## 2 Design and Analysis of the Device
### Magnetic Harmonic Drive Gear  
Magnetic gears are mechanisms for the conversion of torque/speed. Their teeth do not touch each other, but instead they are provided with magnets that exert force from a certain distance. Magnetic gears have several advantages over conventional mechanical gears: lack of wear, silent operation, reduced vibration, no need of lubrication, reduced maintenance, and improved reliability. Furthermore, they provide intrinsic overload protection and have no connection between the parts [15; 16; 17]. A few years before the HD was invented, [PERSON] patented an electric motor [18] in which magnetic teeth and a Nonius or Vernier principle was used. In 1967, [PERSON] [19] patented a coaxial magnetic gearing arrangement comprising a high-speed axle and a permanent magnetized low-speed rotor with many magnetic teeth. Despite the fact that the invention could be used as a speed-reducing gear with a reduction ratio (_i_) of 50:1, the transmitted torques were small because only one of the parts had permanent magnetization. In 2001, [PERSON] and [PERSON] presented what they called a \"high-performance magnetic gearbox\" [20]. Rare-earth magnets were used in a topology consisting of two coaxial, low- and high-speed rotors, with several steel ferromagnetic pole-pieces in between. Using 2D magnetostatic finite element method simulations, a torque density exceeding 100 kN m/m3 was calculated for \(i\) = 5.5:1. One great advantage of their system is that it is a single-stage mechanical gearbox with a relatively high torque, comparable to two- and three-stage helical gearboxes with typical torque densities between 50-150 kNm/m\({}^{3}\). In addition, the direction of rotation could be chosen. In 2003, [PERSON] _et al._[21] presented the first experimental results of a simplified coaxial magnetic gearbox prototype. It produced similar results: \(i\) = 5.5:1, with a torque density of around 90 kNm/m\({}^{3}\), and an efficiency of 81% (55 W energy power losses at 1000 rpm speed). Although there is a large background in magnetic gear developments, they have been mostly optimized in order to obtain a high torque density. Other mechanical aspects of the gear, _i.e._, gearing theory, radial instability, vibration damping, backlash, and reduction ratio availability, have not been taken into account up until now. In the MAGDRIVE prototype described in the present work, the reduction ratio is \(i\) = -20:1. It is the first time such a high ratio is experimentally demonstrated. The working principle of the MAGDRIVE is like that of the Nonius-Vernier motors as shown in Figure 1.  
The elementary mesh unit of a MAGDRIVE is composed of a stator provided with _ns_ permanent magnet teeth uniformly distributed along a length \(L\), a moving part (output) with _no_ teeth made of soft magnetic material, and, finally, a single permanent magnet as an input. The condition for this to be an elementary mesh unit is that _/no-ns_/ _= 1_, as shown in Figure 1.  
Displacement of the input tooth parallel to the length makes the output teeth magnetize and align, one by one, with the stator teeth. In this way, when the input moves along a length \(L\), the output moves  
Figure 1: Scheme of the MAGDRIVE mesh principle. \(L\)/no_. Therefore, the reduction of this movement will be \(i=\)_1/no_ provided _no_ is greater than _ns_. The ratio will be \(i=-\)_1/no_ if _no_ is smaller than _ns_. Several elementary reduction units can be repeated \(N\) times along a circumference in order to build a coaxial gear of the same ratio \(i\). In this case, there will be _Ns_ = _N'ns_ teeth in the stator, _No_ = _N'no_ teeth in the output, and \(N\) = _Ns-No_ teeth in the input member. The reduction ratio for the coaxial gear can be also calculated as:  
\[i=\frac{N_{o}-N_{s}}{N_{o}} \tag{1}\]  
This formula is valid for both positive and negative ratios. Additionally, like in any other kind of mechanism, a kinematic inversion can also be taken exchanging the roles of \"input\", \"output\", and \"stator\" (or mechanical ground).