Fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random load

Abstract

The invention discloses a fatigue damage assessment method under a uniform modulation non-stationary non-Gaussian random load, which relates to the field of random vibration fatigue damage assessment and comprises the following steps: acquiring a non-stationary random process based on a uniform modulation function; a non-stationary random process obtained by modulating the stationary non-Gaussian random process is defined as a real non-stationary non-Gaussian random process; on the basis of an EPSD decomposition method used for non-stationary Gaussian fatigue damage evaluation, a stationary non-Gaussian fatigue damage evaluation spectrum method is fused to carry out random vibration fatigue damage solving on non-stationary and non-Gaussian stress, the situation of non-stationary and non-Gaussian concurrently in random vibration fatigue damage analysis is further considered, and for the non-Gaussian property, the non-stationary and non-Gaussian random vibration fatigue damage evaluation method is used for solving the random vibration fatigue damage of the non-stationary and non-Gaussian random vibration fatigue damage. More reasonable definition division is carried out on a non-stationary random process, and the problem of random vibration fatigue damage assessment during more complex non-stationary and non-Gaussian concurrently is solved under the condition of a precision reliable spectrum.

Description

Technical Field

[0001] The 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 wind, waves, currents, and aviation equipment subjected to various impact loads. These excitations may lead to fatigue damage. However, the randomness of the loads brings great difficulties to the fatigue damage assessment and fatigue life prediction of these key components of the structure. 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 is usually done by determining fatigue life from stress responses measured in the field or from laboratory tests. However, field measurements and laboratory tests are often costly and time-consuming. In the context of random vibration fatigue, a variety of spectral methods based on power spectral density (PSD) have emerged to facilitate rapid estimation of fatigue damage. These spectral methods eliminate the need for stress response time histories and significantly reduce the time required for fatigue damage assessment.

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

[0005] Although the PSD-based spectral method is effective in many cases, PSD cannot characterize non-Gaussianity. Therefore, when faced with non-Gaussian loads, the stationary Gaussian spectrum method is no longer applicable. In the field of non-Gaussian random process processing and analysis, nonlinear transformation is a very important tool. The core idea of ​​nonlinear transformation is to establish the transformation relationship between non-Gaussian and Gaussian random processes through nonlinear functions. Therefore, the stationary non-Gaussian spectrum method that combines nonlinear transformation with the stationary Gaussian spectrum method is proposed. Typical stationary non-Gaussian spectrum methods include the non-Gaussian NB method for narrowband random processes and the non-Gaussian DK method and non-Gaussian TB method for broadband non-Gaussian random processes.

[0006] In order to solve the problem of non-stationary random vibration, the evolution 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 the non-stationary process into multiple equivalent stationary PSDs, so that non-stationary fatigue damage assessment can be performed using the stationary spectrum method.

[0007] However, actual non-stationary loads are complex, and in many cases the loads exhibit both non-stationary and non-Gaussian characteristics. Non-stationarity describes the nature of the statistical characteristics of a random process that change over time, while non-Gaussianity describes the characteristics of a random process that deviates from a Gaussian probability distribution. Simple non-Gaussian analysis is generally based on a stationary random process, 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 but not non-Gaussianity. Therefore, in the case of both non-stationarity and non-Gaussianity, the fatigue damage calculated by this method will deviate from the actual damage, thereby affecting the estimation of the fatigue life or performance of the engineering structure.

[0008] Therefore, in order to solve the problem of complex non-stationary non-Gaussian random vibration fatigue damage assessment, it is urgent to continue to establish a fatigue damage assessment method based on the evolutionary power spectral density. Summary of the invention

[0009] To solve the above problems, the present application proposes a fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads, comprising the following steps: S1. Based on the amplitude modulation method and uniform modulation function, the stationary non-Gaussian random process is modulated to obtain a non-stationary random process, and it is defined as a true non-stationary non-Gaussian random process; S2. Based on the evolutionary power spectral density decomposition method, the spectral method of stationary non-Gaussian random vibration fatigue analysis is integrated to solve the fatigue damage of non-stationary non-Gaussian random stress;

[0010] Preferably, the specific content of modulating the stationary non-Gaussian random process to obtain the non-stationary random process in S1 includes selecting a slow-varying function that is only related to time as a modulation function (i.e., a uniform modulation function), and obtaining the non-stationary non-Gaussian random process by an amplitude modulation method based on the stationary non-Gaussian random process, and the non-stationary non-Gaussian random process obtained by modulation is characterized by the evolution power spectrum density and the skewness and kurtosis values; Similarly, a uniform modulation function is selected, and the random process obtained by modulating the stationary Gaussian random process is defined as a non-stationary Gaussian random process, and the non-stationary Gaussian random process is characterized by the evolution power spectrum density; Non-stationary Gaussian random processes and non-stationary non-Gaussian random processes together constitute non-stationary random processes.

[0011] Preferably, the modulation function is represented as a slowly varying function related to frequency and time ; in, is the value of the slowly varying function, is the frequency, t For time; When the modulation function does not vary with frequency, it is called a uniform modulation function.

[0012] Preferably, the modulation process of the non-stationary Gaussian random process includes: A zero-mean non-stationary Gaussian random process is expressed as: ; in, Indicates frequency, t Indicates time, is a slowly varying non-uniform modulation function, is a zero-mean orthogonal increment random process, The corresponding zero-mean stationary Gaussian random process is expressed as: ; The relationship between the evolution power spectrum (EPSD) of a non-stationary Gaussian random process and the power spectrum of a stationary Gaussian random process is expressed as: ; The non-stationary Gaussian random process is obtained by multiplying the basic stationary Gaussian random process with the uniform modulation function: ; At this time, the relationship between the evolution power spectrum of the non-stationary Gaussian random process and the power spectrum of the stationary Gaussian random process is expressed as: .

[0013] Preferably, the modulation process of the non-stationary non-Gaussian random process includes: By 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: ; in, 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.

[0014] Stationary non-Gaussian random processZ ( t ) The system is transformed by the Johnson nonlinearity from a stationary Gaussian random process X ( t ) is converted to: ; in, , They are stationary Gaussian random processes X ( t ) and stationary non-Gaussian random processes Z ( t ) in its standardized form; , , , The four parameters of the conversion system are obtained by moment evaluation method, and the selection of SU, SB, and SL is based on a stationary non-Gaussian random process. Z ( t ) is determined by the combined value of skewness and kurtosis.

[0015] Modulated non-stationary non-Gaussian random process No. n Step center distance and stationary non-Gaussian random processes Z ( t ) n Step center distance Satisfy the following relationship: ; in, is a uniform modulation function of n The statistical moment of order, E Express expectations.

[0016] The relationship between the skewness and kurtosis of the modulated non-stationary non-Gaussian random process and the skewness and kurtosis of the basic stationary non-Gaussian random process is as follows: ; in, represents the skewness, Represents kurtosis. Skewness and kurtosis are both characteristic values ​​used to characterize the non-Gaussian nature of a random process. For a Gaussian random process, its skewness is 0 and its kurtosis is 3. , They represent non-stationary non-Gaussian random processes respectively. The skewness and kurtosis of , They represent stationary non-Gaussian random processes respectively. Z ( t )’s skewness and kurtosis, , , They represent stationary non-Gaussian random processes respectively. Z ( t )'s 2nd, 3rd, and 4th order central moments, , , Denote the modulation function The second, third, and fourth order statistical moments of , Defined as the skewness modulation influence coefficient and the kurtosis modulation influence coefficient, both of which depend on the modulation function The statistical moments of the modulation signal are used to characterize the influence of the modulation signal on the skewness and kurtosis of the non-stationary non-Gaussian random process. is the third-order statistical moment of the non-stationary non-Gaussian random process, is the second-order statistical moment of the non-stationary non-Gaussian random process, is the fourth-order statistical moment of a non-stationary non-Gaussian random process.

[0017] Preferably, in S2, based on the evolution power spectral density EPSD decomposition method, the spectral method under the stationary non-Gaussian random load is incorporated to solve the vibration fatigue damage of the non-stationary non-Gaussian random stress, including: Combining the characteristics of non-stationary non-Gaussian random processes characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, the spectral method of typical stationary non-Gaussian fatigue analysis is integrated 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.

[0018] Preferably, in combination with the characteristics of the non-stationary non-Gaussian random process being characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, the spectral method of typical stationary non-Gaussian fatigue damage assessment is integrated, and the specific content for solving the non-stationary non-Gaussian random vibration fatigue damage is: Z ( t ) is a stationary Gaussian random process X ( t ) is a stationary non-Gaussian random process obtained by nonlinear transformation, Z ( t )and X ( t ) has the same power spectral density (PSD), for Z ( t ) a non-stationary non-Gaussian random process obtained by uniform modulation; When the modulation function When the modulus of is 1, The average power spectral density of Z ( t ) remain consistent, and The EPSD is expressed as: ; in, is a non-stationary non-Gaussian random process EPSD, and They are Z ( t )and X ( t )’s PSD.

[0019] When the EPSD of the vibration system excitation and the skewness and kurtosis of the response signal are known, the non-stationary non-Gaussian fatigue damage is calculated based on the EPSD decomposition and the non-Gaussian TB method or the non-Gaussian DK method is used; The calculation process is: (1) The EPSD of the excitation is characterized as a combination of a uniform modulation function and its average PSD. The EPSD of the excitation is discretized in the time domain. The discrete EPSD of the excitation is considered to be a combination of the PSDs at each moment. The PSD at each discrete moment is the product of the modulation function value at that moment and the average PSD of the excitation. The average PSD of the excitation is the PSD corresponding to the non-stationarity. (2) For each discrete moment of PSD, the corresponding system response PSD at each discrete moment is obtained 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 basic stationary non-Gaussian random process; (4) For each discrete response PSD, the fatigue damage in the corresponding discrete time period is calculated using the non-Gaussian NB method, the non-Gaussian DK method, or the non-Gaussian TB method in combination with the skewness and kurtosis values ​​of the underlying stationary non-Gaussian random process corresponding to the response; (5) The fatigue damage calculated in each discrete time period is accumulated to obtain the fatigue damage value in the overall time domain.

[0020] To sum up, the present invention starts with modulating a non-stationary random process, 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 the non-stationary random process.

[0021] Combined with the amplitude modulation method that uses uniform modulation function to characterize non-stationarity, EPSD is used to characterize the time-frequency characteristics of non-stationary non-Gaussian random processes, and skewness and kurtosis indicators are used 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, the spectral method (non-Gaussian NB method, DK method and TB method) of typical stationary non-Gaussian fatigue damage assessment is integrated to solve non-stationary non-Gaussian random vibration fatigue damage.

[0022] Compared with the traditional method of non-stationary Gaussian fatigue damage assessment using EPSD decomposition, the present invention further considers the situation of both non-stationary and non-Gaussian in random vibration fatigue damage analysis, and defines and divides the non-stationary random process more reasonably for non-Gaussianity, and solves the more complex problem of non-stationary non-Gaussian random vibration fatigue damage assessment. In general, compared with the prior art, the present invention better solves the problem of random vibration fatigue damage assessment when both non-stationary and non-Gaussian are present.

[0023] The technical method of the present invention is further described in detail below through the drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 A modulation function used to obtain a non-stationary non-Gaussian random process in the first embodiment of the present invention; Figure 2 is the EPSD verification result of the non-stationary non-Gaussian random process obtained by modulation in the first embodiment of the present invention, Figure 2 (a) in the equation is the estimated value of EPSD. Figure 2 (b) in the figure is the theoretical value of EPSD; Figure 3 is the input and output average PSD spectrum selected in the second embodiment of the present invention, Figure 3 (a) is the average PSD spectrum of acceleration excitation. Figure 3 (b) is the stress corresponding average PSD spectrum; Figure 4 A time domain fatigue damage calculation flow chart for verifying the results of the EPSD-based non-stationary non-Gaussian fatigue damage assessment spectrum method of the present invention; Figure 5 The results of the EPSD-based non-stationary non-Gaussian fatigue damage assessment spectrum method and the time domain method in the narrowband spectrum of the present invention are compared. Figure 5 (a) in the figure is the non-stationary Gaussian case. Figure 5 (b) is the non-stationary non-Gaussian case ( , ), Figure 5 (c) Non-stationary non-Gaussian case ( , ); Figure 6 is the input and output average PSD spectrum used to verify the non-stationary non-Gaussian fatigue damage assessment spectrum method based on EPSD under broadband spectrum in Example 3 of the present invention, Figure 6 (a) is the average PSD spectrum of acceleration excitation. Figure 6 (b) is the stress corresponding average PSD spectrum 1, Figure 6 (c) in the figure is the stress corresponding average PSD spectrum 2; Figure 7 This is a comparison of the broadband non-stationary non-Gaussian fatigue damage spectrum method and the time domain method in Example 3 of the present invention. Figure 7 (a) in the figure is the non-stationary Gaussian case. Figure 7 (b) is the non-stationary non-Gaussian case ( , ), Figure 7 (c) Non-stationary non-Gaussian case ( , ); Figure 8 This is the second comparison 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 8 (a) in the figure is the non-stationary Gaussian case. Figure 8 (b) is the non-stationary non-Gaussian case ( , ), Figure 8 (c) Non-stationary non-Gaussian case ( , ); Fig. 9 The PSD spectra of different bandwidth parameters selected in the third embodiment of the present invention; Fig.10 The results of the EPSD-based non-stationary non-Gaussian fatigue damage assessment spectrum method and the time domain method under different bandwidth conditions are compared. Fig.10 (a) in the figure is the non-stationary Gaussian case. Fig.10 (b) is the non-stationary non-Gaussian case ( , ), Fig.10 (c) Non-stationary non-Gaussian case ( , ). DETAILED DESCRIPTION

[0025] The technical method of the present invention is further described below by means of the accompanying drawings and embodiments. It should be noted that unless otherwise specifically stated, the relative arrangement of the components and steps, numerical expressions and numerical values ​​described in these embodiments do not limit the scope of the present application.

[0026] The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the present application, its application, or uses.

[0027] Technologies, systems, and devices known to ordinary technicians in the relevant art may not be discussed in detail, but where appropriate, the technologies, systems, and devices should be considered part of the specification.

[0028] In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not limiting. Therefore, other examples of the exemplary embodiments may have different values.

[0029] Unless otherwise defined, technical or scientific terms used in the present invention shall have the common meanings understood by one having ordinary skills in the field to which the present invention belongs.

[0030] In order to solve the problem of fatigue damage assessment under non-stationary non-Gaussian random loads, this 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, this application uses EPSD as the analysis basis, 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 perform non-stationary non-Gaussian fatigue damage assessment.

[0031] This application proposes a fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random loads, comprising the following steps: S1. Based on the amplitude modulation method and uniform modulation function, the stationary non-Gaussian random process is modulated to obtain a non-stationary random process, and it is defined as a true non-stationary non-Gaussian random process; Preferably, the specific content of modulating the stationary non-Gaussian random process to obtain the non-stationary random process in S1 includes selecting a slow-varying function that is only related to time as a modulation function (i.e., a uniform modulation function), and obtaining the non-stationary non-Gaussian random process by an amplitude modulation method based on the stationary non-Gaussian random process, and the non-stationary non-Gaussian random process obtained by modulation is characterized by the evolution power spectrum density and the skewness and kurtosis values; Similarly, a uniform modulation function is selected, and the random process obtained by modulating the stationary Gaussian random process is defined as a non-stationary Gaussian random process, and the non-stationary Gaussian random process is characterized by the evolution power spectrum density; Non-stationary Gaussian random processes and non-stationary non-Gaussian random processes together constitute non-stationary random processes.

[0032] Preferably, the modulation function is represented as a slowly varying function related to frequency and time ;in, is the value of the slowly varying function, is the frequency, t For time; When the modulation function does not vary with frequency, it is called a uniform modulation function.

[0033] Preferably, the modulation process of the non-stationary Gaussian random process includes: A zero-mean non-stationary Gaussian random process is expressed as: ; in, Indicates frequency, t Indicates time, is a slowly varying non-uniform modulation function, is a zero-mean orthogonal increment random process, The corresponding zero-mean stationary Gaussian random process is expressed as: ; The relationship between the evolution power spectrum of a non-stationary Gaussian random process and the power spectrum of a stationary Gaussian random process is expressed as: ; The non-stationary Gaussian random process is obtained by multiplying the basic stationary Gaussian random process with the uniform modulation function: ; At this time, the relationship between the evolution power spectrum of the non-stationary Gaussian random process and the power spectrum of the stationary Gaussian random process is expressed as: .

[0034] Preferably, the modulation process of the non-stationary non-Gaussian random process includes: By 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: ; in, 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.

[0035] Stationary non-Gaussian random process Z ( t ) The system is transformed by the Johnson nonlinearity from a stationary Gaussian random process X ( t ) is converted to: ; in, , They are stationary Gaussian random processes X ( t ) and stationary non-Gaussian random processes Z ( t ) in its standardized form; , , , The four parameters of the conversion system are obtained by moment evaluation method, and the selection of SU, SB, and SL is based on a stationary non-Gaussian random process. Z ( t ) is determined by the combined value of skewness and kurtosis.

[0036] Modulated non-stationary non-Gaussian random process No. n Step center distance and stationary non-Gaussian random processes Z ( t ) n Step center distance Satisfy the following relationship: ; in, is a uniform modulation function of n The statistical moment of order, E Express expectations.

[0037] The relationship between the skewness and kurtosis of the modulated non-stationary non-Gaussian random process and the skewness and kurtosis of the basic stationary non-Gaussian random process is as follows: ; in, represents the skewness, Represents kurtosis. Skewness and kurtosis are both characteristic values ​​used to characterize the non-Gaussian nature of a random process. For a Gaussian random process, its skewness is 0 and its kurtosis is 3. , They represent non-stationary non-Gaussian random processes respectively. The skewness and kurtosis of , They represent stationary non-Gaussian random processes respectively. Z ( t )’s skewness and kurtosis, , , They represent stationary non-Gaussian random processes respectively. Z ( t )'s 2nd, 3rd, and 4th order central moments, , , Denote the modulation function The second, third, and fourth order statistical moments of , Defined as the skewness modulation influence coefficient and the kurtosis modulation influence coefficient, both of which depend on the modulation function The statistical moments of the modulation signal are used to characterize the influence of the modulation signal on the skewness and kurtosis of the non-stationary non-Gaussian random process. is the third-order statistical moment of the non-stationary non-Gaussian random process, is the second-order statistical moment of the non-stationary non-Gaussian random process, is the fourth-order statistical moment of a non-stationary non-Gaussian random process.

[0038] S2. Based on the evolutionary power spectral density decomposition method, the spectral method of stationary non-Gaussian random vibration fatigue analysis is integrated to solve the fatigue damage of non-stationary non-Gaussian random stress.

[0039] Preferably, in S2, based on the evolution power spectral density EPSD decomposition method, the spectral method under the stationary non-Gaussian random load is incorporated to solve the vibration fatigue damage of the non-stationary non-Gaussian random stress, including: Combining the characteristics of non-stationary non-Gaussian random processes characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, the spectral method of typical stationary non-Gaussian fatigue analysis is integrated 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.

[0040] Preferably, in combination with the characteristics of the non-stationary non-Gaussian random process being characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, the spectral method of typical stationary non-Gaussian fatigue damage assessment is integrated, and the specific content for solving the non-stationary non-Gaussian random vibration fatigue damage is: Z ( t ) is a stationary Gaussian random process X ( t ) is a stationary non-Gaussian random process obtained by nonlinear transformation, Z ( t )and X ( t ) has the same power spectral density (PSD), for Z ( t ) a non-stationary non-Gaussian random process obtained by uniform modulation; When the modulation function When the modulus of is 1, The average power spectral density of Z ( t ) remain consistent, and The EPSD is expressed as: ; in, is a non-stationary non-Gaussian random process EPSD, and They are Z ( t )and X ( t )’s PSD.

[0041] When the EPSD of the vibration system excitation and the skewness and kurtosis of the response are known, the non-stationary non-Gaussian fatigue damage is calculated based on the EPSD decomposition and the non-Gaussian TB method or non-Gaussian DK method is selected; The calculation process is: (1) The EPSD of the excitation is characterized as a combination of a uniform modulation function and its average PSD. The EPSD of the excitation is discretized in the time domain. The discrete EPSD of the excitation is considered to be a combination of the PSDs at each moment. The PSD at each discrete moment is the product of the modulation function value at that moment and the average PSD of the excitation. The average PSD of the excitation is the PSD corresponding to the non-stationarity. (2) For each discrete moment of PSD, the corresponding system response PSD at each discrete moment is obtained 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 basic stationary non-Gaussian random process; (4) For each discrete response PSD, the fatigue damage in the corresponding discrete time period is calculated using the non-Gaussian NB method, the non-Gaussian DK method, or the non-Gaussian TB method in combination with the skewness and kurtosis values ​​of the underlying stationary non-Gaussian random process corresponding to the response; (5) The fatigue damage calculated in each discrete time period is accumulated to obtain the fatigue damage value in the overall time domain.

[0042] In order to verify the accuracy of the proposed evaluation strategy, a series of non-stationary non-Gaussian stress time series were generated, and the time domain fatigue damage was calculated based on the rain flow counting method and the linear damage accumulation law, and the fatigue damage results obtained by combining the EPSD discretization method with the stationary non-Gaussian spectrum method were compared. In the process of analysis, different factors were considered and verified. The results show that the proposed modeling method of non-stationary non-Gaussian random process can well characterize the coexistence of non-stationarity and non-Gaussianity, and the combination of EPSD discretization method and stationary non-Gaussian spectrum method can produce very reliable results for fatigue damage assessment 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 impact of non-stationarity on fatigue damage. In addition, the combination of non-Gaussian DK method and non-Gaussian TB method with EPSD discretization method is more suitable for non-stationary non-Gaussian fatigue damage assessment, and can provide robust damage results under various bandwidth conditions.

[0043] Embodiment 1 Based on the amplitude modulation method and uniform modulation function, the stationary non-Gaussian random process is modulated to obtain a non-stationary random process, which is 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 are verified.

[0044] Given the power spectral density, the Gaussian random process is obtained by: ; in, n is the frequency point number, n Greater than 100, n The larger it is, the more accurate the description of the Gaussian random process is; Indicates the frequency interval; is the frequency, is the power spectral density, t is the time vector, To be evenly distributed Random phase angle.

[0045] Assume a zero-mean Gaussian random process, whose power spectral density is a flat spectrum, size is 1, and frequency range is [0Hz, 20Hz]. Generate 100 random samples according to the above formula, and convert the Gaussian random samples into non-Gaussian random samples through Johnson nonlinear transformation. The target skewness and kurtosis values ​​of the stationary non-Gaussian random process obtained by transformation and the actual average skewness and kurtosis values ​​of the samples are shown in Table 1. After comparative analysis, the stationary non-Gaussian random samples obtained by transformation can well match the target skewness and kurtosis values.

[0046] Table 1 ;

[0047] Based on the stationary non-Gaussian random samples obtained by nonlinear transformation, we select Figure 1 The modulation function shown in the figure is used to obtain non-stationary non-Gaussian samples. 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 final theoretical and actual average skewness and kurtosis values ​​of the non-stationary non-Gaussian random samples are shown in Table 2. It can be seen from Table 2 that the sample averages of each group can match the theoretical averages very well, proving the accuracy of the calculation theory of skewness and kurtosis adjustment coefficients.

[0048] Table 2 ;

[0049] In addition, since the PSD of the basic stationary Gaussian random process is the same as the PSD of the stationary non-Gaussian random process obtained after nonlinear 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, a non-stationary non-Gaussian random sample is selected to estimate its EPSD. Finally, the theoretical and estimated EPSD of the modulated non-stationary non-Gaussian random process are as follows: Figure 2 As shown, the results show that the estimated values ​​of EPSD have a high degree of match with the theoretical values, proving the accuracy of the EPSD theory related to the modulation function.

[0050] Embodiment 2 In order to verify the effectiveness of combining the EPSD decomposition method with the non-Gaussian NB spectrum method for non-stationary non-Gaussian fatigue damage assessment, this application selects a narrow-band spectrum and verifies it by comparing the results of the time-domain rain flow counting (RFC) method with the spectrum method.

[0051] 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 under fundamental excitation is: ; in, is 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.

[0052] Frequency response function between the relative displacement of the system and the basic acceleration excitation It is expressed as: ; in, Represents an imaginary unit.

[0053] The relationship between the relative displacement PSD of the system and the basic acceleration excitation PSD can be described by the frequency response function: ; in, is the power spectral density of the system relative displacement response, is the power spectral density of the basic acceleration excitation.

[0054] Assuming stress response X ( t ) and the relative displacement of the system Y ( t ) is proportional to: ; in, is a material-dependent proportional constant. Therefore: ; Let the acceleration excitation be a non-stationary non-Gaussian random process , its average PSD is a flat spectrum as follows: ; The natural frequency of the system ,quality , damping ratio . Proportional constant .

[0055] The average PSD of the system acceleration excitation and the average PSD of the stress response calculated by the system FRF are as follows: Figure 3 shown.

[0056] In order to obtain non-stationary non-Gaussian random processes of different modulation types, four different modulation functions are used, which are as follows: (1) The sinusoidal modulation function is the first-order modulation function, and its expression is as follows: ; in, p 1 , p 2 as a parameter.

[0057] (2) The step modulation function is the quadratic modulation function, and its expression is as follows: ; in, p 1 , p 2 as a parameter.

[0058] (3) Beta distribution modulation function is the cubic modulation function. It comes from Beta distribution, and its probability density function is as follows: ; in, p 1 , p 2 As parameters, represents the gamma function.

[0059] (4) The random process modulation function is the quartic modulation function, which is expressed as follows: ; in, p 1 , p 2 As parameters, x ( t ) is a stationary Gaussian random process generated by a low-frequency flat power spectral density.

[0060] The above four types of modulation functions all contain two parameters p 1 , p 2 , by continuously adjusting their respective parameters, the initial modulation function with different skewness modulation coefficients and kurtosis modulation coefficients can be obtained. In order to ensure that the modulation function does not change the average PSD of the random process after modulation, the modulus of the modulation function is 1, so it is necessary to scale the obtained initial modulation function in proportion to its root mean square to obtain the final modulation function, and the 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 four modulation functions are kept the same (equal to 1.23), and the kurtosis adjustment coefficient is also (equal to 1.60).

[0061] In addition, for verification purposes, this application assumes that the overall kurtosis and skewness value combinations of the two final non-stationary non-Gaussian stresses are as well as The calculated skewness and kurtosis combination values ​​of the corresponding stationary non-Gaussian stress are [ , ]as well as[ , ].

[0062] The SN curves used for fatigue analysis are selected as follows: ; The process of fatigue damage estimation based on the spectral method is to first select Figure 3The excitation average PSD shown in the figure and the above four modulation functions construct the excitation EPSD, that is, the excitation EPSD is characterized as a combination of a uniform modulation function and its average PSD; secondly, the excitation EPSD is discretized in the time domain. The discrete excitation EPSD can be considered as a combination of 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 each discrete moment of PSD, the corresponding system response PSD at each discrete moment is obtained through the frequency response function of the vibration system; thirdly, by The statistical moment of the modulation function obtains the skewness and kurtosis modulation influence coefficient, and combines 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 aimed at the narrowband spectrum, for each discrete response PSD, combined with the skewness and kurtosis values ​​of the basic stationary non-Gaussian random process corresponding to the response, the non-Gaussian NB (narrowband) method is used to calculate the fatigue damage in the corresponding discrete time period; finally, the fatigue damage calculated in each discrete time period is accumulated to obtain the fatigue damage value in the overall time domain.

[0063] In order to verify the accuracy of the spectral method, that is, the accuracy of the non-stationary non-Gaussian narrow-band fatigue damage assessment results combined with the EPSD decomposition method and the non-Gaussian NB method, the time domain fatigue analysis method is selected to obtain the time domain fatigue damage. The calculation process is as follows: Figure 4 As shown. According to the steps shown, first, a stationary Gaussian stress is generated based on the average PSD of the response, and the above four modulation functions are selected to obtain non-stationary Gaussian stress. The rain flow counting method and the linear damage accumulation law are used to solve the non-stationary Gaussian fatigue damage in the time domain for non-stationary Gaussian stress. Secondly, the skewness and kurtosis modulation influence coefficients are obtained from the statistical moment of the modulation function, and the skewness and kurtosis values ​​of the corresponding basic stationary non-Gaussian random process are obtained by combining the skewness and kurtosis values ​​of the non-stationary non-Gaussian response. Then, based on the skewness value and kurtosis value, and based on the generated stationary Gaussian stress, the stationary non-Gaussian stress is obtained through nonlinear transformation. Similarly, the above four modulation functions are selected to obtain non-stationary non-Gaussian stress. The rain flow counting method and the linear damage accumulation law are used to solve the non-stationary Gaussian fatigue damage in the time domain for non-stationary non-Gaussian stress.

[0064] It should be noted that in order to ensure the reliability of the results, 50 samples are generated each time the random stress is applied, and the final fatigue damage takes the average fatigue damage of the 50 samples as the final result, the same below.

[0065] Finally, the fatigue damage results of the time domain method and the spectral method under the narrow-band average PSD spectrum are as follows Figure 5 shown. , , , , , representing , , , , ,in, is a stationary Gaussian process, is a non-stationary Gaussian process with a modulation function, is a quadratic modulation function non-stationary Gaussian process, is a cubic modulation function non-stationary Gaussian process, It is a non-stationary Gaussian process with a quartic modulation function; , , , , , representing , , , , ,in, is a stationary non-Gaussian process, is a non-stationary non-Gaussian process, is a quadratic non-stationary non-Gaussian process, is a cubic non-stationary non-Gaussian process, It is a quartic non-stationary non-Gaussian process.

[0066] It can be seen from the figure that the fatigue damage 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 narrow band case.

[0067] In addition, regardless of whether the underlying process is Gaussian or non-Gaussian, the fatigue damage of the non-stationary random process after amplitude modulation increases, which proves the necessity of considering non-stationarity in fatigue damage assessment; and the fatigue damage of non-stationary non-Gaussian is higher than that of non-stationary Gaussian, and the fatigue damage of stationary non-Gaussian is also higher than that of stationary Gaussian, which proves the necessity of considering non-Gaussian in fatigue damage assessment.

[0068] For four different forms of modulation functions, the fatigue damage results are not much different. It is proved that the selection of different forms of modulation functions does not affect the accuracy of the method.

[0069] Embodiment 3 In order to verify the effectiveness of the EPSD decomposition method combined with the non-Gaussian DK and TB spectrum 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 rain flow counting (RFC) method with the spectrum method.

[0070] Since the frequency response function of the SDOF system in the second embodiment 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: ; in, represents the basic acceleration excitation; is the mass column vector; are the mass, damping and stiffness matrices respectively; They are relative acceleration, relative velocity, and relative displacement vector respectively.

[0071] The MDOF system j The frequency response function (FRF) between the relative displacement response of the order degree of freedom and the base acceleration excitation can be expressed as: ; in, , represents the angular frequency.

[0072] To get different response PSDs, consider the following two cases: (1) The number of degrees of freedom of the MDOF system is selected to be 2, the mass is 1 kg, and the damping coefficients 1 and 2 are c 1 = 1.2 and c 2 = 0.6, stiffness coefficients 1 and 2 are =6000 N / m and = 1500 N / m. The calculated modal damping ratios 1 and 2 are = 0.012 and = 0.0178.

[0073] (2) The number of degrees of freedom of the MDOF system is selected to be 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.

[0074] The same acceleration excitation average PSD and proportional constant between relative displacement and stress as in Example 2 are used. Finally, the average PSD of input and output in the two cases is as follows: Figure 6 The calculated Vanmarcke bandwidth coefficients in the two cases are 0.471 and 0.478, respectively, which are consistent with the broadband spectrum characteristics (the bandwidth coefficient is less than 0.1).

[0075] In fatigue analysis, the same SN curve, modulation function, and the final skewness and kurtosis values ​​of the non-stationary non-Gaussian random process as those in Example 2 are used. As for fatigue analysis, the analysis procedure of Example 2 is also used. The slight difference is that when the spectrum method is used for fatigue assessment, the stationary non-Gaussian NB method, DK method, TB method and EPSD decomposition method are combined, that is, compared with Example 2, two more stationary non-Gaussian spectrum methods for broadband spectra are selected. The final fatigue damage results of the time domain method and the spectrum method are shown in Figure 2. Figure 7 and Figure 8 shown.

[0076] from Figure 7 and Figure 8 It can be seen that the spectral method results using the non-Gaussian DK method and the non-Gaussian TB method are very close to the fatigue damage results obtained by the time domain method, which proves the accuracy of the EPSD decomposition method, the non-Gaussian DK method and the TB method combined for non-stationary non-Gaussian fatigue damage assessment in the broadband case, and 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.

[0077] However, the fatigue damage results obtained by the spectral method using the non-Gaussian NB method are quite different from those 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 broadband conditions.

[0078] also, Figure 7 and Figure 8 The performance rules are consistent, and the fatigue damage results for the four different forms of modulation functions are not much different, which proves the robustness of the conclusions drawn for different broadband spectra and different forms of modulation functions.

[0079] In order to verify the accuracy of fatigue damage assessment by combining EPSD decomposition method with non-Gaussian spectrum method under different bandwidth coefficients, the above case (1) is selected for further analysis. By adjusting the value of stiffness coefficient k2, different frequency response functions can be obtained, and then the response PSD of different bandwidth parameters can be obtained. The final stress response PSD is as follows: Fig. 9 As shown, the bandwidth parameters corresponding to the selected stiffness coefficient and stress response PSD are shown in Table 3.

[0080] Table 3 ;

[0081] Since different spectra correspond to different fatigue damage levels, the relative error between the fatigue damage calculated by the spectrum method and the time domain method is used as a basis to analyze the accuracy of fatigue damage results obtained by different spectrum methods under various bandwidth parameters. The definition of relative error is as follows: ; in, represents the spectral fatigue damage obtained by combining different spectral methods with the EPSD decomposition method, and It represents the time domain fatigue damage obtained by the RFC method.

[0082] The calculated results are as follows Fig.10 As shown. Fig.10 It can be seen that the relative errors corresponding to fatigue damage 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), such as Fig.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 a lower level represented by the gray area. This result shows that the non-Gaussian NB method is only suitable for non-stationary fatigue damage assessment under narrowband conditions, while the non-Gaussian DK method and the non-Gaussian TB method are suitable for non-stationary non-Gaussian fatigue damage assessment in combination with the EPSD decomposition method under all bandwidth conditions.

[0083] Finally, it should be noted that the above embodiments are only used to illustrate the technical method of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical method of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical method to 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 load, characterized in that: The following steps are involved: S1. Based on the amplitude modulation method and uniform modulation function, the stationary non-Gaussian random process is modulated to obtain a non-stationary random process, and it is defined as a true non-stationary non-Gaussian random process; S2. Based on the evolutionary power spectral density EPSD decomposition method, the spectral method of stationary non-Gaussian random vibration fatigue analysis is integrated to solve the fatigue damage of non-stationary non-Gaussian random stress; The specific contents of modulating the stationary non-Gaussian random process to obtain the non-stationary non-Gaussian random process in S1 include: A slow-varying function that is only related to time is selected as the modulation function, i.e., the uniform modulation function. Based on the stationary non-Gaussian random process, the amplitude modulation method is used to obtain the non-stationary non-Gaussian random process. The non-stationary non-Gaussian random process obtained by modulation is characterized by the evolution power spectrum density and the skewness and kurtosis values. Similarly, a uniform modulation function is selected, and the random process obtained by modulating the stationary Gaussian random process is defined as a non-stationary Gaussian random process, and the non-stationary Gaussian random process is characterized by the evolution power spectrum density; Non-stationary Gaussian random processes and non-stationary non-Gaussian random processes together constitute non-stationary random processes.

2. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random load according to claim 1 is characterized in that: The modulation function is expressed as a slowly varying function related to frequency and time, that is, a non-uniform modulation function. ; in, is the value of the slowly varying function, is the frequency, t For time; When the modulation function does not vary with frequency, it is called a uniform modulation function.

3. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random load according to claim 2 is characterized in that: The modulation process of the non-stationary Gaussian random process includes: A non-stationary Gaussian random process with zero mean It is expressed as: ； in, Indicates frequency, t Indicates time, represents the imaginary unit, is a non-uniform modulation function, is a zero-mean orthogonal increment random process, The corresponding zero-mean stationary Gaussian random process is expressed as: ； Evolutionary Power Spectrum of a Nonstationary Gaussian Random Process and the power spectrum of a stationary Gaussian random process The relationship between them is expressed as: ； Through the basic stationary Gaussian random process With uniform modulation function Multiplying them together gives a non-stationary Gaussian random process: ； At this time, the evolution power spectrum of the non-stationary Gaussian random process is and the power spectrum of a stationary Gaussian random process The relationship between them is expressed as: 。 4. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random load according to claim 2 is characterized in that: The modulation process of non-stationary non-Gaussian random process includes: By 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: ； in, Z ( t ) is a zero-mean stationary non-Gaussian random process, is a uniform modulation function, It is a non-stationary non-Gaussian random process with a mean of 0; Stationary non-Gaussian random process Z ( t ) The system is transformed by the Johnson nonlinearity from a stationary Gaussian random process X ( t ) is converted to: ； in, , They are stationary Gaussian random processes X ( t ) and stationary non-Gaussian random processes Z ( t ) in its standardized form; , , , The four parameters of the transformation system are obtained by moment evaluation method. The selection of SU, SB, and SL is based on a stationary non-Gaussian random process. Z ( t ) is determined by the combined value of skewness and kurtosis; Modulated non-stationary non-Gaussian random process No. n Step center distance and stationary non-Gaussian random processes Z ( t ) n Step center distance Satisfy the following relationship: ； in, is a uniform modulation function of n The statistical moment of order, E express expectations; The relationship between the skewness and kurtosis of the modulated non-stationary non-Gaussian random process and the skewness and kurtosis of the basic stationary non-Gaussian random process is as follows: ； in, represents skewness, Represents kurtosis. Skewness and kurtosis are both characteristic values ​​used to characterize the non-Gaussian nature of a random process. For a Gaussian random process, its skewness is 0 and its kurtosis is 3. , They represent non-stationary non-Gaussian random processes respectively. The skewness and kurtosis of , They represent stationary non-Gaussian random processes respectively. Z ( t )’s skewness and kurtosis, , , They represent stationary non-Gaussian random processes respectively. Z ( t )'s 2nd, 3rd, and 4th order central moments, , , Denote the modulation function The second, third, and fourth order statistical moments of , Defined as the skewness modulation influence coefficient and the kurtosis modulation influence coefficient, both of which depend on the modulation function The statistical moments of the modulation signal are used to characterize the influence of the modulation signal on the skewness and kurtosis of the non-stationary non-Gaussian random process. is the third-order statistical moment of the non-stationary non-Gaussian random process, is the second-order statistical moment of the non-stationary non-Gaussian random process, is the fourth-order statistical moment of a non-stationary non-Gaussian random process.

5. The fatigue damage assessment method under uniformly modulated non-stationary non-Gaussian random load according to claim 1 is characterized in that: In S2, based on the evolutionary power spectral density EPSD decomposition method, the spectral method under stationary non-Gaussian random loads is incorporated to solve the vibration fatigue damage of non-stationary non-Gaussian random stresses. The contents include: Combining the characteristics of non-stationary non-Gaussian random processes characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, the spectral method of typical stationary non-Gaussian fatigue analysis is integrated 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 load according to claim 5 is characterized in that: Combining the characteristics of non-stationary non-Gaussian random processes characterized by EPSD and its skewness and kurtosis values, based on the EPSD decomposition method, the spectral method of typical stationary non-Gaussian fatigue damage assessment is integrated to solve the specific content of non-stationary non-Gaussian random vibration fatigue damage: Z ( t ) is a stationary Gaussian random process X ( t ) is a stationary non-Gaussian random process obtained by nonlinear transformation, Z ( t )and X ( t ) has the same power spectral density PSD, for Z ( t ) a non-stationary non-Gaussian random process obtained by uniform modulation; When the modulation function When the modulus of is 1, The average power spectral density of Z ( t ) remain consistent, and The EPSD is expressed as: ； in, is a non-stationary non-Gaussian random process EPSD, and They are Z ( t )and X ( t )’s PSD; When the EPSD of the vibration system excitation and the skewness and kurtosis of the response signal are known, the non-stationary non-Gaussian fatigue damage is calculated based on the EPSD decomposition and the non-Gaussian TB method or the non-Gaussian DK method is used; The process of calculating non-stationary non-Gaussian fatigue damage is as follows: (1) The EPSD of the excitation is characterized as a combination of a uniform modulation function and its average PSD. The EPSD of the excitation is discretized in the time domain. The discrete EPSD of the excitation is considered to be a combination of the PSDs at each moment. The PSD at each discrete moment is the product of the modulation function value at that moment and the average PSD of the excitation. The average PSD of the excitation is the PSD corresponding to the non-stationarity. (2) For each discrete moment of PSD, the corresponding system response PSD at each discrete moment is obtained 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 basic stationary non-Gaussian random process; (4) For each discrete response PSD, the fatigue damage in the corresponding discrete time period is calculated using the non-Gaussian NB method, the non-Gaussian DK method, or the non-Gaussian TB method in combination with the skewness and kurtosis values ​​of the underlying stationary non-Gaussian random process corresponding to the response; (5) The fatigue damage calculated in each discrete time period is accumulated to obtain the fatigue damage value in the overall time domain.

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