Phase Plane Portraits Research Articles

Chaos in coupled lateral-longitudinal vibration of electrostatically actuated microresonators

The precision and stability of the microresonators can be affected by their nonlinear vibrational behaviors. So, the main objective of this paper is to predict the nonlinear dynamic phenomena in the electrostatically actuated microresonators. In the framework of the Von Karman's theory and the Euler-Bernoulli beam model, the nonlinear governing equations of motion are developed using the Hamilton's principle. Also, the coupling of lateral and longitudinal vibrations is considered. The governing partial differential equations are discretized by means of the single-mode Galerkin's method and solved using the Runge–Kutta method. The influences of various system parameters such as the actuation frequency, lateral-longitudinal coupling, amplitude of the ac voltage and the flexural rigidity on the response of the microresonator are investigated. The bifurcation diagrams, power spectra analysis, Poincare’ map, phase plane portrait, maximum Lyapunov exponent and the Melnikov function are employed to inspect the chaotic behavior of the microresonator. The results indicate that, the vibrational behaviors of the microresonator include 2T-periodic, 4T-periodic, 7T-periodic, quasi-periodic and chaotic motions. Also, at low actuation frequency, considering the longitudinal coupling effects makes a substantial difference in the routes to chaos and vibrational response of microresonator. So, optimization and performance improvements of microresonator can be realized by the results of this paper.

Nonlinear analysis of open-chain flexible manipulator with time-dependent structure

• Nonlinear dynamics analysis of time-varying structure are investigated. • The equations derived by using Hamilton’s principle. • The distinct bifurcation parameters of force & torque inputs have considered. • The effects of excitation frequency, damping coefficient have studied. The analysis of motion equations of flexible-link manipulators indicates that nonlinear terms and behavioral instability exist in their outcomes. In this system, when the differential equations possess structural time-dependent parameters, there is a higher chance for system instability, especially chaos in comparison to time-invariant ones. It is a point for abnormal behavior that is unpredictable and always accompanied by damages. Therefore, the physical parameters and system inputs that cause chaotic responses should be detected and prevented as much as possible. To this aim, the dynamic equations of flexible link manipulator with revolute-prismatic joints as a case of time-variant dynamic system are obtained by Hamilton’s principle with the assumption of nonlinear strains. The discretized motion equations are solved and the results are presented in the forms of bifurcation diagrams (for variation of torque/force amplitude), Poincaré maps, phase-plane portraits, and the largest Lyapunov exponent. Finally, the obtained results are validated with the aid of the experimental setup. The results indicate that although the system response in low torque/force amplitudes is subharmonic (amplitude of 0.01) and quasi-periodic (amplitudes of 0.02–0.03), it becomes quasi-periodic/chaotic with a mild slope with respect to time in high amplitude values. Moreover, since the behavior of modal generalized coordinate is different from the rigid ones, the quasi-periodic and chaotic vibrations do not hurt the joints.

Chaotic vibrations of carbon nanotubes subjected to a traversing force considering nonlocal elasticity theory

The existence of chaos in the lateral vibration of the carbon nanotube (CNT) can contribute to source of instability and inaccuracy within the nano mechanical systems. So, chaotic vibrations of a simply supported CNT which is subjected to a traversing harmonic force are studied in this paper. The model of the system is formulated by using nonlocal Euler–Bernoulli beam theory. The equation of motion is solved using the Rung–Kutta method. The effects of the nonlocal parameter, velocity and amplitude of the traversing harmonic force on the nonlinear dynamic response of the system are analyzed by the bifurcation diagrams, phase plane portrait, power spectra analysis, Poincaré map and the maximum Lyapunov exponent. The results indicate that the nonlocal parameter, velocity and amplitude of the traversing harmonic force have considerable effects on the bifurcation behavior and can be used as effective control parameters for avoiding chaos.

Dynamic analysis of ball screw feed system with abrupt-change-nonlinear drive stiffness

Purpose For ball screw feed system, a sudden start or stop has a great influence on the transmission stiffness, so the axial stiffness mutation of feed system will occur. The purpose of this paper is to study the influence of acceleration on the transmission stiffness and dynamic characteristics of the ball screw feed system. Design/methodology/approach Taking the ball screw feed system as a research object, on the basis of the Hertz contact theory and the mixed element method, axial stiffness model and dynamic model are established. And the system stability was analyzed by the time history diagram and Phase-plane portrait diagram. The feed system was analyzed theoretically and experimentally, the experimental results are in good agreement with the model results. Findings Lead screw lead angle, preload, load and start acceleration affected ball-screw pair, bearing and transmission stiffness. And the load, nut contact stiffness, bearing contact stiffness, preload have a large effect on the transmission stiffness. The results show that a certain acceleration value will make the axial stiffness abrupt change. Originality/value This research provides a useful theoretical support for ensuring a good dynamic for the ball screw feed system.

A Method for Quantifying Automobile Brake Creep Groan Intensity Based on Friction-Induced Vibration and Noise

To quantify the intensity of automobile brake creep groan, both experimental and analytical studies are innovatively conducted on the friction-induced vibration and noise of the disc brake in this paper. Experimentally, three factors, brake disc initial temperature, terrain, and gear position, are comprehensively contemplated to design six different test conditions, based on which a creep groan vehicle road test is conducted. Depending on the subjective evaluation and statistical analysis of the annoyance degree caused by vibration and noise in the starting and braking process under different test conditions, the influence of each factor on creep groan intensity is obtained. By processing the brake caliper and lower suspension arm acceleration and the interior sound pressure signals acquired in the test, the occurrence conditions and transient dynamic characteristics of creep groan are explored. One of the creep groan vibration modes is triggered by the rapid removal of the force exerted on the brake pedal when the vehicle starts at an extremely low speed, which presents intermittent impact and chaotic characteristics. The other vibration mode shows periodic harmonic characteristics, whose energy is concentrated on specific frequencies in the form of frequency doubling. This vibration mode occurs when continuously maintaining a low brake oil pressure or slowly releasing the force exerted on the brake pedal during the vehicle low-speed starting process, the velocity-displacement phase-plane portrait of which shows a stable limit cycle of stick-slip motion. Analytically, four vibration indexes and four noise indexes are established to evaluate the intensity of creep groan, the validity of which is verified by linear regression analysis. Finally, by combining all effective indexes, multiple linear regression analysis is performed, based on which the mapping relationship between the subjective evaluation of creep groan and effective indexes is obtained. The results demonstrate that the combination of the logarithmic vibration dose value of maximum brake caliper acceleration pulse and the loudness of interior noise can accurately describe the annoying degree caused by creep groan. The regression model can quantify and predict creep groan intensity of the test vehicle under different test conditions, which has guiding significance for engineering applications.

Characterization of dynamics and information processing of integrate-and-fire neuron models

The integrate-and-fire (IF) model is the most widely used simple spiking neuron model in neuromorphic computing as well as in artificial neural network algorithms. Here, we characterize the dynamics and information processing of IF models using computer simulations and information-theoretic approaches. Neural dynamics is analysed by means of the time evolution of axonal spikes and the phase-plane portrait, and the coding efficiency of a neuron is estimated by the ratio of mutual information of input and output spikes for the binary hidden neural state. The exponential IF model exhibits higher similarity to the biophysical model in both neural dynamics and coding, compared to other IF type models. Electronic circuit simulations based on the simulation programme with integrated circuit emphasis reveal the nonlinear current–voltage characteristics of the IF neuron models and criticalities in the neural networks. Relevant information-theoretic measures indicate that the computational capabilities of neuromorphic devices largely depend on the neuron models. Such an approach combined with the analysis of neural dynamics provides a useful tool to investigate underlying dynamics of mathematical neuron models; it is applicable to the design and evaluation of neuromorphic models.

A new locally active memristive synapse-coupled neuron model

In this paper, a new type of non-volatile locally active memristor with bi-stability is proposed by injecting appropriate voltage pulses to realize a switching mechanism between two stable states. It is found that the memristive parameters of the new memristor can affect the local activity, which has been rarely reported, and this phenomenon is explained based on mathematical analyses and numerical simulations. Then, a locally active memristive coupled neuron model is constructed using the proposed memristor as a connecting synapse. The parameter-associated dynamical behaviors are revealed by bifurcation plots, phase plane portraits and dynamical evolution maps. Moreover, the bi-stability phenomenon of the new coupled neuron model is disclosed by local attraction basins, and the periodic burster and multi-scroll chaotic burster are found if a multi-level pulse current is used to imitate a periodical external stimulus on the neurons. The Hamiltonian energy function is calculated and analyzed with or without external excitation. Finally, the neuronal circuit is designed and implemented, which can mimic electrical activity of the neurons and is useful for physical applications. The experimental results captured from the analog circuit are consistent well with the numerical simulation results.

Dynamic Characteristics Analysis of the Composite Motion of Curve-Face Gear

Abstract The unified model was proposed for the three configurations of the composite motion of curve-face gear pair. A coupled bending-torsional-axial dynamic model with multi degree-of-freedom transmission system was proposed comprehensively considering nonlinear parameters such as tooth mesh stiffness, static transmission error and backlash. The fourth-order Runge–Kutta numerical method was applied to solve the dimensionless dynamic differential equations of the system, and then the dynamic responses of the system were investigated by time history, phase-plane portrait, poincare map, FFT spectrum and bifurcation diagram. It was found that the dynamic responses of the three configurations were complex. It presented quasi-periodic motion only under certain configuration and parameter conditions. With the increase of meshing frequency, the system state was characterized by the alternation between chaotic motion and periodic motion. Eccentricity ratio of displacement curve of curve-face gear pair mainly affected the rotational frequency of the system. As the eccentricity ratio increased, the dominant frequency of the system gradually changed from meshing frequency to rotational frequency. The correctness of the theoretical model was verified through corresponding experimental studies.

Nonlinear Phenomena in Axially Moving Beams with Speed-Dependent Tension and Tension-Dependent Speed

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

Stationary Shear Flows of Nematic Liquid Crystals: A Comprehensive Study via Ericksen–Leslie Model

We apply the Ericksen–Leslie dynamic model to investigate stationary solutions of planar shear flows for nematic liquid crystals. Nematic material is extremely rich and involves several parameters to characterize the basic local properties. The Ericksen–Leslie model of the nematic material includes Frank’s parameters for Oseen–Frank energy and Leslie’s dynamic parameters. Even in this simple setting of shear flows, the dynamics supported by the material exhibit quick rich behavior. With the aid of a Hamiltonian formulation and phase plane portraits, we are able to provide a rather complete picture about the possible solutions. In particular, we establish the existence of multiple solutions for the boundary value problem in many situations. We also try to explain the physical reasons for some of the results.

Nonlinear Transverse Vibration of a Hyperelastic Beam Under Harmonically Varying Axial Loading

Abstract Nonlinear transverse vibration of a hyperelastic beam under a harmonically varying axial load is analyzed in this work. Equations of motion of the beam are derived via the extended Hamilton's principle, where transverse vibration is coupled with longitudinal vibration. The governing equation of nonlinear transverse vibration of the beam is obtained by decoupling the equations of motion. By applying the Galerkin method, the governing equation transforms to a series of nonlinear ordinary differential equations (ODEs). Response of the beam is obtained via three different methods: the Runge–Kutta method, multiple scales method, and harmonic balance method. Time histories, phase-plane portraits, fast Fourier transforms (FFTs), and amplitude–frequency responses of nonlinear transverse vibration of the beam are obtained. Comparison of results from the three methods is made. Results from the multiple scales method are in good agreement with those from the harmonic balance and Runge–Kutta methods when the amplitude of vibration is small. Effects of the material parameter and geometrical parameter of the beam on its amplitude–frequency responses are analyzed.

Dynamic analysis and chaos control of spur gear transmission system with idler

This study aims to analyze the chaotic dynamics and present a chaos controller for spur gear transmission systems with idler. The chaotic dynamics of the spur gear mechanism has already been investigated. However, the presence of idler gears affects the chaotic behavior and the route to chaos for the nonlinear model of spur gears. This is investigated through the derivation of dimensionless dynamics, defining a Poincare’ section, and extracting the bifurcation diagrams of the system for variations of several parameter models. A nonlinear time-varying dynamic model of a spur gear transmission system with idler is established where backlash, time-varying stiffness, static transmission error, and external excitation are included and a region for the occurrence of chaos is found. The chaotic vibration suppression of the system is done by detecting the unstable periodic orbits embedded in the strange attractors and developing control law by employing sliding mode and adaptive sliding mode control strategy. The controller transmits a chaotic trajectory into the detected unstable periodic orbits. Numerical simulations including phase plane portrait, time histories diagrams, Poincare’ sections, and bifurcation diagrams demonstrate the behavior of the system and confirm the performance of the proposed controller.

Nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading in the subcritical buckling regime

Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton’s principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated.

Case Study of Self-Organization Processes in Information System Caching Components

Most modern Information Systems (IS) are designed with Object-Relational Mapping components (ORM). Such components bring down the number of queries to the database server and therefore increase the system performance. Caching mechanisms in software components are complex dynamic systems of open type. A simulation model of cache-application interaction has been created to assess the critical modes of the system. The authors suggest accumulating the cache elements states during the application run at discrete moments of time and present them as multi-variable time series. This work also suggests a method for reconstruction of phase-plane portrait of the system with multidimensional dynamics of the cache elements states. The article shows self-organization processes in caching components of information systems and illustrates the variability of system states for various initial conditions followed by transition to steady-state conditions. In particular, the paper illustrates dissipative structures as well as deterministic chaos with the complete determinism of queries in simulated information systems.

Three-dimensional parametric resonance of fluid-conveying pipes in the pre-buckling and post-buckling states

The paper presents a model of describing three-dimension parametric vibrations of a simply supported pipe conveying pulsating fluid. The three-dimensional motion equation of the system is a set of two nonlinear partial differential equations developed on the basis of Euler-Bernoulli beam theory, geometric nonlinearity and Kelvin-Voigt damping model. The three-dimensional motion equation is discretized by the Galerkin method and the nonlinear responses are solved by a fourth order Runge-Kutta integration algorithm. The three-dimensional model is validated for the natural frequencies from the pre-buckling state to the post-buckling state. Results of the nonlinear parametric resonance responses are shown in the form of time histories, spectrograms, phase-plane portraits, Poincaré map sections and motion trajectories. The vibration mode of sub-harmonic resonance is similar to a certain order free vibration mode, and the vibration mode of combination resonance is a superposition of two adjacent modes. The motion trajectories indicate the vibration of sub-harmonic resonance is planar but the vibration of combination resonance is non-planar. The lock-in phenomenon occurs in the pre-buckling and post-buckling states when the natural frequencies are locked to the pulsation frequency. The amplitudes of parametric resonance in the post-buckling state is much larger in comparison to parametric resonance in the pre-buckling state. Jump phenomenon is highlighted for the parametric resonance in the post-buckling state in comparison to the pre-buckling state.

Dynamics of inverted flags: Experiments and comparison with theory

The study of the dynamics of cantilevered thin flexible plates in reverse axial flow – also known as inverted flags – has become of significant interest, partly due to their energy harvesting potential. This paper presents fresh experimental results, aiming to enhance our understanding of the dynamics of inverted flags, particularly concerning stability and global dynamics sensitivity to various system parameters, such as the aspect ratio (i.e. height-to-length ratio) and flag material. This is achieved through testing flags of various dimensions, made from different materials. The dynamics of the system is presented in the form of bifurcation diagrams in which the flag tip angle is shown as a function of the dimensionless flow velocity. The frequency content of oscillatory motions is presented in the form of spectrograms. Also, time-histories, phase-plane portraits, as well as power spectral density and probability density function plots are presented for different flow regimes. Interesting dynamical features, such as small- and large-amplitude flapping, static divergence, large-amplitude buckling, and chaotic motions, have been observed experimentally, some for the first time. It is shown that for flags of very low aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of large aspect ratio, on the other hand, the undeflected static equilibrium becomes unstable via a supercritical pitchfork bifurcation leading to static divergence (buckling) at a sufficiently high flow velocity; at higher flow velocities, past the pitchfork bifurcation, a supercritical Hopf bifurcation materializes leading to flapping motion around the deflected static equilibrium; at even higher flow velocities, flapping motion becomes symmetric around the undeflected static equilibrium. Interestingly, it is found that heavy flags may exhibit large-amplitude flapping right after the initial static equilibrium, provided that they are subjected to a sufficiently large disturbance. In addition to experiments, nonlinear analytical models for the two asymptotic cases of zero and infinite aspect ratios are outlined, the results of which are used for comparison with the experimental results. Theory and experiments are found to be in reasonably good agreement.

Nonlinear dynamic behavior and bifurcation analysis of a rotating viscoelastic size-dependent beam based on non-classical theories

In this paper, the free and harmonically forced vibrations of a viscoelastic rotating microbeam have been analyzed by exploiting non-classical theories, such as the Modified Couple Stress Theory and Modified Strain Gradient Theory (MSGT). A continuous model is derived and reduced to a well-known Duffing–Rayleigh system via a Galerkin projection; then, the Multiple Scales Method is used to obtain amplitude equations. Free and forced nonlinear oscillations are studied via classical and non-classical theories, and their corresponding results are compared with each other. An expression for natural frequency is obtained in which the effects of the internal damping and small-scale parameters are obvious. It is observed that the natural frequency obtained using the MSGT has the highest value, while the Classical Theory (CT) predicts the lowest natural frequency value. Also, it is seen that by increasing the hub radius, the influence of rotational speed on natural frequency is increased. According to obtained results, the responses corresponding to the CT become completely stable for the lowest value of the excitation force and the highest value of the external damping coefficient. For MSGT, responses are stable when the values of the excitation force and external damping coefficients are highest and lowest, respectively. It is shown that for the MSGT, the system is stable when the internal damping coefficient is small, while for the CT, the beam is stable for the highest value of the internal damping coefficient. The system parameters—such as the detuning parameter, damping coefficients, and excitation force—affect the solution, such a way that the system may possess one or two stable spirals depending on the values of those parameters, as shown in phase plane portraits.

Linear and Non-linear Dynamic Methods Toward Investigating Proprioception Impairment in Non-specific Low Back Pain Patients.

Central nervous system (CNS) uses vision, vestibular, and somatosensory information to maintain body stability. Research has shown that there is more lumbar proprioception error among low back pain (LBP) individuals as compared to healthy people. In this study, two groups of 20 healthy people and 20 non-specific low back pain (NSLBP) participants took part in this investigation. This investigation focused on somatosensory sensors and in order to alter proprioception, a vibrator (frequency of 70 Hz, amplitude of 0.5 mm) was placed on the soleus muscle area of each leg and two vibrators were placed bilaterally across the lower back muscles. Individuals, whose vision was occluded, were placed on two surfaces (foam and rigid) on force plate, and trunk angles were recorded simultaneously. Tests were performed in eight separate trials; the independent variables were vibration (four levels) and surface (two levels) for within subjects and two groups (healthy and LBP) for between subjects (4 × 2 × 2). MANOVA and multi-factor ANOVA tests were done. Linear parameters for center of pressure (COP) [deviation of amplitude, deviation of velocity, phase plane portrait (PPP), and overall mean velocity] and non-linear parameters for COP and trunk angle [recurrence quantification analysis (RQA) and Lyapunov exponents] were chosen as dependent variables. Results indicated that NSLBP individuals relied more on ankle proprioception for postural stability. Similarly, RQA parameters for the COP on both sides and for the trunk sagittal angle indicated more repeated patterns of movement among the LBP cohort. Analysis of short and long Lyapunov exponents showed that people with LBP caused no use of all joints in their bodies (non-flexible), are less stable than healthy subjects.

Nonlinear vibration and chaos of a moving flexible graphene smart electronic web

Flexible graphene smart electronics are a kind of electronic device made of silver or other inorganic material and a flexible graphene web substrate. Because of their flexibility and excellent electronic properties, flexible graphene smart electronics have become a research topic of interest. However, during the mass production process, the high-speed transmission of the continuous tension state is susceptible to external disturbances, which seriously restricts the improvement of the flexible graphene smart electronic printing speed. Therefore, studying the dynamic characteristics of moving flexible graphene electronic materials is of great significance for improving their quality and production efficiency. This paper considers a roll-to-roll smart RFID label graphene web as the research object. Based on the energy functional variational theory, the Bubnov-Galerkin method is used to eliminate the residual value of the dynamic Von Karman partial differential equation, and a discrete forced nonlinear vibration control equation for a moving flexible graphene smart electronic web is established. The Melnikov method is used to obtain the Melnikov function of the orbit parameter for the vibration control equation and solve the conditions of the chaos criterion. Using the fourth-order Runge-Kutta method, the nonlinear vibration phase-plane portraits, Poincare maps, time history diagram, and power spectra of the moving flexible graphene smart electronic web are obtained. The effects of the web moving speed, external damping, and initial system conditions on nonlinear dynamic behaviour are calculated, and the stable and unstable working regions of the moving flexible graphene smart electronic web are obtained. This research provides a theoretical basis for the dynamic stability and vibration control of a moving graphene smart electronic web.

Stability analysis of DGP brane-world model with agegraphic dark energy

The aim of this work is to apply the dynamical system approach to study the linear dynamics of the normal DGP brane-world model with agegraphic dark energy. The stability analysis of the model will be investigated and the phase plane portrait will be illustrated. The nature of critical points will be analyzed by evaluating the eigenvalues of a linearized Jacobi matrix. Also, the statefinder diagnostic procedure will be applied to show the slight deviation of our model from the ΛCDM model. One of the most interesting results of this work is the great alleviation of the coincidence problem.