Lagrange-Maxwell Equations Research Articles

COMPACT ANALYTICAL MODEL FOR ELECTROMAGNETIC DISK-SHAPED UNDERWATER TRANSDUCER

This paper proposes a mathematical model for a disk-type pulsed electrodynamic transducer operating in the low-frequency range. A well-known architecture of an electromagnetic acoustic radiator with a helical coil and a conducting disk is studied. In the work, the equations of the electromechanical system in the form of the Lagrange-Maxwell equations are built with the use of the Green's functions of a plane axisymmetric acoustic problem to estimate the reaction force of the fluid. A comparison is made between the results of a numerical solution of the obtained equations and direct numerical calculation in the COMSOL finite element analysis software. The resulting model shows good qualitative agreement with the results of finite element calculations while allowing calculations with a variation for all main model parameters required to design the transducer.

Torsional behaviors of motor rotor of permanent magnet semi-direct drive system for aerial passenger device

Considering the influence of motor shaft torsional damping, the torsional dynamics equations of a permanent magnet semi-direct drive rotor system in an aerial passenger device are derived. The derivation relies on the coupling between the mathematical model and the gear dynamics model of the permanent magnet motor, and the Lagrange–Maxwell equations are used for streamlined analysis. To protect the motor under extreme working conditions, the influence of supercritical Hopf bifurcation caused by different torsional dampings of the motor shaft on the torsional vibration of the motor rotor is investigated. Based on the Hurwitz criterion, the influence of the linear control gain and nonlinear gain of the washout filter on the system bifurcation point and the limit cycle amplitude of the system is analyzed. Then, from the perspective of the system instability and oscillation caused by friction damping change of the drive motor and the complex electromechanical coupling behavior in the permanent magnet semi-direct drive cutting drive system of the aerial passenger device, the system stability domain is defined. The results can be applied to effectively suppress the torsional vibration of the shaft system in the permanent magnet semi-direct drive system of the aerial passenger device, providing a theoretical guidance for stable operation of the system.

Creation of an electromechanical model of an exoskeleton link in the form of Lagrange-Maxwell equations for agricultural mechanization

The electromechanical model of exoskeleton link has been developed for agricultural mechanization. First, a differential equation of link motion was compiled without taking into account electric drives in the form of the Lagrange equation of the second kind. For it, the inverse and direct problems of dynamics are solved. Then, this equation takes into account the influence of the rotating rotor and the gear ratio of the gearbox on the dynamics of the link. For this model, the direct problem of dynamics is solved. A significant influence on the results of the numerical solution for taking into account the rotating rotor of the electric drive has been established. Then the system of differential equations, describing dynamics of the controlled motion of the model, in the form of Lagrange-Maxwell equations has been composed. The local system of coordinates has been applied in the model, since the electric drive changes the angles between the links. Direct and inverse dynamics problems have been solved. The comparative analysis of the models with electric drive and without it has been made. It has been established that taking into account the electrical drive system allows achieving good results in modeling accuracy.

Modal analysis of vibrations and evaluation of vibration activity of the platform and propeller of an RW-UAS aerial robot

The quadrocopter used to perform mine surveying and geodetic work, to develop a non-contact ore sampling technology based on the use of an intelligent unmanned aerial vehicle equipped with technical vision and a specialized X-ray fluorescent energy-dispersive device installed on it. In this work, the equations of quadrocopter dynamics under load derived, and the influence of dynamic disturbances on SSS propellers carried out. The analysis of torques on four propellers carried out. The equations of dynamics of 4 rotors of the RW-UAS robot (quadcopter) are written on the basis of the Lagrange-Maxwell equations. An analysis of the control moments carried out, which makes it possible to implement horizontal and vertical flights of the quadrocopter. The simulation carried out using the Maple analytical computing environment. Based on the modal analysis of the equations of dynamics of the air robot, the results obtained where the ranges of wave disturbances on the air vehicle are determined and the SSS of the blades and their elastoplastic displacement obtained. The conducted research shows the need for a detailed study of dynamic processes on an air robot, the behavior of the rotor system is especially important.

Mathematical modeling of typical electrical circuits, oscillators and pendulums

The work is devoted to mathematical and simulation modeling of electromechanical systems, in which oscillations in electrical circuits, oscillators and pendulums are studied under the action of forces of various physical nature. To compile the equations of electromechanical systems, the apparatus of analytical mechanics was used, in which the electromagnetic and mechanical quantities characterizing the system appear as formally equal. The equations of motion are based on the Lagrange-Maxwell equations. The simulation was performed using computer algebra programs.

MODELING OF DYNAMICS OF ELECTRODYNAMIC SPEAKER

This article discusses an electromechanical system in the form of an electrodynamic loudspeaker. To describe the dynamics of a loudspeaker, a finite-dimensional model with a given number of a finite number of independent mechanical and electrical parameters is used. Using the electromechanical analogy force - voltage and using the Lagrange-Maxwell equations, a closed system of two nonlinear differential equations of the second order of motion of the system is compiled. Using modern advances in computer technology, a number of numerical experiments were performed in the Mathcad system with varying values of the input parameters of the system, such as the mass of the moving coil, inductance, ohmic resistance, and graphs of the time variation of the coil movement and current strength were plotted. Phase portraits were also built. The movement of the system has the character of damped oscillations, and the phase portraits have a stable focus. The influence of the varied parameters on the period of damped oscillations and the transition of the system to the state of equilibrium is shown. The study was carried out for a nonlinear model without the use of asymptotic methods, which made it possible to exclude the methodological error of the solution. The model can exhibit complex dynamics. Having a mathematical model and a calculation program, it is possible to carry out further studies of the system under consideration, identifying the positions of stable and unstable equilibrium, modes of self-oscillations, defining areas of periodic and chaotic modes of different nature. The results obtained can be used in the development of technical devices capable of demonstrating complex behavior. Methodologically, the proposed material is interesting for undergraduate and graduate students specializing in electromechanics, in terms of teaching the principles of constructing and analyzing electromechanical systems.

Управление звеном экзоскелета переменной длины

The aim of the study is to develop a spatial electromechanical model of a variable-length link for use in telescopic manipulators, anthropomorphic robots, exoskeletons, and in studying the human musculoskeletal system. The proposed link model has two massive absolutely solid sections at the ends and a weightless section of variable length located between them. The study was carried out using the methods of theoretical mechanics, electromechanics, mathematical modeling, engineering design, numerical methods for solving systems of ordinary differential equations, control theory, nonlinear dynamics, experimental methods, and empirical data on the biomechanical properties of the human musculoskeletal system. The reliability of the obtained results is substantiated by a rigorous use of the above-mentioned methods. As a result of the study, a system of Lagrange-Maxwell differential equations was written, and an electromechanical model of an anthropomorphic system was developed in the Matlab Simulink software package. With the specified geometric and inertial parameters of a variable-length link corresponding to an average person's leg lower part and the time corresponding to the single-support motion phase, the electric motors and reducing gears implementing the human musculoskeletal system link's biomechanical motion fragment are selected. All of the selected motors have a sufficient operating parameters margin. The trajectories of all generalized coordinates along which the anthropomorphic system performs its necessary motion are determined. The mechanism load diagrams are obtained. The control system for the motors is synthesized, and the positioning error is evaluated. The novelty of the approach is that the newly developed electromechanical models of controlled variable-length links have a wide range of applying the obtained results and can be used in designing anthropomorphic robots and comfortable new-generation exoskeletons. Thus, the electromechanical model of a variable-length link with the parameters corresponding to the average person's leg lower part has been developed. The electric drives and transmissions able to implement a motion close to the anthropomorphic one have been selected; its implementation has been demonstrated, and the numerical calculation results are given.

ТЕКТОЛОГІЯ ДИНАМІЧНИХ СИСТЕМ І ЯВИЩЕ ГІПЕРСИЛОВОЇ ВЗАЄМОДІЇ У СТРУКТУРНИХ РІВНЯННЯХ УЗАГАЛЬНЕНОГО ЕЛЕКТРИЧНОГО КОЛА

The paper presents a number of the solutions of the important and interrelated theoretical problems, having technology-specific and general natural significance. Taken together and in the interaction with one another, they lay the theoretical basis for the formation and development of the new trend in classical electrical engineering, characteristic feature of this trend is the construction of the generalized, regarding the number of the degrees of freedom, continuous in time, homogeneous or mixed by their nature dynamic systems, as of purely electrical and combined physical origin, their structural analysis and formalization on the deductive basis of the process of mathematical and physical identification. In the connection of the above-mentioned the author determined and revealed the essence of the unknown general natural phenomenon of the hyperforce (hypervalence) interaction between the elementary structural links which is observed or can be observed in the dynamic systems with the concentrated parameters of the random physical nature and complexity in the process of their motion in the phase space under the impact of the internal and external forces. It is shown and mathematically proved that in general case the internal force interaction between the structural elements of the dynamic system is at the same time multidimensional and corresponds to the dimensionalities of the subspaces of the system topological space, which are determined by all the combinatorial combinations from the number n to the number k, where the number k belongs to the area 2 ≤ k ≤ n, and n is the number of the degrees freedom of this system. In such embedded subspaces the available multidimensional forces of the random dimensionality are independent one from another. Among others, this simplifies the predominant, for grater part of theories and scientific systems, paradigm concerning the possibility of exceptionally binary (k = 2) presentation of the character of the force interaction (and correspondingly mathematical relations) between the structural elements of the dynamic systems during their motion. Accounting the phenomenon of the hyperforce (hypervalence) interaction considerably broadens the classes of the studied dynamic systems. On the example of the electric engineering systems the application of Lagrange-Maxwell equations enabled to solve the number of decomposition problems, which form the basis of one of the fundamental problems of the theoretical electrical engineering – construction of the electric circuit. generalized by the number of the degrees of freedom . The obtained results, namely topological structural diagram of the generalized electrical circuit and structurally determined system of differential equations of its motion (structural equations), today have the highest degree of generalization and deductively embrace wide classes of the electrical circuits and systems – both already known and possible ones.

Electromechanical coupling vibration characteristics of an AC servomotor-driven translational flexible manipulator

The nonstationary transition status of the motor start-up phase creates great threat against the stable operation of the flexible manipulator system. This article investigates the electromechanical coupling dynamics and vibration response characteristics for a flexible manipulator of an alternating current servomotor-driven linear positioning platform with considering the start-up dynamic characteristics of the motor. Based on the constructed global electromechanical coupling effect and the Lagrange–Maxwell equations, the dynamic model of the whole system is established. The electromechanical coupling vibration mechanism of the flexible manipulator is obtained by analyzing the multiphysical process and multiparameter coupling phenomenon of the whole system. The result demonstrates that the nonstationary transition status of the motor initialization phase is mainly manifested during the disturbance of the three-phase stator current. As the speed of the linear positioning platform increases, the current disturbance, arousing the change of the servo driving force of the linear positioning platform, has dominant frequency shift and frequency amplitude decrease. Then, the vibration response of the flexible manipulator is markedly affected and the variation of the high-order modes vibration response is more obvious. The analysis result is significant for improving the dynamic performance of the motor-driven flexible robot manipulator system.

The interrelated modelling method of the nonlinear dynamics of rigid rotors in passive and active magnetic bearings

A method is suggested for building mathematical models of dynamics of rotors in magnetic bearings of different types (passive and active). It is based on Lagrange-Maxwell differential equations in a form identical to that of Routh equations in mechanics. The expressions for magnetic energy and forces in active magnetic bearings with account for control laws for introducing them into the mathematical models have been found by adapting the analytical method of analysing magnetic circuits. This method is based on building equivalent circuits and using the loop flux method to account for dissipation fluxes and magnetic resistances of AMB magnetic circuit sections and ensure noncriticality of the mathematical model to emergence of “zero” gaps and currents. Besides, the mathematical models account for such nonlinearities as nonlinear dependence of magnetic forces on gaps in passive and active magnetic bearings and on currents in the coils of electromagnets, nonlinearities linked to coil inductance, a geometric link between electromagnets in one AMB and links between all AMB in one rotor, which results, among other factors, in connectedness of processes in orthogonal directions. The method’s validity has been confirmed experimentally by a laboratory setup being a prototype of a complete combined magnetic-electromagnetic suspension in small-size rotor machinery. The suggested approach has helped detect in the system and investigate different nonlinear rotor dynamics phenomena such as super- and subharmonic vibrations with determination of resonance modes.

Electromechanical Dynamics Analysis and Simulation on Roll-Forming Equipment Based on Linear Motor

Electromechanical dynamics analysis and simulation on roll-forming equipment based on linear motor is discussed in this paper. By using Lagrange–Maxwell equations and energy method, the relation between mechanical components and electromagnetic components of the system is described. Mechanical dynamics model is established and differential equations are also established. Differential equations of two-phase winding on d–q axis are obtained by C transformation, which comes from three-phase winding equations on A–B–C axis. The whole system is controlled by efficiently controlling the linear motors, using the control model of id = 0, which simplifies the complex problem. By fourth-order Runge–Kutta’s method, discrete solutions of differential equations are obtained. Finally, by using Matlab language, the system is simulated. The results show that this method works very well.

Modeling and Simulation of Electromechanical Coupling Dynamics for Permanent Magnet Synchronous AC Servomotor

Lagrange-Maxwell equations and Park transform are adopted based on electrical and mechanical energy equations consisting of servomotor parameters. Lagrange-Maxwell equations are transformed from three-phase stator reference coordinates to two-phase rotor reference coordinates. Electromechanical coupling dynamics equations of permanent magnet synchronous servomotor in two-phase reference coordinates are obtained. In this dynamical modeling method of electromechanical coupling system, to establish dynamical differential equations needs to measure amplitude of the flux induced by the permanent magnets and the winding's inductance in dq0 reference of the motor but needs not to measure the size of magnetic circuit. The equations deduced is terse, efficient, and the equations easy to use. Electromechanical coupling dynamics system, servomotor is simulated. Simulation results show that the electromechanical coupling dynamics equations deduced for servomotor are correct, and current control schemes are reasonable.

Velocity-dependent symmetries and non-Noether conserved quantities of electromechanical systems

The theory of velocity-dependent symmetries (or Lie symmetry) and non-Noether conserved quantities are presented corresponding to both the continuous and discrete electromechanical systems. Firstly, based on the invariance of Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities, the definition and the determining equations of velocity-dependent symmetry are obtained for continuous electromechanical systems; the Lie’s theorem and the non-Noether conserved quantity of this symmetry are produced associated with continuous electromechanical systems. Secondly, the operators of transformation and the operators of differentiation are introduced in the space of discrete variables; a series of commuting relations of discrete vector operators are defined. Thirdly, based on the invariance of discrete Lagrange-Maxwell equations under infinitesimal transformations with respect to generalized coordinates and generalized charge quantities, the definition and the determining equations of velocity-dependent symmetry are obtained associated with discrete electromechanical systems; the Lie’s theorem and the non-Noether conserved quantity are proved for the discrete electromechanical systems. This paper has shown that the discrete analogue of conserved quantity can be directly demonstrated by the commuting relation of discrete vector operators. Finally, an example is discussed to illustrate the results.

Noether conserved quantities and Lie point symmetries of difference Lagrange–Maxwell equations and lattices

This paper presents a method to find Noether-type conserved quantities and Lie point symmetries for discrete mechanico-electrical dynamical systems, which leave invariant the set of solutions of the corresponding difference scheme. This approach makes it possible to devise techniques for solving the Lagrange–Maxwell equations in differences which correspond to mechanico-electrical systems, by adapting existing differential equations. In particular, it obtains a new systematic method to determine both the one-parameter Lie groups and the discrete Noether conserved quantities of Lie point symmetries for mechanico-electrical systems. As an application, it obtains the Lie point symmetries and the conserved quantities for the difference equation of a model that represents a capacitor microphone.

Unified symmetry of mechanico-electrical systems with nonholonomic constraints of non-Chetaev’s type

The unified symmetry of mechanico-electrical systems with nonholonomic constraints of non-Chetaev’s type are studied, and the definition and the criterion of unified symmetry of mechanico-electrical systems are deduced from the Lagrange-Maxwell equations. The Noether conserved quantity, the Hojman conserved quantity and a new conserved quantity deduced from the unified symmetry are obtained. An example is given to illustrate the application of the results.

Dynamic Optimization Method on Electromechanical Coupling System by Exponential Inertia Weight Particle Swarm Algorithm

Dynamic optimization of electromechanical coupling system is a significant engineering problem in the field of mechatronics. The performance improvement of electromechanical equipment depends on the system design parameters. Aiming at the spindle unit of refitted machine tool for solid rocket,the vibration acceleration of tool is taken as objective function,and the electromechanical system design parameters are appointed as design variables. Dynamic optimization model is set up by adopting Lagrange-Maxwell equations,Park transform and electromechanical system energy equations. In the procedure of seeking high efficient optimization method,exponential function is adopted to be the weight function of particle swarm optimization algorithm. Exponential inertia weight particle swarm algorithm(EPSA),is formed and applied to solve the dynamic optimization problem of electromechanical system. The probability density function of EPSA is presented and used to perform convergence analysis. After calculation,the optimized design parameters of the spindle unit are obtained in limited time period. The vibration acceleration of the tool has been decreased greatly by the optimized design parameters. The research job in the paper reveals that the problem of dynamic optimization of electromechanical system can be solved by the method of combining system dynamic analysis with reformed swarm particle optimization. Such kind of method can be applied in the design of robots,NC machine,and other electromechanical equipments.

Noether symmetries of discrete mechanico–electrical systems

This paper focuses on studying Noether symmetries and conservation laws of the discrete mechanico-electrical systems with the nonconservative and the dissipative forces. Based on the invariance of discrete Hamilton action of the systems under the infinitesimal transformation with respect to the generalized coordinates, the generalized electrical quantities and time, it presents the discrete analogue of variational principle, the discrete analogue of Lagrange–Maxwell equations, the discrete analogue of Noether theorems for Lagrange–Maxwell and Lagrange mechanico-electrical systems. Also, the discrete Noether operator identity and the discrete Noether-type conservation laws are obtained for these systems. An actual example is given to illustrate these results.

Unified symmetry of mechano-electrical systems with nonholonomic constraints

The unified symmetry of mechano-electrical systems with nonholonomic constraints are studied in this paper, the definition and the criterion of unified symmetry of mechano-electrical systems with nonholonomic constraints are derived from the Lagrange–Maxwell equations. The Noether conserved quantity, Hojman conserved quantity and Mei conserved quantity are then deduced from the unified symmetry. An example is given to illustrate the application of the results.

Unified symmetry of mechanico-electrical systems

The unified symmetry of mechanico-electrical systems are studied in this paper. The definition and criterion of unified symmetry of mechanico-electrical systems are derived from the Lagrange-Maxwell equations. The Noether conserved quantity, Hojman conserved quantity and a new conserved quantity deduced from the unified symmetry are obtained. An example is given to illustrate the application of the results.

Mathematical models in natural science and engineering

Dynamic system.- Fluid flowing-out from a vessel.- Equilibrium and auto-oscillations of fluid level in a vessel with simultaneous inflow and outflow.- Transitive process, equilibrium state and auto-oscillations.- Dynamics of water level in a reservoired hydropower station.- Energetic model of a heart.- Soiling a water reservoir with a bay and the Caspian Sea puzzles.- Exponential processes.- Populations coexistence dynamics.- Flow biological reactor.- Mathematical model of an immune response of a living organism to an infectious invastion.- Mathematic model for the community 'Manufacturers-Products-Managers'.- Linear oscillator.- Electromechanical analogies. Lagrange-Maxwell equations.- Galileo-Huygens watch.- Generator of electric oscillations.- Soft and hard regimes of exciting auto-oscillations.- Stochastic oscillator.- Instability and auto-oscillations caused by friction.- Forced oscillations of a linear oscillator.- Parametric excitation and stabilization.- Normal oscillations and beatings.- Stabilizing an inverted pendulum.- Controllable pendulum and a two-legged pacing.- Dynamic models for games, teaching and rational behaviour.- Perceptron and pattern recognition.- Kepler laws and the two-body problem solved by Newton.- Distributed dynamic models in mechanics and physics.- Fundamental solution of the thermal conductivity equation.- Travelling waves and the dispersion equation.- Faraday-Maxwell theory of electromagnetism and Maxwell-Hertz electromagnetic waves.- Wave reflection and refraction.- Standing waves and oscillations of a bounded string.- Microparticle.- Space and time.- Speeding up relativistic microparticles in a cyclotron.- Mathematics as a language and as an operating system and models.- Geometrical, physical, analogous, mathematical and imitative types of modelling.- General scheme of methematical modelling.- Models of vibratory pile driving.- The fundamental mathematical model of the modern science and the theory of oscillations.- Mathematicalmodel as a fruitful idea of research. D-partition.- Idealization, mathematical correctness and reality.- Dynamical interpretation of the least square method and global searching optimization with an adaptive model.- Theoretic game model of the human society.