Stability Of Positive Equilibrium Research Articles

Multimodal vortex-induced vibration mitigation and design approach of bistable nonlinear energy sink inerter on bridge structure

Large-scale structures, e.g., long-span bridge structures, are prone to induce multi-modal vibrations due to their densely spaced low modal frequencies. Due to the limited frequency bandwidth of linear dynamic absorbers, they are incapable of effectively mitigating vibrations across multiple modes. To this end, the bistable nonlinear energy sink inerter (BNESI) is used to mitigate the multimodal vortex-induced vibration (VIV) of the beam structure. The highly nonlinear equilibrium differential equations of the beam-BNESI system are numerically solved, and the simulated annealing (SA) algorithm is employed to determine the optimal VIV reduction ratio and BNESI parameters. In comparison to the cubic-type nonlinear energy sink inerter (CNESI), BNESI is found to possess more stable equilibrium positions, smaller stiffness coefficients, and higher VIV mitigation efficiency. The selection of design modes has been found to influence the efficiency of multimodal VIV mitigation, with the use of the intermediate modal order as the design mode resulting in the highest efficiency for multimodal VIV mitigation. The performance-based multimodal VIV mitigation design can be realized with three parameters, i.e., inertance ratio, damping coefficient, and stiffness coefficient. Moreover, the performance-based multimodal VIV mitigation approach and models proposed in this study demonstrate a high level of precision.

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.

Delay induced Hopf bifurcation and its control in a fractional p53 protein oscillator model

The p53 protein is essential for cancer prevention. Research on p53 mathematical models is popular since its oscillatory dynamics were observed. However, there is little research on fractional-order p53 oscillators, as well as on Hopf bifurcation control. We therefore introduce fractional-order into a p53 oscillator model and propose a delayed state feedback bifurcation controller. The stability of positive equilibrium and the existence of Hopf bifurcation are studied analytically regarding delay as the bifurcation parameter in both controlled and uncontrolled scenarios. Numerical simulations are consistent with the analytical results. We reveal that a smaller fractional-order facilitates the stability of fixed point and a larger feedback gain advances the Hopf bifurcation point. The work may inspire cancer therapy by artificially controlling the occurrence of p53 oscillations.

A Cartesian diver to study oscillations and internal gravity waves in a stratified fluid

We propose a variation of the well-known Cartesian diver experiment where, instead of moving in a uniform fluid, the diver floats in a fluid stratified in density. In contrast to the original experiment, for a given external pressure the diver can stop in a stable equilibrium position within the fluid, at the depth where the surrounding density matches its own. By varying the applied pressure, the density of the diver changes and it moves until it reaches a new stable equilibrium condition at a different depth. When a sudden pressure pulse is applied, the diver, pushed off its equilibrium position, starts oscillating due to a restoring force that depends on the density gradient. The oscillations produce internal gravity waves that are typical of stratified fluids, when a portion of them is displaced and transmits its motion to the surrounding fluid. Although they are extremely difficult to observe, gravity waves are particularly interesting, as they typically occur in the atmosphere and in the stars. We propose a simple experiment and suggest a way to make the internal gravity waves visible. The experiment can be realized by students with easy-to-find household objects and used to improve their understanding of many concepts and laws of hydrodynamics, but also to introduce them to complex phenomena of general interest.

Stability analysis of a pine wood nematode prevention and control model with delay

Pine wilt disease is a destructive forest disease with strong infectivity, a wide spread range and high difficulty in prevention and control. Since controlling Monochamus alternatus, the vector of pine wood nematode (Bursaphelenchus xylophilus) can reduce the occurrence of pine wilt disease efficiently, the parasitic natural enemy of M. alternatus, Dastarcus helophoroides, is introduced in this paper. Considering the influence of parasitic time of D. helophoroides on the control effect, based on the mutualistic symbiosis and parasitic relationship among pine wood nematode, M. alternatus and D. heloporoides, this paper establishes a pine wood nematode prevention and control model with delay. Then, the stability of positive equilibrium and the existence of Hopf bifurcation are discussed. Besides, we obtain the normal form of Hopf bifurcation by applying the multiple time scales method. Finally, numerical simulations with two sets of meaningful parameters selected by means of statistical analysis are carried out to support the theoretical findings. Through the comparative analysis of numerical simulations, the factors affecting the control effect of pine wilt disease are obtained, and some suggestions are put forward for practical control in the forest.

Spatiotemporal Dynamics of a General Two-Species System with Taxis Term

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one.

Theoretical analysis of linear oscillations of bubbles in a polydisperse cluster

The dynamics of a single air bubble in an unbounded liquid are significantly different from the dynamics of an individual bubble in clusters due to the hydrodynamic interaction between the bubbles. Studying the mechanism of this interaction is one of the important aspects in the study of the fundamental nature of acoustic and hydrodynamic cavitation. In this work, to analyze small oscillations of bubbles in a spherical cluster near a stable equilibrium position, the mathematical theory of a linear conservative system with several degrees of freedom is applied to explain the mechanism of interaction between bubbles of different sizes. Using this theory, in the general case, it has been proven that the number of resonance frequencies in a polydisperse cluster coincides with the number of fractions. It is shown that in the regions of the main resonance (at low frequencies) bubbles of different fractions oscillate in phase, and in the regions of secondary resonances (at high frequencies) the phases successively change to the opposite, starting with the fraction containing bubbles of the largest radius, and further changing in order of decreasing. Using the example of a two-fraction cluster, it was found that there is an inertial connection between the bubbles, but there is no force connection; when the number of bubbles of one of the fractions is small, the connection between them and the bubbles of the other fraction is weak, while the interaction between them can be strong. An analysis of the energy transfer between bubbles of different fractions showed that the change in the nature of bubbles oscillations in the fraction with a small radius, while the nature of vibration of bubbles in the other fraction does not change, is the result of dynamic damping.

Design and analysis of a microgripper for trans-scale clamping based on a compliant multistable mechanism.

Piezoelectric actuators commonly used in microgrippers have a small stroke, and their accuracy is reduced by the transmission amplification unit, which leads to a contradiction between the clamping range and the clamping accuracy in existing piezoelectric-actuated microgrippers. This paper proposes a design scheme to divide the total clamping range of the microgripper into segments based on the compliant multistable mechanism (CMM). First, by using the stable equilibrium positions of the CMM, the total clamping range of the microgripper is divided into multiple smaller clamping sub-intervals to accommodate objects of different scales. Then, the theoretical models of the displacement amplification ratio of the microgripper amplification mechanism and the stiffness of the microgripper in different clamping sub-intervals are established, and the force-displacement characteristics of the CMM are analyzed. Next, through finite element simulation, the correctness of the theoretical analyses is verified, and it is shown that objects between 0 µm and 1.650mm can be clamped using four clamping sub-intervals under a five times displacement amplification ratio. Finally, a microgripper of the CMM consisting of two three-segment fully compliant bistable mechanisms connected in series is designed and machined, and microgripper segmented clamping experiments are conducted. The experimental results demonstrate the feasibility of the design scheme proposed in this paper.

Hand-held rolling magnetic-spring energy harvester: Design, analysis, and experimental verification

Aiming at the random and low-frequency characteristics of human motion, a hand-held rolling magnetic-spring energy harvester is proposed to scavenge energy from handshaking. The harvester applies the rolling movement of a magnetically suspended magnet and possesses the merit of low damping. The dynamic model is established and the open-circuit voltages are predicted by combining Faraday's law of electromagnetic induction and finite element analysis. By calculating magnetic flux through coils, the effect of the coil’s position on energy harvesting efficiency is studied through numerical and experimental approaches. Experimental results under harmonic excitations indicate that the coil positioned directly below the stable equilibrium position of the moving magnet has the best output capability, and a maximum open-circuit voltage of 1.33 V and a half-power bandwidth of 1.61 Hz is achieved at 0.3 g. Under handshaking excitation, an instantaneous peak power of 20.15 mW is achieved, along with a maximum average power of 2.22 mW. The corresponding volume and mass power densities are respectively 26.04 μW/cm3 and 23.28 μW/g. Furthermore, handshaking excitation in 5 s to charge a capacitor of 470 μF could enable the watch, time display, timer, and hygrothermograph to operate for 1100 s, 513 s, 86 s, and 35 s respectively, demonstrating the brilliant application foreground of the proposed harvester.

Nonlinear dynamics of 3D superconducting maglevs with six degrees of freedom: dependence on magnetization strategy and disturbance type

Based on the heat diffusion equation, Maxwell’s equations, and translational and rotational dynamic equations, we establish and theoretically validate an electromagnetic-thermal-mechanical coupling model to analyze the levitation performance during normal operation and the nonlinear dynamic behavior under disturbance for 3D maglev systems composed of a six-degree-of-freedom bulk superconductor (SC) and a Halbach-type guideway of permanent magnets (PMs) with different magnetization strategies and different types of disturbances, as well as the change rules of magnetic force and torque during translational or rotational cycle movement. In order to ensure the system security, we propose a generalized electromagnetic restoring force model to theoretically analyze the stability of the SC moving along the directions of various degrees of freedom. The results show that after being disturbed, the SC vibrates along the direction of each degree of freedom, and the vibration center, i.e. equilibrium position, will drift along each vibration direction. With time increasing, the equilibrium position will appear periodically on both sides of the working position. Compared to zero-field cooling magnetization, field cooling magnetization enables the SC to trap more flux in its interior to alleviate the drift phenomenon and reduce the energy loss. This advantage can be further enhanced by adding an extra step of preloading treatment. For the lateral motion, the system has one stable focus point and two unstable saddle points. Whether the system at these saddle points is stable depends on the direction of disturbance-induced velocity. For the rotational motion, the system has only one stable focus point, which means that regardless of the type of disturbance, the SC will finally come back to its stable equilibrium position. Besides, the stability is related to the axis around which the SC rotates, and rotating around the longitudinal axis is more likely to generate larger magnetic force, torque and local temperature rise. Either field cooling magnetization or preloading treatment can effectively improve system stability.

Numerical simulations of the dynamics of a deformable particle in a viscoelastic liquid subjected to Poiseuille flow in a cylindrical microcapillary at non-negligible inertia

The manipulation and control of particles in microfluidic devices through non-intrusive methods is pivotal in many application fields, e.g., cell focusing and sorting. Inertial microfluidics is rapidly gaining attention in the scientific community because of the considerable advantages in terms of throughput. In addition to inertia, other factors can trigger the cross-stream migration of particles in liquids undergoing pressure-driven channel flows, such as the deformability of the particles themselves and/or the viscoelasticity of the carrier fluid. For this reason, the dynamics of an initially spherical elastic particle suspended in a viscoelastic liquid subjected to pressure-driven flow in a cylindrical channel at non-negligible inertia is studied through three-dimensional arbitrary Lagrangian–Eulerian finite-element numerical simulations. The mechanical behavior of the particle is described through the neo-Hookean hyper-elastic constitutive equation, whereas the rheological behavior of the carrier liquid is described through the Giesekus model. The Reynolds number Re, measuring the relative importance of inertial and viscous forces in the tube, the elastic capillary number Cae, measuring the relative importance of liquid viscous stress and solid elastic stress, and the Deborah number De, measuring the ratio of the liquid relaxation time and the flow characteristic time, are varied. The particle migrates transversally to the flow direction until reaching a radial equilibrium position depending on Re, Cae, and De. Different dynamics are observed depending on the interplay among inertia and elasticity of both the liquid and the solid phase: one, two, or even three stable equilibrium positions can be found along the tube radius.

Применение теории катастроф для математического моделирования оползневого процесса на вогнутых склонах горных территорий

Introduction. Analysis of existing theoretical and experimental studies has shown that the model approach is the main method of landslide research. The existing mathematical model of landslides does not meet the requirements of the necessary adequacy. Materials and methods. This study uses a mathematical modeling methodology based on catastrophe theory. Results and discussion. To solve this actual problem, in this article authors developed a mathematical model of the landslide process on the concave slopes of mountainous territories. The developed model contains two components. The first of them is a mathematical model of the stress field in the volume of rocks located inside the slope section. This model uses the framework of fractal and multifractal modeling methods developed by the authors. The results of this model research are final expressions for calculating the stress field used rock pressure and bending stress as the external stress field. The superposition of the field induced by these external stresses gives the stress field in the volume of rocks located inside the slope section. Analysis of program implementation of this model showed that there are two areas in the slope section: compressive and tensile stresses adjacent to each other. At the boundary between these areas, there is a discontinuity of the stress field. A displacement surface passes along this boundary, forming a potentially landslide body. Moreover, it was found that a potentially landslide body on a slope is in a state of local and global instabilities. A potentially landslide body tends to occupy the position of the minimum potential energy. Local instability is expressed as the tendency of movement to a stable equilibrium without changing its location in the rock mass. The tendency of landslide body to move down the slope is a demonstration of global instability. The second mathematical model describes the realization of local instability that leads to the formation of a landslide body. Conclusion. According to the model analysis, it was found that the implementation of instability leads to the formation of a landslide body. At the same time, according to this analysis a landslide body can take up three stable equilibrium positions, allowing it to stay on the slope without global instability. Suggestions for practical application and direction of future research. The research results can be used to predict landslides on concave slopes of mountainous territories and to develop new mathematical models allowing to make the analysis of concave slopes of mountainous territories taking into account fracturing.

Sparse identification of the dynamics of a nonlinear multistable oscillator

AbstractRecently, data‐driven modeling approaches are getting increasingly examined regarding their applicability for nonlinear mechanical or mechatronic systems. With a high data availability and often insufficiently accurate descriptions of complex behavior of real systems using established physical models, statistical models provide promising alternatives. Alongside machine learning techniques like deep neural networks, sparse regression is increasingly used to obtain models from measurement data. With sparse regression, governing equations are estimated from a given function space so that the data are explained with as few terms as possible while maintaining a low model error. This method is implemented in a framework called sparse identification of nonlinear dynamics. This paper demonstrates the application of this method on free and forced vibrations of a two degree of freedom nonlinear rotary oscillator with two stable equilibrium positions. The setup and data acquisition as well as the application of sparse identification are described. The selected function space containing the candidate functions is essential for an accurate representation of the system at hand. Monomials form a commonly used function space because they can approximate a wide variety of nonlinear characteristics. However, in this work, it is shown that monomials alone are insufficient when dry friction appears. Therefore, to account for Coulomb friction, a sign‐function is added as a candidate to the function space of monomials up to the fifth order. Adaptions of the common optimization algorithms turned out to be necessary for the inclusion of Coulomb friction. As a result, it is found that the addition of the sign‐function for Coulomb friction increases the model quality significantly.

Enantioselective transport of chiral spheres using focused femtosecond laser pulses.

Optical tweezers are commonly used for manipulating chiral particles by tailoring the properties of the electromagnetic field or of the particles themselves. Non-linearity provides additional degree of freedom to control the manipulation by changing the trapping conditions. In this work, we leverage the nonlinear optical properties of a medium by illuminating it with a circularly polarized laser pulse, enabling single particle enantioselection for the chiral spheres immersed in it. By adjusting the power of the laser pulses, we demonstrate stable trapping of chiral spheres with one handedness near the focal region, while spheres with the opposite handedness are repelled. This enables the chiral resolution of racemic mixtures. Additionally, we perturbed the stable equilibrium position of the trap by driving the sample stage, leading to the emergence of a new stable equilibrium position achieved under the action of the Stokes force. Here we show that the chirality of each individually trapped particle can also be characterized by the rotation of the equilibrium position. Since the power of the laser pulses can be experimentally controlled, this scheme is practical to perform enantioselection, chiral characterization, and chiral resolution of a single chiral sphere with arbitrarily small chirality parameters.

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.

Stable Equilibrium Rotor Positions for Three-Phase Switched Reluctance Machine

In mobile applications such as eletric vehicles, weight and installation space are the primary limiting factors that affect the system efficiency and consequently the driving range. Reusing the machine windings of an electrical machine as inductors, e. g. in DC-DC converters for onboard chargers, saves weight and space. However, currents in the machine windings when operating as DC-DC converter can result in undesired torque ripple generated by the machine. Therefore, stable rotor positions of a switched reluctance machine have to be found to prevent it from rotating in DC-DC converter operation mode. In this study, stable rotor positions for a wide range of current distributions are determined, enabling flexibility in the choice of currents. A method to analyze such stable equilibrium positions for switched reluctance motors is presented. The proposed method is used to determine the stable positions that facilitate flexible current distribution between phases. Moreover, locus curves are calculated, simulated, and measured to illustrate and evaluate stable rotor positions for the various current distributions.

A Tether System at the L1, L2 Collinear Libration Points of the Mars–Phobos System: Analytical Solutions

This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

Dislocation in a Strained Layer Embedded in a Semi-Infinite Matrix

Abstract The misfit stress in a thin layer embedded in a semi-infinite matrix has been first determined near the free surface of the structure, using the virtual dislocation formalism. From a Peach–Koehler force analysis, the different equilibrium positions (unstable and stable) of an edge dislocation gliding in a plane of the layer inclined with respect to the upper interface and emerging at the point of intersection of the upper interface and this free surface have been then characterized with respect to the lattice mismatch and the inclination angle of the gliding plane. It has been found that the dislocation may exhibit stable equilibrium position near the interface and/or near the free surface. A diagram of the position stability has been then determined versus the misfit parameter and the inclination angle. The energy variation due to the introduction of an edge dislocation from the free surface until the matrix layer interface has been finally determined, when the dislocation is gliding in the plane inclined with respect to the interface horizontal axis. A critical thickness of the layer beyond which the formation of the dislocation in the interfaces is energetically favorable has been finally determined as well as its position with respect to the free surface in the lower interface.