Fatigue Damage Assessment Method under Uniform Modulation of Non-Stationary and Non-Gaussian Random Loads

Abstract

The present invention discloses a fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads, which relates to the field of random vibration fatigue damage assessment. The method includes obtaining a non-stationary random process based on a uniform modulation function, and defining the non-stationary random process obtained by modulating a stationary non-Gaussian random process as a true non-stationary non-Gaussian random process. Based on the EPSD decomposition method for non-stationary Gaussian fatigue damage assessment, the spectral method for stationary non-Gaussian fatigue damage assessment is incorporated to solve the random vibration fatigue damage of non-stationary non-Gaussian stress. The present invention further considers the situation where both non-stationarity and non-Gaussianity coexist in random vibration fatigue damage analysis. For non-Gaussianity, a more reasonable definition and division of the non-stationary random process are carried out, and the problem of random vibration fatigue damage assessment with both non-stationarity and non-Gaussianity is solved under the condition of a reliable precision spectrum.

Description

Technical Field

[0001] The present invention relates to the field of fatigue damage assessment, in particular to a fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads. Background Art

[0002] Many engineering structures are in a vibration environment for a long time and are often subjected to random loads, such as vehicle components excited by road unevenness, wind turbine structures subjected to wind loads, marine structures subjected to the actions of wind, waves, currents, etc., and airborne equipment in aviation affected by various impact loads. These excitations may lead to the occurrence of fatigue damage. However, the randomness of the loads brings great difficulties to the fatigue damage assessment of the key components of these structures and the prediction of fatigue life. Therefore, it is of great significance to quickly and accurately assess the fatigue damage of structures under random vibration loads, especially complex non-stationary non-Gaussian loads.

[0003] Fatigue damage assessment usually determines the fatigue life through stress responses measured on site or through laboratory tests. However, on-site measurement and laboratory tests are often costly and time-consuming. In the context of random vibration fatigue, a variety of spectral methods based on the power spectral density (PSD) have emerged to facilitate the rapid estimation of fatigue damage. These spectral methods eliminate the need for the stress response time history, significantly reducing the time required for fatigue damage assessment.

[0004] Traditional spectral methods based on PSD usually follow the stationary Gaussian assumption of the stress response, and many spectral methods have been successively proposed and are divided into two categories: narrow-band methods and wide-band methods according to the distribution of PSD in frequency.

[0005] Although spectral methods based on PSD are effective in many cases, PSD cannot characterize non-Gaussianity. Therefore, when facing non-Gaussian loads, stationary Gaussian spectral methods are no longer applicable. In the field of non-Gaussian random process processing and analysis, non-linear transformation is a very important tool. The core idea of non-linear transformation is to establish a transformation relationship between non-Gaussian and Gaussian random processes through a non-linear function. Therefore, stationary non-Gaussian spectral methods combining non-linear transformation with stationary Gaussian spectral methods have been proposed. Typical stationary non-Gaussian spectral methods include the non-Gaussian NB method for narrow-band random processes and the non-Gaussian DK method and non-Gaussian TB method for wide-band non-Gaussian random processes.

[0006] To solve the problem of non-stationary random vibration, the Evolutionary Power Spectral Density (EPSD) is proposed as a new analysis tool. EPSD can not only effectively capture the time-frequency characteristics of non-stationary random processes, but also be combined with conventional spectral methods for non-stationary fatigue damage assessment. The established EPSD decomposition method decomposes non-stationary processes into multiple equivalent stationary PSDs, thus using stationary spectral methods for non-stationary fatigue damage assessment.

[0007] However, the actual non-stationary loads are complex, and in many cases, the loads exhibit characteristics of both non-stationarity and non-Gaussianity. Since non-stationarity describes the property of the statistical characteristics of a random process changing with time, while non-Gaussianity describes the characteristic of a random process deviating from the Gaussian probability distribution. Pure non-Gaussianity analysis is generally based on stationary random processes, and the non-stationarity of a random process will cause its overall probability distribution to deviate from Gaussian. The existing EPSD decomposition method only considers non-stationarity and does not consider non-Gaussianity. Therefore, in the case of both non-stationarity and non-Gaussianity, the fatigue damage calculated by this method will deviate from the true damage, thus affecting the prediction of the fatigue life or performance of engineering structures.

[0008] Therefore, to solve the problem of fatigue damage assessment of complex non-stationary non-Gaussian random vibrations, it is urgent to continue to establish a set of fatigue damage assessment methods based on the evolutionary power spectral density. Summary of the Invention

[0009] To solve the above problems, this application proposes a fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads, including the following steps:

[0010] S1. Based on the amplitude modulation method and the uniform modulation function, modulate the stationary non-Gaussian random process to obtain a non-stationary random process, and define it as a true non-stationary non-Gaussian random process;

[0011] S2. Based on the evolutionary power spectral density decomposition method, incorporate the spectral method of stationary non-Gaussian random vibration fatigue analysis to solve the fatigue damage of non-stationary non-Gaussian random stress;

[0012] Preferably, the specific content of modulating the stationary non-Gaussian random process to obtain a non-stationary random process in S1 includes selecting a slow-varying function that is only related to time as the modulation function (i.e., the uniform modulation function), and based on the stationary non-Gaussian random process, obtaining the non-stationary non-Gaussian random process through the amplitude modulation method. The non-stationary non-Gaussian random process obtained by modulation is characterized by the evolutionary power spectral density and the skewness and kurtosis values.

[0013] Similarly, a uniformly modulated function is selected, and the random process obtained by modulating a stationary Gaussian random process is defined as a non-stationary Gaussian random process, which is characterized by the evolving power spectral density;

[0014] The non-stationary Gaussian random process and the non-stationary non-Gaussian random process together constitute the non-stationary random process.

[0015] Preferably, the modulation function is expressed as a slowly varying function related to frequency and time ;

[0016] where, is the value of the slowly varying function, is the frequency, t is the time;

[0017] When the modulation function does not vary with frequency, it is called a uniformly modulated function.

[0018] Preferably, the modulation process of the non-stationary Gaussian random process includes:

[0019] A zero-mean non-stationary Gaussian random process is expressed as:

[0020] ;

[0021] where, represents the frequency, t represents the time, is a slowly varying non-uniform modulation function, is a zero-mean orthogonal increment random process, and the zero-mean stationary Gaussian random process corresponding to is expressed as:

[0022] ;

[0023] The relationship between the evolving power spectrum (EPSD) of the non-stationary Gaussian random process and the power spectrum of the stationary Gaussian random process is expressed as:

[0024] ;

[0025] The non-stationary Gaussian random process is obtained by multiplying the underlying stationary Gaussian random process by the uniformly modulated function:

[0026] ;

[0027] At this time, the relationship between the evolving power spectrum of the non-stationary Gaussian random process and the power spectrum of the stationary Gaussian random process is expressed as:

[0028] .

[0029] Preferably, the modulation process of the non-stationary non-Gaussian random process includes:

[0030] Replacing the underlying stationary Gaussian random process in the modulation process of the non-stationary Gaussian random process with a stationary non-Gaussian random process, the modulation process of the non-stationary non-Gaussian random process can be obtained:

[0031] ;

[0032] Wherein, Z ( t ) is a zero-mean stationary non-Gaussian random process, is a uniform modulation function, is a non-stationary non-Gaussian random process with a mean of 0.

[0033] And the stationary non-Gaussian random process Z ( t ) is transformed from the stationary Gaussian random process X ( t ) through the Johnson non-linear transformation system:

[0034] ;

[0035] Wherein, , are respectively the standardized forms of the stationary Gaussian random process X ( t ) and the stationary non-Gaussian random process Z ( t ); , , , are the four parameters of the transformation system, obtained by the moment evaluation method. The selection of SU, SB, and SL is determined by the skewness and kurtosis combination value of the stationary non-Gaussian random process Z ( t ).

[0036] The -th central moment n of the non-stationary non-Gaussian random process obtained by modulation and the Z ( t )-th central moment n of the stationary non-Gaussian random process satisfy the following relationship:

[0037] ;

[0038] Wherein, is the -th statistical moment of the uniform modulation function n ,E Indicates expectation.

[0039] The relationships between the skewness and kurtosis values of the non-stationary non-Gaussian random process obtained by modulation and those of the basic stationary non-Gaussian random process are as follows:

[0040] ;

[0041] Among them, represents skewness, represents kurtosis. Both skewness and kurtosis are characteristic values used to characterize the non-Gaussianity of a random process. For a Gaussian random process, its skewness is 0 and its kurtosis is 3; , respectively represent the skewness and kurtosis of the non-stationary non-Gaussian random process , , respectively represent the skewness and kurtosis of the stationary non-Gaussian random process Z ( t ), , , respectively represent the second, third, and fourth central moments of the stationary non-Gaussian random process Z ( t ), , , respectively represent the second, third, and fourth statistical moments of the modulation function , , are defined as the skewness modulation influence coefficient and the kurtosis modulation influence coefficient, both of which depend on the statistical moments of the modulation function and respectively characterize the influence degrees of the modulation signal on the skewness and kurtosis of the non-stationary non-Gaussian random process, is the third statistical moment of the non-stationary non-Gaussian random process, is the second statistical moment of the non-stationary non-Gaussian random process, is the fourth statistical moment of the non-stationary non-Gaussian random process.

[0042] Preferably, the content of solving the vibration fatigue damage of the non-stationary non-Gaussian random stress by integrating the spectral method under stationary non-Gaussian random loads based on the evolution power spectral density EPSD decomposition method in S2 includes:

[0043] Combining the characteristics that the non-stationary non-Gaussian random process is jointly characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, integrating the spectral method of typical stationary non-Gaussian fatigue analysis to solve the non-stationary non-Gaussian random vibration fatigue damage;

[0044] The spectral methods for the fatigue analysis of stationary non-Gaussian random vibrations include the non-Gaussian NB method, the non-Gaussian DK method, and the non-Gaussian TB method.

[0045] Preferably, in combination with the characteristics that the EPSD and its skewness and kurtosis values jointly characterize the non-stationary non-Gaussian random process, based on the EPSD decomposition method, the spectral methods for evaluating the fatigue damage of typical stationary non-Gaussian are incorporated to solve the specific content of the non-stationary non-Gaussian random vibration fatigue damage as follows:

[0046] Z ( t ) is a stationary Gaussian random process X ( t ) is a stationary non-Gaussian random process obtained by non-linear transformation, Z ( t ) is consistent with the power spectral density (PSD) of X ( t ); is Z ( t ) is a non-stationary non-Gaussian random process obtained by uniform modulation;

[0047] When the modulus of the modulation function is 1, the average power spectral density of is consistent with Z ( t ), and the EPSD of is expressed as:

[0048] ;

[0049] wherein, is the EPSD of the non-stationary non-Gaussian random process , and are respectively the PSDs of Z ( t ) and X ( t ).

[0050] When the EPSD of the vibration system excitation and the skewness and kurtosis of the response signal are known, based on the EPSD decomposition, and the non-Gaussian TB method or the non-Gaussian DK method is selected to calculate the non-stationary non-Gaussian fatigue damage;

[0051] The calculation process is as follows:

[0052] (1) Represent the excited EPSD in the form of a combination of a uniformly modulated function and its average PSD; discretize the excited EPSD in the time domain. The discretized excited EPSD is considered as a combination of the PSDs at each moment, and the PSD at each discrete moment is the product of the modulation function value at that moment and the excited average PSD. The excited average PSD is the PSD corresponding to the case where its non-stationarity is not considered.

[0053] (2) For the PSD at each discrete moment, obtain the response PSDs at the corresponding discrete moments of the system response through the frequency response function of the vibration system.

[0054] (3) Obtain the skewness and kurtosis modulation influence coefficients from the statistical moments of the modulation function, and combine the skewness and kurtosis values of the non-stationary non-Gaussian response to obtain the skewness and kurtosis values of the corresponding underlying stationary non-Gaussian random process.

[0055] (4) For each discrete response PSD, combine the skewness and kurtosis values of the underlying stationary non-Gaussian random process corresponding to the response, and use the non-Gaussian NB method or the non-Gaussian DK method or the non-Gaussian TB method to calculate the fatigue damage within the corresponding discrete time period.

[0056] (5) Accumulate the fatigue damages calculated within each discrete time period to obtain the fatigue damage value within the overall time domain.

[0057] In summary, starting from the modulated non-stationary random process, the present invention defines the random process obtained by modulating a stationary non-Gaussian random process as a non-stationary non-Gaussian random process, and regards the random process obtained by modulating a stationary Gaussian random process as a non-stationary Gaussian random process, so as to better distinguish non-stationary random processes.

[0058] Combining the amplitude modulation method that characterizes non-stationarity with a uniformly modulated function, using EPSD to characterize the time-frequency characteristics of non-stationary non-Gaussian random processes, and using skewness and kurtosis indexes to characterize the non-Gaussian characteristics of non-stationary non-Gaussian random processes. Based on the EPSD decomposition method for non-stationary Gaussian fatigue damage assessment, integrating the spectral methods (non-Gaussian NB method, DK method, and TB method) for typical stationary non-Gaussian fatigue damage assessment, to solve the non-stationary non-Gaussian random vibration fatigue damage.

[0059] Compared with the traditional method for non-stationary Gaussian fatigue damage assessment using the EPSD decomposition method, the present invention further considers the situation of both non-stationarity and non-Gaussianity in random vibration fatigue damage analysis, and for non-Gaussianity, makes a more reasonable definition and division of non-stationary random processes, and solves the problem of more complex non-stationary non-Gaussian random vibration fatigue damage assessment. Generally speaking, compared with the prior art, the present invention better solves the problem of random vibration fatigue damage assessment when both non-stationarity and non-Gaussianity coexist.

[0060] The technical method of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Description of the Drawings

[0061] Figure 1 It is the modulation function used to obtain the non-stationary non-Gaussian random process in the first embodiment of the present invention;

[0062] Figure 2 It is the verification result of the non-stationary non-Gaussian random process EPSD obtained by modulation in the first embodiment of the present invention. Figure 2 In (a) is the estimated value of EPSD; Figure 2 In (b) is the theoretical value of EPSD;

[0063] Figure 3 It is the input and output average PSD spectra selected in the second embodiment of the present invention. Figure 3 In (a) is the average PSD spectrum of the acceleration excitation; Figure 3 In (b) is the average PSD spectrum of the stress response;

[0064] Figure 4 It is the time-domain fatigue damage calculation flowchart for verifying the result of the non-stationary non-Gaussian fatigue damage assessment spectrum method based on EPSD of the present invention.

[0065] Figure 5 It is the comparison of the results of the non-stationary non-Gaussian fatigue damage assessment spectrum method based on EPSD and the time-domain method of the present invention under a narrowband spectrum. Figure 5 In (a) is the non-stationary Gaussian case; Figure 5 In (b) is the non-stationary non-Gaussian case ( , ) Figure 5 In (c) is the non-stationary non-Gaussian case ( , )

[0066] Figure 6 It is the input and output average PSD spectra adopted to verify the non-stationary non-Gaussian fatigue damage assessment spectrum method based on EPSD under a broadband spectrum in the third embodiment of the present invention. Figure 6 In (a) is the average PSD spectrum of the acceleration excitation; Figure 6 In (b) is the first average PSD spectrum of the stress response; Figure 6 In (c) is the second average PSD spectrum of the stress response;

[0067] Figure 7 It is the comparison of the results of the broadband non-stationary non-Gaussian fatigue damage spectrum method and the time-domain method in the third embodiment of the present invention. Figure 7 In (a) is the non-stationary Gaussian case; Figure 7where (b) is a non-stationary non-Gaussian case ( , ); Figure 7 where (c) is a non-stationary non-Gaussian case ( , );

[0068] Figure 8 This is the second comparison of the results between the broadband non-stationary non-Gaussian fatigue damage spectrum method and the time-domain method in the third embodiment of the present invention. Figure 8 where (a) is a non-stationary Gaussian case, Figure 8 where (b) is a non-stationary non-Gaussian case ( , ), Figure 8 where (c) is a non-stationary non-Gaussian case ( , );

[0069] Figure 9 This is the PSD spectrum with different bandwidth parameters selected in the third embodiment of the present invention.

[0070] Figure 10 This is the comparison of the results between the non-stationary non-Gaussian fatigue damage assessment spectrum method based on EPSD and the time-domain method under different bandwidth conditions of the present invention. Figure 10 where (a) is a non-stationary Gaussian case, Figure 10 where (b) is a non-stationary non-Gaussian case ( , ), Figure 10 where (c) is a non-stationary non-Gaussian case ( , ). Detailed implementation manners

[0071] The technical method of the present invention will be further described below through the accompanying drawings and embodiments. It should be noted that: unless otherwise specifically stated, the relative arrangements of components and steps, numerical expressions and values set forth in these embodiments do not limit the scope of the present application.

[0072] The following description of at least one exemplary embodiment is merely illustrative in nature and in no way limiting of the present application, its application, or its use.

[0073] Technologies, systems, and devices known to those of ordinary skill in the relevant art may not be discussed in detail, but where appropriate, the technologies, systems, and devices should be regarded as part of the specification.

[0074] In all the examples shown and discussed here, any specific value should be construed as merely exemplary and not as a limitation. Therefore, other examples of the exemplary embodiments may have different values.

[0075] Unless otherwise defined, the technical terms or scientific terms used in the present invention shall have the ordinary meanings understood by those with ordinary skills in the field to which the present invention pertains.

[0076] To solve the problem of fatigue damage assessment under non-stationary non-Gaussian random loads, the present application proposes a modeling method based on a stationary non-Gaussian random process and using a uniform modulation function for amplitude modulation to characterize the true non-stationary non-Gaussian random load. In addition, based on EPSD analysis, the present application adopts the EPSD discretization method for non-stationary Gaussian fatigue damage assessment and combines it with the spectral method for stationary non-Gaussian fatigue damage assessment to conduct non-stationary non-Gaussian fatigue damage assessment.

[0077] The present application proposes a fatigue damage assessment method under a uniformly modulated non-stationary non-Gaussian random load, including the following steps:

[0078] S1. Based on the amplitude modulation method and the uniform modulation function, modulate the stationary non-Gaussian random process to obtain a non-stationary random process, and define it as the true non-stationary non-Gaussian random process;

[0079] Preferably, the specific content of modulating the stationary non-Gaussian random process to obtain a non-stationary random process in S1 includes selecting a slow-varying function that is only related to time as the modulation function (i.e., the uniform modulation function), and based on the stationary non-Gaussian random process, obtaining the non-stationary non-Gaussian random process through the amplitude modulation method. The non-stationary non-Gaussian random process obtained by modulation is characterized by the evolutionary power spectral density and the skewness and kurtosis values;

[0080] Similarly, select the uniform modulation function, and define the random process obtained by modulating the stationary Gaussian random process as the non-stationary Gaussian random process, and the non-stationary Gaussian random process is characterized by the evolutionary power spectral density;

[0081] The non-stationary Gaussian random process and the non-stationary non-Gaussian random process together constitute the non-stationary random process.

[0082] Preferably, the modulation function is expressed as a slow-varying function related to frequency and time ; where is the value of the slow-varying function, is the frequency, t is the time;

[0083] When the modulation function does not change with frequency, it is called the uniform modulation function.

[0084] Preferably, the modulation process of the non-stationary Gaussian random process includes:

[0085] A zero-mean non-stationary Gaussian random process is expressed as:

[0086] ;

[0087] Among them, represents frequency, t represents time, is a slowly varying non-uniform modulation function, is a zero-mean orthogonal increment random process, and the zero-mean stationary Gaussian random process corresponding to is expressed as:

[0088] ;

[0089] The relationship between the evolving power spectrum of the non-stationary Gaussian random process and the power spectrum of the stationary Gaussian random process is expressed as:

[0090] ;

[0091] The non-stationary Gaussian random process is obtained by multiplying the underlying stationary Gaussian random process by a uniform modulation function:

[0092] ;

[0093] At this time, the relationship between the evolving power spectrum of the non-stationary Gaussian random process and the power spectrum of the stationary Gaussian random process is expressed as:

[0094] .

[0095] Preferably, the modulation process of the non-stationary non-Gaussian random process includes:

[0096] Replacing the underlying stationary Gaussian random process in the modulation process of the non-stationary Gaussian random process with a stationary non-Gaussian random process, the modulation process of the non-stationary non-Gaussian random process can be obtained:

[0097] ;

[0098] Among them, Z ( t ) is a zero-mean stationary non-Gaussian random process, is a uniform modulation function, is a non-stationary non-Gaussian random process with a mean of 0.

[0099] And the stationary non-Gaussian random process Z ( t ) is obtained by converting the stationary Gaussian random process X ( t ) through the Johnson nonlinear transformation system:

[0100] ;

[0101] Among them, , are the standardized forms of a stationary Gaussian random process X ( t ) and a stationary non-Gaussian random process Z ( t ); , , , are the four parameters of the conversion system, obtained by the moment evaluation method. The selection of SU, SB, and SL is determined by the combined skewness and kurtosis values of the stationary non-Gaussian random process Z ( t ).

[0102] The nth-order central moment of the non-stationary non-Gaussian random process n obtained by modulation and the nth-order central moment Z ( t ) of the stationary non-Gaussian random process n satisfy the following relationship: ;

[0103] ;

[0104] where is the nth-order statistical moment of the uniform modulation function , n and E denotes the expectation.

[0105] The relationships between the skewness and kurtosis values of the non-stationary non-Gaussian random process obtained by modulation and the skewness and kurtosis values of the underlying stationary non-Gaussian random process are as follows:

[0106] ;

[0107] where denotes the skewness, denotes the kurtosis. Both skewness and kurtosis are characteristic values used to characterize the non-Gaussianity of a random process. For a Gaussian random process, its skewness is 0 and its kurtosis is 3; , denote the skewness and kurtosis of the non-stationary non-Gaussian random process respectively, , denote the skewness and kurtosis of the stationary non-Gaussian random process Z ( t ) respectively, , , denote the skewness and kurtosis of the stationary non-Gaussian random process Z ( tThe second, third, and fourth central moments of , , respectively represent the second, third, and fourth statistical moments of the modulation function . , are defined as the skewness modulation influence coefficient and the kurtosis modulation influence coefficient, both of which depend on the statistical moments of the modulation function and respectively characterize the influence degrees of the modulation signal on the skewness and kurtosis of the non-stationary non-Gaussian random process. is the third statistical moment of the non-stationary non-Gaussian random process, is the second statistical moment of the non-stationary non-Gaussian random process, is the fourth statistical moment of the non-stationary non-Gaussian random process.

[0108] S2 is based on the evolutionary power spectral density decomposition method and incorporates the spectral method for stationary non-Gaussian random vibration fatigue analysis to solve the fatigue damage of non-stationary non-Gaussian random stress.

[0109] Preferably, the content of S2 based on the evolutionary power spectral density EPSD decomposition method and incorporating the spectral method under stationary non-Gaussian random loads to solve the vibration fatigue damage of non-stationary non-Gaussian random stress includes:

[0110] Combining the characteristics of the non-stationary non-Gaussian random process jointly characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, incorporating the spectral method for typical stationary non-Gaussian fatigue analysis, and used to solve the non-stationary non-Gaussian random vibration fatigue damage;

[0111] The spectral methods for stationary non-Gaussian random vibration fatigue analysis include the non-Gaussian NB method, the non-Gaussian DK method, and the non-Gaussian TB method.

[0112] Preferably, combining the characteristics of the non-stationary non-Gaussian random process jointly characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, the specific content of incorporating the spectral method for typical stationary non-Gaussian fatigue damage assessment to solve the non-stationary non-Gaussian random vibration fatigue damage is:

[0113] Z ( t ) is a stationary Gaussian random process X ( t ) obtained by non-linearly transforming a stationary non-Gaussian random process, Z ( t ) has the same power spectral density (PSD) as X ( t ), is Z ( t ) obtained by uniformly modulating a non-stationary non-Gaussian random process;

[0114] When the modulus of the modulation function is 1, the average power spectral density of Z ( t ) is consistent with that of and the EPSD of

[0115] ;

[0116] wherein, is the EPSD of the non-stationary and non-Gaussian random process , and are respectively the PSD of Z ( t ) and X ( t ).

[0117] When the EPSD of the vibration system excitation and the skewness and kurtosis of the response are known, based on the EPSD decomposition, and by selecting the non-Gaussian TB method or the non-Gaussian DK method, the calculation of the non-stationary and non-Gaussian fatigue damage is carried out;

[0118] The calculation process is as follows:

[0119] (1) Characterize the EPSD of the excitation as a combination of a uniform modulation function and its average PSD; discretize the EPSD of the excitation in the time domain. The discretized excitation EPSD is considered as a combination of the PSDs at each moment, and the PSD at each discrete moment is the product of the modulation function value at that moment and the excitation average PSD. The excitation average PSD is the PSD corresponding to not considering its non-stationarity;

[0120] (2) For the PSD at each discrete moment, through the frequency response function of the vibration system, obtain the response PSDs at the corresponding discrete moments of the system response;

[0121] (3) Obtain the skewness and kurtosis modulation influence coefficients from the statistical moments of the modulation function, and combine with the skewness and kurtosis values of the non-stationary and non-Gaussian response to obtain the skewness and kurtosis values of the corresponding basic stationary non-Gaussian random process;

[0122] (4) For each discrete response PSD, combine with the skewness and kurtosis values of the corresponding basic stationary non-Gaussian random process of the response, and use the non-Gaussian NB method or the non-Gaussian DK method or the non-Gaussian TB method to calculate the fatigue damage within the corresponding discrete time period;

[0123] (5) Accumulate the fatigue damages calculated for each discrete time period to obtain the fatigue damage value within the overall time domain.

[0124] To verify the accuracy of the proposed evaluation strategy, a series of non-stationary non-Gaussian stress time series were generated, and the rain-flow counting method and the linear damage accumulation rule were used to calculate the fatigue damage in the time domain. The results were compared with those obtained by combining the EPSD discretization method and the stationary non-Gaussian spectrum method. During the analysis, different factors were considered and verified. The results show that the proposed modeling method for non-stationary non-Gaussian random processes can well characterize the coexistence of non-stationarity and non-Gaussianity, and the method of combining the EPSD discretization method and the stationary non-Gaussian spectrum method can produce very reliable results for fatigue damage evaluation under non-stationary non-Gaussian random loads. It is worth noting that the skewness and kurtosis modulation influence coefficients reflected by the uniform modulation function are key indicators of the influence of non-stationarity on fatigue damage. In addition, the combination of the non-Gaussian DK method and the non-Gaussian TB method with the EPSD discretization method is more suitable for non-stationary non-Gaussian fatigue damage evaluation and can provide robust damage results under various bandwidth conditions.

[0125] Example 1

[0126] Based on the amplitude modulation method and the uniform modulation function, a stationary non-Gaussian random process was modulated to obtain a non-stationary random process, which was defined as a true non-stationary non-Gaussian random process. The EPSD, skewness, and kurtosis values of the modulated non-stationary non-Gaussian random process were verified.

[0127] Given the power spectral density, a Gaussian random process is obtained by the following formula:

[0128] ;

[0129] where n is the number of frequency points, n is greater than 100, n the larger, the more accurately the Gaussian random process is described; represents the frequency interval; is the frequency, is the power spectral density, t is the time vector, is uniformly distributed in random phase angles.

[0130] Assume a zero-mean Gaussian random process with a flat power spectral density of 1 and a frequency range of [0 Hz, 20 Hz]. According to the above formula, 100 random samples were generated, and the Gaussian random samples were transformed into non-Gaussian random samples through the Johnson nonlinear transformation. The target skewness and kurtosis values of the transformed stationary non-Gaussian random process and the actual average skewness and kurtosis values of the samples are shown in Table 1. Through comparative analysis, the transformed stationary non-Gaussian random samples can well match the target skewness and kurtosis values.

[0131] Table 1

[0132] ;

[0133] Based on the stationary non-Gaussian random samples obtained by non-linear transformation, the modulation function shown in Figure 1 is selected to obtain non-stationary non-Gaussian samples. And based on the skewness and kurtosis modulation influence coefficients corresponding to the modulation function and the actual skewness and kurtosis values of the basic stationary non-Gaussian random samples, the theoretical average skewness and kurtosis of the non-stationary non-Gaussian random samples are obtained. The theoretical and actual average skewness and kurtosis values of the final non-stationary non-Gaussian random samples are shown in Table 2. It can be seen from Table 2 that the sample average values of each group can well match the theoretical average values, proving the accuracy of the calculation theory of the skewness and kurtosis adjustment coefficients.

[0134] Table 2

[0135] ;

[0136] In addition, since the PSD of the basic stationary Gaussian random process is the same as that of the stationary non-Gaussian random process obtained after non-linear transformation, the theoretical EPSD of the modulated non-stationary non-Gaussian random process is calculated based on the PSD of the basic stationary Gaussian random process and the modulation function. Then, non-stationary non-Gaussian random samples are selected to estimate their EPSD. The theoretical and estimated EPSD of the finally modulated non-stationary non-Gaussian random process are as shown in Figure 2 . The results show that the estimated values of EPSD have a high degree of matching with the theoretical values, proving the accuracy of the EPSD theory related to the modulation function.

[0137] Example 2

[0138] In order to verify the effectiveness of the non-stationary non-Gaussian fatigue damage assessment by combining the EPSD decomposition method with the non-Gaussian NB spectrum method, this application selects a narrowband spectrum and verifies it by comparing the results of the time-domain rainflow counting method (RFC) with the spectrum method.

[0139] For a single-degree-of-freedom (SDOF) vibration system, when the damping is not large, the system is equivalent to a narrowband filter. At this time, the PSD of the system response is a narrowband spectrum. The motion equation of the SDOF system excited by the base is:

[0140] ;

[0141] where is the acceleration excitation; m , c , k are the mass, damping and stiffness of the system respectively; They are relative acceleration, relative velocity, and relative displacement respectively.

[0142] The frequency response function between the relative displacement of the system and the base acceleration excitation is expressed as:

[0143] ;

[0144] where represents the imaginary unit.

[0145] The relationship between the relative displacement PSD of the system and the base acceleration excitation PSD can be described by the frequency response function:

[0146] ;

[0147] where is the power spectral density of the relative displacement response of the system, is the power spectral density of the base acceleration excitation.

[0148] Assume that the stress response X ( t ) is proportional to the relative displacement of the system Y ( t ):

[0149] ;

[0150] where is a proportional constant related to the material. Therefore:

[0151] ;

[0152] Let the acceleration excitation be a non-stationary non-Gaussian random process , and its average PSD is a flat spectrum as follows:

[0153] ;

[0154] The natural frequency of the system , mass , damping ratio . The proportional constant .

[0155] The average PSD of the system acceleration excitation and the average PSD of the stress response calculated from the FRF of the system are as Figure 3 shown.

[0156] In order to obtain non-stationary non-Gaussian random processes of different modulation types, 4 different modulation functions are adopted, which are as follows:

[0157] (1) The sine-type modulation function, i.e., the first-order modulation function, has the following expression:

[0158] ;

[0159] where, p 1 , p 2 are parameters.

[0160] (2) The step-type modulation function, i.e., the second-order modulation function, has the following expression:

[0161] ;

[0162] where, p 1 , p 2 are parameters.

[0163] (3) The Beta-distribution-type modulation function, i.e., the third-order modulation function, the modulation function is derived from the Beta distribution, and its probability density function is as follows:

[0164] ;

[0165] where, p 1 , p 2 are parameters, represents the gamma function.

[0166] (4) The stochastic-process-type modulation function, i.e., the fourth-order modulation function, has the following expression:

[0167] ;

[0168] where, p 1 , p 2 are parameters, x ( t ) is a stationary Gaussian stochastic process generated from a low-frequency flat power spectral density.

[0169] The above four types of modulation functions all contain two parameters p 1 , p 2, by continuously adjusting their respective parameters, initial modulation functions with different skewness modulation coefficients and kurtosis modulation coefficients can be obtained. To ensure that the modulation function does not change the average PSD of the modulated random process, the modulus of the modulation function is 1. Therefore, the obtained initial modulation function needs to be scaled proportionally according to its root mean square to obtain the final modulation function, and this scaling process does not change the corresponding skewness adjustment coefficient and kurtosis adjustment coefficient. For the convenience of comparison, in the analysis of this application, the skewness adjustment coefficients of the 4 modulation functions are kept the same (equal to 1.23), and the kurtosis adjustment coefficients are also the same (equal to 1.60).

[0170] In addition, for verification, this application assumes two combinations of the overall kurtosis and skewness values of the final non-stationary non-Gaussian stresses, which are respectively and . The corresponding skewness and kurtosis combination values of the stationary non-Gaussian stresses obtained by calculation are respectively , and , .

[0171] The S-N curve for fatigue analysis is selected as follows:

[0172] ;

[0173] According to the process of fatigue damage estimation by the spectral method, first, select Figure 3 The excitation average PSD shown and the above 4 modulation functions to construct the excitation EPSD, that is, represent the excitation EPSD in the combined form of a uniform modulation function and its average PSD; secondly, discretize the excitation EPSD in the time domain. The discretized excitation EPSD can be considered as a combination of the PSDs at each moment, and the PSD at each discrete moment is the product of the modulation function value at that moment and the excitation average PSD. For the PSD at each discrete moment, through the frequency response function of the vibration system, obtain the response PSD at each discrete moment of the corresponding system response; thirdly, obtain the skewness and kurtosis modulation influence coefficients from the statistical moments of the modulation function, and combine the skewness and kurtosis values of the non-stationary non-Gaussian response to obtain the skewness and kurtosis values of the corresponding basic stationary non-Gaussian random process; again, since this analysis is for a narrowband spectrum, for each discrete response PSD, combine the skewness and kurtosis values of the corresponding basic stationary non-Gaussian random process, and use the non-Gaussian NB (narrowband) method to calculate the fatigue damage within the corresponding discrete time period; finally, accumulate the fatigue damage calculated in each discrete time period to obtain the fatigue damage value within the overall time domain.

[0174] To verify the accuracy of the spectral method, that is, the accuracy of the fatigue damage assessment results of the EPSD decomposition method combined with the non-Gaussian NB method for non-stationary non-Gaussian narrow-band, the time-domain fatigue analysis method is selected to obtain the time-domain fatigue damage, and the calculation process is as follows Figure 4 shown. According to the steps shown, first, stationary Gaussian stress is generated based on the average PSD of the response. The above-mentioned 4 modulation functions are respectively selected to obtain non-stationary Gaussian stress. The rain-flow counting method and the linear damage accumulation rule are used to solve the time-domain non-stationary Gaussian fatigue damage for the non-stationary Gaussian stress. Secondly, the skewness and kurtosis modulation influence coefficients are obtained from the statistical moments of the modulation function, and combined with the skewness and kurtosis values of the non-stationary non-Gaussian response, the skewness and kurtosis values of the corresponding basic stationary non-Gaussian random process are obtained. Then, based on the skewness value and kurtosis value, and taking the generated stationary Gaussian stress as the basis, stationary non-Gaussian stress is obtained through non-linear transformation. The above-mentioned 4 modulation functions are also selected to obtain non-stationary non-Gaussian stress. The rain-flow counting method and the linear damage accumulation rule are used to solve the time-domain non-stationary Gaussian fatigue damage for the non-stationary non-Gaussian stress.

[0175] It should be noted that to ensure the reliability of the results, 50 samples are generated for each random stress, and the final fatigue damage is the average value of the fatigue damage of 50 samples as the final result, and the same applies hereinafter.

[0176] Finally, the time-domain method and the spectral method fatigue damage results under the narrow-band average PSD spectrum are as follows Figure 5 shown. , , , , , respectively represent , , , , , where is a stationary Gaussian process, is a non-stationary Gaussian process of the first modulation function, is a non-stationary Gaussian process of the second modulation function, is a non-stationary Gaussian process of the third modulation function, is a non-stationary Gaussian process of the fourth modulation function;

[0177] , , , , , respectively represent , , , , , where is a stationary non-Gaussian process, is a first-order non-stationary non-Gaussian process, is a second-order non-stationary non-Gaussian process, is a third-order non-stationary non-Gaussian process, is a fourth-order non-stationary non-Gaussian process.

[0178] It can be seen from the figure that the fatigue damages obtained by the time-domain method and the spectral method are very close, which proves the accuracy of the EPSD decomposition method combined with the non-Gaussian NB method for non-stationary non-Gaussian fatigue damage assessment in the narrowband case.

[0179] In addition, regardless of whether the basic process is Gaussian or non-Gaussian, the fatigue damage of the amplitude-modulated non-stationary random process increases, which proves the necessity of considering non-stationarity in fatigue damage assessment; and the non-stationary non-Gaussian fatigue damage is higher than that of non-stationary Gaussian, and the stationary non-Gaussian fatigue damage is also higher than that of stationary Gaussian, which proves the necessity of considering non-Gaussianity in fatigue damage assessment.

[0180] For the four different forms of modulation functions, the fatigue damage results are not much different. It proves that the selection of different forms of modulation functions does not affect the accuracy of this method.

[0181] Embodiment 3

[0182] In order to verify the effectiveness of the EPSD decomposition method combined with the non-Gaussian DK and TB spectral methods for non-stationary non-Gaussian fatigue damage assessment, this application selects a broadband spectrum and verifies it by comparing the results of the time-domain rainflow counting method (RFC) with the spectral method.

[0183] Since the frequency response function of the SDOF system in Embodiment 2 is equivalent to a narrowband filter, the stress response PSD obtained by the vibration analysis of the SDOF system is generally narrowband. In order to obtain a broadband spectrum, a multi-degree-of-freedom (MDOF) vibration system is selected for analysis. The vibration equation of the MDOF system is as follows:

[0184] ;

[0185] Among them, represents the base acceleration excitation; is the mass column vector; are the mass, damping, and stiffness matrices respectively; are the relative acceleration, relative velocity, and relative displacement vectors respectively.

[0186] The frequency response function (FRF) between the relative displacement response of the j th degree of freedom of the MDOF system and the base acceleration excitation can be expressed as:

[0187] ;

[0188] Among them, , represents the angular frequency.

[0189] In order to obtain different response PSDs, consider the following two cases:

[0190] (1) The number of degrees of freedom of the MDOF system is selected as 2, the masses are all 1 kg, and the damping coefficients 1 and 2 are c 1 = 1.2 and c 2 = 0.6, and the stiffness coefficients 1 and 2 are = 6000 N / m and = 1500 N / m. The modal damping ratios 1 and 2 obtained by calculation are = 0.012 and = 0.0178.

[0191] (2) The number of degrees of freedom of the MDOF system is selected as 10, all masses are set to 1 kg, all stiffness coefficients are set to 5000 N / m, and all modal damping ratios are set to 0.02.

[0192] The same acceleration excitation average PSD and the proportional constant between relative displacement and stress as in Example 2 are adopted , finally, the input and output average PSDs in the two cases are as Figure 6 shown. The Vanmarcke bandwidth coefficients in the two cases obtained by calculation are 0.471 and 0.478 respectively, which conform to the broadband spectrum characteristics (bandwidth coefficient less than 0.1).

[0193] In fatigue analysis, the same S-N curve, modulation function, and the skewness and kurtosis values of the final non-stationary non-Gaussian random process as in Example 2 are adopted. For fatigue analysis, the analysis procedure of Example 2 is also adopted. The only difference is that when using the spectral method for fatigue assessment, the stationary non-Gaussian NB method, DK method, TB method are combined with the EPSD decomposition method respectively, that is, two more stationary non-Gaussian spectral methods for broadband spectra are selected compared with Example 2. The final time-domain method and spectral method fatigue damage results are as Figure 7 and Figure 8 shown.

[0194] From Figure 7 and Figure 8It can be seen that for the spectral method results using the non-Gaussian DK method and the non-Gaussian TB method, they are very close to the fatigue damage results obtained by the time-domain method, which proves the accuracy of the EPSD decomposition method combined with the non-Gaussian DK method and the TB method for non-stationary non-Gaussian fatigue damage assessment in the broadband case. Moreover, the results of the DK method and the TB method are almost the same, indicating that the choice of the two spectral methods can achieve the same effect.

[0195] However, for the spectral method results using the non-Gaussian NB method, there is a large difference from the fatigue damage results obtained by the time-domain method, which proves that it is not feasible to combine the EPSD decomposition method with the non-Gaussian NB method for non-stationary non-Gaussian fatigue damage assessment in the broadband case.

[0196] In addition, Figure 7 and Figure 8 show the same pattern. For the four different forms of modulation functions, the fatigue damage results are not very different, which proves the robustness of the conclusions obtained for different broadband spectra and different forms of modulation functions.

[0197] To verify the accuracy of the EPSD decomposition method combined with the non-Gaussian spectral method for fatigue damage assessment under different bandwidth coefficients, the above-mentioned case (1) is selected for further analysis. By adjusting the value of the stiffness coefficient k2, different frequency response functions can be obtained, and then the response PSDs with different bandwidth parameters can be obtained. The finally obtained stress response PSD is as Figure 9 shown, and the bandwidth parameters corresponding to the selected stiffness coefficient and stress response PSD are shown in Table 3.

[0198] Table 3

[0199] ;

[0200] Since the fatigue damage levels corresponding to different spectra are different, therefore, based on the relative error between the fatigue damage calculated by the spectral method and the time-domain method, it is used to analyze the accuracy of the fatigue damage results obtained by different spectral methods under various bandwidth parameters. The definition of the relative error is as follows:

[0201] ;

[0202] where, represents the spectral method fatigue damage obtained by combining different spectral methods with the EPSD decomposition method, while represents the time-domain fatigue damage obtained by the RFC method.

[0203] The calculated results are as Figure 10 shown. From Figure 10It can be seen that the relative errors corresponding to the fatigue damages obtained by combining the non-Gaussian DK method and the TB method with the EPSD decomposition method remain within a very narrow range under all bandwidth conditions ( 0.1), as shown in the gray area of Figure 10 . The relative error corresponding to the fatigue damage obtained by combining the non-Gaussian NB method with the EPSD decomposition method decreases as the bandwidth parameter decreases. Only when the bandwidth parameter is in a small range, the relative error is at the low level indicated by the gray area. This result shows that the non-Gaussian NB method is only applicable to the assessment of non-stationary fatigue damage under narrowband conditions, while the non-Gaussian DK method and the non-Gaussian TB method are applicable to the combination with the EPSD decomposition method for non-stationary non-Gaussian fatigue damage assessment under all bandwidth conditions.

[0204] Finally, it should be noted that the above embodiments are only used to illustrate the technical method of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify or equivalently replace the technical method of the present invention, and these modifications or equivalent replacements cannot make the modified technical method deviate from the spirit and scope of the technical method of the present invention.

Claims

1. A fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads, characterized in that, It includes the following steps: S1. Based on the amplitude modulation method and the uniform modulation function, modulate the stationary non-Gaussian random process to obtain a non-stationary random process, which is defined as a true non-stationary non-Gaussian random process; S2. Based on the evolutionary power spectral density (EPSD) decomposition method, incorporate the spectral method for stationary non-Gaussian random vibration fatigue analysis to solve the fatigue damage of non-stationary non-Gaussian random stress; The specific content of modulating the stationary non-Gaussian random process to obtain a non-stationary non-Gaussian random process in S1 includes: Select a slow-varying function that is only related to time as the modulation function, i.e., the uniform modulation function. Based on the stationary non-Gaussian random process, obtain the non-stationary non-Gaussian random process through the amplitude modulation method. The non-stationary non-Gaussian random process obtained through modulation is characterized by the evolutionary power spectral density and the skewness and kurtosis values; Similarly, select the uniform modulation function, and define the random process obtained by modulating the stationary Gaussian random process as a non-stationary Gaussian random process, which is characterized by the evolutionary power spectral density; The non-stationary Gaussian random process and the non-stationary non-Gaussian random process together constitute the non-stationary random process.

2. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads according to claim 1, wherein The modulation function is expressed as a slowly varying function related to frequency and time, i.e., a non-uniform modulation function ; wherein, is the value of the slow-varying function, is the frequency, t is the time; When the modulation function does not change with frequency, it is called the uniform modulation function.

3. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads according to claim 2, wherein The modulation process of the non-stationary Gaussian random process includes: A non-stationary Gaussian random process with zero mean is expressed as: ； Among them, represents frequency, t represents time, represents the imaginary unit, is a non-uniform modulation function, is a zero-mean orthogonal increment random process, and the zero-mean stationary Gaussian random process corresponding to is expressed as: ； Evolutionary Power Spectrum of Non-Stationary Gaussian Random Process And the power spectrum of stationary Gaussian random process The relationship between them is expressed as: ； Through a basic stationary Gaussian random process Multiplied by a uniform modulation function To obtain a non-stationary Gaussian random process: ； At this time, the evolving power spectrum of the non-stationary Gaussian random process and the power spectrum of the stationary Gaussian random process are related as follows: 。 4. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads according to claim 2, wherein The modulation process of the non-stationary non-Gaussian random process includes: Replacing the basic stationary Gaussian random process in the modulation process of the non-stationary Gaussian random process with a stationary non-Gaussian random process, the modulation process of the non-stationary non-Gaussian random process can be obtained: ； wherein, Z ( t ) is a zero-mean stationary non-Gaussian random process, is a uniform modulation function, is a non-stationary non-Gaussian random process with a mean of 0; A stationary non-Gaussian random process Z ( t ) is transformed from a stationary Gaussian random process X ( t ) through a Johnson nonlinear transformation system: ； Among them, , are the standardized forms of a stationary Gaussian random process X ( t ) and a stationary non-Gaussian random process Z ( t ), respectively; , , , are the four parameters of the transformation system, obtained by the moment evaluation method. The selection of SU, SB, and SL is determined by the combined values of the skewness and kurtosis of the stationary non-Gaussian random process Z ( t ); The non-stationary and non-Gaussian random process obtained by modulation The n th central moment and the stationary non-Gaussian random process Z ( t ) of the n th central moment satisfy the following relationship: ； Among them, is a uniform modulation function of n the nth-order statistical moment, E denotes the expectation; The relationship between the skewness and kurtosis values of the non-stationary non-Gaussian random process obtained through modulation and the skewness and kurtosis values of the basic stationary non-Gaussian random process is as follows: ； Among them, represents skewness, represents kurtosis. Both skewness and kurtosis are characteristic values used to characterize the non-Gaussianity of a random process. For a Gaussian random process, its skewness is 0 and its kurtosis is 3; , respectively represent the skewness and kurtosis of the non-stationary non-Gaussian random process . , respectively represent the skewness and kurtosis of the stationary non-Gaussian random process Z ( t ). , , respectively represent the second, third, and fourth central moments of the stationary non-Gaussian random process Z ( t ). , , respectively represent the second, third, and fourth statistical moments of the modulation function . , are defined as the skewness modulation influence coefficient and the kurtosis modulation influence coefficient, both of which depend on the statistical moments of the modulation function and respectively characterize the influence degrees of the modulation signal on the skewness and kurtosis of the non-stationary non-Gaussian random process. is the third statistical moment of the non-stationary non-Gaussian random process, is the second statistical moment of the non-stationary non-Gaussian random process, is the fourth statistical moment of the non-stationary non-Gaussian random process.

5. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads according to claim 1, wherein, The content of solving the vibration fatigue damage of non-stationary non-Gaussian random stress by incorporating the spectral method under stationary non-Gaussian random load based on the EPSD decomposition method in S2 includes: Combining the characteristics that the non-stationary non-Gaussian random process is jointly characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, incorporate the spectral method for typical stationary non-Gaussian fatigue analysis to solve the non-stationary non-Gaussian random vibration fatigue damage; The spectral methods for stationary non-Gaussian random vibration fatigue analysis include the non-Gaussian NB method, the non-Gaussian DK method, and the non-Gaussian TB method.

6. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads according to claim 5, wherein The specific content of solving the non-stationary non-Gaussian random vibration fatigue damage by incorporating the spectral method for typical stationary non-Gaussian fatigue damage assessment based on the EPSD decomposition method, combining the characteristics that the non-stationary non-Gaussian random process is jointly characterized by EPSD and its skewness and kurtosis values, is as follows: Z ( t ) is a stationary Gaussian random process X ( t ) is a stationary non-Gaussian random process obtained through non-linear transformation, Z ( t ) and X ( t ) have the same power spectral density PSD, is Z ( t ) a non-stationary non-Gaussian random process obtained through uniform modulation; When the modulus of the modulation function is 1, the average power spectral density of Z ( t ) is consistent with that of, and the EPSD of is expressed as: ； Among them, is a non-stationary and non-Gaussian random process of the EPSD, and are respectively Z ( t ) and X ( t ) of the PSD; When the EPSD of the vibration system excitation and the skewness and kurtosis of the response signal are known, based on the EPSD decomposition, select the non-Gaussian TB method or the non-Gaussian DK method to calculate the non-stationary non-Gaussian fatigue damage; The process of calculating the non-stationary non-Gaussian fatigue damage is as follows: (1) Characterize the excitation EPSD as a combination of a uniformly modulated function and its average PSD; discretize the excitation EPSD in the time domain. The discretized excitation EPSD is considered as a combination of the PSDs at each moment, and the PSD at each discrete moment is the product of the modulation function value at that moment and the excitation average PSD. The excitation average PSD is the PSD corresponding to the case without considering its non-stationarity. (2) For the PSD at each discrete moment, obtain the response PSD at each discrete moment of the corresponding system response through the frequency response function of the vibration system. (3) Obtain the skewness and kurtosis modulation influence coefficients from the statistical moments of the modulation function, and combine the skewness and kurtosis values of the non-stationary non-Gaussian response to obtain the skewness and kurtosis values of the corresponding underlying stationary non-Gaussian random process. (4) For each discrete response PSD, combine the skewness and kurtosis values of the underlying stationary non-Gaussian random process corresponding to the response, and use the non-Gaussian NB method or non-Gaussian DK method or non-Gaussian TB method to calculate the fatigue damage within the corresponding discrete time period. (5) Accumulate the fatigue damage calculated for each discrete time period to obtain the fatigue damage value in the overall time domain.

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