Homework Help: Ansatz for Hysteretic oscillation

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1. Oct 4, 2015

Remixex

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. Oct 5, 2015

rude man

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.

3. Oct 5, 2015

Remixex

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

4. Oct 5, 2015

rude man

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".