## 2 Parametric Accelerated Life Testing for Mechanical Product
### Placing an Entire Parametric ALT Scheme  
Reliability might be described as the capability of a system to function under specified circumstances for a stated interval of time [25]. It has traditionally been illustrated by the \"bathtub curve\" (Figure 2) which has three sections [26]. In the first section, there is a decreasing failure rate in the earlier portion of the product's life (\(\beta<1\)). In the second section, there is a flat failure rate (\(\beta=1\)) during the middle life of the product which follows an exponential distribution. Eventually, there is an increasing failure rate to the termination of the product's life (\(\beta>1\)) which follows a Weibull distribution. Assuming that \(T\) is a random variable specifying the time to failure, the reliability function, the fraction yet surviving at time \(t\), could be expressed as follows:  
\[R(t)=P(T>t) \tag{1}\]  
As the complement of \(R(t)\), the accumulative distribution function (CDF), \(F(f)\), might also be defined as:  
\[F(t)=1-R(t) \tag{2}\]  
For individuals in a population has a density function \(f(t)\). The equivalent distribution function is the fraction of the population failing by the time \(t\). That is,  
\[F(t)=\int_{-\infty}^{t}f(s)ds \tag{3}\]  
The failure rate function \(\lambda(t)\) measures the instantaneous risk, in that \(\lambda(t)\delta t\) is the chance of failing in the following small interval \(\delta t\) given survival to time \(t\). From the relation we can find as follows:  
\[P(\text{survival to }t+\delta t)=P(\text{survival to }t)\,P(\text{survival to }\delta t \,\text{survival to }t) \tag{4}\]  
where \(P\) is probability  
Figure 2: Reliability Index; BX Life (\(L_{B}\)) on the bathtub. So we can obtain the following equations. That is,  
\[1-F(t+\delta t)=\{1-F(t)\}\{1-\lambda(t)\delta(t)\} \tag{5a}\] \[\delta t\Gamma^{\prime}(t)=\{1-F(t)\}\lambda(t)\delta(t) \tag{5b}\]  
So, the failure rate function is expressed as follows:  
\[\lambda(t)=\delta(t)F^{\prime}(t)/\{1-F(t)\}\delta(t)=f(t)/\{1-F(t)\}=f(t)/R(t) \tag{6a}\] \[\text{or}\;\lambda(t)=\frac{f(t)}{R(t)}=\frac{dF(t)/dt}{R(t)}=\frac{(1-R(t)) ^{{}^{\prime}}}{R(t)}=\frac{-R^{\prime}(t)}{R(t)} \tag{6b}\]  
If Equation (6b) is integrated with respect to time, the X% accumulative failure \(F(L_{B})\) at \(BX\) life, \(L_{B}\) may be estimated. \[F=\int\lambda(t)dt=-lnR(t)\] (7a) In other words, we might state \[F(t)=1-R(t) \tag{2}\]0 as: \[F(t)=1-R(t) \tag{2}\]1  
Assume that \(T_{1}\) be the time of the first failure, we may express reliability function \(R(t)\). That is,  
\[F(t)=1-R(t) \tag{2}\]2  
If a commercially manufactured product follows the characteristics of the bathtub curve, it might have difficulties achieving success in the field because of the short lifetime and large failure rates early in its life due to design defects. Manufacturers will clearly improve the product design by subjecting its lifetime targets to (1) removing unexpected failures, (2) minimizing random failures for its working period, and (3) extending product lifetime. As the product design improves, the failure rates from the field decease and the product lifetime is extended. For such situations, the established bathtub curve could be converted to a simple line with low failure and longer life that only increases the failure rates toward the termination of its life. Consequently, because it is less than approximately 20 percent of the cumulative failure rates, the reliability of a mechanical system might be expressed as [27]:  
\[F(t)=1-R(t) \tag{2}\]3  
The reliability of mechanical system therefore might be achieved by estimating the objective product lifetime \(L_{B}\) and failure rate \(\lambda\) after optimally identifying the market failure by parametric ALT and modifying the design flaws (or material) of structures (Figure 3). To achieve the goal of a specific product lifetime by parametric ALT, three cases of the product module were required: (1) a changed module, (2) a newly altered module, and (3) a related module to the former design on the base of field request.