Research Article | Open Access
Chaoshuai Han, Yongliang Ma, Xianqiang Qu, Mindong Yang, "An Analytical Solution for Predicting the Vibration-Fatigue-Life in Bimodal Random Processes", Shock and Vibration, vol. 2017, Article ID 1010726, 18 pages, 2017. https://doi.org/10.1155/2017/1010726
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. Frequency methods are generally adopted to deal with this problem. This paper focuses on bimodal spectra methods, including Jiao-Moan method, Fu-Cebon 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.
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 , 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 [8–10]. 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 , Benasciutti-Tovo method , and Park method [13, 14] can be used to evaluate the vibration-fatigue-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  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  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  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.
2. Theory of Fatigue Analysis
2.1. Fatigue Analysis
The basic curve for fatigue analysis can be given aswhere 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 where is the number of cycles in the stress amplitude , resulting from rainflow counting (RFC) , 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 ; the analytical expression is given as Furthermore, the frequency of rainflow cycles can be replaced by rate of mean zero upcrossing .
In general, bimodal process is a wide-band process; thus, the fatigue damage in bimodal process can be calculated through (6).
2.2. 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
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 [8–10]. 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.
3. A Review of Bimodal Methods
3.1. Jiao-Moan (JM) Method
To simplify the study, , , and are normalized as , , and through the following transformation:and thenwhere
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 towhere and are the envelopes of and , respectively.
The rate of mean zero upcrossing due to can be calculated aswhere
After the approximation, a closed-form solution of the bandwidth correction factor can be then derived 
3.2. 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 . 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 . where and .
The number of cycles of the large cycles is . Thus, the fatigue damage due to the large stress cycles can be expressed by
3.3. 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 . 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, , while the number of cycles corresponding to the small cycles is . 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.
3.4. 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 . 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 and . It is found that (22) becomes closer to (19) with the increase of .
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.
4. Derivation of an Analytical Solution
4.1. Derivation of an Analytical Solution for (25)
Equation (25) can be rewritten as
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
Equation (31) will be divided into five parts.
(1) The First Part. It is as follows:
(2) The Second Part. It is as follows:
(3) The Third Part. It is as follows:
(4) The Fourth Part. It is as follows:
Equation (35) contains a standard Normal cumulative distribution function . It is difficult to get an exact solution directly. Thus, a new variable is introduced, as follows:With a method of variable substitution, (35) can be simplified:By defining(37) becomesand using integration by parts, (38) reduces to
Equation (40) is a recurrence formula; when is an odd number, it becomeswhere is a double factorial function and has an analytical expression which can be derived convenientlyWhen is an even number, iswhereSpecific derivation of (44) can be seen in Appendix A.
In addition, a Matlab program has been written to calculate in Appendix B.
(5) The Fifth Part. It is as follows:
Similarly to the fourth part, the analytical solution of the fifth part can be derived asThe final solution is
5. 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.
5.1. Treatment of Double Integral Based on FC Method
Equation (26) contains a double integral, in which variables and are in the range of and , 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.
5.2. 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 isNote that and for (31) have analytical solution; therefore, only , , and 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 . 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.
5.3. Discussion of Results
It is very convenient to use JM method in the case of , while FC method and MFC method can be used for any case regardless of and . Therefore, the analytical results are divided into two: and .
The solutions calculated by different mathematical methods for and in the case of 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 and . FC’s numerical integral solution matches the exact numerical integral solution very well in most cases. However, for relatively low value of or when 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 and in the case of are shown in Tables 2 and 3. The results indicate that the latter two solutions are always approximately equal for any value of and , while FC’s numerical integral solutions show good agreement with the exact integral solutions only in a few cases. As like the case of , for relatively high or low values of and (i.e., , ), 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 and ) .