Bifurcations and chaotic threshold for a nonlinear system with an irrational restoring force

Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor＇s expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov＇s method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. （Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soe. A 366 635）.

FINITE TIME BLOW UP OF THE SOLUTIONS TO BOUSSINESQ EQUATION WITH LINEAR RESTORING FORCE AND ARBITRARY POSITIVE ENERGY

Finite time blow up of the solutions to Boussinesq equation with linear restoring force and combined power nonlinearities is studied. Sufficient conditions on the initial data for nonexistence of global solutions are derived. The results are valid for initial data with arbitrary high positive energy. The proofs are based on the concave method and new sign preserving functionals.

Restoring force correction based on online discrete tangent stiffness estimation method for real-time hybrid simulation

In real-time hybrid simulation（RTHS）, it is difficult if not impossible to completely erase the error in restoring force due to actuator response delay using existing displacement-based compensation methods. This paper proposes a new force correction method based on online discrete tangent stiffness estimation（online DTSE） to provide accurate online estimation of the instantaneous stiffness of the physical substructure. Following the discrete curve parameter recognition theory, the online DTSE method estimates the instantaneous stiffness mainly through adaptively building a fuzzy segment with the latest measurements, constructing several strict bounding lines of the segment and calculating the slope of the strict bounding lines, which significantly improves the calculation efficiency and accuracy for the instantaneous stiffness estimation. The results of both computational simulation and real-time hybrid simulation show that：（1） the online DTSE method has high calculation efficiency, of which the relatively short computation time will not interrupt RTHS; and（2） the online DTSE method provides better estimation for the instantaneous stiffness, compared with other existing estimation methods. Due to the quick and accurate estimation of instantaneous stiffness, the online DTSE method therefore provides a promising technique to correct restoring forces in RTHS.

Prediction method of restoring force based on online AdaBoost regression tree algorithm in hybrid test

In order to solve the poor generalization ability of the back-propagation(BP)neural network in the model updating hybrid test,a novel method called the AdaBoost regression tree algorithm is introduced into the model updating procedure in hybrid tests.During the learning phase,the regression tree is selected as a weak regression model to be trained,and then multiple trained weak regression models are integrated into a strong regression model.Finally,the training results are generated through voting by all the selected regression models.A 2-DOF nonlinear structure was numerically simulated by utilizing the online AdaBoost regression tree algorithm and the BP neural network algorithm as a contrast.The results show that the prediction accuracy of the online AdaBoost regression algorithm is 48.3%higher than that of the BP neural network algorithm,which verifies that the online AdaBoost regression tree algorithm has better generalization ability compared to the BP neural network algorithm.Furthermore,it can effectively eliminate the influence of weight initialization and improve the prediction accuracy of the restoring force in hybrid tests.

Experimental study on characteristic curves of restoring forces of Populus alba var. pyramidalis

Static tests under cyclic loading were carried out on Populus alba var. pyramidalis to determine its characteristic curves of restoring forces. The results show that P alba var. pyramidalis had good elastic performance. There was degradation of strength and stiffness under cyclic loading and unloading, where characteristic curves of restoring forces ofP. alba var. pyrarnidalis assumed a ＂flat and contrary S-shape＂ form. Simultaneously, P alba vat. pyramidalis showed typical characteristics of brittle destruction under large high peak loading and unloading. Furthermore, dynamic tests were carried out under wind loads to obtain dynamic displacement curves and dynamic strain curves. P alba var. pyramidalis reflected its stochastic dynamic performance in building up its resistance to the stochastic wind loads and its dynamically hysteretic properties. The dynamic response of this species was random and fuzzy under stochastic wind loads. This study has theoretical significance and reference value to research on the static characteristics and wind-induced dynamic performance of forests.

Stochastic response analysis of noisy system with non-negative real-power restoring force by generalized cell mapping method

The stochastic response of a noisy system with non-negative restoring force is investigated. The generalized cell mapping （GCM） method compute the transient and stationary probability density functions （PDFs） real-power is used to Combined with the global properties of the noise-free system, the evolutionary process of the tran- sient PDFs is revealed. The results show that stochastic P-bifurcation occurs when the system parameter varies in the response analysis and the stationary PDF evolves from bimodal to unimodal along the unstable manifold during the bifurcation.

Experimental study of the restoring force mechanism in the self-centering beam （SCB）

In the past, several self-centering （SC） seismic systems have been developed. However, examples of selfcentering systems used in practice are limited due to unusual field construction practices, high initial cost premiums and deformation incompatibility with the gravity framing. A self centering beam moment frame （SCB-MF） has been developed that mitigates several of these issues while adding to the advantages of a typical SC system. The self-centering beam （SCB） is a shop-fabricated, self-contained structural component that when implemented in a moment resisting frame can bring a building back to plumb after an earthquake. This paper describes the SCB concepts and experimental program on five SCB specimens at two-third scale relative to a prototype building. Experimental results are presented including the global force-deformation behavior. The SCBs are shown to undergo 5%-6% story drift without any observable damage to the SCB body and columns. Strength equations developed for the SCB predict the moment capacity well, with a mean difference of 6% between experimental and predicted capacities. The behavior of the restoring force mechanism is described. The limit states that cause a loss in system＇s restoring force which lead to a decrease in the selfcentering capacity of the SCB-MF, are presented.

A Nonlinear Restoring Effect Study of Mooring System and its Application

Mooring system plays an important role in station keeping of floating offshore structures. Coupled analysis on mooring-buoy interactions has been increasingly studied in recent years. At present, chains and wire ropes are widely used in offshore engineering practice. On the basis of mooring line statics, an explicit formulation of single mooring chain/wire rope stiffness coefficients and mooring stiffness matrix of the mooring system were derived in this article, taking into account the horizontal restoring force, vertical restoring force and their coupling terms. The nonlinearity of mooring stiffness was analyzed, and the influences of various parameters, such as material, displacement, pre-tension and water depth, were investigated. Finally some application cases of the mooring stiffness in hydrodynamic calculation were presented. Data shows that this kind of stiffness can reckon in linear and nonlinear forces of mooring system. Also, the stiffness can be used in hydrodynamic analysis to get the eieenfrequencv of slow drift motions.

Introduction to Synchronized Kinematic and Electromagnetic Mechanics

Introduction to fundamental physics according to the parallel harmonization of kinematic and electromagnetic mechanics, in accordance with Wilhelm Wien’s project, which involved the integration in kinematic mechanics of the mass increase of the electron as a function of its velocity, as measured by Walter Kaufmann with his bubble-chamber experiments, and analyzed and confirmed by H. A. Lorentz and all the leading edge physicists who then re-analyzed this data.

Poly-Silicon Micromachined Switch

By using LPCVD SiO 2 and poly silicon as sacrificial layer and cantilever respectively,a poly silicon micromachined RF MEMS(radio frequency microelectronic mechanical system) switch is fabricated.During the fabrication process,the stress of poly silicon is released to prevent poly silicon membrane from bending,and the issue of compatibility between RF switch and IC process technology is also resolved.The low residual tensile stress poly silicon cantilever is obtained by the optimization.The switch is tested,and the preliminary test results show:the pull down voltage is 89V,and the switch speed is about 5μs.The switch provides the potential to build a new fully monolithic integrated RF MEMS for radar and communications applications.

Frame property of unequal storey height with specially shaped columns under cyclic loading

A 1/3-scale reinfored concrete(RC) frame of unequal storey height with specially shaped columns was tested under low frequency cyclic loading.The damage characteristic,bearing capacity,deformation capacity and ductility were analyzed.The restoring force model of the frame was obtained based on the study of the hysteresis curve measured in experiment,and the stiffness degeneration characteristics of every storey of the frame were analyzed.Finally the accumulated damage was analyzed with the damage assessment model.It is shown that the seismic behavior of this frame of unequal storey height with specially shaped columns is generally good,but the bottom of first floor column is a weak part,which should be paid more attention in design,and the restoring force model derived from this experiment can be seen as a valuable guide for design and non-linear finite element analysis for this kind of structure.

Analysis of a kind of Duffing oscillator system used to detect weak signals

The stability of the periodic solution of the Duffing oscillator system in the periodic phase state is proved by using the Yoshizaw theorem, which establishes a theoretical basis for using this kind of chaotic oscillator system to detect weak signals. The restoring force term of the system affects the weak-signal detection ability of the system directly, the quantitative relationship between the coefficients of the linear and nonlinear items of the restoring force of the Duffing oscillator system and the SNR in the detection of weak signals is obtained through a large number of simulation experiments, then a new restoring force function with better detection results is established.

Data-based model free hysteresis identification for a nonlinear structure

In the last two decades, the damage detection for civil engineering structures has been widely treated as a modal analysis problem and most of the currently available vibration-based system identification approaches are based on modal parameters, namely the natural frequencies, mode shapes and damping ratios, and/or their derivations, which are suitable for linear systems. Nonlinearity is generic in engineering structures. For example, the initiation and development of cracks in civil engineering structures as typical structural damages are nonlinear process. One of the major challenges in damage detection, early warning and damage prognosis is to obtain reasonably accurate identification of nonlinear performance such as hysteresis which is the direct indicator of damage initiation and development under dynamic excitations. In this study, a general data-based identification approach for hysteretic performance in form of nonlinear restoring force using structural dynamic responses and complete and incomplete excitation measurement time series was proposed and validated with a 4-story frame structure equipped with smart devices of magneto-theological （MR） damper to simulate nonlinear performance. Firstly, as an optimization method, the least-squares technique was employed to identify the system matrices of an equivalent linear system of the nonlinear structure model basing on the exci- tation force and the corresponding vibration measurements with impact test when complete and incomplete excitations; and secondly, the nonlinear restoring force of the structure was identified and compared with the test measurements fi- nally. Results show that the proposed data-based approach is capable of identifying the nonlinear behavior of engineering structures and can be employed to evaluate the damage initiation and development of different structure under dynamic loads.

NEW ANALYTICAL METHOD ON SPACE OBLIQUE SPRING

The kinematical characteristic analysis of a space oblique spring is carried out.It first points out that the vibration of a space oblique spring can be regarded as the composition of an oscillatory vibration and a straight linear vibration.The formulas are derived for the calculation of restoring force and damping force of the spring according to the idea above.

Correcting Theory of an Ultrasonic Stepping Motor Having Self-correcting Function

An open looped ultrasonic stepping motor with self correction function was studied theoretically. In order to analyse the mechanism for the motor’s function, A linear ultrasonic self correcting stepping motor is designed, and the dynamics of the motor was studied. For clarification, a set of experiment device was also designed, the calculated results were consistent with the experimental ones.