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The procedures used to estimate structural modal parameters as natural frequency, damping ratios, and mode shapes are generally based on frequency methods. However, methods of time-frequency analysis are highly sensible to the parameters used to calculate the discrete Fourier transform: windowing, resolution, and preprocessing. Thus, the uncertainty of the modal parameters is increased if a proper parameter selection is not considered. In this work, the influence of three different time domain windows functions (Hanning, flat-top, and rectangular) used to estimate modal parameters are discussed in the framework of ISO 18431 standard. Experimental results are conducted over an AISI 1020 steel plate, which is excited by means of a hammer element. Vibration response is acquired by using acceleration records according to the ISO 7626-5 reference guides. The results are compared with a theoretical method and it is obtained that the flat-top window is the best function for experimental modal analysis.

Physical behavior of complex engineering systems can be studied through prediction and simulation analysis by means of specialized software [

Because of the importance of modal analysis in the field of structural analysis, well established procedures to obtain proper estimations of modal parameters are required. Although there exists a huge documentation about methods used for modal analysis [

In this paper, a practical implementation of the abovementioned ISO standards with a special emphasis on computing the natural frequency values is demonstrated. Thus, the influence of using three domain window functions (

The procedure used in this paper to estimate the natural frequencies of a modal model is based on the analysis of measurements from Frequency response functions (FRFs). In this sense, the extraction of relevant frequency information is performed by applying spectrum estimation techniques to structural vibrational records. Thus, FRFs are approximated by the cross-power spectral density (PSD) between the vibrational responses.

In this section, the conceptual issues involved in the estimation of structural natural frequencies by means of FRFs are detailed. Also, fundamentals of methods used to estimate the frequency decomposition are presented, focusing on given details about the parameters with great influence for its implementation.

Dynamical model of structures is constructed on physical knowledge and fundamental laws of motion according to [

Moreover to law motion and NExT equations used in methodologies for modal parameter identification, the FRF relationship in structures must be specified. The FRF for one

Typical FRF of an

According to Figure

Classical approach used to estimate modal parameters is often referred to as Peak-Peaking (PP), which is a nonparametric method essentially developed in frequency domain. The main advantages of this method are its user-friendliness, simple use, and fast results obtaining. In this method, average normalized power spectral densities and frequency response functions between all the measurement points of the structure are evaluated [

Determine the natural frequencies by means of the PSD computed from acceleration records by identifying all frequencies present at peaks

Estimate damping ratios

Half power band frequencies.

According to Figure

Estimate the power spectral density matrix at discrete frequencies.

Do a singular value decomposition of the power spectral density.

For an

Frequency response functions between input and output are approximated as cross-power spectral densities between responses while the impulse response functions are approximated by cross-correlations between responses. Cross-PSDs are obtained using Welch method (FFT based method) [

In estimating power spectral density (PSD) of a signal, there are two tradeoffs. One is frequency resolution and the other is noise in the signal. To obtain a good estimation of PSD, we should have large length of the signal but during measurements we have finite length of signal. If we take small block size, bad frequency resolution could introduce leakage in the spectrum. To reduce leakage, signals are windowed, that is, multiplied with a window in time domain. Many windows are available, each one having specific application in signal processing [

Spectral characteristics of recommended ISO-time windows.

According to Figure

The 7626-5 and 18431 ISO standards give recommendations about recording protocol and selection of window function in order to estimate modal parameters [

Considering the selected set of time windows specified in the ISO standards, in this paper the results of applying

The next sections show the results obtained by applying experimental and theoretical procedures. For experimental analysis the three time windows specified in the ISO standard are considered: Hanning, flat-top, and rectangular functions, while the theoretical approach is based on finite element method.

A steel plate was used in order to analyze the influence in the estimation of natural frequencies when different window functions are considered. In Figure

Bench structure.

Measurements from acceleration response were conducted over 4 points of the structure (P6, P7, P8, and P12). Vibration data were recorded under impulse hammer excitation type with sample time

Recorded vibration signals.

In order to calculate modal frequency of different deformation modes, finite element simulation through a numerical model was performed by using ANSYS software. The model includes the plate detailed geometry and it is characterized by density = 7733.75 Kg/m^{3},

Structural finite element model.

The modal shapes studied by means of the simulation software are depicted in Figure

Modal frequency obtained by numerical analysis.

First bending: 636 Hz

First torsion: 671 Hz

Second torsion: 1472 Hz

Second bending: 1640 Hz

In order to evaluate the influence of windowing effect, experimental measurements of acceleration records were processed considering windows described in previous sections. As a first result, the 2 Khz range of interest for cross-power spectral density of the acceleration measurements is depicted in Figure

CPSD for acceleration records.

Also, the frequency response function computed by means of frequency domain decomposition after processing the CPSD of data in Figure

Frequency response function.

A comparison of the modes estimated according to ISO 7626-5 by using the three selected time window functions is presented in Table

Natural frequencies [Hz] estimated by using three different windows.

Vibrational mode | Numerical |
Hanning | Flat TOP | Rectangular |
---|---|---|---|---|

First bending | 636 | 640.9 | 634.8 | 636.3 |

First torsion | 671 | 671.4 | 671.4 | 668.3 |

Second torsion | 1472 | 1465 | 1469 | 1468 |

Second bending | 1640 | 1697 | 1691 | 1694 |

Finally, the percentage errors for each natural frequency with respect to theoretical numerical model are summarized in Table

Percentage error (%) of mode estimation with respect to numerical model.

Vibrational |
Hanning |
Flat-top |
Rectangular |
---|---|---|---|

First bending | 0.7704 | 0.1887 | 0.0472 |

First torsion | 0.0596 | 0.0596 | 0.4024 |

Second torsion | 0.4755 | 0.2038 | 0.2717 |

Second bending | 3.4756 | 3.1098 | 3.2927 |

According to results in Table

Although no meaning differences were found when the three windows specified in the ISO standard were used to estimate natural frequencies, a slight better result is obtained for flat-top function. This implies that for modal parameter estimation purposes the selection of time windowing function has low influence, with major errors for the highest modes. However, the influence of the windowing preprocessing for the analysis of different modal parameters as shape mode and factor participation should be studied. Also, further analysis should be conducted with respect to additional parameters involved in the spectrum estimation such as overlap, FFT length, and segmentation. Moreover, it is recommended to include uncertainty analysis to evaluate the influence of using different time windows.

The authors declare that they have no competing interests.

This work has been developed as part of a collaborative work between researches from Universitaria de Investigación y Desarrollo (UDI) and Universidad Pontificia Bolivariana (UPB), Bucaramanga, Colombia.