Unstable Equilibrium Position Research Articles

Space debris in vicinity of collinear libration points of the Earth–Moon system

Practical space exploration has a priority the issues of safety of space missions. And an important fundamental problem arises — the study of the evolution and the stability of a space debris cloud, consisting of artificial objects remnants, or debris of extra-terrestrial origin.We discuss a new behavior of mechanical system near an unstable equilibrium position — the trajectories located in the selected neighborhood of unstable equilibrium. As an important application, we consider perturbed motion in collinear libration points vicinity of the restricted circular three-body problem.Numerical simulations for the parameters of the Earth–Moon system convincingly illustrate our theoretical study. This mathematical model allows us predicting how long the debris with certain energy can fly in vicinity of unstable collinear libration point, being close to the localized trajectory. This must be taken under consideration when planning the space missions. And also the localized trajectories may be useful when spaceship flies around the collinear libration point.

ВЛИЯНИЕ ТОЧЕК ПОВОРОТА НА ЗАДЕРЖКУ РЕШЕНИЯ ВБЛИЗИ НЕУСТОЙЧИВОГО ПОЛОЖЕНИЯ РАВНОВЕСИЯ

This paper considers a system of equations consisting of three first-order equations of fast variables and one equation of a slow variable. The first approximation matrix of the system of fast variables has different eigenvalues, with one pair of complex conjugates. The eigenvalues have zeros and they are turning points. The system has an equilibrium position, and the stability of the equilibrium position is lost at a certain value of the slow variable. The problem is set to study the delay in solving a system of fast variables near an unstable equilibrium position and the influence of turning points on the delay. The delay of the solution near an unstable position is proven and the influence of turning points is clarified. The proof used geometric constructions using the level line of harmonic functions, methods of successive approximations, Laplace, stationary phase, integration by parts and comparison of infinitesimals.

Formation of a dislocation dipole in the consecutive interfaces of two nanowires

The formation of a dipole of edge dislocations in the consecutive interfaces of two neighboring square-shaped nanowires has been investigated from an energy-based calculation. Assuming the dipole is nucleated in the center of the matrix row between the two nanowires, the energy variation induced by the climbing of the dislocations in the interfaces (one by interface) has been first determined. The energy barrier and unstable equilibrium positions of the dipole have been thus characterized. Considering then the dislocations can also glide into the interfaces, the resulting supplementary gliding energy variation has been considered and the dislocation stable equilibrium positions in the interfaces have been determined, assuming they are distributed symmetrically with respect to the central point of the structure. The effect of the width of the matrix row between the two nanowires on the critical misfit strain required for the introduction of the dislocations in the nanowire interfaces has been finally analyzed.

Establishing mechanisms for nonlinear collector tribodynamics of magnetization under ferroresonance regimes

The paper considers a parallel electromagnetic oscillating circuit with a nonlinear inductance under conditions of excitation of ferroresonances. Physical mechanisms of dynamic self-regulation of the system of spin magnetic moments of ferromagnetic and ferrimagnetic dielectrics in the surrounding magnetic field are established. It is shown that upon entering the magnetic saturation regime, the effect of dynamic antiferroresonance is observed, due to the cyclic reversal of magnetization in the internal magnetic field. This effect has a collector character and corresponds to the maximum potential energy of the magnetic moment in the field and the antiparallel orientation of the moment with respect to the field. Such regimes are realized in thermodynamically non-equilibrium conditions and are correlated with unstable equilibrium positions of the corresponding mechanical analogues. The resulting forms of oscillations correlate with the dynamics of an inverted pendulum and have a significantly non-harmonic character. Autosynchronization of frequencies and nonlinear mixing of such forms with quasi-static modes imitating the time form of external excitation of oscillations were revealed. It is shown that the nonlinearity of ferromagnetic elements self-limits the height of resonance maxima. And on the other hand, it contributes to the cascade transport of energy by the spectrum of disturbances, which can have negative consequences when exciting low-frequency ferroresonances in power grids. The effect of dynamic antiferroresonance has the opposite direction to the known quasi-static behavior of the system of spin magnetic moments in an external field and must be taken into account when calculating and operating electrical systems with nonlinear inductances. Examples of collector self-oscillating modes similar in their physical mechanisms in nonlinear contact tribodynamics systems are given

Perturbation bounds for stable gyroscopic systems in motion about an unstable equilibrium position

Relative perturbation bounds for stable linear gyroscopic systems Mẍ(t)+Gẋ(t)+Kx(t)=0 are considered, where M,K∈Rn×n are symmetric, M is positive definite, K negative definite and G∈Rn×n is skew-symmetric such that |G|>kM−k−1K for some k>0. This means that all of the eigenvalues of the system are purely imaginary and semi-simple. For this kind of systems, we present an upper bound for the relative change in eigenvalues as well as the sinΘ type bound for the corresponding eigenvectors under the perturbation of the system matrices. The performance of obtained bounds is illustrated in numerical experiments.

Asymptotics of Solutions of Autonomous Singularly Perturbed Equations when the Stability of the Equilibrium Position Changes at Several Points

Considers a system consisting of 2n equations with fast variables and one equation with a slow variable. The first approximation matrix of the system of fast variables has 2n pairwise complex conjugate eigenvalues. The stability of the equilibrium position, a system of fast variables, is influenced by all eigenvalues. Early studies considered cases where the stability of the equilibrium position is affected by only one pair of complex conjugate eigenvalues. The problem of delaying the solution of a system of fast variables near an unstable equilibrium position has been solved. The delay time for the decision is determined.

Delay in Solving Autonomous Singularly Perturbed Equations Near an Unstable Equilibrium Position

Delay in Solving Autonomous Singularly Perturbed Equations Near an Unstable Equilibrium Position

Interaction between an edge dislocation and a circular incompressible liquid inclusion

We use Muskhelishvili’s complex variable formulation to study the interaction problem associated with a circular incompressible liquid inclusion embedded in an infinite isotropic elastic matrix subjected to the action of an edge dislocation at an arbitrary position. A closed-form solution to the problem is derived largely with the aid of analytic continuation. We obtain, in explicit form, expressions for the internal uniform hydrostatic stresses, nonuniform strains and nonuniform rigid body rotation within the liquid inclusion; the hoop stress along the liquid-solid interface on the matrix side and the image force acting on the edge dislocation. We observe that (1) the internal strains and rigid body rotation within the liquid inclusion are independent of the elastic property of the matrix; (2) the internal hydrostatic stress field within the liquid inclusion is unaffected by Poisson’s ratio of the matrix and is proportional to the shear modulus of the matrix; and (3) an unstable equilibrium position always exists for a climbing dislocation.

Synthesis of a Bistable Recuperative Pump Powered by Shape Memory Alloys and a Two-Section Involute Cam

The paper discusses the synthesis problem of a bistable piston pump, employing a driving mechanism that comprises shape memory alloy wires, a two-section involute cam, and an energy-recuperating spring. The transition from one stable end position to another in the pump is achieved by heating and subsequently shortening one of the shape memory alloy wires, initiating the motion of the mechanism. This is then followed by the engagement of the recuperative spring to traverse the intermediate unstable equilibrium position and complete the entire stroke. The reversal of motion follows a similar approach, where the second SMA wire shortens while the first wire remains in a cold state. Importantly, the mechanism necessitates a low force within the shape memory alloy wire to initiate motion towards the opposite stable position. This research encompasses the examination of type, geometric, and force synthesis considerations for the pump, leading to the development of fundamental kinematic and force relationships. Moreover, a novel mechanism is proposed and synthesized, incorporating a two-section involute cam with a cusp point, to generate the desired discontinuous moment function produced by the recuperative spring. Further analysis reveals that the thermal time constant, which regulates the dynamic response of the mechanism, is directly proportional to the diameter of the driving SMA and inversely proportional to the square root of the number of SMA branches.

Mechanical Equilibrium of a Nonmagnetic Body Immersed in a Cylindrical Container with a Magnetic Fluid Magnetized by an External Homogeneous Magnetic Field

Purpose. Analytical and numerical description of the magnetohydrodynamic forces acting on a small nonmagnetic spherical body in a cylindrical container with magnetic fluid (magnetofluid dispenser and separator approximation) that determine the hydrostatic mechanical equilibrium in the system.Methods. The numerical study solves the magnetostatic problem by the finite element method in the FEMM program package using the Lua script language. The system of Maxwell’s equations is solved by the standard method in the vector potential formulation. The analytical solution of the magnetostatic problem is obtained by the mirror image method using a simplifying model representation of the linear law of magnetization of a magnetic fluid. The ponderomotive force acting on a body immersed in a magnetic fluid is calculated using the Rosensweig formula and the energy approach.Results. A refined expression for the magnetic ponderomotive force acting on a nonmagnetic sphere immersed in a cylindrical container with magnetized magnetic fluid is obtained. Direct numerical simulation of the laboratory experiment is performed, which allows us to compare the accuracy of the numerical and analytical solutions with the experimental data. Despite violating the limits of applicability of the analytical theory, the new expression correctly describes the nonmonotone coordinate dependence of the force, and the error in determining the coordinate extremums does not exceed 6 % and 26 % in absolute value. The physical justification for the condition of mechanical equilibrium in the model system under study is given.Conclusion. The competition of two oppositely directed magnetic forces leads to the fact that a nonmagnetic sphere in a cylindrical container with magnetized magnetic fluid has one unstable mechanical equilibrium position in the center of the container, so that the body is pressed against the wall, or (additionally) two stable equilibrium positions that allow the body to levitate near the container wall without touching it.

Dislocation equilibrium in a misfitted composite structure with partially coherent interfaces

The effect of misfit strain on the equilibrium positions of two edge dislocations of the same Burgers vector that are symmetrically displayed in a matrix with respect to its two interfaces with a thin film has been theoretically investigated from a Peach–Koehler force-based analysis. Assuming the film/matrix interfaces are partially coherent, the stable and unstable equilibrium positions of the dislocations have been determined as well as the resulting stability diagram versus the misfit strain, the interface inconsistency length and the layer thickness.

Vertical dynamics of high-pressure air cushion bearing module: Stability and auto-oscillation

Unstable static equilibrium and vertical auto-oscillations are problems encountered in the application of a high-pressure air cushion module (ACM) for the indoor transportation of heavy loads. The results of two base methods are compared: linear stability analysis based on the linearized equations of the perturbed motion coupled to airflow fluctuations and numerical evaluation of the nonlinear dynamic response of the system. The main distinguishing feature of the analysis is that it comprehensively examines air compressibility in the cushion plenum and in the inlet and outlet flow zones. Accordingly, the study and its findings are valid for high-pressure and low-flow applications of air cushion transportation. A narrow stability region in the parameter domain is identified when the cushion pressure approximates the supply pressure. The peculiarities of the vertical dynamics of the system near stable and unstable static equilibrium positions are described, and influence of viscous damping is considered. The study results contribute to the generalized representation of the stability and vertical dynamics of ACMs and to the theoretical database for ACM design.

Comparative Cost Functions Analysis in the Construction of a Reference Angular Motion Implemented by Magnetorquers

This paper considers a construction procedure of a satellite reference angular motion in the vicinity of an unstable gravitational equilibrium position. The satellite is stabilized on the reference trajectory by the magnetic coils. The problem is solved in several stages. An optimization problem to be solved by the particle swarm optimization method is formulated at each stage. Cost functions are based on the linearized model analysis. The main stage is the construction of a special reference motion, which provides the minimum control torque projection on the geomagnetic induction vector. Optimal geomagnetic field dipole approximation for a given time interval is considered to reduce tracking errors. The paper compares combinations of different cost functions in terms of the terminal attitude accuracy in the presence of perturbations.

Stochastic resonance of multi-stable energy harvesting system with high-order stiffness from rotational environment

This paper aims to study the stochastic resonance of multi-stable energy harvesting system and enhance the performance of energy harvesting from random rotational environment. The influences of high-order stiffness terms and multi-stable characteristics are analyzed. The spectral amplification is derived to quantify the stochastic resonance for the energy harvesting system with penta-stable configuration driven by noise and periodic forcing. The non-monotonic dependence of spectral amplification on noise intensity indicates the existence of stochastic resonance. In particular, a critical high-order stiffness coefficient is revealed, for which the stochastic resonance is optimum. Meanwhile, the maximum of spectral amplification can be optimized by choosing appropriate unstable equilibrium positions. In addition, it is found that both the average input energy and the overall efficiency in power conversion can be further enhanced under conditions of stochastic resonance. Finally, the penta-stable energy harvester demonstrates a good performance in output power for different noise intensities and damping coefficients.

Global dynamics of the Hořava–Lifshitz cosmology in the presence of non-zero cosmological constant in a flat space

Using the qualitative theory of differential equations, the global dynamics of a cosmological model based on Hořava-Lifshitz gravity is studied in the space with zero curvature in the presence of the non-zero cosmological constant. The study shows that there may be three unstable finite equilibrium positions under the Hořava-Lifshitz cosmology model and that the final evolution of the orbits of the cosmological model in the physical region of interest may tend towards some infinite equilibrium position, which may correspond to the late-time state of the universe.

MOPSO TABANLI LQG SERVO KONTROL YAKLAŞIMI İLE ARAÇ ÜZERİNDE KONUMLU TERS SARKACIN KONTROLÜ

The Inverted Pendulum, located on the vehicle, is widely used in the academic sense for the application of various control methods and comparison of their performance. An unstable and non-linear Inverted Pendulum is sensitive and fragile to system disturbances and measurement noises. Exposure to disturbances and sensor measurement noise adversely affects the performance of control systems and causes a decrease in control quality. One of the methods used as a solution to the mentioned problem is the control design method known as Linear Quadratic Gaussian (LQG), which is a combination of Kalman Filter and LQR control.
 
 In this study, for the Inverted Pendulum located on the vehicle, which is exposed to system disturbances and sensor noise, while the vehicle carrying the pendulum follows a given reference, the pendulum is required to maintain its unstable vertical equilibrium position. System disturbances and sensor noises are chosen as Gaussian White Noise. LQG servo control approach is adopted to provide reference tracking and maintain stability around the balance point. It is necessary to optimize the performance of the controller to be used in order to best meet the control requirements. For this purpose, the performance criterion weight matrices for the LQI block within the LQG servo controller have been optimized with the help of the Multi-Objective Particle Swarm Optimization Algorithm (MOPSO).

The Optimal Erection of the Inverted Pendulum

The erection of the inverted pendulum is a classic control problem, which has appeared in several variants. One of the most challenging is the minimum-time erection of a pendulum that is mounted on a moving cart. The aim is to erect the pendulum from the ‘straight-down’ (stable equilibrium) to a ‘straight-up’ (unstable equilibrium) position in minimum time. The swing-up maneuver is usually addressed using a pre-defined control strategy, e.g., energy-based control or selecting the switching times in a bang-bang structure. The aim of this paper is to show that the minimum-time solution may have a singular arc, with the optimal control taking a bang-singular-bang form. The singular arc segment of the control law is a feedback law that is derived herein with the solution discussed. A sensitivity analysis of the solution structure is also performed by varying the model parameters. Finally, the time-optimal solution is compared with that obtained using an energy-based control strategy.

Dislocation Dipole in a Strained Nanostructured Layer Buried in an Infinite-Size Matrix

Abstract The equilibrium positions of a dipole of edge dislocations embedded in a nanostructure composed of a strained two-dimensional layer buried in an infinite-size matrix with a rectangular nanowire located at the upper layer–matrix interface have been determined from a Peach–Koëhler force calculation. The location of the unstable and stable equilibrium positions of the dislocations of the dipole gliding in the horizontal plane has been specified in the nanostructure versus the misfit strain and nanowire aspect ratio. The critical misfit below which the dipole has no equilibrium position has been finally determined versus the size of the nanowire.

COEXISTENCE FROM INTERSPECIFIC MATINGS FOR MOSQUITOES WITH STAGE STRUCTURE

Stage-structured models, including groups of larvae and adults, for two interacting mosquito species are studied in this paper. When there are no interspecific matings, the model dynamics follows the competitive exclusion principle such that only one of the two species can survive and there is no coexistent positive equilibrium. As interspecific matings take place, the interactive dynamics become complex. One of the two species can still dominate and drive the other species extinct with no existence of positive equilibrium, either species can survive or die out depending on initial sizes of the species where there exists an unstable positive equilibrium, or the two species coexist with a locally asymptotically stable positive equilibrium under certain conditions. Other dynamical features can occur as well. Detailed mathematical analysis and numerical examples are provided. Brief biological discussions are also given.

A novel approach regarding the fixed points of repelling nature

Although there are various numerical techniques to find the stable equilibria of a nonlinear system in scientific computing, no widely-used computational approach has been encountered to discover the unstable equilibria of a system in the literature. This study aims at presenting a new approach to uncover the equilibrium positions of a dynamical system exhibiting a repelling nature. A newly developed algorithm called the reversed fixed point iteration method (RFPIM) is presented to find the unstable equilibrium positions of a nonlinear system. The current method, keeping the real features of a problem, is able to uncover the behavior of a nonlinear system near the unstable equilibria without facing any conventional drawbacks. In this respect, it is mathematically proven and numerically observed that the present approach has various superiorities against the conventional approach. Some illustrative examples regarding the root-finding problem, population dynamics, integral equations, control problems, and chaotic systems have been utilized to test the current method. The results reveal that in finding the unstable equilibria of a differential operator, the use of relatively larger step sizes together with the RFPIM leads to more accurate results as opposed to common expectations. The findings have clearly shown that less consumption of CPU time and storage space could be succeeded by using larger step sizes via the RFPIM.