame components, and make contact using a spherical or toroidal contact element. However, this approach also introduces two new problems:

    • An additional bending moment is generated when the load is not applied at the center of the contact element's attachment point, the Flexible Beam Frame component. Depending on the length of the contact element, deformations caused by this moment can be greater than the deformation caused by the force itself when the force is applied at the ends of the contact element. Overall, this unwanted moment makes the simulation unrealistic and must be avoided.
    • When the beam bends, a gap (see below) or an overlap is created between adjacent Rectangle components. If there is a gap, the object exerting a force on the beam can fall through it. Overlaps can create differences in dynamic behavior when the radius of curvature of the beam is on the opposite side of the point of contact.

    To avoid these problems, the solution presented here uses an intermediate kinematic chain (encircled in yellow below) that redistributes the contact force on the Rectangle component on two support points (ports to attach Flexible Beam Frame components) in a linear fashion.

     

     

    To address gaps, the contact element (Rectangle) attached to the kinematic chain has the same width as the chain and connects to the adjacent contact elements via multibody frames. In the image below, 10 contact elements are laid on top of a single Flexible Beam component, like a belt made out of tiles. The belt has to be pinned to the flexible beam at one location (highlighted in yellow). The location of this fixed point determines how the flexible beam is loaded by tangential contact forces (friction forces) and should be selected carefully.

     

     

    Some observations on the attached model:

    • Low damping and high initial potential energy of the ball can result in a failed simulation (due to constraint projection failure). Increasing the number of elastic coordinates has a similar effect. Constraint projection can be turned off in the simulation settings to continue simulation.
    • The bouncing ball excites several eigenmodes at once, causing the beam to wiggle chaotically in combination with the varying bouncing frequency of the ball. A similar looking effect can also be achieved with special initial conditions, as demonstrated with Maple in this excellent post on Euler beams and partial differential equations.
    • Repeated simulations with low damping lead to different results (an indication of chaotic behavior; see three successive simulations below (gold) compared to a saved solution(red)). The moment in the animation when the ball travels backward represents a metastable equilibrium point of the simulation. This makes predictions beyond this point difficult, as the behavior of the system is highly dependent on the model parameters. Whether the reversal is a simulation artifact or can happen in reality remains to be seen. Overall, this example could evolve into a nice experimental fun project for students.

    • Setting the gravitational constant for Mars, everything is different. I could not reproduce the fun factor on Earth. A reason more to stay ,-)

    Ball_bouncing_on_a_flexible_beam.msim

     

     

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