Differential equation / Physics problem

[Scrapster]

This came from math class, but also can be considered a physics problem.

This is the equation of a damped mass-spring system with no external force acting on it:

2u'' + (alpha)u' + 3u = 0

For what value of alpha will the mass experience oscillations?

2 is the mass
alpha represents the damping constant
3 is the spring constant

I'm not really sure where to start.

Any suggestions?

 

[AlphaIVT]

you got the equation wrong its:

2u'' + (alphaivt)u' + 3u = 0


haha

 

[Scrapster]

Any OTHER ideas?

 

[Bignate603]

take the problem, right it out ver neatly on a clean sheet of paper, and procede to violently tear it upthen burn the pieces. You won't know the answer, but you won't feel so bad about being stuck...

 

[hendon]

get the characteristic equation and solve.
For
2u'' + (alpha)u' + 3u = 0
characteristic equation is
2r^2 + alpha*r + 3 = 0

Then consider the different cases for alpha, and what solutions you'll get.

You'll want the roots of the characteristic equation to be purely imaginary for pure oscillations. For decaying/growing oscillations, the roots have to be complex.

 

[Ken g6]

Ah, yes, dam*ed harmonic oscillators. I once thought I sort of understood them, but we never used 2nd order differential equations to describe them (thank goodness!)

Maybe Eric can give you some insight. Good luck!

Edit: It's all coming back to me now. Hopefully, Hendon's method will work. If you know what a Fourier transform is, it might apply - though I don't remember how yet. all I remember is there's some way to solve differential equations like this with Fourier transforms.

 

[Scrapster]

get the characteristic equation and solve.

I know all that. I'm just confused on how to tell if alpha will produce an oscillation (not just a pure one or a decaying/growing one, ANY oscillation). We don't use any methods or formulas to solve this thing. Just basic methods of differentials.

Any ideas?

 

[Scrapster]

^

 

[theonlymantis]

Being a senior physics major who is currently studying quantum mechanics, I can tell you with certainty that the best way to solve this problem is to invoke indeterminacy, say that the book is full of crap and this situation is nonexistent, and turn it in.

(at least, that is my approach to electromagnetics-- write "this does not appear possible" and turn it in.)

Goldie



<< One can never hope to understand quantum mechanics, only hope to someday believe it is true. --Feynman (or something to that effect) >>

 

[UWDawg]

Ha!! I wish I would have thought of that.

 

[MereMortal]

Scrapster, hendon has shown you the path. You will get three possible solutions, depending on how the damping coefficient compares to the spring constant. Any condition that leads to sines or cosines (or e^(imx), e^(-imx)) in the general solution exhibits oscillatory behavior.

If you still have problems, just ask.

 

[heat23]

i think we did this exact problem in our Diff eq class

 

[Scrapster]

You will get three possible solutions

I have the characteristic equation:

2r^2 + (alpha)r + 3 = 0

But now I'm not sure what I should be trying for alpha. I'm really not sure what the correlation between damping and oscillations are. Suggestions?

 

[Scrapster]

I'll take it one step further,

To find my roots, here's my quadratic equation.

-alpha + or - sqroot(alpha^2 - 24) all over 4.

If I want to get complex numbers here I would need to make alpha^2 less than 24.

But I'm confused on why I need to do that?

 

[Scrapster]

Well, I know that overdamped (2 real roots < 0) doesn't produce oscillations and critically damped doesn't either. So, do I need to find the alpha that barely makes my quadratic produce 2 complex #'s to get the solution in the form: (e^(real)*cos(imag) + e^(real)*sin(imag))???

Can someone verify this?

 

[Thom]

so you have a second order differential equation.

make it a homogeneous equation of the form au'' + bu' + cu = 0

where a,b,c are constants

u' = dy/dx
u'' = d2x/dy2 (squared not 2)


so, separate the variables so that all x are on one side and all y are on the other,

let y=Ae(to the mx) as a solution

knowing that if y=Ae(to the mx) dy/dx = Ame(to the mx) and d2y/dx2 = Amme(to the mx)

subs these into the original equation

rearrange everything and find it cancels nicely to a quadratic of the form Ae(to the mx) (amm +bm +c) = 0

let Ae(to the mx) not equal 0

therefore amm +bm +c = 0

This is the auxilary equation

now you can use 'the formula' (-b+- root b squared -4ac over 2a etc) to find the roots of your equation

and from there, something else. feck knows if this was any help, but the auxilary equation might be useful. I never did understand ordinary differential equations above first order.

 

[hendon]

Scrapster: yup.. I think you got it

so alpha^2 < 24 => oscillations
alpha^2 >= 24 => no oscillations