Nonlinear Effects in Granular Crystals with Broken Periodicity

When studying physical systems, it is common to make approximations: the contact interaction is linear, the crystal is periodic, the variations occurs slowly, the mass of a particle is constant with velocity, or the position of a particle is exactly known are just a few examples. These approximations help us simplify complex systems to make them more comprehensible while still demonstrating interesting physics. But what happens when these assumptions break down? This question becomes particularly interesting in the materials science community in designing new materials structures with exotic properties In this thesis, we study the mechanical response and dynamics in granular crystals, in which the approximation of linearity and infinite size break down. The system is inherently finite, and contact interaction can be tuned to access different nonlinear regimes. When the assumptions of linearity and perfect periodicity are no longer valid, a host of interesting physical phenomena presents itself. The advantage of using a granular crystal is in its experimental feasibility and its similarity to many other materials systems. This allows us to both leverage past experience in the condensed matter physics and materials science communities while also presenting results with implications beyond the narrower granular physics community. In addition, we bring tools from the nonlinear systems community to study the dynamics in finite lattices, where there are inherently more degrees of freedom. This approach leads to the major contributions of this thesis in broken periodic systems. We demonstrate the first defect mode whose spatial profile can be tuned from highly localized to completely delocalized by simply tuning an external parameter. Using the sensitive dynamics near bifurcation points, we present a completely new approach to modifying the incremental stiffness of a lattice to arbitrary values. We show how using nonlinear defect modes, the incremental stiffness can be tuned to anywhere in the force-displacement relation. Other contributions include demonstrating nonlinear breakdown of mechanical filters as a result of finite size, and the presents of frequency attenuation bands in essentially nonlinear materials. We finish by presenting two new energy harvesting systems based on our experience with instabilities in weakly nonlinear systems.

Range of validity of the method of averaging

Sufficient conditions are derived for the validity of approximate periodic solutions of a class of second order ordinary nonlinear differential equations. An approximate solution is defined to be valid if an exact solution exists in a neighborhood of the approximation.

Two classes of validity criteria are developed. Existence is obtained using the contraction mapping principle in one case, and the Schauder-Leray fixed point theorem in the other. Both classes of validity criteria make use of symmetry properties of periodic functions, and both classes yield an upper bound on a norm of the difference between the approximate and exact solution. This bound is used in a procedure which establishes sufficient stability conditions for the approximated solution.

Application to a system with piecewise linear restoring force (bilinear system) reveals that the approximate solution obtained by the method of averaging is valid away from regions where the response exhibits vertical tangents. A narrow instability region is obtained near one-half the natural frequency of the equivalent linear system. Sufficient conditions for the validity of resonant solutions are also derived, and two term harmonic balance approximate solutions which exhibit ultraharmonic and subharmonic resonances are studied.

Analog methods of nonlinear vibration analysis

This thesis presents methods by which electrical analogies can be obtained for nonlinear systems. The accuracy of these methods is investigated and several specific types of nonlinear equations are studied in detail.

In Part I a general method is given for obtaining electrical analogs of nonlinear systems with one degree of freedom. Loop and node methods are compared and the stability of the loop analogy is briefly considered.

Parts II and III give a description of the equipment and a discussion of its accuracy. Comparisons are made between experimental and analytic solutions of linear systems.

Part IV is concerned with systems having a nonlinear restoring force. In particular, solutions of Duffing's equation are obtained, both by using the electrical analogy and also by approximate analytical methods.

Systems with nonlinear damping are considered in Part V. Two specific examples are chosen: (1) forced oscillations and (2) self-excited oscillations (van der Pol’s equation). Comparisons are made with approximate analytic solutions.

Part VI gives experimental data for a system obeying Mathieu's equation. Regions of stability are obtained. Examples of subharmonic, ultraharmonic, and ultrasubharmonic oscillat1ons are shown.

Corpo carotídeo e hipertensão: alterações celulares e da matriz extracelular

O aumento de doenças cardiovasculares possui como fator primordial e, mais comum, o aumento da pressão arterial (PA). Esta, pode gerar complicações em outros órgãos e acarretar diversas patologias, podendo levar a morte. A hipertensão é uma síndrome multifatorial, cuja maior incidência ocorre em indivíduos obesos, sedentários e consumidores em excesso de bebidas alcoólicas e sal. O corpo carotídeo (CC) é um órgão quimiorreceptor localizado na bifurcação da artéria carótida, formado de estruturas básicas chamadas glomus. Cada glomus carotídeo é constituído de células tipo 1 envoltas por células tipo 2 ou sustentaculares. Este trabalho teve como objetivo analisar as alterações morfofuncionais que ocorrem no CC, causada por hipertensão arterial induzida pelo L-NAME, um inibidor da enzima óxido nítrico sintase. Para isso, o estudo utilizou 20 ratos Wistar divididos em dois grupos: controle (C) e L-NAME (LN). Após a administração de 40mg/kg/dia de L-NAME por 45 dias, o CC foi coletado. Observou-se aumento significativo da pressão arterial a partir da segunda semana de administração de L-NAME. Análise quantitativa mostrou uma redução no número de núcleos do glomus carotídeo e o aumento na área total do órgão no grupo LN. Não foi encontrado diferença significativa no número de núcleos totais do corpo carotídeo entre os grupos. Na análise morfológica do grupo LN, observamos a formação de vacúolos nas células tipo 1 do glomus, bem como uma redução do número total de núcleos das células de cada glomus carotídeo. A análise qualitativa sugeriu um aumento no número de fibras colágenas e fibras do sistema elástico na matriz extracelular e grânulos no grupo LN. Imunomarcações com anticorpo anti VEGF e NF-kB e nNOS mostram-se aumentadas e dispersas por todo CC no grupo LN em relação ao grupo C. Além disso, marcações para Substância-P também foram observadas em maior quantidade nas células tipo 1 do grupo LN. Quanto à marcação para PGP 9.5, houve a redução desta marcação caracterizada dentro do glomus carotídeo grupo LN comparado ao grupo C. O estudo sugere que o corpo carotídeo, em resposta à hipertensão induzida pela inibição da enzima óxido nítrico sintase, gera mudanças morfofisiológicas semelhantes as encontradas em hipóxia.

Robust Sliding Control of SEIR Epidemic Models

This paper is aimed at designing a robust vaccination strategy capable of eradicating an infectious disease from a population regardless of the potential uncertainty in the parameters defining the disease. For this purpose, a control theoretic approach based on a sliding-mode control law is used. Initially, the controller is designed assuming certain knowledge of an upper-bound of the uncertainty signal. Afterwards, this condition is removed while an adaptive sliding control system is designed. The closed-loop properties are proved mathematically in the nonadaptive and adaptive cases. Furthermore, the usual sign function appearing in the sliding-mode control is substituted by the saturation function in order to prevent chattering. In addition, the properties achieved by the closed-loop system under this variation are also stated and proved analytically. The closed-loop system is able to attain the control objective regardless of the parametric uncertainties of the model and the lack of a priori knowledge on the system.

Growth and survival of sockeye salmon (Oncorhynchus nerka) from Karluk Lake and River, Alaska, in relation to climatic and oceanic regimes and indices, 1922–2000

We examined whether the relationship between climate and salmon production was linked through the effect of climate on the growth of sockeye salmon (Oncorhynchus nerka) at sea. Smolt length and juvenile, immature, and maturing growth rates were estimated from increments on scales of adult sockeye salmon that returned to the Karluk River and Lake system on Kodiak Island, Alaska, over 77 years, 1924–2000. Survival was higher during the warm climate regimes and lower during the cool regime. Growth was not correlated with survival, as estimated from the residuals of the Ricker stock-recruitment model. Juvenile growth was correlated with an atmospheric forcing index and immature growth was correlated with the amount of coastal precipitation, but the magnitude of winter and spring coastal downwelling in the Gulf of Alaska and the Pacific Northwest atmospheric patterns that influence the directional bifurcation of the Pacific Current were not related to the growth of Karluk sockeye salmon. However, indices of sea surface temperature, coastal precipitation, and atmospheric circulation in the eastern North Pacific were correlated with the survival of Karluk sockeye salmon. Winter and spring precipitation and atmospheric circulation are possible processes linking survival to climate variation in Karluk sockeye salmon.

Stability analysis and observer design for discrete-time SEIR epidemic models

This paper applies Micken's discretization method to obtain a discrete-time SEIR epidemic model. The positivity of the model along with the existence and stability of equilibrium points is discussed for the discrete-time case. Afterwards, the design of a state observer for this discrete-time SEIR epidemic model is tackled. The analysis of the model along with the observer design is faced in an implicit way instead of obtaining first an explicit formulation of the system which is the novelty of the presented approach. Moreover, some sufficient conditions to ensure the asymptotic stability of the observer are provided in terms of a matrix inequality that can be cast in the form of a LMI. The feasibility of the matrix inequality is proved, while some simulation examples show the operation and usefulness of the observer.

On the Stability and Equilibrium Points of Multistaged SI (n) R Epidemic Models

This paper relies on the concept of next generation matrix defined ad hoc for a new proposed extended SEIR model referred to as SI(n)R-model to study its stability. The model includes n successive stages of infectious subpopulations, each one acting at the exposed subpopulation of the next infectious stage in a cascade global disposal where each infectious population acts as the exposed subpopulation of the next infectious stage. The model also has internal delays which characterize the time intervals of the coupling of the susceptible dynamics with the infectious populations of the various cascade infectious stages. Since the susceptible subpopulation is common, and then unique, to all the infectious stages, its coupled dynamic action on each of those stages is modeled with an increasing delay as the infectious stage index increases from 1 to n. The physical interpretation of the model is that the dynamics of the disease exhibits different stages in which the infectivity and the mortality rates vary as the individual numbers go through the process of recovery, each stage with a characteristic average time.

Phylogenetic study of complete cytochrome b genes in musk deer (genus Moschus) using museum samples

As an endangered animal group, musk deer (genus Moschus) are not only a great concern of wildlife conservation, but also of special interest to evolutionary studies due to long-standing arguments on the taxonomic and phylogenetic associations in this group. Using museum samples, we sequenced complete mitochondrial cytochrome b genes (1140 bp) of all suggested species of musk deer in order to reconstruct their phylogenetic history through molecular information. Our results showed that the cytochrome b gene tree is rather robust and concurred for all the algorithms employed (parsimony, maximum likelihood, and distance methods). Further, the relative rate test indicated a constant sequence substitution rate among all the species, permitting the dating of divergence events by molecular clock. According to the molecular topology, M. moschiferus branched off the earliest from a common ancestor of musk deer (about 700,000 years ago); then followed the bifurcation forming the M. berezouskii lineage and the lineage clustering M. fuscus, M. chrysogaster, and M. leucogaster (around 370,000 years before present), interestingly the most recent speciation event in musk deer happened rather recently (140,000 years ago), which might have resulted from the diversified habitats and geographic barriers in southwest China caused by gigantic movements of the Qinghai-Tibetan Plateau in history. Combining the data of current distributions, fossil records, and molecular data of this study, we suggest that the historical dispersion of musk deer might be from north to south in China. Additionally, in our further analyses involving other pecora species, musk deer was strongly supported as a monophyletic group and a valid family in Artiodactyla, closely related to Cervidae. (C) 1999 Academic Press.

Experimental investigation into parameters governing corner interactions for transonic shock wave/ boundary layer interactions

Experiments were conducted investigating the interaction between a normal shock wave and a corner boundary layer in a constant area rectangular duct. Active corner suction and passive blowing were applied to manipulate the natural corner flows developing in the working section of the Cambridge University supersonic wind tunnel. In addition robust vane micro-vortex generators were applied to the corners of the working section. Experiments were conducted at Mach numbers of M∞=1.4 and 1.5. Flow visualisation was carried out through schlieren and surface oil flow, while static pressures were recorded via floor tappings. The results indicate that an interplay occurs between the corner flow and the centre line flow. It is believed that corner flow separation acts to induce a shock bifurcation, which in turn leads to a smearing of the adverse pressure gradient elsewhere. In addition the blockage effect from the corners was seen to result in a reacceleration of the subsonic post-shock flow. As a result manipulation of the corner regions allows a separated or attached centre line flow to be observed at the same Mach number. Copyright © 2010 by Babinsky, Burton, Bruce.

Triggering in a thermoacoustic system with stochastic noise

This paper explores the mechanism of triggering in a simple thermoacoustic system, the Rijke tube. It is demonstrated that additive stochastic perturbations can cause triggering before the linear stability limit of a thermoacoustic system. When triggering from low noise amplitudes, the system is seen to evolve to self-sustained oscillations via an unstable periodic solution of the governing equations. Practical stability is introduced as a measure of the stability of a linearly stable state when finite perturbations are present. The concept of a stochastic stability map is used to demonstrate the change in practical stability limits for a system with a subcritical bifurcation, once stochastic terms are included. The practical stability limits are found to be strongly dependent on the strength of noise.