Stability Of Periodic Motions Research Articles

Human short-latency reflexes show precise short-term gain adaptation after prior motion.

The central nervous system adapts the gain of short-latency reflex loops to changing conditions. Experiments on biomimetic robots showed that reflex modulation could substantially increase energy efficiency and stability of periodic motions if, unlike known mechanisms, the reflex modulation both acted precisely on the muscles involved and lasted after the motion. This study tests the presence of such a mechanism by having participants repeatedly rotate either their right elbow or shoulder joint, before perturbing either joint. The results demonstrate a mechanism that modulates short-latency reflex gains after prior motion with joint-specific precision. Enhanced gains were observed hundreds of milliseconds after movement cessation, a time scale well-suited to quickly adapt overall periodic motion cycles. A serotonin antagonist significantly decreased these post-movement gains diffusely across joints. But blocking serotonin did not affect the joint-specificity of the gain scaling more than a placebo, suggesting that serotonin sets the overall reflex gain across joints after movement by an effect that is modulated in a joint-specific manner by an unidentified neural circuit. These results confirm the existence of a new, joint-specific, fast, persistent adaptation of short-latency reflex loops induced by motion in human arms.

Diversity and transition of periodic motion of a periodically excited soft-impacting machinery

Dynamics of a periodically excited vibro-impact system with soft impacts is investigated. Essential features of period-one multi-impact motion group and correlated transition characteristics in low-frequency range are discussed in detail by the way of two-parameter bifurcation space providing qualitative domains for different periodic motions. The main focus is given to the effect of sensitive parameters including constraint stiffness k 0, clearance threshold b, and damping parameter ζ on the system response. The low-frequency characteristics in the finite-dimensional parameter space are particularly explored. It is found that the increase of k 0 induces multi-type bifurcation of period-one double-impact symmetrical motion, which induces a rich variety of periodic motions, and period-one multi-impact motion group orbit primarily exist in the small-clearance b and low-frequency ω zone. Based on the evolution irreversibility of adjacent period-one multi-impact orbit, the mechanism of singularies appearing in pairs and two different transition zones (hysteresis and liguliform zones) is studied, the result of which provides a theoretical reference value for the common low-frequency vibration instability phenomenon in the field of mechanical engineering. For small-damping coefficient ζ, period-one multi-impact motion has a large quantity, and the main bridge for the transition of adjacent period-one multi-impact motion is liguliform zone, which embraces period-one multi-impact asymmetrical motion and period-n multi-impact subharmonic motion and a certain chaotic zone. For large-damping coefficient ζ, the amount of period-one multi-impact motion group is reduced, and the main bridge for the transition of adjacent period-one multi-impact motion is hysteresis zone, where adjacent period-one multi-impact orbits can coexist according to initial conditions. As designing and renovating impact mechanical equipment, the reasonable matching law of dynamic parameters can be determined through two-parameter bifurcation space, which is conducive to making the system work in stable periodic motion and obtaining larger instantaneous impact velocity.

Comparison of feed-forward control strategies for simplified vertical hopping model with intrinsic muscle properties

To analyse walking, running or hopping motions, models with high degrees of freedom are usually used. However simple reductionist models are advantageous within certain limits. In a simple manner, the hopping motion is generally modelled by a spring-mass system, resulting in piecewise smooth dynamics with marginally stable periodic solutions. For a more realistic behaviour, the spring is replaced by a variety of muscle models due to which asymptotically stable periodic motions may occur. The intrinsic properties of the muscle model, i.e. preflexes, are usually taken into account in three complexities-constant, linear and Hill-type. In this paper, we propose a semi-closed form feed-forward control which represents the muscle activation and results in symmetrical hopping motion. The research question is whether hopping motions with symmetric force-time history have advantages over asymmetric ones in two aspects. The first aspect is its applicability for describing human motion. The second aspect is related to robotics where the efficiency is expressed in term of performance measures. The symmetric systems are compared with each other and with those from the literature using performance measures such as hopping height, energetic efficiency, stability of the periodic orbit, and dynamical robustness estimated by the local integrity measure (LIM). The paper also demonstrates that the DynIn MatLab Toolbox that has been developed for the estimation of the LIM of equilibrium points is applicable for periodic orbits.

Dynamic Analysis and PD Control in a 12-Pole Active Magnetic Bearing System

This paper conducts an in-depth study on the dynamic stability and complex vibration behavior of a 12-pole active magnetic bearing (AMB) system considering gravitational effects under a PD controller. Firstly, based on electromagnetic theory and Newton’s second law, a two-degree-of-freedom control equation of the system, including PD control terms and gravitational effects, is constructed. This equation involves not only parametric excitation, quadratic nonlinearity, and cubic nonlinearity but also a more pronounced coupling effect between the magnetic poles due to the presence of gravity. Secondly, using the multi-scale method, a four-dimensional averaged equation of the system in Cartesian and polar coordinates is derived. Finally, through numerical analysis, the system’s amplitude–frequency response, motion trajectory, the relationship between energy and amplitude, and global dynamic behaviors such as bifurcation and chaos are discussed in detail. The results show that the PD controller significantly affects the system’s spring hardening/softening characteristics, excitation, amplitude, energy, and stability. Specifically, increasing the proportional gain can quickly suppress the rotor’s motion, but it also increases the system’s instability. Adjusting the differential gain can transition the system from a chaotic state to a stable periodic motion.

RBF Neural Network for Chaotic Motion Control of Collision Vibration System

Aiming at the control problem of chaotic motion of a kind of collision vibration system with gap, a chaotic motion control of collision vibration system based on RBF neural network is proposed. Firstly, the system mechanical model and chaotic motion are introduced, and then a chaotic controller based on RBFNN is designed to control and simulate the chaotic attractor. The model information of the system is not used in the control method. In this paper, the model of the system is used only to generate the input / output data of the system, and it is not used for the design of the controller. The parameters of AHGSA algorithm are set as follows: the population size is 30, the maximum number of iterations is 100,G0, a = 18, and the proportional coefficient P = 0.96. Small disturbances are applied to the controllable parameter ωof the system to suppress the chaotic motion of the system and make the system tend to stable periodic motion. In order to more clearly show the control effect of chaotic motion, the chaotic motion is controlled when the system iterates 400 times. The results show that the chaotic motion can be quickly controlled to periodic 1-1 motion, the phase diagram is a closed curve, and there is a peak in the spectrum diagram. Chaotic motion can be quickly controlled as periodic 2-2 motion, the phase diagram is two closed curves, and two obvious peaks appear in the spectrum diagram. The proposed method can effectively control the chaotic motion of the system, and the expected target can be not only the fixed point of period 1, but also other periodic orbits.

Forced vibration of an axially moving free-free beam with internal resonance

In this paper, the nonlinear governing equations of motion for an axially moving cylindrical shell with free ends which modeled as a free-free beam are established by the generalized Hamilton’s principle for the first time. The Galerkin method and multiple-scale method are adopted to solve the governing equations. Stability of the motion determined by the velocity and axial stiffness is investigated according to the linear equation firstly. Next, multiple-scale approach method is used to investigate the frequency response by adjusting the detuning parameters of the first two natural frequencies considering the internal resonance. Stability of periodic motion is studied through the trajectories of eigenvalues and phase of trajectories with adjusting the detuning parameter. Influences of its velocity and flexible force of the moving beam on the periodic motion and its stability are addressed through bifurcation diagrams and Lyapunov exponents. Correctness of the presented method for investigating dynamic behavior of the moving beam has been validated through Runge–Kutta numerical calculation. The goal of the research lies in the application of the method in the free-free moving beam and revealing the nonlinear dynamic behavior of the beam.

Research on impact vibration response of hinged fluid-conveying pipe with bilateral gap constraints

Factors such as loose supports or barriers over the fluid-conveying pipe may cause bilateral gap constraints. Flow-induced vibration will occur during pumping, which leads to collision between the pipe and the constraints. A piecewise linear spring model with large stiffness is employed to simulate the collision process between the pipe and the constraints, and numerical simulation is carried out for the collision of hinged fluid-conveying pipe with the bilateral gap constraints. The dynamic characteristics of hinged fluid-conveying pipe with pulsating internal flow excitation and bilateral gap constraints during impact vibration are studied. Influences of various parameters including frequency, average flow rate of the pulsed internal flow excitation, position of the bilateral constraints, and gap dimensions on the system are investigated. Three paths are found for the pipe system entering chaotic window from stable periodic motion: entering chaotic window after the occurrence of period-double bifurcation, transitioning into chaotic window after developing almost periodic motion, and jumping into chaotic window directly. If the distance between two constraints is closer, the pipe between the constraints will be subject to stronger constraints during the collision process, which can cause the system to enter chaos easily. Many dynamic phenomena have been observed, such as a sliding bifurcation when periodic incomplete chattering-impact developing periodic complete chattering-impact, strong centrosymmetric properties in the bifurcation diagram and the Poincaré map at the mid-point of the pipe, jumping between stable periods, period-double vibration response, inverse period-double vibration response, grazing motion, and viscous motion.

Rayleigh–Bénard convection with an immersed floating body

The paper presents the results of an experimental and numerical study of turbulent thermal convection in a rectangular box containing an extended immersed free-floating plate. Varying the values of control parameters, such as Rayleigh number, aspect ratio and vertical position of the plate, provides a wide range of possible modes, from immobile and purely periodic to stochastic. We have shown that stable periodic motions occur when the plate floats close to one of the heat exchangers. An increase in the distance between the plate and the heat exchanger breaks the periodic motion and (at moderate Rayleigh numbers) leads to a pronounced asymmetry, when the plate stays close to one of the walls most of the time, makes rare excursions to the opposite wall and immediately returns. As the Rayleigh number increases, the plate motions from one edge of the box to the other reappear, but always have an irregular character. Regarding the dependence of the system behaviour on the geometry of the box, both lower and upper limits of periodic plate motions were found in the experiments. In the numerical simulations, the upper limit was not achieved – the plate moves quasi-periodically through the chain of vortices of different signs even at the largest aspect ratio being considered. The heat-insulating floating plate provides the spatial and temporal variation of the heat flux and reduces the integral heat flux, but the reduction in heat flux depends significantly on the vertical position of the plate.

On the Orbital Stability of Periodic Motions of a Heavy Rigid Body in the Bobylev – Steklov Case

The problem of the orbital stability of periodic motions of a heavy rigid body with a fixed point is investigated. The periodic motions are described by a particular solution obtained by D. N. Bobylev and V. A. Steklov and lie on the zero level set of the area integral. The problem of nonlinear orbital stability is studied. It is shown that the domain of possible parameter values is separated into two regions: a region of orbital stability and a region of orbital instability. At the boundary of these regions, the orbital instability of the periodic motions takes place.

Complexity and Chaos Analysis for Two-Dimensional Discrete-Time Predator–Prey Leslie–Gower Model with Fractional Orders

The paper introduces a novel two-dimensional fractional discrete-time predator–prey Leslie–Gower model with an Allee effect on the predator population. The model’s nonlinear dynamics are explored using various numerical techniques, including phase portraits, bifurcations and maximum Lyapunov exponent, with consideration given to both commensurate and incommensurate fractional orders. These techniques reveal that the fractional-order predator–prey Leslie–Gower model exhibits intricate and diverse dynamical characteristics, including stable trajectories, periodic motion, and chaotic attractors, which are affected by the variance of the system parameters, the commensurate fractional order, and the incommensurate fractional order. Finally, we employ the 0–1 method, the approximate entropy test and the C0 algorithm to measure complexity and confirm chaos in the proposed system.

TeCVP: A Time-Efficient Control Method for a Hexapod Wheel-Legged Robot Based on Velocity Planning.

Addressing the problem that control methods of wheel-legged robots for future Mars exploration missions are too complex, a time-efficient control method based on velocity planning for a hexapod wheel-legged robot is proposed in this paper, which is named time-efficient control based on velocity planning (TeCVP). When the foot end or wheel at knee comes into contact with the ground, the desired velocity of the foot end or knee is transformed according to the velocity transformation of the rigid body from the desired velocity of the torso which is obtained by the deviation of torso position and posture. Furthermore, the torques of joints can be obtained by impedance control. When suspended, the leg is regarded as a system consisting of a virtual spring and a virtual damper to realize control of legs in the swing phase. In addition, leg sequences of switching motion between wheeled configuration and legged configuration are planned. According to a complexity analysis, velocity planning control has lower time complexity and less times of multiplication and addition compared with virtual model control. In addition, simulations show that velocity planning control can realize stable periodic gait motion, wheel-leg switching motion and wheeled motion and the operation time of velocity planning control is about 33.89% less than that of virtual model control, which promises a great prospect for velocity planning control in future planetary exploration missions.

Routes toward chaos in a memristor-based Shinriki circuit.

In this paper, the complex routes to chaos in a memristor-based Shinriki circuit are discussed semi-analytically via the discrete implicit mapping method. The bifurcation trees of period-m (m = 1, 2, 4 and 3, 6) motions with varying system parameters are accurately presented through discrete nodes. The corresponding critical values of bifurcation points are obtained by period-double bifurcation, saddle-node bifurcation, and Neimark bifurcation, which can be determined by the global view of eigenvalues analysis. Unstable periodic orbits are compared with the stable ones obtained by numerical methods that can reveal the process of convergence. The basins of attractors are also employed to analyze the coexistence of asymmetric stable periodic motions. Furthermore, hardware experiments are designed via Field Programmable Gate Array to verify the analysis model. As expected, an evolution of periodic motions is observed in this memristor-based Shinrik's circuit and the experimental results are consistent with that of the calculations through the discrete mapping method.

Study on nonlinear dynamic characteristics of gear system with 3D anisotropic rough tooth surface based on fractal theory

Different from the mainstream research, this study fully considers the anisotropic micro-topography of the tooth surface when studying the dynamic characteristics of gears. An anisotropic 3D fractal rough tooth surface model is proposed based on fractal geometry theory. The major advantage of the novel model is that it contains more surface micro details, especially on the tooth width, which are ignored in general studies. The backlash and time-varying meshing stiffness (TVMS) are then revised on the basis of this model. Finally, a 6-degree-of-freedom nonlinear dynamic model considering surface topography is established and the influence of fractal parameters on gear dynamics is investigated. The results show that the fractal parameters change the backlash and TVMS and have an obvious impact on the dynamics and vibration responses of the transmission system. The rough gear system enters chaos earlier and has better stability at high speeds in comparison with the smooth system. As the fractal dimension increases, the dynamic transmission error amplitude gradually decreases and the system tends to stable periodic motion. The characteristic scale coefficient yields a more moderate but opposite effect on the dynamic response of the system compared with the fractal dimension. The finding of this study can provide theoretical guidance for tooth surface machining from the perspective of tooth surface morphology.

Controlling the dynamics of a COVID‐19 mathematical model using a parameter switching algorithm

In this paper, the dynamics of an autonomous mathematical model of COVID‐19 depending on a real bifurcation parameter is controlled by the parameter switching (PS) algorithm. With this technique, it is proved that every attractor of the considered system can be numerically approximated and, therefore, the system can be determined to evolve along, for example, a stable periodic motion or a chaotic attractor. In this way, the algorithm can be considered as a chaos control or anticontrol (chaoticization) algorithm. Contrarily to existing chaos control techniques that generate modified attractors, the obtained attractors with the PS algorithm belong to the set of the system attractors. It is analytically shown that using the PS algorithm, every system attractor can be expressed as a convex combination of some existing attractors. Interestingly, the PS algorithm can be viewed as a generalization of Parrondo's paradox.

Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted Four-Body Problem

In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the so-called local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third- and fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.

On an origami structure of period-1 motions to homoclinic orbits in the Rössler system.

In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the Rössler system. The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to period-2n motions (n=1,2,…,∞). On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period-1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the period-1 motions to homoclinic orbits, the implicit map of the Rössler system is developed through the discretization of the corresponding differential equations. The Poincaré mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the Rössler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues. Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly.

Periodic motion of microscale cantilevered fluid-conveying pipes with symmetric breaking on the cross-section

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of cantilevered fluid-conveying micropipes with asymmetric cross-section. Fluid-conveying microscale pipes with ring-shaped cross-sections of O(2) symmetry are widely used in engineering. However, their cross-sections usually have a small symmetric breaking owing to manufacturing errors, which can be described by a small parameter. Hence, the dynamic behaviors of such geometrically imperfect micropipes are extremely required to be explored. This study mainly investigates the effects of the symmetric breaking parameter, flow velocity and dimensionless material length scale parameter on the dynamic behaviors of microscale cantilevered fluid-conveying micropipes. Based on the modified couple stress theory (MCST) and Hamilton's principle, the governing equation is derived, where the effects of the asymmetry of cross-section on the equation are reflected. Using the center manifold theory and normal form method, the original governing equation is rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system, and the averaging equation of the reduced system is obtained by combining the averaging method. The existence and stability of the equilibrium point of the averaging equation are studied to obtain the results of the existence and stability of periodic motions of the original system. The results demonstrate that the types of periodic motions and stability when the pipe loses stability from different places on the curve of the critical flow velocity versus mass ratio (critical flow velocity curve) may differ. Meanwhile, the effects of various parameters on these periodic motions are presented. The increase in the dimensionless material length scale parameter increases the section corresponding to stable plane periodic motion and reduces the section corresponding to stable spatial periodic motion on the critical flow velocity curve. When dimensionless material length scale parameter is given, the existence and stability of various periodic motions of the system depend on the ratio of the increment of flow velocity around the critical value to the symmetric breaking parameter. In addition, some new phenomena are found in this paper. Stable 0 planar periodic motion of the macropipe can occur when pipe loses its stability from the hysteresis part of the critical flow velocity curve. Stable 0 planar periodic motion of microscale pipes can occur when pipe loses its stability from the non-hysteresis part of the critical flow velocity curve, and then stable 0 planar periodic motion and stable π planar periodic motion can coexist. Furthermore, stable planar and stable spatial periodic motions of pipes with symmetric breaking can coexist, and stable torus motion has also been observed; this is different from the dynamics of pipes with O(2) symmetry.

Double grazing bifurcations of the non-smooth railway wheelset systems

There are numerous non-smooth factors in railway vehicle systems, such as flange impact, dry friction, creep force, and so on. Such non-smooth factors heavily affect the dynamical behavior of the railway systems. In this paper, we investigate and mathematically analyze the double grazing bifurcations of the railway wheelset systems with flange contact. Two types of models of flange impact are considered, one is a rigid impact model and the other is a soft impact model. First, we derive Poincaré maps near the grazing trajectory by the Poincaré-section discontinuity mapping (PDM) approach for the two impact models. Then, we analyze and compare the near grazing dynamics of the two models. It is shown that in the rigid impact model the stable periodic motion of the railway wheelset system translates into a chaotic motion after the grazing bifurcation, while in the soft impact model a pitchfork bifurcation occurs and the system tends to the chaotic state through a period doubling bifurcation. Our results also extend the applicability of the PDM of one constraint surface to that of two constraint surfaces for autonomous systems.

Breakup dynamics of emulsion droplet and effects of inner interface

The size and the dispersibility of droplets are the significant factors that determine the performance in applications of the emulsion droplets, which are controlled by the breakup characteristics of interfaces. In this paper, a four-phase glass capillary device is used to generate three-layer wrapped (O 1 /O 2 /W/O 3 ) emulsion droplets stably, recording the evolution of interfaces to investigate the breakup characteristics of interfaces under different flow rates. Based on the breakup form of interfaces, the phase diagram which is recorded by the flow rate ratios ( α , β ) shows the distribution of different generation modes, including Jetting Tubing (JT), Dripping Tubing (DT), Dripping (D), Narrow Jetting (NJ) and Widen Jetting (WJ). The interfaces are recorded with time in different generation modes in this study, focusing on the generation details of DJ, D, and NJ mode, where the droplet generates with the stable periodic motion. The inner droplets do not influence the generation modes but provide the interactions of the interfaces in some cases, especially on the neck diameter of the outer interface. In DT mode, a larger flow rate ratio intensifies the fluctuation of the interface, while it pulls the neck of the inner interface and the outer one to the similar position along the flow direction. In NJ mode, the relation between prediction length and the experimental one follows a similar scaling law to single droplets with various coefficients. • Generation modes of four-phase emulsion droplets can be separated in the phase diagram by the flow rate ratios ( α , β ). • The inner phases have no effect on the distribution of formation modes, while the critical boundaries varies in each mode. • Inner droplets have influences on the characteristic parameters of each mode, including the neck diameter.

Higher-Order Complex Periodic Motions in a Nonlinear, Electromagnetically Tuned Mass Damper System

In this paper, the existence and bifurcations of higher-order complex periodic motions in a nonlinear, electromagnetically tuned mass damper system are studied through period-3, period-9, and period-12 motions. For such a nonlinear system, there exist three branches of period-3 motions, one branch of period-9 motions, and one branch of period-12 motion. The corresponding stability and bifurcations of periodic motions are determined. The frequency-amplitude characteristics for bifurcation routes of such higher-order periodic motions are presented. Numerical illustrations are presented for complex periodic motions. Higher-order unstable and stable periodic motions exist in nonlinear systems, which are difficult to control for energy harvesting and vibration reduction. The unstable higher-order periodic motions make the system responses more complicated. The higher-order stable and unstable periodic motions have been studied comprehensively. This study on periodic motions can help people better understand the dynamical behaviors of an electromagnetically tuned mass damper system under different excitation frequencies. The study of higher-order periodic motion stability and bifurcations will also benefit the new development and design of vibration reduction and energy harvesting systems.