Phase Plane Portraits Research Articles

Theoretical analysis of dynamic behaviors of cable-stayed bridges excited by two harmonic forces

To better understand the dynamic behaviors of cable-stayed bridges, this study investigates the dynamic behaviors of a cable-stayed shallow arch subjected to two external harmonic excitations using the analytical approach. First, dimensionless planar vibration equations of the system are obtained by applying the Hamilton principle, and three ordinary differential equations of the arch and the two cables are obtained by using the Galerkin discretization method. Second, modulation equations involving the amplitude and phase of the dynamic response of the system are derived by applying the method of multiple scales. Third, three simultaneous resonance cases are considered. Finally, parametric study results are illustrated through frequency responses, amplitude responses, phase plane and bifurcation diagrams. Chaos phenomenon is also detected and presented. To validate the developed analytical solutions, numerical simulations are conducted by applying the Runge–Kutta method to integrate the original ordinary differential equations. The results demonstrate that acceptable consistency is reached in the results obtained from the analytical solutions and the Runge–Kutta method in the three simulated cases. The obtained results show that the system’s dynamic responses in the three simulated cases exhibit similarities in their frequency and amplitude responses, while some qualitative differences exist in the phase plane portraits (e.g., period-1, period-2, period-3 solutions) and their bifurcation diagrams.

Effects of vibro-medical insoles with and without vibrations on balance control in diabetic patients with mild-to-moderate peripheral neuropathy

The purpose of this study was to investigate the effects of a total-contact insole with and without subthreshold mechanical random noise on the balance control in diabetic patients with mild-to-moderate peripheral neuropathy. Twenty diabetic patients with mild-to-moderate neuropathy was recruited to this study. A total-contact insole was prototyped and vibratory motors were embedded into it. The parameters of the center of pressure (amplitude, velocity, and phase plane portrait) were analyzed after 30-minute walks with the shoe only, the shoe with vibro-medical insole with and without vibrations in eyes open and closed condition. The center of pressure amplitude, velocity, and phase plane portrait in the anterior-posterior and medio-lateral directions were significantly decreased using a vibro-medical insole without vibration in the eyes open condition (p < 0.05), as compared to the shoe, and with vibro- medical insole with vibration in both eyes open and closed conditions (p < 0.05) compared to the shoe. A significant drop was observed in the center of pressure amplitude, velocity and phase plane portrait parameters when the vibro-medical insole with vibration was used compared to vibro- medical insole without vibration in eyes closed condition (p < 0.05). The use of vibro-medical insoles was found to improve the patient’s balance control, as compared to the shoe. In the eyes closed condition, an improvement in the balance control was observed only with vibro-medical insole with vibration rather than vibro-medical insole without vibration or the shoe. Current finding suggest that a combination of the total-contact insole with vibration may improve the balance control remarkably in diabetic patients with mild-to-moderate neuropathy.

Distributed-parameter Dynamic Modeling and Bifurcation Analysis of a Trapezoidal Piezomagnetoelastic Energy Harvester

In this paper, the effect of the bimorph profile on the nonlinear dynamic behavior and performance of a vibratory piezomagnetoelastic energy harvester is investigated. The proposed model is composed of upper and lower piezoelectric layers on a trapezoidal cantilever beam with one attached tip magnet as well as two external magnets. The magnetic field of two external magnets generates magnetic forces and moment on the tip magnet. The bimorph structure is considered as a distributed-parameter system, and the external forces are obtained by analyzing the magnetic field of the external magnets. Equations of motion are obtained using electromagnetic Lagrange equations based on the generalized Hamilton principle and the Euler-Bernoulli beam theory. The proposed model for the bimorph and magnetic forces is validated by previously published experimental results. In order to compare the nonlinear behavior of the rectangular and trapezoidal beam profiles, the bifurcation diagrams are depicted for various control parameters such as the separation distances of the magnets, beam root width, and beam tip width. Verification of the bifurcation diagrams is performed by the phase plane portraits and Poincare maps. Also, the harvested power level is compared for different profiles of the bimorph. Moreover, the simultaneous effects of exciting frequency and bifurcation parameters on the system performance are investigated by the waterfall diagrams. The obtained results show that the trapezoidal beam profile with a lower tip width has higher performance than the rectangular beam. In trapezoidal beam profiles, the subharmonic and chaotic motions have relatively higher output powers than periodic motions.

Nonlinear modal interaction of an electrically actuated microbeam with flexible support

For the electrically actuated microbeam subjected to a combination of DC and AC voltage loadings, we studied the nonlinear dynamical responses and internal resonance analytically. The flexible boundary condition is considered. The modal interaction due to three-to-one internal resonance between the first and second modes is highlighted. The method of multiple scales is employed to get the modulation equation, which describes the amplitude and phases of the involving modes. Then, the equilibrium solutions of the modulation equation are calculated and their stability is examined. The frequency and force response curves reflecting the primary resonance (the first mode) are presented. We find that the first and second modes in the proposed model are coupled, and hence the energy transfer can occur between the involving modes. Moreover, it is found that the response of the system may encounter Hopf bifurcation for some specific parameters. As for simulation, the Galerkin scheme is applied to verify the analytical results in terms of time history, phase-plane portrait, Fourier spectrum, and Poincare section. The simulation results are in good agreement with the analytical ones.

Nonlinear Parametric Vibration of a Fluid-Conveying Pipe Flexibly Restrained at the Ends

In this paper, the nonlinear parametric vibration of fluid-conveying pipes flexibly restrained at both ends and subjected to the pulsation flow excitation is investigated. The nonlinear equation of motion is derived using Hamilton’s principle by considering the Kevin–Voigt viscoelastic damping, the geometric nonlinearity and the translational and rotational springs supported at the ends. The mode functions and eigen-frequencies are determined by the assumed mode method according to the elastic boundary conditions. The Galerkin method is implemented to obtain the natural frequencies and mode shapes of the pipe conveying fluid with different flow velocities. The effects of flexibly restrained conditions on stability of the pipe are analyzed. The nonlinear responses of the pipe under pulsating flow excitation are solved by the direct numerical method. The vibration behaviors are discussed in details, such as time history, frequency spectrum, phase-plane portrait, Poincare map and motion trajectory. The results show that the responses of sub-harmonic resonance and combination resonance can also be reflected in the rigidly supported pipes. The 1/5, 1/8 and 1/13 sub-harmonic resonances can occur at certain excitation frequencies of the nonlinear parametric vibration system. The steady-state response amplitudes increase by a large margin and significantly affect the stability of the pipe. The effects of different spring stiffness coefficients on the parametric resonance responses are presented. For larger translational springs and rotational stiffness coefficients, the resonance frequencies shift to higher regions and the resonance amplitudes may reduce by a certain extent in accordance with the rigid-body motion. This study can provide helpful guidance on the analysis and design of piping systems subject to vibrations.

Dynamic modeling and extended bifurcation analysis of flexible-link manipulator

In this article, the nonlinear dynamic analysis of a flexible-link manipulator is presented. Especially, the possibility of chaos occurrence in the system dynamic model is investigated. Upon the occurrence of chaos, the system dynamical behavior becomes unpredictable which in turn brings about uncertainty and irregularity in the system motion. The importance of this investigation is pronounced in similar systems such as double pendulum and single-link flexible manipulator. What makes this study distinct from previous ones is the increase in the number of links as well as the changing the bifurcation parameters from system mechanical parameters to force and torque inputs. To this aim, the motion equations of the N-link robot, which are derived with the aid of the recursive Gibbs-Appell formulation and the assumed modes method, are used. In the end, the equations of motion are developed for a two-link flexible manipulator, and its nonlinear dynamical behavior is analyzed via numerical integration of discrete equations. The results are presented in the form of bifurcation diagrams (for variation of torque amplitude), time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms. The outcomes indicate that when there is no offset, the decrease in damping results in chaotic generalized modal coordinates. In addition, as the excitation frequency decreases from 2π to π, a limiting amplitude is created at 0.35 before which the behavior of generalized rigid and modal coordinates is different, while this behavior has more similarity after this point. An experimental setup is also used to check the torques as the system input.

Nonlinear dynamic modeling and analysis for aviation axial piston pump in center spring return mechanism

Aviation axial piston pump is an important energy conversion component in aircraft hydraulic system, and the study on the nonlinearity of key operating mechanism is very necessary for improving the reliable operation of the pump. Addressing the aviation axial piston pump, this article analyzes nonlinear dynamic characteristics of the center spring return mechanism by using theoretical analysis and numerical simulation. Firstly, mathematical models of oil-film forces of slipper/swash plate pair and cylinder/port plate pair are derived. Then, based on the nonlinear dynamic theory, some specific theoretical analysis methods such as Lyapunov index spectrum, bifurcation diagram, time series, phase plane portrait, Poincaré map, and power spectrum are used to carry out the numerical simulation of nonlinear dynamic behaviors. The result indicates that characteristics of the center spring can reveal the reason for the nonlinear vibration of the center spring return mechanism. Meanwhile, the behavior of the oil thickness of slipper/swash plate pair is in the periodic and steady motion, and the variation of the oil thickness of cylinder/port plate pair is in chaotic vibration, which establishes the theoretical foundation of the aviation axial piston pump for the traceability and suppression of vibration. The obtained results are of important guidance significance for optimal design and dynamic control of the center spring return mechanism of aviation axial piston pump.

The Phenomenological Model of the Urban Agglomeration Dynamics: Theoretical Aspect

The article provides the phenomenological analysis of the influence of centrifugal and centripetal forces on the agglomerative process. The paper shows the possibility of applying mathematical models for assessing the resulting effect of their impact on the urban agglomeration. The phenomenological description is carried out by means of the phase space, the dynamic space, the space of parameters and a certain mathematical model. The urban agglomeration population as an indicator of the economic activity concentration and the change of population number are taken as the main phase variables. At certain restrictions to the type of functional dependence of change rates of population concentration on its number the following phenomena can be observed: self-oscillations; damped oscillations; fluctuations with an increasing, but limited in the asymptotics amplitude; monotonous achievement of the stable balance fr om above or fr om below (logistic dependence); steady, semi-steady or unstable lim it cycles of various lengths. The characteristic feature of self-oscillations is the lack of external periodic effect. The general qualitative behavior of the system in the whole dynamic space is manifested at t→∞ when its asymptotic characteristics do not depend on entry conditions. The most important solutions are stationary points and lim it cycles. The problem of stability and the nature of stationary points are directly connected with the qualitative image of the phase plane portrait in their environment and its topology. The type of phase plane portraits depends on the model parameters and is defined by the type of the special point. The article provides the theoretical phase plane portraits of the urban agglomeration dynamics for different types of static correlation between growth rates of the agglomeration population and economic activity concentration. The paper presents the current phase plane portrait of the Vladivostok urban agglomeration.

A Theoretical Study of Light Soliton Produced by Semiconductor Quantum Dot Waveguides and Propagation in Optical Fibers

In this paper, the propagation of light soliton is studied in nonlinear optical fiber. We propose the external excitation of SQD waveguides through an optical source that allows the generation of solitary waves that are propagated through a non-linear optical fiber. The soliton formation is studied theoretically from the non-linear interaction between the external optical excitation and SQDs, considering SQDs as a quantum system of three energy levels. In this study, the Fourier Split-Step (FSS) method is used to solve numerically continuous nonlinear Schrodinger equation (NLSE) to evolution of the soliton pulse emitted by the SQDs inside an optical fiber with real physical parameters. The effect of SQDs density and electric field on the pulse width is also studied. Phase plane portraits are drawn for the stability of soliton in fiber and SQDs using software Matcont.

The Amount and Temporal Structure of Center of Pressure Fluctuations During Quiet Standing in Patients With Chronic Low Back Pain.

The characteristics of postural sway were assessed in quiet standing under three different postural task conditions in 14 patients with nonspecific chronic low back pain and 12 healthy subjects using linear and nonlinear center of pressure parameters. The linear parameters consisted of area, the mean total velocity, sway amplitude, the SD of velocity, and the phase plane portrait. The nonlinear parameters included the Lyapunov exponent, sample entropy, and the correlation dimension. The results showed that the amount of postural sway was higher in the patients with low back pain compared with the healthy subjects. Assessing the nonlinear parameters of the center of pressure showed a lower sample entropy and a higher correlation dimension in the patients with low back pain compared with the healthy subjects. The results of this study demonstrate the greater regularity and higher dimensionality of the center of pressure fluctuations in patients with nonspecific chronic low back pain, which suggests that these patients adopt different postural control strategies to maintain an upright stance.

Experimental and artificial intelligence for determination of stable criteria in cyclic voltammetric process of medicinal herbs for biofuel cells

This study proposed an expert system approach on the basis of artificial intelligence (AI) in the modeling of cyclic voltammogram (CV) profiles of green tea extracts. AI approach of artificial neural networks is applied to generate the model phase-plane portraits of current output versus applied voltage through CV scan cycles. The predicted current values were validated using experiments, and generic ability of approach was examined by testing on the CV scan cycles generated from Syzygium aromaticum and Citrus reticulate. It was concluded that AI approach can be employed to reveal stable point (cycle and voltage) in CV profiles for bioenergy applications.

Three-to-One Internal Resonance in MEMS Arch Resonators.

We present an investigation of the nonlinear dynamics of a microelectromechanical system (MEMS) arch subjected to a combination of AC and DC loadings in the presence of three-to-one internal resonance. The axial force resulting from the residual stress or temperature variation is considered in the governing equation of motion. The method of multiple scales is used to solve the governing equation. A four first-order ordinary differential equation describing the modulation of the amplitudes and phase angles is obtained. The equilibrium solution and its stability of the modulation equations are determined. Moreover, we also obtain the reduced-order model (ROM) of the MEMS arch employing the Galerkin scheme. The dynamic response is presented in the form of time traces, Fourier spectrum, phase-plane portrait, and Poincare sections. The results show that when there is an internal resonance, the energy transfer occurs between the first and third modes. In addition, the response of the MEMS arch presents abundant dynamic behaviors, such as Hopf bifurcation and quasiperiodic motions.

Nonlinear characteristics of rubbing rotor–bearing system of locomotive excited by rotating speed of the traction motor

The paper aims to study the nonlinear characteristics excited by rotating speed of the traction motor. Considering the rub-impact force, Hertzian contact force and unbalanced rotor, a typical of dynamic model of rotor-bearing system for locomotive is established, and Runge-Kutta method is adopted to solve the nonlinear response. Based on bifurcation diagrams, axis trajectory curves, phase plane portraits, Poincaré maps and amplitude spectrums, nonlinear characteristics of the system are analysed when rotating speed of the traction motor is changed. The simulation reveals complex dynamic behaviour of the system including periodic, multi-periodic and chaotic motion. The results will help us understand and control the nonlinear behaviour of the proposed traction system for railway locomotive in the future research.

Nonlinear analysis of the locomotive traction system with rub-impact and nonlinear stiffness

Considering rub-impact forces, eccentricity of rotor, and nonlinear stiffness of armature shaft, a dynamic differential equation is built to investigate the bifurcation and chaos behavior of the locomotive traction system. The nonlinear analysis methods are inclusive of bifurcation diagrams, phase plane portraits, Poincaré maps, displacement–time curves, and spectrograms. The simulation results reveal the complex dynamic comprising period-3 to quasi-periodic bifurcation and intermittent bifurcation. Some research results are references for dynamic design and rub-impact fault diagnosis of the locomotive traction system.

Non-linear dynamics of a simply supported fluid-conveying pipe subjected to motion-limiting constraints: Two-dimensional analysis

In this paper, the coupled axial and transverse non-linear vibrations of a simply supported pipe, subjected to motion-limiting constraints, conveying pulsating fluid are investigated. The equations of motion are discretized into a set of coupled ordinary differential equations via Galerkin's method, which are then solved using Houbolt's method combined with Newton-Raphson iterative technique. The bifurcation diagrams are constructed to present the global dynamics of the system. Phase-plane portraits, Poincaré maps and power spectrum diagrams are plotted to show the characteristics of some typical motions, such as quasi-periodic and chaotic motions. Finally, the influence of axial motion on the system dynamics is investigated.

Nonlinear dynamical behaviors of a moving membrane under external excitation

In this paper, the nonlinear vibration characteristics of a moving printing membrane under external excitation are studied. Based on the Von Karman nonlinear plate theory, the nonlinear vibration equation of the axial motion membrane under the external excitation is deduced. The Galerkin’s method is used to discretize the vibration differential equations of the membrane, and then the state equation of the system is obtained. The state equation of the system is numerically solved by the fourth-order Runge–Kutta method. The relationship between the nonlinear vibration characteristics and the amplitude of external excitation, damping coefficient, and aspect ratio of the printing membrane is analyzed by using the time histories, phase-plane portraits, Poincare maps, and bifurcation diagrams. Chaotic intervals and the stable working range of the moving membrane are obtained. This study provides a theoretical basis for predicting and controlling the stability of the membrane.

Order error in the calculation of continuous relative phase

The aims of this study were to demonstrate “order error” in the calculation of continuous relative phase (CRP) and to suggest two alternative methods—(i) constructing phase-plane portraits by plotting position over velocity; and (ii), the Hilbert transform—to rectify it. Order error is the change of CRP order between two degrees of freedom (e.g., body segments) when using the conventional method of constructing phase-plane portraits (i.e., velocity over position). Both sinusoidal and non-sinusoidal simulated signals as well as signals from human movement kinematics were used to investigate order error and the performance of the two alternative methods. Both methods have been shown to lead to correct results for simulated sinusoidal and non-sinusoidal signals. For human movement data, however, the Hilbert transform is superior for calculating CRP.

Nonlinear mechanical behaviour of microshells

This study examines the nonlinear large-amplitude static and dynamic responses of a doubly curved shallow microshell in the framework of the modified couple stress (MCS) theory. To this end, the expressions for the classical and higher-order stresses and strains are consistently derived in an orthogonal curvilinear coordinate system employing the Novozhilov shell theory. The strain energy of the system is then consistently derived utilising the Novozhilov shell formulations in the framework of the MCS theory. The kinetic energy of the microshell is obtained while accounting for all out-of-plane and in-plane displacements. Furthermore, the work of the distributed out-of-plane load is accounted for and the energy dissipation is taken into account via the Rayleigh energy dissipation function. An assumed-mode technique is utilised to expand the out-of-plane and in-plane displacements via series expansions. The Lagrange equations are then utilised to derive the discretised equations of motion in the form of a set of nonlinearly coupled ordinary differential equations (ODEs). This set of nonlinear ODEs is solved making use of a continuation technique (for the nonlinear static and dynamic analyses) as well as an eigenvalue extraction method (for the linear natural frequency analysis). Extensive numerical simulations are carried out for both static and dynamic cases and the effects of different parameters, such as the radius of curvature, the magnitude and direction of the applied distributed load, and the small-scale parameter are investigated. The numerical results are constructed in the form of nonlinear static deflection curves, nonlinear dynamic frequency-amplitude diagrams, time traces, and phase-plane portraits.

ХАОТИЧЕСКИЕ КОЛЕБАНИЯ ГЕОМЕТРИЧЕСКИ НЕЛИНЕЙНЫХ НАНОРАЗМЕРНЫХ ПОЛОГИХ ОСЕСИММЕТРИЧНЫХ ОБОЛОЧЕК

The Hamilton - Ostrogradsky principle is used in the paper to construct a mathematical model of vibrations of geometrically nonlinear nano-dimensional shallow axisymmetric spherical shells. The model is based on the following relations and assumptions: the body of the shell is elastic, homogeneous and isotropic, the Kirchhoff - Love hypothesis, a modified momentary theory of elasticity is used to account for the dependence of elastic behavior of the shell on the dimension-dependent parameter; the shallowness of the shell is determined based on the hypotheses of V.Z. Vlasov, the geometrical nonlinearity is determined using T. Karman's hypothesis. An approach for determining the “true” solution is proposed. The partial differential equation is reduced to a Cauchy problem with the help of the second-order accuracy finite difference method, which is solved using several Runge - Kutta-type methods: the 4th- and 2nd-order Runge - Kutta methods, the 4th-order Runge - Kutta - Fehlberg method, the 4th-order Cash - Karp method, the 8th-order Runge - Kutta - Prince - Dormand method, the implicit 2nd- and 4th-order Runge - Kutta method. An algorithm and a software complex for obtaining numerical results are developed. The convergence of the above methods for a space and time coordinate is studied. The study is based on the qualitative theory of nonlinear dynamics: signals, 2D and 3D phase-plane portraits, power spectrums, Morley's wavelets, deflection curves, Poincare sections and autocorrelation functions are analyzed, the sign of Lyapunov's index is analyzed using several methods: Wolf's, Kantz's, Rosenstein's. An example of analyzing plates and shallow shells is given. The analysis of the obtained results shows that with the increase of the dimension-dependent parameter the vibration character changes from chaotic to harmonic, and the value of the dynamic critical load increases.

Система автоматической регулировки амплитуды колебаний для однокольцевого RC-автогенератора

Harmonic RC oscillators have long been known and are widely used in various fields of radio engineering. They have become widely used in medical equipment, various sensors, measuring equipment, etc. Their main advantages in operating within short to long wave frequency bands are compact dimensions, simplicity and low cost of manufacturing, possibility of widely retuning the frequency, etc. In our previous articles, we examined several versions of such circuits, including single-loop RC oscillators, which are the simplest ones. It was shown that the major disadvantage of such self-oscillators in generating close-to-sinewave signals is the need to operate them with a very small self-excitation margin. An attempt to increase the self-excitation margin inevitably leads to a growth of higher harmonic components, distortion of the output signal, and failure of oscillations. As a result, additional techniques must be used to secure stable steady-state operation. One of such techniques is automatic amplitude control (AAC). With the AAC circuit switched in operation, the generator output signal amplitude is controlled, due to which a larger self-excitation margin is obtained at the initial time instant, and the required amplitude value is maintained during steady-state operation. The article presents a version of such system taking as an example a single-loop self-oscillator with the Wien bridge in the feedback circuit. As is known, the list of important issues to be taken care of in designing AAC systems includes, among other things, an analysis of their stability and the selection of time constants for the system functional units. There are many methods for analyzing the stability of an AAC system, and all of them involve the need to set up differential equations of the system. A version of addressing this objective is presented to produce the system of differential equations describing the operation of a self-oscillator fitted with the AAC system. The obtained system of equations was applied for selecting — using the Routh–Hurwitz stability criterion — the time constant for the low-pass filter (LPF) of the AAC system amplitude detector (AD). It has been shown that the selection of the time constant for the AD LPF is constrained, on the one hand, by the requirements for the amplitude detector operation mode, and on the other hand, by the need to ensure fast response of the AAC system to variations of the generator voltage amplitude. The time histories of the system response to variations of the generator voltage amplitude and the system phase-plane portrait have been obtained.