Oscillation question - vibration topic

[ko_kidd]

oscillation question -- vibration topic

Book is asking me to prove that ξ = 1 corresponds with the smallest value of c such that no oscillation occurs.

Says let λ = -b (b is a positive number)

and that's all I'm given.

I'm not even sure I understand how you can tell there is no oscillation from the characteristic equation alone.

 

[CompuChip]

Well at the very least you'll have to give us some context. Because now it's like asking: "I have a car and the mechanic told me to polish the round part."

What is ξ, what is c, what is λ and what oscillation are we talking about?

Please post the entire question and any relevant equations. Physicists tend to re-use letters over and over, so if you don't say what they mean we can't help you.

 

[Mech_Engineer]

This Wikipedia article should give you a good idea of what you'll need.

http://en.wikipedia.org/wiki/Damping

 

[ko_kidd]

single degree of freedom systems

assuming you start with mx'' + cx' + kx = 0

where '' stands for the dots above x

the solution to that equation would be λ^2 + (c/m)*λ + k/m = 0

the characteristic equation

c --> constant of proportionality
λ --> root of the equation from the solution format x = Ae^λt
ζ --> damping ratio, in this case 1

(meant to say zeta)

 

[CompuChip]

So the border case between oscillation and no oscillation is exactly that where the two roots of the characteristic equation are equal. In that case, the two independent solutions are not $e^{\lambda_1 t}$ and $e^{\lambda_2 t}$ (with $\lambda_{1,2}$ the two roots of the equation, so you'd have $\lambda_1 = \lambda_2$), but you'd have to take $e^{\lambda t}, t e^{\lambda t}$ (you can check that). So the general solution will become $$x(t) = (A + B t) e^{-\lambda t}$$ which you can see is exponentially damped. This case is called critical damping. Below the critical damping, you will have something like $$A e^{-\omega_1 t} + B^{-\omega_2 t}$$ where [itex]\omega_{1,2}[/tex] are two positive real numbers and the system is called over-damped. Above the critical damping, the solutions will be complex and you can use Eulers identity to rewrite the exponentials into sines and cosines, which shows you that there is oscillation.<br /> <br /> Also check out the Wikipedia page linked to by MechEngineer, it has some clarifying graphs.[/itex]