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Homework Help: Find the steady-state oscillation of the mass–spring system modeled by the given ODE.

  1. Sep 10, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the steady-state oscillation of the mass–spring system
    modeled by the given ODE. Show the details of your
    calculations.

    2. Relevant equations

    1. y'' + 6y' + 8y = 130 cos 3t

    2. 4y’’ + 8y’ + 13y = 8 sin 1.5t

    3. The attempt at a solution

    1. cos(3t) at the end means the basic angular frequency is 3 radians per second. Hence the steady-state oscillation frequency is 3/2pi Hz = 0.477 Hz.

    2. solve for homogeneous differential equation,

    4y'' + 8y' + 13y = 0

    propose y = e^(ct)

    y' = ce^(ct)

    y'' = c²e^(ct)

    substitute into mass-spring motion equation,

    4y'' + 8y' + 13y = 0

    4c²e^(ct) + 8ce^(ct) + 13ce^(ct) = 0

    e^(ct)(4c² + 8c + 13) = 0

    of course for unique solution, it must be e^(ct) ≠ 0

    4c² + 8c + 13 = 0

    c = -1 ± (3i)/2


    homogeneous solution is

    y(t) = e^(-t) (A sin (3t/2) + B cos (3t/2))

    where A and B is an arbitrary constants which dependent to boundary conditions
     
    Last edited: Sep 10, 2010
  2. jcsd
  3. Sep 10, 2010 #2

    rock.freak667

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    Homework Helper

    Re: Find the steady-state oscillation of the mass–spring system modeled by the given

    You will need to post your attempt before we can help you.
     
  4. Sep 15, 2010 #3
    Re: Find the steady-state oscillation of the mass–spring system modeled by the given

    please help me
     
  5. Sep 15, 2010 #4

    rock.freak667

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    Homework Helper

    Re: Find the steady-state oscillation of the mass–spring system modeled by the given

    Well you need to get the particular integral for the first one. At steady state means that as t→∞. So you can see that in your calculations, as t→∞, the homogeneous part tends to zero.

    The PI for cos(3t) would be y=Ccos(3t)+Dsin(3t), you need to get 'C' and 'D'.
     
  6. Sep 15, 2010 #5
    Re: Find the steady-state oscillation of the mass–spring system modeled by the given

    You can solve the particular integral as rock.freak suggested.

    A more physically intuitive way of doing it may look like this:

    The problem is basically the steady-state sinusoidal response of a second order LTI (Linear Time Invariant)system. If the system is stable(bounded input bounded output), the output is an attenuated sinusoid which has the same freq as the excitation and a constant phase lag.

    Firstly you need to confirm if the system is stable otherwise there is no stead-state response because it's unbound.

    It's convenient to use complex exponential to express the sinusoidal excitation and the solution. because sine and cos are special case of it.

    The ode
    [tex]y''+2\zeta \omega_{n} y'+\omega_{n}^{2}y=Ae^{j\omega t}[/tex]

    Stability test:
    Given the natural freq [tex]\omega_{n} >0[/tex], the system is stable if and only if [tex]\zeta>0[/tex]. It means the homogeneous solutions die as t approaches infinity.
    The output, or solution [tex]y=Me^{j(\omega t+\theta)}[/tex]

    You then plug y into the ODE, the magnitude M and the phase lag theta can be very easily found. In engineering, [tex]M(\omega)[/tex] is called frequency response, [tex]\theta(\omega)[/tex] is called phase response.
     
  7. Sep 16, 2010 #6
    Re: Find the steady-state oscillation of the mass–spring system modeled by the given

    thank you
     
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