Research Article  Open Access
A Correction Method for the Underwater Shock Signals of Floating Shock Platforms Based on a Combination of FFT and LowFrequency Oscillator
Abstract
Accurate shock loading is required for evaluating and analyzing the shock resistance of warship equipment. However, measured shock acceleration signals contain trend term errors, which cause serious lowfrequency distortion of the shock response spectrum (SRS). We propose a combination method of fast Fourier transform (FFT) and lowfrequency oscillator for correcting the underwater shock signals. Based on the equal displacement line fitted by the measured displacement response data, the Fourier transform spectrum of the shock acceleration signal is corrected for eliminating lowfrequency errors. The results of underwater explosion tests on a floating platform indicate that the average difference between the equal displacement line and SRS in the lowfrequency band (4∼20 Hz) can be reduced from 14.7% to 3.5%, and the midhighfrequency band without the trend term is nearly unaffected. The corrected SRS can faithfully reflect the actual shock environment of the warship equipment at a specific installation location of the floating shock platform.
1. Introduction
With the rapid development of modern science and technology, the explosive power of underwater weapons is constantly improving. Meanwhile, warship equipment has become more and more sophisticated [1]. Therefore, the requirements of shock resistance performance for equipment are gradually increasing [2]. Compared to the shock testing machine, a floating shock platform can realistically simulate the shock environment of an actual warship facing underwater noncontact explosions. For these reasons, floating shock platforms, which were used as standard test devices for evaluating the shock resistance of largescale warship equipment, have been widely used in navies around the world [3, 4].
Shock strength of underwater noncontact explosions on a floating shock platform is usually characterized by the SRS converted from acceleration signals [5, 6]. Due to the highfrequency, transient, and highorder mechanical environment characteristics of underwater noncontact explosions, the requirement for the signal acquisition system is very strict. Acceleration signals are susceptible to complex errors according to measurement system and other environmental factors. Trend term error is one of these types of errors [7, 8]. The existence of a trend term error will cause a great deviation or even a complete distortion in the lowfrequency band of SRS [9]. This will seriously affect the judgment on shock strength and the assessment results of shock tests.
Since the current shock tests cannot eliminate the existence of trend term errors, it becomes necessary to develop some correction techniques to separate the lowfrequency trend term from the measured acceleration data. Several scholars have proposed various methods for correcting the acceleration data. The main methods include the least squares method, a lowpass filtering method, a Fourier analysis method, a discrete wavelet transform (DWT) method, and an empirical mode decomposition (EMD) method. Irvine [10] summarized six factors that produce trend errors in the shock test. Gaberson [11] emphasizes that the acceleration, velocity, and displacement curves must be zero at the end of the shock when drawing the SRS so that the SRS can present theoretical characteristics at each frequency. Grillo [12] used a least squares approach to remove the trend term from the integrating velocity signal and then differentiated the velocity to obtain a corrected acceleration signal. However, this method cannot correct the acceleration signal containing abrupt spikes. Du et al. [13] proposed a method of combining the design shock spectrum with the measured signal via a Fourier transform method; however, the design shock spectrum according to a set of relevant standards cannot reflect the dynamic characteristics of underwater explosion shock signals for actual warships. Smallwood and Cap [14] first used DWT to correct inaccurate shock test data. Edwards [15] proposed a method for selecting corrected wavelet parameters in subsequent studies. Yet, DWT only has a few specific mother wavelets to choose from. Li et al. [16] proposed an adaptive EMD correction method using correlation coefficients of positive and negative SRS as the reconstruction condition. However, the EMD method has its own drawbacks, namely, end effect, modal aliasing, and inability to separate effectively.
Due to the strong nonlinearity and transient nature of shock data, there is no standard specification or a uniform parameter selection method that can automatically and accurately correct the acceleration data [17]. As with most techniques, a common practice is to apply the correction in increasing severity to the acceleration data until the SRS yields physically plausible values [18]. Because there is no independent displacement response measurement to compare, the accuracy of the corrected data is mostly judged by the operator experience. Thus, although the corrected acceleration is considered credible, it is not necessarily correct. And then, the effectiveness of shock resistance performance analysis for warship equipment will be affected. In view of these, a correction method based on a combination of FFT and lowfrequency oscillator is proposed.
Hereinafter, the relation of SRS with Fourier amplitude spectrum (FAS) is discussed. And then, the combination correction method based on the ratio between FAS and the equal displacement line fitted by lowfrequency oscillator data is proposed. In Section 3, the shock test method for the floating shock platform and the method for obtaining the equal displacement line are present and discussed. Finally, the test results show that the corrected shock signal can perfectly reflect the actual shock characteristics.
2. Principle of the Combination Correction Method
2.1. Relation of SRS with FAS
The relative displacement response of a linear undamped singledegreeoffreedom system (SDOF) to a generalized excitation is written as follows [19]:where ω_{n} is the natural angular frequency of the nth SDOF. The above equation can be expanded by a trigonometric function equation as follows:which is of the formwithin which C and S are functions of time t. can still be written aswith
When the function , the response has a maximum for this value. This yields the maximum absolute value of :
In addition, although the actual damping is not zero, the damping effect on maximum relative displacement response is less than 5% when the damping ratio is 0.05 [20]. According to the data obtained from the shock test, the structural damping ratio of the floating platform is generally about 0.035 [21]. Therefore, the damping can be neglected in this work.
On the contrary, the Fourier transform of is calculated as if the shock signals were nonzero with ω_{n}, and the pulsation is written as
Transforming the above equation by the Euler equation, the actual and imaginary parts of can be expressed by the following equation, respectively:
Since equations (9) and (10) correspond to C and S, respectively, we can obtain the following relation by equation (7):
According to the shock spectrum theory, pseudovelocity shock spectrum (PVSS) is ω_{n} times of the maximum relative displacement spectrum . With this representation, the FAS can be considered identical to the PVSS of SRS. Although there are errors between fast Fourier transform and the digital recursion method for computing SRS, the FAS obtained by FFT can be still regarded as equivalent to SRS within the warship shock engineering applications (natural frequency: 2 Hz∼1000 Hz) [22].
2.2. Combination Correction Method
If the equal displacement line fitted by the actual displacement data is equal to the lowfrequency part of SRS , no correction is required. Otherwise, it is necessary to correct the shock acceleration according to . The discrete Fourier transform of the shock acceleration signal can be expressed as
The ratio of the equal displacement line to the lowfrequency part of SRS can be defined as
According to the prediction of the shock environment characteristics of the floating platform, the frequency range of the equal displacement part of SRS is mainly between 4 Hz and 20 Hz [23].
To ensure coincides with , the Fourier transform amplitude of the measured shock acceleration signal is corrected bywhere consists of two parts: the lowfrequency part and the medium and highfrequency part . Because the trend term error mainly concentrates at the lowfrequency part, the medium and highfrequency spectrum is not corrected. Therefore, the values of each frequency in are equal to 1.
And then, the corrected shock acceleration can be obtained by an inverse Fourier transform on , that is,
Obviously, the lowfrequency part of SRS converted from is more approximate to . Finally, repeating the above calibration process can make the acceleration signal correction reach a certain precision requirement.
Moreover, the acquisition method of the equal displacement line is discussed in the next section.
3. Shock Test on Floating Shock Platform
3.1. Test Method for Floating Shock Platform
The length, width, and height of the floating shock platform are 19 m, 9.1 m, and 7 m, respectively. The internal effective height is 6 m, the double bottom height is 1 m, bulkhead height is 0.5 m, and maximum draft is 3 m. The newly developed test equipment (weight 80 ton) is installed in the center of the floating platform bottom. In order to evaluate the shock strength of the test equipment, accelerometers (Bruel & Kjaer, model 8309, Denmark) are installed next to the test equipment and then converted to SRS from the acceleration signals for XYZ directions, respectively. Because the measured shock accelerations contain trend term errors, a lowfrequency oscillator group is installed next to the accelerometers for correcting the SRS by the actual measured displacement response. Before the test, a 150 kg TNT standard explosion source is placed in the horizontal direction of the floating platform. On the premise of safety, the explosion source will be detonated. The test schematic diagram of the floating platform is shown in Figure 1, and the test conditions are listed in Table 1.

3.2. Measurement Method for LowFrequency Oscillator
At the moment of underwater explosion, the shock signals are received by sensors installed on various parts of the floating platform. And then the information is stored in the data acquisition system for subsequent data analysis. The measured physical quantities mainly include the acceleration of the platform bottom, the displacements of the lowfrequency oscillator group, and the structural strain on the test equipment. We only consider the first two physical quantities here.
The lowfrequency oscillator group is divided into vertical, longitudinal, and horizontal directions. The vertical structure diagram of the lowfrequency oscillator group is shown in Figure 2. The oscillators in the other two directions are similar to this structure.
Figure 2 shows that the vertical oscillator system consists of three suboscillator systems with different frequencies. The subsystems exist independently, and the vibration conditions do not affect each other. In addition, the three subsystems are fixed onto the same base to ensure that they are at the same level. The base is rigidly connected to the bottom of the floating platform.
Each suboscillator system consists of two cylinder springs, a block of mass, a sliding bearing, a circular guide plate, and two guide shafts. The two springs are precompressed and installed on both ends of the block of mass. Both springs are always compressed during operation. The smooth cylindrical shaft in the middle of the springs acts as the guide shaft for springs and the block of mass. A sliding bearing is installed inside the block of mass to reduce friction from the coefficient between the block of mass and the guide shaft. To prevent the oscillator from rotating due to the spring force, a circular guide plate has been attached to the block of mass. The plate is designed to cooperate with the auxiliary guide shaft. According to reference [23], the frequency range for equal displacement part of SRS is mainly from 4 to 20 Hz. Thus, the nature frequencies of the three suboscillator systems are adjusted to 6 Hz, 10 Hz, and 20 Hz, by adjusting the weight of the block of mass and the stiffness of the springs.
The motion displacement of each oscillator is measured by a noncontact magnetostrictive displacement sensor (Miran, model MTM3, China); this is more suitable to a shock environment (200 g). Its maximum measurement error is ±0.05% Fs, and an effective measurement range is 0∼80 mm, which meets the requirements of the test. The main body of the displacement sensor is fixed behind each oscillator and is parallel to each guide shaft. The matching magnetic sensing block is fixed onto the each circular guide plate. When the underwater explosion shock occurs, the magnetic sensing block reciprocates in the vertical direction with the block of mass. Thereby, the relative displacement can be measured.
Because each suboscillator system can be considered a singledegreeoffreedom system, the displacement measured by each oscillator represents the shock displacement response at the corresponding frequency. The photograph of the lowfrequency oscillator group can be seen in Figure 3.
3.3. Fitting of the Equal Displacement Line
According to the measured displacement data D of lowfrequency oscillators with different frequencies, the pseudovelocity PV can be calculated by the following equation:
The results of maximum displacements and pseudovelocities at different frequencies and test conditions are listed in Table 2. In this paper, we only analyse the vertical data.

As we can see from Table 2, each test condition exhibits similar displacement characteristics in the lowfrequency band. The displacement terms are 10 mm, 22 mm, and 33 mm. The maximum deviation in each test condition is 1.6 mm, 3.5 mm, and 2.7 mm, which indicates that the measured displacement data exhibit high consistency.
Using a linear regression model, we fitted the data to an equal displacement line based on pseudovelocity data listed in Table 2. The pseudovelocity functional form (equal displacement line) under three different test conditions can be expressed by equations (17)–(19), respectively. The accuracy of regression line (to each function) is high: the correlation coefficients are 0.9998, 0.9995, and 0.9948, respectively:
The pseudovelocity data and the fitted equal displacement line under different test conditions are shown in a four logarithmic coordinate system, as seen in Figure 4.
4. Results of Shock Test Data Correction
In this section, the effectiveness of the combination correction method is verified by evaluating the shock test data. Figure 5 shows the original accelerometer record and its integrations under test condition 2. In the test, the accelerometer displayed no net displacement. Yet, the integration shows a false displacement of approximately 0.6 m. Also, the FAS shown in Figure 6 exhibits too many lowfrequency components associated with the data source. These indicate that the trend term error has been traced to the original the acceleration signal.
Figure 7 shows the comparison between the SRS and the fitted equal displacement line. As shown in Figure 7, the displacement response in the lowfrequency band (4∼20 Hz) should be approximately 20 mm, based on measurement of the lowfrequency oscillator. Yet, the displacement response computed by the acceleration signal can reach 37.6 mm. The maximum difference is up to 69.4%.
Afterwards, the original acceleration is corrected by using the combination correction method introduced in Section 2.2. The corrected acceleration signal and the associated integration results are shown in Figure 8. Figures 5 and 8 show that the acceleration signals exhibit nearly consistently, indicating that the corrected signal still contains a main energy component. Yet, both the integral velocity and integral displacement converge to zero, indicating that the zerodrift phenomenon caused by the trend term error has been eliminated.
Figure 9 shows the comparison of SRS before and after correction. We find that the signal components in the medium and highfrequency bands have nearly no energy loss, while the lowfrequency trend term has been well suppressed. The corrected SRS in Figure 9 is the result of repeating the correction step two times. Figure 9 also shows that the maximum difference between SRS and the equal displacement line in the lowfrequency band is reduced by approximately 55.7% (from 69.6% to 13.9%). The average difference is reduced by approximately 11.2% (from 14.0% to 2.8%). The correction accuracy meets the requirements of shock resistance performance assessment for warship equipment. Moreover, if overcorrected, timedomain acceleration signals will be distorted.
Similarly, for test condition 1, the average difference between the SRS and equal displacement line is reduced by approximately 11.4% (from 13.5% to 2.1%). And for test condition 3, the average difference is reduced by approximately 11.2% (from 14.7% to 3.5%). In addition, the correction method is nearly unaffected for the midhighfrequency band of SRS that do not contain trend term errors. Therefore, the corrected SRS can faithfully reflect actual shock characteristics. These results show that the combination correction method presented in this paper indeed improves the accuracy shock performance testing.
5. Conclusions
Based on traditional shock signal correction technology, we propose a method composed of FFT and lowfrequency oscillator for correcting an underwater shock signal. The validity of the method is verified by observing a floating platform explosion test. The key points of our study are as follows:(1)Underwater explosion shocks exist in harsh mechanical environment of high frequency, transient, and high magnitude, and this makes it difficult to accurately determine shock acceleration signals. The trend term errors contained in the signals will cause severe distortion of the displacement response, which will seriously affect the shock resistance assessment on warship equipment. Therefore, we need to find a way to accurately analyse and correct the validity of the shock signal.(2)The Fourier amplitude spectrum (FAS) of shock acceleration can be considered equal to the SRS within the warship shock engineering applications. By correcting the Fourier transform amplitude of the shock acceleration signal to match the equal displacement line and then operating the inverse Fourier transform, a corrected shock signal without a significant trend term error can be obtained.(3)The lowfrequency oscillator designed in this paper can accurately measure the displacement response at different frequencies. For each explosion condition, the displacement response has high consistency. For example, the maximum displacement deviation is less than 3.5 mm. Also, the goodness of fit for each displacement line equation is higher than 0.9948.(4)The results of underwater explosion tests indicate that the combination correction method can remove the trend items from the shock signal obviously. Compared with the equal displacement line, the maximum and average differences of SRS in the lowfrequency band (4∼20 Hz) can be reduced to 14.3% and 3.5%, respectively. In addition, the SRS of the midhighfrequency band without the trend term is nearly unaffected. The correction effect fully meets the requirement of shock resistance analysis for warship equipment.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (Grant no. 51775354).
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Copyright
Copyright © 2019 Peng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.