Dept. of Appl.Math @ ucc.ie
Systems with Hysteresis
Welcome

Contents

What is `Hysteresis'

Preisach Model

Closed loop systems
   Pendulum in magnetic field
      Equations
   ODE with Relay
   ODE with Weiss model
   ODE with Preisach model

Complex behavior

Other Hysterons

Advanced modelling

Bibliography

Equations

In the absence of a magnetic field, the pendulum dynamics are described (in the linear approximation) by the equation

\begin{displaymath} x''+ax'+x=A \sin(\omega t), \end{displaymath}

which can be integrated explicitly. However, when the magnetic field is present, the situation is much more complex. The iron substance of the pendulum will be magnetized and demagnetized. This process is called ferromagnetic hysteresis. The magnetization will interact with the external magnetic field, and thus for small oscillations the equation will have the form
\begin{displaymath} x''+ax'+x=A\sin(\omega t)+b y(t), \quad y(t)=P x(t), \end{displaymath} (3)

where $x(t)$ is the displacement from its equilibrium position at $x=0$, and the nonlinearity $P x(t)$ describes the interaction between the external magnetic field and the magnetized pendulum itself. We know that, in the first approximation, this interaction can be described by the Preisach transducer with an appropriate measure $\mu$.