Vibration Fatigue Damage Estimation by New Stress Correction Based on Kurtosis Control of Random Excitation Loadings

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Article Vibration Fatigue Damage Estimation by New Stress Correction Based on Kurtosis Control of Random Excitation Loadings

Yuzhu Wang 1,2 and Roger Serra 1,*

1 INSA Centre Val de Loire, Laboratoire de Mécanique G. Lamé E.A.7494, Campus de Blois, Equipe DivS, 3 rue de la Chocalaterie, 41000 Blois, France; [email protected] 2 China Academy of Railway Sciences, Locomotive Car Research Institute, Beijing 100081, China * Correspondence: [email protected]

Abstract: In the pioneer CAE stage, life assessment is the essential part to make the product meet the life requirement. Commonly, the lives of ﬂexible structures are determined by vibration fatigue which accrues at or close to their natural frequencies. However, existing PSD vibration fatigue damage estimation methods have two prerequisites for use: the behavior of the mechanical system must be linear and the probability density function of the response stresses must follow a Gaussian distribution. Under operating conditions, non-Gaussian signals are often recorded as excitation (usually observed through kurtosis), which will result in non-Gaussian response stresses. A new correction is needed to make the PSD approach available for the non-Gaussian vibration to deal with the inevitable extreme value of high kurtosis. This work aims to solve the vibration fatigue estimation under the non-Gaussian vibration; the key is the probability density function of response

stress. This work researches the importance of non-Gaussianity numerically and experimentally. The beam specimens with two notches were used in this research. All excitation stays in the frequency

Citation: Wang, Y.; Serra, R. range that only affects the second natural frequency, although their kurtosis is different. The results Vibration Fatigue Damage Estimation show that the probability density function of response stress under different kurtoses can be obtained by New Stress Correction Based on by kurtosis correction based on the PSD approach of the frequency domain. Kurtosis Control of Random Excitation Loadings. Sensors 2021, 21, Keywords: vibration fatigue test; non-gaussian excitation; kurtosis control 4518. https://doi.org/10.3390/ s21134518

Academic Editors: Len Gelman and 1. Introduction Hong-Nan Li The random vibration load closed to the mechanical components’ natural frequencies will signiﬁcantly affect their fatigue life. It is called vibration fatigue, which has become Received: 23 March 2021 a hot subject in recent years. For the fatigue-life estimation of random vibration, two Accepted: 24 May 2021 Published: 1 July 2021 methods are usually used: time domain and frequency domain [1]. For the time-domain approach, the response-stress distribution and its number of

Publisher’s Note: MDPI stays neutral cycles are estimated through cycle counting. The cumulative fatigue damage is determined with regard to jurisdictional claims in by Miner’s law with the Wohler’s curve. It can be understood that at one stress level, published maps and institutional afﬁl- the number of cycles used is the damage. The total number of cycles that can be used iations. is the fatigue life. The computational power it needs is proportional to the amount of data. The frequency-domain approach described random vibration by power spectral density (PSD), which describes the power distribution along with the frequency content of the process. The second-order spectrum analysis in the frequency domain can better study the mechanical system excited by stochastic stationary Gaussian vibration. In the Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. aspect of signal extrapolation, the frequency-domain approach has advantages, because the This article is an open access article frequency distribution is not affected by signal duration. However, the dynamic behavior distributed under the terms and of the mechanical system is linear in theory, so the stress process also has a Gaussian conditions of the Creative Commons distribution. In this manner, the response–stress distribution can be performed using the Attribution (CC BY) license (https:// stress PSD, which is the output of the dynamic analysis, through one of the frequency creativecommons.org/licenses/by/ methods present in the literature [2–6]. As well-known, the evaluation of an expected value 4.0/). of any random parameter derived from PSD is equivalent to estimating the same parameter

Sensors 2021, 21, 4518. https://doi.org/10.3390/s21134518 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 4518 2 of 22

in the unlimited time history in the random-stress process. Thus, it can be considered that a fatigue-damage estimation performed in the frequency domain significantly reduces the computational times. For this reason, an extension of the preliminary hypotheses of the frequency-domain approach may be well received. In a real case, however, the typical loading is non-Gaussian, and this can cause the response to be non-Gaussian, which may lead to shorter fatigue life. This means that sometimes the frequency-domain approach for fatigue estimation directly performed by stress PSD cannot be applied. For this reason, non-Gaussian vibration fatigue has been the focus. Typical non-Gaussian loads include the excitation of rail irregularity and pressure fluctuations for an aircraft. It is proposed in military environmental standards that the non-Gaussian loads case under extreme conditions need to be considered [7,8]. There are several studies that were published in recent years to show how non-Gaussian excitation impacts vibration fatigue life. In literature, Kihl et al. first proposed that the fatigue damage caused by the stress-time history can be determined by the Gaussian damage Dg, which is obtained from the methods mentioned above and the corrective coefficient of non-Gaussianity λng [9]: Dng = λngDg (1) Winterstein obtained a polynomial to generate non-Gaussian time histories. Then he obtained λng through the Wohler’s curve of the component and parameters from the polynomial [10]. Benasciutti made it so that the non-Gaussian loading was modeled as a transformed Gaussian process and defined λng with kurtosis, skewness and material parameter, also the parameters which are used to generate non-Gaussian excitation [11,12]; Rizzi et al. and Kihm et al. describe how non-Gaussian excitation affects the fatigue life of linear and nonlinear structures [13–15]. Braccesi et al. propose a corrective coefficient to the narrow- band formula in their research. He points the most significant field of the research consists in the evaluation of non-normality indices without stress-time history reconstruction [16]. Nieslony verifies the case of a non-zero mean stress, non-Gaussian loading histories with the use of the spectral method with the corrective factor and obtains successful results. The only difference in λng is the power of the material parameter [17]. Palmieri et al. point out that fatigue life due to burst and non-stationary excitations is significantly shorter [18]. It is a real challenge to obtain the kurtosis of the response stress at the critical position only through the frequency-domain information without constructing the time-domain signal, which makes the use of this method problematic. It is necessary to try to achieve this goal through the kurtosis of the excitation signal rather than the kurtosis of response stress. Although it is pointed out in the above literature that λng is only related to the property of the material and the kurtosis and the skewness of excitation, no other material is used to verify it. The author considers that a single corrective coefficient cannot reflect the influence of non-Gaussian methods on fatigue-damage estimation. At high kurtosis, the extreme value will aggravate the fatigue damage process and further shorten the fatigue life. According to previous studies, kurtosis also has an effect on other links of fatigue- damage estimation. For a more accurate response to fatigue damage, the breakthrough point of the study was chosen to be the cycle counting of response stress. In the present study, a series of finite duration data with the same PSD is designed to demonstrate the effect of non-Gaussianity (kurtosis) on the response-stress distribution. The PSD shape and frequency range never change in this study. The assumption is maintained that the dynamic behavior of the mechanical system is linear and the excitation is stationary. The aim of this study is to experimentally research the improved PSD method in the frequency domain to infer the distribution of response stress under excitation with different kurtoses. The advantage of this method is that it can directly reflect the range of stress changes. Ultimately, more accurate damage can be estimated. Sensors 2021, 21, 4518 3 of 22

This manuscript is organized as follows: in Section2, the PSD approach in the frequency for the response of the dynamic structure under vibration is presented. In Sections3 and4, the corrected PSD method is proposed for non-Gaussian vibration based on kurtosis control and demonstrated. In Section5, this method is veriﬁed under different PSD level signals. In this paper, stress-distribution correction rather than damage correction is used to solve the fatigue-damage estimation for non-Gaussian random vibration. It was veriﬁed that the method proposed in this study can solve the fatigue-damage estimation at different kurtosis levels with higher accuracy than previous methods.

2. PSD Approach in the Frequency Domain Now, when calculating the general method of fatigue damage, according to Wohler’s curve of the material, calculate the damage under the corresponding stress cycle and add it to the total damage suffered by the structure. This result is based on the established Miner’s law [19]. Therefore, obtaining the damage under the corresponding stress cycle is called the key to calculate the fatigue damage. The more commonly used methods are the rainflow counting method based on the time-domain signal (time-domain method) and the PSD method based on the frequency-domain information (frequency-domain method). The power spectral density (PSD) is a description of the power contained in the frequency components of the random vibration signal and is a statistical characteristic in the frequency domain of random vibration [2,20]. In random vibration, due to the time history of vibration being significantly non-periodic, the power spectral density must be used to calculate. The random vibration signal can be regarded as composed of an infinite number of simple harmonic motions, so the power spectrum of the random vibration signal is the sum of the harmonic vibration power in a given frequency range. For a time-series x(t) of a classic random signal with zero mean, using PSD S(ω) to represent its frequency-domain characteristic, Rx(τ) is the autocorrelation function of x(t):

Z ∞ 1 −iωτ Sx(ω) = Rx(τ)e dτ (2) 2π −∞

The power spectral density at the speciﬁed frequency is the average of the mean square values of the signal. For linear structures, the vibration of a multi-degree of a freedom system can be described as [21]: .. . Mx + Cx + Kx = f (3) where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, x is the vector of degrees of freedom and f is the excitation vector. By constructing the modal matrix of the system and using the modal superposition method, the displacement mode shape can be obtained. Then, the frequency-response equation of structural stress is obtained by the relationship between displacement and stress [22–24]:

n ϕσ ϕT Hσ(ω) = i i (4) ∑ + 2 + i=1 ki ω mi jωci

σ where H (ω) denotes the stress-frequency-response function and ki, mi, ci are the i-th modal σ stiffness, modal mass and modal damping, respectively. ϕi is the corresponding modal strain component of the system. In this project, the stress PSD function Sb( f ) at the structural hazard point is determined by the following function:

2 Sb( f ) = Aa( f )Hba( f ) (5)

Aa( f ) is excitation applied to point a and Hba( f ) is the frequency-response function at point b, and the structure excitation is at point a. The N-order moment of inertia is deﬁned as Sensors 2021, 21, 4518 4 of 22

∞ n mn = ∑ fi Sb( fi)∆ f (6) i=1

The root mean square (RMS) values of the stress response σRMS and the input excitation obtained by the 0-order moment of inertia are v √ u+∞ = = u ( ) 2 σRMS m0 t∑ Aa fi Hba( f )∆ f (7) i=1

The frequency domain usually uses the moment of inertia, the approximate estimate, the zero-value positive pass expectation E(0) and the peak expectation E(p): p E(0) = m2/m0 (8) p E(p) = m4/m2 (9) In the frequency domain, the random process is usually described using the spectral irregularity factor α and the spectral width factor ε:

α = E(0)/E(p) (10) p ε = 1 − α2, ε ∈ (0, 1) (11) Dirlik used Monte Carlo to simulate the power-spectral density function in 70 different shapes and proposed an exponential distribution and two Rayleigh distributions to describe the probability density function of approximate rain ﬂow cyclic amplitude. The Dirlik method has high accuracy [4,25,26]. The rainﬂow cycle amplitude probability density estimate is given by:

1 G Z G Z Z2 Z2 ( )DK = √ 1 − + 2 − + − p S exp 2 exp 2 G3Zexp (12) 2 m0 Q Q R 2R 2

where Z is the normalized amplitude and Xm is the mean frequency, as fellow

1 S m1 m2 2 Z = √ and xm = 2 m0 m0 m4

and other parameters are deﬁned as:

2 2 2 xm − α 1 − α − G + G G = 2 , G = 2 1 1 , G = 1 − G − G 1 2 2 + 3 1 2 1 + α2 1 R

α − x − G2 1.25(α − G − G R) = 2 m 1 = 2 3 2 R 2 and Q 1 − α2 − G1 + G1 G1 For Gaussian signals, the stress-probability density equation obtained from the PSD and rainﬂow count results obtained under the time-domain signal should be consistent. From Wohler’s curve of the material, structural damage can be determined. The classical method is the linear damage accumulation principle proposed by Miner’s Law, which is:

n(S) = E(0)p(Si)∆Si (13)

The whole life under the corresponding stress level can be obtained from Wohler’s curve of the material. Through Miner’s law, the damage corresponding to the stress level can be obtained. n(s) D = (14) ∑ N(s) Sensors 2021, 21, 4518 5 of 22

3. Non-Gaussian PSD Method in the Frequency Domain Generally, a random vibration signal x is distinguished as a Gaussian signal if its probability density function P(x) is given as

1 (x−µ)2 P(x) = √ e 2σ2 (15) 2πσ2 where µ is the mean and σ is the standard deviation. The i-th central moment was deﬁned as

n 1 i Mi = ∑ xj − µ (16) n j=1

n is the number of data points related to time and sample rate. Commonly, the 2 second-order central moment (i = 2) M2 is the same as variance σ . Two parameters are used to determine whether random variables follow Gaussian distribution, skewness S and kurtosis K, which are defined as follows: M M S = 3 and K = 4 (17) σ4 σ4 The skewness is an index of asymmetry of the mean value. The kurtosis is an index of sharpness. Generally, the contribution of skewness in fatigue-damage estimation is tiny, and it is a common practice to focus only on kurtosis. As discussed in the first section of this paper, Braccesi’s method is more acceptable than other methods. In this paper, this method is selected for comparison [16]. m is the coefficient of Wohler’s curve:

3 2 ! m 2 K − 3 S λ = exp − (18) ng π 5 4

According to the 1986 Moors’ paper, kurtosis is actually used to represent the dis- persion of the data between the mean ± RMS [27]. High kurtosis values will appear for two reasons: 1. Most data concentrated around the mean, but also contain occasional values far from the mean; 2. The trailing distribution weight is too large. On the contrary, higher kurtosis means that there is a sizeable extreme value. For the same PSD response stress, the higher the kurtosis, the higher the extremum. The fatigue characteristics of materials, Wohler’s curve, has a high sensitivity to stress, especially on the left side of the curve; with the increase of stress, life decreases sharply. Therefore, the author’s view is that a single corrective coefficient cannot accurately reflect the influence of high kurtosis (higher than 5) on the fatigue-damage estimation under non-Gaussian random vibration. Moreover, the maximum stress may exceed the yield limit or even fracture limit at high kurtosis. Besides, the nonlinear damage-accumulation principle needs to be considered due to the influence of the loading sequence at high kurtosis [28]. Therefore, it is necessary to obtain the probability distribution function of response stress under non-Gaussian random vibration. So, the stress distribution function in the non-Gaussian frequency-domain method can be hypothesized as

p(s)NG = αp(β(s − ∆Sc))G, s ∈ (0, pRMS) (19)

The commonly used stress range 0–6 RMS becomes 0–p RMS. Among them, α is the maximum distribution adjustment value, also called enlarge scale; β is the α concentration adjustment coefﬁcient; ∆Sc is the center offset and p is the stress maximum-range coefﬁcient. Sensors 2021, 21, x FOR PEER REVIEW 6 of 23

Sensors 2021, 21, 4518 6 of 22 The commonly used stress range 0–6 RMS becomes 0-p RMS. Among them, α is the maximum distribution adjustment value, also called enlarge scale; β is the αconcentration

adjustment coefficient; ∆푆푐 is the center offset and p is the stress maximum-range coeffi- Throughcient. Through parameter-ﬁtting, parameter-fitting, the relationship the relationship between between these these four parameters four parameters and kurtosis and kur- istosis established. is established.

4.4. Virtual Virtual Experiment Experiment InIn order order to to highlight highlight the the differences differences from from the the kurtosis kurtosis effect effect and and avoid avoid the interferencethe interfer- ofence other of factors,other factors, the virtual the virtual test system test system is used is toused repeat to repeat the response the response process process of specimens of spec- underimens the under excitation the excitation of non-Gaussian of non-Gaussian signals. Thesignals. virtual The test virtual system test uses system actual uses test actual data to build a ﬁnite element specimen and adjusts the material density, stiffness and damping test data to build a finite element specimen and adjusts the material density, stiffness and coefﬁcients to make the entire model match the real situation. Double-notched specimens damping coefficients to make the entire model match the real situation. Double-notched are used as research objects in this study. The experiment system is shown in Figure1. specimens are used as research objects in this study. The experiment system is shown in The material card of the specimen shown in Table1. Through the modal analysis for the Figure 1. The material card of the specimen shown in Table 1. Through the modal analysis specimen both experiment and virtual experiment, the density and modal information of for the specimen both experiment and virtual experiment, the density and modal infor- the specimens are obtained. mation of the specimens are obtained.

FigureFigure 1. 1.The The experiment experiment system system and and the the virtual virtual experiment experiment of of two two notch notch specimens. specimens.

TableTable 1. 1.Material Material card card of of AISI304. AISI304.

DensityDensity Elastic Elastic Modulus Modulus Yields Yields Strength Ultimate Tension Tension Strength Strength mm 7942.27942.2 kg/m3 161161 GPa GPa 365 365 MPa 710.5710.5 MPa MPa 17.9217.92

According to the modal analysis result, the second-order mode 88 Hz is selected and According to the modal analysis result, the second-order mode 88 Hz is selected and the frequency radius of the excitation is 40 Hz. In the beginning, 30 min total length time the frequency radius of the excitation is 40 Hz. In the beginning, 30 min total length time series of acceleration were generated. They have the same PSD level and form but differ- series of acceleration were generated. They have the same PSD level and form but different ent kurtosis. Here, a relatively safe PSD level is selected to facilitate the observation of kurtosis. Here, a relatively safe PSD level is selected to facilitate the observation of damage damage at high kurtosis. The sample rate is 512 Hz according to the Shannon formula. at high kurtosis. The sample rate is 512 Hz according to the Shannon formula. Due to the computational power, the excitation signal with a length of 10 min is intercepted from each datum and it is the minimum that can pass the ergodicity test. Eight different kurtosis

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levels are studied, from three to twenty and two cases for each kurtosis level. The tolerance of the kurtosis is ±0.4. At this stage, a total of 16 time series are created. They have the same frequency domain information, as shown in Table2.

Table 2. Frequency-domain information (g = 9.81 m/s2).

Frequency (Hz) PSD (g2/Hz) PSD (m2/s4)/Hz 48 0.25 24.05 128 0.25 24.05 RMS 4.472136 g 43.87 m/s2

Sensors 2021, 21, x FOR PEER REVIEW 7 of 23 From Figure2, the difference in the maximum values of the signals with different kurtosis is obvious. Figure3 shows that the energy distribution of the signal is consistent in the frequency domain. The details of each signal are shown in Table3. Due to the computational power, the excitation signal with a length of 10 min is inter- In the process of the virtual experiment, through finite element simulation by Abaqus, cepted from each datum and it is the minimum that can pass the ergodicity test. Eight all parameters are verified in the reality modal analysis process with the specimen data. different kurtosis levels are studied, from three to twenty and two cases for each kurtosis Wohler’s curve of the specimen is also checked in order to reduce the possible difference level. The tolerance of the kurtosis is ±0.4. At this stage, a total of 16 time series are created. from manufacturing [29]. They have the same frequency domain information, as shown in Table 2. The maximum response stress occurred at the center section of the first notch. One of theFrom four Figure nodes 2, is selectedthe difference as the in observation the maximum object, values shown of inthe Figure signals4. Althoughwith different it is kurtosismentioned is obv inious. the article Figure by 3 Georgeshows that and the Michalis energy that distribution materials of will the exhibitsignal is a non-linearconsistent inresponse the frequency undernon-Gaussian domain. The details excitation, of each temporarily signal are this shown effect in is ignoredTable 3. in this study [30].

FigureFigure 2. 2. ExcitationExcitation signal signal (10 (10 min) min) with with kurtosis kurtosis (20, (20, 10, 10, 8, 8, 5, 5, 4.5, 4.5, 4, 4, 3.5, 3.5, 3). 3).

Table 2. Frequency-domain information (g = 9.81 m/s2).

Frequency (Hz) PSD (g2/Hz) PSD (m2/s4)/Hz 48 0.25 24.05 128 0.25 24.05 RMS 4.472136 g 43.87 m/s2

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. Figure 3. PSD of excitation acceleration in each kurtosis. Figure 3. PSD of excitation acceleration in each kurtosis. Table 3. RMS and Kurtosis of excitation signals. In the process of the virtual experiment, through finite element simulation by Abaqus,Duration all parameters 10 are min verified in the Sample reality Ratemodal analysis process 512 with the specimen data.No. Wohler’s curve 1of the specimen 2 is 3 also checked 4 in 5 order to 6 reduce 7 the possible 8 differenceRMS (m/sfrom2) manufactur43.84ing [29] 43.73. 43.98 43.76 43.81 43.92 43.83 43.83 Kurtosis 20.02 20.08 10.09 10.04 8.00 8.00 4.98 4.98 Table 3. RMSNo. and Kurtosis of 9 excitation 10 signals. 11 12 13 14 15 16 Sensors 2021, 21, x FOR PEER REVIEW 2 9 of 23 RMS (m/s ) 43.71 43.82 43.93 43.84 43.88 43.84 43.85 43.85 DurationKurtosis 10 min 4.53 4.55Sample 4.02 Rate 4.02 3.505 3.5012 2.99 3.00 No. 1 2 3 4 5 6 7 8 RMS (m/s2) 43.84 43.73 43.98 43.76 43.81 43.92 43.83 43.83 Kurtosis 20.02 20.08 10.09 10.04 8.00 8.00 4.98 4.98 No. 9 10 11 12 13 14 15 16 RMS (m/s2) 43.71 43.82 43.93 43.84 43.88 43.84 43.85 43.85 Kurtosis 4.53 4.55 4.02 4.02 3.50 3.50 2.99 3.00

The maximum response stress occurred at the center section of the first notch. One of the four nodes is selected as the observation object, shown in Figure 4. Although it is mentioned in the article by George and Michalis that materials will exhibit a non-linear response under non-Gaussian excitation, temporarily this effect is ignored in this study [30]. From the response-stress results shown in Table 4, it was found that the response stress’s kurtosis was attenuated. As the acceleration of kurtosis increases, the kurtosis of the response stress decays more significantly. The RMS is 42.86 MPa. There is no significant effect from kurtosis on the RMS and PSD. The details are shown in Figure 5.

Table 4. The RMS and Kurtosis of response stress. FigureFigure 4. 4.Critical Critical node node of of the the FE FE model. model. Duration 10 min Sample Rate 512 FromNo. the response-stress1 results2 shown3 in Table44 , it was5 found6 that the7 response8 stress’s kurtosis was attenuated. As the acceleration of kurtosis increases, the kurtosis of RMS (MPa) 43.32 43.02 43.31 43.02 43.07 43.05 43.17 43.08 the response stress decays more signiﬁcantly. The RMS is 42.86 MPa. There is no signiﬁcant Kurtosis 9.67 9.49 5.63 5.69 4.99 4.84 3.68 3.69 effect from kurtosis on the RMS and PSD. The details are shown in Figure5. No. 9 10 11 12 13 14 15 16 RMS (MPa) 42.97 43.13 43.19 43.15 43.12 43.12 43.10 43.14 Kurtosis 3.55 3.56 3.40 3.38 3.17 3.19 2.98 2.98

. Figure 5. PSD of response stress in each kurtosis.

By comparing the stress-cycle distribution results of different kurtoses, it can be found that as kurtosis increases, the peak of the distribution function increases; the highest point shifts to the low-stress region, and the maximum stress distribution range increases [31]. However, according to research, the maximum and minimum values of response stress are related to the value of the input signal, which means that it is determined by the value of the excitation acceleration. It can be concluded that the magnitude of the response stress is related to the input kurtosis. Therefore, in the relationship between kurtosis and damage in subsequent studies, kurtosis refers to the input kurtosis, that is, the kurtosis of the excitation acceleration. In addition, most of the time, it is more difficult to obtain the response stress than excitation. Through the rainflow counting method, the probability density distribution of response stress can be obtained. Here we also need to pay attention to the maximum stress. It accounts for a tiny proportion but cannot be ignored.

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Table 4. The RMS and Kurtosis of response stress.

Duration 10 min Sample Rate 512 No. 1 2 3 4 5 6 7 8 RMS (MPa) 43.32 43.02 43.31 43.02 43.07 43.05 43.17 43.08 Kurtosis 9.67 9.49 5.63 5.69 4.99 4.84 3.68 3.69 No. 9 10 11 12 13 14 15 16 RMS (MPa) 42.97 43.13 43.19 43.15 43.12 43.12 43.10 43.14 Figure 4. Critical node of the FE model. Kurtosis 3.55 3.56 3.40 3.38 3.17 3.19 2.98 2.98

. Figure 5. PSD of response stress in each kurtosis. Figure 5. PSD of response stress in each kurtosis. By comparing the stress-cycle distribution results of different kurtoses, it can be found thatBy as kurtosiscomparing increases, the stress the peak-cycle of distribution the distribution results function of different increases; kurtos thee highests, it can point be foundshifts that to the as low-stresskurtosis increases, region, andthe peak the maximum of the distribution stress distribution function increases range increases; the highest [31]. pointHowever, shifts accordingto the low- tostress research, region, the and maximum the maximum and minimum stress distribution values of range response increases stress [31].arerelated However, to the according value of theto research, input signal, the whichmaximum means and that minimum it is determined values of by response the value stressof the are excitation related to acceleration. the value of the It can input be signal, concluded which that means the magnitudethat it is determined of the response by the valuestress of is the related excitation to the acceleration. input kurtosis. It can Therefore, be concluded in the that relationship the magnitude between of the kurtosis response and stressdamage is related in subsequent to the input studies, kurtosis. kurtosis Therefore, refers to in the the input relationship kurtosis, between that is, the kurtosis kurtosis and of damagethe excitation in subsequent acceleration. studies, In addition,kurtosis refers most to of the input time, itkurtosis, is more that difficult is, the to kurtosis obtain the of theresponse excitation stress acceleration. than excitation. In addition, most of the time, it is more difficult to obtain the responseThrough stress thethan rainflow excitation. counting method, the probability density distribution of re- sponseThrough stress canthe berainflow obtained. counting Here wemethod, also need the toprobability pay attention density to the distribution maximum of stress. re- sponseIt accounts stress for can a tinybe obtained. proportion Here but we cannot also need be ignored. to pay attention to the maximum stress. It accountsAccording for a tiny to the proportion results shown but cannot in Figure be ignored.6, it can be clearly observed that when the kurtosis is three, the results obtained by the time-domain method and the frequency- domain method are the same. The result from the frequency-domain method clearly finds the effect of kurtosis on the stress-cycle distribution [32].

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According to the results shown in Figure 6, it can be clearly observed that when the Sensors 2021, 21, 4518 kurtosis is three, the results obtained by the time-domain method and the frequency10 of 22-domain method are the same. The result from the frequency-domain method clearly finds the effect of kurtosis on the stress-cycle distribution [32].

FigureFigure 6. Rainflow 6. Rainﬂow counting counting results results (K (K = = 20, 20, 10, 10, 8, 5, 4.5, 4,4, 3.5,3.5, 3). 3).

According to Kihm’s research, the peak-to-valley value of non-Gaussian signals has According to Kihm’s research, the peak-to-valley value of non-Gaussian signals has a corresponding proportional relationship with RMS [15]. And it is also resented in this a corresponding proportional relationship with RMS [15]. And it is also resented in this study, as shown in Table5. Therefore, adjusting the rainﬂow counting results obtained study,in the as frequency-domainshown in Table 5. method Therefore, and adjusting obtaining thethe stressrainflow cycle counting distribution results results obtained at the in

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Sensors 2021, 21, 4518 11 of 22 the frequency-domain method and obtaining the stress cycle distribution results at the corresponding kurtosis according to the kurtosis, stress RMS value and the Wohler’s curve should be considered. Next, based on the yield limit of the material, a decision is madecorresponding as to whether kurtosis to consider according the pl toastic the strain kurtosis, portion stress and RMS then value the damage and the Wohler’sof the elastic curve strainshould and bethe considered. plastic strain Next, are basedcalculated on the separately yield limit. F ofinally, the material, the expected a decision total damage is made as at theto whether current kurtosis to consider is obtained. the plastic Of strain course, portion in the andactual then signal, the damage the distribution of the elastic of high strain- stressand areas the plastic is very strain random are calculateddue to the separately.length and Finally,low probability. the expected Considering total damage that the at the zerocurrent-mean kurtosisrandom issignal obtained. is symmetric Of course, on in both the actualsides of signal, the 0 thepoint, distribution the probability of high-stress and magnitudeareas is very of positive random and due negative to the length occurrences and low are probability. equal, and Considering thus the trend that of the peak zero-mean and random signal is symmetric on both sides of the 0 point, the probability and magnitude of valley values with kurtosis is considered to be the same. The difference Peakk of the peak andpositive valley values and negative is used. occurrences Then, only area curve equal, equation and thus is thecalculated trend of here peak to and fit the valley ratio values of with kurtosis is considered to be the same. The difference Peak of the peak and valley Peakk to RMS. k values is used. Then, only a curve equation is calculated here to ﬁt the ratio of Peakk to RMS. Table 5. Accelerated statistical comparison among different kurtoses. Table 5. Accelerated statistical comparison among different kurtoses. Kurtosis 20.06 10.06 8.00 4.98 4.54 4.02 3.50 3.00 Kurtosis 20.06Mean(m/s 10.062) 0 8.000 4.980 4.540 4.020 0 3.500 3.000 Mean (m/s2) 0 0 0 0 0 0 0 0 RMS(m/s2) 43.78 43.87 43.86 43.83 43.77 43.88 43.86 43.85 RMS (m/s2) 43.78 43.87 43.86 43.83 43.77 43.88 43.86 43.85 Peak/RMS 16.88Peak/RMS 13.87 16.88 14.13 13.87 9.96 14.13 10.059.96 10.05 8.57 8.57 8.63 8.63 4.44 4.44 Valley/RMS −20.05Valley/RMS−15.16 −20.05−13.52 −15.16− 12.20 −13.52 −10.47−12.20 −10.47−9.07 −9.07− 8.25 −8.25 −−4.304.30 P-V/RMS 36.93P-V/RMS 29.03 36.93 27.65 29.03 22.16 27.65 20.5122.16 20.51 17.64 17.64 16.88 16.88 8.73 8.73

AccordingAccording to the to thecontour contour of the of thecurve, curve, the thedouble double exponent exponent is selected is selected as the as basis the ba- function.sis function.

푃푒푎푘푘 Peak=k 12.5 ∗ 푒푥푝 0.0225 ∗ (푘 − 3) − 7.5 ∗ 푒푥푝 (−0.75 ∗ (푘 − 3)) p= p = 12.5 ∗ exp( (0.0225 ∗ (k − 3))) − 7.5 ∗ exp(−0.75 ∗ (k − 3)) (20(20)) 푅푀푆RMS Next, the stress response function of the structure could be used to obtain the stress at differentNext, acceleration the stress response values according function ofto theFigure structure 7. The could changes be usedin the to distribution obtain the stress of theat rain different flow counting acceleration results values should according be considered to Figure in 7the. The next changes stage. Mainly in the distributionfocus on the of the rain ﬂow counting results should be considered in the next stage. Mainly focus on the center offset, the maximum distribution rate and the concentrate change. center offset, the maximum distribution rate and the concentrate change.

FigureFigure 7. Acceleration 7. Acceleration-range-range fit curve ﬁt curve by bykurtosis. kurtosis.

TheThe stress stress probability probability distribution distribution of the of the structure structure in inthe the frequency frequency range range and and the the RMSRMS value value of ofthe the stress stress could could be becalculated calculated by bythe the PSD PSD approach. approach. The The max max distribution distribution positionposition will will move move to the to the negative negative direction direction of the of the x-axis x-axis while while kurtosis kurtosis is increas is increasing.ing. The The speciﬁc value is related to the maximum stress range. It can be known from the previous description that all signal’s mean and root mean square is the same. For the random process with different kurtosis, if the ﬁrst-order or second-order statistic is constant during

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the growth of the maximum distribution range, as the kurtosis grows, the stress value of the maximum distribution rate will decrease, and the distribution rate of the maximum distribution will increase. So, the degree of increase in both the offset and the distribution rate based on the maximum distribution range can be inversely found. According to the characteristic of the Gaussian distribution, the change of the main distribution interval can be calculated, which refers to the distribution range of 1σ stress. In fact, the difference in offset is not very obvious at low kurtosis. At the same time, using the ratio of the maximum probability density distribution under different kurtoses to the maximum probability distribution under the Gaussian signal, the growth rate of the rainflow counting result could be determined. All coefficient fittings are performed using the double-exponential function as the basis function. Center-position offset fit by kurtosis:

4 4 ∆Sc = 5.7 ∗ 10 ∗ exp(−0.03010 ∗ (k − 3)) − 5.7 ∗ 10 ∗ exp(−0.03012 ∗ (k − 3)) (21)

Enlarged-scale ﬁt by kurtosis:

α = 1.048 ∗ exp(0.00651 ∗ (k − 3)) − 0.04663 ∗ exp(−0.6225 ∗ (k − 3)) (22)

Select the middle part of the curve and distribute the same numerical points to investigate the change of the curve span, here the value of 33% was chosen. Enlarged scale ﬁt by kurtosis:

β = 0.006886 ∗ exp(−1.042 ∗ (k − 3)) − 0.9922 ∗ exp(−0.003538 ∗ (k − 3)) (23)

Then, according to this coefficient, combined with Equation (18), the stress distribution density of any kurtosis at this node in the model under the same PSD form can be obtained. Next, fit Figure8 to obtain the relationship between the offset of the center position and the enlarged scale with kurtosis. In this process, it could be found that between 2 RMS stress and 6 RMS stress (86.4~259.2 MPa), the distribution of non-Gaussian signals is actually smaller than the distribution of Gaussian signals. So set within this interval, use a negative offset. This reduction factor is determined by the total number of cycles. Since the total number of cycles is constant, the number of cycles obtained by enlarging the curve growth should be subtracted within this stress range. The curve is processed using a smoothing function when kurtosis is higher than 5. Now we could infer the rainflow counting results of non-Gaussian signals based on the rainflow counting results obtained from the frequency-domain method and Gaus- sian signals. To check the results, use the Bendat method in the narrow-band and the Dirlik method in the wide-band, respectively, as the fundamental equations which are modulated by using the center position offset and the enlarged scale. So in the stress range 0–p RMS : ! β(S − ∆S ) β(S − ∆S )2 ( )nGNB = ∗ c − c p S α 2 exp 2 (24) σRMS 2σRMS

α G Z G Z Z2 Z2 ( )nGDK = √ 1 − + 2 − + · − p S exp 2 exp 2 G3Z exp (25) 2 m0 Q Q R 2R 2 where Z is the normalized amplitude and Xm is the mean frequency, as fellow

1 β(S − ∆Sc) m1 m2 2 Z = √ and xm = 2 m0 m0 m4 Sensors 2021, 21, 4518 13 of 22

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Other parameters are deﬁned as: Other parameters are defined as: Other parameters are defined as: 2x − α2 = m 2 2 G1 2(푥푚 −22 훼2,) 퐺 2=(푥푚1−+훼α2 ) , 퐺 =1 2,2 1 1 +2 훼2 1 + 훼2 2 2 1 − α12−−훼2−G퐺211++퐺1G1 G =퐺 1−=훼2−퐺1+퐺1 , , 2퐺 =2 1+푅, 2 1+1푅+ R G퐺==1 1−−G퐺−−G퐺, , 퐺3 =3 31 − 퐺1 −1 1퐺2, 22 2 α2 − xm − G1 1.25(α2 − G3 − G2R) = 2= R 훼2and−푥푚2 − Q퐺 1.25(훼2−퐺3−퐺2푅) − − +훼2−2푥푚−퐺1 1 1.25G(훼2−퐺3−퐺2푅) 1 α2 RG=1R =G1 2 andQ = Q = 1 1−훼2−퐺21 + 퐺and 퐺1 1−훼2−퐺1+퐺1 1 퐺1 TheTheThe new new results results by bykurtosis kurtosis increase increase comparison comparison with with time time series series as follows as follows:follows: by :the b byy the movingmovingmoving-average-average-average method, method, method, the the thenew new new results results results look look look similar similar to the to the original original rainflow rainﬂowrainflow counting counting results.results.results. The The The results results results for for fora total a a total total of 5 of of kurtosis 5 5 kurtosis kurtosis levels levels are arepresented presented in Figures in Figures 9–14.9 9– –1414..

Figure 8. The stress distribution rate in ±RMS. FigureFigure 8.8. The stress distributiondistribution raterate inin ±±RMS.RMS.

FigureFigure 9. Rainflow 9. Rainﬂow counting counting results results for Kurtosis for Kurtosis = 3. = 3. Figure 9. Rainflow counting results for Kurtosis = 3.

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Figure 10. Rainflow counting results for Kurtosis = 4. FigureFigure 10. 10.RainflowRainﬂow counting counting results results for forKurtosis Kurtosis = 4. = 4.

FigureFigure 11. 11.RainflowRainﬂow counting counting results results for forKurtosis Kurtosis = 5.= 5. Figure 11. Rainflow counting results for Kurtosis = 5.

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Figure 12. Rainflow counting results for Kurtosis = 8. FigureFigure 12. 12.RainflowRainﬂow counting counting results results for forKurtosis Kurtosis = 8. = 8.

FigureFigure 13. 13.RainflowRainﬂow counting counting results results for forKurtosis Kurtosis = 10. = 10. Figure 13. Rainflow counting results for Kurtosis = 10.

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Figure 14. Rainflow counting results for Kurtosis = 20. FigureFigure 14. Rainflow 14. Rainflow counting counting results results for Kurtosis for Kurtosis = 20. = 20. The actual operation method is to sum according to the probability density distribu- TheThe actual actual operation operation method method is isto tosum sum according according to to the the probability probability density density distribu- distribution tion function, which is usually around 98%, and the remaining 1–2% is evenly supple- tionfunction, function, which which is is usually usually around around 98%, 98%, and and the the remaining remaining 1–2% 1–2% is evenly is evenly supplemented supplemented on 6RMS–pRMS. Next, it is only necessary to calculate according to the damage- mentedon 6RMS–pRMS. on 6RMS–pRMS. Next, Next, it is only it is necessary only necessary to calculate to calculate according according to the damage-estimationto the damage- estimation method proposed previously. Errors are unavoidable, especially for random estimationmethod method proposed proposed previously. previously. Errors are Errors unavoidable, are unavoidable, especially especially for random for signals; random only signals; only approximate damage can be estimated. Each adjustment coefficient and the signaapproximatels; only approximate damage candamage be estimated. can be estimated. Each adjustment Each adjustment coefficient coefficient and the and degree the of degree of attenuation of the kurtosis are affected by the material properties and the struc- degreeattenuation of attenuation of the kurtosisof the kurtosis are affected are affected by the by material the material properties properties and the and structure the struc- of the ture of the specimen. Therefore, before applying, the coefficients of the specific structure turespecimen. of the specimen. Therefore, Therefore, before applying, before applying, the coefficients the coefficients of the specific of the specific structure structure need to be need to be verified by actual experiments. needverified to be verified by actual by experiments. actual experiments. By comparing the damage with the different kurtoses levels from the time domain By Bycomparing comparing the thedamage damage with with the thedifferent different kurtos kurtoseses level levelss from from thethe time time domain domain method,method, the the results results obtained obtained by by Braccesi’s Braccesi’s method method and and the the new new method method are are compared, compared, method, the results obtained by Braccesi’s method and the new method are compared, shownshown in in Figure Figure 15. 15 The. The considerable considerable difference difference may may come come from from Wohler’s Wohler’s curve curve of the of shown in Figure 15. The considerable difference may come from Wohler’s curve of the material.the material. material.

FigureFigure 15. 15. DamageDamage results results in in 10 10 min min.. Figure 15. Damage results in 10 min.

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The advantage of this method is that there is no need to simulate the stress time series, but theThe distribution advantage of of this the methodexcitation is time that there series is needs no need to be to made. simulate However, the stress the time probabilityseries, butdensity the distributiondistribution offunction the excitation has a smaller time series probability needs tovalue be made. in a high However,-stress the range.probability When the densitysize of data distribution cannot meet function the requirements, has a smaller the probability results are value greatly in aaffected high-stress by randomness.range. When the size of data cannot meet the requirements, the results are greatly affected bySo, randomness. this method can be summarized as follows: first, according to the requirements, establish So,the thistime method-domain can signal be summarizedunder the demand as follows: kurtosis first,. Through according the to frequency the requirements,-do- mainestablish method, thethe time-domainprobability distribution signal under of the the structural demand response kurtosis. stress Through of the the material frequency- in a domainspecific method,frequency the range probability under distributionthe Gaussian of thesignal structural is obtained. response The stress corresponding of the material relationshipin a specific between frequency acceleration range underand stress the Gaussianat the point signal of interest is obtained. is obtained The corresponding using ex- perimentalrelationship or simulation between methods. acceleration The andfitting stress method at the was point used of to interest infer the is obtainedmaximum using valueexperimental of the stress range, or simulation the maximum methods. distribution The fitting position method and was the used magnification to infer the factor. maximum Finally,value the of stress the stress probability range, thedensity maximum function distribution under the positionGaussian and signal the magnificationobtained by the factor. frequencyFinally, domain the stress method probability is used density to infer function the stress under-probability the Gaussian density signal function obtained under by the the nonfrequency-Gaussian domain signal method at a specific is used kurtosis to infer for the damage stress-probability estimation. It density should functionbe pointed under out thatthe non-Gaussianthe form of the signal coeff aticient a specific is not kurtosisunique; foronly damage the one estimation. with better It adaptability should be pointed is selectedout thatthis time. the form The offlowchart the coefficient is as follows is not unique;in Figure only 16. the one with better adaptability is selected this time. The flowchart is as follows in Figure 16.

FigureFigure 16. Corrective 16. Corrective method method flowchart. flowchart. 5. Verification Experiment 5. Verification Experiment In order to verify the validity of this result, the following signals are used for further verification;In order to verify details the are validity shown of in this Table result,6. In the this following process, the signals length are and used frequency for further range verificationof the non-Gaussian; details are shown signal in and Table PSD 6. In shape this areprocess, consistent the length with thoseand frequency previously range used as illustrated in the Figures 17 and 18. In this case, 0.4 g2/Hz is the highest load that the

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Sensors 2021, 21, 4518 18 of 22 of the non-Gaussian signal and PSD shape are consistent with those previously used as of the non-Gaussian signal and PSD shape are consistent with those previously used as illustrated in the Figures 17 and 18. In this case, 0.4 g2/Hz is the highest load that the shaker illustrated in the Figures 17 and 18. In this case, 0.4 g2/Hz is the highest load that the shaker can accept when kurtosis is 10. As the PSD is increasing, the shaker would overload. How- can accept when kurtosis is 10. As the PSD is increasing, the shaker would overload. How- ever, in the case of Gaussian random vibration, this situation will not occur. ever,shaker in the can case accept of Gaussian when kurtosis random is 10. vibration, As the PSD this is situation increasing, will the not shaker occur. would overload. However, in the case of Gaussian random vibration, this situation will not occur. Table 6. Input loading in the verification experiment. Table 6. Input loading in the verification experiment. Table 6. Input loading in the veriﬁcation experiment. PSD Corrective Method Virtual Experiment PSD Kurtosis Corrective Method Virtual Experiment Experiment LevelPSD Kurtosis (Numerical)Corrective Method (TimeVirtual Domain Experiment) Experiment Level Kurtosis (Numerical) (Time Domain) Experiment 0.45 g2Level/Hz 10 10(Numerical) min 1(Time0 min Domain) 0.45 g2/Hz 10 10 min 10 min 0.6 g2/Hz2 10 10 min 10 min 0.60.45 g2/Hz g /Hz 10 1010 min 10 min 10 min 10 min 0.4 0.6g2/Hz g2/Hz 10 1010 min 10 min10 min 10min Failure 0.4 g2/Hz 10 10 min 10 min Failure 0.4 g2/Hz 10 10 min 10 min Failure

Figure 17. Comparison of time-domain and frequency-domain results. FigureFigure 17. 17. ComparisonComparison of oftime time-domain-domain and and frequency frequency-domain-domain results. results.

FigureFigure 18. 18.ComparisonComparison of time of time-domain-domain and and frequency frequency-domain-domain results. results. Figure 18. Comparison of time-domain and frequency-domain results.

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AA totally totally new new specimen specimen is is applied applied to to non-Gaussian non-Gaussian random random vibration vibration excitation excitation until until failurefailure by by the the shaker. shaker. In In the the process, process, the the failure failure standard standard of of the the virtual virtual fatigue fatigue test test is is based based onon the the failure failure of of the the node; node thus,; thusΣ, ΣDD = = 1. 1. The The actual actual fatigue fatigue test test standard standard is is based based on on the the appearanceappearance of of visible visible cracks cracks at criticalat critical locations. locations. According According to the to establishedthe established ﬁnite finite element ele- virtualment virtual model, model, the load the of theload same of the level same is appliedlevel is toapplied Abaqus, to andAbaqus, the response and the stressresponse of thestress critical of the node critical is recorded. node is recorded. AfterAfter 1.5 1.5 h, h visible, visible cracks cracks appeared appeared in in the the middle middle of of the the ﬁrst first notch notch of of the the specimen. specimen. TheThe location location of of the the crack crack generation generation is is shown shown in in Figure Figure 19 19,, which which is is consistent consistent with with the the 2 expectedexpected location. location. In In fact, fact, the the lifetime lifetime of of Gaussian Gaussian random random vibration vibration at at a PSDa PSD of of 0.7 0.7 g /Hzg2/Hz isis the the same same asas thethe aboveabove damage.damage. It It is is clear clear that that the the harm harm of of kurtosis kurtosis is is more more worthy worthy of ofattention. attention.

Figure 19. Non-Gaussian fatigue test. Figure 19. Non-Gaussian fatigue test.

InIn order order to to reduce reduce the the simulation simulation time, time, the durationthe duration time time of the of input the input signal signal is selected is se- aslected 10 min. as 10 The min non-Gaussian. The non-Gaussian random random vibration vibration excitation excitation with the with same the characteristicssame character- asistics the experimentas the experiment is applied is applied to the to virtual the virtual specimen. specimen. By recording By recording the responsethe response stress stress of criticalof critical nodes nodes within within 10 min,10 min the, the damage damage value value during during this this period period can can be be estimated; estimated the; the processprocess is is shown shown in in Figures Figures 20 20 and and 21 21.. According to the material property and the above rainﬂow counting results, it can be calculated that the damage is about 0.067 within 10 min. Therefore, when the damage is equal to 1, the life of the specimen is 149.25 min or 2.48 h. With the same PSD level, the average life of the Gaussian signal is 71.4 h. The results of other methods are shown in Table7. Relatively speaking, this method further improves the accuracy of the calculation.

Table 7. The result of the fatigue of validation experiment.

PSD Level Kurtosis Corrective Method Braccesi’s Nieslony’s Experiment 0.45 g2/Hz 10 2.48 h 70.9 h 66.9 h 1.5 h

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Figure 20. Response stress of the PSD approach. FigureFigure 20. 20.ResponseResponse stress stress of the of thePSD PSD approach. approach.

FigureFigure 21. Rainflow 21. Rainﬂow result result of kurtosis of kurtosis = 10. = 10. Figure 21. Rainflow result of kurtosis = 10. 6. Discussion of Results According to the material property and the above rainflow counting results, it can be calculatedAccordingBy that comparing the to thedamage thematerial rain is about countproperty 0.067 results, and within threethe above 10 fundamental min rainflow. Therefore, parameters counting when results, shouldthe damage it be can focused beis equalcalculatedon: to maximum 1, thethat life the stress, of damage the maximum specimen is about distribution is 0.067 149.25 within min value or10 and2.48 min locationh. .Therefore, Withof the the samewhen maximum PSDthe damagelevdistributionel, the is . averageequal to Bylife 1, Figurestheof thelife Gaussian9of– 14the, itspecimen is signal conﬁrmed isis 71.4149.25 that h. theminThe new orresults 2.48 correction ofh. Withother method themethods same can PSDare well shown lev inferel, thein the Tableaveragefatigue 7. Relatively life damage of the speaking, ofGaussian the structure this signal method underis 71.4 further different h. The improves results kurtosis ofthe signalother accuracy excitationsmethods of the are calculation. with shown the samein PSD level. A comparison using two methods as basic functions, Dirlik and Bendat, reveals Table 7. Relatively speaking, this method further improves the accuracy of the calculation. little difference. This means that this correction can be applied to both narrow-band and wide-band. Unfortunately, this study was not available to validate wide-band.

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In this method, the strain-stress transfer function of the structure is used. With Figures 17 and 18, it can be conﬁrmed that this transfer function is only related to the structure. When the structure is constant, the results obtained by this method can be applied to different PSD classes. That is, it is possible to use low PSD levels to obtain a correction relationship to solve fatigue damage for high PSD levels. Note that random vibrations with high PSD peaks have very high equipment requirements for laboratory simulations. However, using this method, the study will become possible. Based on the validation tests, it can be determined that the method further improves the accuracy of damage calculation, shown in Table7. At the same time, it can be found that a larger kurtosis has a signiﬁcant effect on fatigue damage, making the fatigue damage under excitation at low PSD levels reach the damage under Gaussian excitation at high PSD levels.

7. Conclusions In this paper, the dynamic behavior of double-notch specimens was studied in order to ﬁnd how the kurtosis of the excitation affects the response stress. The relation of kurtosis with the excitation and response stress of the critical position is found, which makes fatigue estimation with excitation kurtosis possible. In fact, the new method has good performance under high kurtosis, which was proved by experiments. Additionally, it applies to all PSD levels and is not limited to the PSD level used to obtain the model. So, it is possible to predict fatigue damage at a high PSD level by using a lower PSD level in the same structure, especially when the test conditions are limited. At present, this method can be close to the real situation in high kurtosis; the accuracy now is 66%, much better than the previous. Both the ﬁtting method and the choice of basis function affect accuracy. In this paper, the more usable double exponential function model is used. However, this result can be further optimized.

Author Contributions: Conceptualization, Y.W.; methodology, Y.W. and R.S.; software, Y.W. and R.S. validation, Y.W. and R.S.; formal analysis, Y.W. and R.S.; investigation, Y.W.; resources, R.S.; data curation, Y.W. and R.S; writing—original draft preparation, Y.W.; writing—review and editing, Y.W. and R.S.; visualization, Y.W.; supervision, R.S.; project administration, R.S.; funding acquisition, R.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by China Scholarship Council program with the grant number No. 201604490078. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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