The paper describes the simulating Helmholtz coils and improved Helmholtz coils in COMSOL. Improved Helmholtz coils allows to create a magnetic field with inhomogeneity is not more than 1 % at a distance of half the radius of the geometric center of the coils system. The simulation technique is described.

The paper describes the design of axial 8-coils system based on Helmholtz coils, which allows to create a magnetic field with inhomogeneity is not more than 0.1% at a distance of half the radius of the geometric center of the coils system. The obtained results are confirmed by experiment.

Purpose: Technological capabilities of manufacturing microelectromechanical system (MEMS) gyroscopes are still insufficient if compared to manufacturing high-efficient gyroscopes and accelerometers. This creates weaknesses in their mechanical structure and restrictions in the measurement accuracy, stability and reliability of MEMS gyroscopes and accelerometers. This paper aims to develop a new architectural solutions for optimization of MEMS gyroscopes and accelerometers and propose a multi-axis MEMS inertial module combining the functions of gyroscope and accelerometer. Design/methodology/approach: The finite element modeling (FEM) and the modal analysis in FEM are used for sensing, drive and control electrode capacitances of the multi-axis MEMS inertial module with the proposed new architecture. The description is given to its step-by-step process of manufacturing. Algorithms are developed to detect its angular rates and linear acceleration along three Cartesian axes. Findings: Experimental results are obtained for eigenfrequencies and capacitances of sensing, drive and control electrodes for 50 manufactured prototypes of the silicon electromechanical sensor (SES). For 42 SES prototypes, a good match is observed between the calculated and simulated capacitance values of comb electrodes. Thus, the mean-square deviation is not over 20 per cent. The maximum difference between the calculated and simulated eigenfrequencies in the drive channel of 11 SES prototypes is not over 3 per cent. The same difference is detected for eigenfrequencies in the first sensing channel of 17 SES prototypes. Originality/value: This study shows a way to design and optimize the structure and theoretical background for the development of the MEMS inertial module combining the functions of gyroscope and accelerometer. The obtained results will improve and expand the manufacturing technology of MEMS gyroscopes and accelerometers.

Preference aggregation as a problem of a single consensus ranking determination, using Kemeny rule, for m rankings, including ties, of n alternatives is considered in the paper. The Kemeny Ranking Problem (KRP) may have considerably more than one optimal solutions (strict orders or permutations of the alternatives) and, hence, special efforts to deal with this phenomenon are needed. In the paper, there is proposed an efficient formal rule for convolution of the N multiple optimal permutations, the output profile β(N, n), into an exact single final consensus ranking, which can include ties. The convolution rule is as follows: in the final consensus ranking, alternatives are arranged in ascending order of their rank sums (total ranks) calculated for the output profile β; some two alternatives are considered to be tolerant if they have the same rank sums in β. The equivalent convolution rule can be also applied as follows: in the final consensus ranking, alternatives are arranged in descending order of row sums (total scores) calculated for a tournament table built for β; some two alternatives are deemed to be tolerant if they have the same row sums. It is shown that, for any alternative, its total rank and total score are equal in sum to the output profile dimension N×n. The convolution rules are validated using Borda count.