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.

Also check out the Wikipedia page linked to by MechEngineer, it has some clarifying graphs.