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Coefficient determination for the underdamped oscillator

  1. Oct 21, 2008 #1
    1. The problem statement, all variables and given/known data

    The general solution of the underdamped oscillator is given by

    x(t) = exp(-Bt)*[(A1)cos{(w1)t} + (A2)sin{(w1)t}]

    Solve for x0 = x(t=0) and v0 = v(t=0) in terms of A1 and A2. Then solve for A1 and A2 in terms of x0, v0 , and w1.

    2. Relevant equations

    w1 = sqrt{ (w0)^2 - B^2 }

    3. The attempt at a solution

    At t = 0, x(0) = x0 = A1.

    Now taking the derivative w.r.t time gives,

    v(t) = -Bexp(-Bt)*[(A1)cos{(w1)t} + (A2)sin{(w1)t}] + exp(-Bt)*[(A2)(w1)cos{(w1)t} - (A1)(w1)cos{(w1)t}]

    So, v(0) = v0 = -B(A1) + A2(w1).

    Am I being retarded somewhere, or is it impossible to solve for A1 and A2 in terms of x0, v0, and w1 only? The problem never indicates anything about using approximations, such as B is close to zero, so v0 can be approximated as v0 = A2(w1). Does this seem like the only route to go with? Thanks.

    By the way, Latex isn't working, so I had to use this convoluted notation.
     
  2. jcsd
  3. Oct 21, 2008 #2
    B does not depend on initial conditions. It's a parameter of the system(as w is).
    B is a damping factor, depends on viscosity of the medium, shape of the oscillator... will be the same no matter what initial conditions.
     
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