Thank you so much for these amazing videos. Your hard work is very much appreciated by me and my fellow students. Your explainations are intuitive and easy to understand. Greetings from Germany :)
The job I'm interviewing for tomorrow is a vibrations analysis job. The round 1 interviewer kept telling me it was difficult. I've done systems analysis with springs and dampeners before, but not necessarily oscillatory motion. At least I recognize some of it though. My weakness tends to be direction.
This is a very good question... The answer is that, in general, the constants c1 and c2 are COMPLEX numbers. In particular, it can be shown that if c1 and c2 are complex conjugates, then this will result in a real function. If you want to show this, my suggestion is to work backwards. Start with x(t) = a1 cos ωt + a2 sin ωt. Then, from Euler's Formula, it can be shown: cos ωt = [exp(ωt) + exp(-ωt)] / 2 and sin ωt = [exp(ωt) - exp(-ωt)] / 2i. Substitute this into the equation of x(t) and group the co-efficients of exp(ωt) and exp(-ωt). You will find that these co-efficients (which are the constants c1 and c2) are complex conjugates.
@@Freeball99 Its a bit confusing. If c1 and c2 are complex and coefficients a1 & a2 turn out to be real, then we will miss the phase information in x(t). And evaluation of phase difference between displacement, velocity and acceleration will not be possible. You are requested to kindly explain this part.
I have just seen your video on harmonic motion. I think it is only a representation thing. Correct me if I am wrong. But I feel complex representation is better.
Good question. I didn't elaborate on this because it is very much a beginner's video and I didn't want to confuse people. The constants, in general, can be complex numbers. So I absorbed the 'i' into a2. Had I not done this, the the constant a2 would have been complex. However, by writing the sin term without the 'i' it ensures that a2 is always real.
@@Freeball99 thank you a lot! I did some math here and, knowing the x(t) must be a real function and changing the function to x(t)=(c1+c2)*cos wt +i*(c1-c2)*sin wt, its clear that the complex pair c1 and c2 must be a conjugate pair. Thats what you mean by absorbed right?
Question about the algebra done at 6:34. How does the m in the square root and the m in the denominator cancel out? I know it's simple arthmatic but I can't seem to convince myself that it works for some reason. Anyways, great videos. My course lectures focus more on the mechanical aspect than the mathematical, so this is really useful. Thank you.
It doesn't cancel out completely. I have SQRT(m) in the numerator AND m in the denominator - which is the same as SQRT(m) x SQRT(m) The SQRT(m) in the numerator cancels ONE of the SQRT(m) in the denominator. What I am left with is SQRT(m) in the denominator - which is what I have done in the video.
The two properties would be the mass of the system and the elastic/restoring forces. Generally the latter is due to the elasticity of the material being used. In the case of a structure like a building, the mass is primarily derived from the mass of the floors (usually wood or concrete) while the elasticity comes from the beams holding up the floors - generally either wood or metal, both of which are flexible.
man you are awesome, I'm following along the material because i stumbled upon it in my recommendations and saw that you have some python coding ! I'm a fresh mechanical graduate trying to learn python for engineering applications, to eventually get a job as a engineer working for an aerospace company like Firefly, SpaceX, Airbus, Barrios, or Parragon . If you have any advice or can maybe show some more python coding videos that would be awesome^2! thanks
Thank you. This is great to hear! With regards to learning Python (which I highly recommend)...you might not be aware that Python was invented by Guido van Rossum who for many years worked at Google (until recently). Partly as a result of this, Python is used extensively at Google (Google's search engine was originally written in Python as is the backend for RUclips). Consequently, there is a lot of instructional material that Google has released on RUclips - these are a good place to start for the fundamentals. Beyond that, you can check out sites like Udacity, Codecademy and Udemy - which have a bunch of free material on Python.
BTW - I do have several other Python coding videos. Look through the playlists below and any of the thumbnails that have a black background are typically coding videos. ruclips.net/p/PL2ym2L69yzkZJ1fY3SQ1JCyvZIoJYXQGZ (mechanical vibrations) ruclips.net/p/PL2ym2L69yzkaue8Ly2Oz51LALRzUV8LZ0 (finite element method)
The short answer is that you haven't been told that gravity exists in the problem (you can think of this mass vibrating horizontally on a frictionless surface). The slightly longer answer is that for translational systems like this one, the effect of gravity is that it simply shifts the equilibrium position about which the mass is oscillating. So, if we assume that the mass is vibrating about its equilibrium position, then we can ignore the effects of gravity. Here is a video explaining it: ruclips.net/video/TpZMTo-Lig4/видео.html
Why do we write F=-kx, when it is already acting opposite to direction of x? In that case ma-kx=0 would be equation of motion. Where 'a' is acceleration. I'm confused in this part. Please clarify it sir.
Yes, you're right. I should have drawn the force vector pointing downward on the free-body diagram. Once I used that F=-kx, I should have flipped the direction of the arrow.
This method yields a 2nd order differential equation. State-space form required the equation(s) to be 1st order equations and is generally a useful form if you are going to be using numerical integration to solve the problem - a common method of solving such equations. One can convert a set of equations to state-space form using simple substitution. Every 2nd order equation that is converted to state-space form, results in two 1st order equations. So in reducing the order of the equation, we double the number of equations.
This goes back to my initial assumption that x(t) = Ce^(rt) - I didn't number this unfortunately. Based on that, I showed that for a second order differential equation one will get two solutions (because the characteristic equation has two roots). So, I expanded my definition of x(t) to be more general so it would include both solutions.
This goes back to fundamentals of differential equations. I talk about it here: ruclips.net/video/S2-26LR8_Es/видео.html The basic idea is that in trying to solve a 2nd order ordinary differential equation (ODE) which states that a linear combination of a function and its derivatives add up to zero, one will quickly come to the conclusion that the exponential function is a good candidate BECAUSE derivative of the exponential function is itself - the exponential function is equal to its derivative d/dx e^x = e^x. In order to generalize it a little more, we add use e^(rx) because differentiating this results in multiplying it by r. This allows us to handle the fact that the terms of the ODE can have different co-efficients. We add the 'c' in front because this allows us to handle different initial conditions.
@@Freeball99 so you tell me that there is no exact solution for ODE , i took a course (numerical method ) it solve the differential eq but not in exact way can you tell why is that or just send me a vid talk about that , thank you
First of all let me address the non-periodic functions. In the event that the function is simply a constant, then this can be treated as a periodic function in the limit as the angular frequency, ω --> 0. For truly non-periodic functions, using Fourier transforms is possible, but very tedious and messy. I would NOT recommend Fourier transforms for this. Rather use a convolution integral instead. As for the treatment of periodic functions, I will have to make a video on this because it's very mathematical and this comments section makes it very difficult to explain things with equations. In the meantime, here is a video on the subject. I think this guy does a good job explaining it. Hopefully this will clear it up for you: ruclips.net/video/x04dnqg-iPw/видео.html
The textbook I used when taking the class was by Singiresu Rao. I'm guessing that many of the examples in my class notes came from this book though I am not sure. I haven't really found a textbook that I think explains everything well.
I think at like 5:32 you mislabeled equations 4 and 5. There are actually six different ones at this point and we'd be using 4 and 6 (for xo and x-double dot)
Yeah, you're right, the label for equation "4" should have been written one line higher...next to the expression for x rather than x_dot. Good catch! Thanks for the feedback.
10:55 you did write solution for x(t) was a1 coswt + a2 sinwt ,,,, i think both should have contained cosine terms as we are concerned with real parts.
@@Freeball99 can I say, while you are combining {c1 exp (iwt) + c2 exp(-iwt).}........ and then taking real part together that ends up with {c1 cos(wt) + c2 cos (wt)},?
@@sayanjitb Here, take a look at this. This is a result of Euler's formula. Constants can, in general, be complex and when multiplied by the "sin" part will make it real. www.dropbox.com/s/y490k7qluq2ic1h/RUclips_Videos_2020-09-09_11.05.32.pdf?dl=0
You are in the vertical direction I think you need to take into consideration the gravitational force the differential equation should have three parts Mg-kx = Ma ==> Ma + kx = Mg which the second order equation with Mg as external force acting on the spring mass system
Typically when examining the simple harmonic oscillator, gravitational effects are ignores. There is no assumption made that it is oscillating vertically and can similarly be thought of as oscillating in the horizontal plane on a frictionless surface. I have made a separate video that examines the effects of gravity and under what conditions the gravitational effects may be ignored. ruclips.net/video/TpZMTo-Lig4/видео.html
How did your midterm go? I don't mean to be a buzz kill, but this is the sort of material that can only be mastered by working many example problems - it's hard to figure it all out at the last minute. I'm happy to answer any question you might have if I can be of help. Is there anything in particular that you find the most confusing? Is it the application of or the solving of the math that's throwing you?
Thank you so much for these amazing videos. Your hard work is very much appreciated by me and my fellow students. Your explainations are intuitive and easy to understand. Greetings from Germany :)
The job I'm interviewing for tomorrow is a vibrations analysis job. The round 1 interviewer kept telling me it was difficult. I've done systems analysis with springs and dampeners before, but not necessarily oscillatory motion. At least I recognize some of it though. My weakness tends to be direction.
How did you apply euler law in eq at 10:48 then the out come was without complex i?
This is a very good question...
The answer is that, in general, the constants c1 and c2 are COMPLEX numbers. In particular, it can be shown that if c1 and c2 are complex conjugates, then this will result in a real function.
If you want to show this, my suggestion is to work backwards. Start with x(t) = a1 cos ωt + a2 sin ωt.
Then, from Euler's Formula, it can be shown:
cos ωt = [exp(ωt) + exp(-ωt)] / 2 and sin ωt = [exp(ωt) - exp(-ωt)] / 2i.
Substitute this into the equation of x(t) and group the co-efficients of exp(ωt) and exp(-ωt). You will find that these co-efficients (which are the constants c1 and c2) are complex conjugates.
@@Freeball99 Its a bit confusing. If c1 and c2 are complex and coefficients a1 & a2 turn out to be real, then we will miss the phase information in x(t). And evaluation of phase difference between displacement, velocity and acceleration will not be possible. You are requested to kindly explain this part.
I have just seen your video on harmonic motion. I think it is only a representation thing. Correct me if I am wrong. But I feel complex representation is better.
10:30 would you mind explaining how the "i" multiplying sin(wt) disappears? I mean, a2 is a real quantity or am i missing something?
Good question. I didn't elaborate on this because it is very much a beginner's video and I didn't want to confuse people. The constants, in general, can be complex numbers. So I absorbed the 'i' into a2. Had I not done this, the the constant a2 would have been complex. However, by writing the sin term without the 'i' it ensures that a2 is always real.
@@Freeball99 thank you a lot! I did some math here and, knowing the x(t) must be a real function and changing the function to x(t)=(c1+c2)*cos wt +i*(c1-c2)*sin wt, its clear that the complex pair c1 and c2 must be a conjugate pair. Thats what you mean by absorbed right?
Question about the algebra done at 6:34. How does the m in the square root and the m in the denominator cancel out? I know it's simple arthmatic but I can't seem to convince myself that it works for some reason. Anyways, great videos. My course lectures focus more on the mechanical aspect than the mathematical, so this is really useful. Thank you.
It doesn't cancel out completely.
I have SQRT(m) in the numerator AND m in the denominator - which is the same as SQRT(m) x SQRT(m)
The SQRT(m) in the numerator cancels ONE of the SQRT(m) in the denominator. What I am left with is SQRT(m) in the denominator - which is what I have done in the video.
Define and explain the two intrinsic properties of physical system which are responsible for oscillating behaviour of oscillating system?
The two properties would be the mass of the system and the elastic/restoring forces. Generally the latter is due to the elasticity of the material being used. In the case of a structure like a building, the mass is primarily derived from the mass of the floors (usually wood or concrete) while the elasticity comes from the beams holding up the floors - generally either wood or metal, both of which are flexible.
man you are awesome, I'm following along the material because i stumbled upon it in my recommendations and saw that you have some python coding ! I'm a fresh mechanical graduate trying to learn python for engineering applications, to eventually get a job as a engineer working for an aerospace company like Firefly, SpaceX, Airbus, Barrios, or Parragon . If you have any advice or can maybe show some more python coding videos that would be awesome^2! thanks
Thank you. This is great to hear!
With regards to learning Python (which I highly recommend)...you might not be aware that Python was invented by Guido van Rossum who for many years worked at Google (until recently). Partly as a result of this, Python is used extensively at Google (Google's search engine was originally written in Python as is the backend for RUclips). Consequently, there is a lot of instructional material that Google has released on RUclips - these are a good place to start for the fundamentals. Beyond that, you can check out sites like Udacity, Codecademy and Udemy - which have a bunch of free material on Python.
BTW - I do have several other Python coding videos. Look through the playlists below and any of the thumbnails that have a black background are typically coding videos.
ruclips.net/p/PL2ym2L69yzkZJ1fY3SQ1JCyvZIoJYXQGZ (mechanical vibrations)
ruclips.net/p/PL2ym2L69yzkaue8Ly2Oz51LALRzUV8LZ0 (finite element method)
great, but I just want to ask why do not we study also the gravity force (mg) ?
The short answer is that you haven't been told that gravity exists in the problem (you can think of this mass vibrating horizontally on a frictionless surface).
The slightly longer answer is that for translational systems like this one, the effect of gravity is that it simply shifts the equilibrium position about which the mass is oscillating. So, if we assume that the mass is vibrating about its equilibrium position, then we can ignore the effects of gravity. Here is a video explaining it: ruclips.net/video/TpZMTo-Lig4/видео.html
Why do we write F=-kx, when it is already acting opposite to direction of x?
In that case ma-kx=0 would be equation of motion. Where 'a' is acceleration. I'm confused in this part.
Please clarify it sir.
Yes, you're right. I should have drawn the force vector pointing downward on the free-body diagram. Once I used that F=-kx, I should have flipped the direction of the arrow.
Thanks! you are saving my life
Could you kindly let me know the difference between this method and the space state condition?
This method yields a 2nd order differential equation. State-space form required the equation(s) to be 1st order equations and is generally a useful form if you are going to be using numerical integration to solve the problem - a common method of solving such equations. One can convert a set of equations to state-space form using simple substitution.
Every 2nd order equation that is converted to state-space form, results in two 1st order equations. So in reducing the order of the equation, we double the number of equations.
thanks for the awesome video. at 9:10 you write out the equation for x(t). Could you please tell me how you got that equation?
This goes back to my initial assumption that x(t) = Ce^(rt) - I didn't number this unfortunately. Based on that, I showed that for a second order differential equation one will get two solutions (because the characteristic equation has two roots). So, I expanded my definition of x(t) to be more general so it would include both solutions.
great vid , but can you can you tell me why when x(t) = c e^rt , why is that in every solution of second order linear equation ?
This goes back to fundamentals of differential equations. I talk about it here: ruclips.net/video/S2-26LR8_Es/видео.html
The basic idea is that in trying to solve a 2nd order ordinary differential equation (ODE) which states that a linear combination of a function and its derivatives add up to zero, one will quickly come to the conclusion that the exponential function is a good candidate BECAUSE derivative of the exponential function is itself - the exponential function is equal to its derivative d/dx e^x = e^x. In order to generalize it a little more, we add use e^(rx) because differentiating this results in multiplying it by r. This allows us to handle the fact that the terms of the ODE can have different co-efficients. We add the 'c' in front because this allows us to handle different initial conditions.
@@Freeball99 so you tell me that there is no exact solution for ODE , i took a course (numerical method ) it solve the differential eq but not in exact way can you tell why is that or just send me a vid talk about that , thank you
@@Freeball99 can you answer my question , thank you
how to calculate the discerete fourier transform of a harmonics function and what to do with non-periodic function?
First of all let me address the non-periodic functions. In the event that the function is simply a constant, then this can be treated as a periodic function in the limit as the angular frequency, ω --> 0. For truly non-periodic functions, using Fourier transforms is possible, but very tedious and messy. I would NOT recommend Fourier transforms for this. Rather use a convolution integral instead.
As for the treatment of periodic functions, I will have to make a video on this because it's very mathematical and this comments section makes it very difficult to explain things with equations. In the meantime, here is a video on the subject. I think this guy does a good job explaining it. Hopefully this will clear it up for you: ruclips.net/video/x04dnqg-iPw/видео.html
What text book do you go by?
The textbook I used when taking the class was by Singiresu Rao. I'm guessing that many of the examples in my class notes came from this book though I am not sure. I haven't really found a textbook that I think explains everything well.
I think at like 5:32 you mislabeled equations 4 and 5. There are actually six different ones at this point and we'd be using 4 and 6 (for xo and x-double dot)
Yeah, you're right, the label for equation "4" should have been written one line higher...next to the expression for x rather than x_dot. Good catch! Thanks for the feedback.
10:55 you did write solution for x(t) was a1 coswt + a2 sinwt ,,,, i think both should have contained cosine terms as we are concerned with real parts.
We are just interested in the real parts, however, the constants c1 & c2 can, in general, be complex numbers.
@@Freeball99 can I say, while you are combining {c1 exp (iwt) + c2 exp(-iwt).}........ and then taking real part together that ends up with {c1 cos(wt) + c2 cos (wt)},?
@@sayanjitb Here, take a look at this. This is a result of Euler's formula. Constants can, in general, be complex and when multiplied by the "sin" part will make it real. www.dropbox.com/s/y490k7qluq2ic1h/RUclips_Videos_2020-09-09_11.05.32.pdf?dl=0
@@Freeball99 Thank you so much sir, i was little bit fuzzy with my Orthodox convention. I got your interpretation now!👍
You are in the vertical direction I think you need to take into consideration the gravitational force the differential equation should have three parts
Mg-kx = Ma ==> Ma + kx = Mg which the second order equation with Mg as external force acting on the spring mass system
Typically when examining the simple harmonic oscillator, gravitational effects are ignores. There is no assumption made that it is oscillating vertically and can similarly be thought of as oscillating in the horizontal plane on a frictionless surface. I have made a separate video that examines the effects of gravity and under what conditions the gravitational effects may be ignored. ruclips.net/video/TpZMTo-Lig4/видео.html
Thank you !
im gonna die during my midterm tomorrow.... :(
How did your midterm go? I don't mean to be a buzz kill, but this is the sort of material that can only be mastered by working many example problems - it's hard to figure it all out at the last minute. I'm happy to answer any question you might have if I can be of help. Is there anything in particular that you find the most confusing? Is it the application of or the solving of the math that's throwing you?
@@Freeball99 he couldn't reply because unfortunately he passed away to midterm induced death.
thanks. this is really helpful.
thanks
That British accent is music to my ears , finally a non Indian STEM video
brilliant.
Also, good explanation lol
Good but you waste lots of time
yet he explains it better than my vibrations class, in 1/10 time.