Poincare Map Research Articles

Nonlinear dynamic characteristics of smart FG-GPLRC sandwich varying thickness truncated conical shell with internal resonance for first three order modes

This paper examines the 1:1:1 internal resonant nonlinear dynamic characteristic of the simply supported varying thickness functionally graded graphene platelets reinforced composite (FG-GPLRC) smart truncated sandwich conical shell subject to the combined effects of transverse load and in-plane force. The truncated smart sandwich conical shell is composed of an FG-GPLRC varying thickness core and two magneto-electro-elastic face layers, whose material properties and constitutive relations are individually identified by the rule of mixture, improved Halpin-Tsai approach and generalized Hooke's law. Utilizing the first-order shear deformation theory (FSDT), von Karman's geometrical nonlinearity, Hamilton's principle and Galerkin technique, the 3DOF dimensionless nonlinear dynamic formulations for the truncated smart FG-GPLRC conical shell are established. The multiple-scale technique is applied to developing the averaged equations for the truncated smart FG-GPLRC conical shell under combined resonance. The frequency-response and force-response curves, Poincare maps, phase portraits, time history diagrams, bifurcation and maximum Lyapunov exponent diagrams can be portrayed by the nonlinear equation solver and Runge-Kutta approach. The effects of the damping and tuning parameters, transverse and in-plane forces on the 1:1:1 internal resonant nonlinear dynamic characteristic of truncated smart varying thickness FG-GPLRC sandwich conical shell are examined.

Periodic, quasi-periodic, chaotic waves and solitonic structures of coupled Benjamin-Bona-Mahony-KdV system

Abstract In this paper, we mainly focus on studying the dynamical behaviour and soliton solution of the coupled Benjamin-Bona-Mahony-Korteweg-de Vries (BBM-KdV) system, which characterizes the propagation of long waves in weakly nonlinear dispersive media. The paper utilizes different tools to detect chaos, such as time series analysis, bifurcation diagrams, power spectra, phase portraits, Poincare maps, and Lyapunov exponents. This analysis helps in more accurate predictive modeling of the systems. This understanding can aid in the design of control strategies, resulting in enhancements in prediction, control, optimization, and design. Additionally, we construct the system's solitary wave structures using the Jacobi elliptic function (JEF) method. We identify periodic wave solutions expressed in terms of rational, hyperbolic, and trigonometric functions. Certain parameter values can lead to periodic wave solutions, solitary waves (bell-shaped solitons), shock wave solutions (kink-shaped soliton solutions), and double periodic wave solutions.

Combining Artificial Neural Networks and Mathematical Models for Unbalance Estimation in a Rotating System under the Nonlinear Journal Bearing Approach

Rotating systems are essential components and play a critical role in many industrial sectors. Unbalance is a very common and serious fault that can cause machine downtime, unplanned maintenance, and potential damage to vital rotating machines. Accurately estimating unbalance in rotor–bearing systems is crucial for ensuring the reliable and efficient operation of machinery. This research paper presents a novel approach utilizing artificial neural networks (ANNs) to estimate the unbalance masses in a multidisk system based on simulation data from a nonlinear rotor–bearing system. Additionally, this study explores the effect of various operating parameters on oil film stability and vibration response through a combination of bifurcation diagrams, spectrum cascades, Poincare maps, and orbit and FFT plots. This study demonstrates the effectiveness of ANNs for unbalance estimation in a fast and accurate way and discusses the potential of ANNs in smart online condition monitoring systems.

Flow topology and mixing in alveolar edema: Unsteady flow in interconnected cavities with moving walls

The study of alveolar fluid mechanics is critical for comprehending respiratory function and lung diseases, particularly in cases of alveolar lesions that result in significant structural and fluid dynamic changes. This study investigates the flow topology and chaotic mixing within both normal and edematous alveoli, where the alveoli in the edematous model are interconnected by pores. To numerically simulate alveolar flow, a mathematical model is developed to ascertain the key parameters of Reynolds number (Re) and alveolar expansion ratio. Subsequently, the flow fields are analyzed to determine wall shear stress (WSS) and to identify WSS critical points and critical points of velocity vector, with a thorough presentation of the various flow topologies corresponding to these critical points. Moreover, a dynamic mode decomposition-based method is introduced to track particle trajectories, and the exploration of chaotic mixing is conducted through tracer advection, Poincare map, and the calculation of finite-time Lyapunov exponents. Results indicate that the edematous model exhibits a higher Re and higher WSS due to the fluid properties. Within the alveoli, high WSS is usually localized at the pores. The pores increase critical points and alter flow topologies, significantly changing chaotic mixing. Additionally, Re and alveolar locations also affect mixing patterns. These findings are crucial for understanding alveolar physiology and designing inhaled drugs for lung diseases, considering the role of chaos in particle transport in the lung acini.

Nonlinear effect on stable state and snap-through bistability of square composite laminate

Bistable structures with the elastic instability caused by buckling are confirmed to perform significantly in micro-electromechanical systems, metamaterials, energy harvester, vibration isolation and morphing structures. The target of this paper is to explore the dynamic responses of cross-ply bistable composite laminate, focusing on the nonlinear effect of the single- and double-well vibration. Both theoretical and finite element (FE) methods are employed to simulate the nonlinear vibration and dynamic snap-through of bistable composite laminate under the foundation excitation at the center. In the theoretical model, the governing equations are established via Lagrange's equation based on the first-order shear deformation theory, von Karman nonlinear strain-displacement relation and Rayleigh-Ritz method. The fourth-order Runge-Kutta method is adopted to solve the governing equations, and the numerical results are validated by FE model. Subsequently, the details of the dynamic responses are analyzed to identify the nonlinear effects in the form of bifurcation diagram, phase portrait, time history, Poincare maps and amplitude spectrum. The dynamic responses are examined for a series of excitation parameters in both time and frequency domain. Through fixed frequency and frequency sweep tests, the nonlinear phenomenon of the single- and double-well vibrations are analyzed including superharmonic resonance, stiffness softening, hysteresis phenomenon, various periodic and chaotic vibration. The diverse responses to external inputs contribute significantly to efficiently predicting mechanical behaviors in real-world conditions, thereby offering indispensable theoretical support for structural design applications.

Analysis of fractional solitary wave propagation with parametric effects and qualitative analysis of the modified Korteweg-de Vries-Kadomtsev-Petviashvili equation

This study explores the fractional form of modified Korteweg-de Vries-Kadomtsev-Petviashvili equation. This equation offers the physical description of how waves propagate and explains how nonlinearity and dispersion may lead to complex and fascinating wave phenomena arising in the diversity of fields like optical fibers, fluid dynamics, plasma waves, and shallow water waves. A variety of solutions in different shapes like bright, dark, singular, and combo solitary wave solutions have been extracted. Two recently developed integration tools known as generalized Arnous method and enhanced modified extended tanh-expansion method have been applied to secure the wave structures. Moreover, the physical significance of obtained solutions is meticulously analyzed by presenting a variety of graphs that illustrate the behaviour of the solutions for specific parameter values and a comprehensive investigation into the influence of the nonlinear parameter on the propagation of the solitary wave have been observed. Further, the governing equation is discussed for the qualitative analysis by the assistance of the Galilean transformation. Chaotic behavior is investigated by introducing a perturbed term in the dynamical system and presenting various analyses, including Poincare maps, time series, 2-dimensional 3-dimensional phase portraits. Moreover, chaotic attractor and sensitivity analysis are also observed. Our findings affirm the reliability of the applied techniques and suggest its potential application in future endeavours to uncover diverse and novel soliton solutions for other nonlinear evolution equations encountered in the realms of mathematical physics and engineering.

Deformation theory and nonlinear dynamic behavior of PVC gel actuators

Polyvinyl chloride (PVC) gel generates complex nonlinear vibration behavior under an alternating current voltage excitation, which has potential application as a dynamic electromechanical actuator. However, there are few studies on the deformation theory of PVC gel actuators, especially the dynamic nonlinear response theory. In this paper, a complex dynamic model is established according to the electrodeformation mechanism of PVC film, and the nonlinear dynamic behavior of the actuator is numerically studied by a differential equation. The effects of applied voltage amplitude, voltage frequency, dibutyl adipate content, mechanical tension, and bias voltage on the dynamic properties of PVC film were analyzed under the condition of equal biaxial tension. The variation of amplitude and the generation and disappearance of the beat frequency during vibration are analyzed by using time-domain characteristics. The degree of PVC actuator nonlinearity as well as vibration stability and periodicity is also reflected based on the phase path and Poincare map. Finally, the law of influence of external condition parameters on the dynamic response of the PVC actuator is obtained.

Effect of tooth surface wear on dynamic characteristics of spur gear transmission system

The meshing stiffness and system dynamic characteristics caused by gear wear fault are studied. Firstly, potential energy method was used to model and solve the meshing stiffness of gears and the effect of wear on the meshing stiffness and static transfer error was analyzed. A 6-dof spur gear dynamics model was established considering various nonlinear factors, and the dynamic responses of the spur gear system under various wear degrees were analyzed by using time domain, spectrum, phase plane, and Poincare mapping. Finally, the visual test platform of the gear box is built, and the wear condition and dynamic response of the gear box under different wear cycles are analyzed. Results show that the meshing stiffness of gear decreases due to wear, and the meshing stiffness of double tooth decreases more than that of single tooth. In addition, the static transmission errors caused by wear changes periodically with meshing frequency. Wear causes the vibration response of gear system to become complex gradually, and the nonlinear of the system is enhanced, which changes from single period motion to quasi period motion. Results can provide theoretical support for gear wear reduction, life extension, vibration reduction, and noise reduction and lubrication improvement.

Research on Dynamics Characteristics of Grounded Damping Nonlinear Energy Sink

The nonlinear energy sink (NES) system mainly dissipates energy through damping elements, and changing the position of the damping element will also change the performance of the NES. In this paper, a grounded damping NES is proposed by grounding the damping element of the traditional cubic stiffness NES. The complex dynamics of this two-degree-of-freedom system are investigated. The slow-varying equations of the system under 1:1 internal resonance are derived by using the complexification-averaging (C×A) method, based on which the influence of the primary structure’s damping on a bifurcation is analyzed. The conditions for the existence of strongly modulated response (SMR) are studied, and the accuracy of the results is verified using the slow invariant manifold (SIM), Poincare mapping, and time-history diagrams. This provides a means of verifying the analytical findings. The vibration suppression effects of the grounded damping NES, as compared to the cubic stiffness NES, are thoroughly studied under both pulse and harmonic excitations. The results indicate that the main structure damping affects the stability of the system and the occurrence of the SMR. Most previous studies of the NESs have overlooked the effect of main structure damping, which may influence the selection of structural parameters. Moreover, under relatively large pulse or harmonic excitations, the vibration suppression effectiveness of the grounded damping NES surpasses that of traditional cubic stiffness NES. This finding has important practical significance for improving the vibration suppression effect and robustness of the NES, and provides a reference for the structural design of the NES.

Dynamical sensitivity on thermal fluctuations of torsional homogeneous and nonhomogeneous microsystems under the influence of Casimir and electrostatic torques: Nonlinear actuation towards chaotic motion

We have studied how the mechanical Casimir torque between the components of a torsional double beam microdevice under the influence of thermal fluctuations could play significant role on the dynamics of the device, and under what conditions can even lead to chaotic behavior. The influence of thermal fluctuations on the dynamical behavior of torsional microsystems, whether homogeneous or nonhomogeneous, has been revealed through bifurcation and phase diagrams, and Poincare maps. For nonhomogeneous microsystems, the device with low conductivity reveals significant sensitivity to thermal fluctuations. Moreover, it is shown the significant dependence of the optimal voltage distribution, which leads to a significant increase in the stable operation range of the device, on the ambient temperature conditions and the optical properties of the constituent materials. For nonconservative torsional microsystem, the use of the Melnikov method and Poincaré plots have shown how thermal effects increase the probability of chaotic behavior. This is because different temperature enhances the effective Casimir torque and the possibility of chaotic motion. Furthermore, it is illustrated how the use of the optimal voltage distribution, by taking into consideration thermal fluctuations and optical properties, can keep the device away from chaotic behavior. This is because by dividing the electrostatic torques the influence of the effective Casimir torque is reduced. Finally, it is demonstrated that by transforming a homogeneous system into a heterogeneous one, the response of the system to thermal fluctuations is reduced making it possible to avoid to increase the possibility for chaotic behavior by changing the temperature.

Chaotic vibration of a curved CNT conveying magnetic fluid in the thermo-magnetic field considering the surface effects

The nonlinear chaotic vibration of curved single-walled carbon nanotube (CSWCNT) conveying magnetic fluid is studied based on the nonlocal Euler-Bernoulli beam model. The governing equation of CSWCNT is presented considering the axial thermo-magnetic load and the surface effect. By employing the Galerkin decomposition approximation method along with the admissible beam shape function satisfying the cantilevered beam boundary conditions, the nonlinear partial differential equation of the system is reduced to a nonlinear ordinary differential equation and numeric integration procedures are used to solve it. By considering the non-dimensional damping, cubic and quadratic terms, and amplitude of external force as controlling parameters, the bifurcation diagram and the largest Lyapunov exponent are employed to distinguish the chaotic, periodic, and quasi-periodic critical parameters of the curved CNT dynamics and validate the predicted results. Also, the phase plane and Poincare map for these critical parameters are provided. The results show that decreasing values of damping coefficient and quadratic term leads to quasi-periodic or chaotic motion, while system has periodic behaviour for smaller values of the external force and nonlinear cubic term. Also, magnetic field intensity and length of the CSWCNT help preventing the chaotic motion. Furthermore, adverse effects of higher values of temperature change, flow velocity and its correction factor are seen based on the obtained results.

Analysis of Shafting System Vibration Characteristics for Mixed-Flow Hydropower Units Considering Sand Wear on Turbine Blades

Addressing the issue of increased shaft-system vibration in high-altitude mixed-flow hydropower generating units due to sand wear on turbine blades, a three-dimensional model of a specific mixed-flow water turbine was constructed. CFD numerical simulations were employed to analyze the fluid exciting force acting on the turbine runner under varying degrees of blade wear. An approximate analytical model was then established for the variation of fluid exciting force in the turbine runner system using the Fourier harmonic analysis method. A multi-degree-of-freedom mathematical model of flexural and inclined coupling vibration of a hydropower unit’s shafting, considering blade wear, was constructed. The nonlinear dynamic model was numerically calculated by the Runge–Kutta method. The vibration responses of the shafting of hydropower units under different wear degrees were obtained by means of a time-domain diagram, frequency-domain diagram, axis-locus diagram, phase-locus diagram, and Poincare mapping. Based on the formula for calculating the wear amount of the blade material, the runner amplitude degradation trajectory model was established, and the pseudo-failure time of turbine blades was determined according to the allowable value of amplitude.

Model of Close Orbit Dynamics Around Asteroid (338) Budrosa

Abstract In this paper, we revisited the model of gravitational potential generated by irregular shapes. A finite straight segment model was used to investigate the motion of particles around a massive straight segment of length 2l, mass M, and uniformly rotates with angular velocity ω. We extended this model by converting it to the cylindrical coordinate and applied this model to the asteroid (338) Budrosa. We obtained the position of equilibrium points in this system. We also estimate the zero-velocity curve and determine the Poincare map and its periodic orbit around equilibrium points.

Generation and analysis of the chaos phenomenon in the molecular-distillation-Navier–Stokes (MDNS) nonlinear system

Introduction: For the Navier-Stokes equation, one of the most essential tasks should be to study its completeness of the complex nonlinear systems. Also, its nature and physical practical applications would be depth explored. Moreover, as one of the routes to chaos, this equation with an external force has been investigated numerically in 1989. Recently, some information is worth noting that when the high symmetry was imposed on the velocity field, the complex nonlinear motions should occur even lead to the chaos phenomenon. However, most of the published papers are based on theoretical studies and rarely deal with the above results, which lost of the match between them and the integrity of the scientific system.Methods: This study analyzed the molecular distillation process in detail based on the basic theory of nonlinear chaotic systems. Then, the mathematical model for the process of molecular distillation with one brushless DC motor (BLDCM) is built and named the Molecular-Distillation-Navier-Stokes (MDNS) equation. Also, its complex and potentially chaotic behaviors and chaotic processes are first discovered and demonstrated, such as chaotic attractors, chaotic co-attractors, phase portraits, time-domain waveforms, Lyapunov exponent spectrums, Poincare maps, the bifurcation diagrams, and so on.Results: The good agreement among theoretical analysis, simulation and experimental results verifies the practicability and flexibility of the configured model.Discussion: The related conclusions have supplemented and improved the theoretical system for the Navier Stokes equations. Also, it reflects the significance in molecular distillation processes. Meanwhile, the novel research direction for the fields of the chaotic nonlinear and complex industrial systems have been explored and discovered.

A high-order target phase approach for the station-keeping of periodic orbits

A novel high-order target phase approach (TPhA) for the station-keeping of periodic orbits is proposed in this work. The key elements of the TPhA method, the phase-angle Poincare map and high-order maneuver map, are constructed using differential algebra (DA) techniques to determine station-keeping epochs and calculate correction maneuvers. A stochastic optimization framework tailored for the TPhA-based station-keeping process is leveraged to search for fuel-optimal and error-robust TPhA parameters. Quasi-satellite orbits (QSOs) around Phobos are investigated to demonstrate the efficacy of TPhA in mutli-fidelity dynamical models. Monte Carlo simulations demonstrated that the baseline QSO of JAXA’s Martian Moons eXploration (MMX) mission could be maintained with a monthly maneuver budget of approximately 1 m/s.

Nonlinear dynamics of a circular curved cantilevered pipe conveying pulsating fluid based on the geometrically exact model

Due to the novel applications of flexible pipes conveying fluid in the field of soft robotics and biomedicine, the investigations on the mechanical responses of the pipes have attracted considerable attention. The fluid-structure interaction (FSI) between the pipe with a curved shape and the time-varying internal fluid flow brings a great challenge to the revelation of the dynamical behaviors of flexible pipes, especially when the pipe is highly flexible and usually undergoes large deformations. In this work, the geometrically exact model (GEM) for a curved cantilevered pipe conveying pulsating fluid is developed based on the extended Hamilton’s principle. The stability of the curved pipe with three different subtended angles is examined with the consideration of steady fluid flow. Specific attention is concentrated on the large-deformation resonance of circular pipes conveying pulsating fluid, which is often encountered in practical engineering. By constructing bifurcation diagrams, oscillating shapes, phase portraits, time traces, and Poincare maps, the dynamic responses of the curved pipe under various system parameters are revealed. The mean flow velocity of the pulsating fluid is chosen to be either subcritical or supercritical. The numerical results show that the curved pipe conveying pulsating fluid can exhibit rich dynamical behaviors, including periodic and quasi-periodic motions. It is also found that the preferred instability type of a cantilevered curved pipe conveying steady fluid is mainly in the flutter of the second mode. For a moderate value of the mass ratio, however, a third-mode flutter may occur, which is quite different from that of a straight pipe system.

Oblique impact dynamic analysis of wedge friction damper with Dankowicz dynamic friction

<abstract> <p>Aiming at the wedge friction damper for freight train bogie, considering Dankowicz dynamic friction, the mechanical model of a three-degree-of-freedom inclined impact vibration system with gap and dynamic friction, is simplified. The mechanical model of the system is established, the motion equation of the system is obtained, and the motion state and conditions of the system are analyzed. The Poincare map is constructed by selecting a fixed collision section, and the response of the system is solved by the fourth-order Runge-Kutta numerical method with variable step size. The transition process of the system motion and the phenomenon of sticking and chatter are analyzed by numerical simulation when the external excitation frequency changes. The results show that: 1) Under certain parameters, with the change of excitation frequency, the system undergoes periodic doubling bifurcation, inverse periodic doubling bifurcation, Grazing bifurcation and Hopf bifurcation, and there is a "periodic bubble" phenomenon in the system motion. When the system excitation frequency is between 3.35–3.55, 4.425–6.12, and 7.34–7.758, the system motion has chatter and sticking phenomena; when the system excitation frequency is between 1.75–3.24 and 3.92–4.425, the sticking phenomenon disappears, and only the chatter phenomenon exists. 2) When other parameters remain unchanged, and the mass ratio decreases from 1.15 to 0.85, nonlinear dynamic phenomena such as the transition between periodic bubbles and chaotic bubbles will be found. In this paper, the bifurcation and chaos characteristics of the impact vibration system of the wedge friction damper are studied, and the rich friction-induced vibration forms such as chatter and sticking are revealed, which provides a reference for improving the stability of vehicle operation and the selection of parameters in vehicle vibration reduction design in engineering practice.</p> </abstract>

Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models

We demonstrate a basic technique for simplifying time-periodic competition models, which is based on the utilization of periodic delta functions as population growth rates. We show that the Poincare mapping splits into a sequence of one-dimensional mappings. The study of the corresponding stable equilibria allows us to make conclusions concerning the coexistence and selection of the family of competitors. In particular, in “all vs. all” systems, for one of the populations to dominate, it is enough to surpass the others with a certain margin, and the correspondent stock constant does not depend on the number of competitors. We present paradoxical examples, where (1) a low-productive population can displace a highly productive one, (2) the displacement is non-transitive, (3) the coexistence is non-transitive. We also show how the delta functions can be utilized for the analysis of a “predator–prey” system.

Numerical analysis and experimental validation of nonlinear broadband monostable and bistable energy harvesters

Vibration based piezoelectric energy harvesting has been receiving more and more attention for supplying usable electrical energy to low-power electronic devices. This type of energy can produce the desired voltage to power any low electronic device or wireless sensor. But most of them provide lower voltage and insufficient power. In this paper, the nonlinear magnetic field is introduced to broaden the effective resonant bandwidth for overcoming this issue. The dynamic linear and nonlinear model of magnetic coupling piezoelectric vibration energy harvester is established based on the Hamilton principle, Euler-Bernoulli beam theory, piezoelectric theory, and Kirchhoff's law and law of mechanics. Numerical solutions of nonlinear energy harvesting systems are studied by using the Runge-Kutta method. The bifurcation diagram, Poincare map, power spectrum, and phase trajectory of voltage output are investigated with different excitation amplitudes and frequencies. Simulation of linear and nonlinear model were illustrated at given excitation levels. Simulation results show the harvesting performance can be improved by proper external magnetic coupling and potential energy function. The theoretical harmonic balance analysis results verify the numerical solution and influence the mechanism of excitation levels and interface resistances. Experimental verification is carried out to measure the nonlinear restoring force and examine the performance of nonlinear bistable and monostable energy harvesters. The experiment results show the nonlinear bistable and monostable energy harvesters can provide larger voltage and more power.

Topological structures of vibration responses for dual-rotor aeroengine

Many advanced aeroengines are typically designed with coaxial coupling dual-rotor structures. As a result, the vibration responses of the rotor system are characterized by complex multifrequency oscillations. In the paper, the topological structures of the nonlinear vibration responses of the dual-rotor system are revealed, providing a better understanding of the complex structural dynamics of many advanced aeroengines. A nonlinear dynamic model of the structural system of a dual-rotor-aeroengine experimental rig is established using the finite element (FE) method. This model takes into the blade-casing rubbing, misalignment, and the nonlinearities of the supporting rolling element bearings. The component mode synthesis (CMS) method is used to reduce the order of the dynamic model with high-degrees of freedom (DOFs) in order to enhance the computational efficiency. We first discover that long periodic structures exist in the multi-harmonic vibration responses of the dual-rotor system. The structures of Poincare mappings and bifurcation diagrams provide clearer and more concise information when obtained through long periodic sampling. The numerical periodic vibration response structures matched the measurements. These results help uncover the inherent structural characteristics of the complex nonlinear vibrations of a dual-rotor aeroengine.