Identification of mechanical systems with local nonlinearities through discrete-time Volterra series and Kautz functions

peer reviewedMathematical modeling of mechanical structures is an important research area in structural dynamics. One of the goals of this area is to obtain a model that accurately predicts the dynamics of the system. However, the nonlinear eff ects caused by large displacements and boundary conditions like gap, backlash or joint are not as well understood as the linear counterpart. This paper identifies a non-parametric discrete-time Volterra model of a benchmark nonlinear structure consisting of a cantilever beam connected to a thin beam at its free end. Time-domain data experimentally measured are used to identify the Volterra kernels, which are expanded with orthogonal Kautz functions to facilitate the identification process. The nonlinear parameters are then estimated through a model updating process involving optimization of the residue between the numerical and experimental kernels. The advantages and drawbacks of the Volterra series for modeling the behavior of nonlinear structures are finally indicated with suggestions to overcome the disadvantages found during the tests

Evaluation of the force x elongation curves dispersion of descending colon of rats using Boltzmann's sigmoidal model

PURPOSE: to evaluate the dispersion of Force x Elongation curves (FE) that represents the mechanical behavior of rat's colon. MATERIALS AND METHODS: ten descending colon segments were submitted to the Total Energy of Rupture Test. Each curve generated by this test was fitted to Boltzmann model that correlates the values of force and elongation through the combination of the parameters A1, A2, x0 and d x. Then, for each parameter were calculated the mean, standard deviation and their correlations. Later, the resultant dispersion was determinated in the whole deformation process by an analysis that considers the correlations between the greatnesses based on the propagation of uncertainties law. The resultant dispersion was multiplied by a coverage factor, considering a normal distribution, to determinate an interval in which 95% of force values will be present. RESULTS: the mean, standard deviation and correlation were determined. The resultant dispersion of force values was expanded drawing limits inside which FE curves will be for a confidence interval of 95%. CONCLUSION: this methodology will be possibly used to evaluate variables that act on the intestinal mechanical behavior.OBJETIVO: avaliar a dispersão de curvas Força x Elongação (FE) representativas do comportamento mecânico de alça cólica íntegra de ratos. MATERIAIS E MÉTODOS: dez segmentos de cólon descendente de ratos machos Wistar foram submetidos ao teste biomecânico Energia Total de Ruptura. Cada curva gerada por esse ensaio foi ajustada ao modelo de Boltzmann, o qual correlaciona os valores de força e elongação por meio da combinação dos parâmetros A1, A2, x0 e d x. Nesse contexto, para cada parâmetro, foram calculadas as médias, os desvios padrão e correlações. Após, determinou-se a dispersão resultante da força em todo processo de deformação através de uma análise que considera as correlações entre as grandezas, com base na lei de propagação das incertezas. Para determinar um intervalo no qual estarão contidos 95% dos valores de força, a dispersão resultante foi multiplicada por um fator de abrangência considerando-se uma distribuição normal. RESULTADOS: os valores da média, do desvio padrão e das correlações foram determinados. A dispersão resultante dos valores de força foi expandida, delineando-se limites dentro dos quais estarão contidas as curvas FE para um intervalo de confiança de 95%. CONCLUSÃO: essa metodologia poderá auxiliar na avaliação de variáveis que interfiram no comportamento mecânico intestinal.515

Application of Volterra series in nonlinear mechanical system identification and in structural health monitoring problems

Nonlinear structures are frequent in structural dynamics, specially considering screwed components, with joints, clearance or flexible components presenting large displacements. In this sense the monitoring of systems based on classical linear methods, as the ones based on modal parameters, can drastically fail to characterize nonlinear effects. This thesis proposed the use of Volterra series for nonlinear system identification aiming applications in damage detection and parameter quantification. The property of this model of representing the linear and nonlinear components of the response of a system was used to formulate damage features to make clear the need of nonlinear modeling. Also metrics based on the linear and nonlinear residues of the terms of the Volterra model were employed to identify parametric models of the structure. The proposed methodologies are illustrated in experimental setups to show the relevance of nonlinear phenomena in the structural health monitoring.Estruturas com comportamento não-linear são frequentes em dinâmica estrutural, principalmente considerando componentes parafusados, com juntas, folgas ou estruturas flexíveis sujeitas à grandes deslocamentos. Desse modo, o monitoramento de estruturas com métodos lineares clássicos, como os baseados em parâmetros modais, podem falhar drasticamente em caracterizar efeitos não-lineares. Neste trabalho foi proposta a utilização de séries de Volterra para identificação de sistemas mecânicos não-lineares em aplicações de detecção de danos e quantificação de parâmetros. A propriedade deste modelo de representar separadamente os componentes de resposta linear e não-linear do sistema foi aplicada para se construir índices de dano que evidenciam a necessidade de modelagem não-linear. Além disso métricas de resíduo linear e não-linear dos termos do modelo de Volterra são empregadas para identificar modelos paramétricos da estrutura. As metodologias propostas são ilustradas em bancadas experimentais de modo a evidenciar a importância de fenômenos não-lineares para o monitoramento de estruturas.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Updating of a Nonlinear Finite Element Model Using Discrete-Time Volterra Series

Abstract In this study, the discrete-time Volterra series are used to update parameters in a nonlinear finite element model. The main idea of the Volterra series is to describe the discrete-time output of a nonlinear system using multidimensional convolutions between the Volterra kernels represented in a Kautz orthogonal basis and the excitations. A metric based on the residue between the experimental and the numerical Volterra kernels is used to identify the parameters of the numerical model. First, the identification of the linear parameters is performed using a metric based only on the first order Volterra kernels. Then the nonlinear parameters are identified through a metric based on the higher-order kernels. The originality of this nonlinear updating method stems from the decoupling of linear and nonlinear parameters and the use of global nonlinear model. In order to put in light the applicability of this technique, this work focus on the identification of the parameters in a nonlinear finite element model of a beam that was preloaded by compression mechanism. This work shows that the updated numerical model was able to represent the behaviour observed in the experimental measurements

Identification of nonlinear structures using discrete-time Volterra series

Mathematical modeling of mechanical structures is an important research area in structural dynamics. The goal is to obtain a model that accurately predicts the dynamics of the system. However, the nonlinear effects caused by gaps, backlash, joints, as well as large displacements are not as well understood as the linear counterpart. In this sense, the Volterra series is an interesting tool for the analysis of nonlinear systems, since it is a generalization of the linear model based on the impulse response function. This paper applies the discrete-time Volterra series expanded in orthonormal Kautz functions to identify a model of a nonlinear benchmark system represented by a Duffing oscillator. The input and output data are used to identify the Volterra kernels of the structure. After the identification of the model, the linear and nonlinear components of the response of the system can be analyzed separately. The paper concludes by indicating the main advantages and drawbacks of this technique to model the behavior of nonlinear systems.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

Damage detection in nonlinear structures using discrete-time volterra series

Structural damage identification is basically a nonlinear phenomenon; however, nonlinear procedures are not used currently in practical applications due to the complexity and difficulty for implementation of such techniques. Therefore, the development of techniques that consider the nonlinear behavior of structures for damage detection is a research of major importance since nonlinear dynamical effects can be erroneously treated as damage in the structure by classical metrics. This paper proposes the discrete-time Volterra series for modeling the nonlinear convolution between the input and output signals in a benchmark nonlinear system. The prediction error of the model in an unknown structural condition is compared with the values of the reference structure in healthy condition for evaluating the method of damage detection. Since the Volterra series separate the response of the system in linear and nonlinear contributions, these indexes are used to show the importance of considering the nonlinear behavior of the structure. The paper concludes pointing out the main advantages and drawbacks of this damage detection methodology. © (2013) Trans Tech Publications

Non-parametric identification of a non-linear buckled beam using discrete-time Volterra Series

The consideration of nonlinearities in mechanical structures is a question of high importance because several common features as joints, large displacements and backlash may give rise to these kinds of phenomena. However, nonlinear tools for the area of structural dynamics are still not consolidated and need further research effort. In this sense, the Volterra series is an interesting mathematical framework to deal with nonlinear dynamics since it is a clear generalization of the linear convolution for weakly nonlinear systems. Unfortunately, the main drawback of this non-parametric model is the need of a large number of terms for accurately identifying the system, but it can be overcomed by expanding the Volterra kernels with orthonormal basis functions. In this paper, this technique is used to identify a Volterra model of a nonlinear buckled beam and the kernels are used for the detection of the nonlinear behavior of the structure. The main advantages and drawbacks of the proposed methodology are highlighted in the final remarks of the paper.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

Challenges for Structural Health Monitoring: Nonlinearities and Uncertainties

International audienceImplementing structural health monitoring (SHM) techniques with a high Technology Readiness Level (TRL) is still challenging due to several practical requirements and assumptions to apply the fundamental methods. Between them, two issues earn some special attention: the linearity hypothesis and the robustness to the natural variability of data. The first point to overcome is that many structural engineering systems inherently behave nonlinearly during operation, even in a healthy state. Here, the assumption of linearity is typically inaccurate, eliminating large classes of feature extraction techniques. This issue is more complicated when the Damage also induces additional nonlinearities, e.g., cracking. The second aspect is the need to quantify the parameters' variation and uncertainties and signal data to interrogate the structural state. This chapter proposes introducing these challenges and some examples of addressing them in this context