scholarly journals Multiple scales analysis of slow–fast quasi-linear systems

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180630 ◽  
Author(s):  
G. Michel ◽  
G. P. Chini
Keyword(s):  
Linear Systems ◽  
Multiple Scales ◽  
Slow Variable ◽  
Angular Frequency ◽  
Model Systems ◽  
Marginal Stability ◽  
Strongly Coupled ◽  
First Case ◽  
Multiple Scales Analysis ◽  
Fast Waves

This article illustrates the application of multiple scales analysis to two archetypal quasi-linear systems; i.e. to systems involving fast dynamical modes, called fluctuations, that are not directly influenced by fluctuation–fluctuation nonlinearities but nevertheless are strongly coupled to a slow variable whose evolution may be fully nonlinear. In the first case, fast waves drive a slow, spatially inhomogeneous evolution of their celerity field. Multiple scales analysis confirms that, although the energyE, the angular frequencyωand the modal structure of the waves evolve, the wave actionE/ωis conserved in the absence of forcing and dissipation. In the second system, the fast modes undergo an instability that is saturated through a feedback on the slow variable. A new multi-scale analysis is developed to treat this case. The key technical point, confirmed by the analysis, is that the fluctuation energy and mode structure evolve slowly to ensure that the slow field remains in a state of near marginal stability. These two model systems appear to be generic, being representative of many if not all quasi-linear systems. In each case, numerical simulations of both the full and reduced dynamical systems are performed to highlight the accuracy and efficiency of the multiple scales approach. Python codes are provided as electronic supplementary material.

The Small Scale Effect on Linear Vibration of Beam-Like Nanostructures

2012 ◽  
Vol 446-449 ◽  
pp. 3432-3435
Author(s):  
Cheng Li ◽  
Lin Quan Yao
Keyword(s):  
Axial Load ◽  
Vibration Mode ◽  
Multiple Scales ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Small Scale ◽  
Transverse Displacement ◽  
Linear Vibration ◽  
Small Scale Effect ◽  
Multiple Scales Analysis

Transverse free dynamics of a beam-like nanostructure with axial load is investigated. The effects of a small size at nano-scale unavailable in classical mechanics are presented. Explicit solutions for natural frequency, vibration mode and transverse displacement are obtained by separation of variables and multiple scales analysis. Results by two methods are in close agreement.

scholarly journals Chatter Control in Turning Process with a Nonlinear Energy Sink

2013 ◽  
Vol 698 ◽  
pp. 89-98 ◽  
Author(s):  
Etienne Gourc ◽  
Sébastien Seguy ◽  
Guilhem Michon ◽  
Alain Berlioz
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Energy Sink ◽  
Turning Process ◽  
Strongly Coupled ◽  
Different Types ◽  
Chatter Control ◽  
Energy Sink ◽  
Nonlinear Energy ◽  
Cubic Stiffness

This paper presents the interest of an original absorber of vibration in order to reduce chatter vibration in turning process. The device is composed of a linear oscillator corresponding to a flexible cutting tool subject to chatter strongly coupled to a Nonlinear Energy Sink (NES), with purely cubic stiffness. The novelty of this work is the use of a nonlinear cutting law, more accurate for modeling the cutting process. The delayed equations of motion are analyzed using a combination of the method of multiple scales and harmonic balance. Different types of responses regimes are revealed such as periodic response and also Strongly Modulated Response (SMR). Analytic results are then compared with numerical simulations. Finally, the potential of the NES is demonstrated to control chatter in turning process.

Hydrodesulfurization (HDS) Model Systems. Opening, Hydrogenation, and Hydrodesulfurization of Dibenzothiophene (DBT) at Iridium. First Case of Catalytic HDS of DBT in Homogeneous Phase

1995 ◽  
Vol 14 (5) ◽  
pp. 2342-2352 ◽  
Author(s):  
Claudio Bianchini ◽  
M. Victoria Jimenez ◽  
Andrea Meli ◽  
Simonetta Moneti ◽  
Francesco Vizza ◽  
...  
Keyword(s):  
Model Systems ◽  
Homogeneous Phase ◽  
First Case

scholarly journals Theoretical and Experimental Study of an Harmonically Forced Vibro-Impact Nonlinear Energy Sink

2013 ◽  
Author(s):  
Etienne Gourc ◽  
Guilhem Michon ◽  
Sébastien Seguy ◽  
Alain Berlioz
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Energy Sink ◽  
Small Mass ◽  
Mass Ratio ◽  
Strongly Coupled ◽  
Electrodynamic Shaker ◽  
Energy Sink ◽  
Nonlinear Energy ◽  
Good Agreement

Recently, it has been demonstrated that a Vibro-Impact type Nonlinear Energy Sink (VI-NES) can be used efficiently to mitigate vibration of a Linear Oscillator (LO) under transient loading. In this paper, the dynamic response of an harmonically forced LO, strongly coupled to a VI-NES is investigated theoretically and experimentally. Due to the small mass ratio between the LO and the flying mass of the NES, the obtained equation of motion are analyzed using the method of multiple scales in the case of 1 : 1 resonance. It is shown that in addition to periodic response, system with VI-NES can exhibit Strongly Modulated Response (SMR). Experimentally, the whole system is embedded on an electrodynamic shaker. The VI-NES is realized with a ball which is free to move in a cavity with a predesigned gap. The mass of the ball is less than 1% of the mass of the LO. The experiment confirms the existence of periodic and SMR response regimes. A good agreement between theoretical and experimental results is observed.

Nonlinear Stability of Thermoelastic Sliding Contact

2005 ◽  
Vol 74 (3) ◽  
pp. 595-598
Author(s):  
Jason D. Miller ◽  
D. Dane Quinn
Keyword(s):  
Multiple Scales ◽  
Dynamical Behavior ◽  
Stable State ◽  
Nonlinear Behavior ◽  
Sliding Contact ◽  
Stability Criteria ◽  
Multiple Scales Analysis ◽  
Two Equilibria

A model for sliding contact of a thermoelastic rod is considered and is subjected to a multiple scales analysis to uncover its nonlinear behavior near a neutrally stable state. The analysis reveals a combination of the contact resistance and frictional intensity that describes the generic unfolding of this critical state and its associated bifurcations. In particular, the system can describe how two equilibria coalesce in a saddle-node bifurcation and generalizes stability criteria that have been presented previously in the literature for this model. Moreover, this analysis describes the role of the initial deformation of the rod on its long-term dynamical behavior.

scholarly journals Input retrieval in finite dimensional linear systems

1982 ◽  
Vol 23 (4) ◽  
pp. 357-382 ◽  
Author(s):  
P. G. Howlett
Keyword(s):  
Linear Systems ◽  
Type System ◽  
System Matrix ◽  
Fundamental Idea ◽  
Finite Dimensional ◽  
First Case ◽  
Direct Linkage ◽  
Explicit Formulae ◽  
Output Only ◽  
General Method

AbstractFor finite dimensional linear systems it is known that in certain circumstances the input can be retrieved from a knowledge of the output only. The main aim of this paper is to produce explicit formulae for input retrieval in systems which do not possess direct linkage from input to output. Although two different procedures are suggested the fundamental idea in both cases is to find an expression for the inverse transfer function of the system. In the first case this is achieved using a general method of power series inversion and in the second case by a sequence of elementary operations on a Rosenbrock type system matrix.

A Multiple-Scales Analysis of Nonlinear, Localized Modes in a Cyclic Periodic System

1993 ◽  
Vol 60 (2) ◽  
pp. 388-397 ◽  
Author(s):  
A. Vakakis ◽  
T. Nayfeh ◽  
M. King
Keyword(s):  
Vibration Isolation ◽  
Multiple Scales ◽  
Algebraic Equations ◽  
Localized Modes ◽  
Mode Localization ◽  
Nonlinear Mode ◽  
Weakly Coupled ◽  
Passive Vibration Isolation ◽  
Coupling Stiffness ◽  
Multiple Scales Analysis

In this work the nonlinear localized modes of an n-degree-of-freedom (DOF) nonlinear cyclic system are examined by the averaging method of multiple scales. The set of nonlinear algebraic equations describing the localized modes is derived and is subsequently solved for systems with various numbers of DOF. It is shown that nonlinear localized modes exist only for small values of the ratio (k/μ), where k is the linear coupling stiffness and μ is the coefficient of the grounding stiffness nonlinearity. As (k/μ) increases the branches of localized modes become nonlocalized and either bifurcate from “extended” antisymmetric modes in inverse, “multiple” Hamiltonian pitchfork bifurcations (for systems with even-DOF), or reach certain limiting values for large values of(k/μ) (for systems with odd-DOF). Motion confinement due to nonlinear mode localization is demonstrated by examining the responses of weakly coupled, perfectly periodic cyclic systems caused by external impulses. Finally, the implications of nonlinear mode localization on the active or passive vibration isolation of such structures are discussed.

Parametric Vibrations in Discs: Point-Wise and Distributed Loads, Including Rotating Friction

1995 ◽  
Author(s):  
H. Ouyang ◽  
S. N. Chan ◽  
J. E. Mottershead ◽  
M. I. Friswell ◽  
M. P. Cartmell
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Transverse Load ◽  
Transverse Mass ◽  
Follower Load ◽  
Parametric Resonances ◽  
Spring System ◽  
Multiple Scales Analysis ◽  
Mass Spring ◽  
Rotating Load

Abstract This paper is concerned with the parametric resonances in a stationary annular disc when excited by a rotating load system. Two forms of the load system are considered. In the first, the load consists of a discrete transverse mass-spring-damper system and a frictional follower load. Secondly, a distributed mass-spring system (without friction) is studied. In both cases the transverse load is rotated at a uniform speed around the disc. Equations of motion are developed for the two cases, and the results of a multiple scales analysis are presented. The disc is found to exhibit many parametric resonances at subcritical speeds when friction is present.

scholarly journals Multiple scales analysis of water hammer attenuation

2011 ◽  
Vol 69 (4) ◽  
pp. 677-690 ◽  
Author(s):  
S. Y. Han ◽  
D. Hansen ◽  
G. Kember
Keyword(s):  
Multiple Scales ◽  
Water Hammer ◽  
Multiple Scales Analysis
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