Solution to the Damped Harmonic Oscillator using LAPLACE TRANSFORMS!
HTML-код
- Опубликовано: 12 сен 2024
- 10-15% Off all my Merch :) Use Code 42069 over on papaflammy.myt...
10% or more Off ALL Stirling Engines! =D stemerch.com/c...
Advent Calendar Playlist: • Advent Calendar 2020
Laplace Transform Playlist: • Laplace transform: The...
Damped Oscillator: • Second Order Linear Ho...
Today we are going to solve the equation for a damped pendulum, using Laplace transforms. We also explore the phase space and turn it into a simplified equation that uses phase shifts by Trigonometric identities. Enjoy! =)
Help me create more free content! =)
stemerch.com/
/ mathable
papaflammy.myt...
Merch :v - papaflammy.myt...
www.amazon.com...
shop.spreadshi...
Become a Member of the Flammily! :0 / @papaflammy69
2nd Channel: / @npcooking69
--------------------------------------------------------------------------------
Wanna send me some stuff? lel:
Postfach 11 15
06731 Bitterfeld-Wolfen
Saxony-Anhalt
Germany
--------------------------------------------------------------------------------
Twitter: / flammablemaths
Instagram: / uncomfortably_cursed_m...
Flammy's subreddit: / flammybois
Facebook: / flammablemaths
Want to know more about me? Watch my QnA! =D • Question and Answer Ti...
![A Rope Sliding Down a Table [Lagrangian Formalism Approach]](http://i.ytimg.com/vi/qBeywEsVePg/mqdefault.jpg)
Your inverse Laplace transform doesn't match up with the identity you provided. You had (s+γ)/((s+γ)^2+ω^2), but the identity on screen had (s+γ)/((s-γ)^2+ω^2). Did I miss something?
I'm thinking exactly the same
My bad, that was a typo! The numerstpr must have a - too!! Sry for that ^^'
@@PapaFlammy69 Don't worry, it was a nice vid anyways. I might start using Laplace transforms to solve equations now, just so that the initial conditions appear in the solution! Very nice
Yup, definitely extremely powerful! :) Thank you for noticing and glad you liked the video
Yeah, the identity graphic had a typo. The sign of b is supposed to match for the damped cosine.
1:35 "now what's a damped harmonic oscillator? Well, imagine you have a pendulum..."
> Harmonic oscillator
> Pendulum
Papa's gone full engineer, he can't be saved anymore.
Isn't a pendulum a harmonic oscillator though? Just not a simple harmonic one.
always excited for another episode of 'papa flammy's seventh counter-order O Y A A A A A A ha, are you here'
this is the proper way to type out the intro
Hey you, thanks for watching
Papa flammy advent calendar
*Oooaeeeeeeeeeaheyyeyeyeyey*
Or you know, guess it to be or the form of ae^(bt)+ce^(dt) or something
You could, but a priori, you would have no reason to guess this unless you already knew the solution is a simple function. Of course, we stand on the shoulders of giants, so it is well-known that these equations are solved by linear combination of exponential functions, but just guessing would be against the spirit of trying to systematically find a solution with the same intuition that our predecessors used a long-time ago. Developing that intuition is useful, since guessing only works for certain types of equation, and certainly not every equation we would be interested in.
God Phase Space + Laplace Transforms are Such and Awesome Combination!!
these transforms make me so damped 🥵
Laplace transform is, in fact, my favorite method to solve all three basic types of oscillation (simple harmonic, damped and driven). =)
The cool thing about the harmonic equation in particular is that you can use it to practice a method you've just learned to solve differential equations (I know four in particular: Euler, Laplace, Frobenius and variational calculus)
“pendulum is basically a swinging delta function”
papa flammy, u are so good i understand this whole derivation, and i havent even reached uni yet-(3 years away :D)
My man snorts a line of cocaine before every video
Love the energy and content
Laplace,
*TRANSFOOOORMMMMMM*
honestly... i still do laplace transforms for fun.... it's more fun to me than probably it should be haha
If you wind the spring your pendulum can swing even one month
You assume that (omega_0 - gamma)(omega_0 + gamma) > 0
In my opinion this should be explained why this assumption is possible
I literally did this today to find the model I need to fit my data to, and I get it recommended
AAAAAAAAAAAAAH, been long time since i heard that intro TTS
this is cool thanks
Papa flammy you came in my dream!!
The figures and the graph in the thumbnail look Amazing, which software are you using?
thanks you , I understand everything but in a research I have the solutions was in cosine it's the same right ! and thank you alot
Ah yes, another 2 to 5 minutes long video :p
Every god damn year xD
@@PapaFlammy69 ye, but I guess no one would complain about too much content ^^
@@PapaFlammy69 Yo chill with the advanced mathematics. Give our rookies also a chance
11:26 Shouldn't there be s-gamma in nominator in order for inverse Laplace transform to work?
yup! Made a typo! ^^'
I love Laplace transform
Papa flammy, what a great video.
have you heard of just solving the quadratic
Shouldn't it be x(0) * (s - gamma)/((s+gamma)^2 + omega^2) as you identify s + gamma as s - (-gamma) so in the numerator it would be s + (-gamma)?
Ah good memories of learning vibrations
Hey can I contact you anywhere to get help on a discrete problem I have? I have some work done on it and can show you my best solution so far but i need some advice on how to continue it
That inversed my mind 😅😅
At the begining equation, for a damped pendulum the "realistic" equation have the form:
\ddot{x} + A*\dot{x} + B*sin(x) = 0
¿there is someone have solve this equation without using approximatioms?
Thank you dear. Can you provide notes of this Question.. ? It's very helpful for me. Love from 🇮🇳 ❣️
may I do the same thing to the damped driven harmonic oscillator ?
I like shrek 3
Danke Papa!
But how about positive feedback for turning a damper oscillator to an in damper oscillator? :p In a feedback system F(s)=H(s)*G(S)/(1+H(s)*G(s))
Sorry I’m an electrical engineer...
Nice!
NICE some ALPHA engineering MATHS
Activa los subtitulos en español.
Physics professors just never use laplace transform and this annoys me a lot
They don't?
Try solving the QHO using Laplace transform. You'll get your fill of gamma functions and Hermite polynomials :)
what is QHO?
@@complex_variation Quantum Harmonic Oscillator.
@@davidlenir7517 thank you!! I'm always trying to learn new thinga
The math is not that difficult if you have someone to guide you through it
@@Anankin12 any advice? I'm currently halfway of electrodynamics Griffiths if that gives a notion of my math background
Yas
Can you help me solve arcsin(x)=1/sin(x)
And integral of sec(x)
I think the solution to the equation is transcendental and cannot be expressed using standard functions.
To antidifferentiate sec(x), rewrite it as 1/cos(x), and multiply by 1, but rewriting 1 as cos(x)/cos(x). In other words, 1/cos(x) = 1/cos(x)·1 = 1/cos(x)·cos(x)/cos(x) = cos(x)/cos(x)^2. Now, notice that cos(x)^2 = 1 - sin(x)^2, so the integrand is equal to cos(x)/[1 - sin(x)^2]. Also, notice that d[sin(x)] = cos(x)·dx. Therefore, let y = sin(x), so cos(x)·dx = dy, resulting in an integrand of 1/(1 - y^2).
1/(1 - y^2) = 1/[(1 - y)·(1 + y)] = 2/[2·(1 - y)·(1 + y)] = [(1 - y) + (1 + y)]/[2·(1 - y)·(1 + y)] = (1/2)/(1 + y) + (1/2)/(1 - y), and the principal antiderivative of this with respect to y is ln(1 + y)/2 - ln(1 - y)/2 = ln(sqrt[(1 + y)/(1 - y)]) = ln[sqrt([1 + sin(x)]/[1 - sin(x)])] = ln[sqrt([1 + sin(x)]^2/[1 - sin(x)^2])] = ln(|[1 + sin(x)]/cos(x)|) = ln[|sec(x) + tan(x)|].
Before adding constants of integration to get the full family of antiderivatives, you must take into consideration that sec(x) has poles at x = π/2 + nπ for every integer n. Therefore, for every discontinuity, there is a different arbitrary constant of integration. Therefore, to get the full family of antiderivatives, you add Σ{C(n)·sgn(x - π/2 - nπ)}. As such, the antiderivatives of sec(x) are given by ln[|sec(x) + tan(x)|] + Σ{C(n)·sgn(x - π/2 - nπ)}. You can verify for yourself that differentiating this will give sec(x) everywhere in its domain.
Had the previous math video on 2x speed. and then lightspeed was achieved
Wow
Interesting
11:20 why does father flamboyant look so smug
Desribe elliptical integral in laymans term.
pappa Flammmmy 😃😃😃
Cat has it
Not first
:(
1:39
Omg1!1!1!1!1!1!1!!!1111
Flammable math did nono German salute let's cancel him
(This is a joke, I know it's not the Nazi salute)
Wow that's an expensive subject..
vErY eBic
Bro should’ve just completed the square and u get the same answer
ايلافيوه (:
Soo.. whats a laplace transform😐
I unsubscribed for that unclear upside-down meme
sooner you will be physicist