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Finding the undamped natural frequency of 2nd order system

  1. Mar 24, 2015 #1
    the following 2nd order differential equation is given:
    2y'' + 4y' +8y=8x.....................................(1)
    i want to find damping ratio, undamped natural frequency, damping ratio coefficient and time constant for the above system.
    solution:
    comparimg (1) with general system equaion

    SysDyn2A1.gif
    (veriable can be exchanged)
    {where: x(t) = Response of the System,
    u(t) = Input to the System,
    z = Damping Ratio,
    wn=Undamped Natural Frequency,
    Gdc= The DC Gain of the System.}
    damping ratio z or zeta:

    2zw=2
    w=2 so z=2/4=0.5

    undamped natural frequency w or omega:
    w=2 but correct ans is 1. any help?
     
  2. jcsd
  3. Mar 25, 2015 #2

    Simon Bridge

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    Using: http://en.wikipedia.org/wiki/Harmonic_oscillator

    Putting the DE familiar form: ##\ddot x + 2\dot x + 4x = 4t## would be the equivalent right?
    Compare with ##\ddot x + 2\zeta \omega_0 \dot x + \omega_0^2 x = f(t)## I get ##2\zeta\omega =2## like you did, and ##\omega_0^2=4\implies \omega_0=2 \implies \zeta = 1/2## ...

    Are you sure the answer you quote as "w" is the undamped frequency?


     
  4. Mar 25, 2015 #3
    http://www.facstaff.bucknell.edu/mastascu/eControlHTML/SysDyn/SysDyn2.html yes it is undamped natural frequency
     
  5. Mar 25, 2015 #4
     
  6. Mar 25, 2015 #5
    another way is to use laplace transformation as:

    SysDyn2A1.gif

    • Then, Laplace transforming both sides and solving for the transfer function - the ratio of the transform of the output to the transform of the input, we find the transfer function to be.
    SysDyn2A2.gif
    but you still get wn=2
     
  7. Mar 25, 2015 #6
    I agree the undamped w = 2
    Why do you think the correct answer is .1?
    Taking damping into consideration w = 1.73
     
  8. Mar 25, 2015 #7
    please tell me how you find w=1.73....
    w=0.1 ans is given in book Electronics and communication engg (OT) by Handa
     
  9. Mar 25, 2015 #8
    The auxiliary equation is: 2m^2 + 4m + 8 = 0
    Use the quadratic formula to solve for the roots = -1 +/- i 1.73
    That leads to the general solution form of e^-t*(A Cos 1.73t + B Sin 1.73t)
    A damped oscillation where w = 1.73
    To solve for the undamped case just disregard the coefficient of the m term which represents the damping resistance. The roots then are +/- i 2 purely imaginary
    An undamped oscillation where w = 2.0

    I don't know where that 0.1 could have come from, a typo maybe?
     
  10. Mar 25, 2015 #9
    ty for the help. can you please tell me about damping co-efficient and time for this particular question?
     
  11. Mar 25, 2015 #10

    Simon Bridge

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    ... please show how you have attempted to answer the rest of the problem, then we can help you with it.
     
  12. Mar 25, 2015 #11
    Zeta=a/2w which implies that a=2 by letting w=2 and zeta=0.5 time constant is 1/a which is 0.5. I dont know about damping coefficient and hope u will help me out
     
  13. Mar 25, 2015 #12

    Simon Bridge

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