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Math Help - Damped Oscillator equation. Finding Energy and general solution for coupled oscillato

  1. #1
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    Damped Oscillator equation. Finding Energy and general solution for coupled oscillato

    Hi! I can do (a) easily enough but (b) has me totally stumped altogether! I can't seem to figure out how they get it in that form! Also (c) I'm having the same problem. I have two exams tomorrow and this is the first one so im juggling between the two subjects going through the past papers so if anyone could help with a solution for this it would be very helpful! Thanks


    1. (a) The damped oscillator equation

    m d2y/dt2 + dy/dt + ky = 0 has a solution of the form
    y(t) = Ae−α t cos(wt − ϕ ).
    Determine α and ϕ.

    (b) Show that the energy of the system in (a) given by E = 1/2mẋ^2 + 1/2kx^2
    satisfies dE/dt = −mv

    (c) Two coupled oscillators are described by the equations
    m1 = −kx1 + k(x2 − x1)
    m2 = −k(x2 − x1) − kx2.
    Construct the general solution (normal modes) for this system.
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  2. #2
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    Re: Damped Oscillator equation. Finding Energy and general solution for coupled oscil

    Quote Originally Posted by paddy543211 View Post
    Hi! I can do (a) easily enough but (b) has me totally stumped altogether! I can't seem to figure out how they get it in that form! Also (c) I'm having the same problem. I have two exams tomorrow and this is the first one so im juggling between the two subjects going through the past papers so if anyone could help with a solution for this it would be very helpful! Thanks


    1. (a) The damped oscillator equation

    m d2y/dt2 + dy/dt + ky = 0 has a solution of the form
    y(t) = Ae−αt cos(wt − ϕ ).
    Determine α and ϕ.

    (b) Show that the energy of the system in (a) given by E = 1/2mẋ^2 + 1/2kx^2
    satisfies dE/dt = −mv

    (c) Two coupled oscillators are described by the equations
    m1 = −kx1 + k(x2 − x1)
    m2 = −k(x2 − x1) − kx2.
    Construct the general solution (normal modes) for this system.
    (b) What is x relative to y? What is v relative to y?

    (c) Try a trial solution of the form

    x_{1}(t)=B_{1}e^{i\omega t}

    x_{2}(t)=B_{2}e^{i\omega t},

    and solve for the \omega's.
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