G. Muscolino

Improved dynamic analysis of structures with mechanical uncertainties under deterministic input

This paper addresses the dynamic analysis of linear systems with uncertain parameters subjected to deterministic excitation. The conventional methods dealing with stochastic structures are based on series expansion of stochastic quantities with respect to uncertain parameters, by means of either Taylor expansion, perturbation technique or Neumann expansion and evaluate the first- and second-order moments of the response by solving deterministic equations. Unfortunately, these methods lead to significant error when the coefficients of variation of uncertainties are relatively large. Herein, an improved first-order perturbation approach is proposed, which considers the stochastic quantities as the sum of their mean and deviation. Comparisons with conventional second-order perturbation approach and Monte Carlo simulations illustrate the efficiency of the proposed method. Applications are discussed in order to investigate the influence of mass, damping and stiffness uncertainty on the dynamic response of the system.

Linear systems excited by polynomial forms of non-Gaussian filtered processes

Abstract A method for the evaluation of the statistical moments of linear one-dimensional or multi-dimensional systems subjected to non-Gaussian input in polynomial forms of filtered normal or non-normal delta-correlated processes is presented. The proposed procedure allows one to avoid the solution of a set of non-linear differential equations, requiring the solution of three sets of linear differential equations. The latter sets are: the equations governing the evolution of the moments of the response forced by input-output cross moments; the equations useful for the evaluation of the input-output cross-moments and the equations governing the evolution of the statistical moments of the filtered input.

Carbon nanotubes and nanosensors: vibration, buckling and ballistic impact

The main properties that make carbon nanotubes (CNTs) a promising technology for many future applications are: extremely high strength, low mass density, linear elastic behavior, almost perfect geometrical structure, and nanometer scale structure. Also, CNTs can conduct electricity better than copper and transmit heat better than diamonds. Therefore, they are bound to find a wide, and possibly revolutionary use in all fields of engineering. The interest in CNTs and their potential use in a wide range of commercial applications; such as nanoelectronics, quantum wire interconnects, field emission devices, composites, chemical sensors, biosensors, detectors, etc.; have rapidly increased in the last two decades. However, the performance of any CNT-based nanostructure is dependent on the mechanical properties of constituent CNTs. Therefore, it is crucial to know the mechanical behavior of individual CNTs such as their vibration frequencies, buckling loads, and deformations under different loadings. This title is dedicated to the vibration, buckling and impact behavior of CNTs, along with theory for carbon nanosensors, like the Bubnov-Galerkin and the Petrov-Galerkin methods, the Bresse-Timoshenko and the Donnell shell theory.

Differential moment equations of FE modelled structures with geometrical non-linearities

Abstract In the framework of the finite element (FE) method, by using the “total Lagrangian approach”, the stochastic analysis of geometrically non-linear structures is performed. To this purpose the deterministic equations of motion are written wording the non-linear contribution in an explicit representation as pseudo-forces. Then the equations of moments of the response for external Gaussian white noise processes are obtained by extending the classical Itos rule to vectors of random processes. The deterministic equations of motion and the equations of moments, here obtained, show a perfect formal similarity. By using this similarity a very effective computational procedure to evaluate the moments of any order of the response is proposed. It is also shown that the proposed formulation can be considered a very important step towards the actual solution of multidegree-of-freedom systems under Gaussian white processes.

Static and Dynamic Analysis of Non-Linear Uncertain Structures

The procedures usually adopted in the evaluation of the stochastic response of structures with uncertain parameters, the so-called stochastic structures, are affected by some limits. Namely, the major drawbacks are: the conspicuous computational time required for the analysis of many degree of freedom systems and the loss of accuracy in the case of large uncertainty in the parameters. A method able to reduce the previous inconveniences was recently introduced in the study of statically loaded linear structures. The method, named improved perturbation method, was also extended to the field of linear dynamics. In both cases, the improved perturbation method provides a good approximation, even in the case of moderately large deviation of the uncertain parameters, and the computational time required is comparable to conventional first order perturbation. The present paper intends to apply the improved perturbation method in the second order analysis of geometrically non-linear uncertain systems subjected to static and dynamic deterministic forces.

RESPONSE STATISTICS OF LINEAR STRUCTURES WITH UNCERTAIN-BUT-BOUNDED PARAMETERS UNDER GAUSSIAN STOCHASTIC INPUT

Uncertainty plays a fundamental role in structural engineering since it may affect both external excitations and structural parameters. In this study, the analysis of linear structures with slight variations of the structural parameters subjected to stochastic excitation is addressed. It is realistically assumed that sufficient data are available to model the external excitation as a Gaussian random process, while only fragmentary or incomplete information about the structural parameters are known. Under this assumption, a nonprobabilistic approach is pursued and the fluctuating properties are modeled as uncertain-but-bounded parameters via interval analysis. A method for evaluating the lower and upper bounds of the second-order statistics of the response is presented. The proposed procedure basically consists in combining random vibration theory with first-order interval Taylor series expansion of the mean-value and covariance vectors of the response. After some algebra, the sets of first-order ordinary differential equations ruling the nominal and first-order sensitivity vectors of response statistics are derived. Once such equations are solved, the bounds of the mean-value and covariance vectors of the response can be evaluated by handy formulas. To validate the procedure, numerical results concerning two different structures with uncertain-but-bounded stiffness properties under seismic excitation are presented.

Monte Carlo simulation in the stochastic analysis of non-linear systems under external stationary Poisson white noise input

A method for the evaluation of the probability density function (p.d.f.) of the response process of non-linear systems under external stationary Poisson white noise excitation is presented. The method takes advantage of the great accuracy of the Monte Carlo simulation (MCS) in evaluating the first two moments of the response process by considering just few samples. The quasi-moment neglect closure is used to close the infinite hierarchy of the moment differential equations of the response process. Moreover, in order to determine the higher order statistical moments of the response, the second-order probabilistic information given by MCS in conjunction with the quasi-moment neglect closure leads to a set of linear differential equations. The quasi-moments up to a given order are used as partial probabilistic information on the response process in order to find the p.d.f. by means of the C-type Gram–Charlier series expansion.

Gaussian and non-Gaussian stochastic sensitivity analysis of discrete structural system

Abstract The derivatives of the response of a structural system with respect to the system parameters are termed sensitivities. They play an important role in assessing the effect of uncertainties in the mathematical model of the system and in predicting changes of the response due to changes of the design parameters. In this paper, a time domain approach for evaluating the sensitivity of discrete structural systems to deterministic, as well as to Gaussian or non-Gaussian stochastic input is presented. In particular, in the latter case, the stochastic input has been assumed to be a delta-correlated process and, by using Kronecker algebra extensively, cumulant sensitivities of order higher than two have been obtained by solving sets of algebraic or differential equations for stationary and non-stationary input, respectively. The theoretical background is developed for the general case of multi-degrees-of-freedom (MDOF) primary system with an attached secondary single-degree-of-freedom (SDOF) structure. However, numerical examples for the simple case of an SDOF primary–secondary structure, in order to explore how variations of the system parameters influence the system, are presented. Finally, it should be noted that a study of the optimal placement of the secondary system within the primary one should be conducted on an MDOF structure.

Clamped-Free Single-Walled Carbon Nanotube-Based Mass Sensor Treated as Bernoulli–Euler Beam

In this study, we investigate the vibrations of the cantilever single-walled carbon nanotube (SWCNT) with attached bacterium on the tip in view of developing the sensor. This sensor will be able to help to identify the bacterium or virus that may be attached to the SWCNT. Two cases are considered: These are light or heavy bacteria attached to the nanotube. The problem is solved by the exact solution, the finite difference method, and the Bubnov-Galerkin method.

Probability density function of MDOF structural systems under non-normal delta-correlated inputs

Abstract A method to approximate the probability density function of MDOF linear systems under non-normal delta-correlated input is presented. The method requires: (i) the evaluation of the response cumulants up a given order, by solving the set of cumulant differential equations; (ii) the evaluation of the quasi-moments of the response by means of recursive relationships, once the response cumulants are known; (iii) the evaluation of the coefficients of the C-type Gram-Charlier series expansion of the response probability density function, by solving a set of linear algebraic equations, whose known terms depend on the quasi-moments of the response up to a given order. The numerical application shows the versatility and the accuracy of the proposed method.