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.