Maintenance Strategy and Condition Monitoring

Problem Statement: **Primary relation:** `Simplified cost rate for planned replacement at age τ: [C_p R(τ)+C_f(1-R(τ))]/∫_0^τ R(t)dt, assuming renewal at planned or first failure.` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** If planned replacement costs 100 units and failure replacement costs 1,000, the tradeoff depends on Weibull shape; constant hazard alone provides little rationale for age replacement. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Plan a falsification test A good explanation can be disproved. State an observation that would contradict the proposed failure mechanism, and arrange measurements so another engineer could reach an independent conclusion rather than merely agreeing with the initial diagnosis. ## 3. Where the model can fail The stated expression assumes the replacement restores an as-new condition, includes the chosen failure cost and measures operating exposure; prove these assumptions before optimization. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.

Problem Statement: **Primary relation:** `Mean downtime per event=total counted outage duration/number of counted restoration events.` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** A failed motor takes 4 hours to replace but 42 hours to receive. If the outage clock includes logistics, the service restoration duration is 46 hours, not 4. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Define the denominator Write down the reference population and the interval across which the observation has meaning. An event per machine-day differs from an event per operating hour if production schedules differ. A simulation that silently uses an assumed denominator produces a neat but potentially false comparison. ## 3. Where the model can fail Restoration can be incomplete if a bypass is left in place or functional testing is skipped; define an acceptable return-to-service criterion. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.

Problem Statement: **Primary relation:** `For discrete centered samples x_i, RMS=sqrt((1/n)Σ x_i²); units depend on whether x is displacement, velocity or acceleration.` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** Four velocity samples 1, -1, 2, -2 mm/s have RMS=sqrt(2.5)=1.581 mm/s. This number alone does not identify the machine condition. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Keep the physical mechanism visible An apparent trend in a chart is evidence about recorded variables, not automatically the explanation of a component failure. Ask which loads, contamination pathways, temperatures, controls, maintenance practices or environmental stresses could produce the observed pattern. ## 3. Where the model can fail

Problem Statement: **Primary relation:** `Temperature rise ΔT=T_location-T_reference is meaningful only when emissivity, load, ambient and viewing conditions are comparable.` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** A bearing housing is 15 °C warmer than an adjacent identical housing. Compare actual loads, insulation and sensor settings before declaring imminent bearing failure. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Separate requirement from measurement An engineer should be able to state the acceptance requirement before seeing the measurement. Otherwise thresholds tend to move after results are known. Preserve the original requirement, report any exception and obtain authorized approval for changes. ## 3. Where the model can fail An infrared camera reports apparent radiance, not direct internal material temperature. Inaccessible or reflective surfaces create misleading readings. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.

Problem Statement: **Primary relation:** `A linear degradation extrapolation t_remaining=(limit-current)/slope is illustrative and valid only while the trend assumptions remain plausible.` The relation should first be read in words and translated into a dimensional statement. Identify which quantities are probabilities, time or cycle measures, rates, costs, loads, engineering scores or empirical indicators. If an input is a probability, its feasible interval is from zero to one; if it is an event intensity, it must be nonnegative and accompanied by a unit such as failures per operating hour. The output should be interpreted at the same stated boundary and over the same mission or reporting period as its inputs. **Reasoning sequence.** (i) Define the asset and the function; (ii) write all inputs and their units; (iii) choose the simplest physically defensible model; (iv) perform the computation using an independent check; (v) compare the result with limiting cases; (vi) state uncertainty and model boundaries before recommending maintenance. For example, the limit of a no-failure event probability as an assumed event rate approaches zero should be consistent with ordinary experience. A result outside its physically possible range suggests either an arithmetic fault, incompatible units or a model used outside its assumptions. **Worked engineering example.** A normalized vibration indicator rises from 2 to 5 in 100 hours and alarm limit is 8. A linear slope 0.03 units/hour predicts 100 hours to threshold, ignoring uncertainty and regime change. After evaluating the arithmetic, report what the result means and—equally important—what it does not mean. Distinguish a hypothetical prediction from a measured population statistic. If a numerical example uses a notional fleet or synthetic data, it must not appear in an operator's work package as a measured site result. For any policy comparison, repeat the calculation using at least two plausible ranges for its most uncertain input and report whether the choice changes. ### Distinguish data from decisions A probability is an input to a decision rather than a decision by itself. Two alternatives with equal failure probability can lead to different responses when consequences, restoration time and regulatory obligations are different. ## 3. Where the model can fail Triggering replacement on every noisy excursion creates false positives and unnecessary downtime; hysteresis or sustained exceedance may help but must not hide dangerous events. An important diagnostic practice is to seek evidence against the simplest explanation. A repair which temporarily restores output can disguise an uncorrected design, installation or operating error. If the apparent rate improves after a maintenance change, normalize against exposure and examine whether production load, weather, equipment population or reporting conventions changed in the same period. Correlation between a signal and a failure does not establish that intervening on the signal will prevent the failure.