Optimal design theories of tuned mass dampers with nonlinear viscous damping

Optimal design theory for linear tuned mass dampers （TMD） has been thoroughly investigated, but is still under development for nonlinear TMDs. In this paper, optimization procedures in the time domain are proposed for design of a TMD with nonlinear viscous damping. A dynamic analysis of a structure implemented with a nonlinear TMD is conducted first. Optimum design parameters for the nonlinear TMD are searched using an optimization method to minimize the performance index. The feasibility of the proposed optimization method is illustrated numerically by using the Taipei 101 structure implemented with TMD. The sensitivity analysis shows that the performance index is less sensitive to the damping coefficient than to the frequency ratio. Time history analysis is conducted using the Taipei 101 structure implemented with different TMDs under wind excitation. For both linear and nonlinear TMDs, the comfort requirements for building occupants are satisfied as long as the TMD is properly designed. It was found that as the damping exponent increases, the relative displacement of the TMD decreases but the damping force increases.

Sliding mode H_∞ control for a class of uncertain nonlinear state-delayed systems

A new proportional-integral （PI） sliding surface is designed for a class of uncertain nonlinear state-delayed systems. Based on this, an adaptive sliding mode controller （ASMC） is synthesized, which guarantees the occurrence of sliding mode even when the system is undergoing parameter uncertainties and external disturbance. The resulting sliding mode has the same order as the original system, so that it becomes easy to solve the H∞ control problem by designing a memoryless H∞ state feedback controller. A delay-dependent sufficient condition is proposed in terms of linear matrix inequalities （LMIs）, which guarantees the sliding mode robust asymptotically stable and has a noise attenuation level γ in an H∞ sense. The admissible state feedback controller can be found by solving a sequential minimization problem subject to LMI constraints by applying the cone complementary linearization method. This design scheme combines the strong robustness of the sliding mode control with the H∞ norm performance. A numerical example is given to illustrate the effectiveness of the proposed scheme.

H_∞tracking design for a class of decentralized nonlinear systems via fuzzy adaptive observer

A novel H∞ tracking-based decentralized indirect adaptive output feedback fuzzy controller for a class of uncertain large-scale nonlinear systems is developed. By virtue of the proper filtering of the observation error dynamics, the observer-based decentralized indirect adaptive fuzzy control scheme is presented for a class of large-scale nonlinear systems using the combination of H∞ tracking technique, a fuzzy adaptive observer and fuzzy inference systems. The output feedback and adaptation mechanisms are both robust and implementable indeed owing to their freedom from the unavailable observation error vector. All the signals of the closed-loop largescale system are guaranteed to stay uniformly bounded and the output errors take on H∞ tracking performance. Simulation results substantiate the effectiveness of the proposed scheme.

Simplified nonlinear theory of the dielectric loaded rectangular Cerenkov maser

To rapidly and accurately investigate the performance of the dielectric loaded rectangular Cerenkov maser, a simplified nonlinear theory is proposed, in which the variations of wave amplitude and wave phase are determined by two coupled first-order differential equations. Through combining with the relativistic equation of motion and adopting the forward wave assumption, the evolutions of the forward wave power, the power growth rate, the axial wave number, the accumulated phase offset, and the information of the particle movement can be obtained in a single-pass calculation. For an illustrative example, this method is used to study the influences of the beam current, the gap distance between the beam and the dielectric surface, and the momentum spread on the forward wave. The variations of the saturated power and the saturation length with the working frequency for the beams with different momentum spreads have also been studied. The result shows that the beam wave interaction is very sensitive to the electron beam state. To further verify this simplified theory, a comparison with the result produced from a rigorous method is also provided, we find that the evolution curves of the forward wave power predicted by the two methods exhibit excellent agreement. In practical applications, the developed theory can be used for the design and analysis of the rectangular Cerenkov maser.

Linear and nonlinear torsional free vibration of functionally graded micro/nano-tubes based on modified couple stress theory

The linear and nonlinear torsional free vibration analyses of functionMly graded micro/nuno-tubes （FGMTs） are analytically investigated based on the couple stress theory. The employed non-classical continuum theory contains one material length scale parameter, which can capture the small scale effect. The FGMT model accounts for the through-radius power-law variation of a two-constituent material. Hamilton＇s principle is used to develop the non-classical nonlinear governing equation. To study the effect of the boundary conditions, two types of end conditions, i.e., fixed-fixed and fixed-free, are considered. The derived boundary value governing equation is of the fourthorder, and is solved by the homotopy analysis method （HAM）. This method is based on the Taylor series with an embedded parameter and is capable of providing very good approximations by means of only a few terms, if the initial guess and the auxiliary linear operator are properly selected. The analytical expressions are developed for the linear and nonlinear natural frequencies, which can be conveniently used to investigate the effects of the dimensionless length scale parameter, the material gradient index, and the vibration amplitude on the natural frequencies of FGMTs.

THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS

We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented in the paper. Factors or the physics influencing dynamic response and dynamic bifurcation for TVES with rheology are identified and are also demonstrated through model problem studies. A simple model problem consisting of a rod (1D) of TVES material with memory fixed at one end and subjected to harmonic excitation at the other end is considered to study nonlinear dynamics of TVES with rheology, frequency response as well as dynamic bifurcation phenomenon.

Nonlinear dynamics of the reversed shear Alfvén eigenmode in burning plasmas

In a tokamak fusion reactor operated at steady state,the equilibrium magnetic field is likely to have reversed shear in the core region,as the noninductive bootstrap current profile generally peaks off-axis.The reversed shear Alfvén eigenmode(RSAE)as a unique branch of the shear Alfvén wave in this equilibrium,can exist with a broad spectrum in wavenumber and frequency,and be resonantly driven unstable by energetic particles(EP).After briefly discussing the RSAE linear properties in burning plasma condition,we review several key topics of the nonlinear dynamics for the RSAE through both wave-EP resonance and wave-wave coupling channels,and illustrate their potentially important role in reactor-scale fusion plasmas.By means of simplified hybrid MHD-kinetic simulations,the RSAEs are shown to have typically broad phase space resonance structure with both circulating and trapped EP,as results of weak/vanishing magnetic shear and relatively low frequency.Through the route of wave-EP nonlinearity,the dominant saturation mechanism is mainly due to the transported resonant EP radially decoupling with the localized RSAE mode structure,and the resultant EP transport generally has a convective feature.The saturated RSAEs also undergo various nonlinear couplings with other collective oscillations.Two typical routes as parametric decay and modulational instability are studied using nonlinear gyrokinetic theory,and applied to the scenario of spontaneous excitation by a finite amplitude pump RSAE.Multiple RSAEs could naturally couple and induce the spectral energy cascade into a low frequency Alfvénic mode,which may effectively transfer the EP energy to fuel ions via collisionless Landau damping.Moreover,zero frequency zonal field structure could be spontaneously excited by modulation of the pump RSAE envelope,and may also lead to saturation of the pump RSAE by both scattering into stable domain and local distortion of the continuum structure.

Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method

The size-dependent effect on the biaxial and shear nonlinear buckling analysis of an isotropic and orthotropic micro-plate based on the surface stress, the modified couple stress theory （MCST）, and the nonlocal elasticity theories using the differential quadrature method （DQM） is presented. Main advantages of the MCST over the classical theory （CT） are the inclusion of the asymmetric couple stress tensor and the consideration of only one material length scale parameter. Based on the nonlinear von Karman assumption, the governing equations of equilibrium for the micro-classical plate consid- ering midplane displacements are derived based on the minimum principle of potential energy. Using the DQM, the biaxial and shear critical buckling loads of the micro-plate for various boundary conditions are obtained. Accuracy of the obtained results is validated by comparing the solutions with those reported in the literature. A parametric study is conducted to show the effects of the aspect ratio, the side-to-thickness ratio, Eringen＇s nonlocal parameter, the material length scale parameter, Young＇s modulus of the surface layer, the surface residual stress, the polymer matrix coefficients, and various boundary conditions on the dimensionless uniaxial, biaxial, and shear critical buckling loads. The results indicate that the critical buckling loads are strongly sensitive to Eringen＇s nonlocal parameter, the material length scale parameter, and the surface residual stress effects, while the effect of Young＇s modulus of the surface layer on the critical buckling load is negligible. Also, considering the size dependent effect causes the increase in the stiffness of the orthotropic micro-plate. The results show that the critical biaxial buckling load increases with an increase in G12/E2 and vice versa for E1/E2. It is shown that the nonlinear biaxial buckling ratio decreases as the aspect ratio increases and vice versa for the buckling amplitude. Because of the most lightweight micro-composite materials with high strength/weight and stiffness/weight ratios, it is anticipated that the results of the present work are useful in experimental characterization of the mechanical properties of micro-composite plates in the aircraft industry and other engineering applications.

Robust H_∞ controller design for a class of nonlinear fuzzy time-delay systems

The problem of fuzzy modeling for state and input time-delays systems with a class of nonlinear uncertainties by fuzzy T-S model is addressed.By using the linear matrix inequality（LMI） method, the problem of fuzzy robust H ∞ controller design for the system is studied.Assuming that the nonlinear uncertain functions in the model considered are gain-bounded, a sufficient condition for the robustly asymptotic stability of the closed-loop system is obtained via Lyapunov stability theory.By solving the LMI, a feedback control law which guarantees the robustly asymptotic stability of the closed-loop system is constructed and the effect of the disturbance input on the controlled output is ruduced to a prescribed level.

Nonlinearity analysis of piezoelectric micromachined ultrasonic transducers based on couple stress theory

This paper studies the static deformation behavior of a piezoelectric micromachined ultrasonic transducer （PMUT） actuated by a strong external electric field. The transducer membrane consists of a piezoelectric layer, a passive layer and two electrode layers. The nonlinearities of the piezoelectric layer caused by electrostriction under a strong electric field are analyzed. Because the thickness of the transducer membrane is on the microscale, the size dependence of the deformation behavior is evaluated using the couple stress theory. The results show that the optimal ratio of the top electrode diameter and the membrane diameter is around 0.674. It is also found that this optimal value does not depend on any other parameters if the thicknesses of the two electrodes are negligible compared with those of the piezo- electric and passive layers. In addition, the nonlinearities of the piezoelectric layer will become stronger along with the increase of the electric field, which means that softening of the membrane stiffness occurs when a strong external electric field is applied. Meanwhile, the optimal thickness ratio for the passive layer and the piezoelectric layer is not equal to 1.0 which is usually adopted by previous researchers. Because there exists size dependence of membrane deforma-tion, the optimal value of this thickness ratio needs to be greater than 1.0 on the microscale.

Design of H_∞ robust fault detection filter for nonlinear time-delay systems

In this paper, the robust fault detection filter (RFDF) design problems are studied for nonlinear time-delay systems with unknown inputs. First, a reference residual model is introduced to formulate the RFDF design problem as an H∞ model-matching problem. Then appropriate input/output selection matrices are introduced to extend a performance index to the time-delay systems in time domain. The reference residual model designed according to the performance index is an optimal residual generator, which takes into account the robustness against disturbances and sensitivity to faults simultaneously. Applying robust H∞ optimization control technique, the existence conditions of the RFDF for nonlinear time-delay systems with unknown inputs are presented in terms of linear matrix inequality (LMI) formulation, independently of time delay. An illustrative design example is used to demonstrate the validity and applicability of the proposed approach.

Weak Fault Feature Extraction of the Rotating Machinery Using Flexible Analytic Wavelet Transform and Nonlinear Quantum Permutation Entropy

Addressing the challenges posed by the nonlinear and non-stationary vibrations in rotating machinery,where weak fault characteristic signals hinder accurate fault state representation,we propose a novel feature extraction method that combines the Flexible Analytic Wavelet Transform(FAWT)with Nonlinear Quantum Permutation Entropy.FAWT,leveraging fractional orders and arbitrary scaling and translation factors,exhibits superior translational invariance and adjustable fundamental oscillatory characteristics.This flexibility enables FAWT to provide well-suited wavelet shapes,effectively matching subtle fault components and avoiding performance degradation associated with fixed frequency partitioning and low-oscillation bases in detecting weak faults.In our approach,gearbox vibration signals undergo FAWT to obtain sub-bands.Quantum theory is then introduced into permutation entropy to propose Nonlinear Quantum Permutation Entropy,a feature that more accurately characterizes the operational state of vibration simulation signals.The nonlinear quantum permutation entropy extracted from sub-bands is utilized to characterize the operating state of rotating machinery.A comprehensive analysis of vibration signals from rolling bearings and gearboxes validates the feasibility of the proposed method.Comparative assessments with parameters derived from traditional permutation entropy,sample entropy,wavelet transform(WT),and empirical mode decomposition(EMD)underscore the superior effectiveness of this approach in fault detection and classification for rotating machinery.

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube （SWCNT） subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.

Monitoring the change in horizontal stress with multi-wave time-lapse seismic response based on nonlinear elasticity theory

Monitoring the change in horizontal stress from the geophysical data is a tough challenge, and it has a crucial impact on broad practical scenarios which involve reservoir exploration and development, carbon dioxide (CO_(2)) injection and storage, shallow surface prospecting and deep-earth structure description. The change in in-situ stress induced by hydrocarbon production and localized tectonic movements causes the changes in rock mechanic properties (e.g. wave velocities, density and anisotropy) and further causes the changes in seismic amplitudes, phases and travel times. In this study, the nonlinear elasticity theory that regards the rock skeleton (solid phase) and pore fluid as an effective whole is used to characterize the effect of horizontal principal stress on rock overall elastic properties and the stress-dependent anisotropy parameters are therefore formulated. Then the approximate P-wave, SV-wave and SH-wave angle-dependent reflection coefficient equations for the horizontal-stress-induced anisotropic media are proposed. It is shown that, on the different reflectors, the stress-induced relative changes in reflectivities (i.e., relative difference) of elastic parameters (i.e., P- and S-wave velocities and density) are much less than the changes in contrasts of anisotropy parameters. Therefore, the effects of stress change on the reflectivities of three elastic parameters are reasonably neglected to further propose an AVO inversion approach incorporating P-, SH- and SV-wave information to estimate the change in horizontal principal stress from the corresponding time-lapse seismic data. Compared with the existing methods, our method eliminates the need for man-made rock-physical or fitting parameters, providing more stable predictive power. 1D test illustrates that the estimated result from time-lapse P-wave reflection data shows the most reasonable agreement with the real model, while the estimated result from SH-wave reflection data shows the largest bias. 2D test illustrates the feasibility of the proposed inversion method for estimating the change in horizontal stress from P-wave time-lapse seismic data.

Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

In this paper,the stability and bifurcation analysis of symmetrical and asymmetrical micro-rotating shafts are investigated when the rotational speed is in the vicinity of the critical speed.With the help of Hamilton's principle,nonlinear equations of motion are derived based on non-classical theories such as the strain gradient theory.In the dynamic modeling,the geometric nonlinearities due to strains,and strain gradients are considered.The bifurcations and steady state solution are compared between the classical theory and the non-classical theories.It is observed that using a non-classical theory has considerable effect in the steady-state response and bifurcations of the system.As a result,under the classical theory,the symmetrical shaft becomes completely stable in the least damping coefficient,while the asymmetrical shaft becomes completely stable in the highest damping coefficient.Under the modified strain gradient theory,the symmetrical shaft becomes completely stable in the least total eccentricity,and under the classical theory the asymmetrical shaft becomes completely stable in the highest total eccentricity.Also,it is shown that by increasing the ratio of the radius of gyration per length scale parameter,the results of the non-classical theory approach those of the classical theory.

Vibration response analysis of 2-DOF locally nonlinear systems based on the theory of modal superposition

Many important vibration phenomena which simultaneously contain quadratic nonlinear stiffness and damping exist in the complicated vibrating systems under practical circumstances. In this paper, we established a 2-degree-of-freedom (DOF) nonlinear vibration model for such a system, deduced the differential equations of motion which govern its dynamics, and worked out the solutions for the governing equations by the principle of superposition of nonlinear normal modes (NLNM) based on Shaw’s theory of invariant manifolds. We conducted numerical simulations with the established model, using superposition of nonlinear normal modes and direct numerical methods, respectively. The obtained results demonstrate the feasibility of the proposed method in that its calculated data varies in a similar tendency to that of the direct numerical solutions.

Unified Control Theory from PID to ACPID

To address the challenge of achieving unified control across diverse nonlinear systems, a comprehensive control theory spanning from PID (Proportional-Integral-Derivative) to ACPID (Auto-Coupling PID) has been proposed. The primary concept is to unify all intricate factors, including internal dynamics and external bounded disturbance, into a single total disturbance. This enables the mapping of various nonlinear systems onto a linear disturbance system. Based on the theory of PID control and the characteristic equation of a critically damping system, Zeng’s stabilization rules (ZSR) and an ACPID control force based on a single speed factor have been designed. ACPID control theory is both simple and practical, with significant scientific significance and application value in the field of control engineering.

Acceleration Feedback Enhanced H_∞Disturbance Attenuation Control for a Class of Nonlinear Underactuated Vehicle Systems

在这篇论文，一个概括加速反馈控制(声频抗流圈) 设计方法，命名声频抗流圈提高了 H ∞控制器，为两个被建议完整激活并且在激活的非线性的自治车辆系统下面。声频抗流圈基于已知的动力学作为柔韧的改进被设计到正常控制。首先，以便拒绝不确定性和外部骚乱，线性 prefilter 在新声频抗流圈设计方法被使用在正常声频抗流圈代替高获得。然后，背走算法被用于 AFC 设计在激活的系统下面。两个的分析在有限获得 L2 稳定性显示出的频率领域和输入产量的骚乱变细新控制器设计方法是适用的。最后，模拟关于无人的模型直升飞机的追踪的控制被进行。结果与没有声频抗流圈，追踪的控制获得验证新方法的可行性的那些相比。

Nonlinear Kinetic Theory and Pulse Interactions in Phase Transition

The kinetics of nucleation of phase transition is a phenomenal theory.Some new technologies of preparation of nanomaterials,for example,by shock wave and by electropulsing,are pulse interactions.Based on the known nonlinear theories of phase transition,the nonlinear kinetics of phase transition is discussed,and a soliton-like model is proposed. This mathematical method can not only explain the basic characteristics of pulse interactions and suddenness of phase transition, and possesses a consistency of mechanism for nucleation and growth.