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Direct yaw moment control (DYC) and differential drive-assist steering (DDAS) for distributed-drive vehicles are both realized by allocating the in-wheel motor torque. To address the interference caused by overlapping control objectives, this paper proposes a multilayer control strategy that integrates DYC and DDAS, consisting of an upper controller, a coordinated decision layer, and a torque distribution layer. The upper controller, designed based on the vehicle's dynamic characteristics, incorporates an adaptive fuzzy control DYC system and a dual PID control DDAS system. The coordinated decision layer is developed utilizing a phase-plane dynamic weighting method, delineating region boundaries by applying the double-line and limit cycle methods. The torque distribution strategy is formulated considering motor peak torque and road adhesion conditions. Multi-condition joint simulation experiments indicate that the proposed multilayer control strategy, integrating the advantages of DYC and DDAS, reduces peak steering wheel torque by approximately 10%, peak yaw rate by around 25%, peak sideslip angle by roughly 29%, and peak sideslip angle rate by about 19%, significantly improving driving stability and maneuvering flexibility. [ABSTRACT FROM AUTHOR]

Objective. The purpose of the study is to develop a new method of analyzing seismic vibrations of seismically isolated structures.Method. The structure is modeled by a system with one degree of freedom with a nonlinear stiffness characteristic. In the absence of damping an analytical solution of oscillation equations on the phase plane is obtained. In this case the connection between speeds and displacements is accurate, and the law of uniformly variable motion is used to calculate displacements.Result. To take into account the attenuation, the law of change in mechanical energy at each integration interval is used.Conclusion. An example of calculating a seismically isolated system with a sliding belt and a spring with a linearly varying characteristic of stiffness is considered. The influence of dry friction and viscous damping on the displacement and acceleration of the system is analyzed.

For the sake of enhancing the handling and stability of distributed drive electric vehicles (DDEVs) under four-wheel steering (4WS) conditions, this study proposes a novel hierarchical control strategy based on a phase plane analysis. This approach involves a meticulous comparison of the stable region in the phase plane to thoroughly analyze the intricate influence of the front wheel angle, rear wheel angle, road adhesion coefficient, and longitudinal speed on the complex dynamic performances of DDEVs and to accurately determine the critical stable-state parameter. Subsequently, a hierarchical control strategy is presented as an integrated solution to achieve the coordinated control of maneuverability and stability. On the upper control level, a model predictive control (MPC) motion controller is developed, wherein the real-time adjustment of the control weight matrix is ingeniously achieved by incorporating the crucial vehicle stable-state parameter. The lower control level is responsible for the optimal torque allocation among the four wheel motors to minimize the tire load rate, thereby ensuring a sufficient tire grip margin. The optimal torque distribution for the four wheel motors is achieved using a sophisticated two-level allocation algorithm, wherein the friction ellipse is employed as a judgement condition. Finally, this developed control strategy is thoroughly validated through co-simulation utilizing the CarSim 2019 and Simulink 2020b commercial software, demonstrating the validity of the developed control strategy. The comparative results indicate that the presented controller ensures a better tracking capability to the desired vehicle state while exhibiting improved handling stability under both the double lane shifting condition and the serpentine working condition. [ABSTRACT FROM AUTHOR]

To improve the handling stability of distributed drive electric vehicles, this paper introduces an electronic differential control strategy based on the stability criterion of the phase plane method. The strategy first plots the distributed electric vehicle's center of mass side angle and center of mass angular speed on the phase plane, and then it analyzes the vehicle's stability under various working conditions to determine the parameters that ensure the stability performance. Subsequently, an adaptive fuzzy control strategy is employed to achieve fast and accurate distribution of the torque to each wheel, thereby enhancing the vehicle's stability. A joint simulation platform is constructed using MATLAB/Simulink and CarSim. A comparison with the traditional electronic differential strategy demonstrates that the proposed distribution strategy based on phase plane stability exhibited excellent stability. [ABSTRACT FROM AUTHOR]

With the rapid development of today's society, the traffic environment has become more and more complex. As an essential part of intelligent vehicles, trajectory tracking has attracted significant attention for its stability and safety. It is prone to poor accuracy and instability in extreme working conditions like high speed. In this paper, a trajectory tracking control strategy to ensure lateral stability at a high speed and low attachment limit conditions is proposed for distributed drive vehicles. The model predictive controller (MPC) was used to control the front wheel angle, and the particle swarm optimization (PSO) algorithm was designed to optimize the MPC control parameters adaptively. The sliding mode controller controls the rear wheel angle, and the vehicle instability degree is judged by analyzing the β − β ˙ phase plane. The controllers of different instability degrees are designed herein. Finally, a torque divider is designed to consider the actuation anti-slip. The designed controller is verified by Carsim and MATLAB-Simulink co-simulation. The results show that the trajectory tracking controller designed in this paper effectively improves the tracking accuracy under the premise of ensuring stability. [ABSTRACT FROM AUTHOR]

To ensure precise phase estimation within the q-axis of the phase-locked loop (PLL), integrating a filter into the q-axis loop is essential to mitigate grid-voltage harmonics. Nevertheless, the intrinsic delay characteristics of this filter impede PLL synchronization during significant grid disturbances. This study begins by developing mathematical models for three types of filters—moving-average filter (MAF) for eliminating odd harmonic components, dq-frame cascaded delayed signal cancellation (dqCDSC) filter, and notch filter (NF). Following the reduction in filter orders, a third-order nonlinear large-signal model of the PLL, incorporating an additional q-axis internal filter, is formulated. Using phase plane analysis, this study investigates the transient synchronism of the grid-following converter (GFL) and explores the influence of delay time constants from the three PLL filters on its behavior while delineating the boundaries of their basins of attraction. Theoretical findings indicate that, relative to the traditional SRF-PLL, incorporating an internal filter into the PLL compromises the transient synchronous stability of GFL. Specifically, greater filter delay time constants exacerbate the GFL's vulnerability to transient instability amid substantial grid disturbances. Hence, careful consideration is essential when using MAF-PLL and NF-PLL in situations demanding high synchronization stability. The theoretical analyses are validated using Matlab/Simulink to verify their accuracy. [ABSTRACT FROM AUTHOR]

Intelligentization is the development trend of the future automobile industry. Intelligentization requires that the dynamic control of the vehicle can complete the trajectory tracking according to the trajectory output of the decision planning the driving state of the vehicle and ensure the driving safety and stability of the vehicle. However, trajectory limit planning and harsh road conditions caused by emergencies will increase the difficulty of trajectory tracking and stability control of unmanned vehicles. In view of the above problems, this paper studies the trajectory tracking and stability control of distributed drive unmanned vehicles. This paper applies a hierarchical control framework. Firstly, in the upper controller, an adaptive prediction time linear quadratic regulator (APT LQR) path following algorithm is proposed to acquire the desired front-wheel-steering angle considering the dynamic stability performance of the tires. The lateral stability of the DDAUV is determined based on the phase plane, and the sliding surface, in the improved sliding mode control (SMC), is further dynamically adjusted to obtain the desired additional yaw moment for coordinating the path following and lateral stability. Then, in the lower controller, considering the slip and the working load of four tires, a comprehensive cost function is established to reasonably distribute the driving torque of four in-wheel motors (IWMs) for producing the desired additional yaw moment. Finally, the proposed control algorithm is verified by the hardware-in-the-loop (HIL) experiment platform. The results show the path following and lateral stability can be coordinated effectively under different driving conditions. [ABSTRACT FROM AUTHOR]

In order to improve transport efficiency and enhance the safety of the crane system, it is required that the trolley can reach the desired position fast enough, while the load swing needs to be within the bounded limits. To achieve these goals, a novel double-step acceleration strategy with equal acceleration amplitude and acceleration time, and unequal acceleration amplitude and acceleration time is proposed for input shaping anti-sway control based on phase plane trajectory planning. Specifically, the desired double-step acceleration signal is designed using an input shaper, and then the phase plane trajectory of the load swing angle is planned using the phase plane method. The geometric constraints of the phase plane trajectory and the physical constraints of the crane allow the amplitude and switching time of the double-step acceleration to be solved. The final simulation results show that the proposed control method achieves an effective anti-sway effect on the container crane. [ABSTRACT FROM AUTHOR]

Lateral control is an essential safety control technology for autonomous vehicles, but the effectiveness of lateral control technology relies heavily on the precision of vehicle motion state judgements. In order to achieve accurate judgements of the vehicle motion state and to improve the control effectiveness of vehicle maneuverability and the stability controller, this paper starts with an analysis of phase plane stability. A simulation analysis is conducted to investigate the effect of the vehicle steering angle of the front wheels, the longitudinal velocity, and the tire–road adhesion coefficient on the boundary of the stability area. The stable area of the phase plane was partitioned using the proposed novel quadrilateral method, and we established a stability area regression model using machine learning methods. We analyzed the inherent connection between the lateral tire forces and the principles of vehicle maneuverability and stability control, indirectly combining the characteristics of tire forces with vehicle maneuverability and stability control. An allocation algorithm for maneuverability and stability control was designed. A co-simulation indicates that the vehicle stability controller not only accurately assesses the motion state of the vehicle but also demonstrates a considerably better performance in maneuverability and stability control compared to a controller using the traditional partitioning method of stable regions. The suggested allocation method enhances vehicle maneuverability and stability by enabling a seamless transition between the two and improving the effectiveness of stability control. [ABSTRACT FROM AUTHOR]

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For sample return missions from deep space, a thin aeroshell capsule was proposed aimed at mitigating the aerodynamic heating during its reentry into the earth's atmosphere. The functional requirements of the capsule mandate that it aerodynamically decelerate sufficiently and land in the appropriate attitude without a parachute. To evaluate its motion attitude during transonic conditions in a wind tunnel, we employed the 1-degree of freedom (1-DOF) oscillation method. Furthermore, to predict its motion during an actual flight based on the wind tunnel test results, we examined the impact of the moment of inertia (MOI) on the motion. In all the MOI cases investigated, the limit cycle oscillation (LCO) was observed with a constant amplitude of angle. The variations in the MOI resulted in differences in the amplitude of the angle and time to reach the LCO. This study focused on the growth phase until the LCO and evaluated the characteristics on the phase plane. Consequently, a gradual approach to acceleration attributed to the static moment was observed in the growth history. A new modeling approach, centered on the nullcline during the growth phase, was adopted. Here, an approximation using a third-order function of angular velocity was applied to reproduce the nullcline obtained from the experiment. Upon non-dimensionalization of this model equation using the motion period, it was confirmed that the non-dimensional parameters were affected by the motion period, while this model focused solely on the growth phase. Furthermore, upon parameter tuning to replicate the amplitude of the angle and time to reach the LCO, and using these parameters, we predicted the impact of the MOI. The predicted results qualitatively reproduced larger MOI leading to larger amplitudes of angle and longer times to reach the LCO, which was consistent with the experimental results.

For the sake of enhancing the handling and stability of distributed drive electric vehicles (DDEVs) under four-wheel steering (4WS) conditions, this study proposes a novel hierarchical control strategy based on a phase plane analysis. This approach involves a meticulous comparison of the stable region in the phase plane to thoroughly analyze the intricate influence of the front wheel angle, rear wheel angle, road adhesion coefficient, and longitudinal speed on the complex dynamic performances of DDEVs and to accurately determine the critical stable-state parameter. Subsequently, a hierarchical control strategy is presented as an integrated solution to achieve the coordinated control of maneuverability and stability. On the upper control level, a model predictive control (MPC) motion controller is developed, wherein the real-time adjustment of the control weight matrix is ingeniously achieved by incorporating the crucial vehicle stable-state parameter. The lower control level is responsible for the optimal torque allocation among the four wheel motors to minimize the tire load rate, thereby ensuring a sufficient tire grip margin. The optimal torque distribution for the four wheel motors is achieved using a sophisticated two-level allocation algorithm, wherein the friction ellipse is employed as a judgement condition. Finally, this developed control strategy is thoroughly validated through co-simulation utilizing the CarSim 2019 and Simulink 2020b commercial software, demonstrating the validity of the developed control strategy. The comparative results indicate that the presented controller ensures a better tracking capability to the desired vehicle state while exhibiting improved handling stability under both the double lane shifting condition and the serpentine working condition.

In this study, a model of two nonlinear spring masses subjected to a smooth friction-velocity curve is examined. The second order governing system of motion is obtained in view of the system friction forces. This system is transformed to another suitable one of first order to investigate the points of equilibrium using Hurwitz's theory. The excitation and critical stick-slip (SS) speeds are determined in accordance with the characteristic equation for the Jacobian matrix of the system. The stability and behavior of the system motion, along with the behavior of the SS movement, are examined. The effect of various parameters, such as excitation speed, damping and fraction coefficients, linear and nonlinear stiffness of springs, and masses that affect the motion and stability of the system is analyzed and studied. This research has numerous practical applications in a variety of industries, including airports, modern car brakes, well drilling, explaining and understanding seismic events, mechanical transportation, friction in bridges, and civil systems, which all exhibit impacted friction in dynamics and stick-slip phenomena. [ABSTRACT FROM AUTHOR]

To improve the handling stability of distributed drive electric vehicles, this paper introduces an electronic differential control strategy based on the stability criterion of the phase plane method. The strategy first plots the distributed electric vehicle’s center of mass side angle and center of mass angular speed on the phase plane, and then it analyzes the vehicle’s stability under various working conditions to determine the parameters that ensure the stability performance. Subsequently, an adaptive fuzzy control strategy is employed to achieve fast and accurate distribution of the torque to each wheel, thereby enhancing the vehicle’s stability. A joint simulation platform is constructed using MATLAB/Simulink and CarSim. A comparison with the traditional electronic differential strategy demonstrates that the proposed distribution strategy based on phase plane stability exhibited excellent stability.

With the rapid development of today’s society, the traffic environment has become more and more complex. As an essential part of intelligent vehicles, trajectory tracking has attracted significant attention for its stability and safety. It is prone to poor accuracy and instability in extreme working conditions like high speed. In this paper, a trajectory tracking control strategy to ensure lateral stability at a high speed and low attachment limit conditions is proposed for distributed drive vehicles. The model predictive controller (MPC) was used to control the front wheel angle, and the particle swarm optimization (PSO) algorithm was designed to optimize the MPC control parameters adaptively. The sliding mode controller controls the rear wheel angle, and the vehicle instability degree is judged by analyzing the β − β˙ phase plane. The controllers of different instability degrees are designed herein. Finally, a torque divider is designed to consider the actuation anti-slip. The designed controller is verified by Carsim and MATLAB-Simulink co-simulation. The results show that the trajectory tracking controller designed in this paper effectively improves the tracking accuracy under the premise of ensuring stability.

Intelligentization is the development trend of the future automobile industry. Intelligentization requires that the dynamic control of the vehicle can complete the trajectory tracking according to the trajectory output of the decision planning the driving state of the vehicle and ensure the driving safety and stability of the vehicle. However, trajectory limit planning and harsh road conditions caused by emergencies will increase the difficulty of trajectory tracking and stability control of unmanned vehicles. In view of the above problems, this paper studies the trajectory tracking and stability control of distributed drive unmanned vehicles. This paper applies a hierarchical control framework. Firstly, in the upper controller, an adaptive prediction time linear quadratic regulator (APT LQR) path following algorithm is proposed to acquire the desired front-wheel-steering angle considering the dynamic stability performance of the tires. The lateral stability of the DDAUV is determined based on the phase plane, and the sliding surface, in the improved sliding mode control (SMC), is further dynamically adjusted to obtain the desired additional yaw moment for coordinating the path following and lateral stability. Then, in the lower controller, considering the slip and the working load of four tires, a comprehensive cost function is established to reasonably distribute the driving torque of four in-wheel motors (IWMs) for producing the desired additional yaw moment. Finally, the proposed control algorithm is verified by the hardware-in-the-loop (HIL) experiment platform. The results show the path following and lateral stability can be coordinated effectively under different driving conditions.

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. [ABSTRACT FROM AUTHOR]

We investigated the thermal buckling temperature and nonlinear free vibration of functionally graded fiber-reinforced composite laminated (FG-FRCL) beams. The governing nonlinear partial differential equations were derived from the Euler–Bernoulli beam theory, accounting for the von Kármán geometrical nonlinearity. Such equations were then reduced to a single equation by neglecting the axial inertia. Thus, the Galerkin method was applied to discretize the governing nonlinear partial differential equation in the form of a nonlinear ordinary differential equation, which was then solved analytically according to the He's variational method. Three different boundary conditions were selected, namely simply, clamped and clamped-free supports. We also investigated the effect of power-index, lay-ups, and uniform temperature rise on the nonlinear natural frequency, phase trajectory and thermal buckling of FG-FRCL beams. The results showed that FG-FRCL beams featured the highest fundamental frequency, whereas composite laminated beams were characterized by the lowest fundamental frequency. Such nonlinear frequencies increase for an increased power index and a decreased temperature. Finally, it was found that FG-FRCL beams with [0/0/0] lay-ups featured the highest nonlinear natural frequency and the highest thermal buckling temperature, followed by [0/90/0] and [90/0/90] lay-ups, while a [90/90/90] lay-up featured the lowest nonlinear natural frequency and critical buckling temperature. [ABSTRACT FROM AUTHOR]

To address the time-optimal trajectory planning (TOTP) problem with joint jerk constraints in a Cartesian coordinate system, we propose a time-optimal path-parameterization (TOPP) algorithm based on nonlinear optimization. The key insight of our approach is the presentation of a comprehensive and effective iterative optimization framework for solving the optimal control problem (OCP) formulation of the TOTP problem in the (s , s ˙) -phase plane. In particular, we identify two major difficulties: establishing TOPP in Cartesian space satisfying third-order constraints in joint space, and finding an efficient computational solution to TOPP, which includes nonlinear constraints. Experimental results demonstrate that the proposed method is an effective solution for time-optimal trajectory planning with joint jerk limits, and can be applied to a wide range of robotic systems. [ABSTRACT FROM AUTHOR]

The Chen-Lee -Liu model has many applications in assorted fields, particularly in the study of nonlinear dynamics, chaos theory, circuit design, signal processing, secure communications, encryption and decryption of chaotic signals, as well as cryptography. The modified extended auxiliary equation mapping method has been applied to the Chen-Lee-Liu model in this article and explores new wave profiles, such as singular periodic solutions, periodic solutions, and kink-type soliton solutions. The complex wave conversion is considered to make a simple differential equation. Three- and two-dimensional images are plotted using Mathematica and MATLAB, and their dispersion and nonlinearity effects are discussed. We also discuss the bifurcation analysis of the studied model. The stability of the equilibrium points is studied, and the phase portrait of the system is presented graphically. The obtained wave profiles might play an important role in telecommunication systems, fiber optics, and nonlinear optics.

The rapid development of computer biometrics over the past 2-3 decades is largely due to the development and widespread introduction into clinical practice of new methods of studying human body health, including pulse methods. It is possible to judge changes in hemodynamic characteristics, heart rate, and blood flow rate in the studied part of the body based on the parameters of the pulse wave signal. At the same time, the physical processes of the formation of the pulse wave shape have not been fully studied, although the number of biophysical models of blood circulation is quite significant. The development of such a model will allow us to effectively apply modern developments in digital signal processing to the pulse wave and increase its diagnostic value. A qualitative model of pulse signal can be entrusted to the development of the base unit of the biotechnical system as a type of edge device. The work is devoted to the improvement of methods of rapid diagnosis of the cardiovascular system based on the analysis of model pulse-grams. An adequate mathematical model of the pulse wave, which corresponds to real pulse signals in different states of the human body and contains mathematical relationships between the main parameters of pulse-grams, has been refined. The algorithm of express diagnostics with the established criteria of the analysis of pulsograms is offered.

There are presented the results of analysis of some irregularities in the series of displacement coordinates on the Earth’s surface for the geographic North Pole in order to relate them temporally to some global jerks in the geomagnetic field. The calculations were made using the International Earth Rotation and Reference Systems Service (IERS) data on the average daily motions of the North Pole from 1962 to 2021 and the information on global jerks in the geomagnetic field measured by the magnetic observatories throughout the world. The identification and analysis of the irregularities in the pole displacement along the Earth’s surface were performed by the Fourier Method and wavelet methods for time series analysis, methods of fixed threshold determination and minimax analysis of the non-Gaussian noise, phase and pseudo-phase space methods, and stroboscopic methods of construction of Poincare maps.The analysis of local Fourier spectra and wavelet-spectra reveals irregularities in the displacement of the North Pole in the time intervals from 1967.04.09 to 1967.11.30, from 1974.03.29 to 1974.09.12, and from 2005.11.03 to 2006.03.07. It is supposed that energy reconstructions which caused short-term pole deviations from the trajectory and starting to follow its previous trajectory at the return points are related to the specific features of oscillatory interactions in the process of the translational-rotational motion of the Earth’s in the Solar System. The time of marking specific points on graphs of the displacement of the pole along the surface of the Earth in 1967 and 1974 is ahead of global geomagnetic jerk occurrence times.

Basic Oxygen Furnace (BOF) steelmaking is an important way for steel production. Correctly recognizing different blowing periods and abnormal refining states is significant to ensure normal production process, while accurately predicting the end-point time helps to increase the first-time qualification rate of molten steel. Since the decarburization products CO and CO2 are the main compositions of off-gas, information of off-gas is explored for BOF steelmaking control. However, the problem is that most of the existing research directly gave the proportions of CO and CO2 as model input but barely considered the variation information of off-gas to describe the production state. At the same time, the off-gas information can be expected to recognize the last blowing period and predict the end-point time earlier than the existing methods that are based on sub-lance or furnace flame image, but little literature makes an attempt. Therefore, this work proposes a new method based on functional data analysis (FDA) and phase plane (PP), defined as FDA-PP, to describe and predict the BOF steelmaking process from the metallurgical dynamics viewpoint. This method extracts the total proportion of CO and CO2 and its first-order derivative as dynamics features of steelmaking process via FDA, which indicate the reaction velocity and acceleration of decarburization reaction, and describes the evolution of dynamics features via PP. Then, the FDA-PP method extracts the features of phase trajectories for production state recognition and end-point time prediction. Experiments on a real production dataset demonstrate that the FDA-PP method has higher production state recognition accuracy than the classical phase space, SVM, and BP methods, which is 87.78% for blowing periods of normal batches, 90.94% for splashing anomaly, and 81.29% for drying anomaly, respectively. At the same time, the FDA-PP method decreases the mean relative prediction error (MRE) of the end-point time prediction for abnormal batches by about 10% compared with the SVM and BP methods. [ABSTRACT FROM AUTHOR]

The article considers dynamical systems on the plane, defined by continuous piecewise smooth vector fields. Such systems are used as mathematical models of real processes with switching. An important task is to find the conditions for the generation of periodic trajectories when the parameters change. The paper describes the bifurcation of the birth of a periodic trajectory from the loop of the separatrix of a sewn saddle-node — an analogue of the classical bifurcation of the separatrix loop of a saddle-node of a smooth dynamical system. Consider a one-parameter family $\{ X_\varepsilon \} $ of continuous piecewise-smooth vector fields on the plane. Let $z^0 $ be a point on the switching line. Let's choose the local coordinates $x,y$ in which $z^0 $ has zero coordinates, and the switching line is given by the equation $y = 0$. Let the vector field $X_0 $ in a semi-neighborhood $y \ge 0$ ($y \le 0$) coincide with a smooth vector field $X_0^ + $ ($X_0^ - $), for which the point $z^0 $ is a stable rough node (rough saddle), and the proper subspaces of the matrix of the linear part of the field in $z^0 $ do not coincide with the straight line $y = 0$. The singular point $z^0 $ is called a sewn saddle-node. There is a single trajectory $L_0 $ that is $\alpha $-limit to $z^0 $ — the outgoing separatrix of the point $z^0 $. It is assumed that $L_0 $ is also $\omega $-limit to $z^0$, and enters $z^0 $ in the leading direction of the node of the field $X_0^ + $. For generic family, when the parameter $\varepsilon $ changes, the sewn saddle-node either splits into a rough node and a rough saddle, or disappears. In the paper it is proved that in the latter case the only periodic trajectory of the field $X_\varepsilon $ is generated from the contour $L_0 \cup \{ z^0 \} $ — a stable limit cycle.

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.

We investigated the thermal buckling temperature and nonlinear free vibration of functionally graded fiber-reinforced composite laminated (FG-FRCL) beams. The governing nonlinear partial differential equations were derived from the Euler–Bernoulli beam theory, accounting for the von Kármán geometrical nonlinearity. Such equations were then reduced to a single equation by neglecting the axial inertia. Thus, the Galerkin method was applied to discretize the governing nonlinear partial differential equation in the form of a nonlinear ordinary differential equation, which was then solved analytically according to the He’s variational method. Three different boundary conditions were selected, namely simply, clamped and clamped-free supports. We also investigated the effect of power-index, lay-ups, and uniform temperature rise on the nonlinear natural frequency, phase trajectory and thermal buckling of FG-FRCL beams. The results showed that FG-FRCL beams featured the highest fundamental frequency, whereas composite laminated beams were characterized by the lowest fundamental frequency. Such nonlinear frequencies increase for an increased power index and a decreased temperature. Finally, it was found that FG-FRCL beams with [0/0/0] lay-ups featured the highest nonlinear natural frequency and the highest thermal buckling temperature, followed by [0/90/0] and [90/0/90] lay-ups, while a [90/90/90] lay-up featured the lowest nonlinear natural frequency and critical buckling temperature.

To address the time-optimal trajectory planning (TOTP) problem with joint jerk constraints in a Cartesian coordinate system, we propose a time-optimal path-parameterization (TOPP) algorithm based on nonlinear optimization. The key insight of our approach is the presentation of a comprehensive and effective iterative optimization framework for solving the optimal control problem (OCP) formulation of the TOTP problem in the (s,s˙)-phase plane. In particular, we identify two major difficulties: establishing TOPP in Cartesian space satisfying third-order constraints in joint space, and finding an efficient computational solution to TOPP, which includes nonlinear constraints. Experimental results demonstrate that the proposed method is an effective solution for time-optimal trajectory planning with joint jerk limits, and can be applied to a wide range of robotic systems.

Nonlinear synaptic integration in dendrites is a fundamental aspect of neural computation. One such key mechanism is the Ca2+ spike at the apical tuft of pyramidal neurons. Characterized by a plateau potential sustained for tens of milliseconds, the Ca2+ spike amplifies excitatory input, facilitates somatic action potentials (APs), and promotes synaptic plasticity. Despite its essential role, the mechanisms regulating it are largely unknown. Using a compartmental model of a layer 5 pyramidal cell (L5PC), we explored the plateau and termination phases of the Ca2+ spike under input current perturbations, long-step current-injections, and variations in the dendritic high-voltage-activated Ca2+ conductance (that occur during cholinergic modulation). We found that, surprisingly, timed excitatory input can shorten the Ca2+ spike duration while inhibitory input can either elongate or terminate it. A significant elongation also occurs when the high-voltage-activated Ca2+ channels (CaHVA) conductance is increased. To mechanistically understand these phenomena, we analyzed the currents involved in the spike. The plateau and termination phases are almost exclusively controlled by the CaHVA inward current and the Im outward K+ current. We reduced the full model to a single-compartment model that faithfully preserved the responses of the Ca2+ spike to interventions and consisted of two dynamic variables: the membrane potential and the K+-channel activation level. A phase-plane analysis of the reduced model provides testable predictions for modulating the Ca2+ spike and reveals various dynamical regimes that explain the robust nature of the spike. Regulating the duration of the Ca2+ spike significantly impacts the cell synaptic-plasticity window and, as we show, its input-output relationship. [ABSTRACT FROM AUTHOR]

Basic Oxygen Furnace (BOF) steelmaking is an important way for steel production. Correctly recognizing different blowing periods and abnormal refining states is significant to ensure normal production process, while accurately predicting the end-point time helps to increase the first-time qualification rate of molten steel. Since the decarburization products CO and CO2 are the main compositions of off-gas, information of off-gas is explored for BOF steelmaking control. However, the problem is that most of the existing research directly gave the proportions of CO and CO2 as model input but barely considered the variation information of off-gas to describe the production state. At the same time, the off-gas information can be expected to recognize the last blowing period and predict the end-point time earlier than the existing methods that are based on sub-lance or furnace flame image, but little literature makes an attempt. Therefore, this work proposes a new method based on functional data analysis (FDA) and phase plane (PP), defined as FDA-PP, to describe and predict the BOF steelmaking process from the metallurgical dynamics viewpoint. This method extracts the total proportion of CO and CO2 and its first-order derivative as dynamics features of steelmaking process via FDA, which indicate the reaction velocity and acceleration of decarburization reaction, and describes the evolution of dynamics features via PP. Then, the FDA-PP method extracts the features of phase trajectories for production state recognition and end-point time prediction. Experiments on a real production dataset demonstrate that the FDA-PP method has higher production state recognition accuracy than the classical phase space, SVM, and BP methods, which is 87.78% for blowing periods of normal batches, 90.94% for splashing anomaly, and 81.29% for drying anomaly, respectively. At the same time, the FDA-PP method decreases the mean relative prediction error (MRE) of the end-point time prediction for abnormal batches by about 10% compared with the SVM and BP methods.

Annihilating of unwanted vibrations of mechanical systems is one of the necessary requirements of designers over years. This current paper investigates dynamical behavior of a clamped-clamped beam attached to a Nonlinear Energy Sink (NES) under the harmonic force with different amplitudes. The nonlinear absorber consists of a linear damper and an essentially nonlinear stiffness. The considered system is a simple model of offshore structures subjected to the external fluid flow. In order to simulate the primary system, Euler-Bernoulli beam theory has been applied. Occurrence of Strongly Modulated Responses (SMR) under sinusoidal load reflects the efficiency of the absorber. Analytical approach (complexification-averaging method) and numerical solution (Runge-Kutta method) are applied for comparison and validation of the results. The boundaries of occurrence of Saddle-node and Hopf bifurcations and relaxation oscillations are obtained. The results show that the NES damping has substantial influence on the NES performance. Furthermore, by approaching the NES location to the beam supports, the probability of occurrence of the SMR decreases and the transient responses of the system lasts longer.

This study develops a high-speed protection method for a modular multilevel converter (MMC)-based DC grids. The method utilizes the single-end derivative of common-mode (CM) and differential-mode (DM) currents to construct a protection phase-plane that divides different DC fault types into different regions. When the protection starts, the first sampling point that enters the fault regions of the protection phase-plane has the largest value. When five sampled data satisfy the corresponding fault conditions, the protection signal will be sent. This scheme ensures the rapidity of protection. CM and DM circuits of MMC, DC transmission line, and fault transition resistance under different fault types are derived following the principle of modulus decomposition. Finally, the synthesized equivalent circuits are obtained. On this basis, the expressions of CM and DM currents are derived to analyze the characteristics of fault currents on the protection phase-plane and provide the rules for distinguishing different fault types. A four-terminal MMC-based DC grid simulation model is established in PSCAD/EMTDC. The rapidness and reliability of the proposed protection method are then verified using different simulation cases. The proposed method is also validated on a scaled-down hardware setup in the laboratory. [ABSTRACT FROM AUTHOR]

A method for controlling a shale-shaker electrical drive system that enables trajectory changes of the shaker frame to be minimized is described. The control method is based on using a self-synchronization effect. The trajectories of the shaker frame are determined indirectly by the phase shift angles of the unbalanced rotation. The phase shift angle of unbalanced vibration exciter rotation was controlled by changing the effective phase stresses. The proposed control method could significantly expand the technological capabilities of drilling shale shakers. [ABSTRACT FROM AUTHOR]

Under consideration is the mathematical model of a clock frequency generator in which some high-frequency oscillations of a movable electrode are excited by a sequence of concentrated electrostatic pulses; wherein the times of pulse action are coordinated with the oscillations of the movable electrode by analogy with the well-known theory of a trigger clock. The results of studying the mathematical model provide a fairly complete understanding of the properties of the oscillations that are excited in the generator. [ABSTRACT FROM AUTHOR]

The results of computational modeling of relay control of a vibrating screen drive system in the phase plane are presented. [ABSTRACT FROM AUTHOR]

Under consideration is some mathematical model of a clock frequency generator, a device of the MEMS class (microelectromechanical systems). We numerically study the solution of the corresponding second-order ordinary differential equation with nonlinear right-hand side and show that there is a region of the model parameters in which the bounded solutions tend to a stable limit cycle in the phase plane and, therefore, the periodic oscillations are stable with respect to the external perturbations. To determine the boundary of the region, we use the parameter continuation method of the solution of the boundary value problem defining the limit cycle. The model leads to the numerical identification of the region of generator operability. [ABSTRACT FROM AUTHOR]

The problem of vibrations cause the collapse of the systems and cause significant economic losses if not avoided. For this reason, the researchers have dealt with this issue in all years, but the problem is not over. In this paper will highlight the problem of controlling the vibrations resulting from fluid flow inside the pipes, in addition to the study of reducing the vibrations of these pipes. The ideal control is designed based on, the state space theory and according to the linear behavior of time, because the stability of the pipes are found from linear change. Different factors were studied on the controller's performance and then the optimal factors were obtained to obtain the best performance. The research included the study of the response and the natural frequency and the study of the dynamic behavior of different types of stabilization of the pipes in the presence with no hydraulic damping (active control) and monitoring the response and stability of each case of stabilization. Where, the control investigation include using the analytical solution of general equation of motion for pipe by using state space technique. There, the parameters studied for active control pipe vibration included the effect of hydraulic damper position, base width of hydraulic damper, damping, and the flow Pressure, on the pipe vibration response, with various pipe boundary conditions supported. [ABSTRACT FROM AUTHOR]

The article aims to document PhaPl that’s a teaching software to plot and research phase portraits of autonomous systems of 2 differential equations on a plane. Interactive computer teaching materials allow to demonstrate tasks describing large number of states of investigated systems clearly and involving students into solving. Plotting and research of phase portraits of autonomous system of 2 differential equations is an important task in “Differential equations” course and other courses that use dynamic systems. The software allows to visualize phase portraits and to perform analysis easily. Plotting of phase portraits needs a lot of routine computations. The software allows teacher to focus on analytical research of autonomous systems of 2 differential equations. The software supports linear and nonlinear autonomous systems of 2 differential equations. The software differs is very different compared with previously known programs: it has very easy graphical user interface and it gives clarity because it demonstrates all steps of solution. To get the full solution, it is enough to just enter a system to research. Initial conditions to plot phase trajectories are chosen automatically. Graphical representation of the phase plane is interactive and allows user to draw additional trajectories with specified initial conditions by mouse hovering over the phase plane. The software is based on popular Free Software (Maxima, Qt4, LaTeX) and it is Free Software itself, thus it is accessible to wider audience, including online students. The software is portable and works on Windows and Linux operating systems. The article describes advantages, disadvantages and peculiar properties of the software, and some aspects of teaching experience. The software is deployed in Moscow State University of Economics, Statistics, and Informatics (MESI), Lomonosov Moscow State University (MSU) since 2013, and in Plekhanov Russian University of Economics since 2016. Further development will be focused on reduction of binary size of the software, portability improvements, and number of analytical results showed.

Today the problem of analysis of biomedical signals with chaotic properties is an important task. Its decision is important for the recognition of signals with varying severity of non-linear component. The paper presents a method for biomedical signal processing, tapping on the pseudo-phase portrait differences from the normal rhythm of some of its violations. The algorithm is intended for medical computer systems and implemented in MATLAB software environment.

Blood cell platelets form aggregates upon vessel wall injury. Under certain conditions, a disintegration of the platelet aggregates, called “reversible aggregation”, is observed in vitro. Previously, we have proposed an extremely simple (two equations, five parameters) ordinary differential equation-based mathematical model of the reversible platelet aggregation. That model was based on mass-action law, and the parameters represented probabilities of platelet aggregate formations. Here, we aimed to perform a nonlinear dynamics analysis of this mathematical model to derive the biomedical meaning of the model’s parameters. The model’s parameters were estimated automatically from experimental data in COPASI software. Further analysis was performed in Python 2.7. Contrary to our expectations, for a broad range of parameter values, the model had only one steady state of the stable type node, thus eliminating the initial assumption that the reversibility of the aggregation curve could be explained by the system’s being near a stable focus. Therefore, we conclude that during platelet aggregation, the system is outside of the influence area of the steady state. Further analysis of the model’s parameters demonstrated that the rate constants for the reaction of aggregate formation from existing aggregates determine the reversibility of the aggregation curve. The other parameters of the model influenced either the initial aggregation rate or the quasi-steady state aggregation values.

Tire longitudinal forces of electrics vehicle with four in-wheel-motors can be adjusted independently. This provides advantages for its stability control. In this paper, an electric vehicle with four in-wheel-motors is taken as the research object. Considering key factors such as vehicle velocity and road adhesion coefficient, the criterion of vehicle stability is studied, based on phase plane of sideslip angle and sideslip-angle rate. To solve the problem that the sideslip angle of vehicles is difficult to measure, an algorithm for estimating the sideslip angle based on extended Kalman filter is designed. The control method for vehicle yaw moment based on sliding-mode control and the distribution method for wheel driving/braking torque are proposed. The distribution method takes the minimum sum of the square for wheel load rate as the optimization objective. Based on Matlab/Simulink and Carsim, a cosimulation model for the stability control of electric vehicles with four in-wheel-motors is built. The accuracy of the proposed stability criterion, the algorithm for estimating the sideslip angle and the wheel torque control method are verified. The relevant research can provide some reference for the development of the stability control for electric vehicles with four in-wheel-motors.

This study investigates the discretisation effects of the zero‐order holder (ZOH) on DC–DC buck converters under the direct ON/OFF control of sliding mode (SM), and the two core issues of stability and periodicity for the discrete system are focused on. The optimal parameter for the SM controller is first given on the basis of stability and response speed in the continuous‐time domain, following its ZOH discretisation. Restricted by the existence condition of discrete SM, the guaranteed stability is investigated and compared with that of the continuous system in the phase plane, proving that the sampling time of ZOH affects its stability region. Meanwhile, the periodic dynamics, including the system state and the symbolic sequence of switching control law, is further studied via analysing the steady limit cycle of state in phase plane; and the existence condition of multi‐periodicity is also deduced. Innovatively this study establishes the influence relationship of the periodic dynamics with the performances of the steady‐state error of output voltage and switching frequency. Simulation and experiment results are presented to validate the theoretical and practical value of the proposed research. [ABSTRACT FROM AUTHOR]

The path of attitude motion for a rigid body in fully nonlinear Lyapunov periodic orbits is planned in the restricted three-body problem. The attitude dynamic characteristics in phase plane are analyzed, and the path constraint conditions are proposed to avoid the divergence of phase trajectories. Then, the path planning method for the three-body problem is proposed by employing the Pseudospectral method. Based on this, solving strategy of attitude-orbit coupled periodic solutions is obtained by adding event constraint conditions. Compared with previous work, the proposed method effectively avoids the divergence of attitude motion and saves a large amount of work on determining the initial guess. To verify the effectiveness of the proposed method, a large number of planning work are carried out. Besides, the paper addresses the question that how translational motions and configuration of the spacecraft affect the planned results. [ABSTRACT FROM AUTHOR]