Bar Finite Element - Deriving the Mass and Stiffness Matrices
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- Опубликовано: 11 окт 2024
- Download notes for THIS video HERE: bit.ly/3oDyPeK
Download notes for my other videos: bit.ly/37OH9lX
Introduction to the finite element method. Derivation of the bar element for axial vibrations.
One of the best work that I have found online.
So many thanks for your words of encouragement. Very nice to hear!
Still it is the best work I can find on RUclips... Such a great explanation 😊
Excellent video on finite element method. Easy to follow and nicely organized. Thanks!
Thanks for the feedback. It's certainly rewarding to hear.
awesome, elegant derivations I found them quite clear. Aerospace engineers process information the same way it appears, keep it up!
Thank you very much for the feedback and the encouragement. Much appreciated!
At about 25 minutes in, right hand side of the page for strain energy, looks like you're missing a transpose on the q at the end, when you plug 15 into 21 and rewrite. The exercise was for finding the stiffness matrix, so it wasn't important, but I figured I would mention it anyway. Thank you for this excellent learning material.
I am very new to FEM. Your video makes it easier to understand it. Thanks!
Thank you. Feedback like this makes it all worth my while.
High quality educational video! Thank you!
Man, that's fucking amazing. Please, continue with your videos about FEM.
I think this is a really excellent series and that you have presented the material in a clear and understandable way. The thing that still confuses me is that the next steps in Finite Elements usually involve this definition of weak solutions, and Galerkin methods, etc. I was hoping you could explain the connection between the material you have presented so far, and the subsequent ideas. I looked at the Rao book you suggested, and he does not explain the additional ideas like Galerkin methods, etc. If you know of any good references for those additional ideas, could you please suggest them. Thanks.
I definitely need to make a videos on Galerkin and Rayleigh-Ritz methods. In the meantime, try Dym & Shames, Solid Mechanics: A Variational Approach.
You are an amazing person, you explain the concept and code it.
Awesome video..
Everything which i did not find elsewhere were in this video... Thank you🙏🙏
I wish I could see you to say thank you in person. You have no idea how much these videos help your subcribers. RUclips should give educational videos more credit to be fair.
Thank you once more sir
amazing explanations
I went through your answer for [K] through MatLab and found that the answer was [-1, 0 ; 0, 1] I might be wrong but I did make sure I differentiated the Shape Functions before Integrating them again with the limits from 0 to L. I could send you my MATLAB code if you want to, in order to verify. Otherwise I found this video very very helpful in understanding these concepts for my Dynamics module for my degree.
Cheers boet! B)
I somehow missed this until now...
The [K] matrix as I have it is correct. This is a well-known result that can be found in many texts. Additionally, your diagonal elements of the stiffness matrix MUST be positive values else it means that when stretching the bar, you are releasing energy instead of storing it.
I believe there is something wrong with your MATLAB implementation. These shape functions and integrals are very simple for the bar element, so my suggestion is to solve them by hand - it will probably take you less than a minute.
Brilliant!
Awesome. Thank you very much! You are incredible.
thank you very much
Hi! When solving partial differential equations with the finite element method, do you need to assume some structure of the solution before solving? For example, here, we are assuming that the solution has a form that can be generated using the separation of variables method. If we assumed a different structure for the solution, would the finite element method produce different results?
I am not aware of any other method for solving these equations.
Didn't you use phi instead of phi' (transposed) to calculate Q at 30:28?
really helpful !!
What software is that you are using for writing all this? I give online classes myself and am currently struggling to find a decent software.
Video is amazing BTW, thanks very much for the content and patience. Greetings from Argentina 🇦🇷🇦🇷
Software is "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil. Video recording using Quicktime.
I don't see the connection between the wave equation and equation 2. Equation 2 was assumed to be the governing equation of motion for the bar, and the assumed linear function for the nodal displacements, equation 3, was ultimately applied to that. I don't really see how the wave equation factored in at all.
The finite element equation of motion is essentially a discretized, numerical representation of the wave equation. It breaks down the complex, continuous problem into a series of simpler, discrete problems that can be solved with linear algebra. This connection allows us to analyze the behavior of physical systems using computational tools, even when an analytical solution to the wave equation is not possible.
We know from the use of Lagrange's equations that at the heart of the equations of motion are the expressions for the kinetic and potential energies (equations 17 & 21) and we use these to derive the consistent mass and stiffness matrices by substituting our assumed, discretized displacement functions. They are so-called "consistent" because for the same displacement and velocities at each end, both the actual bar and the finite element of the bar will store the same amounts of kinetic and potential energy. This is the connection between the two equations.
perfect
Hi! Can you explain why this method does not find the additional solutions u=+-ax, a = constant and u = +- at , a = constant
I'm not sure I understand the question.
@@Freeball99 those functions should also satisfy the wave equation?
We satisfy the spacial requirements of the equation (approximately) by using appropriate shape functions (ie ones that satisfy the boundary conditions and a few other requirements). This produces a set of 2nd order ODE's in time. Since these equations are 2nd order, there will be two solutions and the result will be a linear combination of these two solutions (based on initial conditions).
Sir, which book would you recommend for vibrations ?
The textbook I used (a long time ago) was "Mechanical Vibrations" by Singiresu Rao (www.amazon.com/Mechanical-Vibrations-5th-Singiresu-Rao/dp/0132128195) - which is okay. I don't think it is a great textbook because it is a little hard to follow (and has some errors), but it is pretty complete in terms of the information it contains. FYI - I haven't seen too many great introductory textbooks out there. Most of them are very expensive and are often not general enough. For this reason, many professors use their own notes and handouts when teaching this class.
Many textbooks use different notation and many times it comes down to a personal preference - some textbooks are just easier to read than others and sometimes it's easier to use a book with the same notation as the teacher is using. From my experience, this is a subject that most students find very difficult to learn on their own from a textbook because it test one's understanding of both math and physics simultaneously. Having someone explain it really helps a lot - which is why I made the video.
+Freeball thanks sir for your reply. We just started this course. And we are studying from daniel j inman. Your lectures are really helpful
How is Daniel's book?
classical lecture from a sound Instructor
shit you got vibration you got fem. my whole semester is here. any propulsion lessons?
any fortran or pseudocode lessons?