Approximate Solutions - The Ritz Method

[@Alina-ek4jr]

After years of engineering you’re the first one to properly explain to me how and why to choose a certain shape function and not just throw it in the excercise. Never thought much about it but that just gave me so much clarity!!

[@SMD1999]

I watch these to go to bed. No disrespect of course, you’re teaching is absolutely wonderful

[@soroushasadian9100]

One of the best instructions I've ever had. Thank you sir

[@Freeball99]

Glad it helped

[@Sjbak_5mountain]

Now i can digest a bit and see the fundamental workframe of FEM. I appreciate

[@Omar-sj7wl]

Always great to see a new video from you.

[@jv2781]

Keep up the great videos! hopefully in the future a video on plates and shells can be done, I feel that there isn't enough satisfactory material on this available.

[@ergonautus4099]

Very interesting! You are very good with explaining notation in all your videos. It helps a lot when trying to understand transformations and such. Thanks!

[@GranHerrmanno]

15:39: Why there is no factor "1/2" in the work of the external force P?

[@Freeball99]

I believe you are confusing two different concepts. Work is defined as force x displacement. 1. In the case of a tip load, the force P is CONSTANT throughout the displacement. When the tip displacement is 0, the force is still P and when the displacement reaches w(L), the force is still P. So no 1/2 needed here. The work is P x w(L) 2. On the other hand if we consider the work done by internal, linear elastic forces we are, in effect, saying that the elastic load is 0 initially when the displacement is 0 AND LINEARLY APPROACHES P when the displacement reaches d, so we can think of the average load as 1/2 P. The work is then 1/2 P x d. This is when get get the factor of 1/2 . Typically, we will see the 1/2 when dealing with energy (strain energy or kinetic energy), but not when dealing with external work.

[@simonlourenco6398]

Your video was amazing ! Thanks ! I'm french but still, i understood everythings you've said. It was very clear, everythings was defined and explained. I feel very lucky to have found this video. Many thanks !!

[@Alliban59]

If I try to derive an equation in general, I will always apply the input (here force P) in positive direction in order to avoid confusion. This can also be achieved by using an appropiate coordinate system. Otherwise, the sign of the input quantity has to be contrary to what follows from the coordinate system which may give rise to error.

[@talalsyed9761]

nicely explained thank you very much...

[@Freeball99]

Glad it was helpful!

[@andcunsan]

Piece of gold finaly I have a feel on one of the "ingredients" of finite element method

[@teny238]

Thank you very good video! I was surprised you didn't talk about the Rayleigh quotient, which is really useful to find the approximate natural frequencies of a system, but then again the video was more focused on static problem

[@Freeball99]

This is such a broad topic that I could not fit it all into one video. This one was already on the longer side. So I started with a simple static problem. There will be additional videos for sure which will deal with Rayleigh's Quotient and the dynamic case.

[@fawgawtten9515]

Thank you

[@nabin6976]

Do you have a textbook or a set of textbooks you refer to while making these videos? Thanks for these videos. Much appreciated.

[@Freeball99]

Much of it comes from my class notes that I took back in the day, however, my "go-to" textbook for the more introductory material is "Mechanical Vibrations" by S.S. Rao and for the more advanced material tends to be Dym & Shames "Solid Mechanics: A Variational Approach". In addition, I look at various online articles and papers and use these for insights. For some topics, Wikipedia is also a useful reference.

[@mathunt1130]

Excellent video. Is there a way to choose a shape function? You seemed to pluck the one you gave from thin air.

[@Freeball99]

I didn't just pluck it out of thin air. I needed to satisfy 4 boundary conditions and I knew that for this I could use a polynomial with 4 constants (ie a 3rd degree polynomial) and apply the 4 BC's to solve for the four constants. Because I wanted to use the Ritz Method, I added a degree to the polynomial (4th degree) so that I have a 5th constant to solve for. Solving for the 5th constant required application of the Ritz method. That said, there are literally entire books that have been written on shape functions. You can generally pick these from a book. And, yes, some shape functions work better than others. The closer the shape function is to approximating the actual mode shape, the more accurate the approximate results that are produced. Two things to take away from this: 1) you can get shape functions from a book 2) you can always use a polynomial of higher-enough degree to satisfy the required boundary conditions and then just add a degree (or more) to that.

[@mcastanonf]

Great! You are the best, can you please run one of this examples using Newton’s second law?, simple supported beam with a uniform load along the entire beam. Thanks in advance

[@Freeball99]

Will add it to my list.

[@vinayakvarma4887]

Can we use Ritz Method for Dynamic problems as well, like a free vibrating Cantilever Beam?

[@Freeball99]

Yes, you certainly can. I will make a future video showing that.

[@steveshaver4000]

Hi! Can you explain how your variational method is a “true” variational method, when it is not derived from an Euler Lagrange equation?

[@Freeball99]

It's a variational method because in invokes the Principle of Minimum Potential Energy which tells us that the potential energy for a system in equilibrium in an extremum. This implies that the variation of the potential energy (the functional) must be zero which, in turn, leads to equation 5. In applying this condition, we are able to find the values for the Ritz Coefficient(s) which minimize the potential energy.

[@Eidachsans]

What happens when you use a cubic polynomial as a shape function? Does this lead to the exact analytical solution for the beam?

[@Andy-hy8px]

Correct. If you substitute the actual mode shape for the shape function, then you'll get the exact result. In general a "better" choice of shape function will cause a more rapid convergence of the approximate result to the exact result.

[@jimbojones1243]

Hi Freeball - love your videos! Just wondering if you're pf Zimbabwean or South African origin. I hear a bit of it in your accent - I'm from Zim originally myself!

[@Freeball99]

Yes, grew up in Durban...a long time ago.

[@mrkaplan6062]

At the start of your videos, bell sound is too loud, specially when using headset, could you do something about it?

[@Freeball99]

Sorry, but it can't be edited at this stage. Will pay closer attention to this is future videos. Thanks for the feedback.

[@raduandreimatei6863]

If I could add some constructive criticism you talked a lot in terms of the "Ritz method" and instead of calling things by their practical name, e.g., saying "we apply the Ritz condition" instead of we just extremize the potential energy functional, things became a bit confusing and unrelated to variational calculus, which from what I understand is a core principle of this method. Another thing I would like to mention is that you already know the steps you take but we don't when we solve problems so by calculating W,xx in advance because you know you're going to need it afterwards, it kind of breaks the natural flow of the solution of the problem that a person who doesn't know the results in advance has to experience and it becomes confusing so sticking to the process that a person who is solving it for the first time would is less confusing in my opinion. Great content, I hope this comment doesn't make it seem otherwise.

[@Freeball99]

Thanks for this and so noted. I always appreciate getting feedback from my viewers... There was a lot of material that was crammed into this one video (really requires additional videos to better describe the topic). As a result of trying to fit all the material into the slides, it was useful to derive W,xx just before I used it on the next slide.

[@umangkumarpatel3661]

How displacement is upward at tip at x=l ?? Both force p and point at x=l is same direction i.e. downwards

[@Freeball99]

Not sure which equation you are referring to, but eqn 24 shows the tip deflection to be negative.

[@SourabhBhat]

The positive direction of displacement is upwards.

[@portablesoup1266]

@@SourabhBhat I was wondering the same thing as Umangkumar. But this makes sense he's talking about the coordinate system

[@mihkelKaalto]

At 15:17 video says, that when displacement and force are in opposite direction then the work is negative. That is correct. But it still seems to me that they are both in negative direction, hence work should be positive. However, at 15:10 you say that displacement is positive upward, but eq.(15) kinda gives that displacement is downwards, not upwards. Sorry for my confusion.

[@sattaralmarzouk9944]

Use Rayleigh - Ritz method to find an approximate solution of the problem y"-y+4xe = 0, y(0) - y(0) = 1, y'(1)+y(1) = -e.