The equilibrium stability for a smooth and discontinuous oscillator with dry friction

Li, Zhixin; Cao, Qingjie; Léger, Alain

doi:10.1007/s10409-015-0481-y

Cited by 10 publications

(4 citation statements)

References 26 publications

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“…However, for a rough hill with a static friction coefficient μ (Figure B), a straightforward static analysis reveals that the ball can remain stationary if it is placed on a slope with a local inclination angle smaller than

\arctan μ

. Therefore, the stable configurations of the ball on the rough hill expand to the whole set of region illustrated in pink in Figure B, including the hilltops . In other words, in conservative systems without friction, maxima of potential energy correspond to unstable equilibrium solutions; while in nonconservative systems with friction, equilibrium solutions can be stable even when potential energy reaches a maximum.…”

Section: Resultsmentioning

confidence: 99%

Programmable Granular Metamaterials for Reusable Energy Absorption

Zhao

Jin

2019

Adv Funct Materials

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Designing metamaterials with programmable features has emerged as a promising pathway for reusable energy absorption. While the current designs of reusable energy absorbers mainly exploit mechanical instability of flexible beams, here is created a new kind of metamaterial for reusable and programmable energy absorption by integrating rigid granular materials and compliant stretchable components. In each unit cell of the metamaterial, the stretchable components connect the granular particles to maintain the integrity and control the deformation pattern of the material. When the metamaterial is subjected to an external load, the input energy is partially trapped as elastic energy in the stretchable components, and partially dissipated by friction between the granular particles, forming hysteresis between the loading and unloading force-displacement curves. Through tuning the structural design of the metamaterial, the pretension and stiffness of the stretchable components, and the size of and friction between the particles, a vast design space is achieved to program the mechanical behavior of the metamaterial, such as the load-displacement curve, the multistability, and the amount of energy dissipation. Experimental impact tests on a thin glass panel confirm energy-absorbing capability of the proposed metamaterial. This design strategy opens a new avenue for creating reusable energy-absorbing metamaterials.The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adfm.201901258. and a long working distance. Most commercial energy-absorbing products adopt plastic deformation or fragmentation of foams, [3] metals, [4] and ceramics, [5] to gain the desired hysteresis behavior, and introduce artificial defects, such as prefolding of origami, [6][7][8] to control the deformation mode. In the products with plastic deformation or fragmentation, dislocations and bond breakage occur at the molecule level, and a large amount of energy can be absorbed. However, they are only for one-time usage, since the materials are permanently damaged.In order to overcome the shortcoming of one-time usage, reusable energyabsorbing materials have been proposed by developing damage-tolerant micro [9] or nano lattices, [5] architectured composite materials, [10,11] and mechanical metamaterials. [12][13][14][15] Mechanical metamaterials are materials with microstructures, which give rise to unique mechanical properties that are otherwise hard to achieve. [12][13][14] By designing the microstructures, researchers have developed energy-absorbing metamaterials mainly through exploiting mechanical instability of the deformable microstructures. [16][17][18][19][20] A well-utilized strategy to achieve reusability is making use of bistability of curved beams and tilted straight beams under buckling, [21][22][23] since during the working process, deformation of these constituent structures is elastic. Although the energyabsorbing capability of these metamaterials is currently not comparable to those...

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\arctan μ

Section: Resultsmentioning

confidence: 99%

Programmable Granular Metamaterials for Reusable Energy Absorption

Zhao

Jin

2019

Adv Funct Materials

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“…which is out of the range of ( 22), so λ 1 is not an attractive sliding mode but it can be repelling sliding mode at a set of points where it satisfies (23). For λ = λ 2 , from (25), by adding π…”

Section: Dynamics Inside the Switching Layermentioning

confidence: 99%

“…For a specific system parameter, the model represents an oscillating mass supported by a parallel damper and spring. By using Filippov's theory, the equilibrium stability of this self-excited SD oscillator [25], the evolution of the sliding region with Coulomb friction characteristic [26], the local and global bifurcations behaviours due to the Stribeck friction characteristic [27] were investigated. The codimension-two bifurcation of the nonlinear damped SD oscillator at the trivial equilibrium has been studied in [28].…”

Section: Introductionmentioning

confidence: 99%

Hidden Dynamics of a Self-excited SD Oscillator

Bandi,

TAMADAPU

2024

Preprint

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The present study explores the nonlinear dynamics of a self-excited smooth and discontinuous (SD) oscillator with geometric nonlinearity at the switching surface. Using a novel framework called hidden dynamics, introduced by Jeffrey, the research addresses the challenge posed by dry friction oscillators where the static friction coefficient is greater than the kinetic friction coefficient, which is ignored in Filippov’s theory. By modelling the belt friction in the SD oscillator as Coulomb friction, we investigate the consequences of the discontinuity in the friction model. The sliding regions were determined analytically and validated through numerical simulations. The system's behaviour is analyzed through the examination of bifurcation diagrams and phase portraits, and a comparison is conducted with Filippov’s theory. Some interesting bifurcation phenomena are highlighted, including a novel phenomenon involving the collision and merging of two degenerate boundaries and the bifurcation of a sliding homoclinic orbit to a saddle. Furthermore, the system's response to harmonic excitation is analyzed, wherein the oscillator displays stick-slip limit cycles, pure slip limit cycles and the emergence of chaotic solutions through periodic doubling bifurcation.

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“…4,5 Hao and Cao 6 designed a quasizero - stiffness vibration isolator based on the SD oscillator and demonstrated its local and global bifurcations using double-parameter bifurcation diagrams and the cell-mapping method. Li et al 7,8 utilized the SD oscillator model to represent a kind of system with dry friction. Yang et al 3,9,10 investigated the primary resonance phenomenon of a quasizero-stiffness SD oscillator with a time delay under the feedback control of velocity and displacement, as well as the stochastic resonance under harmonic excitation and Gaussian white noise.…”

Section: Introductionmentioning

confidence: 99%

Subharmonic resonance and fixed-range asymptotic stability of the fractional-order SD oscillator

Cui-yan,

Ming-hao,

En-li

et al. 2024

Advances in Mechanical Engineering

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The smooth and discontinuous (SD) oscillator is a typical system with strong nonlinear characteristics, and it is widely used in low-frequency vibration isolation and energy harvesting. A fractional damping model denoted by the Caputo model is introduced into the SD oscillator to adjust the property of the secondary resonance and evaluate the stability of the system. The influence of the fractional damping model on the one-third subharmonic resonance and the fixed-range asymptotic stability is studied. Residue theory and the Laplace transform are used to solve the fractional damping model. The amplitude–frequency response function and the existence conditions are derived by means of the averaging method. Lyapunov theory is used to determine the stable criteria of steady-state solutions. The cell-mapping method is ameliorated and used to calculate the fixed-range asymptotic stability of the one-third subharmonic resonance. The main results are as follows: a gap in the excitation amplitude occurs in the region of the existence condition of the one-third subharmonic resonance when the smooth parameter is smaller than 1. The generation of one-third subharmonic resonance is totally avoided for all frequencies when the excitation amplitudes are within the gap. The width of the gap, as well as the amplitude of the one-third subharmonic resonance, is affected by the parameters of the fractional damping term. The fixed-range asymptotic stability of the one-third subharmonic resonance is weak when the fractional damping parameters are large, which indicates a low resistance of the one-third subharmonic resonance to the external disturbance. The tuning effects of the fractional damping model on the one-third subharmonic resonance and fixed-range asymptotic stability are beneficial for the applications of SD oscillators.

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The equilibrium stability for a smooth and discontinuous oscillator with dry friction

Cited by 10 publications

References 26 publications

Programmable Granular Metamaterials for Reusable Energy Absorption

Programmable Granular Metamaterials for Reusable Energy Absorption

Hidden Dynamics of a Self-excited SD Oscillator

Subharmonic resonance and fixed-range asymptotic stability of the fractional-order SD oscillator

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