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Abstract The triboelectric nanogenerator (TENG) reflects a prospering field, that uses Maxwell's displacement current as the driving force to transform mechanical energy into electricity. Based on the law of energy conservation, many theoretical models of TENGs are proposed to provide a detailed insight into how energy flows and transformation in the energy harvesting system, while ignoring a hidden but extremely important point about TENG's momentum transfer and conservation. Here a series of analysis is presented for the momentum transfer and conservation in TENGs based on Maxwell equations and stress tensor. Using a time‐dependent 3D mathematical model, it is elaborated that how the time‐ and spatial‐dependent momentum current is influenced by the field and the dielectric materials, demonstrating that momentum is overall conserved for a TENG. In other words, the TENG device can not only convert mechanical energy into electricity, but it is also able to transfer momentum. Momentum transfer is another important characteristic of TENGs, and finally, the essential differences and similarities among the momentum transfer, energy transfer, and energy transformation in TENGs are systematically discussed. This study will certainly serve as a new starting point for exploring momentum transfer and conservation in the TENG momentum transfer system.

The main aim of the present work is to investigate the role of the Maxwell stress tensor in the study of active materials. Despite the importance of this tensor in modeling mechatronic devices used in sophisticated applications, its non–symmetry still generates controversies in the literature, probably because classical continuum mechanics assumes a symmetric Cauchy stress, although the sum of Cauchy and Maxwell stresses is non–symmetric. In the framework of generalised continuum mechanics–a more advanced formalism than the classical one–, each material point has an associated microstructure so that the micro–rotations of the electric/magnetic dipoles present in real active materials may be simulated. To this end, a modified total stress formulation, including an angular momentum balance, is developed and implemented into a finite element research code using a complex–step formulation. It is concluded that generalised mechanics allows for incorporating both symmetric and non–symmetric contributions of the Maxwell tensor. Consequently, the rotations generated by the electromagnetic field may be analysed. The influence of the complete Maxwell tensor in a magnetostrictive actuator is studied by several magneto–mechanical numerical experiments of a Terfenol–D rod encircled by air, and several conclusions are highlighted. [ABSTRACT FROM AUTHOR]

Electrostatic layer jamming represents a lightweight, low energy consumption, electrically tunable, and cost-effective variable stiffness structure. Flexible parallel electroadhesive structures are the simplest form of electrostatic layer jamming. There is a lack of comprehensive and experimentally validated theoretical variable stiffness models of flexible parallel electroadhesive structures. Here we present the first variable stiffness model of flexible parallel electroadhesive structures under three-point bending, cantilever beam bending subjected to tip concentrated forces, and cantilever beam bending subjected to uniformly distributed forces, using the Euler–Bernoulli beam theory and considering friction and slip between layers by integrating the Maxwell stress tensor into the model. We find that: (1) three-point bending and cantilever beam bending under tip concentrated forces only have pre-slip and full-slip, whereas cantilever beam bending under uniformly distributed forces has an additional partial-slip which can be used for stiffness modulation; (2) the stiffness during the pre-slip stage is four times larger than the stiffness in the full-slip stage; and (3) increasing the voltage, dielectric permittivity, and coefficient of friction can elongate the pre-slip stage, thus enhancing the structural load capability. A customized three-point bending and a cantilever beam bending experimental setup were developed and the experimental deflection–force curve agreed relatively well with the theoretical one. The model, which considered electrode thickness and Young's modulus, and the results presented in this work are useful insights for understanding the variable stiffness mechanism of electroadhesive layer jamming and are helpful for their structural optimization towards practical applications. [ABSTRACT FROM AUTHOR]

The switched reluctance motor (SRM) is a promising candidate for electric propulsion systems. In the design process of an SRM, the motor geometry needs to be determined. Using the finite element method (FEM) might be time-consuming for the sizing of the motor geometry. As an alternative, electromagnetic models based on a magnetic equivalent circuit (MEC) can be utilized for the sizing of an SRM. MEC models require fewer computational resources and can help determine the electromagnetic performance with reasonable accuracy. Using the conventional MEC method for SRM sizing might be challenging since the flux pattern inside the motor should be changed for different motor dimensions. In order to address this challenge, this paper applies a reluctance mesh-based MEC technique to determine the geometry of a three-phase 12/16 SRM for a high-lift motor in the NASA Maxwell X-57 electric aircraft. A comprehensive reluctance mesh-based MEC model is developed for this purpose. Both the static and dynamic characteristics of the SRM geometry are evaluated using the reluctance mesh-based MEC method. The determined geometry is verified using the results computed from FEM. [ABSTRACT FROM AUTHOR]

The propagation matrix method and the Maxwell stress tensor are developed to calculate radiation pressure on an inhomogeneous active chiral structure. Based on boundary conditions, a 4 × 4 propagation matrix connecting two forward and two backward plane waves in adjacent chiral media is derived. This propagation matrix method allows us to access the net, co‐ and cross‐polarized radiation pressure, which are expressed by co‐ and cross‐polarized reflection and transmission coefficients, exerted on magnetoelectric coupling inhomogeneous chiral slabs via the derivation of the Maxwell stress tensor for arbitrary polarized plane waves incidence. The accuracies of the methods are examined using the finite difference time domain method and the Lorentz force densities. The radiation pressure on a homogeneous or inhomogeneous active chiral slab affected by the angle of incidence, thickness, chirality parameter, and polarization states of incident waves are investigated. This work is believed to provide insights for detecting chirality parameters of inhomogeneous materials by optical forces. [ABSTRACT FROM AUTHOR]

This paper presents an innovative method for performing the initial sizing of tubular linear actuators (TLAs). Based on the Maxwell Stress Tensor (MST), an analytical expression is developed that allows the calculation of the force produced by a TLA in a quick and simple way. This expression can be employed for determining dimensional parameters considering the electric and magnetic loadings usually found in such electromagnetic devices. This technique can be applied to design slotless or slotted tubular linear permanent magnet synchronous motors (TLPMSM). In order to validate the developed method, four topologies of TLPMSM are considered. The calculated results of the developed analytical expression are presented and compared with results obtained by means of a magnetostatic finite element analysis (FEA). Additionally, the test results of a TLA prototype were used to validate experimentally the expression of force calculation by means of the MST method. [ABSTRACT FROM AUTHOR]

Acoustic noise and vibration is one of the shortcomings of a switched reluctance machine (SRM). Harmonics of the radial force waveform in the airgap excite the stator structure at different vibration modes with specific frequencies. The radial force density in the airgap should be calculated before analyzing and mitigating acoustic noise and vibration. This paper proposes a reluctance mesh-based magnetic equivalent circuit (MEC) model to calculate the airgap radial force density. Reluctance mesh-based MEC models are developed for 3-phase 6/4, 6/16, 12/8 and 4-phase 8/6, 8/10, and 16/12 SRMs. A technique for the dynamic modeling of SRMs is proposed using the reluctance mesh-based MEC method. Dynamic currents are calculated using the proposed technique and, then, the radial and tangential flux density in the airgap are calculated. The radial force density in the airgap is calculated by applying the Maxwell Stress Tensor method. Fourier series is used to calculate the harmonics of the radial force density. The results obtained from the MEC model are verified using finite element method (FEM) models. The implemented MEC-based dynamic modeling method is validated using experimental results.

This article proposes an improved tubular single-phase permanent magnet linear oscillation actuator (SP-PMLOA) considering eliminating detent force for direct drive compressor applications. First, the flux tube method and Maxwell stress tensor are used to derive the end force and cogging force formulations based on the magnetic flux leakage of the magnet rings. Then, a novel model with minimum detent force is presented by finite element parametric sweeping. Finally, the electromagnetic performance of the three SP-PMLOAs is compared using 2-D finite element analysis (FEA), whose merits are evaluated, and the thermal distribution of the optimal PMLOA is analyzed using 3-D FEA. The predicted results show that the proposed structure not only can significantly reduce the detent force and the mover axial length but also can increase the thrust force-moving magnet mass ratio. [ABSTRACT FROM AUTHOR]

To study the clamping device in reluctance force launcher, the Maxwell stress tensor method is extended to contact force calculations. It is generally believed that the Maxwell stress tensor method is only suitable for objects surrounded by air or vacuum. However, in this article, the Maxwell stress tensor applicable to ferromagnetic materials is re-derived, and the classical Maxwell stress tensor is successfully reproduced. Its integral region is determined as the outer surface of the interface between the object and the outside world, which means that the Maxwell stress tensor method can be used for contact force calculations. Subsequently, the Maxwell stress tensor method was used for force studies in the clamping device. The needed clamping force of the clamping device is calculated. The reducing effect of ferromagnetic materials on the clamping force and the effect on the acceleration force of the projectile are analyzed. Finally, based on the determined integral region, the reason why the integral result of Maxwell stress tensor method is related to the integral path is successfully explained, which provides another theoretical basis for the contact force calculation method in this article. [ABSTRACT FROM AUTHOR]

The Maxwell stress tensor has a great significance, for example for the calculation of forces in electromagnetic energy converters, and thus is standard content of advanced lectures during electrical engineering study. However, the derivation and understanding of it is difficult and there is standard literature that is not consistent and thus increases the difficulties. In order to gain a consistent and vivid derivation of the stress tensor for use in electrical engineering lectures, two common variants are analysed from a didactic as well as scientific point of view. It is shown that one variant is consistent whereas the other one can lead to contradictions in case of materials with permeability gradients and thus is not recommended for lectures. Moreover, the resulting tensors of both variants yield different force distributions. For the consistent variant, methods for clarifying the respective force distribution for use in lectures are derived. Finally, a brief consideration on the experimental provability of the results is carried out. [ABSTRACT FROM AUTHOR]

Installing a radial passive magnetic bearing that consists of a pair of permanent magnet rings may be a successful technical solution for a number of devices as it eliminates wear and friction and is inexpensive. The low load capacity and radial stiffness of this design limit its scope of application. This article proposes four alternative designs of a passive magnetic bearing for the rotor (impeller) of a circulatory support pump. A finite element model is constructed, verified, and used in a theoretical study of magnetic fields of a radial passive magnetic bearing. The arrangement that provides the greatest radial stiffness and load capacity at given size restrictions is identified. The obtained force—radial displacement dependence is used to model the dynamics of a circulatory support pump impeller. [ABSTRACT FROM AUTHOR]

We have shown that when sharply focusing a linearly polarized optical vortex with topological charge 2, in the near-axis region of the focal plane, not only does a reverse energy flow (the negative on-axis projection of the Poynting vector) occur, but also the right-handed circular polariza-tion of light. Moreover, due to spin-orbital angular momentum conversion, the on-axis polarization vector and the transverse energy flow rotate around the optical axis in the same direction (counter-clockwise). If an absorbing spherical microparticle is put in the focus on the optical axis, it will rotate around the axis and around its center of mass counterclockwise. Numerical simulation results confirms the theoretical predictions.

This contribution introduces a vector interpretation to unify the description of torque development in rotating electrical machines. The motivation behind this attempt is the necessity of applying different physical interpretation and calculation methods, such as Lorentz force, Maxwell stress tensor, co-energy methods etc., for predicting and estimating different torque components, e.g. synchronous torque, oscillating torque, cogging torque, reluctance torque etc., in different rotating electrical machines. The unified vector model describes and estimates the different torque components in rotating electrical machines with an apprehensible concise formulation. Beyond that, the other most frequently used tools and terms in the field of electrical machines, such as Park and Clark transformations, phasor diagram etc., can be derived directly from this model, which has also a simplifying didactic contribution to a conventional course of electrical machines.

Linear Flux Switching Machines (LFSMs) are suitable candidates for long stroke applications as they confines all excitation sources to primary thus leaving completely passive, robust, and low cost secondary. Permanent Magnet LFSMs (PMLFSMs) enables high thrust force density and efficiency. However, deficiency of controllable air-gap flux, risk of PM demagnetization, and increasing cost of rare earth PM materials diverted researchers towards Field Excited LFSMs (FELFSMs). FELFSMs wiped out aforementioned PMLFSM’s shortcomings at the cost of low thrust force density. In this paper, merits of PMLFSM and FELFSM are combined by proposing a novel Hybrid Excited LFSM (HELFSM). Proposed machine is excited by PMs, Field Excitation Coils (FECs), and Armature Windings (AWs). However, complex magnetic circuit of poly-excited HELFSM compels designers to adopt FE Analysis (FEA) for design, analysis, and optimization. To decrease dependency on computationally complex and time consuming FEA, an analytical model combining lumped parameter magnetic equivalent circuit, Fourier analysis, Laplace equation, and Maxwell Stress Tensor method is proposed to predict open-circuit flux linkage, B-EMF, normal and tangential components of no-load and on-load magnetic flux density, detent, and thrust force performance. Finally, predictions of the developed analytical model are validated with corresponding FEA and experimental results.

The switched reluctance motor (SRM) is a promising candidate for electric propulsion systems. In the design process of an SRM, the motor geometry needs to be determined. Using the finite element method (FEM) might be time-consuming for the sizing of the motor geometry. As an alternative, electromagnetic models based on a magnetic equivalent circuit (MEC) can be utilized for the sizing of an SRM. MEC models require fewer computational resources and can help determine the electromagnetic performance with reasonable accuracy. Using the conventional MEC method for SRM sizing might be challenging since the flux pattern inside the motor should be changed for different motor dimensions. In order to address this challenge, this paper applies a reluctance mesh-based MEC technique to determine the geometry of a three-phase 12/16 SRM for a high-lift motor in the NASA Maxwell X-57 electric aircraft. A comprehensive reluctance mesh-based MEC model is developed for this purpose. Both the static and dynamic characteristics of the SRM geometry are evaluated using the reluctance mesh-based MEC method. The determined geometry is verified using the results computed from FEM.

Shunt reactors are usually used in electrical systems to imbibe reactive powers created by capacitive powers on the lines when the system is operating on low or no loads. Moreover, they are also used to balance reactive powers and maintain the stability of a specified voltage. In general, the air gaps of a magnetic circuit shunt reactor are arranged along the iron core to reduce the influence of fringing and leakage fluxes. Therefore, non-magnetic materials made of ceramics or marbles are often used in air gaps to separate the iron core packets. The direction of the fringing flux is perpendicular to the laminations, so the core packets of the shunt reactor are generally made from radially laminated silicon steels. Due to the alternating electromagnetic field through the core, a periodically altered electromagnetic force is produced between the core packets, tending to compress the ceramic spacers. This electromagnetic force causes vibration and noise in the core. In this research, a finite element approach based on the Maxwell stress tensor was developed to compute the magnetic flux density and the electromagnetic forces appearing in a shunt reactor. [ABSTRACT FROM AUTHOR]

This article describes numerical and experimental modeling of the electro-adhesion force and its application for the electrostatic sorting of a metal/polymer granular mixture, using an interdigitated electrode network covered with a dielectric layer. Based on Maxwell stress tensor formulation, a numerical simulation is carried out to study the properties of the electro-adhesion force in the presence of conductive and dielectric particles, for different electrical and geometrical parameters. Moreover, experimental analysis was performed to study the effect of the electrical factors on the electro-adhesion force. Numerical results indicate that the adhesion force applied on the metal particles is at least two decades higher than the one applied on the dielectric particles. Furthermore, the experimental study shows that the electro-adhesion actuator attracts the conductive granules without any effect on the dielectric particles when a square ac signal is applied. [ABSTRACT FROM AUTHOR]

We show that a 70-nm dielectric nanoparticle placed on the optical axis near the surface (at a distance less than 100 nm) of a high-NA gradient microlens made of silicon, which is illuminated by a laser beam of 1.55 μm wavelength, is attracted to the lens surface with a piconewton force. The profile of the lens refractive index is described by a hyperbolic secant function. If a cut-out is made in the lens output surface, then the nanoparticle will be pulled into this cut-out, producing a kind of 'optical magnet'. If a reverse energy flow is to be generated on the optical axis near the output surface of such a gradient lens, this will lead to an absorbing dielectric nanoparticle being pulled toward the surface with a greater force than a similar non-absorbing particle. In the absence of a reverse flow, both absorbing and non-absorbing particles will be attracted to the surface with an equal force. The electromagnetic fields involved are calculated using a finite difference time domain (FDTD) method and the acting forces are calculated using a Maxwell stress tensor.

Shunt reactors are usually used in electrical systems to imbibe reactive powers created by capacitive powers on the lines when the system is operating on low or no loads. Moreover, they are also used to balance reactive powers and maintain the stability of a specified voltage. In general, the air gaps of a magnetic circuit shunt reactor are arranged along the iron core to reduce the influence of fringing and leakage fluxes. Therefore, non-magnetic materials made of ceramics or marbles are often used in air gaps to separate the iron core packets. The direction of the fringing flux is perpendicular to the laminations, so the core packets of the shunt reactor are generally made from radially laminated silicon steels. Due to the alternating electromagnetic field through the core, a periodically altered electromagnetic force is produced between the core packets, tending to compress the ceramic spacers. This electromagnetic force causes vibration and noise in the core. In this research, a finite element approach based on the Maxwell stress tensor was developed to compute the magnetic flux density and the electromagnetic forces appearing in a shunt reactor.

There are three principal sources of noise and vibration in electrical machines: electromagnetic sources, mechanical sources, and aerodynamic sources. Nowadays, one of the major advantages of permanent-magnet synchronous machines is their torque density. This density is achieved through a high magnetic flux density in the air gap, which is achieved through hard magnets. Unfortunately, in these machines, electromagnetic forces have been identified as the main source of vibration and noise, and high magnetic flux densities make these vibrations and noises more significant. With the aim of better understanding the relationship between electromagnetic forces and design variables, this article, which is the continuation of previous work, firstly describes a study of the sources of magnetic forces in permanent-magnet synchronous machines. Subsequently, an analytical model for the computation of the radial forces originating from electromagnetic sources in permanent-magnet synchronous machines is stated. This model analyzes the forces on both the rotor surface and the base of the stator tooth. The analytical results were corroborated through simulations using the finite element method (FEM) and also by experimental tests performed over two prototypes. The results achieved by the analytical model show good agreement with both FEM results and experimental measurements. [ABSTRACT FROM AUTHOR]

The basic continuum model for polar fluids is deceptively simple. The free energy integral consists of four terms: the coupling of polarisation to an external field, the electrostatic energy of the induced electric field interacting with itself and the stored polarisation energy quadratic in the polarisation. A local function of density accounts for the mechanical state of the fluid. Viewed as a non-equilibrium free energy functional of number density and polarisation, minimisation in these two densities under constraints of the Maxwell field equations should lead to the correct equilibrium state. The alternative is a continuum mechanics approach in which the mechanical degree of freedom is extended to full deformation. We show that the continuum electromechanics method leads to a force balance equation which is consistent with the density functional equilibrium equation. The continuum mechanics procedure is significantly more demanding. The gain is a well-defined pressure tensor derived from deformation of total energy. This resolves the issue of the uncertainty in the pressure tensor obtained from integration of the force density, which is the conventional method in density based thermomechanics. Our derivation is based on the variational electrostatics approach developed by Ericksen (Arch. Rational Mech. Anal. 183 299 (2007)). [ABSTRACT FROM AUTHOR]

In this paper, an indirect analytical method is presented to analyze and optimize the electromagnetic performances, such as air-gap flux density, back EMF, cogging torque, and etc. of the V-shaped interior permanent magnet (PM) synchronous machine (IPMSM). The new method provides an indirect calculation model of the V-shaped IPMSM, by transferring the rotor into an equivalent linear one in the polar coordinate system. Based on the reconstruction of the V shaped PM, two part of magnetization calculation models are established, and the magnetic field distribution of the motor is analyzed by utilizing the magnetic scalar potential. The Maxwell stress tensor method with Schwarz-Christoffel (SC) transformation is subsequently utilized to analyze the magnetic field in detail and predict the cogging torque performances of the V-shaped IPMSM accurately. The results are compared with that obtained by the finite element analysis (FEA) method. Moreover, a prototype is built and tested, and the results validate the accuracy and validity of the analytical computation results.

Magnetic gears offer significant advantages such as low noise and vibration level, lower maintenance and higher reliability compared to mechanical gears and are suitable for many applications in the industry. The coaxial magnetic gear has been extensively discussed in the literature, since it achieves higher torque densities amongst other magnetic gear configurations. The magnetic field is generated by permanent magnets mounted on the two rotors and a modulator between them. The modulator consists of ferromagnetic segments that are typically encased in a resin in order to increase its stiffness without compromising the generated magnetic field. However, due to the development of radial forces, oscillations of the ferromagnetic segments occur, which lead to torque ripples that affect the operation of the coaxial magnetic gear drive in applications where accuracy is required. This work introduces a computationally lightweight analytical 2D model in order to determine the applied radial force on the ferromagnetic segments at each angle of rotation of the two rotors and henceforth calculate the displacement of these segments using FEA. In this way it is possible to assess the variation of the torque (ripple) versus the angle of rotation of the input or output shaft. A parametric investigation examining the influence of the ferromagnetic segment thickness on the resulting torque ripple of a specific drive was carried out illustrating the benefits of the analytical models developed herein.

In the research of torque ripple of permanent magnet toroidal motor (PMTM), the planet gears are affected by the end-effect force during its rotation due to the special structure of the central worm stator. The existence of the end-effect force will cause vibration and noise, which will lead to poor operation accuracy of the planet gears. In view of the above problems, the formation mechanism of the torque ripple is analyzed. We point out that the sudden change in the magnetic induction intensity at the end of the central worm stator is the main source of the end-effect force. The Schwarz–Christoffel transformation method is used to analyze the magnetic field distribution at the end of the stator. The mathematical model of the end-effect force of PMTM is derived by applying Maxwell stress tensor method. The influence of the structural parameters on the end-effect force of PMTM is analyzed by finite-element simulation. The results show that the output fluctuation of PMTM can be effectively reduced by choosing the optimal wrap angle of the central worm stator by calculating. In addition, choosing the appropriate polar arc coefficient of the planet gears can further reduce the influence of the end-effect force. These research results can provide important reference for the structural optimization of PMTM. [ABSTRACT FROM AUTHOR]

There are three principal sources of noise and vibration in electrical machines: electromagnetic sources, mechanical sources, and aerodynamic sources. Nowadays, one of the major advantages of permanent-magnet synchronous machines is their torque density. This density is achieved through a high magnetic flux density in the air gap, which is achieved through hard magnets. Unfortunately, in these machines, electromagnetic forces have been identified as the main source of vibration and noise, and high magnetic flux densities make these vibrations and noises more significant. With the aim of better understanding the relationship between electromagnetic forces and design variables, this article, which is the continuation of previous work, firstly describes a study of the sources of magnetic forces in permanent-magnet synchronous machines. Subsequently, an analytical model for the computation of the radial forces originating from electromagnetic sources in permanent-magnet synchronous machines is stated. This model analyzes the forces on both the rotor surface and the base of the stator tooth. The analytical results were corroborated through simulations using the finite element method (FEM) and also by experimental tests performed over two prototypes. The results achieved by the analytical model show good agreement with both FEM results and experimental measurements.

Giant magnetostrictive actuators are suitable for applications requiring large mechanical displacements under low magnetic fields; for instance Terfenol-D made out of rare earth-iron materials can produce important strains. But these actuators exhibit hysteretic non-linear behavior, making it very difficult to experimentally characterize them. Therefore, sophisticated numerical algorithms to develop computational tools are necessary. In this work, theoretical and numerical formulations within the finite element method are developed to simulate magnetostriction. Theoretically, within the framework of non-equilibrium thermodynamics, the hysteresis is introduced by the Debye-memory relaxation. Numerically, the main novelty is the time integration, coupled Newmark- β (for mechanical) and convolution integrals (for magnetic constitutive equations); the non-linearity is solved with the standard Newton–Raphson algorithm. Constitutive non-linearities are incorporated with the Maxwell stress tensor, quadratically dependent on the magnetic field. The numerical code is validated using analytical and experimental solutions; several examples are presented to demonstrate the capabilities of the present formulation. [ABSTRACT FROM AUTHOR]

We present a shape design optimization method for interdigitated electrodes in an electro-adhesive device. In the finite element analysis of electrostatics using linear basis functions, a finite node displacement method is used for the accurate electro-adhesive force by integrating the electric field along the boundary surface. The floating potential boundary for conductive objects is handled by a charge conservation law in the electrostatic analysis as well as the shape design sensitivity analysis. In numerical examples, the structural shape of interdigitated electrodes is optimized to maximize the electro-adhesion force per unit area for both conductive and non-conductive objects. It turns out that the electro-adhesive force is mainly induced by an electrostatic induction for conductive objects and by an electric polarization for non-conductive ones. There is an optimal ratio of electrode width and air gap thickness for non-conductive objects but no limit for conductive objects. [ABSTRACT FROM AUTHOR]

Objective: The efficiency of vibrational energy harvesting systems that consist of a cantilever beam with attached piezoceramic layers can be increased by intentionally introducing nonlinearities. These nonlinearities are often implemented in the form of permanent magnets near the beam's free end. The influence of these magnets is typically assumed to be a single transverse force that depends cubically on the displacement of the beam tip. The coefficients of a corresponding single degree of freedom model are often found heuristically, without an explicit modeling of the magnetoelastic force. Methods: In this paper, we present and assess the validity of a procedure to determine the magnetoelastic forces acting on the beam from physical laws of the magnetic field with corresponding parameters. The paper outlines the method itself, describing initially how the magnetic field is computed by a Finite Element simulation. In the second step, the total transverse force on the beam is determined from the magnetic field by means of a numerical evaluation of the Maxwell stress tensor. The required minimum degree of a suitable polynomial force approximation of the numeric values is discussed briefly. The validity of this model is then considered by investigating its bifurcation behavior with respect to mono-, bi-, and tristability for different distances between the magnets and comparing the findings to results found by experiments. Results: While the model's predictions of the number of equilibrium positions for any magnet distance are generally in good agreement with results of experiments, there are deviations when it comes to the exact positions of the equilibria. With respect to these findings, the limitations of a two-dimensional magnetic field modeling with only linear material models are addressed. Conclusion: The paper concludes that the method outlined here is a step toward the deduction of a detailed model based on physical laws of the magnetic field with corresponding parameters replacing simple heuristic polynomial magnetic force laws. [ABSTRACT FROM AUTHOR]

We examine the translation and rotation of an uncharged spheroidal colloid in polar solvents (water) near a charged flat surface. We solve the nonlinear Poisson-Boltzmann equation outside of the colloid in two dimensions for all tilt angles θ with respect to the surface normal. The colloid's size is assumed to be comparable to the Debye's length and hence field gradients are essential. The Maxwell stress tensor, including the ideal gas pressure of ions, is integrated over the colloid's surface to give the total force and torque on the colloid. From the torque we calculate the effective angular potential U eff (θ). The classical behavior where the colloid tends to align in the direction perpendicular to the surface (parallel to the field, θ = 0) is retrieved at large colloid-surface distances or small surface potentials. We find a surprising transition whereby at small separations or large potentials the colloid aligns parallel to the surface (θ = 90 °). Moreover, this colloid orientation is amplified at a finite value of the aspect ratio. This transition may have important consequences to flow of colloidal suspensions or as a tool to switch layering of such suspensions near a surface. [ABSTRACT FROM AUTHOR]

We propose and theoretically demonstrate, through numerical simulations, a model for a dielectric microsphere capable of being manipulated translationally and rotationally by linearly polarized plane electromagnetic waves. By the use of realistic material parameters and finite-element-method simulations of the Maxwell stress tensor, we illustrate how transverse radiation forces and radiation torques could be applied to the spherical particle through a proper control of the frequency and polarization direction of the incident linearly polarized plane wave. [ABSTRACT FROM AUTHOR]

The Maxwell Tensor (MT) method is widely used to compute global forces or local surface forces for vibroacoustic design of electrical machines under electromagnetic excitation. In particular, the air-gap MT method is based on a cylindrical shell in the middle of the air gap. This article proposes to quantify the differences between the air-gap MT and the magnetic force wave experienced by the stator. In particular, the air gap to stator transfer and the tooth mechanical modulation effects are studied. A new formula is proposed to extend the tooth modulation effect to tangential forces. A numerical application is performed with a turbo-alternator to illustrate the respective and combined effects of both phenomena. This article highlights that the tooth mechanical modulation is relevant even for electrical machines with a high number of teeth. In addition, the combination of the two phenomena has a clear impact on the calculated surface force. Therefore, it is recommended to take into account the air-gap transfer for any study of the tooth mechanical modulation effect. [ABSTRACT FROM AUTHOR]

Rotating electrical machines are becoming ubiquitous as we move towards a more electrified and sustainable world. Torque calculation is an essential task in the design process of rotating machines. In the frame of finite element analysis, the Maxwell stress tensor method is a common technique to calculate the torque exerted on a rigid body. However, the computed torque is strongly mesh-dependent and can be inaccurate. The popular Arkkio method can overcome this problem. But Arkkio's formulation can only calculate the axial torque of conventional radial-flux electrical machines and is unsuitable for axial-flux motors and generators, which are attracting increasing interest due to the power density and efficiency improvements that the axial topology can enable. This article presents an easy-to-implement formulation for calculating the axial torque of axial-flux machines in finite element analysis. • We present a novel torque calculation method for axial-flux machines. • The formulation is easy-to-implement in the frame of finite element analysis. • It overcomes the accuracy issues of the Maxwell stress tensor method. • Our approach is analogous to Arkkio's formulation for radial-flux machines. [ABSTRACT FROM AUTHOR]

Highlights • The effect of nanoscale surface roughness on the optical trapping properties of nanoparticle near the composite gold film is investigated in detail. • Surface roughness in the calculation model is simulated by utilizing the Monte-Carlo method. • Surface plasmon polaritons are excited by the interaction between the incident Gaussian beam and the periodic structure of gold film. • The effects of root mean square height and correlation length of rough surface on optical trapping force are analyzed in detail. Abstract Based on the three-dimensional dispersive finite difference time domain method and Maxwell stress tensor equation, the effect of nanoscale surface roughness on the optical trapping properties of nanoparticle in a vicinity of the composite gold film with periodic structure is investigated numerically. The periodic structure is observed as circular holes which can excite the surface plasmon polaritons on the metal-dielectric interface with particular emphasis on its crucial role in tailoring the optical force acting on a nearby nanoparticle. Utilizing the Monte-Carlo method, the surface roughness is added into the calculation model of the proposed method to accurately investigate the optical performance of the film-tuned nanoparticle system. Selected calculations on the effects of root mean square height and correlation length of rough surface are analyzed in detail to demonstrate that the negative effect of the surface roughness on the optical trapping force can be eliminated when the ratio of correlation length to root mean square height is equal to 10. Accurate investigation of optical trapping properties of nanoparticle in a vicinity of the composite gold film could provide guidelines for further research on the optical system design and manipulation of arbitrary composite nanoparticles. [ABSTRACT FROM AUTHOR]

Local force distribution is required to evaluate the deformation and vibration of a core in an electrical machine using the mechanical analysis. In the local force calculation, the Minkowski and Chu models with linear and nonlinear energy expressions are often used for the Maxwell stress tensor in linear or nonlinear condition, respectively. In the nonlinear magnetic field analysis, the volume forces, which move a magnetic body, obtained by using the above two models coincide with each other, whereas their local forces, which deform the magnetic body, are different. In this paper, a new expression of Maxwell stress tensor derived from Fleming’s left-hand rule is proposed. Then, the local forces of a simple magnetic body obtained by using various expressions of Maxwell stress tensor are compared with that obtained from the equivalent magnetizing current method. The surface force obtained from the proposed Maxwell stress tensor coincides with the equivalent magnetizing current method, and it is much different from those obtained by using the Minkowski and Chu models in this simple analytical model. Finally, the local force distribution in a 2-D finite element model is also demonstrated to show the necessity of the proposed model. [ABSTRACT FROM AUTHOR]

Electromagnetic vibration disturbing forces in different variants of the rotor displacement from an axis of the stator bore is carried out. Investigation for ТG type ТGV-200-2 by finite element method in COMSOL Multiphysics is carried out. The field mathematical model of static and dynamic eccentricity is described. The amplitude vibration disturbing forces are greatest, when a static eccentricity direction coincides with an axis of the stator winding phase is shown. The diagnostic features static and dynamic eccentricities are formulated. The most value of forces in the point with minimal air gap is shown. The diagnostic features static and dynamic eccentricities and the method of diagnostic eccentricity are formulated. Diagnostic feature of static eccentricity is to change the amplitude Maxwell stress tensor is established. The dynamic eccentricity diagnostic features are appearance in the spectrum of vibration disturbing forces rotating and multiple harmonics.

The considerable electric-induced shape change, together with the attributes of lightweight, high efficiency, and inexpensive cost, makes dielectric elastomer, a promising soft active material for the realization of actuators in broad applications. Although, a number of prototype devices have been demonstrated in the past few years, the further development of this technology necessitates adequate analytical and numerical tools. Especially, previous theoretical studies always neglect the influence of surrounding medium. Due to the large deformation and nonlinear equations of states involved in dielectric elastomer, finite element method (FEM) is anticipated; however, the few available formulations employ homemade codes, which are inconvenient to implement. The aim of this work is to present a numerical approach with the commercial FEM package COMSOL to investigate the nonlinear response of dielectric elastomer under electric stimulation. The influence of surrounding free space on the electric field is analyzed and the corresponding electric force is taken into account through an electric surface traction on the circumstances edge. By employing Maxwell stress tensor as actuation pressure, the mechanical and electric governing equations for dielectric elastomer are coupled, and then solved simultaneously with the Gent model of stain energy to derive the electric induced large deformation as well as the electromechanical instability. The finite element implementation presented here may provide a powerful computational tool to help design and optimize the engineering applications of dielectric elastomer.

The paper presents an analysis of local and global forces acting on the ferromagnetic material of a modulator in a co-axial magnetic gear, taking several design variants and the impact of loading into account. The analyses include a modulator with cores manufactured from a soft-magnetic composite material and two variants made from electrical steel with laminations stacked in different directions. Variations of local forces acting on individual pole pieces of the modulator are analyzed at different loads, showing that the force spectra are subject to significant variation with an increasing load. The presented magnetostatic analyses are extended by structural analysis that provides estimation of stress and displacement for the modulator assembled from additively manufactured acrylonitrile butadiene styrene (ABS)-plastic parts. The analysis carried out for the least favorable design case of the magnetic circuit of the modulator shows that an application of the technology is significantly restricted by the magnetic gear torque volumetric density. Some changes to the modulator mechanical design are proposed in the paper to mitigate the drawbacks of this technology.