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Ansatz for Hysteretic oscillation

  1. Oct 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Given
    X'' + iG^2 X + w^2 X = 0
    G^2 = h/m
    w^2= k/m
    What would be a good "educated guess" to solve that differential equation?
    My oscillations and waves teacher asked this on a test and since I didn't see anything depending on the speed of the object X assumed it was a really weird kind of Simple Harmonic Motion.
    We were all wrong x.x
    I actually don't know if to ask this here or in the math forums
    2. Relevant equations
    3. The attempt at a solution

    Ansatz= Acos(ϑt + ϒ)

    It was wrong, i tried to factor iG^2 and w^2 into one big "harmonic" frequency and solve like an SHM
     
  2. jcsd
  3. Oct 5, 2015 #2

    rude man

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    You need to explain what the system is you're describing with the equation.
    What is X? displacement?
    what is G? what does h/m mean? I know m is mass but what is h?
    is i = √(-1)?
    Is "Ansatz" your guess as to the solution? not a commonly used term, at least not in the U.S.
     
  4. Oct 5, 2015 #3
    Ansatz means educated guess, you "guess" the solution to the differential equation

    a more general form would be
    mX'' + hXi + kX = 0 with m mass, h hysteretic coefficient and k spring constant (all are real constants)
    I just don't know how to solve a differential equation with a imaginary (i) number within it, every time i search it takes me into Real damped systems with imaginary roots (but not imaginary coefficients)
    i tried solving the system like a harmonic oscillation and it was wrong
     
  5. Oct 5, 2015 #4

    rude man

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    You may need to take a course in solving differential equations. I don't see that the fact that one of the coefficients of a linear ordinary differential equation (ODE) with constant coefficients is imaginary should be a problem. There is formal math for handling complex numbers.

    Anyway, your Ansatz is wrong; it applies to a system without damping, i.e. h = 0.

    Since you may not have a background in solving ODE's, look up the solution to the second-order system with damping, i.e. x'' + bx' + kx = 0. You also need a finite initial condition if you want the non-trivial solution (trivial is x = 0).

    You also haven't described your system verbally which sort of ties our hands. It's hard for me to imagine that making the damping term imaginary suffices to describe a system with hysteresis. A system with hysteresis requires very advanced math, called "describing functions".
     
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