What is the function x(t) for an underdamped oscillating system

[Damian]

Homework Statement

Homework Equations

and the attempt at a solution[/B]
Approach: Use the solution for the damped oscillating system provided in the formula sheet. We must use the given initial conditions to find the unknown phase $\phi$ and that will give us an expression for $x$ in time. Could use the 'general' solution with the unknowns $C_1$ and $C_2$ but the math seems much harder, so we can use the form below to simplify the calculation.

Since it's underdamped, $x(t) = A_0 e^{\frac{-t}{\tau}} cos(\omega't+\phi)$

Initial conditions: $t=0, x = A_0$ and $t=0, \dot x=0$

Using initial conditions: $A_0 = A_0 cos\phi$ so that means $\phi = 0$

But when using velocity, $\dot x = 0 = A_0 (-\frac{1}{\tau}cos(0) - sin(0) \cdot \omega'$ which would mean that the amplitude and/or damping rate are zero when the parts are stationary.

Does this mean $x(t) = A_0 e^{\frac{-t}{\tau}} cos(\omega't)$?

Thanks in advance for any help, hints or comments :)

 

[jambaugh]

The e^{-t/\tau} factor occurs due to the dampening it is the exponential decay of the amplitude as the dampening dissipates the energy. You should leave it out (effectively \tau \to \infty) for the undampened case.

 

[Damian]

Thanks for your reply jambaugh.

In this question, it said the system was underdamped - I thought that mean the amplitude does decay exponentially over time. So should I still leave out the e^{-t/\tau} factor?

 

[ehild]

The e^{-t/\tau} factor occurs due to the dampening it is the exponential decay of the amplitude as the dampening dissipates the energy. You should leave it out (effectively \tau \to \infty) for the undampened case.

It was underdamped , not undamped.
http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html

 

[ehild]

Thanks for your reply jambaugh.

In this question, it said the system was underdamped - I thought that mean the amplitude does decay exponentially over time. So should I still leave out the e^{-t/\tau} factor?

No, you need the exponential factor. But you should give τ and ω' in terms of γ and ω₀.

 

[jambaugh]

Oh, My bad eyesight! I read "under" as "un-". Very different case and your approach looks correct qualified with what ehild said. I apologize for my misreading your question. Did that twice now recently.