Nondestructive testing technology refers to a method of inspection and testing that relies on modern technology and instruments. It involves analyzing changes in the response of materials, parts, and equipment without causing damage or affecting the performance and internal structure of the tested object. Vibration is inherent in various aspects of daily life and is particularly prevalent in mechanical fields such as automobile and ship manufacturing (Festa et al., 2021). Recognizing the significance of this phenomenon and the need to mitigate its negative effects, numerous scholars have delved into the study of mechanical vibration and vibration systems. As an illustration, let’s consider a single-degree-of-freedom system, where the primary forces acting upon it are depicted in Figure 1.

In Figure 1, k is the stiffness of the spring; F t indicates external incentive force; x represents the displacement of the object; c represents the damping of the object; m indicates the mass of the object. When the object in the system in Figure 1 is in the equilibrium position, the resonance force formula can be obtained through the equilibrium conditions, as shown in Eq. 1. ∑ F x = 0 , m g − k δ t = 0 (1)

In Eq. 1, δ t is the spring static deformation. At the same time, when the object is excited by the outside, the system will deviate from the equilibrium position. At this moment, the formula of the system is m x . . . Assuming that there is no damping in the system, p n = k m , the vibration differential formula of undamped SDOF system can be obtained in Eq. 2. x . . + p n 2 x = 0 (2)

In Eq. 2, p represents the natural frequency obtained by system vibration. Equation 3 is the calculation of natural frequency based on Eq. 2. f n = 1 T = p n 2 π = 1 2 π k m (3)

In Eq. 3, f n is the natural frequency. Integrating the above formulas, the differential motion formulas of the undamped system with n freedom degrees can be obtained. It is assumed that the special solution of the differential formula of motion for the free vibration of an undamped system of degrees of freedom is x i = A i ⁡ sin p t + φ i = 1 , 2 , 3 , ⋯ , n or x = A ⁡ sin p t + φ , where A = A 1 , A 2 , ⋯ , A n T . K A − p 2 M A = 0 can be obtained through the matrix M x . . + x K = 0 . Frequency formula of the system is shown in Eq. 4. K − p 2 M = 0 (4)

In Eq. 4, the obtained frequency formula is actually a p 2 polynomial of n degree, and natural frequencies n can be obtained by calculation (Tsalera et al., 2021). The vibration system with corresponding n degrees of freedom has a n natural frequency, and all frequencies are sorted from small to large as follows 0 ≤ p 1 ≤ p 2 ≤ ⋯ ≤ p n . Among the above frequencies, p 1 refers to the first natural frequency of system vibration, and p 2 refers to the second natural frequency. The relationship between the object studied by this method and the outside world is shown in Figure 2.

In Figure 2, it can be observed that during experimental modal analysis, it is essential to collect complete sets of data pertaining to system excitation and response. By utilizing this comprehensive input and output information for parameter identification, the frequency response function can be obtained, followed by the determination of modal parameters. Equation 5 is the specific expression representing the relationship. H ω = ∑ i = 1 n ϕ i ϕ i T K i − ω 2 M i + j ω C i (5)

In Eq. 5, M i is the modal mass matrix; K i is the modal stiffness matrix; C i is the modal damping coefficient matrix. The study uses the hammering method to obtain the natural frequencies of ship parts. The experimental system diagram of the hammering method is shown in Figure 3.

On the basis of all the above experiments, ANSYS, abaqus, and other simulation software are used to carry out modal analysis experiments. The ultimate purpose is to obtain the natural frequency of ship reliability (Sudo et al., 2021). For a continuous initial signal, the calculation obtained by Short-Time Fourier Transform (STFT) is shown in Eq. 6. F t , f = ∫ x τ ω τ − t e − j 2 π f t d τ (6)

In Eq. 6, x τ represents the signal waiting for analysis; ω τ represents the window function; F t , f represents the spectrum distribution of the signal x τ at the time t . However, during the actual manufacturing and operation of ships, computers and embedded devices can only read and process discrete digital signals, and the expression of discrete STFT is shown in Eq. 7. X n , m = ∑ − ∞ ∞ x m ω n − m e − j ω m (7)

In Eq. 7, x m represents the input signal; ω m represents the window function; X n , m represents a two-dimensional function defined on the sample time and frequency; n indicates the number of samples; Finally, the time-frequency diagram formula of input signal is obtained as S n , ω = X n , ω 2 .

The research conducted on the Vibroacoustic nondestructive testing reliability platform, which utilizes the hammering method to obtain the natural frequency of parts, focuses on analyzing the sound signals captured by high-precision microphones.

However, during the actual quality inspection of ship parts, various types of interference may be present, necessitating the implementation of suitable methods to mitigate noise and enhance the quality of acoustic vibration signals. In this regard, SS is introduced as a noise reduction technique. Considering that the acoustic signal generated by knocking in the production process of ship parts constitutes a non-stationary and time-varying signal, it is necessary to perform framing to appropriately process the signal. It is assumed that in one frame, y n is the input initial sound signal; x n is pure sound signal; S n indicates a noise signal. At the same time, FT is used to express the relationship in Eq. 8. y n = x n + s n Y ω = X ω + S ω (8)

In Eq. 8, Y ω is initial sound signal; X ω is pure sound signal; S ω is a noise signal in the environment. The above relationship is optimized by power spectrum in Eq. 9. Y ω 2 = X ω 2 + S ω 2 + X ω S ω − + X ω − S ω Y ω 2 = X ω 2 + S ω 2 + 2 Re X ω S ω − (9)

In Eq. 9, S ω − and S ω − represent the complex conjugate of S ω and X ω ; X ω 2 , S ω 2 represent the short-time power spectrum of pure sound signal and noise signal; 2 Re X ω S ω − represents a cross item. When s n represents zero mean, s n and x n are not related to two parameters, the cross term is 0. This relationship is simplified in Eq. 10. Y ω 2 = X ω 2 + S ω 2 X ω 2 = Y ω 2 − S ω 2 (10)

In the process of ship parts manufacturing and production, sound signal quality is closely related to frequency amplitude spectrum generated by the parts, and there is no obvious relationship with frequency phase diagram. Therefore, this study retains the frequency phase spectrum of initial sound signal to replace frequency phase spectrum of pure sound signal, and then estimates power spectrum of the noise. The expression of noise reduction processing for sound signal is shown in Eq. 11. X ∧ ω 2 = Y ω 2 − S ∧ ω 2 Y ω = Y ω e − j φ y ω S ω = S ω e − j φ s ω (11)

In Eq. 11, S ω 2 is power spectrum of noise signal estimated as S ∧ ω 2 ; X ∧ ω 2 is that of pure sound signal; Y ω is amplitude spectrum of noisy signal; φ y ω represents phase spectrum; S ω is the amplitude spectrum; φ s ω is the phase spectrum. After the pure power spectrum is squared, pure sound signal of this frame can be obtained by FT combined with the frequency and phase spectrum φ y ω of noisy sound signal, and pure sound signal after noise reduction of the whole sound signal can be obtained by iterating all the obtained frames. Basic reduction principle is shown in Figure 4.

In Figure 4, during the production of ship parts, when applying the SS method for noise reduction, certain challenges may arise, such as the occurrence of noise residue due to overestimation of the noise spectrum. Additionally, issues like “music noise” can occur when the noise level is too low. To address these concerns, the parameters are optimized based on an enhanced SS method and a platform noise reduction module that effectively mitigates noise interference is developed.

To overcome the problems of noise residue and “music noise”, an improved SS is proposed. The formula for resolving the negative value issue resulting from excessive noise reduction using the improved SS is presented in Eq. 12. D X ω = X ∧ ω 2 , D Y ω = Y ω 2 , D S ω = S ∧ ω 2 l e t D X ω = D Y ω − D S ω D X ′ ω = D X ω   i f , D X ω > 0 0   o t h e r w i s e (12)

In the process of noise reduction using improved method, over subtraction factor and threshold parameter of the lower spectral limit are introduced to reduce residual noise and reduce “music noise” perception. Equation 13 is the noise reduction formula after introducing factors. l e t D X ω = D Y ω − D S ω D X ′ ω = D S ω   i f , D S ω > β D S ω β D S ω   o t h e r w i s e w i t h α ≥ 1 , 0 < β ≪ 1 (13)

In Eq. 13, α represents the over subtraction factor; β indicates the lower limit threshold parameter of the spectrum. In the implementation of improved method, over subtraction factor value is not fixed, and the factor value must be determined according to SNR of each audio frame in Eq. 14. α = 5   f o r , S N R ≤ − 5 α = − 0.16 * S N R + 4.2   f o r ≤ S N R ≤ 20 α = 1   f o r , S N R ≥ 25 (14)

In Eq. 14, S N R stands for signal-to-noise ratio. When SNR is low, α is larger; When small, α is small. Threshold parameter of the lower spectrum limit depends on SNR of input signal. Equation 15 is the specific calculation. β = 0.06   f o r , S N R ≤ − 5 β = − 0.011 * S N R + 0.005   f o r , − 5 ≤ S N R ≤ 0 β = 0.003 * S N R + 0.005   f o r , 0 ≤ S N R ≤ 5 β = 0.02   f o r , S N R ≥ 25 (15)

By introducing the over subtraction factor to increase the subtraction of noise spectrum, and limiting minimum value of spectrum to β D s ω by β , the trough between the peaks began to level off, and finally successfully reduced the perception of “music noise”. The improved spectral subtraction process obtained is shown in Figure 5. In Figure 5, during the implementation process of improved spectral subtraction, the value of the super-subtraction factor is not fixed and needs to be adjusted according to each audio frame. The signal-to-noise ratio is used to determine the appropriate value. When the signal-to-noise ratio is low, the over-reduction factor is larger; when the signal-to-noise ratio is high, the over-reduction factor is larger. The reduction factor is smaller.

A ship is a complex floating structure that encounters various dynamic and static loads, including waves, wind, cargo loads, machinery, and more. The condition of individual structural components within a ship plays a vital role in ensuring its overall reliability. To enhance the detection and assessment of a ship’s structure and reliability, the experiment involves gathering historical data related to maintenance, repairs, and accidents of the ship. Through the application of failure mode and effect analysis, key structures and potential failure modes of the ship are identified, enabling a comprehensive understanding of its integrity.

After the optimization of the reliability level measurement system for ship parts detection, it is necessary to compare the performance of the research algorithm in the actual operation process to test its effectiveness. To ensure fairness and rationality of overall process, Spectral Subtraction (Spectral Subtraction, SS), the Improved Wavelet Transform (IWT), and the methods and research methods in literature (Liu et al., 2021b) were selected for performance comparison. Except for the special experimental parameters, the experimental parameters of all other algorithms are the same as those of the research method. The number of iterations of all algorithms is set to 150, and then the performance of different models is analyzed based on the German dagm-2007 data set and fabric data set. The convergence of four different algorithms is shown in Figure 6.

Figure 6A shows the convergence comparison on the dagm-2007 data set in Germany. As iterations of the system increased, the fitness of all algorithms showed an increase in varying degrees, accompanied by the upward and downward fluctuation of the curve. When the iteration runs to 150 times, SS, IWT and literature (Liu et al., 2021b) do not reach a stable change trend. When the iteration runs to 91 times, the maximum fitness value of the research method is 97.2, and then it keeps stable operation. Figure 6B shows the experiment of convergence on the fabric data set. During the operation of the system, the fitness values of all algorithms change rapidly and tend to be stable within 100 generations of iteration. When the iteration runs 15 times, the fitness value of the research method starts to approach 99.9; When the operation times are 105, 110 and 89 respectively, the fitness values of IWT, literature (Liu et al., 2021b) and SS methods begin to tend to be stable, which are less than the research methods. In summary, the research method has the highest fitness value and faster convergence.

Then, the German dagm-2007 data set is used as the main data set to expand the performance test, and the accuracy and recall rates of the four algorithms are compared. The specific PR (Precision Recall) line is shown in Figure 7.

The comparison of PR curves in Figure 7 shows that when the accuracy rate of the algorithm is 0.900, the recall rate of the corresponding research method is 0.712; The recall rates of SS, IWT and literature (Liu et al., 2021b) were 0.532, 0.585, and 0.632, respectively. On the contrary, when the recall rate of all models is 0.800, the accuracy rate of the corresponding research method is 0.821; The accuracy rates of SS, IWT and literature (Liu et al., 2021b) were 0.667, 0.698, and 0.724, respectively. The results above show that when the accuracy rate or recall rate is the same value, recall rate and accuracy rate of the research method are the largest. This also means that the research method has a higher accuracy in the optimal selection of the reliability level measurement system for ship parts detection, and can provide a more comprehensive service for ship parts detection. At the same time, a higher recall rate can improve the overall utilization rate of ship reliability level measurement in the market.

The research compares the reliability detection accuracy, uncertainty, sensitivity and confidence interval of different models under the fabric (Fabric) data set test, as shown in Table 1.

As shown in Table 1, under the fabric (Fabric) data set test, the accuracy of the research method is 98.54%; the uncertainty is 5.58; the sensitivity detection is 0.26 μm; the confidence intervals of the four algorithms are all 95%. Among the other three algorithms, the IWT algorithm has the best performance. The accuracy of this algorithm is 97.78%, the uncertainty is 6.94, and the sensitivity is 0.25 μm. The accuracy of the literature [21] and the SS method are 97.89% and 96.77% respectively, which is far less than the accuracy of the method constructed in the study. From the comparison, it can be seen that the research method has a higher accuracy in detecting ship parts, and the correct prediction of parts detection accounts for a larger proportion, which is consistent with the change trend of the PR curve, which further proves that the research method can maintain good robustness during operation.

To more comprehensively verify the robustness of the constructed method, the Fabric data set was then used as the main task data set to analyze the F1 values of the four algorithms for comparison. The results are shown in Figure 8.

As can be seen in Figure 8, as the number of iterations increases, the F1 value of the research method has always been at the maximum, approaching 1.00. And the F1 value is greater than 0.5 at the beginning of the iteration. The F1 values of the remaining three algorithms have always been smaller than the research method, and the values fluctuated greatly. The final F1 value was also far smaller than the F1 value of the research method. This shows that the error in ship parts detection under the operation of the research method is small and has excellent robustness.

To simulate environmental noise during the actual operation of the ship, this study mixes the blower, voice dialogue and low amplitude high-frequency noise to briefly simulate the industrial noise that will appear on the assembly line of the ship parts quality inspection workshop, and uses the research method for noise reduction. To carry out regular testing and evaluation of ship parts smoothly, this experiment selects Ultrasonic Testing (UT) and Radiographic Testing (RT) to assist in testing the reliability level of ship parts. In the reliability level detection of ship parts, the application of high-frequency eigenmode or modal analysis can provide engineers and maintenance personnel with the location of ship cracks and defects, ship material degradation, contact and friction problems, and loosening of bolt joints Interference with parts, etc. In conclusion, high-frequency eigenmode or modal analysis provides an effective tool for reliability level detection of ship components and can help diagnose various potential structural and material problems. The research method is applied to noise reduction in the actual environment. The sound signal waveform changes before and after SS noise reduction are shown in Figure 9.

Figure 9A is a noise-free speech signal; Figure 9B is a noisy speech signal; Figure 9C is the spectrogram of the noise-reduced speech signal. Comparing the three spectrograms, it can be found that the spectrogram obtained after using the improved spectral subtraction method to de-noise the sound signal has fewer residual yellow spots and almost no excess noise remains. This shows that the research method is significantly better than the noise speech signal processing effect. This may be because this research method uses vibroacoustic principles to preprocess noise before noise reduction.

Meanwhile, to more accurately show the advantages of the research method, improved SS method can compare spectrograms of noise-reduced sound signals, as shown in Figure 10.

Figure 10A is a noise-free speech signal; Figure 10B is a noisy speech signal; Figure 10C is the spectrogram of the speech signal after noise reduction. Comparing the three spectrograms, it can be found that the spectrogram obtained after using the improved spectral subtraction method to de-noise the sound signal has fewer residual yellow spots and almost no excess noise remains. This shows that the research method is significantly better than the noise speech signal processing effect. This may be because this research method uses vibroacoustic principles to preprocess noise before noise reduction. Using the sound function in MATLAB to monitor and detect the signal, it can be found that the sound signal after the improved SS de-noising basically has no noise residue and “music noise”. Results show that improved SS has high feasibility in noise reduction. Figure 11 shows the time domain comparison of three kinds of knocking signals.

Figure 11A is the amplitude diagram of the noiseless tapping signal; Figure 11B is the amplitude diagram of the noise tapping signal; Figure 11C is the amplitude diagram of the noise-reduced tapping signal. It can be found that the amplitude of the noise in the sound signal after noise reduction processing has been reduced to a very low level, and there is basically no difference compared with the noiseless tapping signal. This means that the research method has eliminated most of the noise signals. From the above results, research method has superior detection performance for ship parts detection, and can realize the accurate detection of parts through noise reduction, so as to ensure the calculation of ship reliability level. Finally, the modal simulation data of ship parts are compared with the experimental data through the experimental platform. Based on the above experimental results, this experiment generates simulation data by fitting the model to the existing data and using the model to predict the future situation.

In the experiment, six structural elements including ship deck, mast, keel, cabin, propeller and double bottom were selected for natural frequency detection. The specific results obtained are shown in Table 2.

As shown in Table 2, the difference between the natural frequencies obtained by the simulation data and the experimental data is small, and the smallest natural frequency difference is the keel component of the ship, and the natural frequency difference is only 0.01; and the largest natural frequency difference is for the double bottom element of the ship, the natural frequency difference is 60.00. At the same time, due to the inevitable errors in the process of modeling, it can be considered that the ship parts detection platform under the research method can obtain the accurate natural frequency of parts and ensure structure reliability level in the process of ship operation.

The natural frequency obtained in the experiment is relatively high. For ship components, the high natural frequency may mean that the component is not easy to enter the resonance state when it is subjected to common external disturbances (such as waves, wind, etc.), which is beneficial. However, that doesn't mean detecting tiny flaws on ship components isn't useful. Here are a few reasons: First, the natural frequency of a part may change over time due to environmental factors, material aging, or other reasons. These changes can be accelerated if a tiny defect exists on a part. Second, although natural frequency is an important parameter, there are many other factors that need to be considered when evaluating the performance and integrity of ship components. This also means that even if the natural frequencies of the experimentally detected ship parts are high, it still makes sense to regularly inspect the parts and repair minor defects.