Multi-scale continuum mechanics: from global bifurcations to noise induced high-dimensional chaos

Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.     

Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints

A method for the modal analysis of continuous gyroscopic systems with nonlinear constraints is developed. This method assumes that the nonlinear constraint can be expressed as a piecewise linear force-deflection profile located at an arbitrary position within the domain. Using this assumption, the mode shapes and natural frequencies are first found for each state, then a mapping method based on the inner product of the mode shapes is developed to map the displacement of the system between the in-contact and out-of-contact states. To illustrate this method, a model for the vibration of a traveling string in contact with a piecewise-linear constraint is developed as an analog of the interaction between magnetic tape and a guide in data storage systems. Five design parameters of the guide are considered: flange clearance, flange stiffness, symmetry of the force-deflection profile in terms of flange stiffness and offset, and the guide's position along the length of the string. There are critical bifurcation thresholds, below which the system exhibits no chaotic behavior and is dominated by period one, symmetric behavior, and above which the system contains asymmetric, higher periodic motion with windows of chaotic behavior. These bifurcation thresholds are particularly pronounced for the transport speed, flange clearance, symmetry of the force deflection profile, and guide position. The stability of the system is sensitive to the system's velocity, and, compared to stationary systems, more mode shapes are needed to accurately model the dynamics of the system.     

A modeling technique for active control design studies with application to spacecraft microvibrations

Microvibrations, at frequencies between 1 and 1000 Hz, generated by on board equipment, can propagate throughout a spacecraft structure and affect the performance of sensitive payloads. To investigate strategies to reduce these dynamic disturbances by means of active control systems, realistic yet simple structural models are necessary to represent the dynamics of the electromechanical system. In this paper a modeling technique which meets this requirement is presented, and the resulting mathematical model is used to develop some initial results on active control strategies. Attention is focused on a mass loaded panel subjected to point excitation sources, the objective being to minimize the displacement at an arbitrary output location. Piezoelectric patches acting as sensors and actuators are employed. The equations of motion are derived by using Lagrange's equation with vibration mode shapes as the Ritz functions. The number of sensors/actuators and their location is variable. The set of equations obtained is then transformed into state variables and some initial controller design studies are undertaken. These are based on standard linear systems optimal control theory where the resulting controller is implemented by a state observer. It is demonstrated that the proposed modeling technique is a feasible realistic basis for in-depth controller design/evaluation studies.     

Towards Realistic Vibration Testing of Large Floor Batteries for Battery Electric Vehicles (BEV)

This contribution shows an analysis of vibration measurement on large floor-mounted traction batteries of Battery Electric Vehicles (BEV). The focus lies on the requirements for a realistic replication of the mechanical environments in a testing laboratory. Especially the analysis on global bending transfer functions and local corner bending coherence indicate that neither a fully stiff fixation of the battery nor a completely independent movement on the four corners yields a realistic and conservative test scenario. The contribution will further show what implication these findings have on future vibration & shock testing equipment for large traction batteries. Additionally, it will cover an outlook on how vibration behavior of highly integrated approaches (cell2car) changes the mechanical loads on the cells.     

Does the Realistic Interpretation of a Mathematical Superposition of States Imply a Mixture?

Contrary to conventional view, it is shown that, for an ensemble of either single-particle systems or multi-particle systems, the realistic interpretation of a mathematical superposition of states that mathematically describes the ensemble does not imply that the ensemble is a mixture. Therefore it cannot be argued, as is conventionally done, that the realistic interpretation is wrong on the basis that some predictions derived from the mixture are different from the corresponding predictions derived from the mathematical superposition of states.     

INVERSE SYNCHRONIZATION OF CHAOTIC SYSTEMS IN AN ERBIUM-DOPED FIBRE DUAL-RING LASER USING THE MUTUAL COUPLING METHOD

Inverse synchronization of chaos is a type of synchronization in which the dynamical variables of two chaotic systems are inversely equal. In this paper, we present a scheme for inverse synchronization of two chaotic systems in an erbium-doped fibre dual-ring laser using the mutual coupling method. For realistic values of the systems, we demonstrate two kinds of results, as follows. (1) Two independent identical chaotic systems can go into inversely synchronized chaotic oscillation for coupling greater than 0.03. (2) When some parameter of one system varies, the state of the coupled systems could go into some periodic states directly or by inverse bifurcation. Simultaneously, they will lose the synchronization as the parameter changes.     

Stable nonequilibrium probability densities and phase transitions for meanfield models in the thermodynamic limit

A nonlinear Fokker-Planck equation is derived to describe the cooperative behavior of general stochastic systems interacting via mean-field couplings, in the limit of an infinite number of such systems. Disordered systems are also considered. In the weak-noise limit; a general result yields the possibility of having bifurcations from stationary solutions of the nonlinear Fokker-Planck equation into stable time-dependent solutions. The latter are interpreted as non-equilibrium probability distributions (states), and the bifurcations to them as nonequilibrium phase transitions. In the thermodynamic limit, results for three models are given for illustrative purposes. A model of self-synchronization of nonlinear oscillators presents a Hopf bifurcation to a time-periodic probability density, which can be analyzed for any value of the noise. The effects of disorder are illustrated by a simplified version of the Sompolinsky-Zippelius model of spin-glasses. Finally, results for the Fukuyama-Lee-Fisher model of charge-density waves are given. A singular perturbation analysis shows that the depinning transition is a bifurcation problem modified by the disorder noise due to impurities. Far from the bifurcation point, the CDW is either pinned or free, obeying (to leading order) the Grüner-Zawadowki-Chaikin equation. Near the bifurcation, the disorder noise drastically modifies the pattern, giving a quenched average of the CDW current which is constant. Critical exponents are found to depend on the noise, and they are larger than Fisher's values for the two probability distributions considered.     

The decoupling of damped linear systems in free or forced vibration

The purpose of this paper is to extend classical modal analysis to decouple any viscously damped linear system in non-oscillatory free vibration or in forced vibration. Based upon an exposition of how exponential decay in a system can be regarded as imaginary oscillations, the concept of damped modes of imaginary vibration is introduced. By phase synchronization of these real and physically excitable modes, a time-varying transformation is constructed to decouple non-oscillatory free vibration. When time drifts caused by viscous damping and by external excitation are both accounted for, a time-varying decoupling transformation for forced vibration is derived. The decoupling procedure devised herein reduces to classical modal analysis for systems that are undamped or classically damped. This paper constitutes the second and final part of a solution to the “classical decoupling problem.” Together with an earlier paper, a general methodology that requires only the solution of a quadratic eigenvalue problem is developed to decouple any damped linear system.     

On the fold-Hopf bifurcation for continuous piecewise linear differential systems with symmetry

In this paper a partial unfolding for an analog to the fold-Hopf bifurcation in three-dimensional symmetric piecewise linear differential systems is obtained. A particular biparametric family of such systems is studied starting from a very degenerate configuration of nonhyperbolic periodic orbits and looking for the possible bifurcation of limit cycles. It is proved that four limit cycles can coexist after perturbation of the original configuration, and other two limit cycles are conjectured. It is shown that the described bifurcation scenario appears for appropriate values of parameters in the celebrated Chua's oscillator.     

Vibration control of two degrees of freedom system using variable inertia vibration absorbers: Modeling and simulation

Variable inertia vibration absorbers (VIVA) are previously used for the vibration control of single degree of freedom (dof) primary systems. The performance of such absorbers is studied in many investigations. This paper presents the dynamic modeling and simulation of a proposed modified design of such VIVA’s for the vibration control of two dof primary systems. Lagrange formulation is used to obtain its dynamic model in an analytical form. This model, which is highly nonlinear, is used to develop a computational algorithm to study the absorber performance characteristics. This algorithm is programmed and simulated in Matlab. The obtained results are numerically verified using SAMS2000 software. The effect of mass and stiffness of the proposed VIVA on its performance and tuning is discussed. An optimization algorithm is developed to select the best absorber parameters for vibration suppression of a specific primary system. The obtained results show a good agreement with those obtained using similar techniques. In addition, a linearized model of VIVA dynamics is developed, tested and simulated for the same data used in its nonlinear model. The relative deviation between results of the linear and nonlinear models is less than 1%, which confirms the realistic use of this linearized model. The experimental testing and verification of the simulation results of the proposed VIVA is the subject of another paper.     

Stabilization and utilization of nonlinear phenomena based on bifurcation control for slow dynamics

Mechanical systems may experience undesirable and unexpected behavior and instability due to the effects of nonlinearity of the systems. Many kinds of control methods to decrease or eliminate the effects have been studied. In particular, bifurcation control to stabilize or utilize nonlinear phenomena is currently an active topic in the field of nonlinear dynamics. This article presents some types of bifurcation control methods with the aim of realizing vibration control and motion control for mechanical systems. It is also indicated through every control method that slowly varying components in the dynamics play important roles for the control and the utilizations of nonlinear phenomena. In the first part, we deal with stabilization control methods for nonlinear resonance which is the 1/3-order subharmonic resonance in a nonlinear spring-mass-damper system and the self-excited oscillation (hunting motion) in a railway vehicle wheelset. The second part deals with positive utilizations of nonlinear phenomena by the generation and the modification of bifurcation phenomena. We propose the amplitude control method of the cantilever probe of an atomic force microscope (AFM) by increasing the nonlinearity in the system. Also, the motion control of a two link underactuated manipulator with a free link and an active link is considered by actuating the bifurcations produced under high-frequency excitation. This article is a discussion on the bifurcation control methods presented by the author and co-researchers by focusing on the actuation of the slowly varying components included in the original dynamics.     

A comparison of various nonlinear models of cochlear compression

The vibration response of the basilar membrane in the cochlea to sinusoidal excitation displays a compressive nonlinearity, conventionally described using an input-output level curve. This displays a slope of 1 dB/dB at low levels and a slope m < 1 dB/dB at higher levels. Two classes of nonlinear systems have been considered as models of this response, one class with static power-law nonlinearity and one class with level-dependent properties (using either an automatic gain control or a Van der Pol oscillator). By carefully choosing their parameters, it is shown that all models can produce level curves that are similar to those measured on the basilar membrane. The models differ, however, in their distortion properties, transient responses, and instantaneous input-output characteristics. The static nonlinearities have a single-valued instantaneous characteristic that is the same at all input levels. The level-dependent systems are multi-valued with an almost linear characteristic, for a given amplitude of excitation, whose slope varies with the excitation level. This observation suggests that historical attempts to use functional modeling (i.e., Wiener of Volterra series) may be ill founded, as these methods are unable to represent level-dependent nonlinear systems with multi-valued characteristics of this kind.     

A superconvergent universality induced by non-associativity

The star products in symbolic dynamics, as effective algebraic operations for describing self-similar bifurcation structure in classical dynamical systems, are found to have either associativity or non-associativity. In this Letter, non-associative star products in trimodal iterative dynamical systems are considered. As the left and right operations have different effects, right-associative star products break the conventional Feigenbaum's metric universality. Through high precision parallel computation, it is found that period-p-tupling bifurcation processes described by right-associative star products exhibit a superconvergent universality of double exponential form.     

Universal occurrence of the phase-flip bifurcation in time-delay coupled systems

Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of pi as a function of the coupling strength or the time delay. This phase-flip is accompanied by discontinuous changes in the frequency of the synchronized oscillators, and in the largest negative Lyapunov exponent or its derivative. Here we illustrate the phenomenology of the bifurcation for several classes of nonlinear oscillators, in the regimes of both periodic and chaotic dynamics. We present extensive numerical simulations and compute the oscillation frequencies and the Lyapunov spectra as a function of the coupling strength. In particular, our simulations provide clear evidence of the phase-flip bifurcation in excitable laser and Fitzhugh-Nagumo neuronal models, and in diffusively coupled predator-prey models with either limit cycle or chaotic dynamics. Our analysis demonstrates marked jumps of the time-delayed and instantaneous fluxes between the two interacting oscillators across the bifurcation; this has strong implications for the performance of the system as well as for practical applications. We further construct an electronic circuit consisting of two coupled Chua oscillators and provide the first formal experimental demonstration of the bifurcation. In totality, our study demonstrates that the phase-flip phenomenon is of broad relevance and importance for a wide range of physical and natural systems.     

Codimension-two bifurcation of axial loaded beam bridge subjected to an infinite series of moving loads

A novel model is proposed which comprises of a beam bridge subjected to an axial load and an infinite series of moving loads. The moving loads, whose distance between the neighbouring ones is the length of the beam bridge, coupled with the axial force can lead the vibration of the beam bridge to codimension-two bifurcation. Of particular concern is a parameter regime where non-persistence set regions undergo a transition to persistence regions. The boundary of each stripe represents a bifurcation which can drive the system off a kind of dynamics and jump to another one, causing damage due to the resulting amplitude jumps. The Galerkin method, averaging method, invertible linear transformation, and near identity nonlinear transformations are used to obtain the universal unfolding for the codimension-two bifurcation of the mid-span deflection. The efficiency of the theoretical analysis obtained in this paper is verified via numerical simulations.     

Alternans and the influence of ionic channel modifications: Cardiac three-dimensional simulations and one-dimensional numerical bifurcation analysis

Cardiac propagation is investigated by simulations using a realistic three-dimensional (3D) geometry including muscle fiber orientation of the ventricles of a rabbit heart and the modified Beeler-Reuter ionic model. Electrical excitation is introduced by a periodic pacing of the lower septum. Depending on the pacing frequency, qualitatively different dynamics are observed, namely, normal heart beat, T-wave alternans, and 2:1 conduction block at small, intermediate, and large pacing frequencies, respectively. In a second step, we performed a numerical stability and bifurcation analysis of a pulse propagating in a one-dimensional (1D) ring of cardiac tissue. The precise onset of the alternans instability is obtained from computer-assisted linear stability analysis of the pulse and computation of the associated spectrum. The critical frequency at the onset of alternans and the profiles of the membrane potential agree well with the ones obtained in the 3D simulations. Next, we computed changes in the wave profiles and in the onset of alternans for the Beeler-Reuter model with modifications of the sodium, calcium, and potassium channels, respectively. For this purpose, we employ the method of numerical bifurcation and stability analysis. While blocking of calcium channels has a stabilizing effect, blocked sodium or potassium channels lead to the occurrence of alternans at lower pacing frequencies. The findings regarding channel blocking are verified within three-dimensional simulations. Altogether, we have found T-wave alternans and conduction block in 3D simulations of a realistic rabbit heart geometry. The onset of alternans has been analyzed by numerical bifurcation and stability analysis of 1D wave trains. By comparing the results of the two approaches, we find that alternans is not strongly influenced by ingredients such as 3D geometry and propagation anisotropy, but depends mostly on the frequency of pacing (frequency of subsequent action potentials). In addition, we have introduced numerical bifurcation and stability analysis as a tool into heart modeling and demonstrated its efficiency in scanning a large set of parameters in the case of models with reduced conductivity. Bifurcation analysis also provides an accurate test for analytical theories of alternans as is demonstrated for the case of the restitution hypothesis.     

Prebifurcational noise rise in nonlinear systems

The phenomenon of prebifurcational noise increase in nonlinear systems in the process of period-doubling bifurcation is investigated. The study is conducted for a discrete system (quadratic mapping); how-ever, many of the laws discovered apply to more general systems. Estimates of the fluctuation variance are obtained both for the linear (away from the bifurcation threshold) and for the nonlinear mode (in the vicinity of the bifurcation threshold). It is shown that the variance of forced fluctuations in the strongly nonlinear mode is proportional to the root-mean-square of the noise intensity rather than to the variance. The possibility of measuring the noise in nonlinear systems on the basis of the prebifurcational noise amplification factor is demonstrated.     

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.     

Novel design for bifurcation control in a delayed fractional dual congestion model

This letter puts forward an ingenious feedback control method with parametric delay to manipulate bifurcation control for a delayed fractional dual congestion model. By employing time delay as a bifurcation parameter, the local dynamics involving stability and Hopf bifurcation is examined. The control efforts can be realized with or without time delay in the strength of feedback control. It suggests that the stability performance is consumedly elevated by exploiting the parametric delay feedback controller, yet Hopf bifurcation engenders ahead of time in the event of the absence of the controller. Moreover, the impact of the order or linear feedback gain on the bifurcation point is numerically discussed via aborative calculation. Numerical simulations are eventually actualized to corroborate the proposed scheme.     

一类高维相对转动非线性动力系统的Lyapunov-Schmidt约化与奇异性分析

研究一类高维相对转动非线性动力系统的降维与分岔特性. 在考虑转动系统中间隙非线性影响因素的基础上, 基于广义耗散系统拉格朗日原理, 建立了一类高维相对转动非线性系统动力学模型.采用Lyapunov-Schmidt(LS)约化方法, 通过对高维非线性动力系统进行降维处理, 得到能够揭示系统非线性动力特性与系统参数之间规律的低维等价分岔方程. 运用奇异性理论对分岔方程进行普适开折, 分析了系统的分岔特性.结合实例参数, 对分岔特性进行仿真分析, 得到相对转动非线性动力系统发生动力失稳的参数区域及系统参数对动力失稳的影响规律.