Phase Plane Plots Research Articles

Dynamic modeling and analysis of viscoelastic hard-magnetic soft actuators with thermal effects

Hard-magnetic soft materials (HMSMs) are a class of magnetoactive smart polymers capable of sustaining a high residual magnetic flux density and can undergo large actuation strains under an external magnetic excitation. Due to these exceptional characteristics, soft actuators based on HMSMs hold enormous potential for remote-controlled applications. The temperature and viscoelasticity considerably influence the performance of these materials during the operation. This work aims to develop an analytical framework for modeling the dynamic behavior of a planar hard-magnetic soft actuators (HMSA) considering temperature and viscoelastic effects. The constitutive behavior of the viscoelastic HMSA is described by employing an incompressible neo-Hookean model in conjunction with a Zener rheological model and the Rayleigh dissipation function. The dynamic governing differential equations of motion are derived by utilizing the non-conservative form of the Euler–Lagrange equation. This study delves into the collective influence of temperature and viscoelastic properties on the stability, periodicity, and resonance characteristics of nonlinear vibrations exhibited by HMSM-based planar actuator subjected to dynamic magnetic loading, presenting the findings through time-history responses, Poincaré maps, and phase-plane plots. The presented results can help in the efficient and robust design of HMSM-based actuators and can also serve as an initial step toward the development of advanced actuators exposed to dynamic loading under variable temperatures for diverse applications in the fields of engineering and medicine.

A memristive neural network with features of asymmetric coexisting attractors and large-scale amplitude control

It is a universally acknowledged fact that memristor is widely used in neural networks owing to its memory functions similar to synapses. This paper aims to construct a memristive neural network (MNN) with special dynamic behaviors and structure, which consists of four cyclic neurons and one unidirectional memristive synapse. In this study, we explored the dynamic behaviors, including asymmetric coexisting attractors and parameter-relied large-scale amplitude control. Specially, we found that there are four different types of asymmetric coexisting attractors, namely coexisting double-point (or periodic or chaotic) attractors and coexisting periodic and chaotic attractors. In order to reveal the characteristics of large-scale amplitude control, we used analysis methods such as phase plane plots and time sequences. The existence of this phenomenon is closely related to system parameters and initial values. Meanwhile, a specific circuit experiment is implemented to verify the feasibility of our designation.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

Dynamic modeling of hard-magnetic soft actuators: Unraveling the role of polymer chain entanglements, crosslinks, and finite extensibility

In recent years, there has been increasing interest in hard-magnetic soft materials (HMSMs) due to their ability to retain high residual magnetization and undergo large deformations under external magnetic loading. The performance of these materials in the dynamic mode of actuation is significantly influenced by internal properties, such as entanglements, crosslinks, and the finite extensibility of polymer chains. This article presents a theoretical framework for modeling the dynamic behavior of a hard-magnetic soft material-based planar actuator. A physics-based nonaffine material model is utilized to consider the inherent properties of polymer chain networks. The governing equation for dynamic motion is derived using Euler–Lagrange’s equation of motion for conservative systems. The devised dynamic model is utilized to examine the dynamic response, stability, periodicity, and resonance properties of a planar hard-magnetic soft actuator for different values of polymer chain entanglements, crosslinks, and finite extensibility parameters. The Poincaré maps and phase-plane plots are presented to analyze the stability and periodicity of the nonlinear vibrations of the actuator. The results reveal that transitions between aperiodic and quasi-periodic oscillations occur when the density of polymer chain entanglements and cross-linking changes. The findings from the present investigation can serve as an initial step towards the design and manufacturing of remotely controlled actuators for various futuristic applications.

Bifurcation analysis of ion-acoustic superperiodic waves in ultra-relativistic dense plasma

Abstract Bifurcation analysis of ion-acoustic solitary, periodic and superperiodic waves in ultra-relativistic quantum degenerate electron-positron-ion plasma is investigated based on Sagdeev’s pseudopotential approach. The phase plane plots for the superperiodic, periodic, and homoclinic trajectories for the dynamical system are developed. Rarefactive and compressive solitary waves are demonstrated to exist. The influence of plasma parameters on the characteristic properties of the periodic and superperiodic waves is studied. Also, the effect of different plasma parameters on the Sagdeev pseudopotential profile is investigated. The results of this study may be used to shed light on the fundamental properties of nonlinear waves in dense objects (e.g., white dwarfs and neutron stars). These results also may be beneficial in describing laboratory plasma (e.g., interaction experiments of intense laser-solid matter), and plasma applications such as microelectronic devices.

An Evaluation of Mathematical Models and Stability Analysis of Learning Based on Reaction Kinetics

Human cognition and consciousness are perhaps the most confounding mystery. Somehow it has a linkage to the process of learning and storage of short-term and long-term memory in the form of knowledge. This paper examines a brief background of early models in learning presented by Atkinson and Shriffrin (1965) and related stochastic models utilizing probability functions. Each of these learning models capture certain facets of the learning process but are ineffective in describing the physical basis in which learning occurs. For this reason, this paper explores analogous mathematical models based on reaction kinetics that have been shown to represent chemical reactions found in nature. Six learning models are presented of unitary, binary, reversible binary, reversible binary with mass action, and enzyme learning model reactions with and without decay. Preliminary analysis of time series plots, phase line diagrams, and phase plane plots were conducted to illustrate equilibrium conditions and stability of the models. Each model is examined in terms of its limitations in the philosophy and inability to capture certain elements that are understood about the learning process. Finally, this paper concludes that the feasibility of understanding behavior such as stability through the tools of applied mathematics and thereby illuminating certain layers of human cognition and learning is a useful tool in examining the suitability of a possible deterministic model that could describe the learning process. Further analysis with empirical data would validate the suitability of the presented models.

Non-linear dynamics of a stiffened composite laminated panel with debonds

Damage or flaws in aeronautical structures compromise performance and safety; therefore, it is very important to identify approaches that can be used for damage detection. In this paper, a composite laminated plate with bonded aluminium stiffeners, which is part of a representative wing structure, is analysed using the finite element method. Debonding is assumed to exist in the connection between the plate and one stiffener, an area found to be critical in previous numerical and experimental static analyses, and several dynamic analyses are carried out. First, it is found that the modes of vibration are only marginally affected by the debonds considered. Then loads that vary harmonically in time are applied to the stiffened panel and contact restrictions between damaged surfaces are imposed. Time histories, phase plane plots, frequency spectra and Poincaré sections of the response in diverse points are extracted. It is verified that these non-linear dynamics based tools allow for the detection of detachments in the connection between stiffeners and composite laminated plates. The response is analysed at several points to see if any are better suited to detect detachment. Different excitation frequencies are attempted and it is shown that excitation at half a resonance frequency has advantages.

Effects of Lamination Schemes on the Periodic Response of the Annular Sectorial Plates

A broad variety of aeronautical, biomedical, and military applications make use of structures in the form of annular sectorial plates. When in use, they are subjected to stress fluctuations and vibrations with considerable amplitudes. Geometric non-linearity is necessary for efficient design of such components, as the vibration responses obtained using the linear strain–displacement relation are often conservative and can only be utilized to approximate the genuine response. This study examined the influence of different lamination schemes on a non-linear/linear steady-state forced vibration response of annular sectorial plates. Both the modified shooting approach and the arc-lengthalgorithm technique were used for solving the governing equations of motion in the time domain. Frequency-response curves were drawn in their entirety, including both the stable and unstable portions. Sectorial plates of varying layer numbers are studied by analyzing their non-linear dynamic behavior by means of frequency response curves, periodic stress variation, phase plane plots, and frequency spectra. The number of layers shown to have an impact on steady-state responsiveness. In a hardening non-linear manner, the annular sector plates show a peak linear displacement amplitude that is significantly larger than the corresponding values in the non-linear analysis.

Non-linear dynamics and stability of rolling bearing–axle coupling system of a railway freight wagon with the change of axle rotating speed

PurposeThe purpose of this paper is to investigate the non-linear characteristics and stability of the rolling bearing–axle coupling system under the excitation of the axle/wheel speed of railway freight cars, so as to put forward a rationale for judging the vibration law and running stability of railway freight wagon.Design/methodology/approachConsidering the effects of eccentric force of the railway wagon axle, the non-linear resistance of the wagon and non-linear support forces of axle box rolling bearings, a centralized mass model of rolling bearing-axle coupling system of railway freight wagon is presented on the basis of the theory of rotor dynamics and non-linear dynamics. Then the Runge-Kutta method is adopted to solve the non-linear response of the proposed system, and numerical simulation including bifurcation diagrams, axis trajectory curves, phase plane plots, Poincaré sections and amplitude spectras are analysed when the axle rotating speed is changed. Meantime, the relation curve between Floquet multiplier and axle rotating speed, which affects the stability of coupling system, is plotted by numerical method based on the Floquet theory and method.FindingsThe simulation results of the dynamic model reveal the abundant dynamic behaviour of the coupling system when the axle rotating speed changes, including single period, quasi period, multi-period and chaotic motion, as well as the evolution law from multi-period motion to chaotic motion. And especially, the bearing–axle coupling system is in stable state with a single period motion when the axle rotating speed changes from 410 rpm to 510 rpm, in which the running speed of railway freight wagon is changed from 62 km/h to 80 km/h, the vibration displacement of the coupling system in X direction is between 1.2 mm and 1.8 mm, and the vibration displacement of the coupling system in Y direction is between 1.0 mm and 1.45 mm. Meanwhile, the influence law of axle rotating speed on the stability is obtained by comparing the bifurcation diagram and Floquet multiplier graph of the coupling system.Originality/valueThe numerical simulation data obtained in this study can provide a theoretical evidence for designing the running speed of railway freight wagon, utilizing or controlling the non-linear dynamic behaviours of the proposed coupling system, and ensuring the stability of railway freight wagons.

Dynamical behaviors, circuit design, and synchronization of a novel symmetric chaotic system with coexisting attractors

In this paper, we introduce a novel three-dimension chaotic system with strange characteristic by applying construction of a 3D chaotic circuit method. Multiple equilibria and abundant coexisting attractors exist in this system. A mathematical model is developed and detailed stability analyses for equilibrium points are executed with obtaining significant results of the period-doubling bifurcation patterns confirmed by phase plane plots and Lyapunov exponent spectra. By varying the initial value and unique controlled parameter, the double-scroll chaotic attractor is broken up into a pair of symmetric singular attractors. Then, the local basins of attraction are investigated concerning the initial condition. Next, the circuit synthesis results generated by Multisim simulation tool validate the self-excitation characteristics of this system. Finally, the feedback control technique is used to study difference synchronization of this system. Main conclusions prove the validity and reliability of difference synchronization.

Experimental investigation on vortex-induced vibration of a flexible pipe with single/bi-frequency parametric excitation

Vortex-induced vibration (VIV) of marine risers is a complicated fluid-structure interaction problem in ocean engineering. The parametric excitation caused by the heaving of the platform complicates the VIV of risers. In this paper, a series of model tests were conducted in a towing tank to investigate the VIV characteristics of a flexible pipe with single/bi-frequency parametric excitation. The linear motor was connected to the end of the pipe through a spring, and the movement of the linear motor provided various parametric excitations for the pipe. With parametric excitation, additional frequency components such as excitation frequency, the multiples of excitation frequency and the sum and difference frequencies are excited. Parametric excitation has more significant effect on VIV in IL direction than that in CF direction. With the increase of excitation amplitude, the maximum RMS strain in CF direction decreases while that in IL direction increases in most cases. The phase plane plot and motion trajectory of the flexible pipe becomes less regular with parametric excitation. When bi-frequency parametric excitation is applied, more types of additional frequencies such as sum and difference frequencies between the two excitation frequencies and other frequencies can be observed in the response, and the parametric excitation component with larger weight has more significant effect on the VIV of the flexible pipe.

Memristor synapse-coupled piecewise-linear simplified Hopfield neural network: Dynamics analysis and circuit implementation

Electromagnetic induction current is generated between the adjacent neurons in neural network caused by the existence of membrane potential difference. Memristor is the fourth fundamental electric element, which can mimic the behavior of neural synapses and simulate the electromagnetic induction effect. In this paper, in order to simplify practical implementation, the commonly used hyperbolic tangent activation function of Hopfield neural network (HNN) is replaced by a piecewise-linear function, and a simple bi-neuron-based memristor synapse-coupled HNN is proposed. Theoretical analysis and numerical simulation results illustrate that the memsirstive HNN possesses five equilibria including one unstable saddle-focus, two unstable saddle points, and two stable node points (or node-foci). Local attraction basins and phase plane plots show that the memristive HNN model behaves multistability of coexisting chaos, periodic limit cycles, and stable point attractors. Various system dynamical behaviors affected by parameters of the piecewise-linear activation function, memristor coupling strength, and initial conditions of the neurons are investigated by numerical simulations. Furthermore, an analog circuit of the memristive HNN model is simply designed, which is easier for hardware implementation as the piecewise-linear activation function can be implemented by simple op-amp-based module. Finally, the experimental results verify the correctness of the design and analyses.

Nonlinear flutter of composite laminates with curvilinear fibres using a full linearized aerodynamic theory

This paper studies nonlinear flutter of rectangular variable stiffness composite laminates (VSCLs) subjected to a low supersonic airflow. The VSCLs are formed by different individual laminae where a polymeric matrix is reinforced by curvilinear carbon fibres. The plate’s displacement field follows a Third-order Shear Deformation Theory (TSDT); a p-version finite element is employed to discretize the displacement components. A full linearized (inviscid, potential flow) aerodynamic theory is used to approximate the aerodynamic pressure. The self-exciting vibrational model is formed using the principle of virtual work. The equations of motion representing the model are solved using Newmark method. Limit cycle oscillations (LCOs) of these plates are studied using time histories, phase-plane plots and FFT spectra. A damage onset criterion is utilized to control the structural health of the laminates during nonlinear flutter. Effects of curvilinear fibre path angles on linear and nonlinear flutter behaviour, as well as the damage onset in VSCLs are investigated.

Accelerated Aging Test Modeling for Polycarbonate: Effects of Light Source

AbstractThis work addresses to propose a model to reproduce accelerated aging results for polycarbonate. Four accelerated aging conditions are studied, including several light sources. In this study, aging behaviors are monitored by the yellowing index. Obtained simulations are well‐tracked the aging data for four accelerated aging conditions. Besides, the features of accelerating environments by a phase plane plot are challenged to visualize.

Analytical Approximate Solutions of a Magnetic Spherical Pendulum: Stability Analysis

Abstract Purpose Under certain conditions, the governing equation of motion of magnetic spherical pendulum results in a cubic-quintic Duffing equation. The current work aims to achieve an analytical bounded procedure of this equation. Methods This may be accomplished by grouping nonlinear expanded frequency, Homotopy perturbation method (HPM), and Laplace transforms. Therefore, this technique helps disregard the appearance of the source of secular terms. Results To validate the obtained explanation, based on the method of Runge–Kutta of the fourth order (RK4), the numerical calculation is performed. On the other hand, the linearized stability analysis is carried out to explore stability neighbouring the fixed points. Moreover, the time history of the attained solution and the corresponding phase plane plots are obtained to expose the influence of the affecting factors in the behavior of motion. Conclusions A comparison between both solutions gives a good matching between them, which explores the worthy accuracy of the approach in question. Several phase portraits are planned toward illustrating the different types of stability and instability near the equilibrium points, where the relation between the expanded and the cyclotron frequency (that are generated by the magnetic field) is characterized for diverse standards of the azimuthal angular velocity.

Analytical solution for the motion of a pendulum with rolling wheel: stability analysis

The current work focuses on the motion of a simple pendulum connected to a wheel and a lightweight spring. The fundamental equation of motion is transformed into a complicated nonlinear ordinary differential equation under restricted surroundings. To achieve the approximate regular solution, the combination of the Homotopy perturbation method (HPM) and Laplace transforms is adopted in combination with the nonlinear expanded frequency. In order to verify the achievable solution, the technique of Runge–Kutta of fourth-order (RK4) is employed. The existence of the obtained solutions over the time, as well as their related phase plane plots, are graphed to display the influence of the parameters on the motion behavior. Additionally, the linearized stability analysis is validated to understand the stability in the neighborhood of the fixed points. The phase portraits near the equilibrium points are sketched.

Modelling of the rotational motion of 6-DOF rigid body according to the Bobylev-Steklov conditions

This work demonstrates the 3D motion of 6-DOF rigid body around the main inertial axes relative to a fixed point. The novelty of this work is to examine the influence of the gyrostatic moment vector and the magnetic field on the gyrostatic motion besides the action of the Newtonian field in accordance with the Bobylev-Steklov conditions. This motion is characterized by the possibility of placing an acceptable constraint on the governing system of motion in the form of imposing a large value of the projection of the angular velocity around the axial axis of the body. The importance of this restriction is that the governing system in addition to the first-integrals can be reduced to an appropriate system of two semi-linear differential equations from second-order and one first integral. The new analytic approximate solutions of the governing equations are achieved using the small parameter technique of Poincaré. Euler’s angles are determined to evaluate the motion at any instance. The realized solutions are represented graphically for different values ​​of the gyrostatic moment and the point charge causing the magnetic field to estimate the impact of these values ​​on the rotational motion. Moreover, the phase plane plots which illustrate the stability behavior of the body are portrayed. The significance of this work goes back to its several applications in practical life and in engineering application like satellites, submarines, and aircraft.

Simulation of nonlinear characteristics of vertical vibration of railway freight wagon varying with train speed

<abstract> <p>Traditional vertical vibration models of rail vehicle usually have high degrees of freedom, which affects the efficiency of numerical simulation. Regardless of the coupling effect between the vehicle and rail, a six degree of freedom (6DOF) quarter model with vertical displacement and rotation angle is selected as the dynamic model. This accomplished by comparing the simulation results of half model and quarter model of the railway freight wagon, and the vertical vibration characteristics of the railway freight wagon when the wagon speed is changed. To further illustrate the nonlinear vibration characteristics and evolution laws of the car body and bogie frame of the freight wagon, the bifurcation diagrams, maximum Lyapunov exponent curves, axis trajectory curves, phase plane plots, Poincaré sections, and amplitude spectras are drawn and adopted to research the dynamic responses. The simulations reveal the complex vibration behavior such as periodic, quasi-periodic, multi-periodic, and chaotic motion. Some research results can help the industry to better design the speed limits of such railway freight wagons, and deeply understand or utilize the vertical vibration law of railway freight wagon in future research.</p> </abstract>

Dynamic divergence of circular cylindrical shells conveying airflow

Experimental studies have shown that circular cylindrical shells, supported at both ends, conveying internal fluid flow can lose stability by dynamic divergence when the shell is highly pliable. This is an instability phenomenon starting as a divergence, with amplitude comparable to the shell radius, that largely constrains the flow. This results in pressure building up and reopening the shell, triggering a dynamic instability. The characteristics of dynamic divergence instability are studied in-depth in this paper for the first time to elucidate the nature and characteristics of this phenomenon. Experiments have been conducted on an elastomer (silicone rubber) thin circular cylindrical shell clamped at both ends and subjected to internal airflow. Bifurcation diagrams have been obtained by varying the flow velocity as the control parameter, exhibiting strong subcritical behaviour and large hysteresis in the flow velocity for the onset and cessation of dynamic instability. The possible existence of a chaotic component in the oscillations was firstly discerned by looking at high-resolution photos taken with a high-speed camera. The existence of chaos in the dynamic response following the initial divergence was then confirmed by means of several qualitative and quantitative measures and criteria for chaos, such as phase plane plots, Poincaré maps, power spectra, the largest Lyapunov exponent, autocorrelation, and probability density function. All these measures have shown that the chaotic nature of dynamic divergence may be intensified or weakened depending on the flow velocity. The results demonstrate that generally at higher flow velocities the oscillations display more complex nonlinear dynamics.

Diagnosis of Friction on an Unbalanced Rotor by Phase-Shift Empirical Mode Decomposition Integration and Recurrence Plot

The friction and imbalance of components in rotating machines are some of the most recurrent failures that significantly increase vibration levels, thus affecting the reliability of the devices, the shelf life of its elements, and the quality of the product. There are many publications related to the different techniques for the diagnosis of friction and imbalance. In this paper, an alternative and new phase-shift empirical mode decomposition integration (PSEMDI) method is proposed to transform the acceleration into its velocity and displacement in order to construct the phase plane and recurrence plot (RP) and analyze the friction. The focus of PSEMDI and RP is to analyze nonlinear failures in mechanical systems. In machinery fault diagnosis, the main reason for using RP is to solve the integration of acceleration, and this can be achieved by phase-shifting the intrinsic mode function (IMF) with the empirical mode decomposition (EMD). Although the highest IMFs contain some frequencies, most of them have very few; thus, by applying the phase shift identity, the integration can be carried out maintaining the nonlinearities. The proposed method is compared with Simpson’s integration and detrending with the EMD method (here referred to as SDEMDI). The experimental RP results show that the proposed method gives significantly more information about the velocity and displacement spectra and it is more stable and proportional than the SDEMDI method. The results of the proposed integration method are compared with vibration measurements obtained with an interferometer.