For small pressure delta values both linear and cubic forms can fit the polytropic law. The parameter air β can be estimated by extracting the pressure delta from the equation (9) through few simple steps obtaining: + = 1 4 3 2 2 r r air c d c hd πµ β (17) Fig. 8 (b) shows the values of air β for radial clearance from 0.01 to 0.1 mm at different temperatures for a moving magnet of 5 mm diameter and 5 mm height. Comparison of different non-linear effects In Fig. 9 are shown the effects of the different non-linear components on the moving magnet behavior at different vehicle speed. Two black dashed lines indicate the position of end-stroke bumpers. Starting from a system with only electromagnetic forces and bumpers have been added the effects of friction and air initially separated and then coupled. Fig. 9 (a) shows the behaviors of the moving magnet at 30 km/h. At low speed it takes several oscillations to dissipate its energy and the equilibrium point of magnetic and radial forces is at about half of the stroke. Fig. 9 (b) shows that at 60 km/h there is only one relevant oscillation of the moving magnet. The small oscillations outside the footprint are not useful in energy recovering. The equilibrium point is noticeably lower in respect to the previous case. In Fig. 9 (c) at 100 km/h there is only one oscillation of the magnet and the moving magnet have to deform the bumper to reach an equilibrium point. Shown results can vary a lot modifying choices about friction and clearance between magnet and guide. With a clearance of several tenth of millimeter on the diameter the pneumatic effect is almost negligible. In order to improve the energy scavenger performance in a wide range of vehicle velocities, two complementary dynamic behaviors are taken into account: • for low velocities, the non-linear elastic magnetic force is used for tuning the adaptive mechanical resonance of the energy scavenger and for increasing the oscillations of the floating magnet around the stationary state; • for intermediate and high velocities, the resonant contribution is progressively less important and the typical onestroke behavior is adequate for the power recovery, as a consequence to the increasing number of revolutions in time of the wheel. Experimental validation of the model In Fig. 10 the block-oriented model and the experimental outcomes are compared. The test is conducted on a shaker and reproducing the wheel profile without the mean centrifugal acceleration at 40 km/h. Due to the absence of a mean value of force, the prototype has been modified with symmetric preload obtained adding a second fixed magnet with repulsive force with respect to the floating magnet. A good agreement is found in the comparison between theoretical model and experiments. Conclusions This paper analyses through an integrated block oriented methodology the dynamic behavior of the moving magnet of an electro-magnetic energy scavenger. The effects of the different non-linearities on the system have been compared. The paper underlines the separated and integrated effects of magnetic, friction, dead-zone end-stops and pneumatic effects of the floating magnet sliding into a calibrated guide. A convenient choice of clearance between moving and fixed parts can be used to create an effective air brake preventing or softening shocks with end stops and to modify system dynamic. The equivalent parameters for a simplified Maxwell model with a polynomial spring and a linear dashpot of the pneumatic effect have been calculated as function of the device geometry. The optimization design of these parameters demonstrates effective performance of the energy scavenger. Simulations and some numerical comparisons try to empathize these results. 347
RkJQdWJsaXNoZXIy MTMzNzEzMQ==