How to Solve for x(t)/x0 in a Damped Oscillator with Initial Values?

[kraigandrews]

Homework Statement

The equation for a damped oscillator is d2x/dt2+2βdx/dt +ω02 x = 0. Let ω0=1.0 s−1 and β = 0.54 s−1. The initial values are x(0) = x0 and v(0)=0.
Determine x(t)/x0 at t = 2π/ω0.

Homework Equations

the solution to equation is given by;

x(t)=e^-$\betat$(A₁e^t$\mu$+A₂e^-t$\mu$)

where $\mu$=$\sqrt{\beta²-\omega_o²}$

The Attempt at a Solution

A₁=1/2(x_o+(x_o$\beta$)/$\mu$)
A₂=1/2(x_o-(x_o$\beta$)/$\mu$)

The problem I am running into is that the parameter I defined as $\mu$ is imaginary for this case, which keeps throwing me off. My only guess is to ignore the term multiplied by A₁ because it is not real, then use only the A₂ term and its multiplier because of the -t in its exponent making -i =1. I do not know if this correct and also even the constants A₁ and A₂ have an i in them as wel.

 

[vela]

The fact that μ is imaginary tells you you have an underdamped system.

Apply the initial conditions to solve for A₁ and A₂. The coefficients will be complex values. Then use Euler's formula, $e^{i\theta} = \cos \theta + i \sin \theta$, to express the solution in terms of sines and cosines. You'll find everything works out so the i's cancel and x(t) is real.