Phenomenal, concise, and graspable derivation and explanation! Loved the use of the complex analysis and how it all fit right in the end. This channel is severely underrated.
My god, I've been looking for a good vibes video on the derivation for damped systems for so long now. Finally found it! Absolutely great job on this and all your other videos. They're absolute gold.
You are a lecturer many students wish they had! Amazing explanation. Quick explanation of complex numbers made things significantly easier to understand!
Thanks a lot for your explanation. I'm now very clearly understand about damped system and other videos from your channel. This was absolutely awesome brother.
Your contents are great Sir. You are doing a great service to the Mechanical Engineering or Aerospace Engineering community. I was curious to know if you if you are a South African by any chance. Sorry about this I know this is obviously not the platform to ask all this
Great job! Two suggestions: 1) cast the final form in terms of r = w/w_n and zeta, which is easier to intuit - especially the resonance peaks; 2) derivation is nice, but could be a bit more concise using Eqn 5 to solve for magnitude and phase of X.
this video helped me a lot in understanding this chapter of my my dynamics module. please make some more videos. will it bee possible for you to make videos on Plane Kinetics of Rigid Bodies? Like example problems would be great!
Hello, in my class I keep getting problems where my harmonic force is in the form Fo*cos(wt). I like the method you do here because it seems easier, but am a little fuzzy on the process of converting cos(wt) into the exponential form using Euler's Formula. Specifically with regards to whether or not the exponential to the 'e' gets an i, even though the cos(wt) is just a real component of a complex number. Does the exponential get an i if its just a real harmonic force?
The idea with using complex notation is that it conveniently captures BOTH the Sin and Cos forcing functions. For a Sin forcing function, we just consider the imaginary part while for a Cos forcing function, we just consider the real part - so all you would need to do is consider the real part (i.e. the terms that are not multiplied by "i"). To get there, you need to use Euler's Formula: e^(i·θ) = Cos θ + i·Sin θ OR e^(-i·θ) = Cos θ - i·Sin θ So, the response would simply be: x(t) = X·(Cos ωt + i·Sin ωt) where X is given in equation. Multiply this all out and group your real terms separately from you imaginary terms. Ignore the imagery part (just cross it out). The remaining real part is your answer. ALTERNATIVELY, FORGET COMPLEX ANALYSIS and go through my video using Cos ωt as your forcing function. Perform all the math step-by-step exactly as I have done, but with your forcing function. This will be a very good exercise if you really want to get a solid understanding of this - rather that just plugging into a formula. Once you do this, I think the results from the complex analysis will make more sense.
If you expand e^(ix) in a Taylor series, you will find it will be a combination of the Taylor series expansion for cos(x) plus i times the Taylor expansion for sin(x).
Can you explain to me if my forcing function is sin? I kind still dont get jt tho huhuhu. And can you show me, if possible, how it is done to solve the response using the forced function sin? Having difficulty on how will I proceed there
Using Euler's Formula, we can write A·e^iωt = A(cos ωt + i·sin ωt). So, if you are using the forcing function of sin ωt, then this is just the IMAGINARY part of e^iωt (ie just that part that is multiplied by "i"). Since the forcing function you wish to use is just the imaginary part of e^iωt, then we are also only interested in the imaginary part of of the response (if we use cos as the forcing function, then we'd use the real part only). In order to find this, take the final equation in the video (I didn't number it, but it should be eqn 9) and apply Euler's formula: e^(iωt-φ) = cos (ωt-φ) + i·sin (ωt-φ) Then, just consider the imaginary part of the response and you're done.
Thanks for the videos, extremly nice. Just to verify: xp(t) represents the particular solution of the system, right? In order to plot x(t) we would also need the homogeneous solution and could then say x(t) = xh(t) + xp(t), where xh(t) is fixed by the initial conditions. Is that correct?
Amplitudes of the peaks will remain constant ONLY if there is no damping. Viscous damping will cause the amplitude of successive peaks to decay exponentially - i.e. if you draw a curve connecting all the successive peaks, your curve will be that of exponential decay (assuming c > 0).
Using Euler's Formula, we can write A·e^iωt = A(cos ωt + i·sin ωt). So, if you are using the forcing function of A·sin ωt, then this is just the IMAGINARY part of A·e^iωt (ie just that part that is multiplied by "i"). Since the forcing function you wish to use is just the imaginary part of A·e^iωt, then we are also only interested in the imaginary part of of the response (if we use cos as the forcing function, then we'd use the real part only). In order to find this, take the final equation in the video (I didn't number it, but it should be eqn 9) and apply Euler's formula: e^i(ωt-φ) = cos (ωt-φ) + i·sin (ωt-φ) Then, just consider the imaginary part of the response and you're done. Also NOTE that in my final equation, I have F0 instead of the amplitude, A, that you have given. These are the same things (ie the magnitude of the forcing function).
Phenomenal, concise, and graspable derivation and explanation! Loved the use of the complex analysis and how it all fit right in the end. This channel is severely underrated.
My god, I've been looking for a good vibes video on the derivation for damped systems for so long now. Finally found it! Absolutely great job on this and all your other videos. They're absolute gold.
You are a lecturer many students wish they had! Amazing explanation. Quick explanation of complex numbers made things significantly easier to understand!
Thanks a lot man! The way you explained is easier to understand compared to my lecturer
Thanks a lot for your explanation. I'm now very clearly understand about damped system and other videos from your channel. This was absolutely awesome brother.
You are most welcome. Glad it was helpful!
Great, presented in a very interesting mode.. Greetings
This was awesome brother what you explained in the last 2 minutes of the video really crystallized everything for me. Thanks!
Your contents are great Sir. You are doing a great service to the Mechanical Engineering or Aerospace Engineering community. I was curious to know if you if you are a South African by any chance. Sorry about this I know this is obviously not the platform to ask all this
Yep. Was born and raised in Durban...but that was a long time ago!
@@Freeball99 Thanks for your reply, Sir.
Great job! Two suggestions: 1) cast the final form in terms of r = w/w_n and zeta, which is easier to intuit - especially the resonance peaks; 2) derivation is nice, but could be a bit more concise using Eqn 5 to solve for magnitude and phase of X.
Thanks for your feedback. much appreciated. Agreed, it's probably better to use the reduced frequency.
Thank you from Mongolian University of Science and Technology
Thank you so much man! Very clear and concise
this video helped me a lot in understanding this chapter of my my dynamics module. please make some more videos.
will it bee possible for you to make videos on Plane Kinetics of Rigid Bodies? Like example problems would be great!
Great Job!!!
would it be possible to get your notes? to condense them into an equation sheet?
There you go: bit.ly/37OH9lXcart
@@Freeball99 awesome!
Hello, in my class I keep getting problems where my harmonic force is in the form Fo*cos(wt). I like the method you do here because it seems easier, but am a little fuzzy on the process of converting cos(wt) into the exponential form using Euler's Formula. Specifically with regards to whether or not the exponential to the 'e' gets an i, even though the cos(wt) is just a real component of a complex number. Does the exponential get an i if its just a real harmonic force?
The idea with using complex notation is that it conveniently captures BOTH the Sin and Cos forcing functions. For a Sin forcing function, we just consider the imaginary part while for a Cos forcing function, we just consider the real part - so all you would need to do is consider the real part (i.e. the terms that are not multiplied by "i").
To get there, you need to use Euler's Formula:
e^(i·θ) = Cos θ + i·Sin θ OR
e^(-i·θ) = Cos θ - i·Sin θ
So, the response would simply be:
x(t) = X·(Cos ωt + i·Sin ωt) where X is given in equation.
Multiply this all out and group your real terms separately from you imaginary terms. Ignore the imagery part (just cross it out). The remaining real part is your answer.
ALTERNATIVELY, FORGET COMPLEX ANALYSIS and go through my video using Cos ωt as your forcing function. Perform all the math step-by-step exactly as I have done, but with your forcing function. This will be a very good exercise if you really want to get a solid understanding of this - rather that just plugging into a formula. Once you do this, I think the results from the complex analysis will make more sense.
If you expand e^(ix) in a Taylor series, you will find it will be a combination of the Taylor series expansion for cos(x) plus i times the Taylor expansion for sin(x).
Can you explain to me if my forcing function is sin? I kind still dont get jt tho huhuhu. And can you show me, if possible, how it is done to solve the response using the forced function sin? Having difficulty on how will I proceed there
Using Euler's Formula, we can write A·e^iωt = A(cos ωt + i·sin ωt).
So, if you are using the forcing function of sin ωt, then this is just the IMAGINARY part of e^iωt (ie just that part that is multiplied by "i").
Since the forcing function you wish to use is just the imaginary part of e^iωt, then we are also only interested in the imaginary part of of the response (if we use cos as the forcing function, then we'd use the real part only).
In order to find this, take the final equation in the video (I didn't number it, but it should be eqn 9) and apply Euler's formula:
e^(iωt-φ) = cos (ωt-φ) + i·sin (ωt-φ)
Then, just consider the imaginary part of the response and you're done.
Thanks for the videos, extremly nice. Just to verify: xp(t) represents the particular solution of the system, right? In order to plot x(t) we would also need the homogeneous solution and could then say x(t) = xh(t) + xp(t), where xh(t) is fixed by the initial conditions. Is that correct?
Yes, what you have stated is correct.
Thanks for your video! Will the amplitude of response keep constant？Will the damping reduce the amplitude?
Amplitudes of the peaks will remain constant ONLY if there is no damping. Viscous damping will cause the amplitude of successive peaks to decay exponentially - i.e. if you draw a curve connecting all the successive peaks, your curve will be that of exponential decay (assuming c > 0).
"Complex analysis is foreign to Mech. Engineers" totally agree!
God bless mate!
What application is that? Very neat presentation and clear voice. Thanks for uploading
I use a iPad Pro (13 inch) with an Apple Pencil. Software is called "Paper" by 53. My microphone is a Blue Yeti - which is amazing!
Hello! I have a question. What if the harmonic loading is something like some amplitude A times sin(wt)?
Great videos btw thanks alot!
Using Euler's Formula, we can write A·e^iωt = A(cos ωt + i·sin ωt).
So, if you are using the forcing function of A·sin ωt, then this is just the IMAGINARY part of A·e^iωt (ie just that part that is multiplied by "i").
Since the forcing function you wish to use is just the imaginary part of A·e^iωt, then we are also only interested in the imaginary part of of the response (if we use cos as the forcing function, then we'd use the real part only).
In order to find this, take the final equation in the video (I didn't number it, but it should be eqn 9) and apply Euler's formula:
e^i(ωt-φ) = cos (ωt-φ) + i·sin (ωt-φ)
Then, just consider the imaginary part of the response and you're done.
Also NOTE that in my final equation, I have F0 instead of the amplitude, A, that you have given. These are the same things (ie the magnitude of the forcing function).
thanks