Abstract
This study evaluates the performance of passively controlled steel frame building under dynamic loads using time series analysis. A novel application is utilized for the time and frequency domains evaluation to analyze the behavior of controlling systems. In addition, the autoregressive moving average (ARMA) neural networks are employed to identify the performance of the controller system. Three passive vibration control devices are utilized in this study, namely, tuned mass damper (TMD), tuned liquid damper (TLD), and tuned liquid column damper (TLCD). The results show that the TMD control system is a more reliable controller than TLD and TLCD systems in terms of vibration mitigation. The probabilistic evaluation and identification model showed that the probability analysis and ARMA neural network model are suitable to evaluate and predict the response of coupled building-controller systems.
1. Introduction
In order to maintain structures safety and service, the application of controllers has been an important option for skyscrapers and important structure all over the world. Controlling systems are designed to protect buildings; bridges; and industrial plants from severe vibrations in seismically active regions [1]. There are different controlling systems, also known as earthquake protective systems, such as simple passive devices and fully active and hybrid systems [1–6].
Passive control systems are being progressively used in vibration control of structures subjected to dynamic loads [7]. The most commonly used passive control systems are tuned mass damper (TMD), tuned liquid damper (TLD), and tuned liquid column damper (TLCD). Tuned liquid column damper, proposed by Sakai et al. [8], combines the liquid mass and orifice damping effect to minimize the vibration. The TLD also uses similar concept to reduce the structural vibration. The advantages and disadvantages of using each passive control system for vibration mitigation are presented in [1, 3, 7, 9]. Chen et al. [10] presented a passive control system for a high-rise building in Beijing. They found that the passive control is an economic system that can be used to decrease the displacement response of structures under seismic or dynamic loads. In addition, Kasai et al. [11] concluded that the most common passive control systems are used in Japan and they found that the performance of passive control dampers is higher with earthquake effects. Bigdeli and Kim [3] and Bigdeli [9] have applied different cases of simulation and experimental studies for different cases of passive control systems. They applied the TMD, TLD, and TLCD controller systems and found that the TMD performance system is better to reduce the vibration of structures in frequency domain. Moreover, many papers are published on the performance investigation of these controlling systems in time and frequency domains [3, 9–15]. This study applies a novel analysis in time and frequency domains based on probabilistic and identification theories.
The time series analysis was used widely to evaluate the performance and identify the damage of structures in real monitoring measurements and experimental studies [15–20]. Kaloop and Hu [17] and Kaloop and Kim [18] have used the statistical analysis to evaluate the performance of bridges under moving loads and damage effects. The traditional probabilistic evaluation was used to assess and evaluate the seismic effects and risks due to earthquakes [21–23]. It, also, can be used to evaluate structures lifespan [24]. In addition, the probabilistic evaluation method was used as an additional procedure to assess the behavior of structures under seismic effects; moreover, the behavior and damage level of structures were expressed using the probability of exceeding the limit state by site-specific earthquakes during design service life of structures [21]. Kralik [25] utilized the probability analysis to estimate the containment failure of concrete structures. Matos et al. [26] evaluated the behavior of simple beam experimentally and numerically using a linear and nonlinear probability analyses and concluded that the nonlinear probability analysis can be used to estimate the behavior of structures.
Performance models identification was used to predict the behavior of structures under dynamic loads. The predicting models were designed based on input and output or output only of the input and response performance of structures [27, 28]. This study is focused on input-output models based on experimental studies. The artificial neural networks (ANN) are one of the most accurate and widely used predicting models that have fruitful applications in many engineering problems [28, 29]. The basics of ANN are presented in [28, 29]; in addition, the application of ANN to predict the performance and damage detection of structures showed that the ANN is a good tool that can be used in this area [18, 19]. Bigdeli and Kim [12] used ANN to predict the behavior of irregular building-controller system under dynamic loads and found that the ANN can be used to identify the performance of building structures. In addition, Li et al. [30] found that the ANN is feasible to model the linear and nonlinear performance of structures under seismic loads. Autoregressive models gave the best performance results when coupled with ANN [28]. Khashei and Bijari [27] introduced an integrated ANN with autoregressive integrated moving average (ARIMA) to identify the time series problems and found that it is more accurate than ANN. However, in this study, the autoregressive and autoregressive moving average are utilized and evaluated with ANN to develop a prediction model that can be used to identify the behavior of structures controllers under dynamic loads.
The main objective of this study is to investigate the probability analysis and prediction models to evaluate the behavior of an experimental study for a three-story building attached with TMD, TLD, and TLCD controlling systems. Also, a novel application for the probability in time and frequency domains is evaluated and discussed for the controlling systems. Furthermore, an autoregressive moving average-neural network model identification response is designed to predict and investigate the behavior of structures under dynamic loads.
2. Materials and Methods
2.1. Response Theory and Experimental Setup
The mechanical configuration of a controlling system consists of a spring and a mass attached on the top of a multistory structure. This system is used as a vibration absorber in order to mitigate structural vibrations. The dominant general equation of motion used for the coupled structure and the absorber for a three-dimensional (3D) structure is given as follows [3, 31]where , , and denote mass, damping, and stiffness matrices of the structure, respectively. The , , and represent the acceleration, velocity, and displacement response of the structure, respectively. The excitation signal is indicated by and which are defined as the displacement and the ground influence vectors, respectively.
A scaled structural model of a 3D 3-story moment-resisting steel frame building structure is selected to conduct the experiments; see Figure 1(a). This study evaluates a steel structure system, widely used in high-risk areas [6, 9]. The characteristics of the structure are as follows: the total height is 930 mm and the natural frequency is 2.7 Hz. The diameter of the cross-section area of every column is 5 mm and the story height equals 310 mm. All degrees of freedom (DOFs) are restricted at the base level. Each floor configuration includes a rigid steel square plate of 300 mm × 300 mm which weights 3,436 gr with three DOFs for translations in - and -directions as well as a rotation around a vertical line passing through the center of mass at every floor. The total number of DOFs (42) after application of the boundary conditions (i.e., rigid diaphragm, Guyan reduction of vertical DOFs, and rotational DOFs around - and -axes) equals 9. The input signal generated artificially through shaking table [29] generates an acceleration signal having a waveform in which parameters such as strong motion duration, ramp times, and number of strong motion peaks are variables. The input and output signals generation and measurements are presented in [3, 31–33].
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Three passive controllers included in this study, TLD, TLCD, and TMD systems, were used as shown in Figures 1(b), 1(c), and 1(d), respectively. TLCD and TLD devices are fabricated by using a light fiberglass material, while the TMD system is fabricated by using steel material. The total weight of each device is less than 3% of the whole steel structure. The controllers’ weight was kept constant during the conduction of the experiments for all systems. The water inside the containers was in a direct contact with the atmosphere and it also could freely move inside the container. In order to compare the results properly, the following points are considered: firstly, all the three devices have the same primary mass, which means that TLCD and TLD and TMD are identical in weight. Secondly, the same amount of solid or water mass was added to the system including water mass to TLCD and TLD and solid mass to TMD at each step of the tests. Moreover, the total mass used in this study is 1.80 Kg. Lastly, the dimensions of the devices are designed in a way to be identical in order to ensure the minimum effects on results.
Figure 2 represents the experimental setup for the controlling system. As shown in Figure 2, a regular structure system is used and the accelerometer sensor is used to measure the response of the three-floor roofs. In this study, the maximum response for the third floor is evaluated.
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2.2. Statistical Analysis
The statistical analysis for the controlling system is applied in two ways; conventional (Mean, Skewness, and Kurtosis) and advanced (probability analysis). The statistical moments are used previously to assess the behavior of structures and evaluate structures’ monitoring systems [17, 18], and also the probability analysis is used to analyze the monitoring system of structures, wind energy, and aerospace evaluation [17, 20, 34, 35]. In this study, the two statistical methods are used to evaluate and assess the behavior of the controlling systems. It is noteworthy that no studies have implemented the probability analysis in control system analysis in time and frequency domain.
In time domain, the Weibull distribution (WD) analysis is widely used to evaluate the behavior of structures [20]. The WD is a probability distribution used to measure the reliability of structures due to its versatility and relative simplicity [36–38], while the reliability of structures can be evaluated by analyzing the interaction between two probability distributions [39]. In WD analysis, the shape () and scale () parameters are used to extract the Weibull probability density function (PDF) and cumulative distribution function (CDF) [20, 34]. These parameters can determine the optimum performance of the behavior of structure-controller system [20, 40]:where is the time series of the structure behavior measurements (acceleration or displacement), is the dimensionless shape parameter, and is the scale parameter in the measurement unites. Several methods are used to estimate the WD parameters [40]. In this study, the least square method is applied as it is more efficient than other methods [40]. The estimation of the WD parameters in (2) refers to probability changes of the response of structures [36]. For instance, the value measured the failure of the structure response [20]. The smallest value of parameters refers to more stability and long life [20, 36, 41]. In addition, the parameters changes refer to the changes of the peak of PDF curve distribution; it means that the structure behavior should be changed [20, 36–38].
In frequency domain, Trauth [42] presented the Lomb-Scargle (LS) periodogram considered a modify power spectrum with an exponential theoretical probability distribution with mean unit, and it is used to estimate the power spectrum density of the measurements signals [39, 42]. In this study, the LS normalized periodogram (, ) is calculated using the time series acceleration measurements for each control system based on least square fit of harmonic function of acceleration data [42]. Therefore, the probability of extracted can be found in between some positive quantities and being [42]. With assuming the independent frequency contents, the probability of none of them having a larger value than is [17, 42]. To compute the false-alarm probability of the null hypothesis, the insignificant probability for the peak of the periodogram can be extracted by [42]
Herein, to estimate the significant frequency, two steps are processed; first: the LS periodogram is used to normalize the signal frequency; then the significance level of any peak in the power spectrum can be calculated [42]. The probability calculation in (3) indicates the false-alarm probability of the null hypothesis. Therefore, a low probability indicates a high significant power spectrum peak. However, the false-alarm probability peaks are referred to high significance of the structures’ dynamic behavior [17].
2.3. Model Identification Approach
In this study, the proposed model is built based on feedforward neural network model with one hidden layer to provide the third-floor behavior, as presented in Figure 3. The parametric identification models are used to detect the behavior of structures [43–45]. The autoregressive (AR) and autoregressive moving average (ARMA) are among the common and simplified parametric models [28, 43]. In this study, the two methods are evaluated with feedforward neural network solution. The components of the proposed model are described and explained with details in [43]. A multi-input single-output (MISO) three-layer model is designed for building behavior analysis (Figure 3). The input parameters of the model are the time series of the seismic signal (input shaking table load) and response of structure measurements. The hyperbolic tangent () and linear activation () functions are used for the hidden and output layers, respectively. This model is designed to detect the nonlinearity of the structure behavior [28, 44]. The ARMA neural network (ARMANN) process is a time series consisting of a random error component and a nonlinear combination of inputs values and past residuals [43]. The ARNN model is a special case of the ARMANN model. However, in general the ARMANN process can be summarized in the following equations [43].
The regression vector isand prediction iswhere is a vector containing the input dynamic load (), output behavior (), and model errors (), is a vector containing the weights, and is the function realized by NN. is the prediction vector, is the past output used for determining the prediction, is the past input, is the time delay, and is the past residuals.
The relationship between the output and inputs has the following mathematical representation [43]:where , andare representing the model parameters often called connection weights; refers to the input nodes of the model errors only; is the number of autoregressive (AR) input nodes; is the number of moving average (MA) parameters input nodes; and is the number of hidden nodes; and are the bias parameters; refers to the AR inputs parameters (; ).
The main purpose of this model is to determine a mapping from the selected input data to the set of possible weights. So that the network will produce predictions vector, which in some sense are close to the true output vector. The determination of the optimal weight values may be completed by using a trial and error process on an error function (see (7)), which is called loss function or mean square error () of the training process. The loss function is a criterion defined in the mathematical approach as below ( is the number of observations):
The presented strategy tries to minimize the loss function through the model design. By following so, the provided output becomes closer to the controller response measurements (optimum outputs). Eventually, the optimum outputs are going to be predicted after having the input data received through enough number of trial and error efforts. Considering the desired output as a control signal, the network will be known as a controller. In addition, to assess the performance of the model, the mean absolute error () is determined with to measure predicting performance of the proposed model in comparison to other predicting models.
3. Results and Discussions
3.1. Response Statistical Analysis
The time series analyses in time and frequency domains are presented in this section. Figure 4 illustrates the time series of seismic shaking table signals and acceleration responses for the uncontrolled system and also for the controlled systems by using TMD, TLD, and TLCD where the entire system was under seismic load effects.
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From Figure 4, the maximum seismic excitation occurred at 11.53 seconds, while the maximum acceleration response of the uncontrolled, TMD, TLD, and TLCD systems occurred at 12.71, 7.85, 10.40, and 10.95 seconds, respectively. The TMD structure response is delayed by 4.86 seconds, and it means that the TMD control system is more sensitive to the loads than TLD and TLCD systems. In addition, the peak displacement reductions are significant with 16.9, 15.0, and 16.0% for the TMD, TLD, and TLCD controlling systems, respectively. Table 1 reports the moments of statistical analyses for the acceleration and computed displacements of the three models. The mean, maximum, Skewness, and Kurtosis are often calculated in order to examine the raw time series data obtained for the excitations and responses of a structure. Herein, the statistical analysis is performed for the absolute measurements acceleration and calculations of displacements.
The acceleration absolute mean shows that the structure vibration is decreased by 22.7, 27.3, and 18.2% with TMD, TLD, and TLCD, respectively. The maximum acceleration and displacement response of the structure with the TLD controller are lower than the TMD and TLCD controllers. Based on the Kurtosis and Skewness results, the TLD probability distribution is better than TMD and TLCD. From Table 1, the statistical acceleration and displacement parameters for the TLD controlling system are more significant, but the TMD controlling system takes 7.85 seconds to control the structure response while the TLD takes 10.40 seconds.
The Weibull distribution of the structure’s displacement of the uncontrolled, TMD, TLD, and TLCD systems is presented in Figure 5. The Weibull probability distribution is applied in this case for the absolute displacement response of each case. Figure 5(a) presents the TMD controller linearized displacement measurements for the Weibull parameters. The probability exceeded for the displacement is presented in Figure 5(b), and the Weibull parameter slope is presented in Figure 5(c). Also, the Weibull parameters for the controlling systems are presented in Table 2.
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The calculation of the autocorrelation coefficients for the fitting lines of the TMD, TLD, and TLCD is found 0.99, 0.97, and 0.95, respectively; the TMD fitting is presented in Figure 5(a). It means that the hypothesis testing for the TMD controller response with WD is high goodness-of-fit. In addition, the probability exceeded of the displacement shows that the TLD displacement is lower than TMD and TLCD. Also, the CDF for the TLD and TMD are very close. In addition, the Weibull slope presentation shows that the TMD slope is lower compared to TLD and TLCD of the displacement measurement. Also, the TMD and TLD Weibull slopes are very close. This means that the TMD controlling system is more reliable than other systems [35]. The two control systems TMD and TLD behaviors are close in a time domain performance. The WD parameters in Table 2 show that the scale parameter () is very small and can be neglected, with close values for the three controlling systems. The shape parameter () for the TMD is smaller compared to the TLD and TLCD. It means that the distribution of the structure performance with TMD controlling system is more normal than TLD and TLCD. The value measures the structure failure, and the smallest value of parameters refers to more stability and long life [20, 36, 41]. Therefore, the response of the TMD controlling system is more stable than other control systems.
The probabilities of the frequency contents are illustrated in Figure 6. The Lomb-Scargle power spectrum and probability methods were used to estimate the frequencies contents and the significant frequency components.
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From Figure 6, the three frequency modes of the TMD and TLD controlling systems are 2.2, 2.5, and 5.2 and 2.2, 2.6, and 5.2 Hz, respectively. Two frequency modes are shown only with the TLCD controlling system, 2.6 and 5.2 Hz, whereas, with uncontrolled system, one frequency mode is shown at 2.8 Hz only. The probability exceeded for the frequency contents is between the values of 2.5 and 3.5; 2.3 and 3.0; 2.3 and 3.1; and 2.1 and 3.0 for the uncontrolled, TLD, TLCD, and TMD systems, respectively. These results show that the damping contribution of the TMD and TLD controlling systems is more effective in suppressing the structure displacement. Therefore, the relation between mass effectiveness and damping contribution is presented in Figure 7. The effective masses of 1.8, 2.1, 2.39, and 2.54 kg and 1.8, 2.0, 2.26, and 2.54 kg are utilized for the TMD and TLD, respectively; in addition, the damping contribution is calculated based on the power spectrum calculation by fast Fourier transformation method for each effective mass of the controlling system [9].
From Figure 7, the linear trend fitting for the TMD controlling system is linear with , while the trend of the TLD fitting shows linear fitting with . This can occur because the behavior of lamped mass TMD is more accurate because of solid mass movement while for TLD the liquid moving inside to provide damping shows more complicated behavior and energy dissipation process. The linear fitting of the two systems is considered as a damping index to predict the performance of the controlling system before actual applications. The TMD controlling system has high damping contribution than the TLD and TLCD. Accordingly, it is concluded that the TMD controlling system is more effective in the frequency and time domains than other controlling systems.
3.2. Response Model Identification Analysis
Based on the previous analysis, the structure design prediction model with the TMD controlling system is considered in this section. Two identification models, AR and ARMA, are evaluated using ANN. From (4), the AR model contains input parameters only without model error time delay. The autoregressive neural network (ARNN) model can be defined by the number of input parameters , while ARMANN model can be defined by . The acceleration of the input loads and third-floor response measurements are considered as input parameters. The third-floor response acceleration is utilized as an output parameter. Training the feedforward neural network (like AR model) and recurrent (like ARMA model) neural network is accomplished through iterative adjustments of the free parameters, that is, the weights and bias, of the network till we obtain the optimal values. There exist various learning algorithms, which are fundamental to the design of neural networks. The Levenberg-Marquardt learning algorithm is the most widely used algorithm for feedforward neural networks and recurrent neural network, which is used in this research [28, 43]. Table 3 represents the statistical analysis for six trails of the model design for the AR and ARMA models with 100 iterations and 5 hidden neurons.
The trail models’ performance evaluation show that two design models can be used for the identification of structures’ behavior equipped with passive TMD controlling systems. The and values are shown equally from trails three to six for the AR and ARMA models. The obtained statistical analysis shows that the AR and ARMA are more suitable to use in the current study, while it can be seen that the MAE and RMSE values for the two models are smaller than other models. The time delay is a main factor for the performance of the model design, where models’ performance depends on the seismic inputs variables. The comparison between the AR and ARMA models shows that the ARMA model is better than AR model to predict the structure behavior. The ARMANN model can be used to detect the behavior of the controlled structure. In addition, the results show that the recurrent identification models are more effective than the feed forward to predict the behavior of the controlled structures. Figure 8 represents the measurements, prediction, and errors for the ARMANN model. From this figure, it can be seen that the maximum error for the model is 0.013 g, and the correlation coefficient between the measurements and prediction signals is high (0.98). This means that the ARMANN model is suitable for use as a tool to predict the behavior of controlled structures under dynamic loads.
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4. Conclusions
Structural vibration control is a challenging and important problem in the structural engineering. The passive and active controlling systems are being used for the vibration control of buildings under various seismic excitations. The time series, frequency contents, and statistical analyses were discussed for TMD, TLD, and TLCD coupled with building structure systems in terms of vibration mitigation. A novel application for the probabilistic evaluation is utilized to analyze the behavior of structures coupled with controlling systems. The results show that the probability analysis of the performance of the TMD and TLD controlling systems is seen closed in time and frequency domains. In addition, the vibration damping of the TMD controlling system is highest than TLD and TLCD. Therefore, the TMD control system is more reliable controller than the TLD and TLCD systems.
Two autoregressive models (i.e., AR and ARMA) are used with neural networks solution to predict the behavior of structure coupled with TMD controlling system. The performances of two models are evaluated and the results showed that the ARMA model is more effective. The recurrent identifying models predict the behavior of the controlled structures in more accurate way. Evaluating the performance of the ARMANN model through experimental measurements showed that the model is a promising tool to predict the behavior of controlled structures under dynamic loads.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2017R1A2B2010120).