Nonlinear response of an imperfect microcantilever static and dynamically actuated considering uncertainties and noise

Benedetti, Kaio C. B.; Gonçalves, Paulo B.

doi:10.1007/s11071-021-06600-2

Cited by 10 publications

(10 citation statements)

References 60 publications

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“…The Ulam method, a classical discretization method of flows in phase-space, was considered. It is the most natural discretization method within the uncertainty framework and has been applied to stochastic dynamical systems [10,30,55]. However, it displays a numerical diffusion due to the phase-space discretization.…”

Section:  mentioning

confidence: 99%

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 1: Adaptative Phase-Space Discretization Strategy

Benedetti

Gonçalves

Lenci

et al. 2022

Preprint

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An adaptative phase-space discretization strategy for the global analysis of stochastic nonlinear dynamical systems with competing attractors considering parameter uncertainty or noise is proposed. The strategy is based on the classical Ulam method. The appropriate transfer operators for a given dynamical system are derived and applied to obtain and refine the basins of attraction boundaries and attractors distributions and densities. A review of the main concepts of parameter uncertainty and stochasticity from a global dynamics perspective is given, and the necessary modifications to the Ulam method are addressed. The stochastic basin of attraction definition here used replaces the usual basin concept. It quantifies the probability of the response associated with a given set of initial conditions converging to a particular attractor. The phase-space dimension is augmented to include the extra dimensions associated with the parameter space for the case of parameter uncertainty, being a function of the number of uncertain parameters. The expanded space is discretized, resulting in a collection of transfer operators that enable obtaining the required statistics. Finally, a Monte Carlo procedure is conducted for the stochastic case to construct the proper transfer operator. The present formulation is employed in Part II to investigate the global dynamics of two archetypal nonlinear oscillators.

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Section:  mentioning

confidence: 99%

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 1: Adaptative Phase-Space Discretization Strategy

Benedetti

Gonçalves

Lenci

et al. 2022

Preprint

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“…The same time-integration parameters are adopted, with time-step T/200. However, the noise requires specialized integrators, so a stochastic version of the fourth-order Runge-Kutta method is adopted for the construction of the flow T  [36], with time-step T/200, where T = 2π/Ω and Ω is the forcing frequency. The transfer matrix of the noise-driven system is the Foias operator, and the projected operator is (see eq.…”

Section: Effects Of Additive White Noisementioning

confidence: 99%

“…30. The influence of noise in such case was previously studied in [36] where a box covering of the attractors is adopted, instead of the present operator framework. Here a distinction between chaotic and nonchaotic solutions is observed in the attractors' distributions.…”

Section: The Case Of a Chaotic Attractor Under White Noisementioning

confidence: 99%

“…Poon and Grebogi [35] showed that noise might move the system away from the neighborhood of the attractor towards the basin boundary and over a nearby saddle point to another basin of attraction, increasing the competition between the attractors. Noise may also cause the basins to merge or disappear [36], and when the noise amplitude is above a critical value, the distinction between two coexisting attractors is lost. In addition, noise may lead to stochastic bifurcations and a shift of the bifurcation point [37][38][39].…”

Section: Introductionmentioning

confidence: 99%

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Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 2: Influence of Uncertainties and Noise on Basins/Attractors Topology and Integrity

Benedetti

Gonçalves

Lenci

et al. 2022

Preprint

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This paper explores the influence of uncertainties and noise on the global dynamics of nonlinear systems, with emphasis on the basins and attractors’ topology and on the dynamic integrity of coexisting solutions. For this, the adaptive phase-space discretization strategy proposed in Part I [1] is employed. Two well-known archetypal oscillators found in engineering applications, the Helmholtz and Duffing oscillators under harmonic or parametric excitation with uncertain parameters or added load noise, are studied. The results demonstrate the adaptive capability of the proposed method, increasing the resolution without increasing the computational cost. Mean basins of attraction and attractors’ distributions are obtained. The time-dependency of stochastic responses is demonstrated, with long-transients influencing the short-term behavior. Finally, the effect of uncertainties and noise on the basins areas, attractors distributions and basin boundaries is discussed, which can be used to evaluate the dynamic integrity of the competing basins.

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“…Recent trends in nonlinear dynamics is a collection of emerging topics tackling parametric resonance and pattern selection in MEMS [ 16 ], the effect of uncertainties and noise on the response of microcantilevers [ 17 ], enhanced sensitivity of measurements by virtually coupling a resonator with a virtual resonator [ 18 ], highly tunable vibration isolation devices exploiting negative stiffness and shape memory material hysteresis [ 19 ].…”

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confidence: 99%

Preface to the special issue NODYCON 2021, Second International Nonlinear Dynamics Conference, Feb. 16–19, 2021

Hajj

Chen²,

Chen

et al. 2022

Nonlinear Dyn

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Nonlinear response of an imperfect microcantilever static and dynamically actuated considering uncertainties and noise

Cited by 10 publications

References 60 publications

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 1: Adaptative Phase-Space Discretization Strategy

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 1: Adaptative Phase-Space Discretization Strategy

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 2: Influence of Uncertainties and Noise on Basins/Attractors Topology and Integrity

Preface to the special issue NODYCON 2021, Second International Nonlinear Dynamics Conference, Feb. 16–19, 2021

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