In this paper the axial symmetry of the magnetic field generated by a permanent magnet of helicoidal toroidal kind is shown. In the first part of the paper we illustrate the shape of the magnet and the number of areas where the field is calculated to demonstrate the symmetry. We define quantitatively the size of the toroidal helical magnet and the regions where the magnetostatic field is evaluated. The field is carried out for each angular sector that represents the regions where the magnetic flux density is computed. This calculation is performed with reference to a matrix of points belonging to each sector. Two sets of evaluations are performed. The first one is referred to a less dense matrix of points relative to all the regions. The aim of this computation is to demonstrate the axial symmetry of the field. The second set of calculations concerns the field evaluation by using a much higher dense matrix of points. By using this data we are able to interpolate the same field with a high precision. This second evaluation of the field is carried out with reference only to the flat region facing the first coil of the helical toroidal magnet. The use of an interpolation surface through the final points of the magnetic induction vectors previously computed allows a very fast evaluation of the field virtually in all the infinite points of the angular sector. The symmetry enables us to drastically reduce the time computation of the magnetostatic field in the points of interest

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This article presents an analytical numerical procedure to compute the\ud magnetostatic field generated from permanent magnets shaped by a helicoidal geometry of a toroidal kind. The transversal section of these magnets is pseudorectangular, and the magnetization model that has been used is based on the equivalent magnetic charge. The modulus of the magnetization vector M has a constant value in all points of the magnet case study. M has the same direction as the binormal in each\ud point of the barycentric curve relative to the helicoidal magnet of a toroidal kind. The procedure proposed to evaluate the magnetic field is shown by computing the distribution of the magnetic flux B on a surface parallel to one of the four surfaces that surround the magnet. This evaluation has been performed with reference to a discrete set of points belonging to the same surface. Successively, a bidimensional interpolant function is used to virtually evaluate B in each point of the previous surface that contains the points where B has been actually computed. The results are presented using a three-dimensional representation of vector B obtained by the computation

In this paper we show that magnetostatic models of permanent magnets, based on distributions of magnetic charge and shaped by a helicoidal geometry of cylindrical type, have the volume charge density ρM equal to zero. This result is valid (i) when the modulus of the vector physical magnetization density M has a constant value in all the points of the helicoidal magnet and (ii) M has the same direction of an oriented straight line having a constant angle with respect to the normal line of a cylindrical helix. Another case study concerns the permanent magnets with conchospiral geometry and magnetization M. For this kind of magnet we show that, in general, ρM≠0. Furthermore, in relation to the nonobvious result ρM=0 obtained for the cylindrical helicoidal permanent magnets, some geometrical physical considerations are illustrated. With reference to these observations, in order to understand if ρM is equal to zero or not without considering divergence computation, the possibility of defining a criterion based only on the geometry and magnetization of the magnet is discussed. Finally, an application of the results obtained from the previous analysis is shown. After drawing an analytical formulation of the surface charge density σM relative to a cylindrical helicoidal magnet, a complete evaluation of the field distribution around this magnet is performed. The results are presented by using cylindrical polar graphics of the magnetic flux B

In this paper some developments concerning the possibility of generating a rectilinear motion of bodies partially or totally submerged subject to vibration, without the use of propellers, are presented. The motion is obtained by a device equipped with counterrotating masses installed in the vessel that vibrates along the longitudinal direction. The hull has a suitably shaped stern. The study considers an analysis for evaluating the energy that the propulsion system consumes in relation to its performances. A further objective was to maximize the speed of the system while keeping certain parameters unchanged relating to the equations of motion of the device and suitably allocating the counterrotating masses. This result is obtained by using elliptical gears to transmit the motion from the driving motor to a double pair of counterrotating masses. Such a solution allows us to reach the variability of the angular velocity of the counterrotating masses during each revolution in accordance with certain laws that maximize the thrust applied to the vessel preferentially along a direction in respect of the opposite one, all being equal. Finally, a formulation to compute the propulsive efficiency of the device study and the results of the numerical simulations carried out are illustrated