Using the Mass Ratio to Induce Band Gaps in a 1D Array of Nonlinearly Coupled Oscillators

2012 ◽  
Author(s):  
Brian P. Bernard ◽  
Jeffrey W. Peyser ◽  
Brian P. Mann ◽  
David P. Arnold
Keyword(s):  
Coupled Oscillators ◽  
Stable Equilibrium ◽  
Equations Of Motion ◽  
Band Gaps ◽  
Dimensional System ◽  
Mass Ratio ◽  
Frequency Range ◽  
One Dimensional ◽  
Propagation Zone ◽  
The Masses

A one dimensional system of nonlinearly coupled magnetic oscillators has been studied. After deriving the equations of motion for each oscillator, the system is linearized about a stable equilibrium and studied using an assumed solution form for a traveling wave. Wave propagation and attenuation regions are predicted by reducing the system of equations to a standard eigenvalue problem. Through evaluating these equations across the entire irreducible Brillouin zone, it is determined that when the masses of each oscillator are identical, the entire frequency range of the system is a propagation zone. By varying the masses comprising a unit cell, band gaps are observed. It is shown that the mass ratio can be used to guide both the size and location of these band gaps. Numerical simulations are performed to support our analytical findings.

GEOMETRICALLY INDUCED NONLINEAR DYNAMICS IN ONE-DIMENSIONAL LATTICES

2008 ◽  
Vol 18 (08) ◽  
pp. 2471-2476
Author(s):  
M. HAMILTON ◽  
O. F. DE ALCANTARA BONFIM
Keyword(s):  
Coupled Oscillators ◽  
Domain Walls ◽  
Equations Of Motion ◽  
Geometric Constraints ◽  
One Dimensional ◽  
Long Wavelength ◽  
Simple Lattice ◽  
A Chain ◽  
The Stability ◽  
The Masses

We present a simple lattice model consisting of a chain of coupled oscillators, where their masses are interconnected by linear springs and allowed to move along a common axis, as in a monorail. In the transverse direction each mass is also attached to two other springs, one on each side of the mass. The ends of these springs are kept at fixed positions. The nonlinearity in the model arises from the geometric constraints imposed on the motion of the masses, as well as from the configuration of the springs, where in the transverse directions the springs are either in the extended or compressed state depending on the position of the mass. Under these conditions we show that solitary waves (domain walls) are present in the system. In the long wavelength limit an analytical solution for these nonlinear waves is found. Numerical integrations of the equations of motion in the full discrete system are also performed to analyze the stability of the domain wall solution. Nonlinear supratransmission is also shown to exist in the model and a discussion of mechanism is presented.

Effect of Double Defects on Transmission Spectra of One-Dimensional Photonic Crystals

2010 ◽  
Vol 663-665 ◽  
pp. 725-728 ◽  
Author(s):  
Yuan Ming Huang ◽  
Qing Lan Ma ◽  
Bao Gai Zhai ◽  
Yun Gao Cai
Keyword(s):  
Photonic Crystals ◽  
Band Gap ◽  
Theoretical Calculations ◽  
Stop Band ◽  
Band Gaps ◽  
Characteristic Matrix ◽  
Frequency Range ◽  
Defect Band ◽  
One Dimensional ◽  
The One

Considered the model of the one-dimensional photonic crystals (1-D PCs) with double defects, the refractive indexes (n2’, n3’ and n2’’, n3’’) of the double defects were 2.0, 4.0 and 4.0, 2.0 respectively. With parameter n2=1.5, n3=2.5, by theoretical calculations with characteristic matrix method, the results shown that for a certain number (14 was taken) of layers of the 1-D PCs, when the double defects abutted, there was a defect band gap in the stop band gap, while when the double defects separated, there occurred two defect band gaps in the stop band gap; besides, with the separation of the two defects, the transmittance of the double defect band gaps decreased gradually. In addition, in this progress, the frequency range of the stop band gap has a little increase from 0.092 to 0.095.

Band gaps in the terahertz frequency range in quasiperiodic one-dimensional magnonic crystals

2010 ◽  
Vol 150 (47-48) ◽  
pp. 2325-2328 ◽  
Author(s):  
C.H.O. Costa ◽  
P.H.R. Barbosa ◽  
F.F. Barbosa Filho ◽  
M.S. Vasconcelos ◽  
E.L. Albuquerque
Keyword(s):  
Band Gaps ◽  
Terahertz Frequency ◽  
Frequency Range ◽  
Terahertz Frequency Range ◽  
One Dimensional ◽  
Magnonic Crystals

scholarly journals Analytical Calculation of Power Flow in Three Coupled Oscillators and its Application to One Dimensional Crystal

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Chifu Ebene Ndikilar ◽  
Sofwan I. Saleh ◽  
Hafeez Yusuf Hafeez ◽  
Lawan Sani Taura
Keyword(s):  
White Noise ◽  
Coupled Oscillators ◽  
Power Flow ◽  
Equations Of Motion ◽  
Analytical Calculation ◽  
Asymptotic Value ◽  
One Dimensional ◽  
Two Systems ◽  
Dimensional Crystal ◽  
Equations Of Motions

The motion of a system consisting of three coupled oscillators of three masses attached together by four springs is studied analytically. The system is used as a model to describe the interactions between atoms in a one dimensional crystal with spring-like forces under white noise excitations.  Two different cases are considered and the frequencies of oscillations are obtained as well as the equations of motion.  The equations of motions are used to determine the power flow in the systems. The power flow determined is used to describe the effects of substitution impurities in a crystal. The power flow of the two systems studied decreases exponentially with increase in frequency to an asymptotic value.

Negative Effective Mass Density of One-Dimensional Hierarchical Metacomposite

2015 ◽  
Vol 82 (3) ◽  
Author(s):  
Xiyue An ◽  
Fangfang Sun ◽  
Peishi Yu ◽  
Hualin Fan ◽  
Shiping He ◽  
...  
Keyword(s):  
Effective Mass ◽  
Mass Density ◽  
Rigid Bodies ◽  
Band Gaps ◽  
Mass Ratio ◽  
One Dimensional ◽  
Negative Effective Mass ◽  
Specific Frequency ◽  
The Relationship ◽  
Internal Resonators

A theoretical model of one-dimensional (1D) hierarchical metacomposite with internal resonators was proposed to generate negative effective mass over specific frequency ranges. Different from the single-resonator microstructure, the current hierarchical metamaterial with multilevel resonators was constructed by a series of springs and rigid bodies. The general formula of the current hierarchical metamaterial model was induced to reveal the relationship between the effective mass and the forcing frequency. It is found that the hierarchical metamaterial with multilevel resonators generates multifrequency band gaps with negative effective masses. The number of the band gaps equals to the order of the hierarchy. The total bandwidth for the negative effective mass increases with the hierarchy, meanwhile increasing the mass ratio can also obviously increase the bandwidth generating negative effective mass.

scholarly journals A group theoretic approach to generalized harmonic vibrations in a one dimensional lattice

1986 ◽  
Vol 9 (1) ◽  
pp. 131-136 ◽  
Author(s):  
J. N. Boyd ◽  
P. N. Raychowdhury
Keyword(s):  
Force Constant ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Theoretic Approach ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Cosine Series ◽  
Harmonic Vibrations ◽  
Wave Forms ◽  
The Masses

Beginning with a group theoretical simplification of the equations of motion for harmonically coupled point masses moving on a fixed circle, we obtain the natural frequencies of motion for the array. By taking the number of vibrating point masses to be very large, we obtain the natural frequencies of vibration for any arbitrary, but symmetric, harmonic coupling of the masses in a one dimensional lattice. The result is a cosine series for the square of the frequency,fj2=1π2∑ℓ=0sa(ℓ)cosℓβwhere0<β=2πjN≤2π,j∈{1,2,3,…,N}anda(ℓ)depends upon the attractive force constant between thej-th and(j+ℓ)-th masses. Lastly, we show that these frequencies will be propagated by wave forms in the lattice.

scholarly journals The Dynamics of Digits: Calculating Pi with Galperin’s Billiards

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 509
Author(s):  
Xabier M. Aretxabaleta ◽  
Marina Gonchenko ◽  
Nathan L. Harshman ◽  
Steven Glenn Jackson ◽  
Maxim Olshanii ◽  
...  
Keyword(s):  
Irrational Number ◽  
Type Model ◽  
Hard Wall ◽  
Body System ◽  
Mass Ratio ◽  
Specific Mass ◽  
One Dimensional ◽  
Number Π ◽  
The Masses ◽  
Three Body

In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number π . This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of π in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be π itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls’ positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of π with Galperin billiards, including curious cases with irrational number bases.

scholarly journals Violent Relaxation and Mixing in 1-D Gravitational Systems

1987 ◽  
Vol 127 ◽  
pp. 523-524
Author(s):  
Marc Luwel
Keyword(s):  
Surface Density ◽  
Gravitational Force ◽  
Equations Of Motion ◽  
Dimensional System ◽  
Double Precision ◽  
One Dimensional ◽  
Gravitational Systems ◽  
Gravitational Model ◽  
The One

The one dimensional gravitational model consists of N mass sheets with surface density mi, parallel to the (y, z)–plane and constrained to move along the x-axis under influence of their mutual gravitational force Fij = −2πGmimj sgn(xi – xj). in order to study the evolution of this one–dimensional system, the N Newtonian equations of motion are integrated numerically, using an “exact” double precision algorithm.

Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

1998 ◽  
Vol 63 (6) ◽  
pp. 761-769 ◽  
Author(s):  
Roland Krämer ◽  
Arno F. Münster
Keyword(s):  
Reaction Diffusion ◽  
Dominant Mode ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Perturbation ◽  
Dimensional System ◽  
One Dimensional ◽  
Brusselator Model ◽  
The One ◽  
Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

scholarly journals Topological phase transition between a normal insulator and a topological metal state in a quasi-one-dimensional system

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Milad Jangjan ◽  
Mir Vahid Hosseini
Keyword(s):  
Phase Transition ◽  
Dimensional System ◽  
Inversion Symmetry ◽  
Topological Phase ◽  
Edge States ◽  
Zero Energy ◽  
One Dimensional ◽  
Topological Phase Transition ◽  
Metal State ◽  
Topological Metal

AbstractWe theoretically report the finding of a new kind of topological phase transition between a normal insulator and a topological metal state where the closing-reopening of bandgap is accompanied by passing the Fermi level through an additional band. The resulting nontrivial topological metal phase is characterized by stable zero-energy localized edge states that exist within the full gapless bulk states. Such states living on a quasi-one-dimensional system with three sublattices per unit cell are protected by hidden inversion symmetry. While other required symmetries such as chiral, particle-hole, or full inversion symmetry are absent in the system.

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