The measured frequency response functions (FRFs) in the modal test are usually contaminated with noise that significantly affects the modal parameter identification. In this paper, a modal peak-based Hankel-SVD (MPHSVD) method is proposed to eliminate the noise contaminated in the measured FRFs in order to improve the accuracy of the identification of modal parameters. This method is divided into four steps. Firstly, the measured FRF signal is transferred to the impulse response function (IRF), and the Hankel-SVD method that works better in the time domain rather than in the frequency domain is further applied for the decomposition of component signals. Secondly, the iteration of the component signal accumulation is conducted to select the component signals that cover the concerned modal features, but some component signals of the residue noise may also be selected. Thirdly, another iteration considering the narrow frequency bands near the modal peak frequencies is conducted to further eliminate the residue noise and get the noise-reduced FRF signal. Finally, the modal identification method is conducted on the noise-reduced FRF to extract the modal parameters. A simulation of the FRF of a flat plate artificially contaminated with the random Gaussian noise and the random harmonic noise is implemented to verify the proposed method. Afterwards, a modal test of a flat plate under the high-temperature condition was undertaken using scanning laser Doppler vibrometry (SLDV). The noise reduction and modal parameter identification were exploited to the measured FRFs. Results show that the reconstructed FRFs retained all of the modal features we concerned about after the noise elimination, and the modal parameters are precisely identified. It demonstrates the superiority and effectiveness of the approach.

In modal testing, the measured FRF signals are usually contaminated with noise [

Generally speaking, the noised high-dimensional data, like the image signal [

In recent years, Yang and Tse [

Instead of using the biggest different point of the singular values to select the number of useful components, some new feature selection methods are based on the decomposed component signals from Hankel-SVD. Qiao and Pan [

In this paper, a novel modal peak-based Hankel-SVD (MPHSVD) method is proposed to eliminate the noise and retain the whole concerned modal features of the FRF signals whose modal peak frequencies and modal peak amplitudes can be selected by observation. The MPHSVD is an iteration Hankel-SVD filter, which contains the accumulation iteration and reselection iteration. The accumulation is to get the component signals of the transformed IRF signal that covers the whole concerned modal features, but these component signals will also cover some noise. The reselection is based on the accumulation result, which aims to separate the component signals which belong to the modal modes from the component signals of the noise and add them up to complete noise reduction thoroughly. Afterwards, the modal identification based on the rational fraction polynomial (RFP) method is conducted to the noise-reduced FRF signal to extract the modal parameters. The simulated and two experimental FRF signals of a flat plate with heavy noise are utilized to verify the approach. The results show the validity and superiority of the proposed method.

The rest of the paper is divided into four sections. In Section

In this section, the modal identification based on MPHSVD is described in detail, and the MPHSVD method is the Hankel-SVD filter with the modal peak-based component signal selection method, which is used for eliminating the noise of the single FRF signal.

As the transformed impulse response function (IRF) signal is better in the noise reduction than the FRF signal, the FRF signal

After the transformed IRF signal is obtained, the Hankel-SVD is conducted to decompose the IRF signal into component signals. For the length ^{th} singular value. The dimension of ^{th} component matrix of the component matrix set. The dimension of ^{th} row and the ^{th} column in matrix ^{th} antidiagonal line of matrix

Then, because Hankel matrix

Finally, the IRF signal

The above derivation is based on the assumption that component matrices still contain the Hankel structure after SVD, but the decomposition result of SVD does not actually follow the assumption. Thus, we usually average the antidiagonal lines of the component matrices such that they contain the Hankel structure and get the component signals. The antidiagonal averaging of the component matrix ^{th} antidiagonal line means adding up the elements whose coordinate summation in

Based on the Hankel-SVD technique, the frequencies and the amplitudes at modal peaks of the FRF signal can be decomposed to the component signals [

For the experimental FRF signal ^{th} iteration, the ^{th} component signal ^{th} accumulated signal ^{th} accumulated signal

Afterwards, the energy spectrum of the accumulated signal

Then, the differences between modal peak amplitudes ^{th} difference vector, and ^{th} modal peak amplitude

Finally, ^{th} iteration is below or equal to the threshold

After the accumulation ends, the whole modal modes will recover, and the number of the selected component signals is recorded as

The peak frequencies of the component signals correlated with the modes are near the modal peak frequencies, while the peak frequencies of the component signals correlated with the residue noise will be away from the modal peak frequencies in some degrees. Thus, if some narrow frequency bands are set near the peak frequencies, the residue noises can be eliminated by reselecting the component signals whose peak frequencies are in the bands from the accumulation result for noise reduction.

For there are

Since there are ^{th} iteration is shown below.

Firstly, for the ^{th} component signal ^{th} component signal

If

After the noise reduction, the noise-reduced FRF signal

In summary, the framework of the modal identification by MPHSVD is given in Figure

Step 1: transform the noised FRF signal to noised IRF signal by IFFT and decompose it to

Step 2: accumulate the component signals in the order of the corresponding singular values from the largest to the smallest to get the first

Step 3: reselect

Step 4: identify the modal parameters from the noise-reduced FRF signal by modal identification

The flowchart of the proposed method.

To verify the proposed method, a simulation of a steel flat plate is carried out using the finite element (FE) analysis. An FRF signal of one point at the middle left of the plate is simulated. The calculated frequency band is [0∼3200] Hz with an interval of 0.5 Hz. Therefore, the FRF signal contains 6400 points with a maximum frequency of 3200 Hz. The corresponding impulse response function (IRF) is attained by the inverse Fourier transform from the FRF signal and contains 12800 points. The simulated FRF signal and its corresponding IRF signal are shown in Figure

(a) Simulated FRF signal. (b) Simulated IRF signal.

In Figure

(a) FRF signal with the added noise. (b) IRF signal with the added noise.

Obviously, the FRF signal in Figure ^{th} mode, “abs” means the absolute value,

Curve fitting results of the noised and noise-free FRF signals. (a) Curve fitted energy spectrum of original FRF. (b) Curve fitted phase spectrum of original FRF. (c) Curve fitted energy spectrum of noised FRF (d) Curve fitted phase spectrum of noised FRF.

Modal parameters of the noised FRF signal.

Mode | Modal frequency (Hz) (original) | Modal frequency (Hz) (noised) | Frequency difference (Hz) | Damping factor (original) | Damping factor (noised) | Damping factor difference (%) |
---|---|---|---|---|---|---|

1 | 63.67794 | 72.34572 | 8.66778 | 0.000674 | −0.00682 | 1111.957 |

2 | 399.3322 | 398.8602 | 0.472005 | 0.000295 | 0.000769 | 160.5308 |

3 | 428.0756 | 428.581 | 0.505418 | 0.000667 | 0.000515 | 22.80982 |

4 | 1119.442 | 1119.783 | 0.340427 | 0.000866 | 0.000991 | 14.50155 |

5 | 1334.573 | 1328.813 | 5.760097 | 0.000943 | 0.000941 | 0.250468 |

6 | 2195.12 | 2195.789 | 0.66933 | 0.001474 | 0.00135 | 8.443071 |

7 | 2381.407 | 2381.004 | 0.402907 | 0.001588 | 0.001627 | 2.482244 |

To eliminate the noise contaminated in the FRF signal, the corresponding noised IRF signal in Figure ^{th} iteration turn, and the first 80 component signals are accumulated to form the accumulated signal. Then, the accumulated signal is transformed to the frequency domain to acquire the correlated FRF signal, and the overlay between the FRF signal from the accumulated signal and the FRF signal from the transformed noise-free IRF signal is given in Figure

Noise reduction result of the component signal accumulation. (a) Energy spectrum. (b) Phase spectrum.

Because seven selected modal peaks are supposed to be kept after noise recovery, seven frequency bands with the bandwidth 35 Hz near the seven modal peak frequencies are set as the threshold for the further noise removal. The component signals, that the frequency of the biggest peak amplitude is in the set frequency bands, are reselected for the further noise reduction. Then, for the 80 component signals selected from accumulation, the component signal reselection is conducted for them to separate the component signals correlated with the modal modes for recovery and abandon the component signals of the random harmonic noises. Finally, 20 component signals are reselected, and they are added up to form the noise-reduced IRF signal. The overlays of the noise-reduced IRF signal against the noised IRF signal and the noise-reduced IRF signal against the noise-free IRF signal are given in Figures

(a) Overlay of the noised IRF signal. (b) Overlay of the noise-free IRF signal.

It can be found that most parts of the noise are removed, and there is mostly no difference between the noise-reduced and the noise-free IRF signals, except the part of the red circle in the figure. Then, a noise removal rate is calculated by the following equation

Noise reduction result of component signal reselection. (a) Energy spectrum. (b) Phase spectrum.

For more detailed analysis of the relationship between the modal modes, the harmonic noises, and the combination of component signals, a part of the first 80 component signals are investigated, and the result is shown in Figure ^{st} and 2^{nd} modes. It can be found that the first mode at 65 Hz is decomposed into two adjacent component signal pairs of component signals 75, 76, 77, and 78. The pair of component signals 75 and 76 is dominant. Mode 2 is decomposed into two component signal pairs, and the dominant pair of component signals 55 and 56 takes 88% of the energy of the mode. Figures ^{rd} and 4^{th} modes. The 3^{rd} mode is decomposed into a dominant pair of component signals 41 and 42 and a nondominant pair of component signals 63 and 64. Mode 4 is decomposed into the pair of component signals 37 and 38. Figures ^{th} and 12^{th} component signals forms a harmonic noise at 2572 Hz. The peak frequencies of these noise component signals are not in the cared frequency bands, and they are abandoned as the noise.

The modes and the corresponding component signals. (a) Component signals of mode 1 at 65 Hz. (b) Component signals of mode 2 at 399 Hz. (c) Component signals of mode 3 at 428 Hz. (d) Component signals of mode 4 at 1119 Hz. (e) Component signals of modal 5 at 1334 Hz. (f) Component signals of modal 6 at 2195 Hz. (g) Component signals of mode 7 at 2381 Hz. (h) Component signals of noise at 2750 Hz. (i) Component signals of the noise at 2572 Hz.

The noise reduction result and the component signal analysis show that the reselection iteration with the bandwidth 35 Hz can separate the component signals of the modal modes from the component signals of the residue noises precisely. Then, in order to investigate the effect of the reselection with different pass bandwidths on denoising, the noise-reduced FRFs obtained from MPHSVD with bandwidths 2 Hz, 5 Hz, 10 Hz, 20 Hz, 45 Hz, 60 Hz, and 80 Hz are plotted in Figure

The results with different pass bandwidths. (a, c, e) Energy spectrum comparison. (b, d, f) Phase spectrum comparison.

After the MPHSVD is implemented, the curve fitting with the rational fraction polynomial (RFP) method is conducted to extract the modal parameters. Modal modes are identified piecewise, and the modal frequencies with the correlated damping factors are extracted for each mode. The curving fitting of the noise-reduced FRF signal is plotted in Figure

Curve fitting result of the noise-reduced FRF signal. (a) Curve fitted energy spectrum of denoised FRF. (b) Curve fitted phase spectrum of denoised FRF.

Modal parameters of the noised and noise-reduced FRF signals.

Mode | Modal frequency (Hz) (original) | Modal frequency (Hz) (denoised) | Frequency difference (Hz) | Damping factor (original) | Damping factor (denoised) | Damping factor difference (%) |
---|---|---|---|---|---|---|

1 | 63.6779 | 63.61561 | 0.062331 | 0.000674 | 0.001091 | 61.78994 |

2 | 399.332 | 399.2953 | 0.036889 | 0.000295 | 0.000393 | 32.93652 |

3 | 428.076 | 428.108 | 0.032425 | 0.000667 | 0.000553 | 17.06583 |

4 | 1119.44 | 1119.429 | 0.013394 | 0.000866 | 0.000788 | 8.984838 |

5 | 1334.57 | 1334.775 | 0.202722 | 0.000943 | 0.000892 | 5.423373 |

6 | 2195.12 | 2195.505 | 0.385319 | 0.001474 | 0.001456 | 1.267132 |

7 | 2381.41 | 2380.844 | 0.562656 | 0.001588 | 0.001563 | 1.541009 |

In order to further investigate the effectiveness of the MPHSVD to deal with various degrees of the noise, four degrees of the random Gaussian noise, listed as 3%, 5%, 7%, and 9% ratios between the amplitudes of the added Gaussian noise and the max amplitude of the noise-free IRF signal, respectively, are added to the simulated noise-free IRF signal. Five simulations of each ratio are conducted, and their corresponding mean SNRs are 7.68 dB, 3.25 dB, 0.31 dB, and −1.89 dB, respectively. Then, the noise removal rates are calculated by equation (

Noise reduction result.

Simulation time | Noise removal rate (SNR = 7.68 dB) | Noise removal rate (SNR = 3.25 dB) | Noise removal rate (SNR = 0.31 dB) | Noise removal rate (SNR = −1.89 dB) |
---|---|---|---|---|

1 | 0.947717 | 0.902006 | 0.858859 | — |

2 | 0.945414 | 0.903764 | 0.881271 | — |

3 | 0.941209 | 0.916029 | 0.845278 | 0.837777 |

4 | 0.942136 | 0.916785 | — | |

5 | 0.947392 | 0.906086 | — | 0.815477 |

Successful rate | 100% | 100% | 60% | 60% |

Overlays of the FRFs with various SNR values. (a) Overlay of the FRFs with SNR = 7.68 dB. (b) Overlay of the FRFs with SNR = 3.25 dB. (c) Overlay of the FRFs with SNR = 0.31 dB. (d) Overlay of the FRFs with SNR = −1.89 dB.

Figures

To further verify the proposed method, an experimental test is undertaken. The force hammer typed as PCB086C03, manufactured by PCB, is used for excitation. The UM-40-AB-HF electromagnetic heater and the heating coil, produced by UIHM, are used for heating the plate. The 3D scanning laser Doppler vibrometers (SLDVs) typed with PSV-400-3D for response measurement are manufactured by Polytec. The experimental setup is shown in Figure

(a) The experimental setup. (b) The measurement points.

In the modal test, the flat plate is firstly grounded at the one end to a metal baffle tightened with four bolts, and the baffle is connected to the test bench to monitor the rigid support. Then, the heating coil is connected to the bench behind the plate and connected to the electromagnetic heater for heating the plate. The force hammer is then connected to the computer for excitation and measuring the input force. Subsequently, the SLDVs are placed in front of the test plate for the noncontact measurement. The measuring points are evenly distributed on the plate, and totally 45 points are selected.

The test starts when the temperature of the plate is heated and stabled at the 200 centigrade degrees. The analysis frequency range is 6400 Hz, and it takes 2 seconds once for each measuring point. After testing, the FRF signals are obtained for these 45 measuring points.

The measured FRF signals of Point 40 and Point 1, marked as the red circle in Figure

Overlay of the FRFs of Point 1 and Point 40. (a) Energy spectrum comparison. (b) Phase spectrum comparison.

It can be found from Figure

For the transformed IRF signals of Point 40 and Point 1, the MPHSVD is conducted to eliminate the noise and retain the concerned modal features. The amplitude difference threshold of accumulation is set as 0.15, and the component signal accumulations of the two transformed IRF signals are conducted. The accumulation of the transformed IRF signal of Point 1 stops at the 43 iteration turn, and the first 43 component signals which cover the whole concerned modal modes are selected. Then, as there are ten modal peaks selected from the energy spectrum of IRF signal 1, ten frequency pass bands with the bandwidth 10 Hz near the modal peak frequencies are set for the reselection of the component signals of the modal modes. Finally, 32 component signals of the first 43 component signals are reselected, and they are added up to form the noise-reduced IRF signal and the noise-reduced FRF signal. For the transformed IRF signal of Point 40, the first 1119 component signals are selected in the accumulation. Then, as there are 8 modal peaks selected from IRF 40, eight frequency bands with the bandwidth 10 Hz are set for the component signal reselection, and 25 component signals from the first 1119 component signals of IRF 40 are reselected for the construction of the noise-reduced FRF signal. Finally, the overlay between the denoised FRF signals and the FRF signals from the transformed IRF signals is given in Figure

Noise reduction result of Points 1 and 40. (a) Energy spectrum of Point 1. (b) Phase spectrum of Point 1. (c) Energy spectrum of Point 40. (d) Phase spectrum of Point 40.

From Figures

(a) Noise modes at 1127 and 1149 Hz. (b) Noise mode at 1014 Hz.

Component signals of noise modes at 1014, 1127, and 1149 Hz. (a) Component signals 78, 80, and 81. (b) Component signals 79, 80, and 81. (c) Component signals 275, 276, and 277. (d) Component signals 278, 279, and 280. (e) Component signals 281, 282, 283, and 284. (f) Component signals 548 and 549. (g) Component signals 1009, 1010, and 1011. (h) Component signals 1012, 1013, and 1014. (i) Component signals 1015, 1016, 1018, and 1019.

After the MPHSVD is done, the RFP method is conducted to extract the modal parameters of the two denoised FRF signals. The curve fitting result and the extracted modal parameters of the concerned modes are shown in Figure

Curve fit of the noise-reduced FRF signal. (a) Energy spectrum of Point 1. (b) Phase spectrum of Point 1. (c) Energy spectrum of Point 40. (d) Phase spectrum of Point 40.

The extracted modal parameters.

Mode | Modal frequency (Hz) (Point 1) | Modal frequency (Hz) (Point 40) | Frequency difference (Hz) | Damping factor (Point 1) | Damping factor (Point 40) | Damping factor difference (%) |
---|---|---|---|---|---|---|

1 | 64.63889 | 64.58773 | 0.05116 | 0.001152 | 0.001335 | 15.87176 |

2 | 392.3132 | 393.5316 | 1.2184 | 0.00645 | 0.005472 | 15.16743 |

3 | 459.1516 | 459.0252 | 0.1264 | 0.000667 | 0.000651 | 2.477419 |

4 | 1015.351 | — | — | 0.00753 | — | — |

5 | 1125.492 | 1126.48 | 0.988 | 0.00459 | 0.007716 | 68.11871 |

6 | 1147.168 | 1148.179 | 1.011 | 0.003201 | 0.00217 | 32.22889 |

7 | 1252.44 | — | — | 0.005597 | — | — |

8 | 1419.851 | 1419.687 | 0.164 | 0.000415 | 0.000545 | 31.51598 |

9 | 2175.132 | — | — | 0.002125 | — | — |

10 | 2524.38 | 2523.689 | 0.691 | 0.000578 | 0.000676 | 16.9878 |

In this paper, a novel noise reduction method based on the modal peak-based Hankel-SVD technique (MPHSVD) is proposed, which aims to acquire the precise FRF signal from the noised FRF signal for the modal identification. The proposed method contains one accumulation iteration to select the component signals which cover the whole modal modes and some residue noise and one reselection iteration to separate the component signals of the modal modes from the residue noise for completing the noise reduction. The advantage of the method is that the component signals correlated with the modal modes can be determined by the two iterations. The simulated FRF of a plate with the added heavy noise is taken as an example to show how the denoising method works. The noise removal rate reaches above 93%, and the further modal analysis shows that the modal frequencies and modal damping factors are extracted precisely. Besides, the detailed analysis of the correlation between the component signals and the modal modes and effect of different bandwidths for the noise reduction are also investigated. The simulation analysis of FRFs contaminated with various degrees of Gaussian noise shows that the noise reductions are fully successful for FRFs with SNRs of 7.68 dB and 3.25 dB, respectively. The application of two experimental FRF signals of a flat plate in the high-temperature environment also shows the superiority and effectiveness of the proposed method. However, some amplitudes of the modal modes contaminated with too much noise decrease after the noise elimination. Nevertheless, the method shows the practical potential to engineering application in the modal test.

Data are available on request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully appreciate the financial support for this work provided by the National Natural Science Foundation of China and the National Safety Academic Foundation of China (no. U1730129). The supports from the Jiangsu Province Key Laboratory of Aerospace Power System and the Key Laboratory of Aero-Engine Thermal Environment and Structure, Ministry of Industry and Information Technology, are also gratefully acknowledged.