Probabilistic characterization of nonlinear systems under Poisson white noise parametric input via complex fractional moments

In this paper the probabilistic characterization of a nonlinear system enforced by parametric Poissonian white noise in terms of complex fractional moments is presented. In fact the initial system driven by a parametric input could be transformed into a system with an external type of excitation through an invertible nonlinear transformation. It is shown that by using Mellin transform theorem and related concepts, the solution of the Kolmogorov-Feller equation for the system with external input may be obtained in a very easy way.

Laplace’s Method of Integration in the Path Integral Approach for the Probabilistic Response of Nonlinear Systems

In this paper the response of nonlinear systems under stationary Gaussian white noise excitation is studied. The Path Integral (PI) approach, generally employed for evaluating the response Probability Density Function (PDF) of systems in short time steps based on the Chapman-Kolmogorov equation, is here used in conjunction with the Laplace’s method of integration. This yields an approximate analytical solution of the integral involved in the Chapman-Kolmogorov equation. Further, in this manner the repetitive integrations, generally required in the conventional numerical implementation of the procedure, can be circumvented. Application to a nonlinear system is considered, and pertinent compa…

Innovative modeling of tuned liquid column damper controlled structures

In this paper a different formulation for the response of structural systems controlled by Tuned Liquid Column Damper (TLCD) devices is developed, based on the mathematical tool of fractional calculus. Although the increasing use of these devices for structural vibration control, it has been demonstrated that existing model may lead to inaccurate prediction of liquid motion, thus reflecting in a possible imprecise description of the structural response. For this reason the recently proposed fractional formulation introduced to model liquid displacements in TLCD devices, is here extended to deal with TLCD controlled structures under base excitations. As demonstrated through an extensive expe…

Inverse Mellin Transform to characterize the nonlinear system PDF response to Poisson white noise

Poisson white noise parametric input and response by using complex fractional moments

Abstract In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.

Numerical and experimental validation of a simplied formulation for the design of TLCD

Smartphone-based bridge monitoring through vehicle-bridge interaction: analysis and experimental assessment

AbstractIn this study, the results of a vast experimental campaign on the applicability of a smartphone-based technique for bridge monitoring are presented. Specifically, the vehicle–bridge interaction (VBI)-based approach is exploited as a cost-effective means to estimate the natural frequencies of bridges, with the final aim of possibly developing low-cost and diffused infrastructure monitoring system. The analysis is performed using a common hybrid vehicle, fully equipped with classical piezoelectric accelerometers and a smartphone MEMS accelerometer, to record its vertical accelerations while passing over the bridge. In this regard, the experimental campaign is carried out considering t…

Complex fractional moments for the characterization of the probabilistic response of non-linear systems subjected to white noises

In this chapter the solution of Fokker-Planck-Kolmogorov type equations is pursued with the aid of Complex Fractional Moments (CFMs). These quantities are the generalization of the well-known integer-order moments and are obtained as Mellin transform of the Probability Density Function (PDF). From this point of view, the PDF can be seen as inverse Mellin transform of the CFMs, and it can be obtained through a limited number of CFMs. These CFMs’ capability allows to solve the Fokker-Planck-Kolmogorov equation governing the evolutionary PDF of non-linear systems forced by white noise with an elegant and efficient strategy. The main difference between this new approach and the other one based …

A Novel Solution to Find the Dynamic Response of an Euler–Bernoulli Beam Fitted with Intraspan TMDs under Poisson Type Loading

This contribution considers a virtual experiment on the vibrational response of rail and road bridges equipped with smart devices in the form of damping elements to mitigate vibrations. The internal damping of the bridge is considered a discontinuity that contain a dashpot. Exact complex eigenvalues and eigenfunctions are derived from a characteristic equation built as the determinant of a 4 x 4 matrix

An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems

The recently developed approximate Wiener path integral (WPI) technique for determining the stochastic response of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems has proven to be reliable and significantly more efficient than a Monte Carlo simulation (MCS) treatment of the problem for low-dimensional systems. Nevertheless, the standard implementation of the WPI technique can be computationally cumbersome for relatively high-dimensional MDOF systems. In this paper, a novel WPI technique formulation/implementation is developed by combining the “localization” capabilities of the WPI solution framework with an appropriately chosen expansion for approximating the system response PDF…

First-passage problem for nonlinear systems under Lévy white noise through path integral method

In this paper, the first-passage problem for nonlinear systems driven by $$\alpha $$ -stable Levy white noises is considered. The path integral solution (PIS) is adopted for determining the reliability function and first-passage time probability density function of nonlinear oscillators. Specifically, based on the properties of $$\alpha $$ -stable random variables and processes, PIS is extended to deal with Levy white noises with any value of the stability index $$\alpha $$ . Application to linear and nonlinear systems considering different values of $$\alpha $$ is reported. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the results.

Rocking of rigid block on nonlinear flexible foundation

Abstract The two prime models used currently to describe rocking of rigid bodies, the Housner’s model and the Winkler foundation model, can capture some of the salient features of the physics of this important problem. These two models involve either null or linear interaction between the block and the foundation. Hopefully, some additional aspects of the problem can be captured by an enhanced nonlinear model for the base-foundation interaction. In this regard, what it is adopted in this paper is the Hunt and Crossley’s nonlinear impact force model in which the impact/contact force is represented by springs in parallel with nonlinear dampers. In this regard, a proper mathematical formulatio…

Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators endowed with fractional derivatives elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes which rely on a discrete version of the…

Random vibration TLCD controlled

Optimal design of tuned liquid column damper inerter for vibration control

Abstract In this paper, the use of a novel passive control device defined as Tuned Liquid Column Damper Inerter (TLCDI) is studied to control the seismic response of structural systems. The TLCDI, recently introduced as an enhanced version of the conventional Tuned Liquid Column Damper, may achieve improved seismic performances by exploiting the mass amplification effect of the so-called inerter device. For this purpose, an optimization procedure for the design of the TLCDI based on a statistical linearization technique and the minimization of the structural displacement variance is proposed. Notably, by assuming a white noise base excitation and considering some additional approximations, …

An Innovative Ambient Identification Method

Ambient modal identification, also known as Operational Modal Analysis (OMA), aims to identify the modal properties of a structure based on vibration data collected when the structure is under its operating conditions, i.e., no initial excitation or known artificial excitation. This procedure for testing and/or monitoring historic buildings, is particularly attractive for civil engineers concerned with the safety of complex historic structures. However, since the external force is not recorded, the identification methods have to be more sophisticated and based on stochastic mechanics. In this context, this contribution will introduce an innovative ambient identification method based on appl…

Structural vibration control through Tuned Liquid Column Dampers: theoretical and experimental analysis

Path Integral approach via Laplace’s method of integration for nonstationary response of nonlinear systems

In this paper the nonstationary response of a class of nonlinear systems subject to broad-band stochastic excitations is examined. A version of the Path Integral (PI) approach is developed for determining the evolution of the response probability density function (PDF). Specifically, the PI approach, utilized for evaluating the response PDF in short time steps based on the Chapman–Kolmogorov equation, is here employed in conjunction with the Laplace’s method of integration. In this manner, an approximate analytical solution of the integral involved in this equation is obtained, thus circumventing the repetitive integrations generally required in the conventional numerical implementation of …

Analysis of Rectangular Orthotropic Membranes for Mechanical Properties Identification through Load-Displacement Data

In this paper, an innovative procedure is introduced for the identification of the mechanical properties of orthotropic membranes based on load-displacement data. To this end, novel functional forms of the displacement components for rectangular membranes are appropriately introduced. Unknown coefficients of these displacement functions are determined, minimizing the total potential energy of the membrane. The energy method is then combined with an optimization procedure to estimate the elastic constants of the membranes in a straightforward manner. Specifically, a genetic algorithm is used to minimize a properly defined objective function directly related to the sought mechanical propertie…

Path Integral Method for Nonlinear Systems Under Levy White Noise

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under…

Optimal tuning of tuned liquid column damper systems in random vibration by means of an approximate formulation

Passive control devices are often added to slender and flexible systems in order to increase their structural safety. Several types of devices have been proposed in order to reduce the dynamic responses of different kind of structural systems. Among them, the tuned liquid column damper (TLCD) proved to be very effective in reducing vibration of various type of structures by means of a combined action which involves the motion of the liquid mass within the tube. The restoring force, in particular, is produced by the force of gravity acting on the liquid and the damping effect is generated by the hydrodynamic head losses that arise during the motion of the liquid inside the TLCD. Since the in…

The TLCD Passive Control: Numerical Investigations vs. Experimental Results

Very recently the tuned liquid column damper (TLCD) is receiving an increasing interest from researchers concerned with vibration control, to be considered an alternative device with respect to the tuned mass damper (TMD), since the former has low cost, easy adjustment, flexible installation. However, in recent studies the authors [1] have pointed out that for TMD the analytical formulation provides results that are in good agreement with the experimental ones, while for TLCD it has been deducted that the analytical formulation needs further investigation. In fact using the classical formulation of the problem, numerical results are very different from the experimental results obtained by t…

Path integral method for first-passage probability determination of nonlinear systems under levy white noise

In this paper the problem of the first-passage probabilities determination of nonlinear systems under alpha-stable Lévy white noises is addressed. Based on the properties of alpha-stable random variables and processes, the Path Integral method is extended to deal with nonlinear systems driven by Lévy white noises with a generic value of the stability index alpha. Furthermore, the determination of reliability functions and first-passage time probability density functions is handled step-by-step through a modification of the Path Integral technique. Comparison with pertinent Monte Carlo simulation reveals the excellent accuracy of the proposed method.

Nonlinear rocking of rigid blocks on flexible foundation: Analysis and experiments

Abstract Primarily, two models are commonly used to describe rocking of rigid bodies; the Housner model, and the Winkler foundation model. The first deals with the motion of a rigid block rocking about its base corners on a rigid foundation. The second deals with the motion of a rigid block rocking and bouncing on a flexible foundation of distributed linear springs and dashpots (Winkler foundation). These models are two-dimensional and can capture some of the features of the physics of the problem. Clearly, there are additional aspects of the problem which may be captured by an enhanced nonlinear model for the base-foundation interaction. In this regard, what it is adopted in this paper is …

Assessment of the tuned mass damper inerter for seismic response control of base‐isolated structures

In this paper, the hybrid control of structures subjected to seismic excitation by means of tuned mass damper inerter (TMDI) and base-isolation subsystems is studied with the aim of improving the dynamic performance of base-isolated structures by reducing the displacement demand of the isolation subsystem. The seismic performance of TMDI hybrid controlled structures is investigated in a comparative study, considering simple isolated systems and systems equipped with other absorber devices such as the tuned mass damper (TMD) and the tuned liquid column damper (TLCD). The TMDI has been optimized by performing a simplified approach based on minimizing the base-isolation subsystem displacement …

Base-isolated structure equipped with tuned liquid column damper: An experimental study

Abstract In this study, a novel passive vibration control strategy is investigated experimentally, where a Tuned Liquid Column Damper protects a base-isolated structure. The Tuned Liquid Column Damper is attached to the base, in contrast to typical attachment points of passive energy dissipation devices in high-rise buildings at elevated levels. Experiments on a base-excited small-scale three-story shear frame are conducted in order to study effects of both control devices – base-isolation and Tuned Liquid Column Damper – on the structural model. The dynamic properties of the stand-alone shear frame and the base-isolation subsystem are derived using standard dynamic test methods based on di…

Mechanical-fluid dynamics coupled model for profiled Ion Exchange Membranes design

In this work, we developed an advanced model useful for the design of profiled IEMs, based on the coupled simulation of local mechanical deformations and of fluid dynamics and associated mass transport phenomena within deformed channels

Dynamic response of equivalent orthotropic plate model for stiffened plate: numerical-experimental assessment

Abstract Over the last two decades, homogenization-based modeling techniques have attracted considerable attention. In fact, through these methods, structures such as corrugated or stiffened plates, commonly referred to as structurally orthotropic plates, can be approximately studied as equivalent flat plates with orthotropic behavior. Specifically, these homogenization techniques allow for the direct determination of the equivalent flexural and torsional rigidities which appear in the governing equation for the deflection of the equivalent orthotropic plate. It is worth noting that, the determined equivalent material properties retain the dependence on the geometric parameters of the origi…

Smart structures through nontraditional design of Tuned Mass Damper Inerter for higher control of base isolated systems

Abstract This paper introduces a smart structure design through the definition of an innovative passive control strategy, referred to as New Tuned Mass Damper Inerter (New TMDI), coupled with a base isolation system (BI), to control displacements in base isolated structures under seismic excitations. The herein proposed New TMDI comprises a recently developed nontraditional Tuned Mass Damper (known as New TMD), in which a secondary mass system is connected to the base plate of the BI system by a spring and to the ground by a dashpot, and of an inerter device placed in parallel with the damper. An optimization procedure which minimizes the base displacement variance of the BI system, conside…

Optimal design of tuned liquid column dampers for seismic response control of base-isolated structures

In this paper, the use of a tuned liquid column damper (TLCD) as a cost-effective means to control the seismic response of a base-isolated structure is studied. A straightforward direct approach for the optimal design of such a device is proposed, considering a white noise model of the base excitation. On this base, a direct optimization procedure of the TLCD design parameters is performed and optimal design charts are presented as a ready-to-use practical design tool. Comparison with the optimal parameters obtained considering a classical iterative statistical linearization technique proves the reliability of the proposed approach. The performance of the base-isolated TLCD-controlled struc…

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 excitat…

Design Of TLCD under random loads: a new formulation

Direct evaluation of the equivalent linear damping for Tuned Liquid Column Damper systems in random vibration for pre-design purposes

Probabilistic characterization of nonlinear systems under parametric Poisson white noise via complex fractional moments

A novel fluid-structure 2D modelling tool for the assessment of membrane deformation effects on electrodialysis units performances

Fractional visco-elastic Timoshenko beam deflection via single equation

SUMMARY This paper deals with the response determination of a visco-elastic Timoshenko beam under static loading condition and taking into account fractional calculus. In particular, the fractional derivative terms arise from representing constitutive behavior of the visco-elastic material. Further, taking advantages of the Mellin transform method recently developed for the solution of fractional differential equation, the problem of fractional Timoshenko beam model is assessed in time domain without invoking the Laplace-transforms as usual. Further, solution provided by the Mellin transform procedure will be compared with classical Central Difference scheme one, based on the Grunwald–Letni…

Fractional visco-elastic Timoshenko beam deflection via single equation

This paper deals with the response determination of a visco-elastic Timoshenko beam under static loading condition and taking into account fractional calculus. In particular, the fractional derivative terms arise from representing constitutive behavior of the visco-elastic material. Further, taking advantages of the Mellin transform method recently developed for the solution of fractional differential equation, the problem of fractional Timoshenko beam model is assessed in time domain without invoking the Laplace-transforms as usual. Further, solution provided by the Mellin transform procedure will be compared with classical Central Difference scheme one, based on the Grunwald-Letnikov appr…

Fractional viscoelastic behaviour under stochastic temperature process

Abstract This paper deals with the mechanical behaviour of a linear viscoelastic material modelled by a fractional Maxwell model and subject to a Gaussian stochastic temperature process. Two methods are introduced to evaluate the response in terms of strain of a material under a deterministic stress and subjected to a varying temperature. In the first approach the response is determined making the material parameters change at each time step, due to the temperature variation. The second method, takes advantage of the Time–Temperature Superposition Principle to lighten the calculations. In this regard, a stochastic characterisation for the Time–Temperature Superposition Principle method is p…

Earthquake Excited Base-Isolated Structures Protected by Tuned Liquid Column Dampers: Design Approach and Experimental Verification

Abstract In this contribution a direct approach for optimal design of a Tuned Liquid Column Damper (TLCD) device attached to the base slab of a base-isolated structure is presented, aiming at reducing the seismic displacement demand of the base-isolation subsystem. Assuming white noise base excitation, for a wide parameter range a direct optimization procedure yields design charts for optimal TLCD quantities. The performance of the base-isolated structure equipped with optimally tuned TLCD device in comparison to the simple base-isolated one is evaluated both numerically and experimentally. In a numerical study the system is subjected to the 44 records of the FEMA P-695 far-field ground mot…

Nontraditional configuration of tuned liquid column damper inerter for base-isolated structures

In this paper, the concept of a novel passive control device, namely the Nontraditional Tuned Liquid Column Damper Inerter (NT-TLCDI), is investigated in combination with seismic base isolation (BI), to control lateral displacement demands in base-isolated structures during seismic events. The considered NT-TLCDI is a revision of the ordinary configuration of the recently proposed Tuned Liquid Column Damper Inerter (TLCDI). Unlike the traditional TLCDI layout, which involves a secondary liquid mass in a U-shaped tank coupled with a grounded inerter and connected to the isolation system by a spring-dashpot system, in the NT-TLCDI configuration, the damper is in parallel with the inerter rath…

Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on Laplace’s method of integration

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman–Kolmogorov (CK) equation governing the evolution in time of the non-stationary respon…

A novel identification procedure from ambient vibration data for buildings of the cultural heritage

Ambient modal identification, also known as Operational Modal Analysis (OMA), aims to identify the modal properties of a structure based on vibration data collected when the structure is under its operating conditions, i.e., no initial excitation or known artificial excitation. This procedure for testing and/or monitoring historic buildings, is particularly attractive for civil engineers concerned with the safety of complex historic structures. However, since the external force is not recorded, the identification methods have to be more sophisticated and based on stochastic mechanics. In this context, this contribution will introduce an innovative ambient identification method based on appl…

OMA: From Research to Engineering Applications

Ambient vibration modal identification, also known as Operational Modal Analysis (OMA), aims to identify the modal properties of a structure based on vibration data collected when the structure is under its operating conditions, i.e., when there is no initial excitation or known artificial excitation. This method for testing and/or monitoring historical buildings and civil structures, is particularly attractive for civil engineers concerned with the safety of complex historical structures. However, in practice, not only records of external force are missing, but uncertainties are involved to a significant extent. Hence, stochastic mechanics approaches are needed in combination with the iden…

Hybrid Passive Control Strategies for Reducing the Displacements at the Base of Seismic Isolated Structures

In this paper, the use of hybrid passive control strategies to mitigate the seismic response of a base-isolated structure is examined. The control performance of three different types of devices used for reducing base displacements of isolated buildings is investigated. Specifically, the Tuned Mass Damper (TMD), the New Tuned Mass Damper (New TMD) and the Tuned Liquid Column Damper (TLCD), each one associated to a Base Isolated structure (BI), have been considered. The seismic induced vibration control of base-isolated structures equipped with the TMD, New TMD or the TLCD is examined and compared with that of the base-isolated system without devices, using real recorded seismic signals as e…

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to …