An Analytical Solution for Predicting the Vibration-Fatigue-Life in Bimodal Random Processes

Predicting the vibration-fatigue-life of engineering structures subjected to random loading is a critical issue for. Frequencymethods are generally adopted to deal with this problem.This paper focuses on bimodal spectra methods, including Jiao-Moanmethod, FuCebon method, and Modified Fu-Cebon method. It has been proven that these three methods can give acceptable fatigue damage results. However, these three bimodal methods do not have analytical solutions. Jiao-Moan method uses an approximate solution, Fu-Cebon method, and Modified Fu-Cebon method needed to be calculated by numerical integration which is obviously not convenient in engineering application. Thus, an analytical solution for predicting the vibration-fatigue-life in bimodal spectra is developed. The accuracy of the analytical solution is compared with numerical integration. The results show that a very good agreement between an analytical solution and numerical integration can be obtained. Finally, case study in offshore structures is conducted and a bandwidth correction factor is computed through using the proposed analytical solution.


Introduction
Engineering structures from different fields (e.g., aircrafts, wind energy utilizations, and automobiles) are commonly subjected to random vibration loading.These loads often cause structural fatigue failure.Thus, it is significant to carry out a study on assessing the vibration-fatigue-life [1,2].
Vibration fatigue analysis commonly consists of two part processes: structural dynamic analysis and results postprocessing.Structural dynamic analysis provides an accurate prediction of the stress responses of fatigue hot-spots.Once the stress responses are obtained, vibration fatigue can be successfully performed.Existing technologies such as operational modal analysis [3], finite element modeling (FEM), and accelerated-vibration-tests are mature and applicable to obtain the stress of structures [4,5].Therefore, the crucial part of a vibration fatigue analysis focuses on results postprocessing.
The postprocessing is usually used to calculate fatigue damage based on known stress responses.When the stress responses are time series, fatigue can be evaluated using a traditional time domain method.However, the stress responses of real structures are mostly characterized by the power spectral density (PSD) function.Thus, frequency domain method becomes popular in vibration fatigue analysis [6,7].
A bimodal spectrum is a particular PSD in the random vibration stress response of a structure.For some simple structures, the stress response of structures will show explicit characterization of two peaks.One peak of the bimodal spectrum is governed by the first-order natural frequency of the structure; another is dominated by the main frequency of applied loads.Therefore, some bimodal methods for fatigue analysis can be adopted [8][9][10].Moreover, several experiments (e.g., vibration tests on mechanical components) and numerical studies (e.g., virtual simulation of dynamic using FEM) also obtain have shown that the stress PSD is a typical bimodal spectrum [5,7,11].However, for some complex flexible structures, the PSD of the stress response of structures usually is a multimodal and wide-band spectrum.For this situation, existing general wide-band spectral methods such as Dirlik method [12], Benasciutti-Tovo method [10], and Park method [13,14] can be used to evaluate the vibrationfatigue-life.Recently, Braccesi et al. [15,16] proposed a bands method to estimate the fatigue damage of a wide-band random process in the frequency domain.In order to speed up the frequency domain method, Braccesi et al. [17,18] developed a modal approach for fatigue damage evaluation of flexible components by FEM.
For fatigue evaluation in bimodal processes, some specific formulae have been proposed.Jiao and Moan [8] provided a bandwidth correction factor from a probabilistic point of view, and the factor is an approximate solution derived by the original model.The approximation inevitably leads to some errors in certain cases.Based on an similar idea, Fu and Cebon [9] developed a formula for predicting the fatigue life in bimodal random processes.In the formula, there is a convolution integral.The author claimed that there is no analytical solution for the convolution integral which has to be calculated by numerical integration.Benasciutti and Tovo [10] compared the above two methods and established a Modified Fu-Cebon method.The new formula improves the damage estimation, but it still needs to calculate numerical integration.
In engineering application, the designers prefer an analytical solution rather than numerical methods.Therefore, the purpose of this paper is to develop an analytical solution to predict the vibration-fatigue-life in bimodal spectra.

Theory of Fatigue Analysis
2.1.Fatigue Analysis.The basic - curve for fatigue analysis can be given as where  represents the stress amplitude;  is the number of cycles to fatigue failure, and  and  are the fatigue strength coefficient and fatigue strength exponent, respectively.The total fatigue damage can then be calculated as a linear accumulation rule after Miner [19] where   is the number of cycles in the stress amplitude   , resulting from rainflow counting (RFC) [20], and   is the number of cycles corresponding to fatigue failure at the same stress amplitude.
When the stress amplitude is a continuum function and its probability density function (PDF) is   (), the fatigue damage in the duration  can be denoted as follows: where ]  is the frequency of rainflow cycles.
For an ideal narrowband process,   () can be approximated by the Rayleigh distribution [21]; the analytical expression is given as Furthermore, the frequency of rainflow cycles ]  can be replaced by rate of mean zero upcrossing ] 0 .According to (3), an analytical solution of fatigue damage [22] for an ideal narrowband process can be written as where Γ(⋅) is the Gamma function.
For general wide-band stress process, fatigue damage can be calculated by a narrowband approximation (i.e., (5)) first, and bandwidth correction is made based on the following model [23]: In general, bimodal process is a wide-band process; thus, the fatigue damage in bimodal process can be calculated through (6).

Basic Principle of Bimodal Spectrum
Process.Assume that a bimodal stress process () is composed of a low frequency process (LF)   () and a high frequency process (HF)   ().
where   () and   () are independent and narrow Gaussian process.
The one-sided spectral density function of () can be summed from the PSD of LF and HF process.
The th-order spectral moments of () are defined as The rate of mean zero upcrossing corresponding to   () and   () is The rate of mean zero upcrossing of () can be expressed as According to (5), (9), and (11), narrowband approximation of bimodal stress process () can be given as Shock and Vibration 3 Equation ( 12) is known as the combined spectrum method in API specifications [24].
The existing bimodal methods proposed by Jiao and Moan, Fu and Cebon, and Benasciutti and Tovo are based on the idea: two types of cycles can be extracted from the rainflow counting, one is the large stress cycle, and the other is the small cycle [8][9][10].The fatigue damage due to () can be approximated with the sum of two individual contributions.
where   represents the damage due to the large stress cycle and   denotes the damage due to the small stress cycle.and then

A Review of Bimodal Methods
where Jiao-Moan points out that the small stress cycles are produced by the envelope of the HF process, which follows the Rayleigh distribution.The fatigue damage due to the small stress cycles can be obtained according to (5).
While the large stress cycles are from the envelop process, () (see Figure 1), the amplitude of () is equal to where   () and   () are the envelopes of  *  () and  *  (), respectively.
The distribution of () can be written as a form of a convolution integral     () and   () obey the Rayleigh distribution; therefore, (18) has an analytical solution which is given [8] The rate of mean zero upcrossing due to () can be calculated as where An approximation was made by Jiao and Moan for (19) as follows [8]:

PDF of amplitude JM
] 0, ⋅  Eq. ( 22)  After the approximation, a closed-form solution of the bandwidth correction factor can be then derived [8] Finally, the fatigue damage can be obtained as ( 6) and ( 12).

Fu-Cebon (FC) Method. Similarly to JM method, Fu and
Cebon also considered that the total damage is produced by a large cycle (  +   ) and a small cycle (  ), as depicted in Figure 2. The small cycles are from the HF process, and the distribution of the amplitude    () is a Rayleigh distribution, as shown in (4).However, the number of cycles associated with the small cycles   is different from JM method and equals (] 0, − ] 0, ) ⋅ .According to (5), the damage due to the small cycles is The amplitude of the large cycles   can be approximated as the sum of amplitude of the LF and HF processes, the distribution of which can be expressed by a convolution of two Rayleigh distributions [9].
The number of cycles of the large cycles is   = ] 0, ⋅ .Thus, the fatigue damage due to the large stress cycles can be expressed by Equation ( 26) can be calculated with numerical integration [9,10].Therefore, the total damage can be obtained according to (13).

Modify Fu-Cebon (MFC) Method.
Benasciutti and Tovo made a comparison between JM method and FC method and concluded that using the envelop process is more suitable [10].Thus, a hybrid technique is adopted to modify the FC method.More specifically, the large cycles and small cycles are produced according to the idea of FC method.The number of cycles associated with the large cycles is defined similarly to JM method.That is,   = ] 0, ⋅ , while the number of cycles corresponding to the small cycles is   = (] 0, − ] 0, ) ⋅ .The total damage for MFC method can be then written according to (13).
Although the accuracy of the MFC method is improved, the fatigue damage still has to be calculated with numerical integral.

Comparison of Three Bimodal
Methods.Detailed comparison of the aforementioned three bimodal methods can be found in Table 1.In all methods, the amplitude of the small cycle obeys Rayleigh distribution, and the corresponding fatigue damage has an analytical expression as in (5); the distribution of amplitude of the large cycle is convolution integration of two Rayleigh distributions, and the relevant fatigue damage can be calculated by (26).
Because of complexity of the convolution integration, several researches assert that (26) has no analytical solution [9,10].To solve this problem, Jiao and Moan used an approximate model (i.e., (22)) to obtain a closed-form solution [8].
However, the approximate model may lead to errors in some cases as in Figure 3 which illustrates the divergence of ( 19) and (22) for different values of  * 0, and  * 0, .It is found that (22) becomes closer to (19) with the increase of  * 0, .For FC and MFC methods, (26) was calculated by numerical technique.Although the numerical technique can give a fatigue damage prediction, it is complex and not convenient when applied in real engineering.In addition, the solutions in some cases are not reasonable.In Section 4, an analytical solution of (26) will be derived to evaluate the fatigue damage, and the derivation of the analytical solution focuses on the fatigue damage of the large cycles.(25).Equation ( 25) can be rewritten as

Derivation of an Analytical Solution for
Equation ( 27) will be divided into two items.
(1) The First Item.It is as follows: (2) The Second Item.It is as follows: The analytical solution of ( 25) can be then obtained Note that when  0, +  0, = 1, (30) is just equal to (19) derived by Jiao and Moan [8].Therefore, ( 19) is a special case of (30).

Derivation of an Analytical Solution for (26) Based on (30).
The derivation of an analytical solution for (26) is on the basis of (30), as Equation (31) will be divided into five parts.
In addition, a Matlab program has been written to calculate ( 0, ,  0, , ) in Appendix B.

Numerical Validation
In this part, the accuracy of FC method and JM method and the derived analytical solution will be validated with numerical integration.Transformation of (26) will be carried out first.

Treatment of Double Integral Based on FC Method.
As pointed out by Fu and Cebon and Benasciutti and Tovo [9, 10], FC's numerical integration can be calculated as the following processes.Equation ( 26) contains a double integral, in which variables  and  are in the range of (0, ∞) and (0, ), respectively.Apparently, the latter is not compatible with the integration range of Gauss-Legendre quadrature formula.Therefore, by using a integration transformation the integral part of ( 26) can be simplified to Equation ( 49) can be thus calculated with Gauss-Legendre and Gauss-Laguerre quadrature formula.

Treatment of Numerical Integral for (31).
Direct calculation of (31) may lead to some mathematical accumulative errors.To obtain a precise integral solution, (31) has to be handled with a variable substitution, and the result is Note that  1 and  2 for (31) have analytical solution; therefore, only  3 ,  4 , and  5 are dealt with in (50).The solution of (50) can be obtained through Gauss-Laguerre quadrature formula.
The accuracy of the numerical integral in ( 49) and (50) depends on the order of nodes and weights which can be obtained from handbook of mathematics [25].The accuracy increases with the increasing orders.However, from the engineering point of view, too many orders of nodes and weights will lead to difficulty in calculation.The integral results of (50) are convergent when the orders of nodes and weights are equal to 30.Therefore, the present study takes the order of 30.

Discussion of Results.
It is very convenient to use JM method in the case of  0, +  0, = 1, while FC method and MFC method can be used for any case regardless of  0, and  0, .Therefore, the analytical results are divided into two:  0, +  0, = 1 and  0, +  0, ̸ = 1.The solutions calculated by different mathematical methods for  = 3 and  = 4 in the case of  0, +  0, = 1 are plotted in Figures 4 and 5.It turns out that the  proposed analytical solution gives the same results with the exact numerical integral solution for any value of  0, and  0, .FC's numerical integral solution matches the exact numerical integral solution very well in most cases.However, for relatively low value of  0, or when  0, tends to 1, it may not be right.JM's approximate solution is close to the exact numerical integral solution only in a small range.FC's numerical integral solutions, the exact numerical integral solutions, and the analytical solutions for  = 3 and  = 4 in the case of  0, +  0, ̸ = 1 are shown in Tables 2 and  3.The results indicate that the latter two solutions are always approximately equal for any value of  0, and  0, , while FC's numerical integral solutions show good agreement with the exact integral solutions only in a few cases.As like the case of  0, +  0, = 1, for relatively high or low values of  0, and  0, (i.e.,  0, = 100,  0, = 1 × 10 −5 ), FC's integral solutions may give incorrect results.This phenomenon is in accord with the analysis of Benasciutti and Tovo (e.g., FC's numerical integration may be impossible for too low values of  0, and  0, ) [10].
In a word, for any value of  0, and  0, , the analytical solution derived in this paper always gives an accurate result.Furthermore, it can be solved very conveniently and quickly with the aid of a personal computer through a program given in Appendix B.

Case Study
In this section, the bandwidth correction factor is used to compare different bimodal spectral methods.
A general analytical solution (GAS) of fatigue damage for JM, FC, and MFC method can be written as can be obtained according to (47), and   and   can be chosen as defined in Table 1, which represent different bimodal spectra methods, that is, JM, FC, and MFC method.
According to (6), the analytical solution of the bandwidth correction factor is Likewise, a general integration solution (GIS) as given in FC and MFC method can be written as can be obtained from (49).The integration solution of the damage correction factor is 6.1.Ideal Bimodal Spectra.Different spectral shapes have been investigated in the literatures [5,26].Bimodal PSDs with two rectangular blocks are used to carry out numerical simulations, as shown in Figure 6.Two blocks are characterized by the amplitude levels   and   , as well as the frequency ranges   −   and   −   .  and   are the central frequencies, as defined in (55) 1 and  2 represent the areas of block 1 and block 2 and are equal to the zero-order moment, respectively.For convenience, the sum of the two spectral moments is normalized to unity; that is,  1 +  2 = 1.
To ensure that these two spectra are approximately narrow band processes,   /  =   /  = 1.1.
Herein, two new parameters are introduced The parameter values in the numerical simulations are conducted as follows:  = 0.1, 0.4, 1, 2, and 9;  = 6 and 10 which ensure two spectra are well-separated;  = 3, 4, and 5 which are widely used in welded steel structures.  does not affect the simulated bandwidth correction factor  [10,27].Thus,   can be taken as the arbitrary value in the theory as long as   > 0; herein,   = 5 rad/s.
In the process of time domain simulation, the time series generated by IDFT [28] contains 20 million sample points and 150-450 thousand rainflow cycles, which is a sufficiently long time history, so that the sampling errors can be neglected.Figure 7 is the result of JM method.The bandwidth correction factor calculated by (23) is in good agreement with the results obtained from (52).For  = 3, JM method can provide a reasonable damage prediction.However, for  = 4 and  = 5, this method tends to underestimate the fatigue damage.
Figures 8 and 9 display the results of FC method and MFC method, which are both calculated by the proposed analytical solution (see ( 52)) and FC's numerical integration solution (see ( 54)).The bandwidth correction factor calculated by ( 52) is very close to the results obtained from (54).Besides, the FC method always provides conservative results compared with RFC.The MFC method improves the accuracy of the original FC method to some extent.However, it may underestimate the fatigue damage in some cases.
As has been discussed above, the bandwidth correction factor calculated by the analytical solution has divergence with that computed from RFC.The disagreement is not because of the error from the analytical solution but arises from the fact that the physical models of the original bimodal methods and the rainflow counting method are different.

Real Bimodal Spectra.
In practice, the rectangular bimodal spectra in previous simulations cannot represent real spectra encountered in the structures.Therefore, more realistic bimodal stress spectra will be chosen to predict the vibration-fatigue-life by using the analytical solution proposed in this paper.For offshore structures subjected to random wave loading, the PSDs of fatigue stress of joints always exhibit two predominant peaks of frequency.Wirsching gave a general expression to characterize the PSD, and the model has been widely used in a few surveys [10,23,29].The analytical form is where  is a scale factor,   is the significant wave height,   is the dominant wave period,   is the natural angular frequency of structure, and  is the damping coefficient. and   do not affect the value of the bandwidth correction factor  [10,27].Thus, they are chosen to be equal to unity for simplicity.The value of other parameters can be seen in Table 4.
Real stress spectra corresponding to group 1 and group 2 can be seen in Figure 10.() is a double peak spectrum, the first peak is produced by the peak of random wave spectrum, and the second one is excitated by the first mode response of structures.
Under the real stress spectra, the zero-order spectral moments corresponding to low frequency and high frequency are very small.JM method provides acceptable damage estimation compared with RFC.The results computed by (54) for FC and MFC method may be not correct in some cases.The incorrect results are mainly caused by (49).However, the proposed analytical solution (see (52)) can give a satisfactory damage prediction and always provide a conservative prediction.More importantly, (52) is more convenient to apply in predicting vibration-fatigue-life than numerical integration.

Conclusion
In this paper, bimodal spectral methods to predict vibrationfatigue-life are investigated.The conclusions are as follows: (1) An analytical solution of the convolution integral of two Rayleigh distributions is developed.Besides, this solution is a general form which is different from that proposed by Jiao and Moan.The latter is only a particular case.
(2) An analytical solution based on bimodal spectral methods is derived.It is validated that the analytical solution shows a good agreement with numerical integration.More importantly, the analytical solution has a stronger attraction than numerical integration in engineering application.
(3) For JM method, the original approximate solution (see (23)) is reasonable in most cases; the analytical solution (see (52)) can give better prediction.
(4) In ideal bimodal processes, JM method and MFC method may overestimate or underestimate the fatigue damage, while in real bimodal processes, they give a conservative prediction.FC method always provides conservative results in any cases.Therefore, FC method can be recommended as a safe design technique in real engineering structures. t

Figure 2 :
Figure 2: The large cycles and small cycles for a random stress process.

Figure 11 :
Figure 11: The bandwidth correction factor of different bimodal methods for (a) group 1 and (b) group 2.

Table 1 :
Comparison of large cycles and small cycles for different spectral methods.

Table 4 :
Parameters for real bimodal spectra.