Shear Building Research Articles

Proportional distribution pattern and modal principle of tuned viscous mass dampers for multi-degree-of-freedom structures

A tuned viscous mass damper (TVMD) is an inerter-based damper that has been adopted in several high-rise buildings in earthquake-prone Japan recently to protect the structures from seismic-induced vibrations. However, as a TVMD contains a mass element, it adds degrees of freedom to the controlled structure, making the system’s eigenmodes complex. Inspired by the Rayleigh damping matrix, this study proposes a mass-proportionally distributed TVMD (MPD-TVMD) system, which enhances the efficiency of TVMD utilization compared to the previously proposed stiffness-proportionally distributed TVMD (SPD-TVMD) system. It is theoretically proven in this study that when the arrangements of the TVMDs are proportional to the stiffness or mass distribution of the primary structure, the equations of motion of a TVMD controlled shear building can be decomposed into the eigenmodes of the uncontrolled structure. This modal principle allows structural engineers to estimate the response of the controlled structure by understanding the modal characteristics of the uncontrolled system, which is highly beneficial for preliminary design decisions and the retrofitting of existing structures. Furthermore, given that the damping properties at each story level are typically adjusted by the number of dampers installed, it is often unrealistic in practical design to maintain strict proportionality in damper distribution. Through numerical examples, this study demonstrates that TVMD systems with realistic distribution patterns that are slightly different from a strictly proportional distribution can still perform as effectively as strictly proportional systems, providing a comprehensive design basis for the application of inerter-based damper in practical engineering.

Vibration control of two portal frames type shear buildings through self-synchronous dynamics of two non-ideal sources indirectly coupled

The present paper proposes a new device where it is observed the self-synchronization between two unbalanced DC motors with a limited power supply when they are indirectly coupled. The vibrating system studied consists of two portal frames type shear buildings coupled, carrying each an unbalanced DC motor. The engines are supposed to rotate in the same direction and act on each as external excitation. The dynamics investigations are done with analytical and numerical methods to achieve this purpose. The synchronous solutions are derived and their stability conditions are also explored using the averaging method. The results of this analytical investigation are confirmed later by numerical simulations. The effects of some physical parameters on the self-synchronization of DC motors are presented. The impact of the nonlinear coupling between the floors on the DC motors dynamics is also explored. The Sommerfeld effect appearing in the system is reduced by taking into account the damping coming from the environment and the coupling between the portal frame. It is observed that in-phase synchronization of the non-ideal sources assures a low amplitude of vibration in the different floors compared to opposite-phase synchronization.

A Comparative Study of Single-Chain and Multi-Chain MCMC Algorithms for Bayesian Model Updating-Based Structural Damage Detection

Bayesian model updating has received considerable attention and has been extensively used in structural damage detection. It provides a rigorous statistical framework for realizing structural system identification and characterizing uncertainties associated with modeling and measurements. The Markov Chain Monte Carlo (MCMC) is a promising tool for inferring the posterior distribution of model parameters to avoid the intractable evaluation of multi-dimensional integration. However, the efficacy of most MCMC techniques suffers from the curse of parameter dimension, which restricts the application of Bayesian model updating to the damage detection of large-scale systems. In addition, there are several MCMC techniques that require users to properly choose application-specific models, based on the understanding of algorithm mechanisms and limitations. As seen in the literature, there is a lack of comprehensive work that investigates the performances of various MCMC algorithms in their application of structural damage detection. In this study, the Differential Evolutionary Adaptive Metropolis (DREAM), a multi-chain MCMC, is explored and adapted to Bayesian model updating. This paper illustrates how DREAM is used for model updating with many uncertainty parameters (i.e., 40 parameters). Furthermore, the study provides a tutorial to users who may be less experienced with Bayesian model updating and MCMC. Two advanced single-chain MCMC algorithms, namely, the Delayed Rejection Adaptive Metropolis (DRAM) and Transitional Markov Chain Monte Carlo (TMCMC), and DREAM are elaborately introduced to allow practitioners to understand better the concepts and practical implementations. Their performances in model updating and damage detection are compared through three different engineering applications with increased complexity, e.g., a forty-story shear building, a two-span continuous steel beam, and a large-scale steel pedestrian bridge.

Exploring the Structural Response of High-Rise Buildings using Fuzzy Logic and Genetic Algorithms

Exploring the dynamic behaviour of high-rise buildings presented a critical challenge. In this study, Fuzzy Logic and Genetic Algorithm (FLGA) were employed to comprehensively investigate the structural response of high-rise buildings (HRBs) for adaptive vibration control systems employing semiactive damper devices. The study integrated a fuzzy logic inference system with a Genetic Algorithm (GA) to create FLGA, which was then employed in a 3-story shear building under real earthquake conditions and multi-dimensional wind loads. The analytical model for a semi-active tuned mass damper (STMD) was examined and evaluated before integrating it into the proposed FLGA system. The results indicated that FLGA reduced story acceleration by an average of 20% and story drift by approximately 50% while efficiently controlling the structure within allowable movement limits and resident comfort criteria under multi-directional wind loads. The suggested optimal controller mitigated both displacement and acceleration responses by 15 to 20% without amplification. The study demonstrated FLGA's effectiveness as a model-free control strategy for managing structural response uncertainties.

Control of a building system with sudden nonlinearity using the sliding mode technique

In civil engineering application such as the seismic resistant structural design, a building system often enters into its nonlinear range to minimize the force transfer to the superstructure. An ideal control design should enhance the usefulness of the nonlinear behavior by diminishing its adverse effects at a minimal control input. This study developed a unified feedback mechanism based on the sliding mode control approach that fostered the multiple selected modal control of a structural system undergoing linear and nonlinear stages of hysteretic deformations. The ideal sliding surface for controlling a specific mode was designed based on the interaction characteristics between two modes at two different stages of deformation. An ′ S′ type functional form of the control force was developed on either side of the sliding surface based on the velocity feedback of the system by ensuring reachability to the ideal sliding surface. To demonstrate the effectiveness of the adopted control, a five-story shear building with uniform mass and initial stiffness distribution with concentrated nonlinearity at the base column was considered under 10 earthquake excitations. The system was studied under the proposed control along with the uncontrolled system. In addition to the adopted control, a popular and existing LQG control technique was also implemented based on the equivalent linearized structural system for comparison. The adopted control provided the desired performance of the structural system while keeping the effectiveness of the system’s nonlinearity under earthquake excitations. Considering the system’s responses and the input control force, the efficiency of the adopted mechanism outperformed the LQG control, which was found to amplify the unfavorable effects of the system’s nonlinearity.

Advanced seismic control strategies for smart base isolation buildings utilizing active tendon and MR dampers

This paper investigates the seismic behaviour of a five-storey shear building that incorporates a base isolation system. Initially, the study considers passive base isolation and employs a multi-objective archived-based whale optimization algorithm called MAWOA to optimize the parameters of base isolation. Subsequently, a novel model is proposed, which incorporates an interval type-2 Takagi-Sugeno fuzzy logic controller (IT2TSFLC) utilizing clustering techniques. The building includes Magneto-rheological (MR) dampers installed at the base isolation level and two active tendons positioned on the first and second storeys of the structure. The semi-active control force of the base isolation with MR dampers is determined by the fuzzy system, while the active control force for the active tendons is computed using the linear quadratic regulator (LQR) algorithm, enabling control force provision during seismic events. The primary objective of this model is to enhance the seismic control of the building. Therefore, it is classified as a proposed model. The structural system is subjected to seismic analyses, considering three different structural configurations: uncontrolled, equipped with passive base isolation, and equipped with semi-active base isolation combining MR dampers and active tendons. The findings of the research demonstrate that by considering the optimization of parameters of the passive base isolation based on the white noise scenario and using these parameters as design parameters, during seismic analysis of the structure in some earthquakes, increased structural responses were observed when compared to uncontrolled structure, highlighting a potential risk. Nevertheless, the proposed system effectively addresses this drawback of passive control systems by markedly reducing structural responses, as compared to both passive base isolation and uncontrolled structure. These results suggest that the proposed system is an effective solution for mitigating seismic risks in structural seismic control.

Application of the artificial neural network and enhanced particle swarm optimization to model updating of structures

AbstractAn efficient and accurate two-stage approach, based on the artificial neural network (ANN) and an enhanced particle swarm optimization (EPSO) approach for model updating of structures using incomplete measurements, is proposed in this study. The first stage, preliminary model updating, employs the ANN to quickly learn the mapping relationship between the natural frequencies and stiffness of the structure using a few training, validation, and testing instances. The inputs and outputs of the ANN are the natural frequencies and stiffness of the structure, respectively. The ANN’s training, validation, and testing instances are extracted through Latin hypercube sampling. The ANN-predicted stiffness provides an excellent basis for determining and reducing the search space of the optimal stiffness in the second stage. The second stage, detailed model updating, searches for the optimal stiffness of the structure by using the EPSO approach. The EPSO approach improves particle swarm optimization (PSO) by employing an elite crossover strategy to avoid trapping in the local optimum and premature convergence. The feasibility and effectiveness of the proposed two-stage approach for stiffness updating of shear building structures using incomplete measurements are demonstrated by numerical and experimental examples. The results present that the proposed two-stage approach improves the computational efficiency and solution quality of the GA (Genetic Algorithm) and PSO for stiffness updating of shear building structures.

Research on Structural Damage Identification of Shear Buildings Based on Modal Curvature Change

In the past few decades, various techniques for structural damage identification based on dynamic properties have been widely studied, among which modal curvature is the research hotspot, due to its high sensitivity to local damage. To realize story-level structural damage identification of shear buildings, this paper first derives the patterns of variation in modal curvature of the damaged structure. Then, a new damage index sensitive to local damage called the Modal Curvature Change modified with Softmax (MCCSM for short), is proposed to improve the accuracy of structural damage identification. Afterward, the validity of the damage index is verified by a shaking table test of a scaled RC frame structure model and its corresponding numerical model. The research results show that the damage location indicated by MCCSM corresponds well with the damage phenomenon in the shaking table test and numerical simulation; compared with using the theoretical mode shape, the MCCSM index can give comparable damage identification results using the mode shape obtained through operational modal analysis from vibration time-history data, which proves the feasibility in engineering practices; through numerical simulation, it is proved that the MCCSM index is applicable in a wide range of damage cases with different story position, number of damaged story, and damage extent.

Adaptive Real Time Efficient Control Algorithm for Buildings Under Wind and Earthquake Forces

The study aims to tackle two key issues: the dependency of optimal controllers on available training data and their limited capability in managing multi-component structural responses. Real Time Fuzzy Logic Efficient Control system (RTFLEC) was developed to address these challenges. This system integrates a fuzzy logic inference system with a multi-verse optimization algorithm and utilizes real time training for real-time optimal controller derivation. It incorporates decentralized systems and optimal controller memory concepts and employs a magnetorheological damper (MR) for adjustable vibration control. Two case studies were conducted to evaluate the RTFLEC system's effectiveness. The first study focused on a three-story shear building under seismic loads, while the second analyzed a 76-story building facing multi-directional wind loads. The results showed that RTFLEC system reduced structural drifts by 28.33% and 51.5% during near-field and far-field earthquakes, respectively. It also decreased structural acceleration by 27.66% and 53% during near-field and far-field earthquakes, respectively. For wind-induced structures, it reduced story displacement by 37.5%, 14.5%, and 12% against across-wind, along-wind, and rotational forces, respectively. RTFLEC system demonstrated robust performance against uncertainties in external excitations, providing exceptional structural responses. Its key advantage lies in real-time adaptability, requiring minimal training data and computational processing. These findings highlight its significant potential in enhancing the performance of adaptive vibration control systems under unpredictable real-world loads.

Phenomenological modeling of fiber‐reinforced elastomeric isolators at multiple lateral deformation levels

AbstractUnbonded fiber‐reinforced elastomeric isolators (FREIs) are a cost‐effective seismic isolation technology that uses lightweight fiber‐fabric reinforcement and forgoes the attachment plates connecting the isolators to the supports. These devices exhibit a complex nonlinear mechanical behavior under lateral deformation, which has typically been represented by uniaxial phenomenological models. In this paper, a new model, called Pivot Bouc–Wen model, is proposed to address the shortcomings of existing numerical models and obtain a better prediction of the response over the whole range of motion. The model has been formulated with the objective of providing (a) improved interpretability of the model parameters, (b) adequate energy dissipation prediction at multiple deformation levels, and (c) stable response at large deformations. The model combines a nonlinear elastic spring and a Bouc–Wen element with a modified pivot hysteresis rule to capture the lateral response of the isolators at different deformation amplitudes. Initial values for the model parameters are recommended based on existing analytical formulations of the quasi‐static lateral response of FREIs and data corresponding to 36 cyclic tests from 12 different experimental programs. The proposed and existing models are compared in their ability to predict the lateral cyclic test results from a previous experimental study. The models are further compared via response history analyses of idealized one, two, three and four‐story base‐isolated shear buildings subjected to 30 ground motions at different intensity levels. The results highlight the importance of capturing the hysteretic response of the isolators at multiple deformation levels and not only at the maximum expected displacement.

Anomaly detection in structural dynamic systems via nonlinearity occurrence analysis using video data

Dynamic loads in disaster events, such as earthquakes, often cause structural damages that severely affect serviceability and even reduce the safety of civil structures or render them unsafe. Most of these damage scenarios can cause the transition from a linear to nonlinear behavior in structural dynamics. Therefore, extracting such nonlinear behaviors is an effective strategy for structural anomaly detection. In recent years, the rapid development of media devices has led to the widespread use of video data, primarily because of the ease of acquisition and lack of contact measurement. This study developed a novel method for detecting anomalies due to structural nonlinearity from video data that could capture target dynamic behaviors via non-contact measurement. The method is based on optical flow, extended repulsive force networks, and morphological operations for extracting singularities due to nonlinear events in the estimated motion vector field in video data. Subsequently, shaking table tests on a three-story shear building model containing a novel controllable hinge bearing verified the effectiveness of the developed method. The results of the feature enhancement process demonstrate the accurate localization of anomalous regions and effective mitigation of unwanted interference using the proposed method. The proposed method facilitates the use of video-based technology to evaluate rapid damage conditions post-earthquake.

Frequency response function-based closed-form expression for multi-damage quantification and its application on shear buildings

Vibration-based damage identification techniques have gained popularity immensely in the field of structural health monitoring (SHM). Frequency response function (FRF)-based damage quantification is mainly performed based on iterative model updating techniques. Most of these methods are time-consuming and erroneous in higher modes. Spectral element method (SEM) can overcome some of these limitations and can determine FRF expression directly for a broad class of civil structures. However, a closed-form expression for multiple damage severity is not proposed to date. In this study, an FRF-based formulation for quantifying damage has been proposed with the help of SEM in order to bypass the mode shape extraction for damage identification. To determine the effectiveness of the proposed formulation both numerical and experimental studies have been performed. A numerical simulation has been performed on a 14-storey shear building. Damage has been introduced by reducing the storey stiffness at intermediate and adjacent storeys. It has been perceived that damage severity at any storey can be estimated with only 3 sensors, one at the top storey and two others at the adjacent floors of the desired storey. To verify the effectiveness of the proposed expression, an experimental study has been carried out on a 9-storey miniature model. It has been observed that the first resonating peak amplitudes of the responses recorded simultaneously at desired locations of intact and damaged structures are sufficient to estimate the damage intensity. Thus, the proposed formulation is efficient as an output measurement-based approach. The novelty of this study lies in the proposed closed-form FRF-based multi-damage severity expression involving minimum number of sensors and without any information on the input excitation. The results obtained from both numerical and experimental study showcase the robustness, simplicity and applicability of the proposed formulation.

H hybrid control and MRD in a steel frame building subjected to excessive vibrations caused by the dynamic action of wind and earthquake

The dynamic loads from earthquakes and winds can destroy lives, cause collapse in civil structures, and interrupt basic services provided to the population. In this scenario, structural designs must be developed to decrease the damage induced by these actions. The objective of this work is to design a hybrid controller based on the H¥ optimization via state feedback and the magneto-rheological damper (MRD) to mitigate the excessive vibrations of a three-story steel frame building, represented through the shear building model, subjected to the simultaneous dynamic action of wind and earthquake. All research is based on computational simulation, experimental research and results will not be addressed. In the numerical analysis, digital computer and MATLAB® software are used, and implemented codes generate the expected results based on the mathematical modeling. With the application of the H¥ control technique via state feedback, the displacements were reduced by 77%. With MRD this reduction was 79%. With the hybrid controller, this reduction was 100%. Thus, the verifications in relation to maximum displacements were met for NBR 15421:2006, NBR 8800:2008 and NBR 6118:2014. From the results, it is concluded that the hybrid controller proved to be more efficient and achieved the proposed objective. The exogenous inputs had zero influence on the behavior of the system output.

Modal Parameter Identification of a Structure Under Earthquake via a Wavelet-Based Subspace Approach

This paper introduces a novel wavelet-based methodology for identifying the modal parameters of a structure in the aftermath of an earthquake. Our proposed approach seamlessly combines a subspace method with a stationary wavelet packet transform. By relocating the subspace method into the wavelet domain and introducing a weighting function, complemented by a moving window technique, the efficiency of our approach is significantly augmented. This enhancement ensures the precise identification of the time-varying modal parameters of a structure. The capacity of the stationary wavelet packet transform for rich signal decomposition and exceptional time-frequency localization is harnessed in our approach. Different subspaces within the stationary wavelet packet transform encapsulate signals with distinct frequency sub-bands, leveraging the fine filtering property to not only discern modes with pronounced modal interference, but also identify numerous modes from the responses of a limited number of measured degrees of freedom. To validate our methodology, we processed numerically simulated responses of both time-invariant and time-varying six-floor shear buildings, accounting for noise and incomplete measurements. Additionally, our approach was applied to the seismic responses of a cable-stayed bridge and the nonlinear responses of a five-story steel frame during a shaking table test. The identified modal parameters were meticulously compared with published results, underscoring the applicability and reliability of our approach for processing real measured data.

Sequential simulated annealing for life-cycle optimization of nonlinear stochastic systems via arbitrary polynomial chaos expansion

Quantifying the uncertainties of engineering systems modeled as nonlinear oscillators subject to random excitation is a theoretically complex and computationally demanding task. Consequently, finding an optimal design of such systems considering their life-cycle performance is prohibitive with Monte Carlo simulation methods. In this paper, an efficient performance-based design optimization approach is developed for finding the optimal parameters of engineering systems modeled as nonlinear/hysteretic oscillators subject to stationary and non-stationary excitation. This novel approach utilizes an arbitrary polynomial chaos expansion to estimate the expected cost of failure without performing a computationally expensive integration over the hazard levels. Moreover, we introduce a novel sequential heuristic optimization scheme based on simulated annealing to minimize the total expected cost over the structure life-cycle. Three examples are included in the paper to assess the developed optimization scheme. First, we use the developed framework to optimize a linear single-degree-of-freedom oscillator subject to broadband excitation. Second, a multi-degree-of-freedom oscillator with cubic nonlinearity in damping and stiffness, subject to stationary broadband excitation, is optimized to show the influence of the problem dimensionality in the optimization process. In the last example, a multi-story reinforced concrete shear building modeled as a multi-degree-of-freedom Bouc-Wen oscillator with stiffness and strength degradation and subject to multi-hazards modeled as stationary (wind excitation) and non-stationary (earthquake) stochastic processes, is optimized.

In-Structure Response Spectra of Seismically Isolated Shear Buildings Considering Eccentricity Effect

In-Structure Response Spectra of Seismically Isolated Shear Buildings Considering Eccentricity Effect

Deep learning‐based response spectrum analysis method for building structures

AbstractThe response spectrum method has gained widespread acceptance in practical applications owing to its favorable compromise between accuracy and practical efficiency. The method predicts the peak responses of multi‐degree‐of‐freedom (MDOF) systems by combining modal responses. The Square Root of the Sum of Squares (SRSS) and Complete Quadratic Combination (CQC) rules are commonly used for modal combinations. However, it has been widely known that these rules have limitations in accurately predicting responses influenced by higher modes and cross‐modal correlations. To improve the accuracy of the response spectrum analysis method for building structures, this paper proposes a Deep learning‐based modal Combination (DC) rule by introducing modal contribution coefficients predicted by a deep neural network (DNN) model. The DC rule enhances prediction accuracy by considering the characteristics of ground motion and the dynamic properties of a structural system. The DC rule provides more accurate predictions than the conventional rules, particularly for irregular response spectra and responses affected by higher modes. The efficiency and applicability of the DC rule are demonstrated by numerical investigations of multistory shear buildings and steel frame structures with regular and irregular shapes. The source codes, data, and trained models are available for download at https://github.com/tyongkim/ERD2.

Time-varying modal identification of structures under seismic excitations using a novel time-frequency method

Civil engineering structures are susceptible to damage when subjected to seismic excitations, resulting in presenting nonlinear features and exhibiting time-varying dynamic behavior of structural modal properties. The instantaneous frequency (IF) of structures is thus an essential indicator to evaluate structural damage state. Time-varying filtering empirical mode decomposition (TVF-EMD) holds promise for non-stationary signals decomposition and IF identification due to its robustness against noise interference. However, the application of TVF-EMD is limited by the requirement for manually presetting the decomposition parameters (i.e., B-spline order and bandwidth threshold). In this study, a novel IF identification method based on an improved TVF-EMD algorithm is proposed. In this method, TVF-EMD is firstly improved to accurately decompose the measured responses into mono-component modal responses, which adopts orthogonal coefficient and energy kurtosis as the objective function of salp swarm algorithm (SSA) to adaptively determine the optimal decomposition parameters. Then, the resulting mono-component modal responses are analyzed by local maximum synchrosequencing transform (LMSST) to obtain structural IF under seismic excitations. Examples of non-stationary signals of sinusoidal function, Duffing system, and simulated two-story shear building model under El Centro seismic excitations are used to illustrate the advantages of the proposed method, which proves to be successful in signal decomposition and IF identification. Furthermore, the proposed method is applied to a shaking table test and a real building-Van Nuys hotel under seismic excitations to identify their IF and evaluate structural damage states. The results reveal that the proposed method is efficient for structural IF identification, and can be utilized to qualitatively evaluate the damage severity of structures subjected to strong seismic excitations.

Bayesian Spectral Decomposition for Efficient Modal Identification Using Ambient Vibration

Modal parameter identification via ambient vibration is popular but faces challenges from uncertainties due to unknown inputs and low signal‐to‐noise ratio. Bayesian methods are gaining increasing attention for operational modal identification due to their ability to quantify uncertainties. However, improvements in computational efficiency are needed, particularly when addressing numerous modes and degrees of freedom. To address this challenge, this study proposes an innovative approach, termed the “Bayesian spectral decomposition” method (BSD), employing the decompose‐and‐conquer strategy. This novel method, operating within the frequency domain, identifies each mode individually by exploiting their inherent separated modal characteristics. For each mode, the response spectrum matrix undergoes an eigenvalue decomposition, yielding crucial eigenvalues (incorporating frequency and damping information) and eigenvectors (containing mode shape information). Subsequently, statistical properties of the eigenvalues and eigenvectors are utilized to establish likelihood functions for Bayesian parameter identification. By combining prior information, the posterior probability distribution functions of modal parameters are derived. The optimal solution is then obtained by resolving the maximum posterior probability distribution function problem. To further quantify the uncertainty of modal parameters, Gaussian distributions are employed to approximate the posterior probability distribution functions. The adoption of the decomposition approach circumvents the joint identification of all modal parameters, substantially reducing the parameter dimensions for optimization. Consequently, this strategy leads to decreased computational complexity and significantly improved computational stability. The effectiveness of the BSD is confirmed through simulated data generated from an 8‐story shear building as well as measured data collected from both an experimental shear frame and the Canton Tower. The results demonstrate that the proposed method achieves high accuracy in identifying modal parameters, greatly improves computational efficiency, and effectively quantifies the uncertainties in modal parameters.

Numerical-computational model for modal and transient analysis of shear building using the Scilab program

A vibração está em toda parte em nossa vida cotidiana. Nos dias atuais, com o avanço de métodos computacionais e da ciência dos materiais, estruturas civis estão cada vez mais flexíveis e esbeltas. Uma elevada flexibilidade estrutural leva a grandes amplitudes de vibração que podem ser transmitidas às pessoas nos edifícios, causando desconforto, perda de eficiência no trabalho devido ao cansaço e até graves alterações de saúde. Nesse contexto, uma abordagem numérico-computacional é apresentada para a análise modal e transiente de estruturas do tipo shear building utilizando o programa livre Scilab. O efeito do amortecimento na estrutura é descrito pelo modelo de amortecimento de Rayleigh. Um estudo com o sistema de controle passivo AMS instalado no topo da estrutura é apresentado. O AMS é um sistema massa – mola que serve para reduzir a amplitude de vibração do sistema estrutural principal. A abordagem numérica apresentada vem a auxiliar os estudantes de física e das engenharias no que concerne à análise de vibração de sistemas estruturais. A metodologia descrita para análise dinâmica é relativamente de fácil implementação computacional e os resultados numéricos tiveram boa precisão e ficaram de acordo com os resultados disponíveis na literatura.