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The reliability models of the components under the nonstationary random load are developed in this paper. Through the definition of the distribution of the random load, it can be seen that the conventional load-strength interference model is suitable for the calculation of the static reliability of the components, which does not reflect the dynamic change in the reliability and cannot be used to evaluate the dynamic reliability. Therefore, by developing an approach to converting the nonstationary random load into the random load whose pdf is the same at each moment when the random load applies, the reliability model based on the longitudinal distribution is derived. Moreover, through the definition of the transverse standard load and the transverse standard load coefficient, the reliability model based on the transverse distribution is derived. When the occurrence of the random load follows the Poisson process, the dynamic reliability models considering the strength degradation are derived. These models take the correlation between the random load and the strength into consideration. The result shows that the dispersion of the initial strength and that of the transverse standard load coefficient have great influences on the reliability and the hazard rate of the components.

Reliability is an important index to the design, manufacture, and maintenance of products. Products failure when environment load is higher than strength. Reliability is defined as the probability that the products function successfully in the mission duration. The well-known load-strength interference (LSI) model plays a significant role in the analysis of the reliability of the mechanical components and systems.

In the LSI model, the environment load and the strength are random variables. The probability density function (pdf) of the load and that of the strength are shown in Figure

The schematic illustration of the LSI model.

Some efforts have been made to investigate the dynamic reliability models based on the LSI model. Wen and Chen [

The models in these papers held the hypothesis that the strength was independent of the load and was described as a simple function of time. Besides, the pdf of the random load was assumed to be the same at each moment when the random load applies. However, when strength degradation is taken into account, the strength is the function of the occurrence frequency of the random load and the magnitude of the random load, especially for mechanical components. It may cause errors in the calculation of reliability to neglect the correlation between the random load and the strength. Moreover, in order to develop the dynamic reliability model, further analysis should be made to obtain the relationship between the strength and the time, instead of assuming a simple function of time to describe the strength.

Furthermore, different distributions of random load can be obtained according to different statistic methods. Therefore, it is necessary to define and distinguish the distributions of the random load, which may otherwise lead to the misusage of the reliability models. In addition, components are subject to nonstationary random load sometimes. In such a situation, great differences may exist in the mean value and the autocorrelation function among different samples of the random load, which makes it difficult to satisfy the ergodic conditions. Thus, the pdf of the random load is different at each moment when the random load applies, and it is important to develop an easy-to-use approach to calculating the reliability of the components under the nonstationary random load.

In this paper, we define the distributions of the random load according to different statistic methods. Then, the dynamic reliability models considering the correlation between the random load and the strength are derived based on the LSI model.

In order to obtain the statistic characteristics of the random load, the samples of the random load are recorded as shown in Figure

The schematic illustration of the distributions of the random load.

From the definition of the distribution of the random load, it is easy to see that the longitudinal distribution reflects the probabilities that the loads with different amplitudes occur at a determinate moment, while the transverse distribution reflects the proportion between the occurrence frequencies of the loads with different magnitudes in the mission duration. Most reliability models are based on the longitudinal distribution of the random load. Besides, when the pdf in the LSI model is the longitudinal pdf, the LSI model actually calculates the reliability when the random load applies once. When the pdf in the LSI model is the transverse pdf, the LSI model calculates the reliability when the strength does not degrade.

Provided that the longitudinal pdf at each moment when the random load applies is the same, it is reasonable to assume that the transverse pdf from each sample is the same, which is approximatively identical with the longitudinal pdf. However, the random load is nonstationary sometimes, and great differences may exist in the mean value and the autocorrelation function among different samples of the random load, which makes it difficult to satisfy the ergodic conditions. In this situation, the longitudinal pdf is different at each moment when the random load applies and cannot be obtained from the transverse pdf. Thus, when the number of the samples of the random load is large, a convenient method to calculate the reliability of the components under the nonstationary random load based on the longitudinal pdf will be proposed in the following section. When the number of the samples of the random load is small, only a few transverse pdfs can be obtained from the samples, and it is impossible to obtain the longitudinal pdf. Then the reliability models based on the transverse distribution under the nonstationary random load will be developed in this paper.

Suppose that the initial reliability is

For nonstationary random load, the longitudinal pdf at each moment when the load applies is different. Moreover, it is impossible to obtain the longitudinal distribution of the random load at each moment by tests, which means a daunting amount of work. Besides, the randomness of the load makes it difficult to describe the degradation process of the strength. Therefore, a new approach to converting the nonstationary random load into the random load whose pdf is the same at each moment when the random load applies is proposed based on the Miner damage accumulation rule as follows.

According to the Miner damage accumulation rule, the damage caused by a determinate load

Hence, the mean value of the damage caused by the application of the random load for once can be expressed as

The damage in (

Suppose the equivalent load at the moment when the random load applies for the

According to the Miner damage accumulation rule and (

From (

According to (

Suppose that the total times of application of the random load in the period of test is

According to (

Calculate the longitudinal standard load

Determine a normal distributed pdf with the mean value of

Calculate the reliability of the components as follows:

This is the method to evaluate the reliability of the components under the nonstationary random load based on the longitudinal distribution. In practice, strength is the function of the times of load application and the magnitude of the load. It might cause errors in the analysis of the reliability to neglect the correlation between the random load and the strength. In general, the strength can be expressed as [

When the initial strength is a random variable, the pdf of strength after arbitrary times of application of the random load can be obtained from (

As a matter of fact, it is difficult to obtain the statistical characteristics of the nonstationary random load through tests. Besides, the evaluation of the reliability of the components under nonstationary considering the correlation between the random load and the strength is even more difficult. This paper provided an approximate approach to dealing with these problems.

As described above, the distribution obtained from each sample of the random load is a transverse distribution. The transverse distribution reflects the proportion between the occurrence frequencies of the loads with different magnitudes in the mission duration. Owing to the limitations in the economy, time, or equipments, engineers can obtain only a few samples of the nonstationary random load sometimes. In this situation, it is impossible to acquire the longitudinal distribution of the random load. Therefore, it is necessary to establish the reliability models based on the transverse distribution of the random load.

Suppose that there is a transverse distribution from a sample of the random load that is denoted as

The total damage caused by the random load that applies for

Provided that there exists a equivalence load denoted as

For the limited samples of the random load, we can first obtain the transverse pdfs of them and calculate the equivalent loads of them. Then we can obtain the pdf of the equivalent loads denoted as

For a given sample of the random load, suppose that its transverse pdf is

For all the given samples of the random load, we define the transverse standard load coefficient

The pdf of

It should be noted that one of the most important properties of

Therefore, the reliability of the components based on the transverse distribution can be calculated as follows:

In this section, the dynamic reliability models are developed when the occurrence of the load follows the Poisson process. The Poisson process can be used to describe the times of the application of the load with the intensity

Suppose that

From (^{−1}, and

Reliability curves with different standard deviations of

Hazard rate curves with different standard deviations of

From Figures

According to the total probability formula, (

From (

As a matter of fact, the initial strength has the same influences on the reliability and the hazard rate calculated by (^{−1}. The transverse standard load is 100 MPa and

Reliability curves with different standard deviations of

Hazard rate curves with different standard deviations of

From Figures

The reliability models of the components under the nonstationary random load are developed in this paper. At first, the distributions of the random load are defined. Then the dynamic reliability model based on the longitudinal distribution of the random load is derived. When constructing the reliability model based on the longitudinal distribution of the random load, an approach to converting the nonstationary random load into the random load whose pdf is the same at each moment when the random load applies is proposed. The results show that larger dispersion of the initial strength makes the reliability decrease faster and the hazard rate increase faster.

Furthermore, by defining the transverse standard load and the transverse standard load coefficient, the dynamic reliability model based on the transverse distribution is derived. The results show that the dispersion of the transverse standard load coefficient has different influences on the reliability and the hazard rate in different stage of the service life of the components. Besides, it should be noted that in the wear-out period, the reliability and the hazard rate are quite sensitive to the dispersion of the transverse standard load coefficient.

In practice, strength is always correlative with the load. It is the function of the occurrence frequency and the magnitude of the load. The assumption that load is independent of strength may cause errors in the calculation of reliability. The models proposed in this paper take the correlation between the load and the strength into consideration. Moreover, they are convenient to use and helpful for the lifecycle management of the components and the systems.

All authors have read and approved this version of the article, and due care has been taken to ensure the integrity of the work. Neither the entire paper nor any part of its content has been published or has been accepted elsewhere. It is not being submitted to any other journal.

The authors declare that there is no conflict of interests regarding the publication of this article.

This work was supported by the Natural Science Foundation of China under contract no. 51175240 and the National Science Foundation of China under contract no. 11072123.