Noise influenced elastic cantilever dynamics with nonlinear tip interaction forces

In this work, the authors study the influence of noise on the dynamics of base-excited elastic cantilever structures at the macroscale and microscale by using experimental, numerical, and analytical means. The macroscale system is a base excited cantilever structure whose tip experiences nonlinear interaction forces. These interaction forces are constructed to be similar in form to tip interaction forces in tapping mode atomic force microscopy (AFM). The macroscale system is used to study nonlinear phenomena and apply the associated findings to the chosen AFM application. In the macroscale experiments, the tip of the cantilever structure experiences long-range attractive and short-range repulsive forces. There is a small magnet attached to the tip, and this magnet is attracted by another one mounted to a high-resolution translatory stage. The magnet fixed to the stage is covered by a compliant material that is periodically impacted by the cantilever’s tip. Building on their earlier work, wherein the authors showed that period-doubling bifurcations associated with near-grazing impacts occur during off-resonance base excitations of macroscale and microscale cantilevers, in the present work, the authors focus on studying the influence of Gaussian white noise when it is included as an addition to a deterministic base excitation input. The repulsive forces are modeled as Derjaguin–Muller–Toporov (DMT) contact forces in both the macroscale and microscale systems, and the attractive forces are modeled as van der Waals attractive forces in the microscale system and magnetic attractive forces in the macroscale system. A reduced-order model, based on a single mode approximation is used to numerically study the response for a combined deterministic and random base excitation. It is experimentally and numerically found that the addition of white Gaussian noise to a harmonic base excitation facilitates contact between the tip and the sample, when there was previously no contact with only the harmonic input, and results in a response that is nominally close to a period-doubled orbit. The qualitative change observed with the addition of noise is associated with near-grazing impacts between the tip and the sample. The numerical and experimental results further motivate the formulation of a general analytical framework, in which the Fokker–Planck equation is derived for the cantilever-impactor system. After making a set of approximations, the moment evolution equations are derived from the Fokker–Planck equation and numerically solved. The resulting findings support the experimental results and demonstrate that noise can be added to the input to facilitate contact between the cantilever’s tip and the surface, when there was previously no contact with only a harmonic input. The effects of Gaussian white noise are numerically studied for a tapping mode AFM application, and it is shown that contact between the tip and the sample can be realized by adding noise of an appropriate level to a harmonic excitation.     

Global transient dynamics of nonlinear parametrically excited systems

In this paper we examine the response of a typical nonlinear system that is subjected to parametric excitation. Particular attention is paid to how basins of attraction evolve such that the global transient stability of the system may be assessed. We show that at a forcing level that is considerably smaller than that at which the steady-state attractor loses its stability, there may exist a rapid erosion and stratification of the basin, signifying a global loss of engineering integrity of the system.We also show, for a system near its equilibrium state, that the boundaries in parameter space can become fractal. The significance of such an analysis is not only that it corresponds to a failure locus for a system subjected to a sudden pulse of excitation, but since the phase-space basin is often eroded throughout its central region, the determination of basin boundaries in control space can often reflect the characteristics of the phase-space basin structure, and hence on the macroscopic level they provide information regarding the global transient stability of the system.     

On the dynamics of randomly excited nonlinear systems

Summary The problem of characterising the dynamics of randomly excited systems is examined. It is shown that the probability approach, though conceptually more rigorous, is difficult to apply to statistics other than normal ones. The direct method of Axelby is recalled and applied to a nonlinear system with random excitation characterised by a statistic of great interest in real physical systems. The application is developed parametrically with reference to a second order system for which the calculations are developed and the quantitative results discussed.

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Sommario Nella disamina del problema di caratterizzare il comportamento dinamico di sistemi eccitati da segnali casuali, si mette in rilievo come l'approccio probabilistico, sebbene concettualmente più rigoroso, sia di difficile applicazione ai casi statistici oltre che a quelli normali. Si richiama quindi il metodo diretto di Axelby che viene applicato ad un sistema non lineare eccitato da un segnale casuale. Lo sviluppo dell'applicazione, in termini parametrici, fa riferimento ad un sistema di secondo grado e i relativi risultati numerici vengono discussi nel loro significato quantitativo.

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Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink

Nonlinear behaviors are investigated for a structure coupled with a nonlinear energy sink. The structure is linear and subject to a harmonic excitation, modeled as a forced single-degree-of-freedom oscillator. The nonlinear energy sink is modeled as an oscillator consisting of a mass,a nonlinear spring, and a linear damper. Based on the numerical solutions, global bifurcation diagrams are presented to reveal the coexistence of periodic and chaotic motions for varying nonlinear energy sink mass and stiffness. Chaos is numerically identified via phase trajectories, power spectra,and Poincaré maps. Amplitude-frequency response curves are predicted by the method of harmonic balance for periodic steady-state responses. Their stabilities are analyzed.The Hopf bifurcation and the saddle-node bifurcation are determined. The investigation demonstrates that a nonlinear energy sink may create dynamic complexity.     

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.     

Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam

The sub- and super-critical dynamics of an axially moving beam subjected to a transverse harmonic excitation force is examined for the cases where the system is tuned to a three-to-one internal resonance as well as for the case where it is not. The governing equation of motion of this gyroscopic system is discretized by employing Galerkin’s technique which yields a set of coupled nonlinear differential equations. For the system in the sub-critical speed regime, the periodic solutions are studied using the pseudo-arclength continuation method, while the global dynamics is investigated numerically. In the latter case, bifurcation diagrams of Poincaré maps are obtained via direct time integration. Moreover, for a selected set of system parameters, the dynamics of the system is presented in the form of time histories, phase-plane portraits, and Poincaré maps. Finally, the effects of different system parameters on the amplitude-frequency responses as well as bifurcation diagrams are presented.     

Vibration control of a transversely excited cantilever beam with tip mass

The problem of controlling the vibration of a transversely excited cantilever beam with tip mass is analyzed within the framework of the Euler–Bernoulli beam theory. A sinusoidally varying transverse excitation is applied at the left end of the cantilever beam, while a payload is attached to the free end of the beam. An active control of the transverse vibration based on cubic velocity is studied. Here, cubic velocity feedback law is proposed as a devise to suppress the vibration of the system subjected to primary and subharmonic resonance conditions. Method of multiple scales as one of the perturbation technique is used to reduce the second-order temporal equation into a set of two first-order differential equations that govern the time variation of the amplitude and phase of the response. Then the stability and bifurcation of the system is investigated. Frequency–response curves are obtained numerically for primary and subharmonic resonance conditions for different values of controller gain. The numerical results portrayed that a significant amount of vibration reduction can be obtained actively by using a suitable value of controller gain. The response obtained using method of multiple scales is compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. Numerical simulation for amplitude is also obtained by integrating the equation of motion in the frequency range between 1 and 3. The developed results can be extensively used to suppress the vibration of a transversely excited cantilever beam with tip mass or similar systems actively.     

Fast and slow dynamics in a nonlinear elastic bar excited by longitudinal vibrations

The dynamics of heterogeneous materials, like rocks and concrete, is complex. It includes such features as nonlinear elasticity, hysteresis, and long-time relaxation. This dynamics is very sensitive to microstructural changes and damage. The goal of this paper is to propose a physical model describing the longitudinal vibrations in heterogeneous material, and to develop a numerical strategy to solve the evolution equations. The theory relies on the coupling of two processes with radically different time scales: a fast process at the frequency of the excitation, governed by nonlinear elasticity and viscoelasticity, and a slow process, governed by the evolution of defects. The evolution equations are written as a nonlinear hyperbolic system with relaxation. A time-domain numerical scheme is developed, based on a splitting strategy. The features observed by numerical simulations show qualitative agreement with the features observed experimentally by Dynamic Acousto-Elastic Testing.     

Three-mode interactions in harmonically excited systems with quadratic nonlinearities

An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ₃2₂ and ₂2₁ to a harmonic excitation of the third mode, where the _m are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitudeF of the excitation as a control parameter. As the excitation amplitudeF is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. AsF is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. AsF is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.     

Tracking modal interactions in nonlinear energy sink dynamics via high-dimensional invariant manifold

Nonlinear Dynamics - The work is devoted to the study of a MEMS resonator dynamics under the action of phase-locked and automatic gain control loops. Particular attention is directed to the study...     

Fractional nonlinear dynamics of learning with memory

Nonlinear Dynamics - In this paper, we consider generalization of the Lucas model of learning (learning-by-doing) that is described in the paper Robert E. Lucas (Econometrica 61(2):251–272,...     

Gas dynamics of a selectively excited gas

In order to describe the motion of a selectively excited gas, model kinetic equations are proposed in the present study, and they are used to construct the equations of gas dynamics. There is consideration of the one-dimensional problem of heat transmission between plates across a selectively excited gas. Discontinuities are found in the temperature on the boundaries of the phases. It is shown that a selectively excited gas may be used to transmit heat from a cold body to a hot one. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 151–158, September–October, 1988.     

Long-range prediction of epileptic seizures with nonlinear dynamics

Patients with uncontrolled epilepsy have some significant problems with planning life routines, and thus one goal of the present study was to explore the viability of predicting seizures in time intervals of one week. The second goal was to utilize the principle of dynamic diseases and to assess the viability of a cusp catastrophe model for seizure onset that was proposed by Cerf (2006). A seizure history of 124 weeks from one adult male patient fit both the cusp and fold catastrophe models (R2 = .92 and .88 respectively) reasonably well using the pdf method and more accurately than counterpart linear models. Prediction of future states was possible, but somewhat compromised because of the nonstationary nature of the data and uncertainties regarding the control variables in the catastrophe models. Analyses of lag functions, however, revealed some surprising elements, suggesting that the precursory conditions for a seizure could be building up over a period of several weeks and that a self-correcting effect within the nervous system could have been occurring.     

Response analysis of randomly excited nonlinear systems with symmetric weighting Preisach hysteresis

An approximate method for analyzing the response of nonlinear systems with the Preisach hysteresis of the non-local memory under a stationary Gaussian excitation is presented based on the covariance and switching probability analysis. The covariance matrix equation of the Preisach hysteretic system response is derived. The cross correlation function of the Preisach hysteretic force and response in the covariance equation is evaluated by the switching probability analysis and the Gaussian approximation to the response process. Then an explicit expression of the correlation function is given for the case of symmetric Preisach weighting functions. The numerical result obtained is in good agreement with that from the digital simulation. The project supported by the National Natural Science Foundation of China (19972059) and Zhejiang Provincial Natural Science Foundation (101046)     

Nonlinear interactions in a parametrically excited system with widely spaced frequencies

An investigation is presented into the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system.     

Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments

We study targeted energy transfers (TETs) and nonlinear modal interactions attachments occurring in the dynamics of a thin cantilever plate on an elastic foundation with strongly nonlinear lightweight attachments of different configurations in a more complicated system towards industrial applications. We examine two types of shock excitations that excite a subset of plate modes, and systematically study, nonlinear modal interactions and passive broadband targeted energy transfer phenomena occurring between the plate and the attachments. The following attachment configurations are considered: (i) a single ungrounded, strongly (essentially) nonlinear single-degree-of-freedom (SDOF) attachment—termed nonlinear energy sink (NES); (ii) a set of two SDOF NESs attached at different points of the plate; and (iii) a single multi-degree-of-freedom (MDOF) NES with multiple essential stiffness nonlinearities. We perform parametric studies by varying the parameters and locations of the NESs, in order to optimize passive TETs from the plate modes to the attachments, and we showed that the optimal position for the NES attachments are at the antinodes of the linear modes of the plate. The parametric study of the damping coefficient of the SDOF NES showed that TETs decreasing with lower values of the coefficient and moreover we showed that the threshold of maximum energy level of the system with strong TETs occured in discrete models is by far beyond the limits of the engineering design of the continua. We examine in detail the underlying dynamical mechanisms influencing TETs by means of empirical mode decomposition (EMD) in combination with wavelet transforms. This integrated approach enables us to systematically study the strong modal interactions occurring between the essentially nonlinear NESs and different plate modes, and to detect the dominant resonance captures between the plate modes and the NESs that cause the observed TETs. Moreover, we perform comparative studies of the performance of different types of NESs and of the linear tuned mass dampers (TMDs) attached to the plate instead of the NESs. Finally, the efficacy of using this type of essentially nonlinear attachments as passive absorbers of broadband vibration energy is discussed.     

On the behavior of parametrically excited purely nonlinear oscillators

In this study, parametrically excited purely nonlinear oscillators are considered. Instabilities associated with 2:1, 3:1, and 4:1 subharmonics resonances are analyzed by assuming the solution for motion in the form of a Jacobi elliptic function, the elliptic parameter, and the frequency of which are calculated based on the energy conservation law of the corresponding conservative system. Chirikov??s overlap criterion is used to obtain the approximate critical value of the amplitude of the parametric excitation that causes the transition from local irregular behavior (seen as chaotic) to global chaos. The analytical results derived are compared with numerically results.