Nonlinear Spectral Properties of Elastic Waves Propagating Along a Pantographic Metamaterial With Local Inertia Amplifiers

2021 ◽  
Author(s):  
Valeria Settimi ◽  
Marco Lepidi ◽  
Andrea Bacigalupo
Keyword(s):  
Elastic Waves ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Nonlinear Effects ◽  
Harmonic Waves ◽  
Nonlinear Dispersion ◽  
Asymptotic Approach ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Atomic Cell

Abstract Pantographic mechanisms can be introduced in the cellular periodic microstructure of architected metamaterials to achieve functional effects of local inertia amplification. The paper presents a one-dimensional pantographic metamaterial, characterized by an inertially amplified tetra-atomic cell. An internally constrained two-degrees-of-freedom model is formulated to describe the undamped free propagation of harmonic waves in the weakly nonlinear regime. A general asymptotic approach is employed to analytically determine the linear and nonlinear dispersion properties. Analytical, although asymptotically approximate, functions are obtained for the nonlinear wavefrequencies and waveforms, which show significant nonlinear effects including softening/hardening bending of the backbone curves and synclastic/anticlastic curvatures of the invariant manifolds.

Asymptotic approach to combustion instability

2005 ◽  
Vol 363 (1830) ◽  
pp. 1247-1259 ◽  
Author(s):  
Xuesong Wu
Keyword(s):  
Combustion Instability ◽  
Parametric Instability ◽  
Acoustic Mode ◽  
Sound Waves ◽  
Asymptotic Approach ◽  
Reaction Heat ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Entire Flow ◽  
Asymptotic Formulation

This paper presents an asymptotic approach to combustion instability in premixed flames under the assumptions of large activation energy and small Mach number. The entire flow consists of four distinct yet fully interactive sub-regions, which accommodate the chemical reaction, heat transport, hydrodynamics and acoustics, respectively. A reduced system was derived to describe the intricate coupling between the flame and acoustics that underlies the combustion instability. The asymptotically reduced system was employed to study the weakly nonlinear interaction between the Darrieus–Landau instability and the longitudinal acoustic mode of the combustion chamber. The general asymptotic formulation includes the influence of enthalpy fluctuation in the oncoming mixture. It is shown that one-dimensional enthalpy fluctuation, through its interaction with flame, produces sound waves, and may cause parametric instability of the flame. The mutual coupling between the sound wave and parametric instability is analysed at the instability thresholds.

Ship Motions From Unsteady Maneuvering in Two-Dimensional Waves: Part II—Analysis of Data

2011 ◽  
Author(s):  
Joseph T. Klamo ◽  
Ray-Qing Lin
Keyword(s):  
Wave Field ◽  
Degrees Of Freedom ◽  
Nonlinear Effects ◽  
Ship Motions ◽  
Two Dimensional ◽  
Linear Superposition ◽  
One Dimensional ◽  
Model Response ◽  
Experimental Laboratory ◽  
Dimensional Wave

An experimental test has been conducted to measure the six degrees-of-freedom motions of a remote-controlled model attempting to hold heading while at forward speed in a two-dimensional wave field. During testing, the two underlying components of the wave field were always orthogonal to each other but various relative headings of the model to the dominant wave were explored. Of particular interest is understanding the nonlinear effects of the two distinct underlying wave encounter frequencies on the model response and the severity to which it causes the response in the two-dimensional wave field to differ from the linear summation of responses from equivalent one-dimensional waves. Since the experimental data contains the full wave-wave and wave-ship interactions of the two-dimensional wave field, we will use numerical results from the Digital, Self-consistent Ship Experimental Laboratory (DiSSEL) to generate the necessary one-dimensional wave results. This allows us to compare the predicted ship response motions from linear superposition of two one-dimensional wave field responses to the measured motions in a two-dimensional wave field for various relative wave heading combinations. It will be shown that for waves forward of beam, the predicted pitch results from superposition are fairly accurate while the roll prediction is not. However, for waves aft of beam, the motion predictions from linear superposition of pitch and roll are both poor. In such aft of beam cases, the disagreement can be quite large due to deviations in the ship heading caused by drift forces.

Nonlinear Six-Degree-of-Freedom Dynamic Model for Risers, Pipelines, and Beams

1986 ◽  
Vol 30 (03) ◽  
pp. 177-185
Author(s):  
Michael M. Bernitsas ◽  
John E. Kokarakis
Keyword(s):  
Cross Sections ◽  
Degrees Of Freedom ◽  
Nonlinear Effects ◽  
Nonlinear Formulation ◽  
Marine Risers ◽  
Internal Fluid ◽  
Rotational Degrees Of Freedom ◽  
Six Degree Of Freedom ◽  
Structural Imperfections ◽  
Inertia Forces

A nonlinear model for the dynamic behavior of tubular beams such as marine risers, pipelines, legs of tension leg platforms, and drill strings is developed. The formulation includes three translational degrees of freedom of the riser cross section and three rotational degrees of freedom for shear and torsion. Nonlinear constitutive equations for cross sections of unequal principal stiffnesses and extensible material are derived. Initial structural imperfections which are inherent in long risers are modeled in the form of initial curvature and geometric torsion which do not induce strains. The inertia forces due to the motion of the riser and internal fluid motions are formulated. The external hydrodynamic and hydrostatic forces are integrated on the riser surface as pressure and traction forces. The model is a comprehensive consistent nonlinear formulation of the riser dynamics and can be used for evaluation of the significance of nonlinear effects.

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

scholarly journals Precessional instability of a fluid cylinder

2011 ◽  
Vol 666 ◽  
pp. 104-145 ◽  
Author(s):  
ROMAIN LAGRANGE ◽  
PATRICE MEUNIER ◽  
FRANÇOIS NADAL ◽  
CHRISTOPHE ELOY
Keyword(s):  
Reynolds Number ◽  
Nonlinear Effects ◽  
Intermittent Flow ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Aspect Ratios ◽  
Image Velocimetry ◽  
Precessional Motion ◽  
Weakly Nonlinear Model

In this paper, the instability of a fluid inside a precessing cylinder is addressed theoretically and experimentally. The precessional motion forces Kelvin modes in the cylinder, which can become resonant for given precessional frequencies and cylinder aspect ratios. When the Reynolds number is large enough, these forced resonant Kelvin modes eventually become unstable. A linear stability analysis based on a triadic resonance between a forced Kelvin mode and two additional free Kelvin modes is carried out. This analysis allows us to predict the spatial structure of the instability and its threshold. These predictions are compared to the vorticity field measured by particle image velocimetry with an excellent agreement. When the Reynolds number is further increased, nonlinear effects appear. A weakly nonlinear theory is developed semi-empirically by introducing a geostrophic mode, which is triggered by the nonlinear interaction of a free Kelvin mode with itself in the presence of viscosity. Amplitude equations are obtained coupling the forced Kelvin mode, the two free Kelvin modes and the geostrophic mode. They show that the instability saturates to a fixed point just above threshold. Increasing the Reynolds number leads to a transition from a steady saturated regime to an intermittent flow in good agreement with experiments. Surprisingly, this weakly nonlinear model still gives a correct estimate of the mean flow inside the cylinder even far from the threshold when the flow is turbulent.

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