Deriving Lagrange's Equations
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- Опубликовано: 23 ноя 2020
- Deriving Lagrange's Equations using Hamilton's Principle. Demonstrating how to incorporate the effects of damping and non-conservative forces into Lagrange's equations.
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I opened this video just to say thank you !! I wish I had a professor like you... my professor opens the pdf document and start talking for hours.....
You're the man!!!! Please keep doing content for youtube! Your job it's amazing!
Thank you very much for those lectures. What your are doing is just amazing. I'm trying to build my own FEM software with python in back-end and you did everything I need as fundamental.
You, good Sir, are heaven sent!
Thank you so much!
Hi...Your teaching has helped me a lot in understanding the basics with such clarity. Request you to upload a video on Kane's method for deriving the equations of motion. It will mean a lot.
Thank for this suggestion. I have some other videos to make first, but would be happy to make a video on Kane's Method in the future.
That's one of the best explanations I've seen in a long long time.
Thank you very mich sir!
Would you mind if I show/link it to my Students?
Go for it. That's what they're for.
Outstanding video...
Excellent! Tks a lot
Thank you very much for sharing, sir!
My pleasure!
Thank youuu
Thank you sir! Could you please make a video about Expansion and Enclosure theorems in Continuous Vibration?
I’m not familiar with this. Do you perhaps have some reference material?
@@Freeball99 Yes, I have. I will send some links and screenshots to your e mail
What exactly the gamma constant (11:31) is? Is there any math relation for it?
I'm currently developing a research paper on the structural dynamics of a cantilever beam, and your playlist with the derivations has been essential for my studies. If you could also tell me your references, it would help a lot. Thank you in advance!
γ is a coefficient that quantifies the extent of damping relative to the internal elastic forces. This is a quantity that is typically determined in the lab and is a function of the material of the beam, its geometry and the boundary conditions and the type of excitation. Typical values range from about 0.01 to 0.1. The specific reference I used for the is "Dynamics of Structures" by Hurty & Rubinstein. The book is long since out of print, but you can find a copy at archive.org. Not sure how helpful you'll find it though. For a reference on variational principles, my go-to reference is "Structural Dynamics: A Variational Approach" by Dym & Shames which is an excellent book! You'll likely also find an archived copy somewhere online.
@@Freeball99 thank you very much sir! i'll keep watching
Thank you. Which software have you used to prepare slides?
The app is called "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.
How can substituting the the Lagrangian into equation 11 result in Laplace's equation, seeing how there's no time derivative term for the potential energy? Seems like there isn't a way to get to d/dt(dL/dq_dot).
The time derivative of the potential energy with respect to q_dot will just be zero, so only the kinetic energy will contribute to the first term of equation 13. Also, in many problems, the kinetic energy will not be depend directly on the generalized coordinate, q, and so, in that case, the kinetic energy would have zero contribution to the second term of equation 13. Therefore, this is not a problem and it allows us to write Lagrange's equation is the compact form shown in 13.
Hello, good video, one question, what application did you use for the video?
The app is called "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.
why T=T(qdot, ... q....) depends on q? I thought kinetic energy only depends on velocity
Yes, the kinetic energy can depend on both the velocity AND position, in general. This comes up often in rotating system, for example the elastic pendulum ruclips.net/video/iULa9A00JpA/видео.htmlsi=CAZUfbyWsu-joQsY&t=153
@@Freeball99 Thank you
Hello sir,
What is meant by virtual displacement?
A virtual displacement means displacing the object/surface by an infinitesimal amount - it's the same idea as taking the variation of the path. This is similar to considering differential slices of objects when studying differential calculus.
this lagrangian equation which is eqn 16 in the video, is derived from using hamilton principle, right! But what about lagrangia equation using D'Alembert principle....coz I'm little bit confused between lagrangian eqution from hamilton principle and d'alembert principle. please solve my query asap.!!!!!
hello Sir, please help me out of this query as soon as possible!!!!!
Sorry for the delayed response, but I missed your comments until now...
Both methods will yield the same equations of motion for systems where they are applicable, but they offer different perspectives and may be more convenient for different kinds of problems. Hamilton's Principle was discovered about 100 years after the concept of the Lagrangian. Originally, d'Alembert's Principle was used to derive the Euler-Lagrange equation, but both lead you to the same result?
Hamilton's Principle is generally more abstract and is based on the concept of minimizing action. It often applies in a broader context, including fields beyond classical mechanics.
d'Alembert's Principle is more intuitive and directly connected to the forces in the problem. It is particularly useful when dealing with non-conservative forces.
At 6:06 is the top equation is a Laplace and the bottom equation is a Poisson. Am I wrong?
No, these are not Laplacian operators. The Laplacian operator represents the divergence squared which is not what appears in either of these equations. Also, Laplace's equation typically deals with an equilibrium state (so no time dependency) while these equations are time-dependent.
I gather that the damping forces can be internal and external. Or are they all external? That's the only point I am confused about.
It depends on the type of damping that you have, but usually you can model a damping force as external if you're unsure. Viscous damping can be treated as either.
04.30 why derivative of qi is zero? i couldn't understand
This goes back to an earlier video on Variational Calculus, this is the equivalent of the boundary condition ...the idea is that because we know the state of the system at times t1 and t2, therefore the variation at each of the end-points of the path is zero.
In other words, this conditions says that when considering an optimal path from t1 to t2, all trial paths must be constrained to have the same beginning and ending points as the original path.