Nonlinear Characteristics Of System Research Articles

Identification of the nonlinear characteristics of rubber bearings in model-free base-isolated buildings using partial measurements of seismic responses

Rubber-bearing isolation is one of the most successfully and widely used isolation technologies to provide lateral flexibility and energy dissipation capacity for reducing structural vibration and protecting the superstructure from damage. The seismic performance of the base-isolated structures partly depends on the nonlinear characteristics of the base isolation system. However, it is hard to establish proper mathematical models for the nonlinear hysteretic behaviors of base isolation due to the complexities of nonlinearities. Consequently, it is strongly desired to develop model-free methodologies for the nonlinear hysteretic performance identification with no assumption on the nonlinear hysteretic models of base isolation. In this paper, a novel method is proposed for this purpose. Firstly, the base isolation is in the linear state when the structure is under the weak earthquake, the restoring force is only provided by linear stiffness and viscous damping of base isolation, and the structural physical parameters can be estimated based on the extended Kalman filter approach. Then, the base isolation is in the nonlinear state when the structure is under the strong earthquake. The nonlinear hysteretic restoring forces from base isolation are treated as “unknown fictitious inputs” to the corresponding structural systems without base isolation. The generalized Kalman filter with unknown input algorithm is adopted for the simultaneous identification of the corresponding structural systems and the hysteretic restoring force of base isolation using only partial structural responses. No information about the structure is needed, and the responses at the location of the base isolation are not required, the proposed method is capable of identifying nonlinear characteristics of base isolation by the direct use of partial structural dynamic response. To validate the performances of the proposed method, some numerical simulation examples of identifying nonlinear hysteretic restoring forces of base isolation in different models are used.

Patterns of Gravitational Cooling in Schrödinger Newton System

We study time evolution of Schrodinger-Newton system using the self-consistent Crank-Nicolson method to understand the dynamical characteristics of nonlinear systems. Compactifying the radial coordinate by a new one, which brings the spatial infinity to a finite value, we are able to impose the boundary condition at infinity allowing for a numerically exact treatment of the Schrodinger-Newton equation. We study patterns of gravitational cooling starting from exponentially localized initial states. When the gravitational attraction is strong enough, we find that a small-sized oscillatory solitonic core is forming quickly, which is surrounded by a growing number of temporary halo states. In addition a significant fraction of particles escape to asymptotic regions. The system eventually settles down to a stable solitonic core state while all the excess kinetic energy is carried away by the escaping particles, which is a phenomenon of gravitational cooling.

Free Fluid-Structure Interaction Method for Accurate Nonlinear Dynamic Characteristics of the Plain Gas Journal Bearings

Comprehensive and deep understanding of the accurate nonlinear dynamic characteristics of the gas journal bearings plays an important role for reliable design of the rotor-bearing system. In this paper, the accurate nonlinear dynamics of a flexible rotor supported by two plain gas journal bearings (PGJB) are investigated by the free fluid-structure interaction method (FFSIM). The FFSIM does not make any simplifying assumptions, considers the interactions between the rotor shaft and the bearing sleeve at each transient time and uses the three-dimensional models for calculation. First, accuracy of the FFSIM is verified by the published experimental results. Then, bifurcation analysis, effects of rotating speed and rotor mass on shaft orbits, phase portraits and frequency response curves of the PGJB rotor system are performed, respectively. Results show that, under the same rotor system and boundary conditions, compared with the simplified two-dimensional analytical methods commonly seen in the open literature, the FFSIM could give more detailed and accurate nonlinear dynamic characteristics of the rotor-bearing system. The research of this paper is expected to further and deeply understand the accurate nonlinear dynamic characteristics of the PGJB rotor system.

Nonlinear dynamic response of a spur gear pair based on the modeling of periodic mesh stiffness and static transmission error

• A complete modeling procedure is applied including the modeling of gear mesh stiffness, transmission error and gear pair. • The analytical gear mesh stiffness shows a good agreement with the published result of finite element approach. • The numerical results of proposed dynamic model match well with the published experimental data. • The effects of several essential gear parameters are investigated. • Deeper properties of gear vibration are revealed to guide the actual design and control of gear dynamics. This study is concerned with the nonlinear frequency response characteristics of a spur gear pair system, in which the modeling of time varying mesh stiffness (TVMS) and static transmission error (STE) is stressed. Firstly, a complete modeling procedure is constructed by adopting existing models of TVMS, STE and gear pair, so that the basic gear features and design parameters are contained in the overall gear model. On this basis, a 1 degree-of-freedom (DOF) spur gear pair model is established including the TVMS, STE and nonlinear backlash. Then, to show the validity and reliability, theoretical solutions of the dynamic model are verified by the published numerical results and experimental data. Besides, the stability of the gear system is investigated analytically to obtain the boundaries that separate stable and unstable regions. Finally, parametric studies are conducted to reveal the effects of several key parameters, such as the contact ratio, spacing error, transmitted load and damping ratio. The results show that the modeling of TVMS and STE should be considered and emphasized to obtain precise predictions of system responses and to reveal in depth properties of gear dynamics. Furthermore, the parametric studies and stability analysis offer useful suggestions for researchers and engineers to achieve desirable design and control of dynamic behaviors of gear system.

Education as a Complex System: Conceptual and Methodological Implications

Education is a complex system, which has conceptual and methodological implications for education research and policy. In this article, an overview is first provided of the Complex Systems Conceptual Framework for Learning (CSCFL), which consists of a set of conceptual perspectives that are generally shared by educational complex systems, organized into two focus areas: collective behaviors of a system, and behaviors of individual agents in a system. Complexity and research methodologies for education are then considered, and it is observed that commonly used quantitative and qualitative techniques are generally appropriate for studying linear dynamics of educational systems. However, it is proposed that computational modeling approaches, being extensively used for studying nonlinear characteristics of complex systems in other fields, can provide a methodological complement to quantitative and qualitative education research approaches. Two research case studies of this approach are discussed. We conclude with a consideration of how viewing education as a complex system using complex systems’ conceptual and methodological tools can help advance education research and also inform policy.

An algorithm for diagnosing nonlinear characteristics of dynamic systems with the integrated periodicity ratio and lyapunov exponent methods

The present research aims to develop a novel approach for diagnosing the nonlinear behavior of dynamic systems with an algorithm integrating the Periodicity Ratio (P-R) and Lyapunov exponent methods. The advantages and disadvantages of the nonlinear behavior diagnostic methodologies with separate employment of the two methods are studied and compared. Although the two methods generally show effectiveness in diagnosing the characteristics of nonlinear dynamic systems, in some cases both methods may miss or misinterpret some of the characteristics of the nonlinear systems. In fact, the two methods can be complementary as identified in this research. With a specified integration of the two methods, the proposed algorithm maintains the advantages and overcomes the shortcomings of the two methods. The proposed algorithm therefore provides higher completeness, efficiency and accuracy to diagnose the nonlinear characteristics in dynamic systems compared with the existing methods such as the Lyapunov exponent method and the P-R method. The algorithm of the proposed approach is presented in detail with a case study to demonstrate its application.

Study of the q-Gaussian distribution with the scale index and calculating entropy by normalized inner scalogram

In this paper, we discuss a method based on wavelet analysis for the study of the q-index of the Gaussian distribution. We derive q-index from the scale index, iscale, using the expression; q≡1+2iscale where iscale is a wavelet based tool for measuring the degree of aperiodicity of a dynamical system in the range of 0≤iscale≤1. We show that this expression gives consistent results with the numerical approach of q-Gaussian distribution which determines the degree of non-extensivity of a dynamical system in the range of 1<q<3. We also suggest a new entropy calculation method based on the normalized inner scalogram for studying the chaotic characteristics of nonlinear dynamical systems.

Nonlinear response analysis for an aero engine dual-rotor system coupled by the inter-shaft bearing

This paper focuses on the nonlinear response characteristics of an aero engine dual-rotor system coupled by the cylindrical roller inter-shaft bearing. The motion equations of the system are formulated considering the unbalance excitations of the two rotors, vertical constant forces acting on the rotor system and the gravities. By using numerical calculation method, the motion equations are solved to obtain the nonlinear responses of the dual-rotor system. Accordingly, complex nonlinearities affected by the bearing radical clearance, the vertical constant force and the rotating speed ratio are discussed in detail. The jump phenomenon, hard resonant hysteresis characteristics are shown for a relatively large bearing clearance, and the soft resonant hysteresis characteristics can be observed for a relatively large vertical constant force. Moreover, the super-harmonic frequency components and the combined frequency components caused by the inter-shaft bearing are observed for both rotors. But the corresponding frequency components for the low-pressure rotor are more complex than that for the high-pressure rotor in same condition. These results would be helpful to recognize the nonlinear dynamic characteristics of dual-rotor bearing system.

A 50-T High-Stability Flat-Top Pulsed Magnetic Field Energized by a 100-MW Pulsed Alternator-Rectifier Power Supply With Model Predictive Control

High-stability flat-top pulsed magnetic field, which combines the advantages of pulsed high magnetic field and steady high magnetic field, is widely used in in the fields of physics, biology, and chemistry. The pulsed alternator-rectifier power supply, with controllable output voltage, is suitable to discharge a multicoil magnet to generate a synthetic flat-top magnetic field with high parameters. However, the coupling characteristics of coil current and the time-varying nonlinear characteristics of magnetic field system bring challenges to generate a high-stability flat-top pulsed magnetic field. This paper first adds a decoupling transformer in discharge circuit of the 50-T flat-top pulsed magnetic field system existed at the Wuhan Nation High Magnetic Field Center, Wuhan, China, and hence, the coupling effect of 50-T dual-coil magnet can be eliminated. Then, a control method of high-stability flat-top pulsed magnetic field based on the model predictive control is proposed. A prediction model of rectifier trigger angle is obtained by establishing the mathematical model of the system. A simulation model is built by MATLAB/Simulink to simulate the discharge process, and a 50-T/120-ms flat-top pulsed magnetic field with an accuracy of 400 ppm is generated.

Chaotic motion of piezoelectric material hyperbolic shell under thermoelastic coupling

Piezoelectric material, which exhibits excellent electro-mechanical conversion properties, is widely used in smart sensors and structures for sonar systems, weather detection and remote sensing. Hyperbolic shell structure made of piezoelectric material is liable to break down when it is used in high temperature environment, which is caused by the unexpected chaotic dynamic motion under the coupling effect of thermal filed and force field. Therefore, the chaotic nonlinear dynamic vibration of simply-supported piezoelectric material hyperbolic shell is studied under the combined action of temperature field and simple harmonic excitation. Based on the theory of finite deformation, the non-linear vibration equation and coordination equation of the hyperbolic shell are established. The non-linear dynamic equation of the structure is obtained by the Bubnov-Galerkin principle. The corresponding undisturbed Hamilton system has a homoclinic orbit. Using Melnikov function, the chaotic motion condition of the dynamic system under the criterion of Smale-horseshoe transformation is obtained. Furthermore, the mathematical model is established by Simulink software and the numerical simulations are performed by the fourth-order Runge-Kutta method. The simulation results accord well with those from the Melnikov method. The bifurcation diagram, the Lyapunov exponent diagram, the phase diagram and Poincaré section diagram are acquired to analyze the influence of temperature field on the non-linear characteristic of piezoelectric material hyperbolic shell system. When the temperature is close to 32℃ and 41℃, the Lyapunov index is less than 0 and the corresponding movement of the system is in the periodic zone, which is the same as that for a temperature range from 36℃ to 37℃. When the Lyapunov index is greater than 0, the corresponding movement of the system is in chaos zone. Therefore, the change of temperature has an additional effect on the stiffness of the system which affects the vibration of the system. The chaos and periodic zones of the system alternate with the increase of temperature and the vibration characteristics of the system can be controlled by changing the temperature field. Therefore, adjusting the temperature field can control the motion state of the system, which helps to improve reliability of the structure.

The Kushner-Stratonovich stochastic method for a master robot of a tele-robotic system: beyond extended Kalman filtering

The surgical robotic system, a tele-robotic system, comprises the operator, master-slave configuration in combination with the object. For the estimation and control of the surgical robotic system, the master-slave set up is the subject of greater robotic research activities as well as has proven promising. In this paper, the multi-dimensional stochastic differential equation as well as the higher-order nonlinear stochastic filter beyond extended Kalman filtering for a master robot of the surgical robotic system are developed. The higher-order nonlinear filtering preserves the greater qualitative characteristics of nonlinear stochastic systems by accounting observation correction terms in stochastic evolution equations. First, the Lagrangian mechanics is exploited to develop the vector stochastic model of the master robot. Secondly, this paper utilises the Kushner-Stratonovich equation, for filtering of the master robot noise equations. This paper demonstrates comparison between the higher-order Kushner-Stratonovich filter, extended Kalman filter and true state trajectories as well.

Analysis on complexity of optical variability based on approximate entropy in Sloan digital sky survey quasars

Variability is one of the most important observational features of quasars, and it is still not clear that the different quasars show different characteristic variability patterns. The optical variability of quasar is very complex, and optical variability has the non-linear characteristic of complex system. In this paper, the approximate entropy method is employed to analyze the complexities of variability in the Sloan digital sky survey (SDSS) stripe 82 quasars. Firstly, in order to show that the approximate entropy method has the effective ability to distinguish the different types of time sequences, the approximate entropy of periodic sequence, noise sequence, chaotic sequence and their mixed sequences are calculated by using the analog signals. The approximate entropy method proves to be an effective method to identify different types of time sequences. Then, we present the approximate entropy of optical variability of spectroscopically-confirmed quasars from the SDSS data release 7 spectroscopic catalog, and their complexities are analyzed. The results show that the maximum approximate entropy of quasars’ optical variability is only 0.58. The complexity of quasars’ optical variability is between the complexities of periodic sequence and white noise sequence. For nearly half of the samples, the complexities of their optical variability are basically consistent with the complexity of chaotic sequence. Quasars’ optical variability is neither completely periodic nor completely stochastic.

The Kushner-Stratonovich stochastic method for a master robot of a tele-robotic system: beyond extended Kalman filtering

The surgical robotic system, a tele-robotic system, comprises the operator, master-slave configuration in combination with the object. For the estimation and control of the surgical robotic system, the master-slave set up is the subject of greater robotic research activities as well as has proven promising. In this paper, the multi-dimensional stochastic differential equation as well as the higher-order nonlinear stochastic filter beyond extended Kalman filtering for a master robot of the surgical robotic system are developed. The higher-order nonlinear filtering preserves the greater qualitative characteristics of nonlinear stochastic systems by accounting observation correction terms in stochastic evolution equations. First, the Lagrangian mechanics is exploited to develop the vector stochastic model of the master robot. Secondly, this paper utilises the Kushner-Stratonovich equation, for filtering of the master robot noise equations. This paper demonstrates comparison between the higher-order Kushner-Stratonovich filter, extended Kalman filter and true state trajectories as well.

Crack characteristic analysis of multi-fault rotor system based on whirl orbits

In this paper, the nonlinear crack characteristics of different multi-fault rotor system are analyzed for crack detection based on whirl orbits. The nonlinear crack response can bring instability of the rotating shaft. The identification of the nonlinear characteristics is very important in the crack detection. Considering the interaction of rub-impact forces, time-varying stiffness and nonlinear oil-film forces, the nonlinear dynamic models are established. The numerical simulations put emphasis on the evolution of whirl orbits by studying orbit morphological characteristics near subcritical speed for a rotor system with crack. The bifurcation diagrams, time series and frequency spectra are used to analyze the nonlinear coupled behaviors of the multi-fault rotor system. The typical features and rules of whirl orbits are found in the rotor system with crack. The morphological characteristics of whirl orbit present the number and orientation changes of inner loops during the subcritical speed region, even though multi-fault exist in the rotor system with crack. The experimental whirl orbits of a rotor system with crack during passage through the 1/2 subcritical speeds agree with the theoretical analysis. The orbit morphological characteristics will offer a method for crack detection in multi-fault rotor system.

Dynamic analysis of nonlinear time-varying spur gear system subjected to multi-frequency excitation

Due to the complex working environment, gear systems always suffer from multiple excitations in actual engineering. This paper concerns the frequency response characteristics of a nonlinear time-varying spur gear system subjected to multi-frequency excitation. Firstly, a single degree-of-freedom gear pair model is established with consideration of the gear backlash, time-varying mesh stiffness and multiple harmonic excitations. Then, using the multiple time scales method, a comprehensive theoretical study is conducted to analyze various resonant cases including primary, parametric and combination resonances. Besides, parametric studies are accomplished to reveal the effects of the multi-frequency excitation on gear dynamics and to provide some useful references for reducing the vibration level. With the help of the fifth-order Runge–Kutta method, the numerical results are obtained to verify the validity of the analytical solutions and to emphasize the significances of the multi-frequency excitation. In addition, a comparison is performed between the numerical results and the published experimental results to validate the proposed gear model. Results show that the presence of the multi-frequency excitation will introduce the interaction between different harmonic excitations, which significantly affects the nonlinear vibration characteristics of a spur gear system. The proposed gear model with multi-frequency excitation could be more reliable and universal than that with single-frequency excitation. In addition, the results of parametric study could provide some suggestions to designers and researchers attempting to obtain desirable dynamic behaviors of a gear system subjected to multi-frequency excitation.

Local defect modelling and nonlinear dynamic analysis for the inter-shaft bearing in a dual-rotor system

• A force model for the inter-shaft bearing with a local defect on surface of the outer or inner race. • The system may resonant in some non-critical-speed region due to the local defect. • Parameters affecting the resonant frequencies and vibration amplitudes of the abnormal resonances. • A comprehensive parametric analysis is carried out to show the nonlinear resonant response characteristics. This paper presents a force model for the inter-shaft bearing with a local defect on the surface of the outer race or the inner race, and the nonlinear dynamic characteristics of a dual-rotor system affected by the local defect are investigated. A simplified dual-rotor system is presented with the consideration of the inter-shaft bearing's nonlinearities such as the Hertzian contact force and the radial clearance. The local defect is considered as a regular dent with a constant depth, thus the radial clearance will increase when rolling elements go through the range of the local defect. The motion equations of the system with eight degrees of freedom are formulated by using the Lagrange's equation. The nonlinear vibration responses of the dual-rotor system affected by the local defect are obtained using numerical method. The results show that there exist four abnormal resonances on the amplitude frequency curves of the system due to the effect of the local defect, apart from the couple of primary resonances excited by the unbalance excitations of the two rotors. With the aid of the characteristic defect frequency analysis, it is revealed that one pair of the abnormal resonances are excited by the characteristic defect frequency, and the other pair of the abnormal resonances are excited by the combination frequency. Furthermore, a comprehensive parametric analysis is carried out to give an insight into the nonlinear resonant response characteristics affected by parameters such as the depth and the span of the defect, the rotation speed ratio, the unbalances of two rotors, the stiffness and the damping of the linear elastic spring, and the radial clearance, the stiffness and the roller number of the inter-shaft bearing. The results show that the vibration amplitudes for the abnormal resonances are mainly determined by the depth and the span of the defect, while the resonant frequencies for the abnormal resonances are mainly influenced by the rotation speed ratio and the roller number of the inter-shaft bearing. However, the rotors’ unbalances mainly affect the corresponding primary resonance rather than the abnormal resonances. The obtained results will contribute to a better understanding of the nonlinear resonant response characteristics of dual-rotor systems with a local defect on the inter-shaft bearing, which are helpful for the fault diagnostics of the inter-shaft bearing in a dual-rotor system.

A novel road dynamic simulation approach for vehicle driveline experiments

A dynamic simulation approach for performing emulation experiments on vehicle driveline test bench is discussed in this paper. In order to reduce costs and shorten new vehicle development cycle time, vehicle simulation on the driveline test bench is an attractive alternative at the development phase to reduce the quantity of proto vehicles. This test method moves the test site from the road to the bench without the need for real chassis parts. Dynamic emulation of mechanical loads is a Hardware-in-the-loop (HIL) procedure, which can be used as a supplement of the conventional simulations in testing of the operation of algorithms without the need for the prototypes. The combustion engine is replaced by a electric drive motor, which replicates the torque and speed signature of an actual engine, The road load resistance of the vehicle on a real test road is accurately simulated on load dynamometer motor. On the basis of analyzing and comparing the advantages and disadvantages of the inverse dynamics model and the forward model based on speed closed loop control method, in view of the high order, nonlinear and multi variable characteristics of test bench system, a load simulation method based on speed adaptive predictive control is presented. It avoids the complex algorithm of closed loop speed compensation, and reduces the influence of inaccurate model parameters on the control precision of the simulation system. The vehicle start and dynamic shift process were simulated on the test bench.

Fuzzy control simulation of hot water supply system for ground source heat pump

Aimed to the hysteresis, time-varying, nonlinear characteristics of ground source heat pump systems, a fuzzy controller has been designed and simulation calculation is implemented with Matlab software. The results indicate that compared to the traditional PID control system, the adjustment time of the fuzzy control method is found shorter by 40%. The system with the fuzzy controller can reduce the steady-state error and enhance the resistance to disturbance.

On the dynamics of imperfect shear deformable microplates

This paper investigates the nonlinear forced dynamical behaviour of a geometrically imperfect viscoelastic shear-deformable microplate. The third-order shear deformation plate theory and the Kelvin–Voigt viscoelastic model are utilised in the framework of the modified version of the couple-stress theory to develop a model for the microsystem. The developed model is in the form of partial differential equations (PDEs) and accounts for geometric nonlinearities, damping nonlinearities, micro-scale size effects, and initial imperfection. Five coupled PDEs are derived for the five independent displacements and rotations. These PDEs are truncated to a set of nonlinearly coupled ordinary differential equations via application of a two-dimensional modal decomposition based on the Galerkin technique. The final set of equations consists of quadratic and cubic nonlinear terms for both damping and stiffness. An efficient numerical algorithm based on a continuation scheme is utilised to analyse the nonlinear forced vibration characteristics of such complicated system. The effects imperfection amplitude, damping nonlinearities, and micro-scale size on forced resonant vibration response are highlighted.

Displacement Transmissibility of Nonlinear Isolation System with Magnetorheological Damper

To improve the performance of single-stage vibration isolation system, the influence of the nonlinear magnetorheological (MR) damper on isolating the vibration from the base excitation was investigated. Being different from the traditional single degree of freedom (DOF) vibration isolation model, an unconstrained two DOF vibration isolation model is proposed, which contains the influence of the base mass. A revised Bingham model is adopted to describe the nonlinear damping force of MR damper. After reasonable assumptions, the displacement response under harmonic excitation is obtained using averaging method which is verified by numerical solution. The absolute displacement transmissibility is used to evaluate the dynamic characteristic of nonlinear isolation system. Parametric analysis is conducted to investigate the influence of the system parameters on improving transmissibility. The increase of MR parameters all obviously suppress the primary resonance response but reduce isolation effect in the high frequency. The increase of base mass can improve vibration isolation effect in the isolation frequency region but enlarge the resonance peak. These conclusions are of significant importance in the analysis and design of vibration attenuation system with MR damper.