Shock Signal Trend Term Error Correction Method Based on Discrete Wavelet Transform and Low-Frequency Oscillator Combination

Shock response spectrum (SRS), calculated according to shock loading signal, is the primary metric for assessing the shock resistance ability of shipborne equipment. Nevertheless, the measured shock acceleration signal contains a trend term error that severely distorts the low frequency of the SRS. The accuracy of present correction methods cannot be assured due to a lack of reference. A discrete wavelet transform (DWT) and low-frequency oscillator combination method is proposed for correcting shock signals in this paper. The optimal wavelet parameters can be selected according to a low-frequency spectrum baseline fitted by the measured relative displacement to reject low-frequency trend term errors. Shock machine test results show that the average difference between the low-frequency spectrum baseline and uncorrected SRS is reduced from 89.8% to 3.2%, while the SRS slope rolled off to 5.8 dB/oct after imposing the proposed correction. The corrected SRS can faithfully show the actual shock loading characteristics of shipborne equipment in shock tests.


Introduction
Warships, inevitably, are attacked by underwater weapons in the course of battle. Even if a warship is not directly hit, the shock wave generated by the underwater explosion can seriously damage the ship's equipment and degrade its effectiveness [1]. e shock resistance of shipborne equipment directly affects its combat performance [2]. Underwater explosion tests are the most direct and effective approach to determine the shock resistance of shipborne equipment [3], but they are costly, labor intensive, and dependent on a great deal of material resources; they cannot be simply deployed at will [4,5]. e shock tester is a mechanism that can provide mechanical shock in formation for certain smaller pieces of shipborne equipment. e shock properties are controllable and the shock can be repeated within the appropriate accuracy requirements. Shock test machines thus play an important role in simulating the underwater shock explosion environment [6].
It is first necessary to determine the intensity of the shock loading before the shock resistance can be determined.
According to relevant international standards, shock intensity is usually characterized by a shock response spectrum (SRS) converted from an acceleration signal [7]. However, acceleration signals are susceptible to complex disturbance arising in measurement systems and from environmental factors. Small errors in the experimental data are magnified after integration and then may significantly affect the authenticity of the data. e trend term error is one such type of error [8,9], which occurs in the low-frequency component of a signal whose period is longer than the record length. Once it emerges, it affects the entire time-domain curve. Irvine [10] summarized six factors that can produce trend errors in shock tests. e existence of a trend term error creates substantial error in the signal itself or may even completely distort the SRS low-frequency band [11]. is seriously affects the shock strength judgment and overall shock test results.
Many scholars have developed data correction methods to separate the low-frequency errors from the measured signal and secure more reliable experimental results. Smyth and Wu controlled the integral trend term error by high-pass filtering [12]. However, these methods are not suitable for processing nonstationary and nonlinear shock signals. Grillo [13] used the least squares fit to remove the errors from the integrating velocity and then differentiated it to obtain the corrected acceleration. However, this method can hardly correct the acceleration signal containing abrupt spikes. Du et al. introduced a method of combining the acceleration Fourier amplitude spectrum with the design SRS; however, a designed SRS according to relevant standards could not reflect the actual response characteristics of measured shock acceleration [14]. e DWT method was first introduced by Smallwood and Cap to correct zero drift shock test data, but the choice of wavelet parameters is entirely based on personnel experience [15]. Edward conducted a series of shock verification tests [16]; Wang et al. selected wavelet parameters based on the correlation coefficients of positive and negative SRS [17]. However, even if the shock signal does not contain a trend term, the correlation coefficient of positive and negative SRS drops sharply once system damping exceeds 0.02, affecting the choice of wavelet parameters [18]. Li et al. introduced an adaptive EMD method utilizing the cross-correlation coefficient between the positive and negative SRS as a judgment standard; their method has the same disadvantages [19]. e EMD method also has shortcomings such as end effect and modal aliasing.
Due to the strong transient and nonlinear characteristics of shock testing, there is no uniform parameter selection method for accurately correcting shock signals without manual operation. Corrections of increasing severity are typically applied to the shock signals until SRS is physically reasonable. Due to the lack of independent shock response measurements for comparison, the correction effect is mostly judged by personal experience. Hence, although the correction effect seems reasonable, it may not be accurate credible. is affects the overall effect of shock resistance ability assessment for shipborne equipment.
A shock signal correction method based on DWT and low-frequency oscillator data combination is proposed in this paper. e SRS calculation method and DWT theory are presented below. A method for quantitatively selecting wavelet parameters based on the best Pearson correlation coefficient between the corrected SRS and a low-frequency spectrum baseline is then given. Section 3 reports a series of shock machine tests conducted to obtain the low-frequency spectrum baseline; the corrected shock signal is shown to contain no low-frequency trend term error. e SRS corrected via the proposed method accurately and precisely reflects actual shock loading characteristics.

Calculation of Shock Response Spectrum.
e SRS describes the maximum response of SDOF systems (e.g., a mass on a spring) related to a generalized excitation ( Figure 1). erefore, the SRS effectively reflects the damage potential of the shock environment to equipment with different natural frequencies.
e SRS can be divided into three categories based on the response parameters: the relative displacement response spectrum (Z max − ω), pseudovelocity response spectrum (PV max − ω), and absolute acceleration response spectrum (A max − ω).
For any frequency SDOFs, the kinematic equation of mass block can be expressed as follows: where x is the absolute displacement of the mass block, y is the absolute displacement of the foundation, and m, c, and k represent the mass, damping coefficient, and stiffness of the system, respectively. Defining z as the relative displacement of the mass block to the foundation, it allows the above equation to be rewritten as follows: where ζ � c/2mω n is the damping ratio of the SDOFs and ω n � ��� � k/m √ is their natural frequency. Assuming that the initial velocity and initial displacement are both zero, the response of aperiodic random excitation can be solved using the Duhamel integral according to the transient characteristics of shock excitation [20]: where ω d � ηω n is the damped natural frequency, η � ����� 1 − ζ 2 , and τ is an integral variable. According to the definition of the SRS, the relative displacement response spectrum (Z max − ω) is composed of the maximum relative displacement of each frequency of SDOFs. erefore, it can be calculated by equation (3), and then, the test shock environment is obtained.
In addition, since the pseudovelocity is equal to the system natural frequency multiplied by the relative displacement response, the pseudovelocity response spectrum (PV max − ω) can be obtained as follows: e acceleration is equal to the natural frequency of the system multiplied by the pseudovelocity, so the acceleration response spectrum (A max − ω) can be expressed by Equations (3)- (5) indicate that the accuracy of the shock acceleration load y ″ directly affects the accuracy of the SRS. It also seriously interferes with the shock resistance design of shipborne equipment. erefore, it is necessary to remove the trend term error contained within the shock acceleration data.

Discrete Wavelet Decomposition.
In practical applications, the continuous wavelet transform has a massive computational burden and must be discretized prior to use. e discrete wavelet function can be expressed as follows [21]: where ψ satisfies the perfect reconstruction condition. e discretization equations of dilation parameter a and translation parameter b are a � a m 0 , where m and n range over Z while a 0 > 1 and b 0 > 0 are fixed. e discretized wavelet coefficients can be expressed as After calculating the wavelet coefficients, the original shock signal y ″ (t) can be decomposed into a linear combination of wavelet and scaling functions with different dilation and translation parameters: where φ m,n is the scaling function and c m,n is the scaling coefficient. ese calculation methods are similar to equation (6) and (9), respectively. N m�1 n d m,n ψ m,n (t) is the high-frequency component in the corresponding decomposition. n c N,n φ N,n (t) is the low-frequency trend term component of the last decomposition. Furthermore, Eliminating the influence of the translation parameter allows equation (10) to be simplified: where A N is the low-frequency approximate component of the Nth decomposition, so y a ″ is the trend term to be eliminated. D m is the high-frequency detail component of the m of the decomposition. erefore, y a ″ is the sum of the detailed components of each order, i.e., the corrected signal. After DWT decomposition, the original shock signal must satisfy the following conditions: (1) e corrected signal y a ″ contains the main features of the original signal y ″ (2) e removed low-frequency signal y a ″ contains the trend term error which leads to zero integration drift

Selection of DWT Parameters.
According to equations (6)-(13), the correction effect by DWT mainly depends on the decomposition level N and the wavelet function ψ. As a biorthogonal wavelet function, the dB wavelet contains no truncation error. e selection of the wavelet function can be transformed into the determination of the vanishing moments of the dB wavelet. A set of low-frequency spring oscillators were designed here in order to obtain the proper wavelet parameters (decomposition layers and vanishing moments). e oscillators can accurately measure the relative displacement response of SDOFs with three different frequencies.
According to the literature [21], the low-frequency band (4-20 Hz) of the SRS is characterized by equal displacement.
e Pearson correlation coefficient ρ P,P * of the corrected low-frequency SRS PV(f L ) and the oscillator low-frequency spectrum baseline PV * (f L ) can be expressed as follows: To ensure that trend components y a ″ in the reconstructed signal are completely removed, the threshold value of correlation coefficient was set to 0.98. e optimal wavelet parameters were then selected by comparing the correlation coefficients at different decomposition layers and vanishing moments. Figure 1: SRS principle.

Shock and Vibration 3
In equation (14), PV(f L ) can be obtained through equation (13) by the corrected acceleration y a ″ . PV * (f L ) can be obtained by fitting the displacement data measured from the spring oscillator. e structure of the spring oscillator and the low-frequency spectrum baseline acquisition method are discussed below.

Shock Test Machine Method.
A shock test was carried out with a 500 kg vertical shock test machine developed by the Chinese Naval Research Academy. As shown in Figure 2, the test machine is mainly composed of a basic system, hydraulic system, mechanical system, and measurement and control system. At the beginning of the test, the velocity generator drives the shock hammer to collide with the waveform generator at the bottom of the shock table, thus producing a positive sinusoidal shock wave. After the collision, the hydraulic damping cylinder buffers and limits the shock table to produce a negative sinusoidal shock wave. e test can generate a double sine wave shock signal that satisfies the BV043/85 military standard. e shock load effectively simulates the positive and negative double-wave shock environment during an underwater noncontact explosion [22]. e layout of shock test site is shown in Figure 3. Newly developed small shipborne equipment was dispersedly fixed on the shock table in this setup; the shipborne equipment is blurred in the photo given here for confidentiality. e equipment's shock strength was measured with accelerometer sensor (B&K, Model 8309, Denmark) fixed on the shock table, as shown in Figure 4; their acceleration signals were then converted to SRS information. In order to correct the SRS with trend errors, a low-frequency oscillator group which can measure displacement response accurately was installed next to the accelerometers. e test was divided into three conditions that were carried out successively. e shock loading information is given in Table 1.

Low-Frequency
Oscillator Structure. Figure 5 shows the low-frequency oscillator structure, which consists of three suboscillators with three different natural frequencies. It is generally accepted that the low-frequency range of SRS is between 4 to 20 Hz [23], so the frequencies of the suboscillators were adjusted here to 6, 10, and 20 Hz. Besides, the three independent suboscillators were fixed to the bottom base to ensure the consistency of their shock response. e bottom base was also rigidly fixed onto the shock table.
Each suboscillator consists of a mass, two springs, a main guide shaft, an auxiliary guide shaft, a sliding bearing, and a guide plate. e two cylinder springs were precompressed and fixed on either end of the mass. e main guide shaft has a smooth surface, and the mass slides on its surface through a sliding bearing for reducing the friction coefficient between the mass and the main guide shaft, thereby minimizing system damping. After testing, the damping ratio of the lowfrequency oscillator was found to be only 0.016 at most, which can be ignored. A guide plate and a cooperated auxiliary guide shaft were also designed to prevent the mass from rotating due to the spring force.
Noncontact magnetostrictive displacement sensors (Balluff, Model BTL6, Germany) were used to measure the displacement response of each suboscillator; this effectively minimized system damping as well. e sensor has a linear error of ±0.04% Fs and a measurement range of − 100∼+100 mm, which meets the relevant shock test. e main block of each displacement sensor was placed behind its matching oscillator and parallel to the main guide shaft. e active magnetic black was installed on the guide plate. When shock occurred in tests, the magnetic sensing block reciprocated in the direction with the mass, thereby revealing the displacement response of each suboscillator. As per the Liaoning Institute of Metrology, the maximum error of the relative displacement of the vibrator at full scale is 1.38 mm, which has less than a 1.7% influence on the SRS. Because each suboscillator system can be considered a SDOFs, the displacement of each suboscillator can be considered as the relative shock displacement response at that corresponding natural frequency.

Low-Frequency Spectrum Baseline Fitting.
e maximum pseudovelocity PV max can be calculated as follows: where the displacement response Z was measured by the low-frequency oscillator group. Table 2 lists the maximum displacement and pseudovelocity results at three frequencies and under various test conditions.
As shown in Table 2, each test condition exhibits approximately equal maximum displacement in the low-frequency band, which is consistent with [23]. e maximum deviations under the three test conditions are only 3.36, 3.35, and 3.89 mm. e data was fitted to a low-frequency spectrum baseline via the linear regression model based on the maximum pseudovelocity data given in Table 2. e pseudovelocity functional form (low-frequency spectrum baseline) under each condition was expressed by equations (16)- (18). e goodness of fit R 2 values for each function is 0.9975, 0.9997, and 0.9955: e maximum value of pseudovelocity and the fitted low-frequency spectrum baseline PV * (f L ) under each condition are presented in Figure 6 on a quadruple logarithm coordinates.

Shock Signal Correction Results
and Discussion e effect of our proposed correction method was verified through analyzing the shock test results. Figure 7 shows the time-history record of original acceleration signal and its two integrations under test condition I. Gaberson pointed out that the integral velocity should be zero and the integral displacement should be stable when the shock ends [24]. However, the acceleration curve did not reveal whether the shock signal contained the trend term error in the test, but the integral velocity at the end was − 2.87 m/s and the integral displacement tended toward negative infinity. Both suggest a serious zero drift phenomenon. Figure 8 shows the comparison between the original SRS and the fitted low-frequency spectrum baseline. Based on the low-frequency oscillator measurements, the displacement response of low-frequency band (4-20 Hz) was practically 21.3 mm. However, the displacement response computed by the acceleration data reached 63.2 mm. e maximum relative difference in this case is as high as 196.5%. e proposed method was next applied to correct the original shock signal. e optimal wavelet parameters were determined by comparing the Pearson correlation coefficients of PV(f L ) and PV * (f L ). A larger vanishing moment order not only indicates stronger localization ability in the frequency domain but also weakens the support of the signal in the time domain. In addition, valuable medium-and  Low-frequency oscillator group Acceleration sensor Shock table Shock and Vibration 5 high-frequency components in the signal are affected if the decomposition level is too small. If the level is too large, the low-frequency trend term error cannot be effectively removed. Vanishing moments dB 4-9 and decomposition levels 2-7 were selected to calculate the shock acceleration data shown in Figure 7(a). Table 3 shows the Pearson correlation coefficients ρ P,P * of the different vanishing moments and decomposition levels. As shown in Table 3, as the decomposition level increases, the correlation coefficient between the corrected SRS and the low-frequency spectrum baseline first increases and then decreases. e correlation coefficients reach their respective maxima (all above 0.998) at the decomposition level of 7. Among all the vanishing moments, dB 7 has the strongest ability to remove signal trend terms; the correlation coefficient is 0.9995 under a decomposition level 7. Figure 9 shows the low-frequency approximate component A m and high-frequency detail component D m obtained by applying the decomposition level 7 and the vanishing moment dB 7 to the original shock acceleration data. e corrected acceleration signal and corresponding integration results after discrete wavelet decomposition and reconstruction are shown in Figure 10. e acceleration signals appear to be nearly consistent, yet the integral velocity converges to zero and the integral displacement is stable. In effect, the low-frequency trend term error within the shock signal has been eliminated. e comparison results of SRS before and after the proposed correction is shown in Figure 11. e mediumhigh frequency signal components are almost unaffected, while the low-frequency trend terms are effectively eliminated. e maximum difference between the SRS and the low-frequency spectrum baseline is reduced roughly from 196.5% to 13.5% and the average difference roughly from 87.4% to 2.7%. e slope of the corrected SRS is 5.8 dB/oct, which is within the reasonable range [25]. e correction accuracy meets the requirements for shipborne equipment shock resistance tests.
e results also show that, under test condition II, the maximum difference between the SRS and low-frequency spectrum baseline is reduced from 173.5% to 12.7%. e average difference is reduced to only 2.1%. Under test condition III, the maximum difference is reduced from 184.7% to 13.8%, while the average difference is reduced to only 3.2%. e selection of wavelet parameters in these cases was the same as under test condition I. Furthermore, the medium-high frequency band of SRS which is unaffected by the trend term error has little change. us, the corrected      Shock and Vibration 9 SRS faithfully reflects actual shock loading characteristics. e proposed combination correction method markedly improves the precision of shock resistance tests.

Conclusions
A novel technique comprised of DWT and low-frequency oscillator data was developed in this study for shock test signal correction. e method was validated in a shock machine test. e main conclusions of this work can be summarized as follows: (1) It is difficult to accurately determine shock acceleration signals in shock resistance tests on shipborne equipment. e trend term error that is mixed within the signals severely distorts the low-frequency response and then seriously affects the shock resistance assessment. e actual shock signal must be accurately restored. (2) e low-frequency oscillator group introduced in this study can precisely measure the displacement response at three frequencies (6, 10, and 20 Hz). e displacement responses measured under various test conditions were consistent; the goodness of fit for each displacement line equation was higher than 0.9955. (3) e optimal vanishing moment to support the proposed correction method is dB 7, and the optimal decomposition level is 7. e correlation coefficient of the corrected SRS and the low-frequency spectrum baseline in the low-frequency band can reach 0.9995. e wavelet parameters selected in this study are suitable in processing the actual shock test. (4) Compared with the low-frequency spectrum baseline, the maximum and average differences of the corrected SRS were reduced to 13.8% and 3.2%, respectively. e SRS of the medium-high frequency band without trend term errors was nearly unchanged. e results of shock tests conducted in this study indicate that the proposed method effectively removes the trend term error from shock acceleration signals. Its effect fully satisfies the requirement for shipborne equipment shock resistance test.

Data Availability
e data used to support the findings of the study are restricted because they involve military secrets.

Conflicts of Interest
e authors declare that they have no conflicts of interest with respect to the research, authorship, and/or publication of this article.