## Utilisation of Finite Element Analysis Force Calculations

4.3.1.1 Maxwell Stress

A useful feature of using FEA is the ability to calculate the integral of the Maxwell stress tensor, which provides a simple method of calculating the magnetic forces acting on the machine [75].

Static force curves at a variety of different currents are shown in Figure 4.15. They are all cyclic over the 24 mm rotor tooth pitch. Currents of the same magnitude but opposite direction result in curves which are reflected about the x axis, or 180° out of phase.

Figure 4.15: Values of static force from FEA calculated Maxwell stress The zero current result is the cogging force, which results from the natural tendency of the teeth to align themselves with the magnets: the position of least reluctance. As the current is progressively increased in magnitude and the armature flux starts to dominate, the effect of cogging force decreases as electromagnetic force increases.

Figure 4.15: Values of static force from FEA calculated Maxwell stress The zero current result is the cogging force, which results from the natural tendency of the teeth to align themselves with the magnets: the position of least reluctance. As the current is progressively increased in magnitude and the armature flux starts to dominate, the effect of cogging force decreases as electromagnetic force increases.

It is now possible to investigate the accuracy of the simple shear stress calculations used in the design stage. Substitution of the values given in Table 4-1 into the equation for flux density under the tooth, (4.2), with 10 Amps flowing in the coils gives 0.63 T. The force on each magnet can the be found from Equation (3.12), giving 82 N, corresponding to a phase force of 2.0 kN. Inspection of Figure 4.15 shows that the FEA predicted amount for 10 Amps has two maxima, corresponding to 2.26 and 2.43 kN. The average of these two is 2.3kN, implying an error of 15 % for the simplified method.

The peak force at 15 Amps is around 3.2 kN, which over the total active area of 0.0288 m2 gives a shear stress of 111 kN/m2.

### 4.3.1.2 Normal Force

The magnetic force per unit area between two ferromagnetic surfaces is given by the relationship of (4.14), derived from the change in energy stored in the magnetic field associated with a small change in airgap size.

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