参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞,可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．     

On the refined modeling of the force distribution in a bistable magnetoelastic energy harvesting system due to a magnetic field

Abstract: The problem of a bistable magnetoelastic beam under base excitation was discussed in [1] under the aspect of chaotic behaviour in mechanical systems. Three decades later the system was used in [2] to design an energy harvesting system which performs well under harmonic excitation for a broad range of excitation frequencies due to its bistability. The initial modeling was tailored to obtain a model with one degree of freedom based on the assumption that the magnetic force acts on the beam tip only. A more appropriate model can be found when considering a distributed force along the beam. The authors present the force distribution on a ferromagnetic beam due to the magnetic field of two permanent magnets. A semi-analytic method is used to compute the magnetic field. The force distribution can in future be used to derive a refined nonlinear dynamical model for the ferromagnetic elastic beam. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surface and nonlocal effects on response of linear and nonlinear NEMS devices

Nonlocal and surface effects become important for nanoscale devices. To model these effects on frequency response of linear and nonlinear nanobeam subjected to electrostatic excitation, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. Subsequently, we apply Galerkin’s method with exact mode shape including nonlocal and surface effects to get static and dynamic modal equations. After validating the procedure with the available results, we analyze the variation of pull-in voltage and frequency resonance by varying surface and nonlocal parameters. To do frequency analysis of nonlinear system, we solve nonlinear dynamic equation using the method of multiple scale. We found that the frequency response of nonlinear system reduces for fixed excitation as the surface and nonlocal effects increase. Also, we found that the nature of nonlinearity can be tuned from hardening to softening by increasing the nonlocal effects.     

Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank

This paper aims to study the stability for primary frequency regulation of hydro-turbine governing system with surge tank. Firstly, a novel nonlinear mathematical model of hydro-turbine governing system considering the nonlinear characteristic of penstock head loss is introduced. The nonlinear state equations under opening control mode and power control mode are derived. Then, the nonlinear dynamic performance of nonlinear hydro-turbine governing system is investigated based on the stable domain for primary frequency regulation. New feature of the nonlinear hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing with a linear model, and the effect mechanism of nonlinear characteristic of penstock head loss is revealed. Finally, the concept of critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solution is derived. The combined tuning and optimization method of governor parameters and sectional area of surge tank is proposed. The results indicate that for the primary frequency regulation under opening control mode and power control mode, the nonlinear hydro-turbine governing system is absolutely stable and conditionally stable, respectively. The stability of the nonlinear hydro-turbine governing system and linear hydro-turbine governing system is the same under opening control model and different under power control model. The nonlinear characteristic of penstock head loss mainly affects the initial stage of dynamic response process of power output, and then changes the stability of the nonlinear system. The critical stable sectional area of surge tank makes the system reach critical stable state. The governor parameters and critical stable sectional area of surge tank jointly determine the distributions of stability states.     

Hydrocarbon microtremors interpreted as nonlinear oscillations driven by oceanic background waves

Characteristic low frequency seismic signals have been observed in areas where hydrocarbon reservoirs are present. A possible interpretation is the excitation of hydrocarbon related resonances. Basic models of an oscillating liquid filled porous medium are investigated. Synthetic spectra of the ab initio Navier–Stokes equations and of basic linear and nonlinear one-dimensional oscillators show characteristic features of measured spectra when oceanic background waves around 0.1–0.2 Hz are assumed to be the external driving force.     

Nonlinear response of a beam under distributed moving contact load

In this paper, the nonlinear behavior of a one-dimensional model of the disc brake pad is examined. The contact normal force between the disc brake pad lining and rotor is represented by a second order polynomial of the relative displacement between the two elastic bodies. The frictional force due to the sliding motion of the rotor against the stationary pad is modeled as a distributed follower-type axial load with time-dependent terms. By Galerkin discretization, the equation governing the transverse motion of the beam model is reduced to a set of extended Duffing system with quasi-periodically modulated excitations. Retaining the first two vibration modes in the governing equations, frequency response curves are obtained by applying a two-dimensional spectral balance method. For the first time, it is predicted that nonlinearity resulting from the contact mechanics between the disc brake pad lining and rotor can lead to a possible irregular motion (chaotic vibration) of the pad in the neighborhood of simple and parametric resonance. This chaotic behavior is identified and quantitatively measured by examining the Poincaré maps, Fourier spectra, and Lyapunov exponents. It is also found that these chaotic motions emerge as a result of successive Hopf bifurcations characterized by the torus breakdown and torus doubling routes as the excitation frequency varies. Various aspects of the numerical difficulties in the solution of the nonlinear equations are also discussed.     

Identification and Self Tuning Control for Active Magnetic Bearing Systems: Levitation of Unknown Rotors

The present paper deals with the problem of levitating rotors with unknown characteristics by means of active magnetic bearings whose properties are known. This problem is of interest in a technical setting to shorten the development time of AMB systems, in particular for controller design. Theoretical interest arises from the fact that several issues in the area of identification and self tuning control are addressed for an unstable system. Our aim is to identify the flexible rotor including gyroscopic effects and to automatically design a robustly stabilizing controller for this system that can be used for running the system under regular operating conditions. To this end, a rigid body model of the rotor is identified based on measured step responses from the plant. Then, the bearings are adjusted to have very low stiffness, and a controller with steep roll‐off is designed in order to avoid excitation of the unknown flexible modes of the system. Once the rotor is floating, the identification algorithm from [1] is applied to obtain information on the flexible modes of the system. Based on this extended model, a robust controller allowing for slow rotation of the rotor is designed. With the rotor rotating at a moderate speed, the frequency response functions are measured, and based on these measurements, the gyroscopic matrixof the system is identified, completing the system model and allowing for design of the desired controller for normal operation. The present contribution focusses on identification of the rigid body model of the flexible rotor.     

基于非线性能量阱的双频激励非线性系统减振

针对某型民用航空发动机双频带激励特点,建立了单自由度线性振子耦合非线性能量阱（nonlinear energy sink,NES）的动力学模型.根据典型双转子发动机在巡航状态下低、高特征频率比（1∶4.74）,为系统设定双频带简谐外激励.利用四阶Runge-Kutta算法,研究了耦合NES振子时系统的振动抑制特征,并从外激励频率对系统主振子动能、系统总体能量的影响等方面,与未耦合NES系统、耦合线性动力吸振器两种情况下的数值计算结果进行对比分析.研究结果表明NES对双频带外激励具有更好的振动抑制效果,用NES降低航空发动机振动有可行性.     

Nonlinear low frequency water waves in a cylindrical shell subjected to high frequency excitations – Part I: Experimental study

This paper presents an experimental investigation on nonlinear low frequency gravity water waves in a partially filled cylindrical shell subjected to high frequency horizontal excitations. The characteristics of natural frequencies and mode shapes of the water–shell coupled system are discussed. The boundaries for onset of gravity waves are measured and plotted by curves of critical excitation force magnitude with respect to excitation frequency. For nonlinear water waves, the time history signals and their spectrums of motion on both water surface and shell are recorded. The shapes of water surface are also measured using scanning laser vibrometer. In particular, the phenomenon of transitions between different gravity wave patterns is observed and expressed by the waterfall graphs. These results exhibit pronounced nonlinear properties of shell–fluid coupled system.     

Nonlinear resonances of a semi-infinite cable on a nonlinear elastic foundation

The Multiple Time Scale (MTS) method is applied to the study of nonlinear resonances of a semi-infinite cable resting on a nonlinear elastic foundation, subject to a constant uniformly distributed load and to a linear viscous damping force. The zero order solution provides the static displacement, which is governed by a nonlinear equation which has been solved in closed form. The first order solution provides the linear resonances, which are seen to be functions of the nonlinearity parameter and of the static displacement at the finite boundary only. Although the first-order governing equation is linear, it has non constant coefficients and cannot be solved in closed form, so that a numerical solution is considered; the eigenfrequencies obtained in this way are also compared with the approximate eigenvalues obtained by the WKB method. At the second order of the MTS expansion, we see that the solution is independent of the intermediate time scale; some additional terms are present, including a time-independent shift of the average position of the oscillations. Finally, the nonlinear frequency–amplitude response curves, which are investigated in detail and which represent the main result of this work, are obtained from the solvability condition at the third order.     

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells北大核心CSCD

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

The method of multiple scales for forced oscillators with some real-power nonlinearities in the stiffness and damping force

The steady-state response of forced damped nonlinear oscillators is considered, the restoring force of which has a non-negative real power-form nonlinear term and the linear term of which can be negative, zero or positive. The damping term is also assumed in a power form, thus covering polynomial and non-polynomial damping. The method of multiple scales with a new expansion parameter is presented in order to cover the cases when the nonlinearity is not necessarily small. Amplitude-frequency equations and approximate solutions for the steady-state response at the frequency of excitation are obtained and compared with numerical results, showing good agreement.     

Dynamic characteristic analysis of an electrostatically-actuated circular nanoplate subject to surface effects

This study investigates the influence of surface effect on the nonlinear behavior of an electrostatically actuated circular nanoplate. The Casimir force, surface effects, and the electrostatic force are modelled. In performing the analysis, the nonlinear governing equation of a circular nanoplate is solved using a hybrid computational scheme combining a differential transformation and finite differences. The method is used to model systems found in previous literature using different methods, producing consistent results, thus verifying that it is suitable for treatment of the nonlinear electrostatic coupling phenomenon. The obtained results from numerical methods demonstrated that the relationship between the thickness, radius, and gap size of a circular nanoplate, and its pull-in voltage, is scale-dependent. The model exhibits size-dependent behavior, showing that surface effects significantly influence the dynamic response of circular nanoplate actuators. Moreover, the influence of surface stress on the pull-in voltage of circular nanoplate is found to be more significant than the influence of surface elastic modulus. Finally, the effects of actuation voltage, excitation frequency, and surface effects on the dynamic behavior of the nanoplate are examined through use of phase portraits. Overall, the results show that the using hybrid method here presented is a suitable technique for analyzing nonlinear behavior characteristic of circular nanoplates.     

Nonlinear vibrations of elastic beams with external general visco-elastic devices

Vibration damping for slender beams is achieved by applying devices at external points. The latter consist of general single visco-elastic springpot elements. An approximate nonlinear boundary value problem is found in frequency domain that holds for moderately large vibrations or for linear beams with external nonlinear devices, both in the vicinity of primary resonances. The interaction force of a so-called springpot element is expressed as a sum of two separate forces: the first develops due to the external loading function at the device location, and the second contribution arises due to an imposed time-harmonic support excitation with no other external forces acting on the structure. Finally the nonlinear frequency response function follows from a (nonlinear) algebraic equation where the influence of the springpot element appears as isolated parameter. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

System Identification with piecewise linear models in the frequency domain – application to an servopneumatic actuator

By means of a real world application a system identification method was investigated for nonlinear systems from input‐output measurements. This approach is based on a blended multiple model structure, which describes the global behaviour of the system over the whole operating range. Depending on the operating point twenty local linear blackbox models were identified in the frequency domain from a finite number of measurements of the inputs and outputs. A comparative study was made of a model, which have been derived using physical laws [4] and measurements of several process states to estimate unknown parameters.     

Torsional oscillations in structures subject to ground motion

A multiple-scale technique has been employed to study nonlinear torsional oscillations in single-storey structures with cubic softening stiffness members subject to a single frequency ground excitation. The structures are studied under free oscillation conditions, primary resonance, and combination resonance. Time history analyses are employed to quantify the dynamic behaviour of a single-storey structure. The instabilities are related to the jump in response which exists in single degree-of-freedom softening oscillators subjected to harmonic excitation. It is shown that this jump can be quantified analytically, and appears as a cusp catastrophe in the bending mode. Time history analyses show that the bending mode can exhibit a limit cycle behaviour of constant amplitude.     

On the use of kernel-based methods in sound synthesis by physical modeling

In this paper we discuss an approach to the modeling of acoustic systems that combines prior information, exploited through physical modeling, and nonlinear dynamics reconstruction, exploited through support vector machine regression. We demonstrate our approach on two case studies, both addressing the broad class of acoustic systems for which the sound generation is obtained through the interaction of a linear system (resonator) and a nonlinear system (excitation). The first case is a physically based impact model, where the resonator is described in terms of its normal modes and the nonlinear contact force is modeled through a simplified collision equation and kernel regression. In the second case study, a model of the voice phonation is illustrated in which the vocal folds are represented by a lumped linear mass-spring system and the nonlinear flow component is modeled through simple Bernoulli-based equations and kernel regression.     

A class of mathematical models for alternated-current electrochemical measurements accounting for non-linear effects

The traditional approach to the modelling of alternated-current electrochemical measurements is daunted by three major drawbacks: (i) only linear behaviour of the interface investigated is accounted for; (ii) the physical meaning of many model parameters is not straightforward; and (iii) entangled inverse parameter identification problems are generated. In this paper we address these problems by defining additional mathematical models related to the Fourier expansion of a more general, physically meaningful nonlinear electrochemical kinetic reaction system. The traditional linear model is regained as the truncation at the first harmonic of the series. Analytical expressions are derived for experimental observables relating to the nonlinear electrochemical behaviour that can be measured independently during the same experiment. Eventually, we show that the higher-harmonic models can be used as a general regularization tool to reduce the number of the multiple globally optimum solutions in the nonlinear leastsquares fitting of alternated-current data, highlighted in Bozzini and Sgura [Numerical issues related to the modelling of electrochemical impedance data by nonlinear least squares, Int. J. Nonlinear Mech. 40(4) (2005) 557–570].