Essential Nonlinearity Research Articles

Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure

Nonlinear normal modes for elastic structures have been studied extensively in the literature. Most studies have been limited to small nonlinear motions and to structures with geometric nonlinearities. This work investigates the nonlinear normal modes in elastic structures that contain essential inertial nonlinearities. For such structures, based on the works of Crespo da Silva and Meirovitch, a general methodology is developed for obtaining multi-degree-of-freedom discretized models for structures in planar motion. The motion of each substructure is represented by a finite number of substructure admissible functions in a way that the geometric compatibility conditions are automatically assured. The multi degree-of-freedom reduced-order models capture the essential dynamics of the system and also retain explicit dependence on important physical parameters such that parametric studies can be conducted. The specific structure considered is a 3-beam elastic structure with a tip mass. Internal resonance conditions between different linear modes of the structure are identified. For the case of 1:2 internal resonance between two global modes of the structure, a two-mode nonlinear model is then developed and nonlinear normal modes for the structure are studied by the method of multiple time scales as well as by a numerical shooting technique. Bifurcations in the nonlinear normal modes are shown to arise as a function of the internal mistuning that represents variations in the tip mass in the structure. The results of the two techniques are also compared.

Differences between cylindrical and conical brass instruments; the nonlinear propagation point of view from experiments and simulations

The brightness of the sound generated by brass instruments is due to the essential nonlinearity of the wave propagation in the pipe. The bright instruments such as the trumpet and the trombone are different from brass instruments such as saxhorns or flugelhorns. The bright instruments have a cylindrical pipe segment just downstream of the mouthpiece. The conical bore of the saxhorns implies a faster decay of the wave, which reduces the nonlinear wave steepening. In this talk, theoretical and experimental results of brassiness associated with brass instruments having different bores, from mainly cylindrical ones to mainly conical ones, will be shown. The theoretical results come from simulations of periodic regimes using a theoretical background based on generalized Burger’s equations.

Impulsive periodic and quasi-periodic orbits of coupled oscillators with essential stiffness nonlinearity

We study the impulsive responses of a grounded linear oscillator coupled to a light nonlinear attachment through an essentially nonlinear (nonlinearizable) stiffness. We analyze the periodic and quasi-periodic dynamics of the undamped system forced by a single impulse on the linear oscillator and being initially at rest, by considering separately low-, moderate- and high-energy impulsive motions. The motivation for studying the impulsive dynamics of this system centers on passive targeted energy transfer properties of the corresponding weakly damped one, that is, of the possibility of one-way, irreversible transfer of energy from the linear oscillator to the nonlinear attachment. A rather surprising aspect of this work is the complexity of the analysis required to study the impulsive dynamics of this system, due to its high degeneracy, as it undergoes a co-dimension three bifurcation.

Experimental demonstration of transient resonance capture in a system of two coupled oscillators with essential stiffness nonlinearity

The purpose of this paper is to report an experimental study of transient resonance capture that may occur in a system of two coupled oscillators with essential (i.e., nonlinearizable) nonlinearity. It is shown that during transient resonance capture the two oscillators are in a state of resonance, the frequency of which varies with time, which leads to targeted nonlinear energy transfer. Further evidence of resonance capture is a non-time-like behavior of the phase difference between the oscillators; this quantity is monitored using the Hilbert transform or the Huang–Hilbert transform in the case of multifrequency response signals.

State-Dependent Scaling Problems and Stability of Interconnected iISS and ISS Systems

This paper addresses the problem of establishing stability of nonlinear interconnected systems. This paper introduces a mathematical formulation of the state-dependent scaling problems whose solutions directly provide Lyapunov functions proving stability properties of interconnected dissipative systems in a unified manner. Stability criteria are interpreted as sufficient conditions for the existence of solutions to the state-dependent scaling problems. Computing solutions to the problems is straightforward for systems covered by classical stability criteria. It, however, could be too difficult for systems with strong nonlinearity. The main purpose of this paper is to demonstrate the effectiveness beyond formal applicability by focusing on interconnected integral input-to-state stable (iISS) systems and input-to-state stable (ISS) systems. This paper derives small-gain-type theorems for interconnected systems involving iISS systems from the state-dependent scaling formulation. This paper provides solutions and Lyapunov functions explicitly. The new framework seamlessly generalizes the ISS small-gain theorem and classical stability criteria such as the Lp small-gain theorem, the passivity theorems, the circle, and Popov criteria. State-dependence of the scaling is crucial for effective treatment of essential nonlinearities, while constants are sufficient for classical nonlinearities

Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments

We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency–energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers.

Robustness of nonlinear targeted energy transfer in coupled oscillators to changes of initial conditions

The problem of choosing the appropriate initial conditions for achieving efficient targeted energy transfer (energy pumping) in a system of coupled oscillators with essential nonlinearity is investigated analytically and numerically. The treatment is based on the simplified description of the dynamic flow at the initial stage of motion and up to time scale of O(1). This approach appears to be complimentary to the invariant manifold approach developed in earlier works; hence, the combination of the two approaches allows the complete description of the targeted energy transfer process at its entire time span, and answers the question of robustness of energy transfer to changes of initial conditions.

Variational approach to dequantization

We present a dequantization procedure based on a variational approach whereby quantum fluctuations latent in the quantum momentum are suppressed. This is done by adding generic local deformations to the quantum momentum operator which give rise to a deformed kinetic term quantifying the amount of 'fuzziness' caused by such fluctuations. Considered as a functional of such deformations, the deformed kinetic term is shown to possess a unique minimum which is seen to be the classical kinetic energy. Furthermore, we show that extremization of the associated deformed action functional introduces an essential nonlinearity to the resulting field equations which are seen to be the classical Hamilton–Jacobi and continuity equations. Thus, a variational procedure determines the particular deformation that has the effect of suppressing the quantum fluctuations, resulting in dequantization of the system.

Insight from Economically Optimal Steady-State Operating Policies for Dynamic Plantwide Control

Shortcut methods for the rapid determination of the optimal steady-state operating policy for certain classes of chemical plants have recently been published [Ward et al., Ind. Eng. Chem. Res. 2004, 43, 3957; 2005, 44, 6729]. The purpose of this work is to investigate the implications of these optimal operating policies for the selection of control methodology and control structure design, and to demonstrate that the operating policies can be implemented dynamically. These methods can predict, using basic information about the process chemistry, whether a self-optimizing control structure will give good economic performance, or whether an explicit optimization layer will be required in the control hierarchy. When a self-optimizing control structure is sufficient, this methodology allows the engineer to anticipate which binary pairings between controlled and manipulated variables are optimal. When an optimizing control layer (e.g., MPC or real-time optimizer (RTO)) is required, the method suggests a low-order model that will capture the essential nonlinearities of the process and that can be used for the design of the optimizing controller. Dynamic simulations of two case studies confirm that the control methodologies and control structures, which are predicted best by the methods of Ward et al., do, in fact, give the economically best operation of the process, when implemented dynamically.

Dynamic fibre sliding along debonded, frictional interfaces

The problem is considered of a fibre that is driven dynamically, by compression at one end, into a matrix. The fibre is not initially bonded to the matrix, so that its motion is resisted solely by friction. Prior work based on simplified models has shown that the combination of inertial effects and friction acting over long domains of the fibre–matrix interface gives rise to behaviour that is far more complex than in the well-known static loading problem. The front velocity may depart significantly from the bar wave speed and regimes of slip, slip/stick and reverse slip can exist for different material choices and loading rates. Here more realistic numerical simulations and detailed observations of dynamic displacement fields in a model push-in experiment are used to seek more complete understanding of the problem. The prior results are at least partly confirmed, especially the ability of simple shear-lag theory to predict front velocities and gross features of the deformation. Some other fundamental aspects are newly revealed, including oscillations in the interface stresses during loading; and suggestions of unstable, possibly chaotic interface conditions during unloading. Consideration of the experiments and two different orders of model suggest that the tentatively characterized chaotic phenomena may arise because of the essential nonlinearity of friction, that the shear traction changes discontinuously with the sense of the motion, rather than being associated with the details of the constitutive law that is assumed for the friction. This contrasts with recent understanding of instability and ill-posedness at interfaces loaded uniformly in time, where the nature of the assumed friction law dominates the outcome.

Energy Transfers in a System of Two Coupled Oscillators with Essential Nonlinearity: 1:1 Resonance Manifold and Transient Bridging Orbits

The purpose of this study is to highlight and explain the vigorous energy transfers that may take place in a linear oscillator weakly coupled to an essentially nonlinear attachment, termed a nonlinear energy sink. Although these energy exchanges are encountered during the transient dynamics of the damped system, it is shown that the dynamics can be interpreted mainly in terms of the periodic orbits of the underlying Hamiltonian system. To this end, a frequency-energy plot gathering the periodic orbits of the system is constructed which demonstrates that, thanks to a 1:1 resonance capture, energy can be irreversibly and almost completely transferred from the linear oscillator to the nonlinear attachment. Furthermore, it is observed that this nonlinear energy pumping is triggered by the excitation of transient bridging orbits compatible with the nonlinear attachment being initially at rest, a common feature in most practical applications. A parametric study of the energy exchanges is also performed to understand the influence of the parameters of the nonlinear energy sink. Finally, the results of experimental measurements supporting the theoretical developments are discussed.

Energy Pumping with Various Nonlinear Structures: Numerical Evidences

Numerical investigations are carried out on a linear structure, weakly coupled to a small nonlinear attachment. The essential nonlinearity of the attachment enables it to resonate with any of the linearized modes of the structure leading to energy pumping, i.e. passive, one-way, irreversible transfer of energy from the structure to the attachment. Different nonlinear structures (piecewise linear system, chaotic system) and efficiency of energy pumping are studied in each case in order to be able to apply it to civil engineering. As a specific application, attenuation of vibrations of a building is studied with two building models. In particular, the case of stochastic excitations is analyzed to examine if it is possible to process energy pumping when a seism occurs and an indicator of efficiency has been introduced.

Irreversible Passive Energy Transfer in Coupled Oscillators with Essential Nonlinearity

We study numerically and analytically the dynamics of passive energy transfer from a damped linear oscillator to an essentially nonlinear end attachment. This transfer is caused by either fundamental or subharmonic resonance capture, and in some cases is initiated by nonlinear beat phenomena. It is shown that, due to the essential nonlinearity, the end attachment is capable of passively absorbing broadband energy at both high and low frequencies, acting, in essence, as a passive broadband boundary controller. Complicated transitions in the damped dynamics can be interpreted based on the topological structure and bifurcations of the periodic solutions of the underlying undamped system. Moreover, complex resonance capture cascades are numerically encountered when we increase the number of degrees of freedom of the system. The ungrounded essentially nonlinear end attachment discussed in this work can find application in numerous practical settings, including vibration and shock isolation of structures, seism...

ACTIVE VIBRATION REJECTION IN STEEL ROLLING MILLS

This contribution is concerned with vibration rejection by nonlinear control techniques in steel industry, particularly in rolling mill plants. The high quality requirements of rolled products, especially in relation to the thickness tolerances, mean a challenging task also from a control point of view. The key part of this contribution deals with so-called wrapper rolls as well as chatter vibration phenomena. With regard to the control design a mathematical model based on physical considerations is introduced where the essential nonlinearities of the system are taken into account. For the purpose of vibration rejection two nonlinear control concepts based on energy considerations and the theory of differential flatness are presented.

Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses

The dynamics of a linear oscillator, coupled to an essentially nonlinear attachment of substantially lower mass, is investigated. The essential (nonlinearizable) nonlinearity of the attachment enables it to resonate with the oscillator, leading to energy pumping phenomena, e.g., passive, almost irreversible transfer of energy from the substructure to the attachment. Feasibility of this process for possible applications depends on relative mass of the attachment, the obvious goal being to minimize it while preserving the efficiency of the pumping. Two different models of the attachment coupled to the main single-degree-of-freedom body are proposed and analyzed both analytically and numerically. It is demonstrated that efficient energy pumping may be obtained for a rather small value of the attachment mass. Two mechanisms of energy pumping in the system under consideration are revealed. The first one is similar to previously studied resonance capture; a novel analytic framework allowing explicit account of the damping is proposed. The second mechanism is related to nonresonant excitation of high-frequency vibrations of the attachment. Both mechanisms are demonstrated numerically for a model consisting of a linear chain with a nonlinear attachment.

Transient resonant interactions of finite linear chains with essentially nonlinear end attachments leading to passive energy pumping

We study resonance capture phenomena leading to energy pumping in systems with multiple degrees of freedom (DOF), composed of N linear oscillators weakly coupled to strongly nonlinear attachments possessing essential (nonlinearizable) cubic stiffness nonlinearities. First we present numerical evidence of energy pumping in the systems under consideration, i.e., of passive, one-way (irreversible) transfer of externally imparted energy to the nonlinear attachments, provided that the energy is above a critical level. To obtain a better understanding of the energy pumping phenomenon we reduce the dynamics governing the chain–attachment interaction to a single, nonlinear integro-differential equation that governs exactly the transient dynamics of the strongly nonlinear attachment. By introducing an approximation based on Jacobian elliptic functions we derive an approximate set of two nonlinear integro-differential modulation equations that govern the time evolution of the amplitude and phase of the motion of the attachment. This set of modulation equations is studied both analytically and numerically. We then perform a perturbation analysis in an O( ε ) neighborhood of a 1:1 resonant manifold of the system in order to study the attracting region in the reduced phase space of the system, that is responsible for resonance capture and nonlinear energy pumping. This analysis provides a justification of the numerical findings.

New nonlinear control of three-phase NPC boost rectifier operating under severe disturbances

This paper proposes a new nonlinear control of three-phase three-level neutral point clamped (NPC) boost rectifier operating under severe disturbances. The nonlinear control strategy is employed to take into account the essential nonlinearities of the system and it totally differ from the common approach of linearizing the nonlinear system around a nominal operating point. In order to apply this technique, a nonlinear state-space mathematical model of the rectifier is developed. An input/output feedback linearization is then designed and the linearizing control law is derived. The resulting model is linearized and decoupled. Afterwards, the stabilizing controllers are designed based on linear techniques to control power factor, output and neutral point voltages. This control uses a 3 kHz pulse width modulator (PWM). Simulation results are performed using Simulink/Matlab and PSB tool. The controllers’ performance has been evaluated under diverse parameter variations and input and output disturbances. A low output voltage ripples and line-current THD and also a small overshoot and a fast settling time are obtained.

Stratified flow over asymmetric and double bell-shaped obstacles

We performed experiments on quasi-two-dimensional lee-wave formation behind obstacles towed through a linearly stratified fluid. Initial transient times, amplitudes and wavelengths are measured for five different obstacle shapes at several flow velocities. We found that the wave field is determined primarily by the height and the leeward slope of the obstacle, in agreement with previous works. Differences between measured quantities and predictions of linear theories indicate that nonlinear effects are important in our parameter range. Experiments with a twin obstacle gave surprising results: average wavelengths and amplitudes are systematically lower than that of produced by an isolated obstacle, which we attribute to the dominance of essential nonlinearities such as anomalously strong wave dispersion.

Bogomol'nyi, Prasad, and Sommerfield configurations in smectics.

It is typical in smectic liquid crystals to describe elastic deformations with a linear theory when the elastic strain is small. In smectics, certain essential nonlinearities arise from the requirement of rotational invariance. By employing the Bogomol'nyi, Prasad, and Sommerfield decomposition and relying on boundary conditions and geometric invariants, we have found a large class of exact solutions. We introduce an approximation for the deformation profile far from a spherical inclusion and find an enhanced attractive interaction at long distances due to the nonlinear elasticity, confirmed by numerical minimization.

Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities

We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes—NNMs), as well as, asynchronous periodic motions (elliptic orbits—EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.