Method and system for generating equivalent load based on statistical envelope of shock response spectrum domain

Abstract

The application relates to the technical field of data processing, and discloses a method and system for generating equivalent load based on impact response spectrum domain statistical envelope. The method comprises the following steps: obtaining a double-damping spectrum value pair by performing double-band response spectrum transformation on a measured impact signal through a physical damping ratio and an auxiliary damping ratio; inversely solving a coherence cycle number point by point from a spectrum ratio of a physical spectrum value and an auxiliary spectrum value and arranging the coherence cycle number into a coherence cycle number field, so as to determine a coherent bandwidth aggregation window and an equivalent independent frequency point number to correct a coverage ratio, obtain a joint confidence envelope spectrum, and then configure and synthesize an equivalent load waveform according to a component frequency recursive point and a component half cycle number. The application improves the overall spectrum joint coverage reliability of an equivalent load spectrum and the consistency of a synthesized waveform with respect to a measured impact oscillation duration characteristic.

Description

Technical Field

[0001] This application relates to the field of data processing technology, and in particular to an equivalent load generation method and system based on the statistical envelope of the impact response spectral domain. Background Technology

[0002] The impact environment that equipment experiences during transportation, launch, separation, and drop needs to be transformed into an applicable load condition. In engineering, it is common practice to transform the measured impact acceleration signal to the impact response spectral domain before processing. This transformation constructs a set of single-degree-of-freedom basic excitation elements according to a given damping ratio. The natural frequencies of each element are logarithmically distributed along the analysis frequency band. Using the measured time history as the basic excitation, each element is solved one by one, and the maximum absolute value of the response is taken to obtain the spectral values ​​at each analysis frequency. The spectral values ​​obtained from multiple measurements are arranged into a matrix according to the number of measurements and the analysis frequency. At each analysis frequency, the spectral values ​​of this column are statistically processed, and the high quantile is taken as the design value. Arranging them frequency by frequency yields the envelope spectrum. The envelope spectrum cannot be directly applied to the shaking table or finite element model. It is necessary to use a family of attenuated sinusoidal components or a family of windowed sinusoidal wavelet components to inversely obtain the time-domain waveform. The component frequencies are evenly distributed at several per octave. The amplitude of each component is iteratively corrected until the deviation between the response spectrum and the envelope spectrum of the synthesized waveform enters the tolerance band.

[0003] The statistical steps in the above processing mostly assume a log-normal distribution and extrapolate higher quantiles using the logarithmic mean and logarithmic standard deviation. When the impact signal is composed of multiple mechanisms such as collision, separation, and drop, the overall spectral value is multi-peaked and heavy-tailed. The two moments cannot characterize the tail, resulting in distorted envelope values. Using non-parametric quantiles can get rid of the distribution assumption, but the number of measurements is limited, and the samples at a single analysis frequency are insufficient to support high quantile values. Samples must be merged across adjacent analysis frequencies. The merging window in the current approach is given by the analysis bandwidth of the spectrum estimator, which is a constant independent of the measured signal and invariant along the analysis frequency. The oscillation of the impact signal in the pulse-dominant frequency band lasts only one or two cycles, while it can last for dozens of cycles in the structural ringing-dominant frequency band. The frequency domain coherence widths of the two differ by an order of magnitude. The fixed window incorporates incoherent spectral values ​​in the ringing frequency band, artificially broadening the distribution, and excludes spectral values ​​that could be merged in the pulse frequency band. The merging itself introduces new distortions. Summary of the Invention

[0004] This application provides an equivalent load generation method and system based on the statistical envelope of the impulse response spectral domain. It solves the problems in existing technologies, such as envelope spectrum distortion caused by fixed merging windows and point-by-point quantization, and the failure to converge in the inverse multi-solution process due to the discarding of frequency domain coherence information during the synthesis stage. This application improves the reliability of the full-spectrum joint coverage of the equivalent load spectrum and the consistency of the synthesized waveform with the measured characteristics of impulse oscillation persistence.

[0005] In a first aspect, this application provides an equivalent load generation method based on the statistical envelope of the shock response spectral domain, the equivalent load generation method based on the statistical envelope of the shock response spectral domain includes: Step S1: Perform dual-passband response spectrum transformation on the measured impact signal using the physical damping ratio and the auxiliary damping ratio to obtain a dual-damped spectrum value pair consisting of the physical spectrum value and the auxiliary spectrum value; Step S2: Based on the spectral ratio of the physical spectrum value to the auxiliary spectrum value in the dual-damped spectrum value pair, and according to the correspondence that the response establishment quantity increases monotonically with the number of excitation cycles and tends to a saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit, perform numerical inverse solution point by point for each analysis frequency, obtain the coherent cycle number of each analysis frequency by filtering the inverse solution results, and arrange the coherent cycle number along the logarithmic frequency to form a coherent cycle number field; Step S3: Determine the coherent bandwidth aggregation window and the equivalent independent frequency point number based on the coherent cyclic number field. Perform coverage ratio correction processing on the physical spectrum value aggregated by the coherent bandwidth aggregation window according to the equivalent independent frequency point number to obtain the joint confidence envelope spectrum. Step S4: Perform time-domain synthesis processing on the joint confidence envelope spectrum based on the coherent cyclic number field, using component frequency recursion and component half-cycle configuration, to obtain the equivalent load waveform.

[0006] Secondly, this application provides an equivalent load generation system based on the statistical envelope of the impact response spectral domain, the equivalent load generation system based on the statistical envelope of the impact response spectral domain includes: The transformation module is used to perform dual-passband response spectrum transformation on the measured impact signal by using the physical damping ratio and the auxiliary damping ratio to obtain a dual-damped spectrum value pair composed of physical spectrum value and auxiliary spectrum value; The inverse solution module is used to perform numerical inverse solution on each analysis frequency point by point according to the spectral ratio of the physical spectrum value and the auxiliary spectrum value in the dual-damped spectrum value pair, based on the correspondence that the response establishment quantity increases monotonically with the number of excitation cycles and tends to a saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit. The coherent cycle number of each analysis frequency is obtained by filtering the inverse solution results, and the coherent cycle number is arranged along the logarithmic frequency to form a coherent cycle number field. The correction module is used to determine the coherent bandwidth aggregation window and the equivalent independent frequency point number based on the coherent cyclic number field, and to perform coverage ratio correction processing on the physical spectrum value aggregated by the coherent bandwidth aggregation window according to the equivalent independent frequency point number to obtain the joint confidence envelope spectrum. The synthesis module is used to perform time-domain synthesis processing on the joint confidence envelope spectrum based on the coherent cyclic number field, with component frequency recursion and component half-cycle configuration, to obtain the equivalent load waveform.

[0007] Thirdly, an equivalent load generation device based on the statistical envelope of the shock response spectral domain is provided, comprising: a memory and at least one processor, wherein the memory stores instructions; the at least one processor invokes the instructions in the memory to cause the equivalent load generation device based on the statistical envelope of the shock response spectral domain to execute the above-described equivalent load generation method based on the statistical envelope of the shock response spectral domain.

[0008] Fourthly, a computer-readable storage medium is provided, wherein instructions are stored therein, which, when executed on a computer, cause the computer to perform the above-described method for generating equivalent loads based on the statistical envelope of the impulse response spectral domain.

[0009] In the technical solution provided in this application, a dual-damped spectral value pair is obtained by performing a dual-passband response spectrum transformation on the same measured impact signal using the physical damping ratio and the auxiliary damping ratio. The input quantities of the two branches are completely identical except for the damping ratio. The excitation amplitude, sensor sensitivity, and intercept window position are canceled out when taking the quotient. Only the damping ratio and the coherence of the excitation at the analysis frequency remain in the spectral ratio. Thus, without adding a transform domain or introducing window length or scale parameters, the oscillation persistence characteristics that were originally impossible to observe in the response spectrum domain are transformed into readable quantities. The coherent cycle number is obtained by inversely solving point by point according to the correspondence that the response establishment quantity increases monotonically with the number of excitation cycles and tends to a saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit. Its value comes from the measured signal itself rather than a priori setting. After arranging the coherent cycle number field along the logarithmic frequency, the difference between the pulse-dominated frequency band and the structural ringing-dominated frequency band is reflected frequency by frequency. The physical damping ratio remains constant throughout the process, the auxiliary damping ratio only performs the detection function, the envelope spectrum and waveform remain strictly below the physical damping ratio, and the referential relationship between the response spectrum and the equivalent structural properties of the specimen is not changed. This is the premise that the algorithm features of this application can be established in the application field of shock environment condition formulation.

[0010] The coherent cyclic number field simultaneously constrains both the statistical and synthetic sides, which is the main contribution of the algorithm features in this application to the scheme. The coherent bandwidth aggregation window is given frequency-by-frequency by the coherent cyclic number field, with a narrow window in the ringing frequency band and a wide window in the pulse frequency band. The spectral values ​​obtained by cross-frequency point merging come from repeated readings of the same coherent oscillation, and the distribution shape is not artificially broadened by merging. The equivalent independent frequency point number is obtained by integrating the same coherent cyclic number field along the logarithmic frequency, and the coverage ratio is corrected accordingly. The coverage meaning carried by the envelope spectrum changes from point-by-point edge coverage to joint coverage of the entire spectrum, which is consistent with the aperture required by the impact environment conditions. The component frequency recursive placement on the synthesis side adopts the same ratio as the independent frequency point connection criterion. The total number of components is consistent with the equivalent number of independent frequency points, and the degrees of freedom obtained by inversion are comparable to the number of constraints provided by the envelope spectrum. The half-cycle number of the components is taken as an odd number not less than 3 to make the net velocity and net displacement of each component zero. The value is close to twice the number of coherent cycles, which makes the oscillation of the synthesized waveform at each component frequency consistent with the actual measurement. In addition to the spectral matching, there is an additional constraint from the actual measurement, and the multiple solutions obtained by inversion converge. The coverage meaning of the statistical domain and the oscillation characteristics of the time domain waveform are connected by the same measurand, and the resolution of the two is no longer mismatched. Attached Figure Description

[0011] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0012] Figure 1 This is a schematic diagram of an embodiment of the equivalent load generation method based on the statistical envelope of the impact response spectral domain in this application. Figure 2 This is a schematic diagram of the coherent cyclic number field in an embodiment of this application; Figure 3 This is a schematic diagram of the joint confidence envelope spectrum in an embodiment of this application. Detailed Implementation

[0013] This application provides a method and system for generating equivalent loads based on the statistical envelope of the impact response spectral domain. The terms "first," "second," "third," "fourth," etc. (if present) in the specification, claims, and accompanying drawings of this application are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments described herein can be implemented in a sequence other than that illustrated or described herein. Furthermore, the terms "comprising" or "having" and any variations thereof are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0014] For ease of understanding, the specific process of the embodiments of this application is described below. Please refer to [link / reference]. Figure 1 One embodiment of the equivalent load generation method based on the statistical envelope of the impact response spectral domain in this application includes: Step S1: Perform dual-passband response spectrum transformation on the measured impact signal using the physical damping ratio and the auxiliary damping ratio to obtain a dual-damped spectrum value pair consisting of the physical spectrum value and the auxiliary spectrum value; Specifically, the physical damping ratio is set to 0.05, corresponding to the quality factor of 10 commonly used in the impact response spectrum, and remains unchanged as an equivalent structural property of the specimen. The auxiliary damping ratio is set to 0.25, its sole function being to apply a wider passband to the same impact time history of equal length to obtain a second reading. This value ensures that the ratio of the two is 5, thus making the range of the spectral ratio an open interval between 1 and 5. The reason for setting this ratio to 5 is that if the ratio is too small, the global variation of the spectral ratio is compressed to near the resolution of the numerical inverse solution; if the ratio is too large, the passband width of the auxiliary passband branch will cross the adjacent resonance peak, causing the auxiliary spectral value to lose its frequency orientation. The analysis frequency bands are arranged at 1 / 24 octaves, and the ratio of adjacent analysis frequencies is 2 raised to the power of 1 / 24. This density ensures that the coherence bandwidth aggregation window still contains more than two analysis frequencies within the window when the coherence cycle number reaches the upper limit of 20, preventing aggregation from failing. The aftershock extension time is taken as 20 divided by the lower limit of the analysis frequency band. This means that the single-degree-of-freedom element with the lowest analysis frequency completes 20 free oscillation cycles within the truncation window. When the physical damping ratio is 0.05, its free oscillation decays to less than 4% of the initial amplitude after 20 cycles, and the extreme values ​​are not truncated or discarded. Ramp-invariant discretization is the common digital filtering implementation for impulse response spectrum calculation. Its second-order recursive coefficients are obtained by algebraically combining the analysis frequency, the physical damping ratio or auxiliary damping ratio substituted into the branch, and the sampling interval. The difference between the physical passband branch and the auxiliary passband branch is only that the physical damping ratio or the auxiliary damping ratio is substituted.

[0015] Step S2: Based on the spectral ratio of the physical spectrum value and the auxiliary spectrum value in the dual-damped spectrum value pair, and according to the correspondence that the response establishment quantity increases monotonically with the number of excitation cycles and tends to the saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit, perform numerical inverse solution point by point for each analysis frequency, obtain the coherent cycle number of each analysis frequency by screening the inverse solution results, and arrange the coherent cycle number along the logarithmic frequency to form a coherent cycle number field; Specifically, when a single-degree-of-freedom element is subjected to an excitation with the same natural frequency, the response amplitude does not reach a steady state instantaneously. Instead, it accumulates gradually with each excitation cycle, and the accumulation process tends towards a steady state according to the law of 1 minus an exponential decay term. The exponent contains the product of pi, the damping ratio, and the number of excitation cycles. The steady-state amplification is the reciprocal of twice the damping ratio. When the same excitation acts on two elements with different damping ratios, the element with the smaller damping ratio builds up more slowly but has a higher steady-state value, while the element with the larger damping ratio builds up more quickly but has a lower steady-state value. The factor related to the excitation amplitude in the ratio of their extreme values ​​is completely canceled out, leaving only the damping ratio and the number of excitation cycles. When the number of excitation cycles approaches zero, both elements have only gone through a very short build-up phase, and the ratio tends to 1. As the number of excitation cycles increases, the smaller damping side continues to accumulate while the larger damping side has saturated, and the ratio monotonically increases and tends to the ratio of the damping ratios, i.e., 5. The lower bound of the preset excitation cycle number is set to 1 because when the excitation cycle number is below 1, the difference between the spectral ratio and 1 falls within the dispersion range of the measured spectral ratio and cannot be distinguished from noise. The upper bound of the preset excitation cycle number is set to 20 because when the physical damping ratio is 0.05, the establishment degree of the excitation cycle number reaching 20 is already above 99%, and the derivative of the spectral ratio with respect to the excitation cycle number is so small that the inverse solution cannot distinguish it. The convergence criterion of the binary iteration is that the difference between two adjacent excitation cycle numbers is less than 0.01. This precision corresponds to the third significant digit of the spectral ratio, which is comparable to the confidence digit of the measured spectral ratio. The median is taken instead of the mean along the number of measurements because individual impacts may have zero drift residue or trigger jitter, and their spectral ratio deviates from the overall value. The median is not affected by a small number of deviations.

[0016] Step S3: Determine the coherent bandwidth aggregation window and the equivalent independent frequency point number based on the coherent cyclic number field. Perform coverage ratio correction processing on the physical spectrum value aggregated by the coherent bandwidth aggregation window according to the equivalent independent frequency point number to obtain the joint confidence envelope spectrum. Specifically, the duration of the oscillation component corresponding to the coherent cycle number at the analysis frequency is the coherent cycle number divided by the analysis frequency. Following the reciprocal relationship between time and bandwidth, the frequency broadening of this oscillation component is the analysis frequency divided by the coherent cycle number. The half-width is half of this, so the half-width of the coherent bandwidth aggregation window is twice the analysis frequency divided by the coherent cycle number. The spectral values ​​of each analysis frequency falling within this window come from readings of the same coherent oscillation in adjacent passbands and can be considered as repeated observations of the same population, thus being merged into an aggregated spectral value sample set. Analysis frequencies outside the window correspond to incoherent oscillation components; merging them artificially broadens the distribution and raises the quantile. The larger the coherent cycle number, the narrower the window and the denser the independent readings, with both having the same orientation. The criterion for two adjacent independent frequency points is that their coherence bandwidths are exactly contiguous without overlapping. Based on this, the ratio of the two analyzed frequencies is 1 plus the reciprocal of the coherence cycle number. Taking the natural logarithm gives the span occupied by a single independent frequency point on the logarithmic frequency axis. Integrating the reciprocal of this span along the logarithmic frequency axis and rounding up yields the equivalent number of independent frequency points across the entire frequency band. Since each independent frequency point is approximately uncorrelated, the probability of the entire spectrum being simultaneously covered by the envelope is the equivalent number of independent frequency points raised to the power of the point-by-point coverage ratio. Therefore, the point-by-point coverage ratio is obtained by taking the equivalent number of independent frequency points raised to the power of the target full-spectrum joint coverage ratio. The target full-spectrum joint coverage ratio is set to 0.95, consistent with the commonly used 95% coverage caliber in the formulation of impact environment conditions. The difference lies in the fact that the 95% in this scheme applies to the entire spectrum rather than a single frequency point, thus eliminating the deviation where the actual coverage of the entire spectrum is far lower than the nominal value when taking the point-by-point quantile.

[0017] Step S4: Based on the coherent cyclic number field, perform time-domain synthesis processing on the joint confidence envelope spectrum by recursively placing component frequencies and configuring component half-cycles to obtain the equivalent load waveform.

[0018] Specifically, the sinusoidal modulated wavelet is constructed by multiplying a modulating sine wave by a carrier sine wave. The modulation frequency is the component frequency divided by the component half-cycle number, and the carrier frequency is the component frequency. The domain is half a cycle of the modulating sine wave, and zero is taken outside the domain. Within the domain, the waveform's first integral over time yields the velocity increment, and the second integral yields the displacement increment. Both integrals are zero when the component half-cycle number is odd, and the velocity and displacement return to their initial values ​​at the end of the waveform, with no unidirectional drift in the vibration table's travel and velocity. When the half-cycle number is even, the velocity increment is not zero, and residual velocity remains at the end of the superimposed waveform. The lower limit for the half-cycle number is set to 3 because if it is 1, the carrier wave does not complete a full cycle within the domain, causing the waveform to degenerate into a single pulse and failing to establish itself at its natural frequency. The half-cycle number is chosen from odd numbers that is closest to twice the number of coherent cycles. This means ensuring that the number of carrier cycles completed by the component carrier within the defined domain matches the number of coherent cycles measured at the component frequency. This ensures that the oscillation duration characteristics of the synthesized waveform are consistent with the measured values, avoiding waveforms with matching spectral values ​​but oscillation durations differing by several times when only peak matching is used. The single-component amplification factor used for component amplitude inverse calculation is obtained by multiplying the reciprocal of twice the physical damping ratio by the establishment term. The establishment term is 1 minus an exponent term with a negative exponent equal to the product of pi, the physical damping ratio, and the component half-cycle number. Dividing the envelope spectrum value by this factor yields the initial value of the component amplitude. The amplitude is updated component by component according to the ratio of the envelope spectrum value to the back-calculated spectrum value of the test waveform. The iteration continues until the absolute value of the ratio of the two at each component frequency is no greater than 1.5 in decibels. This criterion is consistent with the tolerance band commonly used in impact test conditions. The iteration process only changes the amplitude, while the component frequency and the number of half-cycles of the component remain unchanged. The odd-number attribute and the total number of components are not affected. The total number of components is equal to the number of equivalent independent frequency points because the recursion ratio of the component frequency and the spacing criterion of the independent frequency points are both 1 plus the reciprocal of the number of coherent cycles.

[0019] In one specific embodiment, step S1 includes: The measured impact signal is subjected to zero drift correction, anti-aliasing filtering and detrending processing to obtain the baseline correction signal; The baseline correction signal is processed by truncating the impact segment and extending the residual vibration according to the trigger threshold to obtain the equal-length impact time history. The duration of the delay after the truncated end is taken as 20 divided by the lower limit of the analysis frequency band. Based on the analysis frequency band, the points are arranged at equal intervals of 1 / 24 octave logarithm. Based on the physical damping ratio and the auxiliary damping ratio, the single-degree-of-freedom basic excitation model is discretized by slope invariance to obtain a dual-passband filter bank composed of physical passband branch and auxiliary passband branch. The ratio of auxiliary damping ratio to physical damping ratio is taken as 5. The equal-length impact time history is input into a dual-passband filter bank for response recursion and absolute value extremum extraction at each analysis frequency. This yields a pair of dual-damped spectrum values ​​consisting of physical spectrum values ​​and auxiliary spectrum values, which are taken from two branches that share the same equal-length impact time history and the same analysis frequency grid.

[0020] Specifically, the measured impact signal is given in the form of an acceleration time history. A period of time without excitation before the impact is taken as the silent period. The average value of all sampling points in this period is used as the DC bias and subtracted from the entire time history. The baseline step generated by the piezoelectric sensor under strong impact is thus eliminated. The entire time history is then fitted with a least-squares straight line and the fitting result is subtracted to remove the slowly varying component caused by sensor temperature drift, thus obtaining the baseline correction signal. Anti-aliasing filtering is performed before bias subtraction. The cutoff frequency is set to twice the upper limit of the analysis frequency band. That is, when the upper limit of the analysis frequency band is 10000 Hz, the cutoff frequency is set to 20000 Hz, leaving a transition band of one octave so that the amplitude at 10000 Hz is not reduced by the roll-off section of the filter. The sampling rate is set to 20 times the upper limit of the analysis frequency band, i.e., 200,000 Hz. This is because the extreme values ​​of the response spectrum are obtained through point-by-point comparisons. At least 20 sampling points must be taken within each carrier cycle to ensure that the reading error of the extreme value is within the thousandths. If the sampling rate of the original record is less than 200,000 Hz, the baseline correction signal is interpolated and supplemented with points in integer multiples before further processing. The trigger threshold is set to 5 times the standard deviation of the silent segment. This multiple ensures that the sensor's background noise is completely excluded from triggering without missing the preceding pulse with a low amplitude. The starting point is taken as the moment when the trigger threshold is first crossed, and then shifted forward by 1 millisecond to include the rising edge below the threshold. The interception point is calculated by extending the duration by 20 times, divided by the lower limit of the analysis frequency band, after the last time the trigger threshold is exceeded. When the lower limit of the analysis frequency band is 10 Hz, this extension is 2 seconds. This means that the 10 Hz single-degree-of-freedom element completes 20 free oscillation cycles within the window. With a physical damping ratio of 0.05, its free oscillation decays to below 4% of the initial amplitude after 20 cycles, ensuring that the extreme response value at the lower limit of the analysis frequency band does not fall outside the window. The interception results from 12 actual measurements have varying lengths. The longest interception is used, and zeros are padded at the end to align them, resulting in an equal-length impact time history.

[0021] The analysis frequency band, from 10 Hz to 10000 Hz, is spaced at equal intervals logarithmic to 1 / 24 octave. The ratio of adjacent analysis frequencies is 2 raised to the power of 1 / 24, or 1.02930, resulting in a total of 241 analysis frequencies across the entire band. This density is not arbitrarily chosen. When the coherent bandwidth aggregation window reaches its upper limit of 20 coherent cycles, its relative half-width is one-fortieth. The window must accommodate at least two analysis frequencies, and the relative spacing of 1 / 24 octaves precisely meets this condition. If 1 / 6 octaves were used instead, only a single analysis frequency would remain within the window in the high-frequency band, resulting in aggregation failure. The single-degree-of-freedom basic excitation model uses the analysis frequency as the natural frequency and the physical damping ratio or auxiliary damping ratio as the damping. Its acceleration-to-acceleration transmission relationship is discretized into a second-order recursive loop using the slope invariance method. This loop contains three forward coefficients and two feedback coefficients. The five coefficients are directly calculated by an algebraic combination of the analysis frequency, the physical damping ratio or auxiliary damping ratio substituted into this branch, and the reciprocal of the sampling interval. The inputs are the current sampling point and the previous two sampling points of the equal-length impact time history, and the outputs are the current value and the previous two values ​​of the response acceleration. The physical damping ratio substituted into the physical passband branch is 0.05, corresponding to the quality factor of 10 common to the impact response spectrum, representing the equivalent structural properties of the specimen. The auxiliary damping ratio substituted into the auxiliary passband branch is 0.25. All inputs for the two branches are identical except for the substituted damping ratios. A ratio of 5 is used to make the spectral ratio range an open interval between 1 and 5. If the ratio drops to 2, the global variation of the spectral ratio is compressed to a level comparable to the dispersion of the measured spectral ratio, making it indistinguishable. If the ratio rises to 10, the half-power bandwidth of the auxiliary passband branch reaches one-fifth of the center frequency, the passband crosses adjacent resonance peaks, and the auxiliary spectral value loses its directional characteristic for the analysis frequency. Equal-length impulse time histories are fed into a dual-passband filter bank, and the recursion of the two branches is run through each of the 241 analysis frequencies. The physical spectral value is the one with the largest absolute value in the physical passband branch output sequence, and the auxiliary spectral value is the one with the largest absolute value in the auxiliary passband branch output sequence. The 12 measurements and the 241 analysis frequencies are interleaved to form two 12x241 matrices, which together are called a dual-damped spectral value pair. Simultaneously, the sampling time when the physical passband branch output reaches its maximum absolute value is recorded. This time is then used as the time delay value of the corresponding component after taking the median of the number of measurements. The two branches share the same impact time history, the same sampling rate, and the same analysis frequency grid. The excitation amplitude, sensor sensitivity, and truncation window position of the same impact have a proportional effect on the output of the two branches. When taking the quotient, these quantities are completely canceled out, leaving only the damping ratio and the coherence of the excitation at the analysis frequency in the quotient.

[0022] In one specific embodiment, based on the spectral ratio of the physical spectral value to the auxiliary spectral value in the dual-damped spectral pair, and according to the correspondence that the response build-up monotonically increases with the number of excitation cycles and tends to a saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit, a numerical inverse solution is performed point by point for each analysis frequency, including: For each pair of dual-damped spectral values, the quotient is obtained by taking the quotient according to the one-to-one correspondence between the actual number of measurements and the analysis frequency. The quotient is then taken with the physical spectral value as the dividend and the auxiliary spectral value as the divisor. The median of the spectral ratio samples along the number of measurements was extracted to obtain the robust spectral ratios for each analysis frequency. The transient establishment law of a single-degree-of-freedom system is amplified by the ratio of physical damping ratio and auxiliary damping ratio to obtain a monotonic saturation correspondence. The monotonic saturation correspondence takes the number of excitation cycles as the independent variable, takes 1 when the number of excitation cycles approaches zero, and takes the ratio of auxiliary damping ratio to physical damping ratio when the number of excitation cycles approaches infinity. Based on the monotonically saturated correspondence, a control spectral ratio sequence is generated with a preset step size. The robust spectral ratio is then subjected to interval clamping and bisection iterative inverse processing according to the control spectral ratio sequence to obtain the inverse cycle number of each analysis frequency. The convergence criterion for bisection iteration is that the difference between the excitation cycle numbers obtained from two adjacent iterations is less than 0.01.

[0023] Specifically, two 12-row, 241-column matrices were obtained by intersecting 12 measured values ​​with 241 analytical frequencies. Each element was then divided element-wise according to its row and column numbers, with the physical spectrum value as the dividend and the auxiliary spectrum value as the divisor, resulting in spectral ratio samples, also in 12-row, 241-column format. Two spectral values ​​for the same element originate from two branches of the same impact time history at the same analytical frequency. Excitation amplitude, sensor sensitivity, and cutoff window position are canceled out during division. Each element in the spectral ratio sample carries only two damping ratios and the coherence of the excitation at that analytical frequency. The median of the 12 spectral ratios was taken along the column direction (i.e., along the number of measured values), compressing the 12 rows into one, resulting in 241 robust spectral ratios. The reason for taking the median is that when individual impacts have residual zero drift or trigger jitter, their overall spectral ratios shift. The mean is affected by these shifts, but the median remains unaffected. Even if five of the 12 readings show a shift, the two middle values ​​still fall within the normal distribution.

[0024] When a single-degree-of-freedom element is subjected to reciprocating excitation with the same natural frequency, the response amplitude accumulates with each excitation cycle. The accumulated amount approaches steady state by subtracting an exponential decay term from 1. For the physical passband branch, this exponent is the negative of the product of pi, the physical damping ratio, and the number of excitation cycles. The steady-state amplification is the reciprocal of twice the physical damping ratio. For the auxiliary passband branch, substituting the auxiliary damping ratio, we can similarly obtain its accumulated amount and steady-state amplification. A smaller physical damping ratio results in a higher steady-state amplification but slower accumulation; a larger auxiliary damping ratio results in a lower steady-state amplification but faster accumulation. Dividing the accumulated amounts of the two branches, the quotient is determined only by the two damping ratios and the number of excitation cycles. This is called the amplification ratioization, and the resulting relationship is called the monotonically saturated correspondence. When the excitation cycle number approaches zero, the cumulative amounts of both branches start with linear segments, and their ratio approaches 1. As the excitation cycle number increases, the auxiliary passband branch saturates first, while the physical passband branch continues to accumulate, and the quotient rises accordingly until the physical passband branch also saturates. The quotient then approaches the ratio of the two steady-state amplification amounts, i.e., the ratio of the auxiliary damping ratio to the physical damping ratio. The derivative of this quotient with respect to the excitation cycle number is positive over the entire domain, and its monotonicity holds. The solution for inversely solving the excitation cycle number using the robust spectral ratio is unique.

[0025] Using a step size of 0.1, values ​​between 1 and 20 of the excitation cycle number are substituted into the monotonically saturated correspondence to calculate 191 control spectral ratios. These are then paired with their corresponding excitation cycle numbers to form a control spectral ratio sequence. This step size ensures that the difference between two adjacent control spectral ratios is within 3% of the low cycle number segment, resulting in a sufficiently narrow interval with no more than ten subsequent iterations. The interval is then used to find two adjacent terms in the control spectral ratio sequence that are exactly less than or exactly greater than the robust spectral ratio. The excitation cycle number paired with these two terms is taken as the initial interval endpoint for the bisection iteration. The bisection iteration takes the midpoint of the interval and substitutes it into the monotonically saturated correspondence to calculate the midpoint spectral ratio. If the midpoint spectral ratio is greater than the robust spectral ratio, the upper end of the interval is moved to the midpoint; if it is less, the lower end is moved to the midpoint. This process of narrowing is repeated until the difference between the excitation cycle numbers obtained from two adjacent iterations is less than 0.01, yielding the inverse solution cycle number for this analysis frequency. The convergence criterion is set to 0.01 because this precision corresponds to the third significant digit of the spectral ratio, which is comparable to the number of reliable digits of the measured spectral ratio. If the criterion were too strict, the iteration would simply loop within the distribution of the measured spectral ratio. The above search and contraction are performed independently for each of the 241 analysis frequencies, resulting in 241 inverse solution loops.

[0026] In one specific embodiment, the coherent cycle number for each analysis frequency is obtained by filtering the inverse kinematics results, and the coherent cycle numbers are arranged along the logarithmic frequencies to form a coherent cycle number field, including: Based on the monotonic saturation correspondence, values ​​are taken at the lower bound of the preset excitation cycle number and the upper bound of the preset excitation cycle number to obtain the spectral ratio cutoff threshold pair; Based on the spectral ratio cutoff threshold, the robust spectral ratio is subjected to out-of-bounds discrimination processing to obtain the out-of-bounds analysis frequency set and the effective analysis frequency set; The inverse solution cycle number is obtained by end value replacement and retention processing based on the out-of-bounds analysis frequency set and the effective analysis frequency set to obtain the coherent cycle number of each analysis frequency. Among them, the analysis frequencies with a robust spectral ratio not greater than the lower threshold in the out-of-bounds analysis frequency set are taken as the lower bound of the preset excitation cycle number, and the analysis frequencies with a robust spectral ratio not less than the upper threshold are taken as the upper bound of the preset excitation cycle number. The coherent cycle numbers are arranged in ascending order of the logarithmic frequency of each analysis frequency, and logarithmic frequency linear interpolation is performed between adjacent analysis frequencies to obtain the coherent cycle number field.

[0027] Specifically, the lower bound of the preset excitation cycle number is set to 1, and the control spectral ratio at that point is calculated using the monotonic saturation correspondence as the lower threshold. The upper bound is set to 20, and the upper threshold is calculated similarly. The two values ​​are paired to form a spectral ratio truncation threshold pair, with the lower threshold being less than the upper threshold. The reason for setting the lower bound to 1 is that when the excitation cycle number is less than 1, the excitation at this analysis frequency has not yet completed a full cycle, and both branches are in the established linear starting phase. The difference between the spectral ratio and 1 falls within the dispersion range of the spectral ratio obtained from 12 actual measurements and cannot be distinguished from noise. The inverse solution is extremely sensitive to small fluctuations in the spectral ratio in this range. The reason for setting the upper bound to 20 is that when the physical damping ratio is 0.05, the cumulative degree of the excitation cycle number reaching 20 is already above 99%. The monotonic saturation correspondence is close to its saturation value at this point, and the derivative of the spectral ratio with respect to the excitation cycle number is so small that it cannot be distinguished by binary iteration. The value obtained by further extrapolation does not have a reliable number of bits.

[0028] Each of the 241 robust spectral ratios is compared with a spectral ratio cutoff threshold. Ratios not exceeding the lower threshold are considered lower bounds, and those not less than the upper threshold are considered upper bounds. These two categories are merged into a bounds analysis frequency set, while the rest are merged into a valid analysis frequency set. The two sets are mutually exclusive, and their union covers all 241 analysis frequencies. For analysis frequencies falling into the bounds analysis frequency set, the inverse loop number is obtained by shrinking the binary iteration at the endpoints of the initial interval. Since the value has reached the interval boundary and is no longer meaningful, it is replaced by an endpoint substitution. For lower bounds, the lower bound of the preset excitation loop number is taken as 1, and for upper bounds, the upper bound of the preset excitation loop number is taken as 20. For analysis frequencies falling into the valid analysis frequency set, the inverse loop number is retained. The two criteria used for substitution—not exceeding the lower threshold and not less than the upper threshold—are identical to the criteria used for bounds detection. The same analysis frequency will not be both judged as bounds and retained as valid, nor will both criteria be applied simultaneously. After substitution and retention, 241 coherent loop numbers are obtained.

[0029] The 241 coherent cyclic numbers are arranged in ascending order of their logarithmic frequencies, with the horizontal axis representing the natural logarithm of the analysis frequencies. Linear interpolation is performed between adjacent analysis frequencies using the logarithmic frequencies to obtain a continuous function with the logarithmic frequencies as the independent variable—the coherent cyclic number field. Interpolation is performed on the logarithmic frequency axis, not the linear frequency axis, because the 241 analysis frequencies are themselves distributed at equal logarithmic intervals of 1 / 24 octave band. On the logarithmic axis, adjacent points are equidistant, and the interpolation weights are directly given by the difference in logarithmic frequencies. On the linear axis, the distance between adjacent points is less than 1 Hz in the low-frequency band and hundreds of Hz in the high-frequency band, resulting in a three-order-of-magnitude difference in the density of the same interpolation rule at both ends. The coherent cyclic number field not only carries the coherent cyclic number values ​​at each analysis frequency but also provides continuous values ​​between two analysis frequencies, eliminating the need for secondary interpolation when performing integration and recursive value retrieval using the logarithmic frequencies.

[0030] Figure 2 This is a schematic diagram of the coherent cyclic number field in an embodiment of this application. Figure 2 The horizontal axis represents the analysis frequency, and the vertical axis represents the coherent cycle number. The solid line represents the coherent cycle number field obtained by completing the robust spectral ratio of the physical spectral value and the auxiliary spectral value through binary iterative inverse solution, end value permutation, and logarithmic frequency linear interpolation. Its value falls around 1.5 in the pulse-dominated low-frequency band and rises to around 14 in the structure-dominated high-frequency band. The light-colored band represents the range of inverse solution values ​​corresponding to the robust spectral ratio dispersion of 12 measured measurements. The dashed line represents the constant value of 10 corresponding to the fixed half-power bandwidth when the physical damping ratio is 0.05. This dashed line intersects the solid line in the mid-frequency band, and the deviation directions on both sides are opposite.

[0031] In one specific embodiment, the robust spectral ratio is subjected to out-of-bounds discrimination processing based on the spectral ratio cutoff threshold to obtain an out-of-bounds analysis frequency set and an effective analysis frequency set, including: The spectral ratio cutoff threshold is calibrated by sub-itemizing the corresponding values ​​of the lower bound and the upper bound of the preset excitation cycle number to obtain the lower threshold and the upper threshold, wherein the lower threshold is less than the upper threshold. Based on the lower threshold, the robust spectral ratio is compared with the lower bound of each analysis frequency to obtain the lower bound marker. The analysis frequencies with a robust spectral ratio not greater than the lower threshold are set as lower bound, and the other analysis frequencies are set as non-lower bound. Based on the upper threshold, the robust spectral ratio is compared with the upper limit of each analysis frequency to obtain the upper limit mark. The analysis frequency with a robust spectral ratio not less than the upper threshold is set as upper limit, and the other analysis frequencies are set as non-upper limit. Based on the lower and upper bounds markers, each analysis frequency is marked, merged, and its complement is extracted to obtain the out-of-bounds analysis frequency set and the effective analysis frequency set. The out-of-bounds analysis frequency set is formed by merging the analysis frequencies set to lower bounds and those set to upper bounds, while the effective analysis frequency set is formed by simultaneously setting the analysis frequencies to neither lower nor upper bounds.

[0032] Specifically, the spectral ratio truncation threshold consists of two sub-terms. The sub-term calibration uses the value obtained by substituting the lower bound of the preset excitation cycle number into the monotonic saturation correspondence as the lower threshold, and the value obtained by substituting the upper bound of the preset excitation cycle number as the upper threshold. The monotonic saturation correspondence is strictly monotonically increasing with respect to the excitation cycle number. Since the lower bound of the preset excitation cycle number is less than the upper bound, the two values ​​obtained after substitution must satisfy the condition that the lower threshold is less than the upper threshold. This relationship does not depend on the specific values ​​of the two endpoints. The robust spectral ratio consists of 241 elements in a row. Each element is compared with the lower threshold; those not greater than the lower threshold are marked as "lower bounds out of bounds," and the rest are marked as "not lower bounds out of bounds." The lower bounds out of bounds marker consists of 241 binary terms of the same length as the robust spectral ratio. Similarly, each element is compared with the upper threshold to obtain the upper bounds out of bounds marker; those not less than the upper threshold are marked as "upper bounds out of bounds," and the rest are marked as "not upper bounds out of bounds." The reason for executing the two comparisons separately is that the end value permutation values ​​corresponding to the boundary overflows on both sides are different. The lower boundary overflow takes the lower bound of the preset excitation cycle number, while the upper boundary overflow takes the upper bound. When combined into a single interval judgment, it can only give two results: inside the interval or outside the interval. It cannot distinguish which side is exceeded, so the end value permutation has no value to take.

[0033] The comparison sign is set to "not greater than" on one side and "not less than" on the other, meaning both boundaries are categorized as belonging to the out-of-bounds side. When the robust spectral ratio is exactly equal to the lower threshold, its inverse iteration number falls at the lower endpoint of the initial interval; when it is exactly equal to the upper threshold, it falls at the upper endpoint. In both cases, the binary iteration has reached the boundary, and the result of the numerical value and endpoint substitution is consistent. Categorizing it to the out-of-bounds side does not change the value and ensures that the criterion has no gaps. The label merging combines the analysis frequencies set as lower out-of-bounds and those set as upper out-of-bounds into the out-of-bounds analysis frequency set. Since the lower threshold is less than the upper threshold, the robust spectral ratio of the same analysis frequency cannot be both not greater than the lower threshold and not less than the upper threshold simultaneously. Merging does not produce duplicate terms. The complement extraction extracts the analysis frequencies that are simultaneously set as neither lower nor upper out-of-bounds to form the effective analysis frequency set. This set is the part of the 241 analysis frequencies excluding the out-of-bounds analysis frequency set. Each analysis frequency has one value on each of the two tags. There are three possible combinations of the two tags: lower bound and non-upper bound, non-lower bound and upper bound, and non-lower bound and non-upper bound. The first two fall into the set of out-of-bounds analysis frequencies, while the third falls into the set of valid analysis frequencies. Therefore, the two sets are mutually exclusive and their union covers all 241 analysis frequencies. There is no analysis frequency that is neither out of bounds nor valid.

[0034] In one specific embodiment, step S3 includes: Based on the coherent cycle number field, the half-width conversion and upper and lower boundary expansion processing are performed on each analysis frequency to obtain the coherent bandwidth aggregation window. The half-width conversion is the quotient obtained by dividing the analysis frequency by twice the coherent cycle number at that analysis frequency as the aggregation window half-width. The upper and lower boundary expansion processing is the analysis frequency minus the aggregation window half-width as the lower boundary of the window and the analysis frequency plus the aggregation window half-width as the upper boundary of the window. Based on the coherent bandwidth aggregation window, the physical spectrum values ​​are processed by in-window frequency point retrieval and cross-frequency point merging to obtain the aggregated spectrum value sample set for each analysis frequency. In-window frequency point retrieval selects the analysis frequency falling between the lower and upper boundaries of the window, and cross-frequency point merging merges the physical spectrum values ​​of all measured times at the selected analysis frequency into the same aggregated spectrum value sample set. Based on the coherent cyclic number field, the logarithmic frequency span of adjacent independent frequency points is taken by adding 1 to the natural logarithm of the reciprocal of the coherent cyclic number, and then the reciprocal of the span is integrated along the logarithmic frequency and rounded up to obtain the equivalent number of independent frequency points. The target whole spectrum joint coverage ratio is corrected by square root of the equivalent independent frequency points to obtain the point-by-point coverage ratio. The aggregated spectrum value sample set is then extracted by quantile and arranged along the analysis frequency according to the point-by-point coverage ratio to obtain the joint confidence envelope spectrum. The square root correction is taken as the root of the equivalent independent frequency points of the target whole spectrum joint coverage ratio.

[0035] Specifically, the duration of the oscillation component corresponding to the coherent cycle number at a certain analysis frequency is equal to the coherent cycle number divided by the analysis frequency. The frequency broadening of an oscillation component with a finite duration is inversely proportional to its duration; therefore, the broadening is equal to the analysis frequency divided by the coherent cycle number. Taking half of this half-width gives the aggregation window's half-width. Subtracting and adding this half-width from the analysis frequency yields the lower and upper bounds of the window. For a coherent cycle number of 1.5 at 30 Hz, the window ranges from 20 Hz to 40 Hz; for 3.82 at 500 Hz, the window ranges from 434.55 Hz to 565.45 Hz; and for 14 at 3000 Hz, the window ranges from 2892.86 Hz to 3107.14 Hz. The higher the coherent cycle number, the longer the oscillation lasts and the more concentrated the frequency domain becomes, resulting in a narrower window. Frequency point retrieval within the window selects frequencies falling between the window boundaries from 241 analysis frequencies. Cross-frequency point merging combines 12 measured physical spectrum values ​​at each selected analysis frequency into a single set, resulting in an aggregated spectral value sample set. The three windows contain 24, 9, and 2 analysis frequencies, respectively, with sample sets containing 300, 108, and 24 physical spectrum values, respectively. The spectral values ​​at each analysis frequency within the window are read from the responses of coherent oscillations in adjacent passbands, representing repeated observations of the same population. Components outside the window are independent, and merging them artificially broadens the distribution.

[0036] Adjacent independent frequency points are categorized based on the fact that their coherence bandwidths are exactly contiguous without overlapping. The ratio of the latter analysis frequency to the former analysis frequency is 1 plus the reciprocal of the coherence cycle number. Taking the natural logarithm, which represents the span occupied by a single independent frequency point on the logarithmic frequency axis, the reciprocal of which is integrated along the logarithmic frequency axis yields the total number of independent frequency points. For example, a coherence cycle number of 1.5 corresponds to a span of 0.51083 in the 10 Hz to 100 Hz range, with a logarithmic span of 2.30259 within the range, resulting in an integral of 4.51. A coherence cycle number of 3.82 corresponds to a span of 0.23237 in the 100 Hz to 10000 Hz range, resulting in an integral of 9.91. A coherence cycle number of 14 corresponds to a span of 0.06899 in the 1000 Hz to 10000 Hz range, resulting in an integral of 33.37. Adding these together yields 47.79, which, rounded up, gives an equivalent number of independent frequency points of 48. The reason for rounding up rather than rounding to the nearest whole number is that the value is included in the exponent of the square root, and a smaller value would result in a lower point-by-point coverage ratio. Since the independent frequency points are approximately uncorrelated, the probability of the entire spectrum being covered simultaneously is equal to the equivalent number of independent frequency points raised to the power of the point-by-point coverage ratio. Therefore, taking the 48th root of 0.95 yields a point-by-point coverage ratio of 0.998932. This 0.95 is consistent with the commonly used 95% coverage standard for impact environmental conditions. When the point-by-point ratio is directly taken as 0.95, the probability of the entire spectrum being covered simultaneously is only 0.95 raised to the power of 48, or 0.0852. Quantile extraction is performed after taking the common logarithm of the aggregated spectral value sample set. The samples are arranged in ascending order, and the index is located by the product of the point coverage ratio and the number of samples. The fractional part of the index is linearly taken between two adjacent index statistics. When the index exceeds the number of samples, the two parameters of shape and scale of the distribution of each transcendent quantity exceeding nine percent are fitted and extrapolated. The obtained logarithmic values ​​are then arranged in ascending order according to the analysis frequency to obtain the joint confidence envelope spectrum.

[0037] Figure 3 This is a schematic diagram of the joint confidence envelope spectrum in an embodiment of this application. Figure 3 The x-axis represents the analysis frequency, and the y-axis represents the response spectrum value. The light-colored thin line represents the response spectrum obtained from 12 actual measurements at the physical damping ratio. The dashed line represents the envelope spectrum obtained by directly taking quantiles point by point according to the target whole spectrum joint coverage ratio of 0.95. The thick solid line represents the joint confidence envelope spectrum obtained after correcting the point-by-point coverage ratio to 0.998932 according to the equivalent independent frequency point number of 48. The spacing between the two envelope spectra in the low-frequency band is smaller than that in the high-frequency band. Figure 2 The difference in the value of the coherent cycle number in the two frequency bands corresponds accordingly.

[0038] In one specific embodiment, step S4 includes: The lower limit of the analysis frequency band is processed by stepwise recursive point placement based on the coherent cycle number field to obtain the component frequency sequence, where the recursion ratio is 1 plus the reciprocal of the coherent cycle number. The component frequency sequence is processed by half-cycle number configuration based on the coherent cycle number field to obtain the component half-cycle number sequence, wherein the component half-cycle number is taken from the odd number not less than 3 that is closest to twice the coherent cycle number. The component amplitudes of the joint confidence envelope spectrum are inversely calculated based on the physical damping ratio and the component half-cycle sequence to obtain the initial component amplitude sequence. Based on the component frequency sequence, component half-cycle number sequence and initial component amplitude sequence, sinusoidal modulation sine wavelet construction, component-by-component superposition and amplitude iterative correction processing are performed to obtain the equivalent load waveform, where the modulation frequency is the quotient of the component frequency and the component half-cycle number, and the carrier frequency is the component frequency.

[0039] Specifically, the component frequency sequence begins with the lower limit of the analysis band at 10 Hz. The recursion ratio is calculated by adding 1 to the reciprocal of the coherent cycle number at that frequency. This recursion ratio is multiplied by the current component frequency to obtain the next component frequency. The last component frequency before exceeding the upper limit of the analysis band (10000 Hz) is the final term. The recursion ratio for the low-frequency band with a coherent cycle number of 1.5 is 1.66667, yielding values ​​of 10 Hz, 16.67 Hz, 27.78 Hz, 46.30 Hz, and 77.16 Hz respectively. The recursion ratio for the mid-frequency band with a coherent cycle number of 3.82 is 1.26178, and the recursion ratio for the high-frequency band with a coherent cycle number of 14 is 1.071429. This recursion ratio is the same as the ratio used in the independent frequency point connection criterion, which is 1 plus the reciprocal of the coherent cycle number. Therefore, the total number of components and the equivalent number of independent frequency points are both 48. The half-cycle number of a component is chosen from among odd numbers not less than 3 that are closest to twice the number of coherent cycles at the component's frequency. Specifically, 3 is chosen for a coherent cycle number of 1.5, 7 for 3.82, and 29 for 14. When the differences are equal, the larger odd number is chosen. The reason for choosing an odd number is that the first and second integrals of the waveform with respect to time within its domain are both zero when the half-cycle number is odd. At the end of the waveform, the velocity and displacement return to their initial values, and the vibration table travel does not exhibit unidirectional drift. Choosing an even number leaves residual velocity at the end. The lower limit is set to 3 because when it is 1, the carrier does not complete a full cycle within the domain, and the waveform degenerates into a single pulse. The value is chosen to approximate twice the number of coherent cycles, ensuring that the number of carrier cycles completed by the component within its domain matches the number of coherent cycles measured at the component's frequency.

[0040] The single-component amplification factor, calculated from the inverse of the component amplitude, is obtained by multiplying the reciprocal of twice the physical damping ratio by the establishment term. The establishment term is 1 minus an exponent term with a negative exponent equal to the product of pi, the physical damping ratio, and the half-cycle number of the component. At 500 Hz, with a half-cycle number of 7 and a physical damping ratio of 0.05, the reciprocal term is 10, the exponent of the exponent term is -1.09956, the establishment term is 0.66704, and the amplification factor is 6.6704. The initial component amplitude of 62.80 is obtained by dividing the value of the joint confidence envelope spectrum at that frequency, 418.9, by 6.6704. The sinusoidal modulated wavelet is constructed by multiplying the modulation sine and the carrier sine. The modulation frequency is the component frequency divided by the component half-cycle number, and the carrier frequency is the component frequency. The domain is defined from zero to the quotient obtained by dividing the component half-cycle number by twice the component frequency, and zero is taken outside the domain. The time delay of each component is the median of the number of times the physical passband branch output reaches its maximum absolute value during sampling. The 48 components are superimposed with their respective amplitudes, frequencies, half-cycle numbers, and time delays to form an experimental waveform. The response spectrum is back-calculated according to the physical damping ratio to obtain the back-calculated spectrum value. The amplitude is updated component by component by dividing the value of the joint confidence envelope spectrum at each component frequency by the back-calculated spectrum value as a correction factor. The first round of back-calculation at 500 Hz yields 445, and the corrected amplitude is 59.11. This process is repeated until the absolute value of the ratio of the envelope value to the back-calculated spectrum value at each component frequency is no greater than 1.5, thus obtaining the equivalent load waveform. The iteration only changes the amplitude, while the component frequency and the number of half-cycles of the component remain unchanged, and the total number of components and the odd number attribute remain unchanged.

[0041] The equivalent load generation method based on the statistical envelope of the impact response spectral domain in this application has been described above. The equivalent load generation system based on the statistical envelope of the impact response spectral domain in this application is described below. One embodiment of the equivalent load generation system based on the statistical envelope of the impact response spectral domain in this application includes: The transformation module is used to perform dual-passband response spectrum transformation on the measured impact signal by using the physical damping ratio and the auxiliary damping ratio to obtain a dual-damped spectrum value pair composed of physical spectrum value and auxiliary spectrum value; The inverse solution module is used to perform numerical inverse solution on each analysis frequency point by point according to the spectral ratio of the physical spectrum value and the auxiliary spectrum value in the dual-damped spectrum value pair, based on the correspondence that the response establishment quantity increases monotonically with the number of excitation cycles and tends to a saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit. The coherent cycle number of each analysis frequency is obtained by filtering the inverse solution results, and the coherent cycle number is arranged along the logarithmic frequency to form a coherent cycle number field. The correction module is used to determine the coherent bandwidth aggregation window and the equivalent independent frequency point number based on the coherent cyclic number field, and to perform coverage ratio correction processing on the physical spectrum value aggregated by the coherent bandwidth aggregation window according to the equivalent independent frequency point number to obtain the joint confidence envelope spectrum. The synthesis module is used to perform time-domain synthesis processing on the joint confidence envelope spectrum based on the coherent cyclic number field, with component frequency recursion and component half-cycle configuration, to obtain the equivalent load waveform.

[0042] This invention also provides an equivalent load generation device based on the statistical envelope of the impact response spectral domain, which can be a server. The device includes a processor, memory, display screen, input device, network interface, and database connected via a system bus. The processor provides computational and control capabilities. The memory includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system, computer programs, and a database. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. The database stores the data corresponding to this embodiment. The network interface communicates with external terminals via a network connection. The computer program is executed by the processor to implement the above method.

[0043] The present invention also provides a computer-readable storage medium, which can be a non-volatile computer-readable storage medium or a volatile computer-readable storage medium, wherein the computer-readable storage medium stores instructions that, when the instructions are executed on a computer, cause the computer to perform the steps of the equivalent load generation method based on the statistical envelope of the impulse response spectral domain.

[0044] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the specific working processes of the systems and units described above can be referred to the corresponding processes in the foregoing method embodiments, and will not be repeated here.

[0045] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause an equivalent load generation device (which may be a personal computer, server, or network device, etc.) based on the statistical envelope of the impulse response spectral domain to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0046] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for generating equivalent loads based on the statistical envelope of the impact response spectral domain, characterized in that, The method includes: Step S1: Perform dual-passband response spectrum transformation on the measured impact signal using the physical damping ratio and the auxiliary damping ratio to obtain a dual-damped spectrum value pair consisting of the physical spectrum value and the auxiliary spectrum value; Step S2: Based on the spectral ratio of the physical spectrum value to the auxiliary spectrum value in the dual-damped spectrum value pair, and according to the correspondence that the response establishment quantity increases monotonically with the number of excitation cycles and tends to a saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit, perform numerical inverse solution point by point for each analysis frequency, obtain the coherent cycle number of each analysis frequency by filtering the inverse solution results, and arrange the coherent cycle number along the logarithmic frequency to form a coherent cycle number field; Step S3: Determine the coherent bandwidth aggregation window and the equivalent independent frequency point number based on the coherent cyclic number field. Perform coverage ratio correction processing on the physical spectrum value aggregated by the coherent bandwidth aggregation window according to the equivalent independent frequency point number to obtain the joint confidence envelope spectrum. Step S4: Perform time-domain synthesis processing on the joint confidence envelope spectrum based on the coherent cyclic number field, using component frequency recursion and component half-cycle configuration, to obtain the equivalent load waveform.

2. The method for generating equivalent loads based on the statistical envelope of the impact response spectral domain according to claim 1, characterized in that, Step S1 includes: The measured impact signal is subjected to zero drift correction, anti-aliasing filtering and detrending processing to obtain the baseline correction signal; The baseline correction signal is processed by truncating the impact segment and extending the residual vibration according to the trigger threshold to obtain the impact time history of equal length, wherein the time delay after the truncated end point is 20 divided by the lower limit of the analysis frequency band. Based on the analysis frequency band, the points are arranged at equal intervals of 1 / 24 octave logarithm, and the single-degree-of-freedom basic excitation model is discretized with slope invariance based on the physical damping ratio and the auxiliary damping ratio to obtain a dual-passband filter bank composed of physical passband branch and auxiliary passband branch, wherein the ratio of the auxiliary damping ratio to the physical damping ratio is 5. The equal-length impact time history is input into the dual-passband filter bank for frequency-by-frequency response recursion and absolute value extremum extraction to obtain a pair of dual-damped spectrum values ​​consisting of physical spectrum values ​​and auxiliary spectrum values. The physical spectrum values ​​and the auxiliary spectrum values ​​are taken from two branches that share the same equal-length impact time history and the same analysis frequency grid.

3. The method for generating equivalent loads based on the statistical envelope of the impact response spectral domain according to claim 1, characterized in that, The step involves performing a point-by-point numerical inverse solution for each analysis frequency based on the spectral ratio of the physical spectral value to the auxiliary spectral value in the dual-damped spectral pair. This is done according to the correspondence that the response build-up monotonically increases with the number of excitation cycles and tends to a saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit. This includes: The quotient of the dual-damped spectral value pairs is obtained by taking the quotient according to the one-to-one correspondence between the actual number of measurements and the analysis frequency. The quotient is then divided by the physical spectral value and the auxiliary spectral value. The median of the spectral ratio sample along the number of measurements is extracted to obtain the robust spectral ratio for each analysis frequency; Based on the physical damping ratio and the auxiliary damping ratio, the transient establishment law of the single-degree-of-freedom system is amplified by ratio processing to obtain a monotonic saturation correspondence. The monotonic saturation correspondence takes the number of excitation cycles as the independent variable, takes 1 when the number of excitation cycles approaches zero, and takes the ratio of the auxiliary damping ratio to the physical damping ratio when the number of excitation cycles approaches infinity. Based on the monotonically saturated correspondence, a control spectral ratio sequence is generated with a preset step size. The robust spectral ratio is then subjected to interval clamping and binary iterative inverse solution processing according to the control spectral ratio sequence to obtain the inverse solution cycle number for each analysis frequency. The convergence criterion for the binary iteration is that the difference between the excitation cycle numbers obtained from two adjacent iterations is less than 0.

01.

4. The method for generating equivalent loads based on the statistical envelope of the impact response spectral domain according to claim 3, characterized in that, The process of obtaining the coherent cycle number for each analysis frequency by filtering the inverse kinematics results and arranging the coherent cycle numbers along the logarithmic frequency to form a coherent cycle number field includes: Based on the monotonically saturated correspondence, values ​​are taken at the lower bound of the preset excitation cycle number and the upper bound of the preset excitation cycle number to obtain the spectral ratio cutoff threshold pair; Based on the spectral ratio cutoff threshold, the robust spectral ratio is subjected to out-of-bounds discrimination processing to obtain the out-of-bounds analysis frequency set and the effective analysis frequency set; The inverse solution cycle number is subjected to end value permutation and retention processing based on the out-of-bounds analysis frequency set and the effective analysis frequency set to obtain the coherent cycle number of each analysis frequency. The analysis frequencies in the out-of-bounds analysis frequency set with a robust spectral ratio not greater than the lower threshold are taken as the lower bound of the preset excitation cycle number, and the analysis frequencies with a robust spectral ratio not less than the upper threshold are taken as the upper bound of the preset excitation cycle number. The coherent cycle numbers are arranged in ascending order of the logarithmic frequency of each analysis frequency, and logarithmic frequency linear interpolation is performed between adjacent analysis frequencies to obtain the coherent cycle number field.

5. The method for generating equivalent loads based on the statistical envelope of the impact response spectral domain according to claim 4, characterized in that, The step of performing out-of-bounds discrimination processing on the robust spectral ratio based on the spectral ratio cutoff threshold to obtain an out-of-bounds analysis frequency set and an effective analysis frequency set includes: The spectral ratio cutoff threshold is calibrated by sub-item calibration based on the corresponding values ​​of the lower bound of the preset excitation cycle number and the upper bound of the preset excitation cycle number, to obtain a lower threshold and an upper threshold, wherein the lower threshold is less than the upper threshold. The robust spectral ratio is subjected to a frequency-by-frequency crossover comparison based on the lower threshold to obtain a crossover flag. Analytical frequencies with a robust spectral ratio not greater than the lower threshold are set as crossover frequencies, and the remaining analytical frequencies are set as non-crossover frequencies. Based on the upper threshold, the robust spectral ratio is subjected to frequency-by-frequency crossover comparison processing to obtain an upper crossover flag. The analysis frequencies with a robust spectral ratio not less than the upper threshold are set as upper crossover, and the remaining analysis frequencies are set as non-upper crossover. Based on the lower bounds marker and the upper bounds marker, each analysis frequency is marked, merged, and its complement is extracted to obtain a set of out-of-bounds analysis frequencies and a set of valid analysis frequencies. The set of out-of-bounds analysis frequencies is formed by merging the analysis frequencies set to lower bounds and the analysis frequencies set to upper bounds. The set of valid analysis frequencies is formed by simultaneously setting the analysis frequencies to neither lower nor upper bounds.

6. The method for generating equivalent loads based on the statistical envelope of the impact response spectral domain according to claim 1, characterized in that, Step S3 includes: Based on the coherent cycle number field, half-width conversion and upper and lower boundary expansion processing are performed on each analysis frequency to obtain a coherent bandwidth aggregation window. The half-width conversion is the quotient obtained by dividing the analysis frequency by twice the coherent cycle number at that analysis frequency as the aggregation window half-width. The upper and lower boundary expansion processing is the analysis frequency minus the aggregation window half-width as the lower boundary of the window and the analysis frequency plus the aggregation window half-width as the upper boundary of the window. According to the coherent bandwidth aggregation window, the physical spectrum value is subjected to in-window frequency point retrieval and cross-frequency point merging to obtain an aggregated spectrum value sample set for each analysis frequency. The in-window frequency point retrieval selects the analysis frequency falling between the lower boundary and the upper boundary of the window, and the cross-frequency point merging merges the physical spectrum values ​​of all measured times at the selected analysis frequency into the same aggregated spectrum value sample set. Based on the coherent cyclic number field, the logarithmic frequency span of adjacent independent frequency points is taken by adding 1 to the natural logarithm of the reciprocal of the coherent cyclic number, and then the reciprocal of the span is integrated along the logarithmic frequency and rounded up to obtain the equivalent number of independent frequency points. The target whole spectrum joint coverage ratio is corrected by square root based on the equivalent independent frequency points to obtain the point-by-point coverage ratio. The aggregated spectrum value sample set is then extracted by quantile and arranged along the analysis frequency according to the point-by-point coverage ratio to obtain the joint confidence envelope spectrum. The square root correction is taken as the root of the equivalent independent frequency points of the target whole spectrum joint coverage ratio.

7. The method for generating equivalent loads based on the statistical envelope of the impact response spectral domain according to claim 1, characterized in that, Step S4 includes: The lower limit of the analysis frequency band is processed by stepwise recursive point placement based on the coherent cycle number field to obtain the component frequency sequence, wherein the recursion ratio is 1 plus the reciprocal of the coherent cycle number. The component frequency sequence is processed by half-cycle number configuration according to the coherent cycle number field to obtain the component half-cycle number sequence, wherein the component half-cycle number is taken from the odd number not less than 3 that is closest to twice the coherent cycle number. Based on the physical damping ratio and the component half-cycle number sequence, the joint confidence envelope spectrum is back-calculated to obtain the initial component amplitude sequence. Based on the component frequency sequence, the component half-cycle number sequence, and the initial component amplitude sequence, sinusoidal modulation sine wavelet construction, component-by-component superposition, and amplitude iterative correction processing are performed to obtain the equivalent load waveform, wherein the modulation frequency is the quotient of the component frequency and the component half-cycle number, and the carrier frequency is the component frequency.

8. An equivalent load generation system based on the statistical envelope of the impact response spectral domain, characterized in that, For implementing the equivalent load generation method based on the statistical envelope of the impact response spectral domain as described in any one of claims 1-7, the equivalent load generation system based on the statistical envelope of the impact response spectral domain comprises: The transformation module is used to perform dual-passband response spectrum transformation on the measured impact signal by using the physical damping ratio and the auxiliary damping ratio to obtain a dual-damped spectrum value pair composed of physical spectrum value and auxiliary spectrum value; The inverse solution module is used to perform numerical inverse solution on each analysis frequency point by point according to the spectral ratio of the physical spectrum value and the auxiliary spectrum value in the dual-damped spectrum value pair, based on the correspondence that the response establishment quantity increases monotonically with the number of excitation cycles and tends to a saturation value with the ratio of the auxiliary damping ratio to the physical damping ratio as the upper limit. The coherent cycle number of each analysis frequency is obtained by filtering the inverse solution results, and the coherent cycle number is arranged along the logarithmic frequency to form a coherent cycle number field. The correction module is used to determine the coherent bandwidth aggregation window and the equivalent independent frequency point number based on the coherent cyclic number field, and to perform coverage ratio correction processing on the physical spectrum value aggregated by the coherent bandwidth aggregation window according to the equivalent independent frequency point number to obtain the joint confidence envelope spectrum. The synthesis module is used to perform time-domain synthesis processing on the joint confidence envelope spectrum based on the coherent cyclic number field, with component frequency recursion and component half-cycle configuration, to obtain the equivalent load waveform.

9. An equivalent load generation device based on the statistical envelope of the impact response spectral domain, characterized in that, The system includes a memory and a processor, wherein the memory stores a computer program that can run on the processor, and the processor executes the computer program to implement the equivalent load generation method based on the statistical envelope of the shock response spectral domain as described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is run by the processor, it causes the processor to execute the equivalent load generation method based on the statistical envelope of the impulse response spectral domain as described in any one of claims 1 to 7.

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