You have made clear so many thoughts I've been having on the history of mathematics and physics and the importance of (in hindsight) such simple concepts. You have sketched in some historical connections that I was unaware of, and provided the clues that opened my mind to the Lagrangian and Hamiltonian.
@nedisawegoyogya3 күн бұрын
A little hard to follow, suppose k = c*int_0^L(phi'*phi'^T), then what does int_0^L(phi'_2*phi'_1*u1 + phi'_2*phi'_2*u2 + phi'_2*phi'_3*u3) represent?
@squirepegg61573 күн бұрын
You have my vote for clarity; it's a great presentation.
@gillesh3335 күн бұрын
I never comment but this time it's worth it, thank you, great video
@FunnyMouth3146 күн бұрын
Will you ever cover hamiltonian mechanics?
@Metallurgicalengineer6 күн бұрын
Hello sir great video! can you tell me please the name of the software you are using to write ? thank you in advance
@Freeball996 күн бұрын
The app is "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.
@tryfonasthemas22209 күн бұрын
Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.
@tryfonasthemas22209 күн бұрын
Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.
@tryfonasthemas22209 күн бұрын
Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.
@zaccandels669510 күн бұрын
Excellent video.
@jeromedubois403812 күн бұрын
Could you recommend me a textbook that would perhaps cover this whole series. I have several books on mechanics of materials but non of them deal with Timoshenko beam theory (and other subjects such as vibration). I also took a finite element analysis class in which the delta operator kept coming back, but my teacher completly flew over explaining its nature and why it does what it does and would also appreciate a good reference textbook for that as well. Feel free to recommend a book for each topic in this series ;) Thanks in advance @Freeball99 This series is an absolute gem by the way.
@Freeball9911 күн бұрын
"Solid Mechanics: Variational Approach" by Dym & Shames tends to be my go-to book for this sort of material. www.google.com/books/edition/Solid_Mechanics/rTw_AAAAQBAJ?hl=en&gbpv=1&printsec=frontcover
@Ivan-mp6ff12 күн бұрын
How to put 5,6,7 into 1 is still beyond me. Good video otherwise, as usual.
@Freeball9911 күн бұрын
At this time, I am examining the particular solution... Eqn 1 contains x and x_ddot. Eqns 5 & 7 give me expressions for x and x_ddot (I don't need eqn 6) I substitute these two expressions into eqn 1, expand it out, then take out cos ωt and sin ωt as common factors.
@Ivan-mp6ff10 күн бұрын
Very kind of you. Didn't expect you to still answer questions years after your good work. I am a medical doctor self learning this information to try to understand how the human body works. And I believe vibration represents the common, most fundamental manifestation of all life forms. All measurable parameters of the human body such as PH, blood flow, oxygenation etc all finally distillate to the ability of the cells to vibrate. I believe the human body runs along the principle of least action, and eigen format. The systems inside us do not procrastinate like we do! They are very measurable and predictable. Engineering knowledge has really opened my eye understanding how medicine works. Thank you for your kind attention. Wish I had an academic buddy like you for brainstorming.
@Abdalrhman_Kilesee12 күн бұрын
Now please do the solution of the bernoullis "the geometrical way"
@dwinsemius13 күн бұрын
It looks at the beginning that you have a lot of linearity but I'm not hearing any mention of that property.
@Freeball9911 күн бұрын
I discuss linearity around the 10:45 mark.
@dwinsemius13 күн бұрын
This is great. I just started the video and I'm holding my breath to see if Oliver Heaviside gets proper credit for inventing this operation. Now as I proceed I'm not seeing evidence that this delta operator is the the same as Dirac's delta operator.
@Freeball9911 күн бұрын
Delta's occurs in many places in math and engineering. In this case, we are referring to the variational operator. It bears no relation other uses like the Dirac delta function or the Kronecker delta.
@supakornsuttidarachai178213 күн бұрын
Is the distributed load the self weight?
@Freeball9911 күн бұрын
In this problem, I have not assumed any gravity is present. I have assumed that the external load is some general function so if you wanted to include the weight of the beam, you could include it in the external load.
@dwinsemius13 күн бұрын
@22:37. "I know this must be setting your mind spinning". Right. I still remember when Dr. Katz laid this out at the very beginning of the sophomore course that I took in the summer of 1968 at the University of Michigan. It was rather unsettling, but once the fog in my brain distilled and I could see its wide applicability it became such a wonderful elixir.
@dwinsemius13 күн бұрын
Great stuff. It's the first time I have heard the word "brachistochrone" actually pronounced. The perspective that the goal is to calculate a function rather than a scalar leads into the need for operators rather than definite integrals very nicely. I wish that I had been prepared for quantum mechanics with this framework.
@luizappel780216 күн бұрын
This series is amazing (symptomatic of the channel as a whole i guess). Quick note. Maybe I missed something, but shouldn't the boundary terms in eqn 25 be negative and included to the integrand of the time integral? They arise from the IBP and are positive eqn 17, then subtracted on the second (strain energy) integrand in the Hamiltonian Principle from eqn 9. Shouldn't they therefore carry the minus sign? Thank you so much for the outstanding work.
@Freeball9914 күн бұрын
Yes, the sign of the last 2 terms of eqn 25 should both be negative (careless error), however, since we're setting each of these terms to zero, it makes no difference in the final analysis and yes these terms should be part of the integrand of the time integral (the dt should be on the line below). Thanks for catching that.
@binoysasmal191616 күн бұрын
Now solve the equations
@Freeball9916 күн бұрын
There you go: kzread.info/dash/bejne/n5mXtsqBeLmalco.html
@aminderichard848819 күн бұрын
Well understood than the past 12 weeks lecturer was in class
@chopinscriabin21 күн бұрын
Does taking the variation of strain energy del U and kinetic energy del T, and plug to hamilton will result in equation of motion of Bernoulli beam in this case? I watched your Timoshenko part 2, and more or less that's what you did to result in equation of motion for Timoshenko beam. Then I was wondering if we can do the same for this simpler case for Bernoulli. Thank you
@Freeball9918 күн бұрын
Yes, it is exactly the same. When deriving things in this video, I glossed over some of the formality in deriving it because I wanted to keep it simple, but then in the Timoshenko video, I wanted to lay it all out since I expect viewers of that video to be a little more familiar with the material. The only difference between the 2 theories is in the strain energies, since the strain energy for the Timoshenko beam includes shear and the EB beam does not. This is consistent with the EB assumption that cross-sections that are normal to the elastic axis before deformation remain normal to the elastic axis after deformation.
@chopinscriabin17 күн бұрын
@@Freeball99 thank you for your reply. I found your video that actually did it (deriving Bernoulli equation of motion from Hamilton), so actually it already answered my original question. Thanks again anyway for explaining it again. Your videos are really awesome.
@ThoTochRMm2826 күн бұрын
Hello I have a question : why here your definition of LG eq. is partial T and not partial L ?
@Freeball9918 күн бұрын
I skipped a step here. Since L = T - V, if I substitute this into Lagrange's Equation, then it reduces to the form I have in the video. It reduces to this because in this problem, the kinetic energy, T, is a function of q_dot only and does not explicitly depend on q.
@CatalinawolffАй бұрын
I am so thankful for your explenations! Question 1: I have one question, I didnt get the background of : at 14:05 when we do the PI on the Right Term - why does the derivative falls apart from the ∂y‘ to ∂y ? Question 2: And why do we want it to do that ? And (a Bit previous) why do we change the sign from ∂(T) (for total diff. i understood that) to ∂(I) I ? And question 3: 15:51 why is everithing zero when we multiply it with del y ? I thought that this ∂y simply Shows is the sign and If its <0 / >0 .. so where do i Take the Information from that multiplied with it = 0 ? Thanks a lot you help so much !! 🎉
@Freeball99Ай бұрын
1. It's not the derivative ∂y‘, but rather the variation 𝛿y' that we are integrating. The formula for integration by parts is: int u dv = uv - int v du. So, in our problem, dv = 𝛿y' therefore integrating gives us v = 𝛿y. Then we plug this into the formula. 2. This is how we integrate by parts. Just plugging into the formula above. Perhaps try to review this technique. 3. The reason we say that the entire expression must be equal to zero is as follows... - If the value of the integral, I, is to be a maximum or minimum, then the integral must not change its sign for all possible variations of 𝛿y. - However 𝛿y is arbitrary (ie can be positive or negative), therefore the part that multiplies 𝛿y equal to 0. - We conclude from this that 𝛿I = 0 is the necessary condition to find an extremal. Hope this makes sense.
@JF17thunder485Ай бұрын
Awesome video
@TwinklingStar0420Ай бұрын
The voice makes it feel like Professor Snape is teaching me this😁
@theo-zj7dmАй бұрын
I am a french student and I had trouble finding good mathematical explanations in French, and then I found your video. This is amazing, very well explained and rigorous. You made my day !
@saurabhsaini1249Ай бұрын
this is the really really cool video start follow you I am waiting for your new videos
@omranalfortei5328Ай бұрын
thank you
@jamestucker1126Ай бұрын
Only one of the best explanations of the Calculus of Variations that I have ever seen or heard.
@omertarkkaraca842Ай бұрын
I couldn't understand substitution of y= C_1*sin^2(omega) where it comes from ?
@Freeball99Ай бұрын
Short Answer: you can now ask Chat GPT for the best substitution to use. Longer Answer: the form of the denominator sqrt(c - y) gives us the clue that a trig substitution is the way to go. From there, you can find a table of useful trig substitutions and look for one with the same form as the integral that we have - ie. sqrt(x) / sqrt(a - x). Old School Alternative: Use a table of integrals to integrate this.
@javadsajedi625Ай бұрын
wonderful explanation, thank you
@brandonfrancis-henry4701Ай бұрын
you are amazing mate!! if i pass my vibrations test tomorrow ill send you a pint from ireland :)
@theo-zj7dmАй бұрын
thanks you so much
@maciejokon7419Ай бұрын
Similar voice to Hans Zimmer ;)
@matija92Ай бұрын
This is amazing. Thank you for your effort
@saurabhsaini1249Ай бұрын
thank you for this
@saurabhsaini1249Ай бұрын
This is classic thankyou make some more also good videos.
@Alina-ek4jrАй бұрын
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!!
@LuisCobb-vx5uoАй бұрын
A COUPLE OF TYPOS TO NOTE: The 2nd term of Equation 8 should have a + sign. It should be + g/l sin θ. ALSO: Equation 20 should read -gy. Agree with the first typo but why should eq. 20 have -gy ?
@Freeball99Ай бұрын
Not sure what I was thinking when I wrote this. As I look at equation 20 now, it seems correct to me. Will edit my comment. Thanks for the feedback.
@sonyaramanАй бұрын
This is the gem, I’ve been struggling to find a good video on derivation of this equation, and there it is. Simply the best 🤝🏻 Additional kudos for bringing in the historical overview of how that used to look like back in time😊
@press2701Ай бұрын
What a voice! Amazing.
@IbrahimBinNazirАй бұрын
Thanks
@rangamurali7667Ай бұрын
Beautiful, word for word, line by line, breaking down the mathematical poem, syntax ..speechless! Brings back memories of college days I wrestled with trying to figure. Can you plz do Maxwell equations? Am sure there are many to catch up, we ask for more and more. Our sincere thanks! Awesome!
@faheemgulzar1Ай бұрын
Excellent videos
@alireza.ghazizadehahsaeiАй бұрын
Your explanation is awesome! Thank you very much
@SurajPrasad-bf9qnАй бұрын
thanks Sir,i understood the concept
@feryfirdaus8192Ай бұрын
im lost
@D_dusze10 күн бұрын
realest comment
@MissPiggyM976Ай бұрын
Why Beta square is assumed bigger than alpha square ?
@Freeball99Ай бұрын
Sorry for the delayed response, but I somehow missed this until now... The condition β*β > α*α physically implies that the spatial frequency components must be real and non-zero to ensure wave-like behavior across the membrane. If this were not true, we would get exponential growth or decay rather than wave patterns.
Пікірлер
Amazing video, thank you for this.
You have made clear so many thoughts I've been having on the history of mathematics and physics and the importance of (in hindsight) such simple concepts. You have sketched in some historical connections that I was unaware of, and provided the clues that opened my mind to the Lagrangian and Hamiltonian.
A little hard to follow, suppose k = c*int_0^L(phi'*phi'^T), then what does int_0^L(phi'_2*phi'_1*u1 + phi'_2*phi'_2*u2 + phi'_2*phi'_3*u3) represent?
You have my vote for clarity; it's a great presentation.
I never comment but this time it's worth it, thank you, great video
Will you ever cover hamiltonian mechanics?
Hello sir great video! can you tell me please the name of the software you are using to write ? thank you in advance
The app is "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.
Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.
Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.
Is there any relationship of what you illustrated with the fourier transform? The steps we used remind me a lot of how one solves pdes in fourier space by converting them to odes.
Excellent video.
Could you recommend me a textbook that would perhaps cover this whole series. I have several books on mechanics of materials but non of them deal with Timoshenko beam theory (and other subjects such as vibration). I also took a finite element analysis class in which the delta operator kept coming back, but my teacher completly flew over explaining its nature and why it does what it does and would also appreciate a good reference textbook for that as well. Feel free to recommend a book for each topic in this series ;) Thanks in advance @Freeball99 This series is an absolute gem by the way.
"Solid Mechanics: Variational Approach" by Dym & Shames tends to be my go-to book for this sort of material. www.google.com/books/edition/Solid_Mechanics/rTw_AAAAQBAJ?hl=en&gbpv=1&printsec=frontcover
How to put 5,6,7 into 1 is still beyond me. Good video otherwise, as usual.
At this time, I am examining the particular solution... Eqn 1 contains x and x_ddot. Eqns 5 & 7 give me expressions for x and x_ddot (I don't need eqn 6) I substitute these two expressions into eqn 1, expand it out, then take out cos ωt and sin ωt as common factors.
Very kind of you. Didn't expect you to still answer questions years after your good work. I am a medical doctor self learning this information to try to understand how the human body works. And I believe vibration represents the common, most fundamental manifestation of all life forms. All measurable parameters of the human body such as PH, blood flow, oxygenation etc all finally distillate to the ability of the cells to vibrate. I believe the human body runs along the principle of least action, and eigen format. The systems inside us do not procrastinate like we do! They are very measurable and predictable. Engineering knowledge has really opened my eye understanding how medicine works. Thank you for your kind attention. Wish I had an academic buddy like you for brainstorming.
Now please do the solution of the bernoullis "the geometrical way"
It looks at the beginning that you have a lot of linearity but I'm not hearing any mention of that property.
I discuss linearity around the 10:45 mark.
This is great. I just started the video and I'm holding my breath to see if Oliver Heaviside gets proper credit for inventing this operation. Now as I proceed I'm not seeing evidence that this delta operator is the the same as Dirac's delta operator.
Delta's occurs in many places in math and engineering. In this case, we are referring to the variational operator. It bears no relation other uses like the Dirac delta function or the Kronecker delta.
Is the distributed load the self weight?
In this problem, I have not assumed any gravity is present. I have assumed that the external load is some general function so if you wanted to include the weight of the beam, you could include it in the external load.
@22:37. "I know this must be setting your mind spinning". Right. I still remember when Dr. Katz laid this out at the very beginning of the sophomore course that I took in the summer of 1968 at the University of Michigan. It was rather unsettling, but once the fog in my brain distilled and I could see its wide applicability it became such a wonderful elixir.
Great stuff. It's the first time I have heard the word "brachistochrone" actually pronounced. The perspective that the goal is to calculate a function rather than a scalar leads into the need for operators rather than definite integrals very nicely. I wish that I had been prepared for quantum mechanics with this framework.
This series is amazing (symptomatic of the channel as a whole i guess). Quick note. Maybe I missed something, but shouldn't the boundary terms in eqn 25 be negative and included to the integrand of the time integral? They arise from the IBP and are positive eqn 17, then subtracted on the second (strain energy) integrand in the Hamiltonian Principle from eqn 9. Shouldn't they therefore carry the minus sign? Thank you so much for the outstanding work.
Yes, the sign of the last 2 terms of eqn 25 should both be negative (careless error), however, since we're setting each of these terms to zero, it makes no difference in the final analysis and yes these terms should be part of the integrand of the time integral (the dt should be on the line below). Thanks for catching that.
Now solve the equations
There you go: kzread.info/dash/bejne/n5mXtsqBeLmalco.html
Well understood than the past 12 weeks lecturer was in class
Does taking the variation of strain energy del U and kinetic energy del T, and plug to hamilton will result in equation of motion of Bernoulli beam in this case? I watched your Timoshenko part 2, and more or less that's what you did to result in equation of motion for Timoshenko beam. Then I was wondering if we can do the same for this simpler case for Bernoulli. Thank you
Yes, it is exactly the same. When deriving things in this video, I glossed over some of the formality in deriving it because I wanted to keep it simple, but then in the Timoshenko video, I wanted to lay it all out since I expect viewers of that video to be a little more familiar with the material. The only difference between the 2 theories is in the strain energies, since the strain energy for the Timoshenko beam includes shear and the EB beam does not. This is consistent with the EB assumption that cross-sections that are normal to the elastic axis before deformation remain normal to the elastic axis after deformation.
@@Freeball99 thank you for your reply. I found your video that actually did it (deriving Bernoulli equation of motion from Hamilton), so actually it already answered my original question. Thanks again anyway for explaining it again. Your videos are really awesome.
Hello I have a question : why here your definition of LG eq. is partial T and not partial L ?
I skipped a step here. Since L = T - V, if I substitute this into Lagrange's Equation, then it reduces to the form I have in the video. It reduces to this because in this problem, the kinetic energy, T, is a function of q_dot only and does not explicitly depend on q.
I am so thankful for your explenations! Question 1: I have one question, I didnt get the background of : at 14:05 when we do the PI on the Right Term - why does the derivative falls apart from the ∂y‘ to ∂y ? Question 2: And why do we want it to do that ? And (a Bit previous) why do we change the sign from ∂(T) (for total diff. i understood that) to ∂(I) I ? And question 3: 15:51 why is everithing zero when we multiply it with del y ? I thought that this ∂y simply Shows is the sign and If its <0 / >0 .. so where do i Take the Information from that multiplied with it = 0 ? Thanks a lot you help so much !! 🎉
1. It's not the derivative ∂y‘, but rather the variation 𝛿y' that we are integrating. The formula for integration by parts is: int u dv = uv - int v du. So, in our problem, dv = 𝛿y' therefore integrating gives us v = 𝛿y. Then we plug this into the formula. 2. This is how we integrate by parts. Just plugging into the formula above. Perhaps try to review this technique. 3. The reason we say that the entire expression must be equal to zero is as follows... - If the value of the integral, I, is to be a maximum or minimum, then the integral must not change its sign for all possible variations of 𝛿y. - However 𝛿y is arbitrary (ie can be positive or negative), therefore the part that multiplies 𝛿y equal to 0. - We conclude from this that 𝛿I = 0 is the necessary condition to find an extremal. Hope this makes sense.
Awesome video
The voice makes it feel like Professor Snape is teaching me this😁
I am a french student and I had trouble finding good mathematical explanations in French, and then I found your video. This is amazing, very well explained and rigorous. You made my day !
this is the really really cool video start follow you I am waiting for your new videos
thank you
Only one of the best explanations of the Calculus of Variations that I have ever seen or heard.
I couldn't understand substitution of y= C_1*sin^2(omega) where it comes from ?
Short Answer: you can now ask Chat GPT for the best substitution to use. Longer Answer: the form of the denominator sqrt(c - y) gives us the clue that a trig substitution is the way to go. From there, you can find a table of useful trig substitutions and look for one with the same form as the integral that we have - ie. sqrt(x) / sqrt(a - x). Old School Alternative: Use a table of integrals to integrate this.
wonderful explanation, thank you
you are amazing mate!! if i pass my vibrations test tomorrow ill send you a pint from ireland :)
thanks you so much
Similar voice to Hans Zimmer ;)
This is amazing. Thank you for your effort
thank you for this
This is classic thankyou make some more also good videos.
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!!
A COUPLE OF TYPOS TO NOTE: The 2nd term of Equation 8 should have a + sign. It should be + g/l sin θ. ALSO: Equation 20 should read -gy. Agree with the first typo but why should eq. 20 have -gy ?
Not sure what I was thinking when I wrote this. As I look at equation 20 now, it seems correct to me. Will edit my comment. Thanks for the feedback.
This is the gem, I’ve been struggling to find a good video on derivation of this equation, and there it is. Simply the best 🤝🏻 Additional kudos for bringing in the historical overview of how that used to look like back in time😊
What a voice! Amazing.
Thanks
Beautiful, word for word, line by line, breaking down the mathematical poem, syntax ..speechless! Brings back memories of college days I wrestled with trying to figure. Can you plz do Maxwell equations? Am sure there are many to catch up, we ask for more and more. Our sincere thanks! Awesome!
Excellent videos
Your explanation is awesome! Thank you very much
thanks Sir,i understood the concept
im lost
realest comment
Why Beta square is assumed bigger than alpha square ?
Sorry for the delayed response, but I somehow missed this until now... The condition β*β > α*α physically implies that the spatial frequency components must be real and non-zero to ensure wave-like behavior across the membrane. If this were not true, we would get exponential growth or decay rather than wave patterns.