(meca_9_8)= # Examples The following paragraphs provide some examples for the application of the methods introduced in {numref}`meca_9_6` and {numref}`meca_9_7`. (meca_9_8_1)= ## Simplified wear modelling: solid lubricant wear In this example, the simplified method presented in {numref}`meca_9_7_1` is utilized to model the reliability of a bearing. The considered {term}`failure mechanism ` is solid lubricant wear. (meca_example_1_given)= `````{dropdown} **Given** A solid lubricated ball bearing is considered. The basic variables, necessary to model the failure mechanism, are modelled as follows: ```{list-table} Variables considered in the example :name: meca-examples-table-1 * -

List of variables		Unit	Distribution	Expected value	CoV
\(V_{\text{limit}}\)	Limiting value (worn volume)	\(m^3\)	Lognormal	\(6\cdot 10^{-8}\)	\(0.2\)
\(n_P\)	Number of phases / time intervals	-	Deterministic	\(1\)	-
\(K_{H,i}\)	Specific wear rate in interval \(i\)	\(Pa^{-1}=N/m^2\)	Lognormal		\(0.6\)
\(\alpha_i\)	Ball-cage interaction in interval \(i\)	\(Nm\)	Lognormal	\(0.033\)	\(0.1\)
\(\text{rev}_i\)	Number of revolutions in interval \(i\)	-	Deterministic	\(10^{6}\)	-
\(\theta\)	Model uncertainty	-	Lognormal	\(1.2\)	\(0.2\)

``` ```{admonition} Note :class: note The values indicated in the following table serve as an illustrative example. ``` In this example, it is assumed that only one mission phase is relevant, i.e. $n_P=1$. ````` (meca_example_1_rel_model)= `````{dropdown} **Reliability model** To estimate the probability of failure with the simplified methods, the following need to be known: * The expected value of the limiting volume, * The CoV of the limiting volume, * The expected value of the volume worn away * The CoV of the volume worn away. According to Eq. {eq}`Equation_3_20`, the expected value of the limiting volume is: ````{admonition} Equation :class: equation ```{math} :label: eq_example_1 \text{E}\left[ X_1 \right] = \text{E}\left[ V_{\text{limit}}\right ] = 6\cdot 10^{-8} ``` ```` According to Eq. {eq}`Equation_3_22`, the coefficient of variation of the limiting value is: ````{admonition} Equation :class: equation ```{math} :label: eq_example_2 v_{X_1}=v_{V_{\text{limit}}}=\frac{\sqrt{\text{Var}\left[V_{\text{limit}}\right]}}{\text{E}\left[V_{\text{limit}}\right]}=\frac{\text{E}\left[V_{\text{limit}}\right]\cdot0.2}{\text{E}\left[V_{\text{limit}}\right]}=0.2 ``` ```` To calculate the CoV of the volume worn away, the covariance of $K_H$ and $\alpha$ is calculated under the assumption of full correlation between $K_H$ and $\alpha$: ````{admonition} Equation :class: equation ```{math} :label: eq_example_3 \text{Cov}\left[K_H,\alpha\right]&= \sigma_{K_H}\cdot\sigma_\alpha=v_{K_H}\cdot \text{E}\left[K_H\right]\cdot v_\alpha\cdot \text{E}\left[\alpha\right]\\ &=0.6\cdot 4\cdot 10^{-15}\cdot 0.1\cdot 0.033=7.92 \cdot 10^{-18} ``` ```` The variance of the product $K_H\cdot\alpha$ is calculated as: ````{admonition} Equation :class: equation ```{math} :label: eq_example_4 \text{Var}\left[K_H\cdot\alpha\right]&=Var_{K_H}\cdot \text{Var}_\alpha+Var_{K_H}E\left[\alpha\right]^2+Var_\alpha E\left[K_H\right]^2\\ &=5.76\cdot10^{-30}\cdot1.1\cdot10^{-5}+5.76\cdot10^{-30}\cdot\left(0.033\right)^2+1.1\cdot10^{-5}\cdot\left(4\cdot10^{-15}\right)^2\\ &=6.51\cdot10^{-33} ``` ```` According to Eq. {eq}`Equation_3_22`, the expected value of the volume worn away is: ````{admonition} Equation :class: equation ```{math} :label: eq_example_5 \text{E}\left[X_2\right]&=\text{E}\left[K_H\cdot\alpha\right]\cdot \text{rev}=\left(\text{E}\left[K_H\right]\cdot \text{E}\left[\alpha\right]+\text{Cov}\left[K_H,\alpha\right]\right)\cdot \text{rev}\\ &=\left(4\cdot10^{-15}\cdot0.033+7.92\cdot10^{-18}\right)10^6\\ &=7\cdot10^{-8} ``` ```` And the CoV of the volume worn away is according to Eq. {eq}`Equation_3_23`: ````{admonition} Equation :class: equation ```{math} :label: eq_example_6 v_{X_2}=\frac{1}{\text{E}\left[X_2\right]}\cdot \text{rev}\cdot\sqrt{Var\left[K_H\cdot\alpha\right]}=\frac{1}{1.4\cdot10^{-10}}\cdot10^6\cdot\sqrt{6.51\cdot10^{-33}}=0.577 ``` ```` Finally, the probability of failure is calculated according to Eq. {eq}`Equation_3_24`: ````{admonition} Equation :class: equation ```{math} :label: eq_example_7 P_f&=P\left[X_1-X_2\cdot\Theta\le0\right]\\ &=\Phi\left(\frac{ln\left(E\left[\Theta\right]\right)-ln\left(E\left[X_1\right]/E\left[X_2\right]\right)+0.5\cdot\left(ln\left(v_{X_1}^2+1\right)-ln\left(v_{X_2}^2+1\right)-ln\left(v_\Theta^2+1\right)\right)}{\sqrt{ln\left(v_{X_1}^2+1\right)+ln\left(v_{X_2}^2+1\right)+ln\left(v_\Theta^2+1\right)}}\right)\\ &=\Phi\left(\frac{ln\left(1.2\right)-ln\left(6\cdot10^{-8}/1.4\cdot10^{-10}\right)+0.5\cdot\left(ln\left(0.2^2+1\right)-ln\left(0.577^2+1\right)-ln\left(0.2^2+1\right)\right)}{\sqrt{ln\left(0.2^2+1\right)+ln\left(0.577^2+1\right)+ln\left(0.2^2+1\right)}}\right)\\ &=\Phi\left(-2.709\right)=0.0034, ``` Where, $\Phi$ denotes the cumulative standard normal distribution. ```` ````` (meca_9_8_2)= ## Updating of reliability estimates derived from structural reliability methods This example is based on the example provided in {numref}`meca_9_8_1`, i.e. it considers a bearing that fails from solid lubricant wear. The structural reliability model is established with the simplified methods in {numref}`meca_9_7_1`. Upon availability of new data on the reliability of the bearing in question, the model is updated making use of the approach described in {numref}`meca_9_6_5`. (meca_example_2_given)= `````{dropdown} **Given** A solid lubricated ball bearing is considered. The basic variables, necessary to establish a reliability model in accordance with the simplified method described in {numref}`meca_9_6_2`, are modelled as follows: From {numref}`meca_9_7_1`, it follows: ````{admonition} Equation :class: equation ```{math} :label: eq_example_8 \text{E}\left[ X_1 \right] = \text{E}\left[ V_{\text{limit}}\right ], ``` ```` And, under the assumption of a single mission phase: ````{admonition} Equation :class: equation ```{math} :label: eq_example_9 \text{E}\left[ X_2 \right] = \text{E}\left[ K_H\cdot\alpha \right]\cdot\text{rev} = (\text{E}\left[ K_H\right]\cdot\text{E}\left[ \alpha\right] + \text{Cov}\left[ K_H, \alpha\right])\cdot\text{rev}. ``` ```` ````` (meca_example_2_prior)= `````{dropdown} **Prior model** The probability of failure in function of the number of revolutions $\text{rev}$ is modelled. According to {numref}`meca_9_6_5`, it can be approximated with the lognormal distribution: ````{admonition} Equation :class: equation ```{math} :label: eq_example_10 \text{rev}\sim\text{Lognormal}(\mu_{\text{rev}, \sigma_{\text{rev}}}). ``` ```` The CoV $v_{X_1}$ and $v_{X_2}$ have already been determined in the example in {numref}`meca_9_8_1`. ````{admonition} Equation :class: equation ```{math} :label: eq_example_11 \sigma_{\text{rev}}=\sqrt{ln\left(v_{X_1}^2+1\right)+ln\left(v_{X_2}^2+1\right)}=\sqrt{ln\left(0.2^2+1\right)+ln\left(0.577^2+1\right)}=0.571. ``` ```` The location parameter $\mu_{\text{rev}}$ is considered uncertain and is modelled with a normal distribution (conjugate prior, see {ref}`Part 2 - Methods `). ````{admonition} Equation :class: equation ```{math} :label: eq_example_12 \mu_{\text{rev}}\sim\text{Normal}(\mu',\sigma') ``` ```` According to {numref}`meca_9_6_5`, $p=\frac{E\left[X_1\right]}{E\left[X_2\right]}$ should be brought to the form $p=k\cdot\frac{1}{rev}$. Using the relations in Eq. {eq}`Equation_3_24` and Eq. {eq}`Equation_3_28`: ````{admonition} Equation :class: equation ```{math} :label: eq_example_13 p=\frac{E\left[X_1\right]}{E\left[X_2\right]}=\frac{E\left[V_{\text{limit}}\right]}{E\left[K_H\cdot\alpha\right]\cdot \text{rev}}, ``` ```{math} :label: eq_example_14 k=\frac{E\left[V_{\text{limit}}\right]}{E\left[K_H\cdot\alpha\right]}=\frac{6\cdot10^{-8}}{1.40\cdot10^{-16}}=4.29\cdot10^8. ``` ```` The prior hyperparameters for the distribution of $\mu_{\text{rev}}$ are estimated according to Eq. {eq}`Equation_3_13` and Eq. {eq}`Equation_3_14`: ````{admonition} Equation :class: equation ```{math} :label: eq_example_15 \mu\prime&=ln\left(k\right)-ln\left(E\left[\Theta\right]\right)+0.5\cdot\left(ln\left(v_\Theta^2+1\right)-ln\left(v_{X_1}^2+1\right)+ln\left(v_{X_2}^2+1\right)\right)\\ &=ln\left(4.29\cdot10^8\right)-ln\left(1.2\right)+0.5\cdot\left(ln\left(0.2^2+1\right)-ln\left(0.2^2+1\right)+ln\left(0.577^2+1\right)\right)\\ &=19.84 ``` ```{math} :label: eq_example_16 \sigma\prime=\sqrt{ln\left(v_\Theta^2+1\right)}=\sqrt{ln\left(0.2^2+1\right)}=0.198 ``` ```` ````` (meca_example_2_data)= `````{dropdown} **Additional data** Additional data on the reliability of the bearing is given in the table below: ```{list-table} Additional data considered in the example :name: meca-examples-table-2 :header-rows: 1 :widths: 20 80 * - Specimen - Revolutions to failure $\widehat{{\text{rev}}_i}$ * - 1 - $2.5\cdot{10}^8$ * - 2 - $2.4\cdot{10}^8$ * - 3 - $2.7\cdot{10}^8$ * - 4 - $3.1\cdot{10}^8$ * - 5 - $3.4\cdot{10}^8$ * - 6 - $3.8\cdot{10}^8$ * - 7 - $5.1\cdot{10}^8$ ``` ````` (meca_example_2_updating)= `````{dropdown} **Updating** With the additional data, the reliability model for the bearing can be updated. The updating is done using the equations for the analytic approach using conjugate priors, given in {ref}`Part 2 - Methods ` of this handbook: ````{admonition} Equation :class: equation ```{math} :label: eq_example_17 \sigma\prime\prime=\frac{1}{\sqrt{\frac{1}{\left(\sigma\prime\right)^2}+\frac{n_{data}}{\left(\sigma_{rev}\right)^2}}}=\frac{1}{\sqrt{\frac{1}{\left(0.198\right)^2}+\frac{7}{\left(0.571\right)^2}}}=0.145, ``` ```{math} :label: eq_example_18 \mu\prime\prime=\left(\sigma\prime\prime\right)^2\cdot\left(\frac{\mu\prime}{\sigma\prime^2}+\frac{\sum_{i=1}^{n_{data}}ln\left({\widehat{rev}}_i\right)}{\sigma_{rev}^2}\right)=0.146\left(\frac{19.84}{\left(0198\right)^2}+\frac{137}{\left(0.571\right)^2}\right)=19.72. ``` ```` The posterior predictive of $\text{rev}$ is also a lognormal distribution. Its distribution function can be calculated with the help of the analytic formulas provided in {ref}`Part 2 - Methods ` of this handbook. ````` (meca_example_2_results)= `````{dropdown} **Results** In {numref}`Figure_mec_example_2_results_1`, the prior and posterior distribution of parameter $\mu_{\text{rev}}$ is represented. In {numref}`Figure_mec_example_2_results_2`, the prior and posterior probability of failure are shown in function of the number of revolutions $\text{rev}$. As can be seen in the figure, the probability of failure increases from the updating. ```{figure} ../../pictures/example_2_results_1.png --- width: 600px name: Figure_mec_example_2_results_1 --- Prior and posterior distribution for parameter $\mu_{\text{rev}}$ ``` ```{figure} ../../pictures/example_2_results_2.png --- width: 600px name: Figure_mec_example_2_results_2 --- Prior and posterior predictive distribution of revolutions to failure $\text{rev}$ ``` ````` (meca_example_3_censored)= ## Updating of structural reliability methods using right censored data In contrast to the example in {numref}`meca_9_8_2`, censored data is often available in practice. In this case, the simplified analytic approach for updating is no longer applicable and numerical methods should be used. In this example, {term}`MCMC` will be used to do the updating for the location parameter. To keep the example simple, the same assumption as in {numref}`meca_9_8_2` is made that the scale parameter is known and will not be updated. This assumption is not necessary using MCMC for the updating and might be relaxed for a more advanced modelling. (meca_example_3_additional_data)= `````{dropdown} **Additional data** Additional, censored, data on the reliability of the bearing is given in the table below ```{list-table} Additional censored data for the reliability of a bearing :name: meca-examples-table-3 :header-rows: 1 :widths: 20 80 * - Specimen - Revolutions to failure $\widehat{{\text{rev}}_i}$ * - 1 - $2.5\cdot{10}^8$ * - 2 - $2.4\cdot{10}^8$ * - 3 - $2.7\cdot{10}^8$ * - 4-20 - $\ge 5.1\cdot{10}^8$ ``` ````` (meca_example_3_updating)= `````{dropdown} **Updating** With the additional data, the reliability model for the bearing can be updated. The updating is done using the MCMC method described in {ref}`Part 2 - Methods `. The data set is right-censored and the likelihood $L$ for a right censored data set is given by: ````{admonition} Equation :class: equation ```{math} L\propto\ \prod_{i=1}^{n}{f\left(\widehat{x_i}|\theta\right)\cdot\prod_{j=1}^{m}\left(1-F\left(\widehat{x_{\text{up}}}|\theta\right)\right)}, ``` ```` Using the data from above ({numref}`meca-examples-table-3`), the likelihood is given by: ````{admonition} Equation :class: equation ```{math} L\propto f\left(2.5\ {10}^8|\theta\right)\cdot f\left(2.4\ {10}^8|\theta\right)\cdot\ f\left(2.7\ {10}^8|\theta\right)\cdot\left(1-F\left(3.4\ {10}^8|\theta\right)\right). ``` ```` Using the additional data, the reliability model for the bearing can be updated. The result of the updating is: ````{admonition} Equation :class: equation ```{math} \sigma\prime\prime&=0.1292,\\ \mu\prime\prime&=20.23 ``` ```` ```{figure} ../../pictures/example_3_results_1.png --- width: 600px name: Figure_mec_example_3_results_1 --- Markov chain and posterior density of the parameter ``` The posterior predictive of $\text{rev}$ is shown in {numref}`Figure_mec_example_3_results_2`. It is seen that the consideration of censored data might have a large impact on the updating and has a large potential especial for end of life decision-making. ```{figure} ../../pictures/example_3_results_2.png --- width: 600px name: Figure_mec_example_3_results_2 --- Prior and posterior distribution for parameter $\mu_{\text{rev}}$ ``` ```{figure} ../../pictures/example_3_results_3.png --- width: 600px name: Figure_mec_example_3_results_3 --- Prior and posterior predictive distribution of revolutions to failure $\text{rev}$ ``` `````