Transient mode localization in coupled strongly nonlinear exactly solvable oscillators

Coupled strongly nonlinear oscillators, whose characteristic is close to linear for low amplitudes but becomes infinitely growing as the amplitude approaches certain limit, are considered in this paper. Such a model may serve for understanding the dynamics of elastic structures within the restricted space bounded by stiff constraints. In particular, this study focuses on the evolution of vibration modes as the energy is gradually pumped into or dissipates out of the system. For instance, based on the two degrees of freedom system, it is shown that the in-phase and out-of-phase motions may follow qualitatively different scenarios as the system’ energy increases. So the in-phase mode appears to absorb the energy with equipartition between the masses. In contrast, the out-of-phase mode provides equal energy distribution only until certain critical energy level. Then, as a result of bifurcation of the 1:1 resonance path, one of the masses becomes a dominant energy receiver in such a way that it takes the energy not only from the main source but also from another mass.     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

Numerical calculation of nonlinear normal modes in structural systems

This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to “artificially destabilize” the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient ε of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for ε = 0 into the manifold existing quasi-statically for instantaneous values of ε. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).     

A very complicated structure of resonances in a nonlinear system with cyclic symmetry: Nonlinear forced localization

The fundamental and subharmonic resonances of a nonlinear cyclic assembly are examined using the asymptotic method of multiple-scales. The system consists of a number of identical cantilever beams coupled by means of weak linear stiffnesses. Assuming beam inextensionality, geometric nonlinearities arise due to longitudinal inertia and the nonlinear relation between beam curvature and transverse displacement. The governing nonlinear partial differential equations are discretized by a Galerkin procedure and the resulting set of coupled ordinary differential equations is solved using an asymptotic analysis. The unforced assembly is known to possess localized nonlinear normal modes, which give rise to a very complicated topological structure of fundamental and subharmonic response curves. In contrast to the linear system which exhibits as many forced resonances as its number of degrees of freedom, the nonlinear system is found to possess a number of additional resonance branches which have no counterparts in linear theory. Some of the additional resonances are spatially localized, corresponding to motions of only a small subset of periodic elements. The analytical results are verified by numerical Poincaré maps, and the forced localization features of the nonlinear assembly are demonstrated by considering its response to impulsive excitations.     

Contribution to efficiency of irreversible passive energy pumping with a strong nonlinear attachment

The present study deals with nonlinear energy pumping which consists in passive irreversible transfer of energy from a linear structure to a nonlinear one. Various results (theoretical, numerical, and experimental) about energy pumping based on recent works are given. Thus, the phenomenon is studied for different excitations: transient and periodical. Moreover, advantages of such a system are carried out in particular efficiency of this phenomenon. That is why the robustness and comparison with classical tuned mass damper are analyzed. An application is considered with physical experiment using a reduced scale building.     

On some performance characteristics of base excited vibration isolation systems with a purely nonlinear restoring force

Base excited vibration isolation systems with a purely nonlinear restoring force and a velocity nth power damper are considered. The restoring force has a single-term power form with the exponent that can be any non-negative real number. Approximations for the steady-state response at the frequency of excitation are obtained by using the Jacobi elliptic function with a changeable elliptic parameter and by applying an elliptic averaging method. The relative and absolute displacement transmissibility of this system are analysed. These performance characteristics are expressed in terms of the damping parameters, but they are also determined for an arbitrary non-negative real power of geometric nonlinearity, which represent new and so far unknown results. Some examples illustrating the effect of the system parameters on these performance characteristics are also presented.     

Nonlinear dynamic analysis of a structure with a friction-based seismic base isolation system

Many dynamical systems are subject to some form of non-smooth or discontinuous nonlinearity. One eminent example of such a nonlinearity is friction. This is caused by the fact that friction always opposes the direction of movement, thus changing sign when the sliding velocity changes sign. In this paper, a structure with friction-based seismic base isolation is regarded. Seismic base isolation can be employed to decouple a superstructure from the potentially hazardous surrounding ground motion. As a result, the seismic resistance of the superstructure can be improved. In this case study, the base isolation system is composed of linear laminated rubber bearings and viscous dampers and nonlinear friction elements. The nonlinear dynamic modelling of the base-isolated structure with the aid of constraint equations, is elaborated. Furthermore, the influence of the dynamic characteristics of the superstructure and the nonlinear modelling of the isolation system, on the total system’s dynamic response, is examined. Hereto, the effects of various modelling approaches are considered. Furthermore, the dynamic performance of the system is studied in both nonlinear transient and steady-state analyses. It is shown that, next to (and in correlation with) transient analyses, steady-state analyses can provide valuable insight in the discontinuous dynamic behaviour of the system. This case study illustrates the importance and development of nonlinear modelling and nonlinear analysis tools for non-smooth dynamical systems.     

Nonlinear normal modes in homogeneous system with time delays

Periodic synchronous regimes of motion are investigated in symmetric homogeneous system of coupled essentially nonlinear oscillators with time delays. Such regimes are similar to nonlinear normal modes (NNMs), known for corresponding conservative system without delays, and can be found analytically. Unlikely the conservative counterpart, the system possesses “oval” modes with constant phase shift between the oscillators, in addition to symmetric/antisymmetric and localized regimes of motion. Numeric simulation demonstrates that the “oval” modes may be attractors of the phase flow. These attractors are particular case of phase-locked solutions, rather ubiquitous in the system under investigation.     

Free vibrations of a thin elastica by normal modes

In this work we investigate the existence, stability and bifurcation of periodic motions in an unforced conservative two degree of freedom system. The system models the nonlinear vibrations of an elastic rod which can undergo both torsional and bending modes. Using a variety of perturbation techniques in conjunction with the computer algebra system MACSYMA, we obtain approximate expressions for a diversity of periodic motions, including nonlinear normal modes, elliptic orbits and non-local modes. The latter motions, which involve both bending and torsional motions in a 2:1 ratio, correspond to behavior previously observed in experiments by Cusumano.     

Bifurcation of nonlinear internal resonant normal modes of a class of multi-degree-of-freedom systems

The method of multiple scales is applied for constructing nonlinear normal modes (NNMs) of a three-degree-of-freedom system which is discretized from a two-link flexible arm connected by a nonlinear torsional spring. The discrete system is with cubic nonlinearity and 1:3 internal resonance between the second and the third modes. The approximate solution for the NNM associated with internal resonance are presented. The NNMs determined here tend to the linear modes as the nonlinearity vanishes, which is significant for one to construct NNM. Greatly different from results of those nonlinear systems without internal resonance, it is found that the NNM involved in internal resonance include coupled and uncoupled two kinds. The bifurcation analysis of the coupled NNM of the system considered is given by means of the singularity theory. The pitchfork and hysteresis bifurcation are simultaneously found. Therefore, the number of NNM arising from the internal resonance may exceed the number of linear modes, in contrast with the case of no internal resonance, where they are equal. Curves displaying variation of the coupling extent of the coupled NNM with the internal-resonance-deturing parameter are proposed for six cases.     

A perturbation method for evaluating nonlinear normal modes of a piecewise linear two-degrees-of-freedom system

The classical Lindstedt–Poincaré method is adapted to analyze the nonlinear normal modes of a piecewise linear system. A simple two degrees-of-freedom, representing a beam with a breathing crack is considered. The fundamental branches of the two modes and their stability are drawn by varying the severity of the crack, i.e., the level of nonlinearity. Results furnished by the asymptotic method give insight into the mechanical behavior of the system and agree well with numerical results; the existence of superabundant modes is proven. The unstable regions and the bifurcated branches are followed by a numerical procedure based on the Poincarè map.     

Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells

Donnell equations are used to simulate free nonlinear oscillations of cylindrical shells with imperfections. The expansion, which consists of two conjugate modes and axisymmetric one, is used to analyze shell oscillations. Amplitudes of the axisymmetric motions are assumed significantly smaller, than the conjugate modes amplitudes. Nonlinear normal vibrations mode, which is determined by shell imperfections, is analyzed. The stability and bifurcations of this mode are studied by the multiple scales method. It is discovered that stable quasiperiodic motions appear at the bifurcations points. The forced oscillations of circular cylindrical shells in the case of two internal resonances and the principle resonance are analyzed too. The multiple scales method is used to obtain the system of six modulation equations. The method for stability analysis of standing waves is suggested. The continuation algorithm is used to analyze fixed points of the system of the modulation equations.     

基于性能的基础隔震结构地震易损性分析

对基础隔震结构进行基于性能的易损性分析。首先,建立了基础隔震结构基于拟力法的能量方程,提出基于变形和隔震层塑性耗能的损伤指标,定义隔震结构的四个损伤性能状态;然后,对隔震结构进行动力非线性分析,计算得到两种损伤指标的损伤值;最后,对损伤值进行线性统计回归分析,推导出结构发生各级破坏的概率计算公式,从而分别得到隔震结构基于两种损伤指标的易损性曲线。研究表明,基于隔震层塑性耗能的损伤指标更能合理反映该结构的损伤程度,为基于性能的隔震结构易损性分析提供了新的思路和方法。     

Fundamental and subharmonic resonances in a system with a ‘1-1’ internal resonance

The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a 1–1 internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses nonlinear normal modes. For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given.     

Stability analysis for nonplanar free vibrations of a cantilever beam by using nonlinear normal modes

Stability analysis of nonplanar free vibrations of a cantilever beam is made by using the nonlinear normal mode concept. Assuming nonplanar motion of the beam, we introduce a nonlinear two-degree-of-freedom model by using Galerkin’s method based on the first mode in each direction. The system turns out to have two normal modes. Using Synge’s stability concept, we examine the stability of each mode. In order to check the validity of the stability criterion obtained analytically, we plot a Poincaré map of the motions neighboring on each mode obtained numerically. It is found that the maps agree with the stability criterion obtained analytically.     

力学反问题的神经网络分析法

在力学中，许多反问题有其重要的工程背景。采用系统控制论的描述，所谓反问题就是逆系统辨识问题。本文利用基于神经网络的逆系统辨识方法对力学反问题进行求解。并给出了材料力学参数反求和裂纹长度的反求两个算例     

Free vibration of two coupled nonlinear oscillators

An analytical investigation is carried out on the free vibration of a two degree of freedom weakly nonlinear oscillator. Namely, the method of multiple time scales is first applied in deriving modulation equations for a van der Pol oscillator coupled with a Duffing oscillator. For the case of non-resonant oscillations, these equations are in standard normal form of a codimension two (Hopf-Hopf) bifurcation, which permits a complete analysis to be performed. Three different types of asymptotic states-corresponding to trivial, periodic and quasiperiodic motions of the original system-are obtained and their stability is analyzed. Transitions between these different solutions are also identified and analyzed in terms of two appropriate parameters. Then, effects of a coupling, a detuning, a nonlinear stiffness and a damping parameter are investigated numerically in a systematic manner. The results are interpreted in terms of classical engineering terminology and are related to some relatively new findings in the area of nonlinear dynamical systems.     

Order reduction of structural dynamic systems with static piecewise linear nonlinearities

A technique for order reduction of dynamic systems in structural form with static piecewise linear nonlinearities is presented. By utilizing two methods which approximate the nonlinear normal mode (NNM) frequencies and mode shapes, reduced-order models are constructed which more accurately represent the dynamics of the full model than do reduced models obtained via standard linear transformations. One method builds a reduced-order model which is dependent on the amplitude (initial conditions) while the other method results in an amplitude-independent reduced model. The two techniques are first applied to reduce two-degree-of-freedom undamped systems with clearance, deadzone, bang-bang, and saturation stiffness nonlinearities to single-mode reduced models which are compared by direct numerical simulation with the full models. It is then shown via a damped four-degree-of-freedom system with two deadzone nonlinearities that one of the proposed techniques allows for reduction to multi-mode reduced models and can accommodate multiple nonsmooth static nonlinearities with several surfaces of discontinuity. The advantages of the proposed methods include obtaining a reduced-order model which is signal-independent (doesn’t require direct integration of the full model), uses a subset of the original physical coordinates, retains the form of the nonsmooth nonlinearities, and closely tracks the actual NNMs of the full model.     

A state-of-the-art review on low-frequency nonlinear vibration isolation with electromagnetic mechanisms

Vibration isolation is one of the most efficient approaches to protecting host structures from harmful vibrations, especially in aerospace, mechanical, and architectural engineering, etc. Traditional linear vibration isolation is hard to meet the requirements of the loading capacity and isolation band simultaneously, which limits further engineering application, especially in the low-frequency range. In recent twenty years, the nonlinear vibration isolation technology has been widely investigated to broaden the vibration isolation band by exploiting beneficial nonlinearities. One of the most widely studied objects is the “three-spring” configured quasi-zero-stiffness (QZS) vibration isolator, which can realize the negative stiffness and high-static-low-dynamic stiffness (HSLDS) characteristics. The nonlinear vibration isolation with QZS can overcome the drawbacks of the linear one to achieve a better broadband vibration isolation performance. Due to the characteristics of fast response, strong stroke, nonlinearities, easy control, and low-cost, the nonlinear vibration with electromagnetic mechanisms has attracted attention. In this review, we focus on the basic theory, design methodology, nonlinear damping mechanism, and active control of electromagnetic QZS vibration isolators. Furthermore, we provide perspectives for further studies with electromagnetic devices to realize high-efficiency vibration isolation.

     

层间隔震结构的能量平衡

从能量观点出发,引入设计用能量谱的概念,分析了地震输入能量在层间隔震结构中的分配和耗散。推导出了基于能量平衡的层间隔震结构的隔震层总剪力、最大变形以及基底剪力的地震响应预测式,分析了隔震层刚度、阻尼以及隔震层的设置位置等参数对层间隔震结构减震性能的影响,并阐明了其自身的减震控制机理。研究表明,基于能量平衡的分析方法是一种预测层间隔震结构地震响应和评价其减震性能的有效手段。