Random Vibration Theory Research Articles

Simulation of strong ground motions modified by attenuation effects under specific soil conditions: modeling the 2017 Puebla, Mexico (Mw 7.1) earthquake

This research presents an updated scheme of simulation for strong ground motions based on stochastic summation methods and the source-to-site attenuation effects under particular soil conditions different from firm sites. Two approaches are proposed to modify and improve the empirical Green’s functions at different source distances from seismic station records: (1) a spectral shape based on far-field point-source spectra models and (2) through specific ground motion prediction equations (GMPEs) and random vibration theory (RVT). The ground motions simulated were compared with the September 19, 2017 Mexico records (normal-fault-intraslab event, Mw 7.1), finding that for soft soils, it will be possible to reproduce some characteristic of spectral peaks appropriately in high frequency compared with available similar approaches or using one-dimensional wave propagation models. On average, the spectral overestimations from available simulation approaches and for high-frequency bandwidth (i.e., 0.2 s and 0.3 s) can be reduced by a factor between 2 and 5 using proper attenuation corrections; this effect could be especially beneficial for small structures and low-rise buildings where the inertial design forces will be potentially reduced.

An Efficient Stochastic Linearisation Procedure for the Seismic Optimal Design of Passive Control Devices

The seismic response of structures is often enhanced by introducing passive control devices that can operate through the dissipation of the input energy or by modifying the dynamic characteristics of the main structure. The inherent non-linearities in the constitutive laws of some of them lead to computation difficulties and have limited the large-scale use and design of these devices. In this study, a procedure for the optimal design of multi passive control devices is proposed. The general case of linear Multi-Degree-Of-Freedom (MDOF) not-classically-damped structural systems controlled by Fluid Viscous Dampers (FVD) are investigated in a stochastic framework. The procedure consists of evaluation of the device optimal pattern by minimizing an objective function related to the dampers cost and subjected to a constraint on the structural behaviour. For each configuration, the complete probabilistic characterization of the response is achieved by employing random vibration theory, Stochastic Linearisation (SL) techniques and a novel analytic model which provides closed-form PSD functions of earthquakes accelerations coherent to response spectra suggested by seismic codes. Exploiting this model, a procedure to speed up the Stochastic Linearisation technique by avoiding any numerical integration is proposed. Applications on MDOF building structures have been carried out to validate the proposed approach in terms of accuracy and reduction of the computational effort and to obtain optimal pattern of the passive control device coherently with the provisions of seismic building codes.

Dynamics and control of spar‐type floating offshore wind turbines with tuned liquid column dampers

This paper investigates the use of tuned liquid column dampers (TLCDs) for vibration control of spar-type floating offshore wind turbines (FOWTs). A 17-degree-of-freedom (17-DOF) aero-hydro-servo-elastic model for the FOWT is first established using multi-body-based formulation and the Euler–Lagrangian equation, taking into consideration the full coupling of the blade-drivetrain-tower-spar vibrations, a collective pitch controller and a generator controller. The outputs of the model are compared with FAST for model verification. Next, a reduced-order 5-DOF model is established for the TLCD-controlled FOWT in-plane vibrations including hydrodynamic added mass as well as stiffness contributions from mooring lines and buoyancy. The model enables revealing the fundamental mechanism of the coupled spar-tower-TLCD system, as well as an efficient procedure for optimal design of FOWT-mounted TLCD. It is found from modal analysis of the spar-tower system that due to the coupling to the spar roll motion, the tower side–side frequency is significantly shifted comparing with the decoupled case, which needs to be accounted for when tuning the TLCD. Based on the 5-DOF model, a frequency-domain (FD) method for TLCD optimization is proposed using random vibration theory, and robust optimization of the TLCD can be performed for given environmental conditions. The optimized damper is then incorporated into the more sophisticated 17-DOF model, and performance of the TLCD is evaluated by means of nonlinear time-domain (TD) simulations. Both FD and TD results indicate that a well-designed TLCD effectively reduces the tower side–side vibration as well as the spar roll motion.

Roughness-induced vehicle energy dissipation from crowdsourced smartphone measurements through random vibration theory

Abstract We propose, calibrate, and validate a crowdsourced approach for estimating power spectral density (PSD) of road roughness based on an inverse analysis of vertical acceleration measured by a smartphone mounted in an unknown position in a vehicle. Built upon random vibration analysis of a half-car mechanistic model of roughness-induced pavement–vehicle interaction, the inverse analysis employs an L2 norm regularization to estimate ride quality metrics, such as the widely used International Roughness Index, from the acceleration PSD. Evoking the fluctuation–dissipation theorem of statistical physics, the inverse framework estimates the half-car dynamic vehicle properties and related excess fuel consumption. The method is validated against (a) laser-measured road roughness data for both inner city and highway road conditions and (b) road roughness data for the state of California. We also show that the phone position in the vehicle only marginally affects road roughness predictions, an important condition for crowdsourced capabilities of the proposed approach.

A Multi Record Based Artificial Near Fault Ground Motion Generation Method

Near fault ground motion is a kind of ground motion with great damage, which is difficult to simulate. The amplitude and frequency functions of near fault ground motion vary greatly with time. When a single ground motion is used for simulation, large errors can easily occur. A method for generating near fault ground motion based on a set of actual ground motions is presented. In order to evaluate these artificially near fault ground motions, artificial ground motions and corresponding natural ground motions are input into the finite element analysis program to obtain the response of the structure. It is found that these artificial near fault ground motions are good simulations of these actual ground motions. The method of near fault ground motion can make up for the shortage of near fault ground motion.•Near Fault ground motions collected from the same earthquake with similar site characteristics are taken as sample group. Each ground motion in the sample group is considered as a record of random event.•Because this method is based on multi ground motion samples, this method can take account of the variability of each ground motion sample on the premise of considering the earthquake characteristics, then the generated ground motion are more close to the real one.•The wavelet technique and random vibration theory are applied to the analysis and simulation of ground motion. Because of the application of these two technologies, the simulation efficiency and accuracy of this method are improved.

A New Tuned Mass Damper Design Method based on Transfer Functions

Tuned mass damper (TMD) is an effective passive device in reducing harmful vibrations as long as they are designed correctly. An optimal TMD design method is proposed based on transfer functions and also differential evolution (DE) algorithm is used. The method includes optimization of all parameters which are the mass, stiffness and damping coefficients of a TMD. By using random vibration theory, the mean-square of top floor absolute acceleration, top floor displacement and the sum of mean-squares of interstorey drifts have been chosen as objective functions to be minimized with respect to upper and lower limits of TMD parameters. In the classical design, the mass is usually chosen by the designer to find the optimum values of the stiffness and damping coefficient. In addition to optimizing stiffness and damping coefficients, in this study TMD mass quantity is also optimized by using DE algorithm to minimize objective functions for TMD design. After that, structure system with TMD is tested under six near fault and three far fault ground motions by evaluating responses of the structure. The results obtained here are compared with the methods available in the literature for verification. Numerical results show that the proposed method is effective for optimal TMD design.

Some significant improvements for interval process model and non-random vibration analysis method

Recently the authors proposed the interval process model for dynamic uncertainty quantification and based on this further developed a kind of non-probabilistic analysis method called ‘non-random vibration analysis method’ to deal with the important random vibration problems, in which the excitation and response are both given in the form of interval process rather than stochastic process. Since it has some attractive advantages such as easy to understand, convenient to use and small dependence on samples, the non-random vibration analysis method is expected to become an effective supplement to the traditional random vibration theory. In this paper, some significant improvements are made for the interval process model and the non-random vibration analysis method, making them not only more rigorous in theory but also more practical in engineering. Firstly, the definitions and relevant conceptions of interval process model are further standardized and improved, and in addition some new conceptions such as the interval process vector and the cross-covariance function matrix are complemented. Secondly, this paper proposes the important conceptions of limit and continuity of interval process, based on which the differential and integral of interval process are defined. Thirdly, the analytic formulation of dynamic response bounds is deduced for both of the linear single degree of freedom (SDOF) vibration system and the multiple degree of freedom (MDOF) vibration system, providing an important theoretical basis for non-random vibration analysis. Fourthly, this paper also gives the formulation and corresponding numerical methods of structural dynamic response bounds based on finite element method (FEM) for complex continuum problems, effectively enhancing the applicability of non-random vibration analysis in engineering. Finally, four numerical examples are investigated to demonstrate the effectiveness of the proposed method.

Closed-form stochastic response of linear building structures to spectrum-consistent seismic excitations

Design codes typically define the seismic action in terms of pseudo-acceleration response spectra, encouraging the use of the modal superposition method for the evaluation of the structural response. For linear structural systems, provided that the ground shaking has been appropriately modelled, the full characterization of the response processes of interest can be achieved with the application of random vibration theory. An analytical model for the power spectral density (PSD) functions consistent with seismic response spectra has been recently proposed. In this paper, taking advantage of this novel PSD model, closed-form approximate expressions of the spectral moments are derived and numerically validated for the seismic response of single- and multi-degree of freedom systems. For demonstration purposes, the proposed formulation is applied to the case of base isolated multi-story buildings, aimed at overcoming the difficulties associated with the non-classical nature of their damping. The paper shows how the proposed approach can be used for a computationally efficient evaluation of the probabilistic distribution of the structural response maxima.

Optimal Design and Distribution of Viscous Dampers for Shear Building Structures Under Seismic Excitations

Viscous dampers (VDs) are effective and widely used passive devices for the protection of civil structures, provided that appropriate design is carried out. For this purpose, an optimal design and optimum distribution of VDs method is presented for a shear building under the critical excitation by using random vibration theory in frequency domain. In the optimization, by using Differential Evolution (DE) algorithm, the and top floor displacement are evaluated as objective functions taking into consideration upper and lower limits of VDs damping coefficients, so that optimal damper placement and properties of shear building can be determined. In this design, VDs-shear building system is tested under three different ground motions being compared to some methods in the literature and uniformly distributed VDs placed at each story. It is shown that, the results of study are both compatible and very successful in reducing the response of structure under the different ground motions.

Energy harvesting from unimorph piezoelectric circular plates under random acoustic and base acceleration excitations

Abstract Piezoelectric materials have been extensively employed in small scale energy harvesting devices. Commonly, the ambient excitation in these energy harvesting systems is considered to be harmonic. While in practical cases, these excitations are random in nature. The objective of the current paper is to use random vibration theory to investigate energy harvesting from a clamped circular plate interconnected to a circular piezoelectric layer. The excitations are assumed to be a random uniform pressure applied to the plate resulted from an acoustic wave and a random base acceleration of the system. Lagrange equations are utilized to derive the coupled equations describing the dynamic behavior of the plate and the circuit voltage. The random inputs are assumed to be band-limited uncorrelated white noises. Using the random vibration theory, the statistical parameters describing the plate’s dynamic and the harvested voltage are derived in terms of random characteristics of the inputs. A parametric study is carried out to investigate the effect of different design parameters on the harvested power. Also, an optimization problem is solved for maximizing the power density in the system. Using the results of the optimization problem, the power density can be significantly improved up to 3.2 mW/m2. The quantitative and qualitative results obtained in this paper can be effectively used for the optimal design of other piezoelectric energy harvesters under stochastic excitations.

Stochastic reduced-order models for stable nonlinear ordinary differential equations

Two methods based on stochastic reduced-order models (SROM) are proposed to solve stochastic stable nonlinear ordinary differential equations. One general method available for the probabilistic characterization of the response of nonlinear systems subjected to random excitation is Monte Carlo (MC), wherein the response of the nonlinear system must be calculated for a large number of samples of the input, which can be very computationally demanding. Random vibration theory is also inadequate for calculating response statistics for both linear systems under non-Gaussian inputs and nonlinear systems subjected to any kind of excitation. The two methods proposed are based on SROM, i.e., stochastic models with a finite number of optimally selected samples. The first method uses a SROM model for the random input. The second method is based on a surrogate model for the response of the nonlinear system defined on a Voronoi tessellation of the input samples. The newly proposed methods are applied for stable nonlinear ordinary differential equations, with deterministic coefficients and stochastic input, that are used in engineering applications: single-degree-of-freedom Duffing and Bouc–Wen systems, and a two-degree-of-freedom nonlinear energy sink system. The numerical results suggest that SROMs are able to estimate statistics of the stochastic responses for these systems efficiently and accurately, results validated by the benchmark MC results.

Suppressing Random Response of a Regular Structure by an Inerter-Based Dynamic Vibration Absorber

AbstractThis paper concentrates on the random vibration suppression of a regular straight beam by using an inerter-based dynamic vibration absorber. For a wideband random point-driven straight beam with an inerter-based dynamic vibration absorber, the distribution of mean-square velocity response along the axis of the straight beam as well as the mean kinetic energy of the whole beam are first analytically derived through the classical linear random vibration theory. Two optimization objectives are established to determine the optimal design parameters: (1) minimizing the maximal mean-square velocity along the axis of the straight beam, which corresponds to the maximal mean kinetic energy density along the axis and (2) minimizing the mean kinetic energy of the whole beam. Numerical search gives the optimal location and the associated optimal parameters of the inerter-based dynamic vibration absorber. Numerical results for a simply supported straight beam illustrate the better performance of an inerter-based dynamic vibration absorber than a traditional dynamic vibration absorber. Parametric sensitivity studies for the robustness analysis of the beam response to deviations from the optimal parameters are conducted. The optimal location locates on the force-excited point, while the suboptimal location locates on its symmetry position. Furthermore, the optimal and suboptimal locations remain invariable regardless of the upper cutoff frequency of band-limited noise, which is fairly important to the location optimization of the inerter-based dynamic vibration absorber.

Non-random vibration analysis for general viscous damping systems

The authors recently developed a kind of non-probabilistic analysis method, named as ‘non-random vibration analysis’, to deal with the important random vibration problems, in which the excitation and response are both given in the form of interval process rather than stochastic process. Since it has some attractive advantages such as easy to understand, convenient to use and small dependence on samples, the non-random vibration analysis method is expected to be an effective supplement of the traditional random vibration theory. In this paper, we further extend the non-random vibration analysis into the general viscous damping system, and formulate a method to calculate the dynamic response bounds of a viscous damping vibration system under uncertain excitations. Firstly, the unit impulse response matrix of the system is obtained by using a complex mode superposition method. Secondly, an analytic formulation of the system dynamic response middle point and radius under uncertain excitations is derived based on the Duhamel's integral, and thus the upper and lower response bounds of the system can be obtained. Finally, two numerical examples are investigated to demonstrate the effectiveness of the proposed method.

A Study of Ground Motion Excitation Based on the Earthquake of September 8, 2017: Evidence that Normal Faults Influence the Stress Parameter

We performed a study of ground motion modeling based on multiple linear regression and random vibration theory using the earthquake sequence of September 8, 2017 (Mw = 8.2). Our results show that there is high attenuation in the region of Oaxaca–Chiapas and central Mexico; however, the peak amplitude excitations of normal fault earthquakes are higher than those of thrust fault earthquakes. The earthquake sequence involved a large number of events, but not all the seismic events were aftershocks because many earthquakes had different rupture styles. The thrust events follow a simple scaling relation with a constant stress parameter, but the normal fault events are difficult to represent with a constant self-similar model. Our best result consists of a $$Q_{0} = 240 \pm 032$$ with a frequency-dependent factor of $$0.63 \pm 0.05$$ . The excitation term is well determined with a stress parameter of 80 bars for the thrust faults, but for the normal faults, the stress parameter is at least 180 bars with a non-self-similar model. Our results show that it is necessary to include new ground motion prediction equations in seismic hazard analysis for “meganormal” faults, which have not previously been considered in subduction zones.

Optimal design of energy harvesting from vibration subject to stochastic colored Gaussian process

The problem treated in this paper is the optimal electromechanical setting of energy harvesters working with vibrations of random nature. We consider a simple system composed of two parts; one electrical circuit with a mechanical single-degree-of-freedom system, coupled to a piezoelectric element. The electric circuit is modeled as a simple one, composed of a resistance and a capacity only, while a linear viscoelastic model is used for the mechanical element. The only joint electro and mechanical element is the piezoelectric device, which is controlled by both the mechanical velocity and the electrical tension. This scheme is treated as dimensionless using the random vibration theory since the base excitation is considered as a stationary white noise Gaussian process. To consider the non-uniform frequency content, that characterizes many real vibration phenomena, the input is properly colored using simple linear filters. System response statistics is evaluated by covariance approach, producing both covariance matrix of system parameters and mean value of electric power under different filters and systems configurations. The optimal ratio between the periods of the mechanical and electric systems is thus defined maximizing the mean power value. This ratio is obtained numerically for some test cases in a graphic representation to evaluate the sensitivity response to selected parameters.

NGA-West2 Empirical Fourier and Duration Models to Generate Adjustable Response Spectra

Adjustment of median ground motion prediction equations (GMPEs) from one region to another region is one of the major challenges within the current practice of seismic hazard analysis. In our approach of generating response spectra, we derive two separate empirical models for a) Fourier amplitude spectrum (FAS) and b) duration of ground motion. To calculate response spectra, the two models are combined within the random vibration theory (RVT) framework. The models are calibrated on recordings obtained from shallow crustal earthquakes in active tectonic regions. We use a subset of NGA-West2 database with M3.2–7.9 earthquakes at distances 0–300 km. The NGA-West2 database expanded over a wide magnitude and distance range facilitates a better constraint over derived models. A frequency-dependent duration model is derived to obtain adjustable response spectral ordinates. Excellent comparison of our approach with other NGA-West2 models implies that it can also be used as a stand-alone model.

Dynamic model of a cylindrical roller on a rough surface, with applications to wind turbine gearbox planetary bearings

Wind turbines of larger power ratings have become increasingly prevalent in recent years, improving the viability of wind energy as a sustainable energy source. However, these large wind turbines have been subjected to higher rates of failure of the wind turbine gearbox, resulting in larger downtime of operation and an increase in cost due to repairs. These failures most frequently initiate in the gearbox’s bearings, especially in the planetary bearings of the planetary stage and high-speed bearings. Currently, most of the research on the detection of planetary bearing faults only addresses the case of localised faults in the outer bearing race, while fewer research considering the detection of distributed bearing faults. The research that does consider distributed bearing faults relies on techniques – such as machine learning for the identification of faulty bearings – that do not account much for the underlying physics of the bearing. In this paper, a model is developed to simulate and analyse the dynamic interaction of a planetary bearing in the presence of surface roughness, which can be used to represent a distributed fault. The model presented uses random vibration theory for simulating the response of the planet bearing induced by distributed faults. The input of the model considers statistical expressions of the roughness geometry using multiple parallel tracks. Numerical simulation of the random vibration of the model is performed using 16 tracks, and the power spectral density of the radial deflection of the roller and the roller–race contact force is determined. The results of the simulation with the multi-track model show that a single-track model significantly overestimates the power spectral densities, and also suggests the stiffness of the bearing race is too high to have an effect on the roller dynamics for a planet bearing.

Evaluation of the site amplification factors estimated by equivalent linear site response analysis using time series and random vibration theory based approaches

The objective of this study is to estimate and compare the site amplification factors (AFs) using two different one dimensional (1-D) equivalent-linear (EQL) site response analysis approaches: the time series (TS) approach and the random vibration theory (RVT) based method. For this purpose, random soil profiles combined with different soil types, EQL soil properties, and unit weights are tested at several input ground motion levels. Analysis results showed that the AFs estimated by the TS-approach are systematically higher than the AFs estimated by the RVT-based method in the short period range (T < 0.5 s), especially when the bedrock peak ground acceleration is higher than 0.2 g. The relative difference between the AFs estimated in both methods is most prominent in granular soils: differences reach up to 35–40% compared to 10–15% for clays. On the other hand, the AFs calculated in this study are in good agreement with the empirical AF models utilized in recent ground motion models, indicating that the RVT-based AF models may be preferred in the future to cover a larger range of scenarios than the empirical datasets.

Seismic duration effect on damping reduction factor using random vibration theory

Damping Reduction Factor plays a key role in scientific literature and Technical Codes, but till now existing formulations present differences and inconsistences probably because obtained by integration of real recorded events, thus sensible to specific used data. This paper investigates the relation between damping reduction factor and earthquake duration by means of random vibration theory. A stochastic process, that is non-stationary and filtered, is used to model a seismic event. The modulation function is suitably chosen to describe earthquakes characterized by different durations. The stochastic process peak theory allows to calculate damping reduction factor after the definition of the probabilistic response of a simple linear visco-elastic oscillator. The variability with seismic duration for different soil conditions and damping ratios is investigated. The study points out that damping reduction factor is more sensitive to seismic duration in the range of high period and on rigid soil with respect to other conditions. The results show that, if damping ratio or effective duration values are increased, the damping reduction factor value diminishes.

Use of RVT for Computation of In-Structure Response Spectra and Peak Responses and Comparison to Time History and Response Spectrum Analysis

The random vibration theory offers a framework for the conversion of response spectra into power spectral densities (PSDs) and vice versa. The PSD is a mathematically more suitable quantity for structural dynamics analysis and can be straightforwardly used to compute structural response in the frequency domain. This allows for the computation of in-structure floor response spectra and peak responses by conducting only one structural analysis. In particular, there is no need to select or generate spectrum-compatible time histories to conduct the analysis. Peak response quantities and confidence intervals can be computed without any further simplifications such as currently used in the response spectrum method, where modal combination rules have to be derived. In contrast to many former studies, the Arias intensity-based definition of strong-motion duration is adopted here. This paper shows that, if the same definitions of strong-motion duration and modeling assumptions are used for time history and RVT computations, then the same result can be expected. This is illustrated by application to a simplified model of a reactor building.