Magnetomechanical Coupling – Physics and FEA Models

Magnetomechanical stuff in FEA and machines both

Hi there! During the last couple of weeks, I’ve been helping some of my colleagues in magneto-mechanical simulations. So, why not utilize all that I’ve learned in a brand new blog post?

So here goes, starting with the obvious.

Magneto-what?

Yeah, I should probably explain what magnetomechanics means in the first place. After all, my work largely deals with electrical machines, and typical electrical machines are very much magnetomechanical systems. Electricity goes in, magnetics happens, and rotation comes out. Or linear motion. Or vice versa, in case of generators.

However, typically this motion part is analysed in a simplified fashion only, like rigid-body rotation around a shaft. We say that the mechanical model is lumped.

Nevertheless, we do not call that magnetomechanics.

Magnetics to mechanics

Instead, proper magneto-mechanical analysis includes elasticity. In other words, how parts of the machine deform due to the forces acting on them. Because we obviously have forces – the machine could hardly produce any torque without them. (Check out this piece on torque computation while you’re here.)

Indeed, torque is produced because the magnetic field in the air-gap creates tangential forces acting on the rotor. And as we know thanks to Newton, similar forces of opposite direction are acting on the stator surface. But as mentioned earlier, torque is only the net sum of these forces. By contrast, machine deformations and vibrations do very much care about the distribution of the forces.

And not all forces generate torque. Indeed, the same air-gap field that generates the torque, also tries to pull the stator and rotor together. These radial forces are often much, much larger than the useful tangential forces – locally. In a properly designed machine, they pretty much cancel each other out. In other words, we don’t see large forces acting on the machine bearings.

Both of these force components are thought to act on the iron surface. And please notice the word “thought” here. The details, like where and how the forces are exactly doing their stuff, are still somewhat under study.

However, something interesting happens inside the iron, too. Indeed, as iron is magnetized, it also changes its shape a little. This phenomenon is called magnetostriction, and has been known since 1842. Magnetostriction can generate audible noise in transformers and electrical machines both, and also results in increased losses and vibrations in general.

Mechanics to magnetics

However – and this might be surprising to some – the magneto-mechanical coupling also works to the opposite direction. Meaning, the mechanical stuff also influences the magnetic stuff.

For example, the magnetic permeability of iron depends on the mechanical stress. If I remember right, a little tension actually increases the permeability a little, at least on moderate flux densities. By contrast, any kind of compression, or a whole lotta tension, leads to a decrease.

Likewise, the iron losses – power losses / heating in iron subjected to a time-varying flux – tend to increase too.

Analysis

Anyways, back to the what we actually did. As you can probably guess, we used finite element analysis on this problem. The magnetics part was modelled with the 2-dimensional vector potential formulation, and coupled to the circuit equations. In other words, the AVI formulation was used. For the mechanics part, a fairly general nonlinear elasticity approach was used. Displacements were assumed to happen purely in the xy-plane (plane-strain model to be exact).

The entire problem was solved in a strongly coupled fashion. Meaning, the vector potential and displacements were both solved at once – more of this later. Finally, the problem was analysed in time-domain, meaning time-stepping analysis was performed.

Implementation details

The same finite element mesh with first-order triangular elements was used for modelling the vector potential and the displacements. I used my SMEKlib library as a template for the analysis. Or rather, a yet-unpublished development version of the library, with classes and other delicious stuff like that.

In any case, the good thing about finite element analysis is that the same fundamental building blocks can be used for wildly different problems. For instance, 2-dimensional magnetic, electric, and thermal problems can be analyzed practically without changing a line of code. Of course, the inputs and outputs will mean different things, but what happens under the hood stays essentially the same.

And the same almost applies to mechanics, too. A single line of code has to be modified in some of the matrix assembly functions, and then some matrices have to be stacked together. But that’s pretty much it.

Simple as that.

Magneto-mechanical material model

Well, simple might not be the word that I’d use to describe the material model itself. After all, it had to combine all the different phenomena together.

Suffice to say, we used something called the Helmholtz free energy, and created an invariant-based expression for it by fitting parameters to measurement data.

In other words, magic.

Results

A picture is worth a thousand words, and an animation is worth half a dozen pictures at least. So enjoy.

The animation shows the stator displacements of a 4-pole induction motor, roughly 20 kW of rated power. Everything is considered – magnetic forces over the airgap as well as magnetostriction in the iron.

Only the magnetic symmetry sector of one pole is shown. However, as expertly noted by Łukasz from Enterfea.com, symmetry doesn’t really work with vibration stuff. Just check out the figure below illustrating the first two vibration modes of another stator. The first one looks roughly elliptical while the second resembles a triangle. Obviously, if there is any symmetricity, it’s completely different between these two. So, we’ll be moving to a full machine cross-section shortly.

But, despite its probable flawfulness, the animation still looks nice. You can see the teeth wobbling quite a bit – that is due to the passing of the rotor teeth. The stator back, on the other hand, contracts in tune with the main flux.

Conclusion

• Magnetics and mechanics are actually coupled to each other.
• Both can be analysed with FEA, using almost identical codes.
• In electrical machines, the coupling primarily causes vibrations.
• The work continues

-Antti

Magnetomechanical Coupling – Physics and FEA Models