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# 数学代写|统计计算作业代写Statistical Computing代考|Fatigue Life Distribution Estimation

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## 数学代写|统计计算作业代写Statistical Computing代考|Abstract Modeling fatigue life

Abstract Modeling fatigue life is complex whether it is applied to structures or experimental programs. Through the years several empirical approaches have been utilized. Each approach has positive aspects; however, none have been acceptable for every circumstance. On many occasions the primary shortcoming for an empirical method is the lack of a sufficiently robust database for statistical modeling. The modeling is exacerbated for loading near typical operating conditions because the scatter in the fatigue lives is quite large. The scatter may be attributed to microstructure, manufacturing, or experimental inconsistencies, or a combination thereof. Empirical modeling is more challenging for extreme life estimation because those events are rare. The primary purpose herein is to propose an empirically based methodology for estimating the cumulative distribution functions for fatigue life, given the applied load. The methodology incorporates available fatigue life data for various stresses or strains using a statistical transformation to merge all the life data so that distribution estimation is more accurate than traditional approaches. Subsequently, the distribution for the transformed and merged data is converted using change-of-variables to estimate the distribution for each applied load. To assess the validity of the proposed methodology percentile bounds are estimated for the life data. The development of the methodology and its subsequent validation is illustrated using three sets of fatigue life data which are readily available in the open literature.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Fatigue Life Data

Fatigue life data are most often presented on an S-N plot which shows the fatigue data for a given load. The load is typically stress or strain. Thus $S-N$ can represent stress-life or strain-life. An additional way in which the fatigue data are presented is on a probability plot. Both of these representations will be used subsequently. Three different sets of fatigue life data are considered for the proposed method.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Data Fusion for Fatigue Life Analysis

The choices of $a$ and $b$ in Eq. (1) are easily determined by simple algebra to be the following:
$$a=\frac{s_{A}}{s_{y}} \text { and } b=N_{A}-\frac{s_{A}}{s_{y}} \bar{y},$$
where $\bar{y}$ is the average and $s_{y}$ is the sample standard deviation for $\left{y_{j}: 1 \leq j \leq n\right}$, and $N_{A}$ and $s_{A}$ are arbitrary values chosen for normalization.

For fatigue data the life times are distributed over several orders of magnitude that the procedure in Eqs. (1) and (2) is applied to the natural logarithm of the life times. Let $m$ be the number of different values of applied stress or strain, i.e., $\left{\Delta \sigma_{k}: 1 \leq k \leq m\right}$ or $\left{\Delta \varepsilon_{k}: 1 \leq k \leq m\right}$. Given $\Delta \sigma_{k}$ or $\Delta \varepsilon_{k}$ the associated life times are $\left{N_{k, j}: 1 \leq j \leq n_{k}\right}$ where $n_{k}$ is its sample size. Let
$$y_{k, j}=\ln \left(N_{k, j}\right)$$
be the transformed life times. Substituting Eq. (2) into Eq. (1) leads to the following:
$$z_{k, j}=\frac{s_{A}}{s_{y, k}}\left(y_{k, j}-\bar{y}{k}\right)+N{A} .$$
Thus, the averages and sample standard deviations of $\left{y_{k, j}: 1 \leq j \leq n_{k}\right}$ and $\left{z_{k, j}: 1 \leq j \leq n_{k}\right}$ are equal to each other. The next step is to merge all the transformed $z_{k, j}$ values from Eq. (4) for $1 \leq j \leq n_{k}$ and $1 \leq k \leq m$. The purpose in using the merged values is to have a more extensive dataset for estimation of the cdf. This is especially critical for estimating the extremes of the cdf more accurately. Subsequently, an appropriate cdf $F_{Z}(z)$ is found that characterizes the merged data. It is assumed that this cdf also characterizes the subsets $\left{z_{k, j}: 1 \leq j \leq n_{k}\right}$ of the merged set. With this assumption and the linear transformation in Eq. (4), the cdfs for $\left{y_{k, j}: 1 \leq j \leq n_{k}\right} F_{y, k}(y)$ can be derived from $F_{Z}(z)$ as follows:
$$F_{y, k}(y)=F_{Z}\left(\frac{s_{A}}{s_{y, k}}\left(y-\bar{y}{k}\right)+N{A}\right) .$$

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|DATA FUSION FOR FATIGUE LIFE ANALYSIS

$$z_{k, j}=\frac{s_{A}}{s_{y, k}}\left(y_{k, j}-\bar{y} {k}\right )+N {A} 。 因此，\left{y_{k, j} 的平均值和样本标准差： 1 \leq j \leq n_{k}\right} 和 \left{z_{k, j}: 1 \leq j \leq n_{k}\right} 彼此相等。下一步是合并来自方程式的所有转换后的 z_{k, j} 值。(4) 对于 1 \leq j \leq n_{k} 和 1 \leq k \leq m。使用合并值的目的是获得更广泛的数据集来估计 cdf。这对于更准确地估计 cdf 的极值尤其重要。随后，找到一个合适的 cdf F_{Z}(z) 来表征合并的数据。假设这个 cdf 也表征了合并集的子集 \left{z_{k, j}: 1 \leq j \leq n_{k}\right}。有了这个假设和方程式中的线性变换。(4)、\left{y_{k, j} 的 cdfs: 1 \leq j \leq n_{k}\right} F_{y,Thus, the averages and sample standard deviations of \left{y_{k, j}: 1 \leq j \leq n_{k}\right} and \left{z_{k, j}: 1 \leq j \leq n_{k}\right} are equal to each other. The next step is to merge all the transformed z_{k, j} values from Eq. (4) for 1 \leq j \leq n_{k} and 1 \leq k \leq m. The purpose in using the merged values is to have a more extensive dataset for estimation of the cdf. This is especially critical for estimating the extremes of the cdf more accurately. Subsequently, an appropriate cdf F_{Z}(z) is found that characterizes the merged data. It is assumed that this cdf also characterizes the subsets \left{z_{k, j}: 1 \leq j \leq n_{k}\right} of the merged set. With this assumption and the linear transformation in Eq. (4), the cdfs for \left{y_{k, j}: 1 \leq j \leq n_{k}\right} F_{y, k}(y) can be derived from F_{Z}(z) as follows: F_{y, k}和=F_{Z}\left(\frac{s_{A}}{s_{y, k}}\left(y-\bar{y} {k}\right)+N {A}\right) 。$$