Analysis on the motion of nonlinear vibration with fractional order and time variable mass

Yu, Yue; Zhou, Wenyao; Zhang, Zhengdi; Bi, Qinsheng

doi:10.1016/j.aml.2021.107621

Cited by 7 publications

(4 citation statements)

References 21 publications

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“…Nonconservative and nonlinear damping forces acting on the resonantly actuated micro-cantilevers can be expressed by the forced Van der Pol oscillator. Dynamic characteristics of the systems can be described by utilizing Van der Pol oscillators for various applications [41][42][43][44][45]. Achieving higher sensitivity, a multi-frequency excitation scheme can be implemented in the presence of hydrodynamical loads.…”

Section: A Multi-modal Nonlinear Dynamic Modelmentioning

confidence: 99%

A multi-modal nonlinear dynamic model to investigate time-domain responses of a micro-cantilever in fluids

Yilmaz

2024

Eng. Res. Express

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In this current work, a new nonlinear dynamic model based on the forced Van der Pol oscillator is introduced to demonstrate the time-domain sensitivities of the micro-cantilever to the varying properties of the surrounding fluids. Effects of diverse multi-frequency excitations on the hydrodynamically forced displacements are investigated for the Glycerol-water solutions with different concentrations. Driving forces at the eigenmode frequencies are applied simultaneously to actuate the micro-cantilever in multi-modal operations. The hydrodynamic force induces notable variations in the observables of high-frequency steady-state vibrations. To illustrate, the frequency of the displacements decreases with increasing dynamic viscosity and density of the fluids (among 55% and 85% Glycerol-water solutions) in bimodal- and trimodal-frequency excitations. Essentially, the observable responses are often used to distinguish the surrounding fluids in which the micro-cantilever operates. In addition, steady-state observables are achieved at only particular eigenmodes in single- and multi-frequency operations. It is highlighted that the periodic oscillations are obtained for the first and second eigenmodes with the highest value of forced Van der Pol parameter (μ =1030). Clearly, higher eigenmodes require different values of the nonlinearity parameter to acquire periodic vibrations in multi-modal operations. In general, achieving steady-state observables is substantially critical in quantifying sensitivity to varying fluid properties. For instance, the vibration frequency of around 7.33 MHz and amplitude of around 0.03 pm are obtained at the first eigenmode for 75% Glycerol-water solution in tetra-modal operations. Note that femtometer amplitudes of deflections can be measured using quantum-enhanced AFM techniques. The frequency responses obtained in this work are compared with the measured ones in the literature and the results show satisfactory agreements. Therefore, a novel multi-modal nonlinear dynamic model enables to quantify observable sensitivity to micro-rheological properties at higher eigenmodes of the micro-cantilever.

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Section: A Multi-modal Nonlinear Dynamic Modelmentioning

confidence: 99%

A multi-modal nonlinear dynamic model to investigate time-domain responses of a micro-cantilever in fluids

Yilmaz

2024

Eng. Res. Express

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“…It can be derived that J 0 01 = based on the generalized integral convergence criterion and equation (14). Further, one can get J 0, 02 = so then J 0.…”

Section: Approximate Equivalent Integer-order Systemmentioning

confidence: 99%

“…Since fractional differential operators involve integrals defined over a time domain, which will lead to significant genetic and memory effects [13]. As a result, fractional calculus has received great attention in the last few decades and has been extensively studied in many fields such as nonlinear oscillators, control systems, diffusion problems, viscoelastic mechanics, rheology and so on [14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Taming chaos in generalized Liénard systems by the fractional-order feedback based on Melnikov analysis

Huang

2023

Phys. Scr.

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The dynamical behavior of Liénard systems has always been a hot topic in nonlinear analysis. In the present study, a simple fractional-order feedback controller is put forward to tame chaos for a class of forced generalized Liénard systems. Adopting harmonic balance method, the first-order approximate equivalent integer-order system of the original fractional-order system is deduced. Then the criterion for taming chaos is established by employing the Melnikov approach. Duffing-Rayleigh chaotic oscillator is taken as an example to illustrate the validity of the proposed method. Firstly, the critical feedback intensity and differential order for taming chaos are obtained by the proposed criterion. Then, multiple numerical indicators such as phase portrait, time history plot, Lyapunov exponent and bifurcation diagram are provided to assist in analyzing theoretical results.

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“…x, τ is the small perturbation function. The procedure for solving (1) is based on the assumption that the approximate solution is the small, perturbed version of the nearly exact solution of the equation with constant parameters [20]. First, the exact solution of the unperturbed, truly nonlinear oscillator in the form of the Ateb function is determined.…”

Section: Model Of Oscillatormentioning

confidence: 99%

Oscillator with Time-Variable Degree of Nonlinearity: Theory and Application in Aging of Polymer Composite Structure

Cveticanin,

Zukovic

2023

Mathematics

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In this paper, a new type of one-degree-of-freedom pure nonlinear oscillator with a time variable degree of nonlinearity is introduced. Namely, the degree of the nonlinearity in the restitution force is not a constant but a ‘slow time’ variable function. The mathematical model is a second-order nonlinear differential equation with time-variable parameters. An approximate solving procedure based on the method of time-variable amplitude, frequency and phase is developed. It is found that the vibration properties depend on initial conditions and the degree of time-variable function. The theoretical results are tested on almost linear, cubic and high-degree nonlinear oscillators. According to theoretical considerations, the process of aging in fiber-reinforced polymer composite structure is investigated. It is found that the amplitude and the frequency of vibration of the reinforced polymer structure decrease over time. The result is in good agreement with already published experimental data. The additional conclusion of the research is that the oscillator model suggested in the paper is suitable for the prediction of mechanic properties of the polymer structure in aging and also in similar destructive processes.

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Analysis on the motion of nonlinear vibration with fractional order and time variable mass

Cited by 7 publications

References 21 publications

A multi-modal nonlinear dynamic model to investigate time-domain responses of a micro-cantilever in fluids

A multi-modal nonlinear dynamic model to investigate time-domain responses of a micro-cantilever in fluids

Taming chaos in generalized Liénard systems by the fractional-order feedback based on Melnikov analysis

Oscillator with Time-Variable Degree of Nonlinearity: Theory and Application in Aging of Polymer Composite Structure

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