Response of linear, viscous damped systems to excitations having time-varying frequency

The response of linear, viscous damped systems to excitations having time-varying frequency is the subject of exact and approximate analyses, which are supplemented by an analog computer study of single degree of freedom system response to excitations having frequencies depending linearly and exponentially on time.

The technique of small perturbations and the methods of stationary phase and saddle-point integration, as well as a novel bounding procedure, are utilized to derive approximate expressions characterizing the system response envelope—particularly near resonances—for the general time-varying excitation frequency.

Descriptive measurements of system resonant behavior recorded during the course of the analog study—maximum response, excitation frequency at which maximum response occurs, and the width of the response peak at the half-power level—are investigated to determine dependence upon natural frequency, damping, and the functional form of the excitation frequency.

The laboratory problem of determining the properties of a physical system from records of its response to excitations of this class is considered, and the transient phenomenon known as “ringing” is treated briefly.

It is shown that system resonant behavior, as portrayed by the above measurements and expressions, is relatively insensitive to the specifics of the excitation frequency-time relation and may be described to good order in terms of parameters combining system properties with the time derivative of excitation frequency evaluated at resonance.

One of these parameters is shown useful for predicting whether or not a given excitation having a time-varying frequency will produce strong or subtle changes in the response envelope of a given system relative to the steady-state response envelope. The parameter is shown, additionally, to be useful for predicting whether or not a particular response record will exhibit the “ringing” phenomenon.

Studies of response to earthquake ground motion

A study is made of the accuracy of electronic digital computer calculations of ground displacement and response spectra from strong-motion earthquake accelerograms. This involves an investigation of methods of the preparatory reduction of accelerograms into a form useful for the digital computation and of the accuracy of subsequent digital calculations. Various checks are made for both the ground displacement and response spectra results, and it is concluded that the main errors are those involved in digitizing the original record. Differences resulting from various investigators digitizing the same experimental record may become as large as 100% of the maximum computed ground displacements. The spread of the results of ground displacement calculations is greater than that of the response spectra calculations. Standardized methods of adjustment and calculation are recommended, to minimize such errors.

Studies are made of the spread of response spectral values about their mean. The distribution is investigated experimentally by Monte Carlo techniques using an electric analog system with white noise excitation, and histograms are presented indicating the dependence of the distribution on the damping and period of the structure. Approximate distributions are obtained analytically by confirming and extending existing results with accurate digital computer calculations. A comparison of the experimental and analytical approaches indicates good agreement for low damping values where the approximations are valid. A family of distribution curves to be used in conjunction with existing average spectra is presented. The combination of analog and digital computations used with Monte Carlo techniques is a promising approach to the statistical problems of earthquake engineering.

Methods of analysis of very small earthquake ground motion records obtained simultaneously at different sites are discussed. The advantages of Fourier spectrum analysis for certain types of studies and methods of calculation of Fourier spectra are presented. The digitizing and analysis of several earthquake records is described and checks are made of the dependence of results on digitizing procedure, earthquake duration and integration step length. Possible dangers of a direct ratio comparison of Fourier spectra curves are pointed out and the necessity for some type of smoothing procedure before comparison is established. A standard method of analysis for the study of comparative ground motion at different sites is recommended.

SteelConverter and Caltech VirtualShaker: Rapid Nonlinear Cloud-Based Structural Model Conversion and Analysis

STEEL, the Caltech created nonlinear large displacement analysis software, is currently used by a large number of researchers at Caltech. However, due to its complexity, lack of visualization tools (such as pre- and post-processing capabilities) rapid creation and analysis of models using this software was difficult. SteelConverter was created as a means to facilitate model creation through the use of the industry standard finite element solver ETABS. This software allows users to create models in ETABS and intelligently convert model information such as geometry, loading, releases, fixity, etc., into a format that STEEL understands. Models that would take several days to create and verify now take several hours or less. The productivity of the researcher as well as the level of confidence in the model being analyzed is greatly increased.

It has always been a major goal of Caltech to spread the knowledge created here to other universities. However, due to the complexity of STEEL it was difficult for researchers or engineers from other universities to conduct analyses. While SteelConverter did help researchers at Caltech improve their research, sending SteelConverter and its documentation to other universities was less than ideal. Issues of version control, individual computer requirements, and the difficulty of releasing updates made a more centralized solution preferred. This is where the idea for Caltech VirtualShaker was born. Through the creation of a centralized website where users could log in, submit, analyze, and process models in the cloud, all of the major concerns associated with the utilization of SteelConverter were eliminated. Caltech VirtualShaker allows users to create profiles where defaults associated with their most commonly run models are saved, and allows them to submit multiple jobs to an online virtual server to be analyzed and post-processed. The creation of this website not only allowed for more rapid distribution of this tool, but also created a means for engineers and researchers with no access to powerful computer clusters to run computationally intensive analyses without the excessive cost of building and maintaining a computer cluster.

In order to increase confidence in the use of STEEL as an analysis system, as well as verify the conversion tools, a series of comparisons were done between STEEL and ETABS. Six models of increasing complexity, ranging from a cantilever column to a twenty-story moment frame, were analyzed to determine the ability of STEEL to accurately calculate basic model properties such as elastic stiffness and damping through a free vibration analysis as well as more complex structural properties such as overall structural capacity through a pushover analysis. These analyses showed a very strong agreement between the two softwares on every aspect of each analysis. However, these analyses also showed the ability of the STEEL analysis algorithm to converge at significantly larger drifts than ETABS when using the more computationally expensive and structurally realistic fiber hinges. Following the ETABS analysis, it was decided to repeat the comparisons in a software more capable of conducting highly nonlinear analysis, called Perform. These analyses again showed a very strong agreement between the two softwares in every aspect of each analysis through instability. However, due to some limitations in Perform, free vibration analyses for the three story one bay chevron brace frame, two bay chevron brace frame, and twenty story moment frame could not be conducted. With the current trend towards ultimate capacity analysis, the ability to use fiber based models allows engineers to gain a better understanding of a building’s behavior under these extreme load scenarios.

Following this, a final study was done on Hall’s U20 structure [1] where the structure was analyzed in all three softwares and their results compared. The pushover curves from each software were compared and the differences caused by variations in software implementation explained. From this, conclusions can be drawn on the effectiveness of each analysis tool when attempting to analyze structures through the point of geometric instability. The analyses show that while ETABS was capable of accurately determining the elastic stiffness of the model, following the onset of inelastic behavior the analysis tool failed to converge. However, for the small number of time steps the ETABS analysis was converging, its results exactly matched those of STEEL, leading to the conclusion that ETABS is not an appropriate analysis package for analyzing a structure through the point of collapse when using fiber elements throughout the model. The analyses also showed that while Perform was capable of calculating the response of the structure accurately, restrictions in the material model resulted in a pushover curve that did not match that of STEEL exactly, particularly post collapse. However, such problems could be alleviated by choosing a more simplistic material model.

Stationary Eddies and Zonal Variations of the Global Hydrological Cycle in a Changing Climate

This thesis advances our physical understanding of the sensitivity of the hydrological cycle to global warming. Specifically, it focuses on changes in the longitudinal (zonal) variation of precipitation minus evaporation (P - E), which is predominantly controlled by planetary-scale stationary eddies. By studying idealized general circulation model (GCM) experiments with zonally varying boundary conditions, this thesis examines the mechanisms controlling the strength of stationary-eddy circulations and their role in the hydrological cycle. The overarching goal of this research is to understand the cause of changes in regional P - E with global warming. An understanding of such changes can be useful for impact studies focusing on water availability, ecosystem management, and flood risk.

Based on a moisture-budget analysis of ERA-Interim data, we establish an approximation for zonally anomalous P - E in terms of surface moisture content and stationary-eddy vertical motion in the lower troposphere. Part of the success of this approximation comes from our finding that transient-eddy moisture fluxes partially cancel the effect of stationary-eddy moisture advection, allowing divergent circulations to dominate the moisture budget. The lower-tropospheric vertical motion is related to horizontal motion in stationary eddies by Sverdrup and Ekman balance. These moisture- and vorticity-budget balances also hold in idealized and comprehensive GCM simulations across a range of climates.

By examining climate changes in the idealized and comprehensive GCM simulations, we are able to show the utility of the vertical motion P - E approximation for splitting changes in zonally anomalous P - E into thermodynamic and dynamic components. Shifts in divergent stationary-eddy circulations dominate changes in zonally anomalous P - E. This limits the local utility of the "wet gets wetter, dry gets drier” idea, where existing P - E patterns are amplified with warming by the increase in atmospheric moisture content, with atmospheric circulations held fixed. The increase in atmospheric moisture content manifests instead in an increase in the amplitude of the zonally anomalous hydrological cycle as measured by the zonal variance of P - E. However, dynamic changes, particularly the slowdown of divergent stationary-eddy circulations, limit the strengthening of the zonally anomalous hydrological cycle. In certain idealized cases, dynamic changes are even strong enough to reverse the tendency towards "wet gets wetter, dry gets drier” with warming.

Motivated by the importance of stationary-eddy vertical velocities in the moisture budget analysis, we examine controls on the amplitude of stationary eddies across a wide range of climates in an idealized GCM with simple topographic and ocean-heating zonal asymmetries. An analysis of the thermodynamic equation in the vicinity of topographic forcing reveals the importance of on-slope surface winds, the midlatitude isentropic slope, and latent heating in setting the amplitude of stationary waves. The response of stationary eddies to climate change is determined primarily by the strength of zonal surface winds hitting the mountain. The sensitivity of stationary-eddies to this surface forcing increases with climate change as the slope of midlatitude isentropes decreases. However, latent heating also plays an important role in damping the stationary-eddy response, and this damping becomes stronger with warming as the atmospheric moisture content increases. We find that the response of tropical overturning circulations forced by ocean heat-flux convergence is described by changes in the vertical structure of moist static energy and deep convection. This is used to derive simple scalings for the Walker circulation strength that capture the monotonic decrease with warming found in our idealized simulations.

Through the work of this thesis, the advances made in understanding the amplitude of stationary-waves in a changing climate can be directly applied to better understand and predict changes in the zonally anomalous hydrological cycle.

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.

Análise da resposta dinâmica experimental de uma passarela tubular mista, aço-concreto, submetida ao caminhar humano.

Este trabalho de pesquisa apresenta como objetivo principal o desenvolvimento de investigação experimental dinâmica sobre estrutura real de uma passarela tubular mista aço-concreto. O sistema estrutural objeto deste trabalho corresponde a uma passarela composta por três vãos (32,5m, 17,5m e 20,0m, respectivamente) e dois balanços (7,50m e 5,0m, respectivamente), com comprimento total de 82,5m. A passarela com estrutura contínua de aço com as ligações soldadas se apoia em quatro pórticos também de aço. Estruturalmente está constituída por duas treliças planas que se interligam através de contraventamentos horizontais fixados na corda superior e inferior da treliça e lajes de concreto, formando um sistema misto com interação completa. A estrutura está submetida correntemente à travessia de pedestres e ciclistas. Testes experimentais foram realizados sobre o sistema estrutural e confrontados com resultados numéricos. Para a modelagem numérica do sistema são empregadas técnicas usuais de discretização, via método dos elementos finitos (MEF), por meio do programa ANSYS. Os resultados experimentais são analisados de acordo com a metodologia desenvolvida, sendo realizada análise modal experimental para a determinação das propriedades dinâmicas: freqüências, modos e taxa de amortecimento, enquanto que os resultados da estrutura, em termos de aceleração de pico, são comparados com os valores limites propostos por diversos autores, normas e recomendações de projeto, para uma avaliação do desempenho da estrutura em relação a vibração quando solicitada pelo caminhar dos pedestres no que diz respeito a critério para conforto humano.

Influência do comportamento semi-rígido de placas de base e de ligações viga-coluna na resposta dinâmica de pórticos de aço.

Tradicionalmente, na análise e dimensionamento de estruturas de aço, assume-se que as ligações viga-coluna são rígidas ou flexíveis (rotuladas). Por outro lado, é de conhecimento geral que a grande maioria das ligações viga-coluna apresenta um comportamento intermediário, ou seja, semi-rígido. Inúmeros trabalhos de pesquisa têm sido desenvolvidos nos últimos vinte e cinco anos, de forma a estudar o comportamento desse tipo de ligação. Um dos principais objetivos desta investigação é o de propor uma metodologia de análise que represente de forma apropriada a influência do comportamento semi-rígido de placas de base e de ligações viga-coluna, sobre a resposta dinâmica (linear e não-linear) de estruturas de aço. Outra contribuição desta dissertação diz respeito à investigação do comportamento dinâmico (linear e não-linear) de pórticos de aço, a partir da consideração de ligações viga-coluna simétricas e não-simétricas e especialmente das placas de base. A análise estrutural é desenvolvida com base no emprego do programa de elementos finitos ANSYS [27]. Nos modelos em elementos finitos foram considerados os efeitos de não-linearidade geométrica (efeitos de segunda ordem), o comportamento não-linear das placas de base e das ligações viga-coluna e, bem como, o efeito de histerese que ocorre quando a estrutura é submetida a cargas cíclicas. Os resultados alcançados indicaram que o fenômeno físico da ressonância não ocorre no que se refere à resposta dinâmica dos modelos semi-rígidos não-lineares. A ressonância não ocorre na resposta dos modelos devido ao fato de que, na análise dinâmica não-linear, o efeito de histerese presente nas ligações (placas de base e viga-coluna), essencialmente com comportamento não-linear, provoca um amortecimento na resposta dinâmica da estrutura.

Studying the influence of substitutional elements on mechanical behavior of Alloy 718

In nickel-based superalloys, substitutional solute species have a strong impact on in service mechanical properties as well as on oxidation and corrosion resistances. In alloy 718, recent studies carried out by tensile tests highlighted the fact that refractory solute species are able to interact strongly with mobile dislocations during plastic deformation, generating dynamic strain ageing, and, in wide ranges of tests temperatures and strain rates, Portevin-Le Chatelier effect. The precise nature of the substitutional element responsible for such a dynamic interaction is still subject to debate. We addressed this question by means of mechanical spectroscopy studies of alloy 718 and various related alloys corresponding to monitored changes in the chemical composition. Only a single internal friction relaxation peak has been observed for all the studied alloys. By analyzing the damping behavior of these alloys at different imposed solicitation frequencies by sweeping a large temperatures range, the activation energies of the relaxation process and the type of mechanism involved have been determined. The process is a "Zener relaxation" in the alloys, i.e. a substitutional atoms dipole reorientation under applied stress. The results tend to prove that Niobium is not involved in the relaxation process whereas Molybdenum content seems to play an important role in the relaxation intensity.

Influence of Humidity on Fiber Bragg Grating Sensors

We demonstrate the influence of the relative humidity (RH) on the wavelength of fiber Bragg grating sensors (FBGS), performing tests with five FBGS at different humidity and temperature conditions. These tests were performed in a climate chamber whose RH changes according to a scheduled profile from 30% to 90%, in steps of 10%. These profiles were repeated for a wide range of temperatures from 10 degrees C to 70 degrees C, in steps of 10 degrees C. Two different types of instrumentation methods have been tested, spot welding and epoxy bonding, in two different materials, steel and carbon fiber reinforced polymer (CFRP). We discuss the results for each type of sensor and instrumentation method by analyzing the linearity of the Bragg wavelength with RH and temperature.

Identificação de danos estruturais utilizando dados no domínio do tempo provenientes de ensaios de vibração

O presente trabalho aborda o problema de identificação de danos em uma estrutura a partir de sua resposta impulsiva. No modelo adotado, a integridade estrutural é continuamente descrita por um parâmetro de coesão. Sendo assim, o Modelo de Elementos Finitos (MEF) é utilizado para discretizar tanto o campo de deslocamentos, quanto o campo de coesão. O problema de identificação de danos é, então, definido como um problema de otimização, cujo objetivo é minimizar, em relação a um vetor de parâmetros nodais de coesão, um funcional definido a partir da diferença entre a resposta impulsiva experimental e a correspondente resposta prevista por um MEF da estrutura. A identificação de danos estruturais baseadas no domínio do tempo apresenta como vantagens a aplicabilidade em sistemas lineares e/ou com elevados níveis de amortecimento, além de apresentar uma elevada sensibilidade à presença de pequenos danos. Estudos numéricos foram realizados considerando-se um modelo de viga de Euler-Bernoulli simplesmente apoiada. Para a determinação do posicionamento ótimo do sensor de deslocamento e do número de pontos da resposta impulsiva, a serem utilizados no processo de identificação de danos, foi considerado o Projeto Ótimo de Experimentos. A posição do sensor e o número de pontos foram determinados segundo o critério D-ótimo. Outros critérios complementares foram também analisados. Uma análise da sensibilidade foi realizada com o intuito de identificar as regiões da estrutura onde a resposta é mais sensível à presença de um dano em um estágio inicial. Para a resolução do problema inverso de identificação de danos foram considerados os métodos de otimização Evolução Diferencial e Levenberg-Marquardt. Simulações numéricas, considerando-se dados corrompidos com ruído aditivo, foram realizadas com o intuito de avaliar a potencialidade da metodologia de identificação de danos, assim como a influência da posição do sensor e do número de dados considerados no processo de identificação. Com os resultados obtidos, percebe-se que o Projeto Ótimo de Experimentos é de fundamental importância para a identificação de danos.

Análise vibratória de fundações de máquinas sobre estacas.

A análise de fundações sob solicitações dinâmicas é algo sempre presente em projetos na área industrial. É um campo pouco explorado na área de engenharia geotécnica, onde existem relativamente poucas informações no Brasil, de maneira geral. O método mais comum de realizar essas análises é a simplificação de modelos estruturais a partir do uso de molas. Sabe-se que esses coeficientes de reação têm uma variação relativamente grande e que esse enfoque de projeto pode, em alguns casos, mostrar-se contra a segurança ou levar a superdimensionamentos desnecessários. Verifica-se, então, a necessidade de uma avaliação mais criteriosa, utilizando a interação solo x estrutura, onde as molas comumente utilizadas nas análises vibratórias convencionais são substituídas pela rigidez real do solo quando concebido como um meio contínuo, através de sua discretização pelo método dos elementos finitos. A presente dissertação analisa o problema através do módulo de dinâmica do programa Plaxis 2D. Neste tipo de análise, além da modelagem do solo como um meio contínuo, torna-se possível introduzir condições de contorno específicas ao problema em estudo, múltiplas camadas de solo, sejam horizontais ou inclinadas, além da introdução de amortecedores capazes de evitar a reflexão espúria das ondas incidentes nos limites da malha de elementos finitos e assim modelar mais adequadamente a perda de energia por radiação. A presente dissertação compara medições experimentais e soluções eficientes de métodos vibratórios clássicos com a resposta obtida pelo MEF, mostrando resultados bastante satisfatórios tanto pelos métodos clássicos quanto pelo MEF.

Análise de vibrações e estudo de conforto humano sobre pisos mistos (aço-concreto) submetidos a ações humanas rítmicas.

Este trabalho tem por objetivo avaliar o comportamento dinâmico de pisos mistos (aço-concreto) sob a ação de cargas provenientes das atividades humanas rítmicas, especificamente a prática de ginástica aeróbica, sob o ponto de vista do conforto humano. Tal avaliação torna-se necessária por crescentes problemas estruturais associados às vibrações excessivas, decorrentes da concepção de sistemas estruturais com baixos níveis de amortecimento e com frequências naturais cada vez mais baixas e bastante próximas das faixas de frequência das excitações associadas às atividades humanas rítmicas. O modelo estrutural investigado baseiase em um piso misto (aço-concreto) submetido a aulas de ginástica aeróbica. A modelagem numérica do piso misto investigado foi realizada com base no emprego do programa ANSYS e foram utilizadas técnicas de discretização por meio do método dos elementos finitos (MEF). As cargas aplicadas sobre o piso, oriundas das atividades aeróbicas, são simuladas através de dois modelos de carregamentos dinâmicos distintos. Uma extensa análise paramétrica foi desenvolvida sobre o modelo estrutural investigado e a resposta dinâmica do sistema foi obtida, em termos dos deslocamentos e das acelerações, e comparada com os limites recomendados por normas e critérios de projeto. A resposta dinâmica do piso estudado viola os critérios de projeto relativos ao conforto humano e indica níveis de vibrações excessivas nos casos de carregamento dinâmicos analisados nesta dissertação.

Formação de nanopadrões em superfícies por sputtering iônico: Estudo numérico da equação anisotrópica amortecida de Kuramoto-Sivashinsky.

Apresenta-se uma abordagemnumérica para ummodelo que descreve a formação de padrões por sputtering iônico na superfície de ummaterial. Esse processo é responsável pela formação de padrões inesperadamente organizados, como ondulações, nanopontos e filas hexagonais de nanoburacos. Uma análise numérica de padrões preexistentes é proposta para investigar a dinâmica na superfície, baseada em ummodelo resumido em uma equação anisotrópica amortecida de Kuramoto-Sivashinsky, em uma superfície bidimensional com condições de contorno periódicas. Apesar de determinística, seu caráter altamente não-linear fornece uma rica gama de resultados, sendo possível descrever acuradamente diferentes padrões. Umesquema semi implícito de diferenças finitas com fatoração no tempo é aplicado na discretização da equação governante. Simulações foram realizadas com coeficientes realísticos relacionados aos parâmetros físicos (anisotropias, orientação do feixe, difusão). A estabilidade do esquema numérico foi analisada por testes de passo de tempo e espaçamento de malha, enquanto a verificação do mesmo foi realizada pelo Método das Soluções Manufaturadas. Ondulações e padrões hexagonais foram obtidos a partir de condições iniciais monomodais para determinados valores do coeficiente de amortecimento, enquanto caos espaço-temporal apareceu para valores inferiores. Os efeitos anisotrópicos na formação de padrões foramestudados, variando o ângulo de incidência.

Torsional Behaviour of Cello Strings

The behaviour of a bowed string depends, among other things, on the frequency, impedance and internal damping of torsional waves on the string. Very little published information is available about these quantities, especially the torsional damping. Measurements of all relevant torsional properties have been made on cello strings of three different constructions. These show that the torsional modes are harmonically spaced to reasonable accuracy, and that the Q factors are approximately equal for all modes of a given string. These torsional Q factors are roughly an order of magnitude smaller than those of the transverse modes of the same string. The torsional wave speed varies somewhat with the tension in the string, decreasing with higher tension. The damping factors are not significantly influenced by tension. These results have been expressed in terms of a novel "reflection function" [1] suitable for direct incorporation into simulations of the bowing process.

Simulation of three-dimensional viscous flow in turbomachinery geometries using a solution-adaptive unstructured mesh methodology

This paper presents a numerical method for the simulation of flow in turbomachinery blade rows using a solution-adaptive mesh methodology. The fully three-dimensional, compressible, Reynolds-averaged Navier-Stokes equations with k-ε turbulence modeling (and low Reynolds number damping terms) are solved on an unstructured mesh formed from tetrahedral finite volumes. At stages in the solution, mesh refinement is carried out based on flagging cell faces with either a fractional variation of a chosen variable (like Mach number) greater than a given threshold or with a mean value of the chosen variable within a given range. Several solutions are presented, including that for the highly three-dimensional flow associated with the corner stall and secondary flow in a transonic compressor cascade, to demonstrate the potential of the new method.