Damped Harmonic Oscillator

  • #1
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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-[itex]\betat[/itex](A1et[itex]\mu[/itex]+A2e-t[itex]\mu[/itex])

where [itex]\mu[/itex]=[itex]\sqrt{\beta2-\omegao2}[/itex]

The Attempt at a Solution



A1=1/2(xo+(xo[itex]\beta[/itex])/[itex]\mu[/itex])
A2=1/2(xo-(xo[itex]\beta[/itex])/[itex]\mu[/itex])

The problem I am running into is that the parameter I defined as [itex]\mu[/itex] 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.
 

Answers and Replies

  • #2
vela
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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, [itex]e^{i\theta} = \cos \theta + i \sin \theta[/itex], 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.
 

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