Stability Of Positive Equilibrium Research Articles

Stabilizing a spherical pendulum on a quadrotor

AbstractIn this article, we design a backstepping control law based on geometric principles to swing up a spherical pendulum mounted on a moving quadrotor. The available degrees of freedom in the control vector also permit us to position the plane of the quadrotor parallel to the ground. The problem addressed here is, indeed, novel and has many practical applications which arise during the transport of a payload mounted on top of a quadrotor. The modeling and control law are coordinate‐free and thus avoid singularity issues. The geometric treatment of the problem greatly simplifies both the modeling and control law for the system. The control action is verified and supported by numerical experiments for aggressive maneuvers starting very close to the downward stable equilibrium position of the pendulum.

Theoretical analysis on the static and dynamic performances of a squeeze film air journal bearing with three separate pads structure

Owing to their distinct non-contact and oil-free characteristics, squeeze film air bearings have been introduced to satisfy ultra-precision, low wear, and ultra-clean requirements. This paper proposes an analytical model of a three-pad squeeze film bearing to study its static and dynamic performance. The bearing force is calculated by integrating the pressure distribution over the bearing surface, which is governed by the Reynolds equation. The stable equilibrium position of the rotor is obtained by the Newton-Raphson method. A numerical method to acquire the bearing dynamic coefficients is originally proposed by considering the vibration of its pad. These dynamic coefficients are determined by solving the perturbation equations derived from the combination of the Reynolds equation and the modified film thickness and pressure. The predicted static and dynamic results show good agreement with experimental results. The parameter study shows that the variation in eccentricity with respect to the rotational speed can be controlled by reasonably adjusting the vibration amplitude or the nominal clearance of the bearing. In addition, the results indicate that the direct stiffness and damping coefficients are increased by decreasing the rotation speed or increasing the vibration amplitude of the bearing pad.

A 2D Membrane MEMS Device Model with Fringing Field: Curvature-Dependent Electrostatic Field and Optimal Control

An important problem in membrane micro-electric-mechanical-system (MEMS) modeling is the fringing-field phenomenon, of which the main effect consists of force-line deformation of electrostatic field E near the edges of the plates, producing the anomalous deformation of the membrane when external voltage V is applied. In the framework of a 2D circular membrane MEMS, representing the fringing-field effect depending on |∇u|2 with the u profile of the membrane, and since strong E produces strong deformation of the membrane, we consider |E| proportional to the mean curvature of the membrane, obtaining a new nonlinear second-order differential model without explicit singularities. In this paper, the main purpose was the analytical study of this model, obtaining an algebraic condition ensuring the existence of at least one solution for it that depends on both the electromechanical properties of the material constituting the membrane and the positive parameter δ that weighs the terms |∇u|2. However, even if the the study of the model did not ensure the uniqueness of the solution, it made it possible to achieve the goal of finding a stable equilibrium position. Moreover, a range of admissible values of V were obtained in order, on the one hand, to win the mechanical inertia of the membrane and, on the other hand, to ensure that the membrane did not touch the upper disk of the device. Lastly, some optimal control conditions based on the variation of potential energy are presented and discussed.

Linear Oscillation of the Inter-Connected Satellites Under the Influence of Magnetic Force Oblateness of the Earth and the Shadow of the Earth Due to Solar Radiation Pressure About the Position of Equilibrium for Small Eccentricity Near the Main Resonance

This paper is devoted to study the non-linear oscillation along the stable position of equilibrium where the system oscillates about the position of equilibrium like a dumb-bell satellite with constant amplitude and phase varying with true anomaly. B.K.M. method has been exploited to get the general solution valid at and near the main resonance n = 1.

Chaotic attitude dynamics of a LEO satellite with flexible panels

This paper deals with the attitude motion of LEO satellites with deployable side panels designed for passive aerodynamic stabilization in a rarefied atmosphere. The influence of the aerodynamic and gravitational torques on the planar attitude motion near the unstable and stable equilibrium positions is studied. The presence of the unstable equilibrium position and small perturbations such as the oscillations of the flexible panels is the cause of chaos. A critical altitude is found above which the chaos is possible. The equations of planar attitude motion of the satellite with deployed flexible panels are obtained. The chaotic behavior of the system is demonstrated through numerical simulations of the attitude motion of a 3U CubeSat. The results of this paper can be used to analyze the applicability of passive aerodynamic stabilization for LEO satellites.

Model of Charged Anisotropic Strange Stars in Minimally Coupled f R Gravity

In the present article, we have investigated a new family of nonsingular solutions of static relativistic compact sphere which incorporates the characteristics of anisotropic fluid and electromagnetic field in the context of minimally coupled f R theory of gravity. The strange matter MIT bag model equation of state (EoS) has been considered along with the usual forms of the Karori–Barua KB metric potentials. For this purpose, we derived the Einstein–Maxwell field equations in the assistance of strange matter EoS and KB type ansatz by employing the two viable and cosmologically well-consistent models of f R = R + γ R 2 and f R = R + γ R R + α R 2 . Thereafter, we have checked the physical acceptability of the proposed results such as pressure, energy density, energy conditions, TOV equation, stability conditions, mass function, compactness, and surface redshift by using graphical representation. Moreover, we have investigated that the energy density and radial pressure are nonsingular at the core or free from central singularity and always regular at every interior point of the compact sphere. The numerical values of such parameters along with the surface density, charge to radius ratio, and bag constant are computed for three well-known compact stars such as CS1 SAXJ 1808 . 4 − 3658 ( x ˜ = 7.07 km , CS2 VelaX − 1 x ˜ = 9.56 km , and CS3 4U1820 − 30 x ˜ = 10 km and are presented in Tables 1–6. Conclusively, we have noticed that our presented charged compact stellar object in the background of two well-known f R models obeys all the necessary conditions for the stable equilibrium position and which is also perfectly fit to compose the strange quark star object.

DNA-assisted assembly of cationic gold nanoparticles: Monte Carlo simulation.

DNA-assisted assembly of ligand-stabilized gold nanoparticles is studied using Monte Carlo simulations with coarse-grained models for DNA and AuNP. Their interaction in a periodic simulation box is described by a combination of electrostatic and pairwise hard core potentials. We first probe the self-assembly of AuNPs resulting in an ordered distribution on a single fixed DNA strand. Subsequently, the effective force calculated between a pair of parallel DNA in the presence of AuNPs shows the attraction between them at short distance associated to a stable equilibrium position. Finally, the osmotic pressure calculated in a compact DNA-AuNP lattice with various amounts of monovalent salt ions shows that an increasing amount of salt prevents aggregate formation.

HOPF BIFURCATION IN A DELAYED PREDATOR-PREY SYSTEM WITH GENERAL GROUP DEFENCE FOR PREY

<abstract> In this paper, a delayed predator-prey system with group defence for prey is investigated. Firstly, in the absence of spatial diffusion and time delay, the stability of positive equilibrium and existence of the Hopf bifurcation are investigated, as well as the direction of the Hopf bifurcation, which is determined by applying the first Lyapunov number. Then, the occurrence of the Hopf bifurcation in the diffusion-driven delayed system is further explored. By using the center manifold reduction and the normal form theorem, the conditions ensuring the occurrence of Hopf bifurcation, its direction and its stability are formulated in terms of different parameters. Finally, some numerical simulations are carried out to verify the theoretical results and the existence of the homogeneous periodic solution is exhibited by setting different values of parameters. Moreover, stable temporal periodic solutions and spatially inhomogeneous periodic solutions are identified from the numerical simulations. The obtained results are also explained and discussed from the practical point of view. </abstract>

On the Almost Sure Stability of Dynamical Systems with Respect to White Noise

We consider a system of ordinary differential equations possessing an asymptotically stable equilibrium position and examine the stability of this equilibrium under permanent stochastic whitenoise-type perturbations. The perturbed system is considered in the form of Stratonovich stochastic differential equations. We also describe restrictions on the parameters of perturbations that guarantee the trajectory-wise stability of the equilibrium with probability 1.

Dielectrophoretic Force Equilibrium of Complex Particles

In contrast to the commonly used spherical Janus particles, here we used engineered Janus particles that are fabricated using photolithography technique for precise control over their geometry and coated regions. Specifically, we studied a lollipop-shaped complex particle where its head is coated with gold while its tail is left bare. Due to their distinct electrical properties (i.e. electrical polarizability) the particle exhibits force equilibrium where opposite dielectrophoretic forces acting on its head and tail exactly cancel each other to yield a stable equilibrium position. This was realized in a quadrupolar electrode array where the equilibrium position of the engineered particle could be tuned by the frequency. This stands in contrast to the standard dielectrophoretic behavior where the particle shifts positions from either the center of the quad to the very edge of the electrodes when shifting from a negative to positive dielectrophoretic response, respectively. This opens new opportunities for positioning control of such complex particles for self-assembly, biosensing, biomimetic spermatozoa and more.

Spatiotemporal Dynamics in a Diffusive Bacterial and Viral Diseases Propagation Model with Chemotaxis

In this article, we study the effect of chemotaxis on the dynamics of a diffusive bacterial and viral diseases propagation model. From three aspects: $$\chi >0$$ , $$\chi =0$$ and $$\chi <0$$ , we investigate the existence of Turing bifurcations and stability of positive equilibrium under Neumann boundary conditions. We find that Turing bifurcations can induced by chemotaxis, which does not occur in the original model. Moreover, for the model with diffusion and chemotaxis, we need explore the new expression of the normal form on Turing bifurcation. By the newly obtained normal form, we can determine the properties of Turing bifurcation. Finally, we perform some numerical simulations to verify the theoretical analysis and obtain stable steady state solutions, spots pattern and spots-strip pattern, which also expand the main results in this article.

Motion and equilibrium position of elliptical and rectangular particles in a channel flow of a power-law fluid

The inertial migration of the elliptical and rectangular particles in a channel flow of a power-law fluid is studied using the lattice Boltzmann method. The numerical method and code are validated by comparing the present results with the previous ones. Effects of power-law index (n) of the fluid, particle shape, particle aspect ratio (α), blockage ratio (k) and Reynolds number (Re) on the particle trajectory and equilibrium position are discussed. The results show that the elliptical and rectangular particles will finally oscillate in a lateral equilibrium position. The distance for the particle from its initial position to the stable equilibrium position is the shortest for the rectangular particle, then followed for the elliptical particle and finally for the circular particle and square particle, the distance is the shortest for the shear-thinning fluid, then followed for the Newtonian fluid and finally for the shear-thickening fluid. This distance for the rectangular particle decreases with increasing Re, but the dependence of the distance on Re for the elliptical particle is not as obvious as that for the rectangular particle. The particle with larger aspect ratio and blockage ratio gets to the equilibrium position faster. The lateral distance from the equilibrium position to the channel centerline is reduced with increasing k, and with decreasing α for both elliptical and rectangular particles. For the particles with larger α, the lateral distance is reduced with the increase of Re, but the relationship between the lateral distance and Re for the particles with smaller α is dependent on n.

Spatial dynamics of a nonlocal bistable reaction diffusion equation

This article concerns a nonlocal bistable reaction-diffusion equation with an integral term. By using Leray-Schauder degree theory, the shift functions and Harnack inequality, we prove the existence of a traveling wave solution connecting 0 to an unknown positive steady state when the support of the integral is not small. Furthermore, for a specific kernel function, the stability of positive equilibrium is studied and some numerical simulations are given to show that the unknown positive steady state may be a periodic steady state. Finally, we demonstrate the periodic steady state indeed exists, using a center manifold theorem.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/84/abstr.html

Parametric Stability of a Pendulum with Variable Length in an Elliptic Orbit

We study the dynamics of a simple pendulum attached to the center of mass of a satellite in an elliptic orbit. We consider the case where the pendulum lies in the orbital plane of the satellite. We find two linearly stable equilibrium positions for the Hamiltonian system describing the problem and study their parametric stability by constructing the boundary curves of the stability/instability regions.

Experimental Characterization of Friction in a Negative Stiffness Nonlinear Oscillator

Nonlinear dissipative phenomena are common features of many dynamical systems and engineering applications, and their experimental characterization has always been a challenge among the research community. Within the wide range of nonlinear damping mechanisms, friction is surely one of the most common, and with a high impact on the dynamical behavior of structures. In this paper, the nonlinear identification of friction in a negative stiffness oscillator is pursued. The structure exhibits a strong nonlinear behavior, mainly due to its polynomial elastic restoring force with a negative stiffness region. This leads to an asymmetric double-well potential with two stable equilibrium positions, and the possibility of switching between them in a chaotic way. Friction plays a crucial role in this context, as it derives from the continuous sliding between the central guide and the moving mass. The system is driven through harmonic tests with several input amplitudes, in order to estimate the variations in the energy dissipated per cycle. The identification of the frictional behavior is then pursed by minimizing the errors between the experimental measurements and the model predictions, using the harmonic balance method in conjunction with a continuation technique on the forcing amplitudes.

Design and modeling of a compliant gripper with parallel movement of jaws

A compliant gripper with parallel movement of jaws is developed. The gripper is composed of a bar linkage mechanism and a bistable mechanism. After a target object is grasped, the embedded bistable mechanism can maintain the gripping of the object with no energy consumption. Unstable fixation due to rotary motion of opposing gripping arms of traditional tweezer-like grippers is alleviated by the proposed gripper. Kinetostatic models of the device based on chained beam constraint model are derived to assist the design of the device. Grasp function of an object is accomplished while the gripper proceeds from its first stable equilibrium position to its second stable equilibrium position, and release of the object is achieved when the gripper returns back to its first stable equilibrium position.

Nonlinear Dynamics of a Duffing-Like Negative Stiffness Oscillator: Modeling and Experimental Characterization

In this paper, a negative stiffness oscillator is modelled and tested to exploit its nonlinear dynamical characteristics. The oscillator is part of a device designed to improve the current collection quality in railway overhead contact lines, and it acts like an asymmetric double-well Duffing system. Thus, it exhibits two stable equilibrium positions plus an unstable one, and the oscillations can either be bounded around one stable point (small oscillations) or include all the three positions (large oscillations). Depending on the input amplitude, the oscillator can exhibit linear and nonlinear dynamics and chaotic motion as well. Furthermore, its design is asymmetrical, and this plays a key role in its dynamic response, as the two natural frequencies associated with the two stable positions differ from each other. The first purpose of this study is to understand the dynamical behavior of the system in the case of linear and nonlinear oscillations around the two stable points and in the case of large oscillations associated with a chaotic motion. To accomplish this task, the device is mounted on a shaking table and it is driven with several levels of excitations and with both harmonic and random inputs. Finally, the nonlinear coefficients associated with the nonlinearities of the system are identified from the measured data.

Coexistence of bistable multi-pulse chaotic motions with large amplitude vibrations in buckled sandwich plate under transverse and in-plane excitations.

This paper researches the dynamic snap-through phenomena and the coexistence of the multi-pulse jumping chaotic dynamics in bistable equilibrium positions and the large amplitude nonlinear vibrations for a simply supported buckled 3D-kagome truss core sandwich rectangular plate under combined transverse and in-plane excitations based on the extended high-dimensional Melnikov method. According to the two-degree-of-freedom non-autonomous nonlinear dynamical system of the buckled truss core sandwich rectangular plate and the theory of normal form, it is found that some nonlinear terms have less effects on the nonlinear dynamic responses than other nonlinear terms. The extended high-dimensional Melnikov method is employed to investigate the dynamic snap-through phenomena and the bistable multi-pulse jumping chaotic vibrations around the upper-mode and the lower-mode for two different cases of buckling in the truss core sandwich rectangular plate. A numerical method is used to detect the bistable multi-pulse jumping chaotic vibrations around two stable equilibrium positions for the non-autonomous buckled truss core sandwich rectangular plate. The obtained theoretical and numerical results indicate that there exists the coexistence of the bistable multi-pulse jumping chaotic vibrations and the large amplitude nonlinear vibrations in the buckled truss core sandwich rectangular plate under combined transverse and in-plane excitations.

Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence

This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out.

Inertial bifurcation of the equilibrium position of a neutrally-buoyant circular cylinder in shear flow between parallel walls

The authors examine the inertial lift on a neutrally-buoyant circular cylinder in confined shear flow. Through two-dimensional lattice Boltzmann simulations, it is shown that the transverse equilibrium position of the cylinder will undergo a pitchfork bifurcation above a critical Reynolds number, with the stable equilibrium position shifting away from the centerline of the channel. The study demonstrates that this critical Reynolds number is dependent on the ratio of particle size to channel width, and occurs below the transition to unsteady flow.