1

I was wondering if someone could please tell me why this below solution models damping well? In particular its amplitude and frequency of the damped oscillation.

$$y=e^{-λt/2m}[A\cos(Λt)+B\sin(Λt)],$$where$$Λ=\sqrt{(ω^2/m)−(λ^2/4m^2)}$$

I know that for simple harmonic motion the frequency of oscillation is $2\pi/\sqrt{k^2/m}$ whereas for the damped motion the frequency is reduced slightly to $2\pi/\sqrt{k^2m-\lambda^2/4m^2}$

Since this is added to the equation for $\Lambda$ is makes it more accurate.

But basically i wondering what the specific strengths and limitations of the model, such as it accurately models damping due to the exponential function, however the model may be limited to small angles due to the small angle approximation.

Cheers

Jr. Mathematician
  • 27
  • Hi and welcome to Physics.SE! Your questions should show us what ou have tried so far and why/where you are stuck. I'd expect the source of this equation (e.g., a physics or calculus textbook) to already give at least a brief explanation of its terms. – stafusa Aug 20 '17 at 13:59
  • @stafusa i know it is an analytical solution of the equation my''+λy'+w^2y, however i don't know how the formula was derived.. I am trying to investigate the strengths and limitations of this analytical solution but have found little info on it – Jr. Mathematician Aug 20 '17 at 14:37
  • Ok, so maybe your question is not so much about the solution, but rather the model, the differential equations it comes from, is that so? Then the main constraint of validity is the damping being exactly linear with the speed, since it's actually in general at least a polynomial. – stafusa Aug 20 '17 at 15:09
  • At this point you have changed the body so that it no longer agrees with the title. The question about the damped frequency has already been asked on the site (just recently, too): https://physics.stackexchange.com/questions/289508/change-in-period-of-damped-pendulum-system. And "Is there any way i can continue to expand on any of this?" is open-ended in a way that is not well handled by the Q&A format of stack exchange and so it off-topic. – dmckee --- ex-moderator kitten Aug 20 '17 at 20:21
  • If depends what you mean by "models well". It is the solution for the case where the damping force is proportional to velocity, but in the real world that is often not remotely accurate. Unfortunately, this tends to get introduced in "Dynamics 101" mainly as a mathematical exercise, and naïve students are left thinking that its "true," in the same way that they think Coulomb's "law" of friction is also true (it isn't, in many situations) - nobody told them the facts! Exercise for the reader: try solving a dynamics problem where Coulomb friction is a damping force and see what happens... – alephzero Aug 20 '17 at 20:56
  • Interesting, but why would this be a polynomial? I therefore assume then this model is not so accurate in the real world since the damping is not actually entirely proportional to velocity.. Is this correct? :) – Jr. Mathematician Aug 20 '17 at 23:20
  • 1
    The damping is proportional to velocity when it is proportional to velocity. That's an easy enough case to generate for lab demos, but it is not the only case that occurs in "real world" examples. It's also easy to find cases where it is roughly constant and other where it is proportional the the square of velocity or even higher powers. But this case in analytically tractable and suffices to demonstrate the important behaviors. – dmckee --- ex-moderator kitten Aug 21 '17 at 00:27

2 Answers2

0

This model works, if you have an exponential decay of the responds $y$, because you can rewrite your expression as $$ y(t) = e^{-\gamma t} A_0 \cos{(\Lambda t + \varphi_0)} $$ where I used the notation $\gamma := \lambda /(2m)$ and $\varphi_0$ is the starting phase of the oscillation. Here a link for the frequency $\lambda$ (link)

Semoi
  • 8,739
  • Could you please explain how you got to this expression? – Jr. Mathematician Aug 20 '17 at 22:59
  • Assume you have an oscillation without damping. At some arbitrary time I say, let's start the experiment. So we set this time to $t=0$. In order to describe an oscillation, you can either use the amplitude and velocity at $t=0$ (which leads to your two unknown parameters $A$ and $B \Lambda$) or, we you can use the amplitude and phase at $t=0$ (which leads to the two unknown parameters I used). If you want to check this use the addition theorem of the cos function. – Semoi Aug 21 '17 at 05:15
0

This appears at first glance to be the standard solution under the assumption that the dissipation is proportional to velocity.

For mechanical oscillators that's pretty accurate when the dissipation is mainly due to fluid drag and the Reynold's number of the moving parts is quite small (more like 1 than 1000) and losses in the solid parts are still smaller.

Making the usual pass to RCL electronic oscillators it's very accurate as long as the resistance is well modeled as Ohmic.

dmckee --- ex-moderator kitten
  • 86,062
  • Cheers, what about in regards to initial conditions? will y=0? – Jr. Mathematician Aug 20 '17 at 14:34
  • Of course $y(0) \ne 0$ if $A \ne 0$. In any case, the initial conditions are independent of the rest of the physics, unless they violate some of the physical assumptions being made in the model being used. – alephzero Aug 20 '17 at 21:02