Very illuminating videos , thank you ... what i would like to see though is a more complicating boundary condition problem with springs or mass for example on both ends ... because in books cantilever is demonstrated but exercises are more like mass - mass b.c. which makes the problem more difficult to solve ... if you could make a video on this topic that would be awesome
Yes indeed, these are difficult problems to solve. Formulating the BCs themselves is still manageble. But the resulting EVP is almost always impossible to solve analytically. For that reason, one must rely on approximation techniques (Rayleigh-Ritz) or numerical techniques for solving the eigenvalue problem. Because in this course, I restricted myself to problems that have an analytical solution, I did not discus these more complicated BCs.
(Possibly stupid question) Shouldn't there be some sort of coefficients in from of each term U(x)\eta(t) so that the sum converges? I'm modeling some vibrations in a beam in MATLAB, and, without these coefficients in front, the first 10 terms appear not to capture ~99% of the variability, which, I would think the first 4 or 5 terms would ordinarily do so. TIA
Thanks for all your nice comments Jeff. The equations of motion of a vibrating string, bar and shaft are in fact all examples of the Wave Equation. The general solution can be written as a Fourier series in space. So in this case, trying a Fourier series will not just be a good approximation, but actually result in the exact solution as presented here. For other continuous problems though, like a vibrating beam, the equation of motion is fourth order in space. Using a Fourier series as general solution will not be able to satisfy all boundary conditions and therefore only approximate the solution. The general solution contains a sine/cosine combo and a sinh/cosh combo. You will see this in the videos about vibrating bars. I hope that once you have arrived there, this will become clear. If not, please ask me again at that time.
Merci monsieur pour votre belle méthode de structurer les idées pour l’apprenant. Une demande s’il te plait, un cours qui porte sur un problème aux valeurs propres de bi-Laplacien, ou des cours ou documents sur le net dans ce sens. Merci infiniment pour vos efforts mon professeur.
Merci pour ton compliment Hafid. Je réponds en Anglais car mon Français n'est pas bon. Eigenvalue problems involving the bi-Laplacian operator can be found in structural mechanics problems like vibrating plates or other problems in which you have 2 spatial directions. A very good text book that I like that contains this is for instance Vibrations of Continuous Systems by Rao. Essentially it means that you have to perform separation of variables in 2 spatial directions and time. So you start with v(x,y,t) = X(x)Y(y)T(t) and similar solution strategy can be applied as discussed in my video. I hope this answers your question.
Very illuminating videos , thank you ... what i would like to see though is a more complicating boundary condition problem with springs or mass for example on both ends ... because in books cantilever is demonstrated but exercises are more like mass - mass b.c. which makes the problem more difficult to solve ... if you could make a video on this topic that would be awesome
Yes indeed, these are difficult problems to solve. Formulating the BCs themselves is still manageble. But the resulting EVP is almost always impossible to solve analytically. For that reason, one must rely on approximation techniques (Rayleigh-Ritz) or numerical techniques for solving the eigenvalue problem. Because in this course, I restricted myself to problems that have an analytical solution, I did not discus these more complicated BCs.
(Possibly stupid question) Shouldn't there be some sort of coefficients in from of each term U(x)\eta(t) so that the sum converges? I'm modeling some vibrations in a beam in MATLAB, and, without these coefficients in front, the first 10 terms appear not to capture ~99% of the variability, which, I would think the first 4 or 5 terms would ordinarily do so. TIA
[OT] How, if at all, can all of this be simplified/streamlined by Laplace/Fourier transforms?
Thanks for all your nice comments Jeff. The equations of motion of a vibrating string, bar and shaft are in fact all examples of the Wave Equation. The general solution can be written as a Fourier series in space. So in this case, trying a Fourier series will not just be a good approximation, but actually result in the exact solution as presented here.
For other continuous problems though, like a vibrating beam, the equation of motion is fourth order in space. Using a Fourier series as general solution will not be able to satisfy all boundary conditions and therefore only approximate the solution. The general solution contains a sine/cosine combo and a sinh/cosh combo. You will see this in the videos about vibrating bars. I hope that once you have arrived there, this will become clear. If not, please ask me again at that time.
@@JurnanSchilder Very good; that absolutely and thoroughly answers the question - thank you.
Merci monsieur pour votre belle méthode de structurer les idées pour l’apprenant.
Une demande s’il te plait, un cours qui porte sur un problème aux valeurs propres de bi-Laplacien, ou des cours ou documents sur le net dans ce sens.
Merci infiniment pour vos efforts mon professeur.
Merci pour ton compliment Hafid. Je réponds en Anglais car mon Français n'est pas bon.
Eigenvalue problems involving the bi-Laplacian operator can be found in structural mechanics problems like vibrating plates or other problems in which you have 2 spatial directions. A very good text book that I like that contains this is for instance Vibrations of Continuous Systems by Rao.
Essentially it means that you have to perform separation of variables in 2 spatial directions and time. So you start with v(x,y,t) = X(x)Y(y)T(t) and similar solution strategy can be applied as discussed in my video. I hope this answers your question.
@@JurnanSchilder thank you very much sir for your answer