Poincare Map Research Articles

Nonlinear dynamic analysis of worn gas foil bearings

Highly nonlinear vibration is one of important factors influencing the failure of gas foil bearing-rotor systems. In this paper, an integrated nonlinear dynamic model is constructed to study the dynamic vibration of worn gas foil bearings. In the wear model, the deformation of foil is considered. The Iordanoff model is used to solve the stiffness of foil, and equivalent viscous damping is introduced to amend the friction factor at the interface between the foil and bearing bracket. The incompressible Reynolds equation is solved by using ADI method, and the Newmark-β method is applied to obtain the trajectory of axis center. In numerical examples, the effects of worn zone on the trajectory of axis center, the Poincare map and bifurcation diagram are discussed. The nonlinear vibration resulting from the worn zone is revealed. Comparison with existing results validates this model.

Double mode model of size-dependent chaotic vibrations of nanoplates based on the nonlocal elasticity theory

In this paper vibrations of the isotropic micro/nanoplates subjected to transverse and in-plane excitation are investigated. The governing equations of the problem are based on the von Kármán plate theory and Kirchhoff–Love hypothesis. The small-size effect is taken into account due to the nonlocal elasticity theory. The formulation of the problem is mixed and employs the Airy stress function. The two-mode approximation of the deflection and application of the Bubnov–Galerkin method reduces the governing system of equations to the system of ordinary differential equations. Varying the load parameters and the nonlocal parameter, the bifurcation analysis is performed. The bifurcations diagrams, the maximum Lyapunov exponents, phase portraits as well as Poincare maps are constructed based on the numerical simulations. It is shown that for some excitation conditions the chaotic motion may occur in the system. Also, the small-scale effects on the character of vibrating regimes are illustrated and discussed.

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 oscillations of electrically driven aniso-visco-hyperelastic dielectric elastomer minimum energy structures

In view of their unique shape morphing behaviour, dielectric elastomer-based minimum energy structures (DEMES) have received an increasing attention in the technology of electroactive soft transduction. Because several of them undergo a time-dependent motion during their operation, understanding their nonlinear dynamic behaviour is crucial to their effective design. Additionally, in the recent past, there has been a growing scientific interest in imparting anisotropy to the material behaviour of dielectric elastomers in view of ameliorating their actuation performance. Spurred with these ongoing efforts, this paper presents an analytical framework for investigating the nonlinear dynamic behaviour of aniso-visco-hyperelastic DEMES actuator with an elementary rectangular geometry. We use a rheological model comprising two Maxwell elements connected in parallel with two single spring elements for modelling the material behaviour of the DE membrane. The governing equations of motion for the underlying non-conservative system are then derived using the Euler–Lagrange equation. The proposed model is used for building insights into the attainable equilibrium states, periodicity of the response as well as the resonant behaviour of the DEMES actuator over a feasible range of anisotropy and viscosity parameters. Our results reveal that the DEMES with hyperelastic material properties exhibits a supercritical pitchfork bifurcation of equilibrium state which is further accelerated in terms of attained equilibrium angle due to membrane anisotropy. A significant enhancement in the equilibrium angle attained by the structure with the extent of membrane anisotropy parameter is observed, indicating a favourable impact of material anisotropy. Poincare maps and phase-portraits are presented for assessing the periodicity of the nonlinear oscillations. The frequency response of the actuator for a combined DC and AC load indicates an upsurge in the resonant frequency with an increase in anisotropy parameter. The underlying analytical model and the trends presented in this study can find their potential use in the design and development of the futuristic anisotropic DEMES actuators subjected to time-dependent actuation.

Chaotic Attractors Generated by a Memristor-Based Chaotic System and Julia Fractal

Abstract A chaotic system has many applications in random number generator, image encryption, and secure communication. It is essential to present abundant chaotic attractors in a single chaotic system. Thus a novel 3D chaotic system based on memristor is designed and analyzed. The system's key advantages are as follows: firstly, both the multi-scroll and extended chaotic attractors are produced by merely modifying system parameters. The complex dynamical phenomena, such as coexisting multiple multi-scroll chaotic attractors, transient phenomenon and state transformation, can be observed; secondly, the proposed system combines with the classical Julia fractal and Julia deformation forms, respectively, and fascinating and various circular chaotic attractors are generated, which take good advantage of the combination of a memristor-based chaotic system and Julia fractal. Meanwhile, by applying both dynamic analysis and numerical simulation, basic behaviors of the system, such as stability of equilibrium point, dissipation and symmetry, Poincare maps, instantaneous phase, 0-1 test and power spectrum are employed to illustrate the superiority of the system. Some rather impressive and promising applications of the proposed chaotic system to chaotic confidential communication and image encryption will be achieved later.

Dynamic Behaviors of a Two-Degree-of-Freedom Impact Oscillator with Two-Sided Constraints

The dynamic model of a vibroimpact system subjected to harmonic excitation with symmetric elastic constraints is investigated with analytical and numerical methods. The codimension-one bifurcation diagrams with respect to frequency of the excitation are obtained by means of the continuation technique, and the different types of bifurcations are detected, such as grazing bifurcation, saddle-node bifurcation, and period-doubling bifurcation, which predicts the complexity of the system considered. Based on the grazing phenomenon obtained, the zero-time-discontinuity mapping is extended from the single constraint system presented in the literature to the two-sided elastic constraint system discussed in this paper. The Poincare mapping of double grazing periodic motion is derived, and this compound mapping is applied to obtain the existence conditions of codimension-two grazing bifurcation point of the system. According to the deduced theoretical result, the grazing curve and the codimension-two grazing bifurcation points are validated by numerical simulation. Finally, various types of periodic-impact motions near the codimension-two grazing bifurcation point are illustrated through the unfolding diagram and phase diagrams.

Nonlinear Dynamic Analysis and Control of FG Cylindrical Shell Fitted with Piezoelectric Layers

In this study, the nonlinear vibration control of functionally graded laminated piezoelectric cylindrical shells under simultaneous parametric axial and radial external excitations is presented. The partial differential equations of shells are derived based on Hamilton’s principle, first-order shear deformation theory (FSDT), and nonlinear von Karman relations. The coupled nonlinear ordinary differential equations are obtained by Galerkin’s procedure and solved by the method of static condensation. Two piezoelectric layers are placed on the outer and inner surfaces of the cylindrical shell each as distributed sensor and actuator. Then the constant-gain negative velocity feedback strategy is employed. Regarding the nonlinear equations of motion, for the first time, the vibration analysis and active vibration control of smart FG cylindrical shells under combined parametric and external excitations are analyzed using the multiple scales approach. The effects of various parameters such as power index, external excitation’s amplitude, and control gain on the dynamic behavior of the system are investigated, using bifurcation diagrams, phase portraits, time histories, and Poincare maps. It is shown that quasi-periodic motion is the most common behavior of the system and controller gain and power index have inevitable effects on enhancing the quasi-periodic response of the system. Care should be exerted in selecting the parameters to have the desired response in the broad range of excitation frequency.

Identification of rotor-stator rub and dependence of dry whip boundary on rotor parameters

Rotor rub is one of the most critical and catastrophic event that occurs in gas turbine aircraft engines and other rotating machineries. Rotor rubbing introduces non-linearity in the system which makes the study of such phenomena difficult. The present study models a Jeffcott rotor along with a stator. This paper aims to develop a rotor rub model with Smoothening Function (SF) which can detect the onset of rotor rub efficiently whilst revealing the properties of Instantaneous Frequencies (IF) of rubbing rotor. First, the Adaptive Chirp Mode Decomposition Method (ACMD) has been used to detect the rub and establish the properties of IF. Next, the benefits of smoothening function in the rotor rub model have been discussed in terms of computation time. Further, the nonlinear dynamical behavior of the model has been studied using Poincare maps and Bifurcation diagram. The results reveal that IF of rotor possesses oscillating property which can be observed using ACMD method and can be used to detect the onset of rub. Moreover, use of SF effectively reduces the computation time by a factor of 11.6 when compared with classic rub model. The rotor orbits, Poincare Maps and Bifurcation diagram clearly shows that the rotor exhibits properties such as full annular rub, partial rub along with quasi-periodic motion. The presence of backward whirl using full spectrum has been observed and boundary for dry whip has been identified. Stability analysis using Floquet Theory reveals that the dry whip makes the rotor unstable. Dry whip boundary’s dependence on the parameters such as rotor stiffness ratio, damping ratio and friction coefficient has been discusses in detail.

Effect of polymer chains entanglements, crosslinks and finite extensibility on the nonlinear dynamic oscillations of dielectric viscoelastomer actuators

Soft materials exhibiting large deformation under external stimuli have gained an increasing attention in the recent past because of their potential applications in soft transducers aimed at achieving biomimetic actuation. This paper theoretically analyzes the effect of internal properties, including entanglements, crosslinks and finite extensibility of polymer chains along with inherent viscoelastic properties of polymers on the performance of Dielectric Elastomer actuator (DEA) in dynamic modes of actuation. A physics based nonaffine material model proposed by Davidson and Goulbourne is used to model the polymer chains entanglements, crosslinks and finite extensibility. To incorprate the viscoelastic properties, a rheological material model based on the additive decomposition of the isotropic strain energy density into equilibrium and viscous parts is implemented. A computationally efficient method, which relies on the principle of least action is used for extracting the governing equation representing the dynamic motion of the DE actuator. The results demonstrate that DEAs with strong entanglements and crosslinks along with small finite extensibility of polymer chains exhibits lower deformation level in DC dynamic modes of actuation. It is inferred that the strong entanglements and crosslinks in polymer chains enhances the resonant frequency, but debilitate the intensity of vibration of viscoelastic DEAs. Further, the periodicity and stability of the nonlinear oscillations exhibited by the viscoelastic DEAs are assessed by employing the Poincare maps and phase portraits. The results of the present investigation can be applied effectively in bridging the mechanism between the microcosmic polymer chains and macroscopic dynamic behavior of viscoelastic DE actuators.

Pathfollowing of high-dimensional hysteretic systems under periodic forcing

The dynamic response and bifurcations of high-dimensional systems endowed with hysteretic restoring forces in all degrees of freedom are investigated. Two types of hysteresis models are considered, namely the Bouc–Wen model and a differential version of the so-called exponential model of hysteresis. The numerical technique tailored for tackling high-dimensional hysteretic systems is based on an enhanced pathfollowing approach based on the Poincare map. In particular, a five-dof mass-spring-damper-like system, with each rheological element described by the Bouc–Wen or the exponential model of hysteresis enriched by cubic and quintic nonlinear elastic terms, is investigated and a rich variety of nonlinear responses and bifurcations is found and discussed.

From Poincaré Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential

In this paper we compare the method of Lagrangian descriptors with the classical method of Poincare maps for revealing the phase space structure of two degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a two degree-of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincare maps are compared for their ability to determine the phase space structures that govern this dynamical process.

Effect of temperature on the nonlinear dynamic behavior of two-lobe non-circular gas-lubricated micro-bearings

In this paper, the molecular gas lubrication model was used to analyze the nonlinear dynamic behavior of two-lobe non-circular gas-lubricated micro-bearings. The effects of temperature rise are taken into account. At high temperatures, in addition to gas rarefaction, its viscosity and friction will also change, and slip across boundaries will occur. The rarefaction of the lubricating gas film caused by the microscale effect at high temperatures was considered. The effects of temperature and rotation speed (with and without rarefaction effect) on the dynamic behavior of the non-circular micro gas bearing were studied. The nonlinear equation governing the gas behavior is discretized using the finite-element method and then solved simultaneously with the dynamic equations of rotor motion using the fourth-order Runge–Kutta method. Center orbit diagrams, phase portraits, Poincare maps, power spectrum, and bifurcation diagrams are used to investigate the dynamic behavior of two-lobe non-circular gas-lubricated micro-bearings. Some results show that with increasing temperature, the rotor behavior changes from T-periodic to quasi-periodic. It was also observed that at high temperatures, with increasing rotational speed, the behavior of the system changes from T-periodic to quasi-periodic, but if the gas is rarefied, this change occurs at a slower speed.

Comparisons of nonlinear vibrations among pure polymer plate and graphene platelet reinforced composite plates under combined transverse and parametric excitations

The comparative researches of the nonlinear vibrations are obtained among the pure polymer plate and three types of graphene platelet reinforced polymer composite plates under combined the transverse and parametric excitations. All edges of the pure polymer and graphene platelet reinforced polymer composite plates are simply supported. Both the uniform and functionally graded distribution forms of the graphene platelets are considered. The differential governing equations of motion for the pure polymer and graphene platelet reinforced polymer composite plates are derived based on the first-order shear deformation plate theory, von Kármán strain displacement relationship and Hamilton principle. The fourth-order Galerkin truncation is employed to discretize the partial differential governing equations of motion to four-degree-of-freedom nonlinear dynamical system. The natural frequencies of the first four vibration modes are examined for the pure polymer and graphene platelet reinforced polymer composite plates with different geometric characteristics. In order to compare the nonlinear vibrations of the prestressed pure polymer and prestressed graphene platelet reinforced polymer composite plates subjected to the transverse point excitation and in-plane uniaxial/biaxial excitations, the time histories, phase portraits and Poincare maps are presented. Some validated results are conducted to present the accuracy of the present approach. The effects of the distribution forms and weight fractions of the graphene platelets on the nonlinear vibrations for the graphene platelet reinforced polymer composite plates are investigated in detail. The research results demonstrate that the nonlinear vibration behaviors of the prestressed pure polymer plate under the complex excitations can be remarkably stabilized by the reinforcement of the graphene platelets.

Application of the piecewise constant method in nonlinear dynamics of drill string

Abstract Aiming at the current development of drilling technology and the deepening of oil and gas exploration, we focus on better studying the nonlinear dynamic characteristics of the drill string under complex working conditions and knowing the real movement of the drill string during drilling. This paper firstly combines the actual situation of the well to establish the dynamic model of the horizontal drill string, and analyzes the dynamic characteristics, giving the expression of the force of each part of the model. Secondly, it introduces the piecewise constant method (simply known as PT method), and gives the solution equation. Then according to the basic parameters, the axial vibration displacement and vibration velocity at the test points are solved by the PT method and the Runge–Kutta method, respectively, and the phase diagram, the Poincare map, and the spectrogram are obtained. The results obtained by the two methods are compared and analyzed. Finally, the relevant experimental tests are carried out. It shows that the results of the dynamic model of the horizontal drill string are basically consistent with the results obtained by the actual test, which verifies the validity of the dynamic model and the correctness of the calculated results. When solving the drill string nonlinear dynamics, the results of the PT method is closer to the theoretical solution than that of the Runge–Kutta method with the same order and time step. And the PT method is better than the Runge–Kutta method with the same order in smoothness and continuity in solving the drill string nonlinear dynamics.

Bifurcation and Number of Periodic Solutions of Some 2n-Dimensional Systems and Its Application

In high dimension, the bifurcation theory of periodic orbits of nonlinear dynamics systems are difficult to establish in general. In this paper, by performing the curvilinear coordinate frame and constructing a Poincare map, we obtain some sufficient conditions of the bifurcation of periodic solutions of some 2n-dimensional systems for the unperturbed system in two cases: one is a decoupled n-degree-of-freedom nonlinear Hamiltonian system and the other has an isolated invariant torus. We use a new method and study new types of systems compared with the existing results. As an application we study the bifurcation and number of periodic solutions of an ice covered suspension system. Under a certain parametrical condition, the number of periodic solutions of this system can be 2 or 1 with the variation of parameter $$p_{2}$$ .

Andronov-Hopf bifurcations, Pomeau-Manneville intermittent chaos and nonlinear vibrations of large deployable space antenna subjected to thermal load and radial pre-stretched membranes with 1:3 internal resonance

Abstract Andronov-Hopf bifurcations, Pomeau-Manneville intermittent chaos and nonlinear vibrations of the large deployable space antenna (LDSA) subjected to the thermal load with the case of 1:3 internal resonance are studied for the first time. The frequencies and vibration modes of the LDSA are analyzed by using the finite element method. It is found that there may exist an approximate threefold relationship between the fourth-order and first-order frequencies of the LDSA. The LDSA is simplified to a composite laminated equivalent cylindrical shell clamped along a generatrix and with the radial pre-stretched membranes at two ends subjected to the thermal load. Considering the case of 1:3 internal resonance, four-dimensional nonlinear averaged equations are obtained by using the method of multiple scales. The amplitude-frequency response curves and amplitude-force response curves are obtained for the LDSA subjected to the thermal load through the prediction-correction continuation algorithm. The fold bifurcation and Andronov-Hopf bifurcation points are located on these resonant response curves. The equivalent model of the LDSA has the hardening spring characteristics. The thermal load has very significant influences on the stability of the LDSA. The nonlinear dynamics of the equivalent model for the LDSA are investigated by using the fourth-order Runge-Kutta algorithm. These nonlinear dynamic behaviors are described by the bifurcation diagrams, waveforms, phase plots, and Poincare maps. The periodic doubling bifurcation and Pomeau-Manneville type intermittent chaos appear in this nonlinear dynamical system. According to the topological evolution of the phase trajectories, the phase-locking phenomenon and a special evolution path of the chaotic attractors are found.

Influence of contact interface friction of bolted disk joint on motion stability of rotor-bearing system

In rotating machines, bolted joints are widely employed to connect adjacent disks to make multiple parts into an integrated one. The contact interface, subjected to tangential force introduced by unbalanced forces during normal operation, influencing the rotor dynamics, especially in the presence of relative speed, is of interest for investigations. However, there are little researches on the rotor dynamics considering the contact interface friction of the bolted disk joint. Thus, a dynamic model of the rotor-bearing system is established, which considers the friction at the contact interface between disks, nonlinear oil-film force, etc. The nonlinear dynamic phenomena of the rotor system were investigated using the fourth-order Runge–Kutta method. Bifurcation diagrams, vibration waveforms, frequency spectrums, shaft orbits, and Poincare maps are used to demonstrate the diversity of the system responses. The results indicate that the rotor system exhibits a rich diversity motion states, such as periodic-1 motion, multiple periodic motion, and quasi-periodic motion. The parametric study about the effect of friction coefficient on rotor dynamics is also conducted in the present work, which shows that a larger friction coefficient will lead the system enters into an unstable motion state at high rotating speed, and make the system enters into multiple periodic bifurcations at a higher rotating speed. The corresponding results can contribute to the further understanding of the nonlinear dynamic characteristics of the bolted joint rotor-bearing system.

Extraction and Visualization of Poincare Map Topology for Spacecraft Trajectory Design

Mission designers must study many dynamical models to plan a low-cost spacecraft trajectory that satisfies mission constraints. They routinely use Poincaré maps to search for a suitable path through the interconnected web of periodic orbits and invariant manifolds found in multi-body gravitational systems. This paper is concerned with the extraction and interactive visual exploration of this structural landscape to assist spacecraft trajectory planning. We propose algorithmic solutions that address the specific challenges posed by the characterization of the topology in astrodynamics problems and allow for an effective visual analysis of the resulting information. This visualization framework is applied to the circular restricted three-body problem (CR3BP), where it reveals novel periodic orbits with their relevant invariant manifolds in a suitable format for interactive transfer selection. Representative design problems illustrate how spacecraft path planners can leverage our topology visualization to fully exploit the natural dynamics pathways for energy-efficient trajectory designs.

Comparative Case Study on Oscillatory Behavior in Power Systems of Marine Vessels With High Power Converters

The chief aim of the proposed paper is to compare the behavior of supply voltage registered onboard three vessels equipped with high power converters during normal operation. The vessels considered are a chemical tanker with shaft generator working via power converter, a research-training ship with bow thruster, and a research vessel with dynamic positioning (DP) and electrical propulsion. In the case of the two latter ships, power electronic loads are present, which include a bow thruster adjustable speed drive equipped with the six pulse uncontrolled rectifier and the adjustable speed drive with an active front end (AFE) for propulsion system. For each ship, instantaneous frequency is determined and subsequently fluctuations of fundamental components and chosen harmonics are calculated using zoom-DFT. The characteristics of these irregular fluctuations are compared and commented on. Next, phase portraits and Poincare map of the registered voltages are calculated and compared. Moreover, the impact of ship rolling and pitching on the oscillatory behavior is assessed for the DP ship. The final conclusions identify reasons for oscillatory behavior in electric power systems of marine vessels containing high power converters on the source side and load side. The feasibility of using voltage trajectories as well as Poincare maps for oscillatory behavior detection is assessed.

Two Action-Angle Surprises on the Sphere

The linearized Poincare map of a periodic orbit of a completely integrable Hamiltonian system is examined in the light of the finer description we get by using coordinate changes in the Lagrangian odd symplectic group. In particular, we obtain non-eigenvalue invariants called moduli. These invariants are surprisingly subtle to calculate even in the case of the geodesic flow on a 2-sphere, and reveal dynamic-geometric information that is otherwise symplectically invisible.