Spectral Amplification Factor Research Articles

Two-dimensional quad-stable Gaussian potential stochastic resonance model for enhanced bearing fault diagnosis

In this study, a two-dimensional quad-stable Gaussian potential stochastic resonance model is explored for the first time. First, the structure of the proposed model is analyzed to have a broader potential field and verified to break through the severe output saturation inherent in the classical two-dimensional quad-stable stochastic resonance model. Then, we analyze the relationship between the structure and parameters of the model and derive the steady-state probability density and the mean first-passage time using adiabatic approximation theory to describe the specific process of the Brownian particle transitions. By combining the adiabatic approximation theory and the probability flow equation, the spectral amplification factor of the model is derived, and the effects of different parameters on the model performance are investigated. Further, a fourth-order Runge-Kutta algorithm was applied to evaluate the model performance in multiple dimensions. Finally, the model parameters were optimized using an adaptive genetic algorithm and applied to complex practical engineering detection. The experimental results show that the proposed model is superior and universal in fault diagnosis. Overall, this study provides important mathematical support for solving various engineering problems and demonstrates a wide range of practical applications.

Entropic stochastic resonance of finite-size particles in confined Brownian transport.

We demonstrate the existence of entropic stochastic resonance (ESR) of passive Brownian particles with finite size in a double- or triple-circular confined cavity, and compare the similarities and differences of ESR in the double-circular cavity and triple-circular cavity. When the diffusion of Brownian particles is constrained to the double- or triple-circular cavity, the presence of irregular boundaries leads to entropic barriers. The interplay between the entropic barriers, a periodic input signal, the gravity of particles, and intrinsic thermal noise may give rise to a peak in the spectral amplification factor and therefore to the appearance of the ESR phenomenon. It is shown that ESR can occur in both a double-circular cavity and a triple-circular cavity, and by adjusting some parameters of the system, the response of the system can be optimized. The differences are that the spectral amplification factor in a triple-circular cavity is significantly larger than that in a double-circular cavity, and compared with the ESR in a double-circular cavity, the ESR effect in a triple-circular cavity occurs within a wider range of external force parameters. In addition, the strength of ESR also depends on the particle radius, and smaller particles can induce more obvious ESR, indicating that the size effect cannot be safely neglected. The ESR phenomenon usually occurs in small-scale systems where confinement and noise play an important role. Therefore, the mechanism that is found could be used to manipulate and control nanodevices and biomolecules.

Floor response spectra and seismic design method of electrical equipment installed on floor in indoor substation

Electrical equipment installed on floors is a typical type of nonstructural component in indoor substations, which could be subjected to earthquake loading during seismic events. To investigate the seismic design method of floor electrical equipment in indoor substations, simulation models of four whole-indoor substations comprising main control buildings and their floor electrical equipment were established. Seismic response spectrum analyses of main-attachment structures were performed with a high number of ground motions as inputs. Using mathematical statistics and data fitting methods, this research suggested the calculation formula of amplification factor of floor acceleration response spectrum. Specifically, when the periodic ratio of electrical equipment to main control building is in the range of 0.9–1.1, the system would produce a strong resonance-like effect with a peak spectral amplification factor. The floor response spectrum for seismic design was influenced by four factors: equipment importance category, floor position, damping ratio of electrical equipment, and torsion effect of main control building. Following that, a seismic design method especially applicable for electrical equipment installed on floor in indoor substation, which combines amplification factor and calculation method of seismic action indicated in the relevant specification, was presented. Lastly, a typical 220 kV whole-indoor substation was taken as an example to validate the accuracy of the design method highlighted in this article.

Rolling Bearing Fault Diagnosis Based on Exact Moment Dynamics for Underdamped Periodic Potential Systems

This research is to build a more general bridge over the model investigation on stochastic resonance (SR) and the laboratory design in rolling bearing faults. To this end, we generalize the derivative matching moment closure method to the underdamped biased periodic potential systems to disclose the non-monotonic evolution of the spectral amplification factor. With the exact moment dynamics available, a two-layer loop algorithm for detecting the incipient bearing faults is then developed. With the outer loop to optimize the output SNR for the best time scale factor and the inner loop to maximize the spectral amplification factor for optimal system parameter, the semi-analytic results are directly related with this laboratory application. The analog and experimental verification based on different datasets show that the proposed method can perform as well as the existing SR method, even under strong noise background. Particularly, the proposed method does not depend on the amplitude of input signal when optimizing parameters, but only on noise intensity and characteristic fault frequency, thus it has higher detection efficiency than the existing simulation based SR methods.

Resonance phenomena caused by mixed coupling in tristable systems

Coupled systems have been widely encountered and have aroused much great interest in improving the response output of the system. Scholars pay more attention to the use of vibration resonance or stochastic resonance to amplify weak signals, but few examine the role of mixed coupling on the system response. The amplifying effect of mixed coupling on signal in tristable oscillators is studied in this paper. The spectral amplification factor, which is used to measure the amplification, shows a resonance peak with the coupling strength. The optimal coupling strength can be found to achieve the maximum output of the system. Moreover, the influence of the depth and position of the potential well on the response output is also analyzed. The results show that the properties of tristable oscillators can change the coupling threshold as well as improve amplification. We speculate that, in addition to the mixed coupling, the intrinsic mechanism of this behavior may be due to the different shapes of the tristable oscillators and their modes of variation. It provides new thoughts for amplifying and detecting weak signals in coupled systems.

Energetic and entropic vibrational resonance with a time-delayed feedback

Energetic and entropic vibrational resonance with a time-delayed feedback is studied by considering a Brownian particle confined in a two-dimensional boundary. Both the time delay and high-frequency signal can optimize the output of the system. The delay time can induce the multimodal resonance with a local optimum peak in the spectral amplification factor. Furthermore, the mechanism corresponded to dynamical transition from a multi-stable state to a monostable state caused by the high-frequency signal is elucidated. In addition, a quantity called the mean entropic Poisson intensity is employed, which can be used to assist in describing the constraint effects of uneven boundaries. It is worth mentioning that this quantity can also properly describe the vibrational resonance of the system. Finally, our work provides some theoretical references for the detection of weak signals in small-scale systems with ubiquitous fluctuation and time delay.

Efficient Path Integration of Nonlinear Oscillators Subject to Combined Random and Harmonic Excitation

Abstract A new path integration (PI) method is studied to improve the efficiency of computing probability density of nonlinear oscillators subject to combined harmonic and random excitation. The new PI method utilizes Fourier series to obtain a spectral presentation of the short time transition probability density (STPD) in the time domain and uses the method of linear least squares to determine the Fourier coefficients. It also utilizes a tensor product spline interpolation method to obtain an accurate representation of the STPD in the state space. The new PI method is applied to the monostable and bistable Duffing oscillators to predict the response statistics, including the time average of asymptotic mean squares and the spectral amplification factor. Specifically, the spectral amplification is used to characterize stochastic resonance of the bistable oscillator. The predictions show good agreement with Monte Carlo simulations and available data in the literature. The new PI method is also used to investigate the influence of noise intensity on stochastic P-bifurcation of the bistable oscillator. Finally, a case study shows that the new PI method reduces the computational time by 1–2 orders of magnitude in comparison with a traditional PI method.

Optimal resonance response of nonlinear system excited by nonlinear frequency modulation signal

Nonlinear frequency modulation (NLFM) signal is widely used in radar, communication and signal processing. The response of nonlinear system excited by this kind of signal has rich information. At the same time, enhancing different types of signals by resonance phenomenon has unique advantages in the field of signal processing. Compared with other signal processing methods, such as empirical mode decomposition, variational mode decomposition, wavelet transform, signal filtering, etc., this kind of method can not only enhance the signal, but also effectively suppress the interference noise. Therefore, it has certain significance to study the nonlinear system optimal response excited by different NLFM signals and enhance the NLFM signal through resonance phenomenon. In this paper, what is mainly studied is the nonlinear system resonance phenomenon excited by different NLFM signals, which is different from in previous studies. Firstly, a real-time scale transformation method is proposed to process the NLFM signals, and its basic principle is to match different NLFM signals by real-time scale coefficients and system parameters. The signal frequency at each time corresponds to the coefficients with different scales and system parameters, thereby realizing the optimal resonance response at each time. In order to describe the optimal resonance response excited by the NLFM signal more accurately, unlike the traditional spectral amplification factor, the real-time spectral amplification factor is introduced as an evaluation index. Then, the influence of system parameters on the optimal system resonance response is discussed, and the optimal resonance region is obtained, which means that the optimal resonance response can be achieved by selecting the parameters in a reasonable range. This method not only greatly enhances the signal characteristics, but also maintains the continuity of signal time-frequency characteristics. Finally, the real-time scale transformation method is compared with the general scale transformation method, showing the superiority of the proposed method in processing NLFM signal. The method and the results of this paper show some potential in dealing with complex NLFM, which provides a reference for NLFM signal enhancement and detection, and has a certain practical significance in signal enhancement. Furthermore, the relevant influence law of the system optimal response excited by the NLFM signal is given, which has a certain reference value for studying the system dynamic behavior under different signal excitations.

Energetic and entropic vibrational resonance

The influence of an energetic bistable potential on the entropic vibrational resonance (EVR) is investigated. We demonstrate the existence of vibrational resonance modulated by the high-frequency signal with energetic and entropic barriers, which is termed energetic and entropic vibrational resonance (EEVR). We use different methods, including the numerical simulation of Langevin equation, the numerical solution of Fokker-Planck equation, and the analytical expressions of two-state approximation theory, to obtain the spectral amplification factor which can be used to quantify EEVR. In addition, there are two mechanisms, the dynamical phase transition of the stable states and the matching between the escape rate and low-frequency signal on time-scale in the EEVR, which are caused by the modulation of the high-frequency signal. Finally, an auxiliary quantity called the mean entropic Poisson intensity is proposed to describe the constraint effects of the uneven boundaries on the Brownian motion. The above research provides some theoretical reference for the detection of weak signals, and the manipulation of motions of an overdamped Brownian particles, under uneven boundary constraints in small-scale stochastic systems.

Investigation of the Performance of Simplified Constitutive Models in Nonlinear 1D Effective Stress Ground Response Analysis

ABSTRACTSeveral building codes, such as the National Building Code of Canada, recommend that an effective stress ground response analysis be performed if a liquefiable stratum is identified within a soil profile. Although, constitutive models for total stress ground response analysis have been well verified against earthquake recordings, existing models for effective stress ground response analysis have yet to be thoroughly validated. This article investigates the predictions of five pore pressure models derived for effective stress ground response analysis. First, a dataset of five downhole arrays and two centrifuge experiments in which a potential of liquefaction was identified is presented. The profiles and ground-motion recordings are selected to represent a broad range of soil properties, ground-motion intensities, and excess-pore pressure generation levels. The differences between predictions of the effective stress models against commonly used 1D ground response total stress equivalent-linear and nonlinear analyses are evaluated. The predicted and measured motions are compared in terms of spectral response and amplification factor. The pore pressure response of all models is evaluated as a function of shear strain and found to agree well with published correlations representing the expected behavior of liquefiable soils. Although, the models show the ability to capture liquefaction triggering, the results indicate that for the selected dataset, total stress simulations were found to be, at least, as precise and accurate as the effective stress simulations. Therefore, simplified models for effective stress ground analysis should be used with caution by practicing engineers to predict surface spectra but can be used confidently to assess the potential for liquefaction triggering.

Establishing site response-based micro-zonation by applying machine learning techniques on ambient noise data: a case study from Northern Potwar Region, Pakistan

Several major earthquakes have jolted Pakistan during the last 30 years, destroyed infrastructure and severe damage to the economy. Despite advancement in data sciences and its linkages with other domains, no such study has been conducted in Pakistan, which incorporates the application of machine learning techniques on ambient noise data. This study presents the application of machine learning techniques on the ambient noise data to establish the micro-zones within the urban settlements of northern Potwar region of Pakistan, based on the site response parameters and seismic vulnerability. The ambient noise data from 148 sites in the study area, are acquired, processed and interpreted. Different clustering techniques namely, K-mean, fuzzy c-means and hierarchical clustering have been applied to the interpreted ambient noise data set. Arc GIS maps of the study area have been developed by making use of the interpretation of the ambient noise data and the resulted clustering solutions. The results showed that the fundamental frequency f0 ranges between 0.5 and 15 Hz, the H/V spectral amplification factor ranges between 0.8 and 5.9; the soft sediment thickness ranges from 1.6 to 316 m, whereas, the soil vulnerability index is observed between 0.1 and 63. These site response parameters indicated that the study area is moderate to highly vulnerable to site amplification, and any seismic event can lead to catastrophe within the study area. The clustering techniques also detected three groups from the interpretation of the ambient noise data set by separating the locations according to their vulnerability due to site amplification. The quality of cluster solutions was evaluated using cluster validity indexes and the results of these techniques were compared. These results present the similarities and dissimilarities among different sites and indicate the sites which are geographically distant but have very similar vulnerability characteristics or vice versa. The Arc GIS tool showed the spatial distribution of site response parameters and three zones were established as zone 1, zone 2 and zone 3 with low, intermediate and high values, respectively. The spatial distribution maps showed that the northeastern and northwestern parts of the study area are more vulnerable to site amplification.

Novel Bearing Fault Diagnosis Algorithm Based on the Method of Moments for Stochastic Resonant Systems

The principle of stochastic resonance (SR) in the noisy Duffing oscillator model has shown benefit for designing novel mechanical fault diagnosis algorithms, where noise is utilized rather than being eliminated. However, there is a clear gap between the model progress and the experimental application. In this article, effort is made trying to narrow the gap by applying the method of moments to obtain the spectral amplification factor within the linear response range to improve the algorithm design, which avoids the conventional time-consuming direct simulations. A strategy for estimating noise, which is critical for programming the algorithm, is proposed and evaluated. Through simulation and experimental data analysis, it is confirmed that the new algorithm has advantages over the overdamped system-based methods as it does not depend on the signal preprocessing techniques such as envelope extraction and high-pass filter. Also, the new method has advantages over the existing underdamped system-based methods as it can decrease the computational time for seeking the optimal parameter by at least one order of magnitude.

Linear frequency modulated signal induced aperiodic resonance

A method is proposed to induce strong aperiodic resonance in a linear frequency modulated signal excited nonlinear system. The phenomenon has application value in weak linear frequency modulated signal enhancement. The spectral amplification factor is used to measure the aperiodic resonance performance. By piece-wise analysis, appropriate system parameters are found. Then, a strong resonance output is achieved. The cross-correlation coefficient between the input and the output is another measurement of aperiodic resonance. However, there is a weakness in analysis of the linear frequency modulated signal. To overcome the shortness, a modified cross-correlation coefficient is proposed. As a result, the error of the cross-correlation coefficient induced by the phase difference of the excited signal is eliminated. In addition, by adding another much faster excitation, aperiodic resonance is further enhanced by vibrational resonance.

Inertial interaction effects analysis on the estimating of dynamic amplification factor for reinforced concrete structures

This work shows the importance of the need for revision of the Algerian seismic regulations RPA 2003 in terms of the dynamic amplification factor for R/C Rigid buildings with period 0.3 < T < 0.35 in stiff site. In most national building codes, formulas for estimating the seismic base shear include many factors such as the dynamic amplification factor D presented in the Algerian and Moroccan seismic codes RPA2003 and RPS2000 respectively, the response coefficient of the building B for the Iranian code ICS1999, and the spectral acceleration amplification factor B0 for the Eurocode 8. To determine this factor, it is necessary to estimate the value of the vibration period of building based on empirical formulas that do not consider the inertial effects. Diagrams representing the variations of this factor show that the latter remains constant over a range from 0 to 0.7 s for the Algerian code corresponding to structures with 1 to 7 levels; 0 to 1 s for the Moroccan, Iranian and Eurocode 8 corresponding to structures with 1 to 10 levels. No difference was noted in the dynamic factor as a result of degradation of the soil bearing capacity in this ranges. The purpose of this paper revolves around two main themes: the effects of Soil Foundation Structures Interaction (SFSI) on the estimation of dynamic amplification factor on the equivalent static analysis, and the elaboration of a new factor DSAF* called “dynamic site amplification factor” which takes into consideration the SFSI effects, the classifications of the seismic zones and the importance of the structures with application of the Algerian seismic code RPA2003. These values are presented in their simple and easy way to the different users. The results show the importance of these safety coefficients, recommended in the seismic regulations under the effect of SFSI, when the RPA2003 placed in the non-conservative side for the structures belongs to the ranges 0 to 0.7 s.

Optimal IMF selection and unknown fault feature extraction for rolling bearings with different defect modes

Rolling bearings are widely used in the rotating machinery. The fault types and fault feature frequencies are usually unknown when rolling bearings fail in the engineering applications, which brings difficulties to accurately recognize the health status of rolling bearings. Realizing the extraction of unknown fault features is of great significance for rolling bearing fault diagnosis in real engineering scenarios. Based on the ensemble empirical mode decomposition (EEMD) and stochastic resonance (SR), an adaptive fault diagnosis method for unknown rolling bearing faults is proposed by designing an effective frequency cut-off criteria and constructing two novel evaluation indicators. For a target signal containing complex frequency components, a frequency cut-off criterion is applied to remove redundant intrinsic mode function (IMF) components and improve the accuracy of unknown fault diagnosis in the process of using EEMD to decompose target signal. For the remaining effective IMFs, the spectral amplification factor (SAF) index is built to estimate the energy of each IMF for selecting the optimal IMF. The adaptive SR with the piecewise mean value (PMV) index is performed to extract the unknown fault features from the optimal IMF. The proposed frequency cut-off criteria obviously reduce the interference of other unknown frequencies. Compared with traditional kurtosis, SAF has good noise immunity and stability. The effectiveness of the proposed method is successfully verified by three rolling bearing experiments with different defect modes. Moreover, the vibration waveform characterizing unknown fault feature is reproduced to some extent based on the adaptive SR response. The proposed method accurately realizes unknown fault diagnosis and provides a reference for condition monitoring of rolling bearings in engineering practice.

The echo chirp signal amplification by the vibrational information fusion method

We use two traditional indices and propose a new fusion index to quantify the echo chirp signal amplification effectively. They are the cross-correlation coefficient, the spectral amplification factor and the fusion index. Through investigating different numerical examples, we verify the validation of the re-scaled vibrational resonance (VR) method for the echo chirp signal by using the two traditional indices. Besides, taking into account that a single index has its own shortness, we combine the advantages of the two traditional indices mentioned and propose a new fusion index based on the information fusion method. Then we verify the superiority of the fusion index. Furthermore, the effect of noise on VR is also investigated. Finally, with the help of the adaptive inertia weight particle swarm optimization (APSO) algorithm, we propose the adaptive VR in noise background and verify its validity.

2D seismic response of the Duzce Basin, Turkey

The existence of the inclined bedrock at the basin edges bring about the concentration of the damages at these areas. The graben-like Duzce Basin with a low inclined subsurface topography is prone to exhibit such effects. In this study, at the first, Duzce Basin edge model is extracted from the data obtained from performed microtremor surveys as well as geotechnical investigations. Using these data, the basin stratification and shear wave velocity profile as well as the geotechnical properties of the layers is identified. Based on these data, the idealized Duzce Basin edge model is proposed. Then, to estimate the dynamic behavior of the Duzce Basin, a fully nonlinear method based on explicit finite difference scheme is performed. The basin is subjected to the collection of 16 strong ground motions with different peak ground accelerations (PGA) levels. Using the recorded acceleration at 12 surface points along the basin edge the variation of the maximum spectral amplification factor is reported. The results show that the Duzce Basin edge behavior is greatly affected by inclined bedrock. Also, the sensitivity of the different points of the basin to different periods is investigated. It is seen that in the case of the motions with the PGA of 0.2 g, this band spans between 0.1s to about 1.7s.

Stochastic Resonance and Bifurcation of Order Parameter in a Coupled System of Underdamped Duffing Oscillators

The long-term mean-field dynamics of coupled underdamped Duffing oscillators driven by an external periodic signal with Gaussian noise is investigated. A Boltzmann-type [Formula: see text]-theorem is proved for the associated nonlinear Fokker–Planck equation to ensure that the system can always be relaxed to one of the stationary states as time is long enough. Based on a general framework of the linear response theory, the linear dynamical susceptibility of the system order parameter is explicitly deduced. With the spectral amplification factor as a quantifying index, calculation by the method of moments discloses that both mono-peak and double-peak resonance might appear, and that noise can greatly signify the peak of the resonance curve of the coupled underdamped system as compared with a single-element bistable system. Then, with the input signals taken from laboratory experiments, further observations show that the mean-field coupled stochastic resonance system can amplify the periodic input signal. Also, it reveals that for some driving frequencies, the optimal stochastic resonance parameter and the critical bifurcation parameter have a close relationship. Moreover, it is found that the damping coefficient can also give rise to nontrivial nonmonotonic behaviors of the resonance curve, and the resultant resonant peak attains its maximal height if the noise intensity or the coupling strength takes the critical value. The new findings reveal the role of the order parameter in a coupled system of chaotic oscillators.

Effects of site\u2013city interaction and polarization of the incident S-wave on the transfer function and fundamental frequency of structures

The variation of 2D fundamental frequency of stand-alone structure on rock ({F}_{{02{text{D}}}}^{text{S}} ) and in a basin ({F}_{{02{text{D}}}}^{text{BS}} ) with the polarization of the incident S-wave is presented. This paper also presents the role of city density and polarization of the exciting S-wave in the site–city interaction effects on the response of structures in a city as well as free-field motion. The seismic responses of the various considered basin and site–city models are computed using SH-wave and P-SV-wave finite-difference programs. A considerable decrease in the value of {F}_{{02{text{D}}}}^{text{S}} of stand-alone structure on rock is obtained with the increase in shape ratio for the SV-wave but not for the SH-wave, as compared to the 1D fundamental frequency of structure ( {F}_{{01{text{D}}}}^{text{S}} ). However, a considerable decrease in the value of {F}_{{02{text{D}}}}^{text{BS}} of a stand-alone structure in basin under double resonance condition is obtained for the SH-wave and not for the SV-wave. The spectral amplification factor at the top of a structure at {F}_{{02{text{D}}}}^{text{BS}} of structure is larger for the SV-wave as compared to the SH-wave. A splitting of the bandwidth of the fundamental mode of vibration of structure and very large reduction in spectral amplification factor at {F}_{{02{text{D}}}}^{text{BS}} of structure is obtained due to the site–city interaction effects under double resonance condition for both the S-wave polarizations. Further, an increase in these inferred effects with an increase in city density and number of structures in a city with a constant city density is obtained. This finding raises the question concerning the validity of ground-motion prediction using soil–structure interaction for the design of structures in an urban environment. Further, the decrease in spectral amplification factor at {F}_{{02{text{D}}}}^{text{BS}} of structure due to the site–city interaction effects is larger in the case of SV-wave as compared to SH-wave. Furthermore, a considerable site–city interaction effects on free-field motion is obtained.

Quantification of Fundamental Frequencies of 3D Basins and Structures and Site–City Interaction Effects on Responses of Structures

Large-scale commercialization in Indian metrocities has made realtors develop new construction sites by filling land depressions with loose soil, which may behave as small three-dimensional (3D) basins. This study presents the development of empirical relations to predict the fundamental frequency of basins ( $$F_{{03{\text{D}}}}^{\text{B}}$$ ) and structures ( $$F_{{03{\text{D}}}}^{\text{S}}$$ ) on rock for different shape ratios to study the effects of site–city interactions (SCIs) on the response of structures under a double resonance condition. The S-wave responses of the various considered basins, structures, and site–city models are simulated using the finite-difference method. Analysis of the simulation results reveals that the $$F_{{03{\text{D}}}}^{\text{B}}$$ of the basin and the $$F_{{03{\text{D}}}}^{\text{S}}$$ of the structure depend strongly on the shape ratio, on average matching with those obtained using the SV-wave response of the section of the respective basin/structure. However, the spectral amplification factor (SAF) obtained at $$F_{{03{\text{D}}}}^{\text{B}}$$ / $$F_{{03{\text{D}}}}^{\text{S}}$$ for the basin/structure is much larger than that obtained using the SV-wave response of a section of the respective basin/structure. Empirical relations are developed to predict the $$F_{{03{\text{D}}}}^{\text{B}}$$ of basins and $$F_{{03{\text{D}}}}^{\text{S}}$$ of structures in terms of the shape ratio and one-dimensional (1D) fundamental frequency of the respective model. Analysis of 3D SCI effects reveals a reduction of the SAF at the fundamental frequency of the structure in the basin ( $$F_{{03{\text{D}}}}^{\text{SB}}$$ ) with an increase of the number of structures as compared with that at the fundamental frequency ( $$F_{{03{\text{D}}}}^{\text{SB}}$$ ) of a standalone structure in the basin, being on the order of about 60 % in the case of a city with 25 structures. This finding indicates the need for 3D SCI studies in urban environments for cost-effective earthquake engineering.