Multiplicative cases from additive cases: Extension of Kolmogorov-Feller equation to parametric Poisson white noise processes

In this paper the response of nonlinear systems driven by parametric Poissonian white noise is examined. As is well known, the response sample function or the response statistics of a system driven by external white noise processes is completely defined. Starting from the system driven by external white noise processes, when an invertible nonlinear transformation is applied, the transformed system in the new state variable is driven by a parametric type excitation. So this latter artificial system may be used as a tool to find out the proper solution to solve systems driven by parametric white noises. In fact, solving this new system, being the nonlinear transformation invertible, we must pass from the solution of the artificial system (driven by parametric noise) to that of the original one (driven by external noise, that is known). Moreover, introducing this invertible nonlinear transformation into the Itˆo’s rule for the original system driven by external input, one can derive the Itˆo’s rule for systems driven by a parametric type excitation, directly. In this latter case one can see how natural is the presence of the Wong–Zakai correction term or the presence of the hierarchy of correction terms in the case of normal and Poissonian white noise, respectively. Direct transformation on the Fokker–Planck and on the Kolmogorov–Feller equation for the case of parametric input are found

On the Stochastic Response of a Fractionally-damped Duffing Oscillator

A numerical method is presented to compute the response of a viscoelastic Duffing oscillator with fractional derivative damping, subjected to a stochastic input. The key idea involves an appropriate discretization of the fractional derivative, based on a preliminary change of variable, that allows to approximate the original system by an equivalent system with additional degrees of freedom, the number of which depends on the discretization of the fractional derivative. Unlike the original system that, due to the presence of the fractional derivative, is governed by non-ordinary differential equations, the equivalent system is governed by ordinary differential equations that can be readily handled by standard integration methods such as the Runge–Kutta method. In this manner, a significant reduction of computational effort is achieved with respect to the classical solution methods, where the fractional derivative is reverted to a Grunwald–Letnikov series expansion and numerical integration methods are applied in incremental form. The method applies for fractional damping of arbitrary order a (0 < a < 1) and yields very satisfactory results. With respect to its applications, it is worth remarking that the method may be considered for evaluating the dynamic response of a structural system under stochastic excitations such as earthquake and wind, or of a motorcycle equipped with viscoelastic devices on a stochastic road ground profile

Comparison among three boundary element methods for torsion problems: CPM, CVBEM, LEM

This paper provides solutions for De Saint-Venant torsion problem on a beam with arbitrary and uniform cross-section. In particular three methods framed into complex analysis have been considered: Complex Polynomial Method (CPM), Complex Variable Boundary Element Method (CVBEM) and Line Element-less Method (LEM), recently proposed. CPM involves the expansion of a complex potential in Taylor series, computing the unknown coefficients by means of collocation points on the boundary. CVBEM takes advantage of Cauchy’s integral formula that returns the solution of Laplace equation when mixed boundary conditions on both real and imaginary parts of the complex potential are known. LEM introduces the expansion in the double-ended Laurent series involving harmonic polynomials, proposing an element-free weak form procedure, by imposing that the square of the net flux of the shear stress across the border is minimized with respect to the series coefficients. These methods have been compared with respect to numerical efficiency and accuracy. Numerical results have been correlated with analytical and approximate solutions that can be already found in literature

Stationary and non-stationary stochastic response of linear fractional viscoelastic systems

A method is presented to compute the stochastic response of single-degree-of-freedom (SDOF) structural systems with fractional derivative damping, subjected to stationary and non-stationary inputs. Based on a few manipulations involving an appropriate change of variable and a discretization of the fractional derivative operator, the equation of motion is reverted to a set of coupled linear equations involving additional degrees of freedom, the number of which depends on the discretization of the fractional derivative operator. As a result of the proposed variable transformation and discretization, the stochastic analysis becomes very straightforward and simple since, based on standard rules of stochastic calculus, it is possible to handle a system featuring Markov response processes of first order and not of infinite order like the original one. Specifically, for inputs of most relevant engineering interest, it is seen that the response second-order statistics can be readily obtained in a closed form, to be implemented in any symbolic package. The method applies for fractional damping of arbitrary order α (0 ≤ α ≤ 1). The results are compared to Monte Carlo simulation data

A Damage Identification procedure based on Hilbert transform: experimental validation

This paper aims at validating the feasibility of an identification procedure, based on the use of the Hilbert transform, by means of experimental tests for shear-type multi-degree-of-freedom systems. Particularly, a three-degree-of-freedom frame will be studied either numerically or experimentally by means of a laboratory scale model built at the laboratory of the Structural, Aerospace and Geotechnical Engineering Department (DISAG) of University of Palermo. Several damage scenarios have been considered to prove the effectiveness of the procedure. Moreover, the experimental tests have been conducted by considering two different input loads: pulse forces, simulated by means of an instrumental hammer, and wide band noise base inputs, by a shake table. In the first section the damage identification procedure, proposed in recent works, is presented. The procedure is based on the minimization of an objective function mathematically based on the properties of the analytical signal and the Hilbert transform. Second section reports the experimental model geometrical data and the data acquisition set-up as built in the DISAG laboratory. In Section 3, the results of the experimental campaigns are presented and discussed having considered three damage scenarios. The validated procedure has been proved to be able to not only detect damage even at early stage but it also needs processing of only few samples of the structural respons

Il Filtro Integrale Auto-Regressivo Continuo (I-ARC) per l’Analisi di Strutture Esposte al Vento

In questo studio viene proposto un metodo per la rappresentazione di processi aleatori Gaussiani e stazionari, utile a modellare la turbolenza della velocità del vento, introducendo la versione integrale del modello auto-regressivo discreto già proposto in precedenza. La rappresentazione di un processo aleatorio di assegnata funzione di correlazione viene condotta integrando un’equazione integro-differenziale in cui viene coinvolto un nucleo, che rappresenta la memoria del processo, in presenza di un rumore bianco Gaussiano. La soluzione dell’equazione rappresenta un campione del processo aleatorio della turbolenza della velocità del vento. E’ stato mostrato che il modello I-ARC fornisce, nei casi in cui effettivamente il processo è filtrato, statistiche coincidenti con quelle del processo obiettivo mentre fornisce risultati molto accurati qualora il processo non è filtrato come nel caso della velocità del vento. Inoltre una formulazione discreta del filtro I-ARC coincide con il modello ARC precedentemente discusso dagli autor

Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results

In capturing visco-elastic behavior, experimental tests play a fundamental rule, since they allow to build up theoretical constitutive laws very useful for simulating their own behavior. The main challenge is representing the visco-elastic materials through simple models, in order to spread their use. However, the wide used models for capturing both relaxation and creep tests are combinations of simple models as Maxwell and/or Kelvin, that depend on several parameters for fitting both creep and relaxation tests. This paper, following Nutting and Gemant idea of fitting experimental data through a power law function, aims at stressing the validity of fractional model. In fact, as soon as relaxation test is well fitted by power law decay then the fractional constitutive law involving Caputo’s derivative directly appears. It will be shown that fractional model is proper for studying visco-elastic behavior, since it may capture both relaxation and creep tests, requiring the identification of two parameters only. This consideration is assessed by the good agreement between experimental tests on creep and relaxation and the fractional model proposed. Experimental tests, here reported are performed on two polymers having different chemical physical properties such that the fractional model may cover a wide range of visco-elastic behavior

Damage identification by a modified Ant Colony Optimization for not well spaced frequency systems

Recently, it has been shown , that a damage detection strategy based on a proper functional calculated on the analytical signal of the structural dynamical response, consents to identify very low damage level. In this regard, they stressed the efficiency of Hilbert Transform to obtain the analytical response representation that shows more sensitivity for predicting damage with respect to the simple signal response. Then, a damage identification procedure based on the minimization of the difference between theoretical and measured data was proposed with satisfactory results. Unfortunately, this procedure, since the need of use of band pass filter around the natural frequency of the system, fails for structures having closed natural frequencies. By the way, performing procedures for sharply detecting damage in not well spaced frequency structure is a hoary problem. Aim of this paper is to extend the aforementioned procedure to these systems. To aim at this, it is desirable to go further insight into optimization algorithms, suitable for this kind of systems. For instance, it will be interesting considering, the ant colony optimization algorithm (ACO) that is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Recently ACO has been extended to continuous domain and labeled as ACOð. A novel aspect of the proposed paper is introducing ACOð into the previous procedure avoiding the use of filters, such that may be available for not well spaced frequency system. Moreover, it will be desirable avoiding the use of Hilbert transform, that means apply the identification procedure directly on the acceleration responses and not on the analytical signal response. Therefore, in this paper it will be introduced a procedure for detecting damage in structures having close frequencies, without using analytical signal response

An Innovative Structural Dynamic Identification Procedure Combining Time Domain OMA Technique and GA

In this paper an innovative and simple Operational Modal Analysis (OMA) method for structural dynamic identification is proposed. It combines the recently introduced Time Domain– Analytical Signal Method (TD–ASM) with the Genetic Algorithm (GA). Specifically, TD–ASM is firstly employed to estimate a subspace of candidate modal parameters, and then the GA is used to identify the structural parameters minimizing the fitness value returned by an appropriately introduced objective function. Notably, this method can be used to estimate structural parameters even for high damping ratios, and it also allows one to identify the Power Spectral Density (PSD) of the structural excitation. The reliability of the proposed method is proved through several numerical applications on two different Multi Degree of Freedom (MDoF) systems, also considering comparisons with other OMA methods. The results obtained in terms of modal parameters identification, Frequency Response Functions (FRFs) matrix estimation, and structural response prediction show the reliability of the proposed procedure