Condition For Equilibrium Point Research Articles

Experimental characterization of a nonlinear mechanical oscillator with softening behaviour for large displacements

This paper presents an experimental insight into the performance of a mechanical oscillator consisting of an X-shaped-spring configuration. This configuration achieves an overall softening characteristic with quasi-zero stiffness behaviour far away from the static equilibrium point. Such a geometrical nonlinear configuration has attracted significant research attention in the last few years, particularly for its application as a vibration isolator with the possibility to extend the quasi-zero-stiffness region beyond that of the classical three-spring nonlinear isolator. However, previous experimental evidence has been limited to small amplitude vibration excitation only. Furthermore, it has been focused mainly on the isolation region, rather than on the large amplitude response, thus circumventing an insight on the damping effects and its modelling. To address this gap, in this paper, both frequency sweeps and random excitations are applied to a prototype device for experimental characterization. A nonlinear stiffness model is developed based on the geometry of the system and a nonlinear damping model is assumed based on experimental observation. The proposed model accurately describes the dynamic behaviour of the system as shown by comparison of theoretical and experimental data.

Local gravity control method for solving load-mismatch issue in isolators

Quasi-zero stiffness (QZS) shows significant advantages in low frequency vibration isolation. However, the isolation performance is heavily dependent on the rated load and prone to load-mismatch. Here, a general local gravity control method (LGCM) is proposed to eliminate these limitations by introducing an expected constant force to adjust the equivalent gravity of the mismatched load. As a result, the isolators can be maintained at an ideal static equilibrium point with low dynamic stiffness. The load-mismatch was investigated and emphasized combined with the spring-link mechanism (SLM). The analysis indicated that the load-mismatch results in stiffness hardening, natural frequency increasing, and vibration isolation band shortening. The principle of the LGCM was fully revealed in the respect of conceptual design, electromechanical coupling analysis, control system and operation mechanism. The harmonic balance method (HBM) was used to model the response of the periodic and sweep excitation. The electromechanical coupling simulation was implemented to theoretically verify adjustment ability of LGCM. A prototype based on SLM and Arduino UNO was fabricated and tested with proportional-integral-derivative (PID) to validate the effectiveness of the LGCM. The results illustrates that the vibration transmissibility in mismatched isolator at resonance frequency is decreased by 50 % with LGCM, which means that the resonance frequency can be decreased efficiently, and the isolation band is significantly broadened. The LGCM can be easily attached to any QZS systems, which can be regarded as a general patch for the bug of load-mismatch and is conducive to broaden the load-variable applications of QZS isolators.

The one-dimensional potential energy function that is analogous to a two-dimensional track

When learning about potential energy functions, students are sometimes told that the potential energy function is analogous to that of a particle sliding along a frictionless roller coaster track or wire confined to a vertical plane with peaks and valleys of the track corresponding to unstable and stable equilibrium points. However, motion along a track with height z(x) is a constrained two-dimensional motion, not a one-dimensional motion, so the exact nature of this analogy may be unclear. We show that the horizontal motion of a point mass sliding along a frictionless track z(x) and subject to a uniform gravitational field is equivalent to the motion of a particle in one dimension characterized by an “analogous potential energy” function UE(x), which generally depends on the total energy of the system (and thus on the initial conditions). We derive a general expression for UE(x) in terms of z(x) and the total energy and show that the equilibrium points of the actual potential energy U(x)=mgz(x) are also static equilibrium points for UE(x) with the same stability. However, UE(x) may have additional dynamic equilibrium points that are not present for U(x). As an example, we derive UE(x) for a double well track and determine the period of oscillations on that track. The results show that in general a single track corresponds to many different analogous potential energy functions, each with a different value for the total energy.

E-Sail Optimal Trajectories to Heliostationary Points

The aim of this paper is to investigate the performance of a robotic spacecraft, whose primary propulsion system is an electric solar wind sail (E-sail), in a mission to a heliostationary point (HP)—that is, a static equilibrium point in a heliocentric and inertial reference frame. A spacecraft placed at a given HP with zero inertial velocity maintains that heliocentric position provided the on-board thrust is able to counterbalance the Sun’s gravitational force. Due to the finite amount of storable propellant mass, a prolonged mission toward an HP may be considered as a typical application of a propellantless propulsion system. In this respect, previous research has been concentrated on the capability of high-performance (photonic) solar sails to reach and maintain such a static equilibrium condition. However, in the case of a solar-sail-based spacecraft, an HP mission requires a sail design with propulsive characteristics that are well beyond the capability of current or near-future technology. This paper shows that a medium-performance E-sail is able to offer a viable alternative to the use of photonic solar sails. To that end, we discuss a typical HP mission from an optimal viewpoint, by looking for the minimum time trajectory necessary for a spacecraft to reach a given HP. In particular, both two- and three-dimensional scenarios are considered, and the time-optimal mission performance is analyzed parametrically as a function of the HP heliocentric position. The paper also illustrates a potential mission application involving the observation of the Sun’s poles from such a static inertial position.

A solution to some problems of Conway and Guy on monostable polyhedra

A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. The aim of this paper is to investigate three questions of Conway, regarding monostable polyhedra, which first appeared in a 1969 paper of Goldberg and Guy. In this note, we answer two of these problems and make a conjecture about the third one. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium points. As another application of this theorem, we prove the existence of a convex polyhedron with only one stable and one unstable point.

On Structural Parameter Optimization Method for Quad Tilt-Wing UAV Based on Indirect Size Estimation of Domain of Attraction

To deal with the flexibility in the electric aircraft design, this study develops a structural parameter optimization method. The quad tilt-wing aircraft exhibits rotary-wing flight and fixed-wing flight modes by tilting its wings. This study focuses on the domain of attraction, which is a class of state vectors that converge to a static equilibrium point after infinite time. The method indirectly estimates the size of domain of attraction of an aircraft by calculating the radius of a sphere that is encompassed by the domain. In addition, after investigating the relationship between change in structural parameters and the size of the domain of attraction, the structural parameters that maximize the size of the domain are surveyed. The effectiveness of the proposed method is demonstrated for the optimization of the quad tilt-wing aircraft. The quad tilt-wing aircraft exhibits rotary-wing flight and fixed-wing flight modes by tilting its wings. As the characteristics of the dynamics of the quad tilt-wing aircraft demonstrate a significant nonlinear change depending on the tilt angle of the wing, we propose a method to optimize the structural parameters for an easy control of the aircraft. Simulation results show that the optimization scheme identifies the parameters that enable the aircraft to perform agile control to address a disturbance moment. Further, although the same controller is used in the original and optimized aircraft, the optimized aircraft can handle a disturbance in a shorter time when compared with the original aircraft.

Mono-unstable polyhedra with point masses have at least 8 vertices

The monostatic property of convex polyhedra (i.e., the property of having just one stable or unstable static equilibrium point) has been in the focus of research ever since Conway and Guy (1969) published the proof of the existence of the first such object, followed by the constructions of Bezdek (2011) and Reshetov (2014). These examples establish F≤14,V≤18 as the respective upper bounds for the minimal number of faces and vertices for a homogeneous mono-stable polyhedron. By proving that no mono-stable homogeneous tetrahedron existed, Conway and Guy (1969) established for the same problem the lower bounds for the number of faces and vertices as F,V≥5 and the same lower bounds were also established for the mono-unstable case (Domokos et al., 2020b). It is also clear that the F,V≥5 bounds also apply for convex, homogeneous point sets with unit masses at each point (also called polyhedral 0-skeletons) and they are also valid for mono-monostatic polyhedra with exactly one stable and one unstable equilibrium point (both homogeneous and 0-skeletons). In this paper we draw on an unexpected source to extend the knowledge on mono-monostatic solids: we present an algorithm by which we improve the lower bound to V≥8 vertices on mono-unstable 0-skeletons. The problem is transformed into the (un)solvability of systems of polynomial inequalities, which is shown by convex optimization. Our algorithm appears to be less well suited to compute the lower bounds for mono-stability. We point out these difficulties in connection with the work of Dawson, Finbow and Mak (Dawson, 1985, Dawson et al., 1998, Dawson and Finbow, 2001) who explored the monostatic property of simplices in higher dimensions.

Control of self-excited rotor disturbances in three-lobe journal bearing space using a couple stress lubricant

In this study, the dynamic behavior of three-lobe noncircular hydrodynamic journal bearings lubricated with couple stress fluid is presented. Currently, application of journal bearings with different geometries such as lobed bearings and using non-Newtonian lubricants, including various additives, are the common suggested solutions by tribology researchers to improve the performance of rotor-journal bearing systems. Due to the incompatibility of new lubricant behaviors including polymer chains or other types of suspended particles with considerable size from the classic Navier–Stokes equations, different models such as couple stress fluid theory have been proposed to evaluate their performance. To investigate the self-excited rotor disturbances caused by various factors, like operating in critical speeds, simultaneous solution of governing Reynolds equation of oil lubricant and the rotor motion equations has been done. The result of dynamic analyses using linear and nonlinear approaches indicates the possibility of enhancing the stiffness and damping ability of the three-lobe bearings by strengthening the couple stress properties of lubricant. It is seen from the results that by increasing the couple stress parameter of oil, the type of rotor perturbations in the bearing clearance space changes from divergent oscillations, as a leading cause of collision between the rotor surface and the bearing shell, to limit cycle motion around the rotor stable position or convergent fluctuations to the static equilibrium point. This improvement in the stability range of three-lobe journal bearings using a couple stress lubricant appears in the results of a linear analysis model in terms of increasing the critical mass parameter and decreasing the whirl frequency ratio. In addition, bifurcation diagram results show that using noncircular three-lobe bearings instead of circular types and replacing the Newtonian lubricant with the couple stress fluid improve the dynamic stability of rotor- bearing supports.

로드셀을 이용한 이륜형 셀프 밸런싱 로봇 주행기기의 실시간 편심 추정 및 보상 기법

When a two-wheeled self-balancing robotic vehicle moves, the rider’s unexpected movement or the posture variation of the integrated mobile manipulator may work as a significant disturbance to the postural stability of the inverted pendulum due to the change of the overall COG (Center of Gravity). In this study, we developed a method to estimate the variation of COG in real time and compensate for the eccentric effect to the posture control system by using load cells. For the lateral eccentricity, the load cells can detect the difference between the normal reactions acting on the two wheels and a proper compensation torque proportional to the difference can be applied. For the longitudinal eccentricity, an equilibrium pitch angle is generated as the reference input to the pitch control loop to keep the robot at its static equilibrium point. The experimental results proved that the posture control performance of a self-balancing robot could be considerably improved by applying the real-time eccentricity compensation technique.

Optimal Spectrum Allocation Based on Primary User Activity Model in Cognitive Radio Wireless Sensor Networks

In Cognitive Radio Wireless Sensor Networks, the licensed spectrum bands are highly dynamic, and their status varies overtime. With the expansion of these networks, regarding the energy constraints and the fact that reallocation of the frequency spectrum is energy-consuming, the problem of controlling the behavior of secondary users in the allocation of the spectrum is of great importance. Providing a method to reduce the number of channel reallocation, which in turn results in reducing energy consumption in such a dynamical network is essential. In this paper, considering the energy constraints, an optimal method for allocating frequency spectrum resources is presented using game theory and Nash equilibrium. By analyzing the activity model of primary users on the frequency spectrum and selecting the appropriate spectrum using the Nash equilibrium, the method reaches the network to a stationary equilibrium point. In these conditions, in addition to reducing interference between primary and secondary users, the number of channel reallocations by cognitive radio users is reduced and thus reduces overall energy consumption in the network and increases its life span.

Primary Resonance Analysis of Permanent Magnet Synchronous Motors Caused by Unbalanced Magnetic Pull

The dynamic behaviors of permanent magnetic synchronous motors considering unbalanced magnetic pull have been extensively investigated. However, the harmonics and cogging effects are ignored and the stator and the rotor are assumed to be smooth cylinders. In this research, an analytical expression of unbalanced magnetic pull considering harmonics and cogging effects is obtained. Based on the model, the motion equations of the rotor system are established. The natural frequency and frequency-response are analyzed by average method, and stability of periodic solution is investigated by Routh-Hurwitz criterion. The analysis indicates that the natural frequency is closely related to stator slot parameters. The frequency-response shows that the amplitude varies between the two static equilibrium points. Besides, when the ratio of slot width to stator slot pitch is close to the peak value, the amplitude tends to be increasing obviously. It is useful for the design of permanent magnet synchronous motors with high stability and low vibration.

Global stability of a fractional order eco-epidemiological system with infected prey

A fractional order eco-epidemiological system with infected prey is proposed and studied. Threshold parameters are obtained to determine the existence and stability conditions of equilibrium points of the proposed fractional order system. In order to clarify the characteristics of the proposed fractional order system, the analysis of existence, uniqueness, non-negativity and boundedness of the solutions, as well as the stability conditions of the system are examined. The global properties of all biologically feasible equilibrium points are studied by constructing suitable Lyapunov functions. It is proved that all equilibrium points of the proposed fractional order system are globally stable whenever they are locally stable. Some numerical simulations are conducted to verify the theoretical results.

Characteristic analysis of a new high-static-low-dynamic stiffness vibration isolator based on the buckling circular plate

The high-static-low-dynamic stiffness vibration isolator has great advantages in vibration isolation because it can decrease the natural frequency of the system while keeping the load capability, but it is usually difficult to implement because of its complex structures and installation space constraints. A high-static-low-dynamic stiffness vibration isolator composed of a buckling circular plate and a traditional linear spring is proposed in this paper. The buckling circular plate works as the negative stiffness corrector paralleled with the linear spring, which can be integrated into the sleeve. If the load is chosen properly, the static equilibrium point will be at the initial quasi-zero stiffness point. However, any changes of the load will lead the equilibrium point deviating from the initial equilibrium point. The nonlinear mathematical model of high-static-low-dynamic stiffness vibration isolator considering load imperfection is developed and its force transmissibility is analyzed with the harmonic balance method and homotopy perturbation method. The influence rule of the system parameters on it is analyzed and the corresponding results show that the force transmissibility will exhibit complicated characteristics, depending on the load imperfection, damper, and excitation force.

Dynamic stability analysis of noncircular two-lobe journal bearings with couple stress lubricant regime

In this research, the steady state and dynamic performances of two-lobe noncircular journal bearings with couple stress lubricant are presented. The lubricating oil, containing additives and contaminants, is modeled as the couple stress fluid. The modified Reynolds equation is obtained using the couple stress lubrication theory and is then solved by finite element method as an efficient numerical technique. The steady-state characteristics of bearings, including the load carrying capacity and attitude angle, are determined for various values of the couple stress parameter. The results show that applying the couple stress fluid improves the efficiency of two-lobe bearings in terms of an increased load carrying capacity and reduced attitude angle. Also, the stability performance of the investigated bearings is studied using rotor motion equations based on linear and nonlinear dynamic methods. The results indicate that any increase in the lubricant couple stress parameter enhances the bearing ability to damp the rotor perturbations. In other words, by varying the lubricant from Newtonian oil to the couple stress type and upgrading its properties, the curves of the critical mass parameter and whirl frequency ratio have an increasing and decreasing trend, respectively. Based on the fourth-order Runge–Kutta method results, the dynamic trajectory of the rotor center in the bearing space changes with increasing the couple stress parameter from diverging disturbances and limits the cycle perturbations around the bearing center to converging oscillations to the static equilibrium point. Moreover, the effect of changing lubricant properties on the two-lobe bearing’s performance is more pronounced at higher values of the couple stress parameter, especially with an increase in the noncircularity of bearings’ geometry.

Nonlinear Aeroelastic Analysis of Heated Curved Panels in Supersonic Air Flow

A nonlinear aeroelastic model for a two-dimensional heated curved panel in supersonic air flow is established by using Galerkin method. The von Karman large deflection theory and the third-order piston theory appended with the static aerodynamic loading are used in the governing equations. The static aero-thermal deflection of the curved panel is firstly obtained by using Newton iterative approach. Then the stability boundary under different temperature elevations is achieved by using Lyapunov’s indirect method. Lastly, the motion equations of a heated curved panel in supersonic air flow are solved by a fourth-order Runge–Kutta numerical scheme. Time history responses, phase plots, Poincare maps and bifurcation diagrams are used for better understanding of the pre/post-flutter responses of a curved panel by varying parameters, including curvatures, dynamic pressures and temperature elevations. The results demonstrate that the flutter boundary drops significantly with increasing temperature elevation for panels with small curvatures. However, the flutter boundary almost keeps the same for panels with large curvatures. The nonlinear flutter characteristics of the curved panel differs from those of flat panels significantly, the nonlinear flutter response of heated curved panels changes from static equilibrium point or LCO to chaos with the increase of temperature elevation or dynamic pressure and the static, LCO motions also exist in the chaotic area. For panels with large curvatures, the chaotic motions cannot occur, and the amplitude of LCO is limited in a certain range.

Effect of multi-frequency parametric excitations on the dynamics of on-board rotor-bearing systems

In the transportation domain such as automotive turbochargers and aircraft turbines, the vibrations of on-board rotors are induced not only by the mass unbalance excitation but also by various movements of their support. The dynamics of an on-board rotor mounted on hydrodynamic finite-length bearings is investigated in the presence of support motions which create multi-frequency parametric excitations. The developed on-board rotor model is based on the gyroscopic Timoshenko beam finite element with two nodes and six degrees of freedom per node for 3D motions (transverse and axial displacements as well as rotations due to the bending and to the torsion). The equations of motion highlight time-varying parametric terms due to the mass unbalance, the support rotations, the coupling between both phenomena and the combination of mass unbalance and support translations. These parametric terms can yield a dynamic instability because they contribute as generators of internal excitation. In the presented applications, single-frequency and multi-frequency parametric excitations are used. Namely, the rotor is excited either by simple and combined sinusoidal support rotations or by a rotating mass unbalance combined with sinusoidal support translations to examine the stability of the static equilibrium point through the Floquet theory.

Linear and nonlinear analysis of mount angle effects on the dynamic stability of micropolar lubricated hydrodynamic lobed journal bearings

Journal bearings are one of the most widely used types of supporting devices for industrial machinery, especially for high-speed rotary systems. Considering the importance of mechanical systems ability to control vibrations and motion disturbances, the design of noncircular journal bearings has been one of the appropriate techniques used in recent years. In the present work, the effects of mount angle as a noncircularity unique feature on the dynamic stability of lobed journal bearings with micropolar lubricant are studied. For this purpose, modified Reynolds equation based on the linear and nonlinear methods is solved by generalized differential quadrature (GDQ) method. Then the static and dynamic characteristics of two-, three- and four-lobe bearings including load carrying capacity, whirl frequency ratio, critical mass parameter and rotor perturbation in terms of bifurcation diagram and Poincare map are presented for different values of mount angle. Nonlinear analysis results indicate that variation of mount angle can change the rotor response from limit cycle oscillations to convergent perturbation to static equilibrium point. Further, against plain circular journal bearings, choosing an appropriate orientation or mounting angle provides the optimal stability of noncircular lobed journal bearings. Also, the linear analysis results for the range of dynamic stability are more cautious than the nonlinear method in most of investigated cases.

Dynamic Prey Predator Model and multiple forms of Harvest of Infected Prey

In this paper, the dynamic of prey predator model was discussed when the relationship between them is functional response type III. In addition, when prey exposure to the disease as nonlinear function. Also the infected prey exposed to harvest as a nonlinear and as linear function. The bounded and positive solutions, periodic, conditions of equilibrium points and the stability were we discussed Some results were illustrated in numerical simulations, and show we can use the linear function of harvesting to control on the dices.

Research on Linear Static Water Resources IO Model Optimization Based on General Equilibrium Method

The static input-output model is limited to analyzing and studying the production process in a certain period, and it is weak in explaining the process of social reproduction. At the same time, the linear assumption of the model is contrary to the actual production. Based on the evolution of the input-output model, the linear static water resources I-O model is extended to the computable nonlinear dynamic water resources I-O model, which overcomes the irreplaceable elements of the input model of the linear water resources I-O model and the CGE model can only solve the market static equilibrium point Shortcomings, more realistic description of the regional economic system and the relationship between water resources system.

Pengaruh Perilaku Merokok pada Pertumbuhan Populasi Penderita Kanker Paru-Paru

This article discusses a model of lung cancer as the effect of smoking behavior on both active and passive smoker. There are four subpopulations in this model, namely susceptible subpopulation, active smoker subpopulation, passive smoker subpopulation, and subpopulation of lung cancer. Dynamical analysis is conducted to determine the equilibrium point, existence condition for equilibrium point, and analyze their stability. Based on analysis result, there are three equilibrium points. First equilibrium point shows that all subpopulations extinct. Second equilibrium point shows that only susceptible subpopulation can survive, and the last equilibrium point shows that all subpopulations can survive. First equilibrium point always exists while the others exist under certain condition. The stability of first equilibrium point can be reached when the intrinsic growth rate is less than the death rate. Whereas, the others equilibrium points will be stable under certain condition. Numerical simulation is performed to illustrate the analysis result. It is shown that numerical results are in accordance with analysis result. These numerical simulations also indicate that the rate of passive smoker plays important role in the growth rate of lung cancer.