Abstract
This chapter describes various methods for reduction of uncertainties in the determination of characteristic values of random quantities (quantiles of normal and Weibull distribution, tolerance limits, linearly correlated data, interference method, Monte Carlo method, bootstrap method).
Keywords
- Random quantity
- uncertainty
- normal distribution
- Weibull distribution
- tolerance limits
- correlation
- interference method
- Monte Carlo method
- bootstrap method
The reliability and safety of engineering objects are mostly formed during the design. Every design process has three stages:
Proposal of conception,
Determination of parameters,
Setting the tolerances.
Here, stages 2 and 3 will be explained in more detail, as they are very important for reliability.
1. Determination of optimum parameters — Robust design
After the concept of the construction (an engine, a bridge, a transmitter, etc.) has been proposed, it is necessary to determine all important parameters. However, input quantities often vary or can attain values different from those assumed in design. Good design ensures that the important output quantities will always lie within the allowable limits. This can be achieved by a suitable choice of nominal values of input quantities and by setting their tolerances.
The nominal values of input quantities form together the
2. Sensitivity analysis
After the design point has been found, the sensitivity analysis could be made to show the influence of the variations of input variables on the variability of the output [8]. The results may be used for setting the tolerances of input quantities to keep the output in the allowable range. The sensitivity analysis can be done using analytical expressions or simulation methods. The analytical expression for the output variable
is known exactly only in simple cases (e.g. resonant frequency of an oscillator or deflection of a beam). Often, the
The sensitivity analysis is usually done in two steps. First, the influence of individual variables is investigated. Several groups of computations are carried out, and in each group, only one variable (
or
the latter expression characterizes the changes of
In this case, linear approximation of the response function may be used, which yields simple expressions. The
For linear approximation, the sensitivity coefficients
where
Generally, two kinds of sensitivity analysis can be made: (1) deterministic, which assumes that the deviations of individual quantities from nominal values have constant magnitude, and (2) stochastic, which assumes the random scatter of individual input quantities around their nominal values.
Both approaches will be illustrated on an example [9]. A cantilever flat spring of rectangular cross-section (Fig. 2) should be used in a precise measuring device. It is necessary to get an idea how the deviations of its individual dimensions and material properties from the nominal values will influence its compliance. The spring compliance
The increments of
where ∂
In our example with the spring, the partial derivative of Equation (6) with respect to the first variable (
and the increment of compliance due to a small increment of the beam length ∆
The formulas for other variables are obtained in a similar way. The resultant expression, involving the changes of all variables, is
and the relative sensitivity of the stiffness is
This formula shows the influence of individual quantities. If the spring will be longer by 1% than the nominal value, the compliance will be higher by 3%; if the elastic modulus
This preliminary analysis reveals which input quantities have very small influence on the variability of the output quantity
The above approach is acceptable if the response function is linear or if the errors due to approximation by linear function are small. If the response function is nonlinear and the investigated ranges of input quantities are not small, the errors will not be negligible (Fig. 3). In such case, it is better to study the influence of deviations of input quantities by modeling the response without simplifications. For example, the influence of
The influence of random variability of input quantities can be investigated using the formula for the scatter of a function of several random variables. For small scatter,
where
the scatter is
The individual components,
The expression obtained by dividing Equation (10) or (12) by the total scatter
The square root of scatter (10) is the standard deviation
the + or – sign corresponds to the upper (or lower) confidence limit and
If Formula (12) is applied on the above example with a spring, one obtains the following expression for the standard deviation of the compliance caused, for example, by random variability of the length
cf. Equation (11). Similar expressions can be written for other variables. The random variability of all input quantities causes the following variability of the spring compliance:
The ratio of the standard deviation of a quantity and its mean is the variation coefficient,
so that the combination of Equations (16) and (17) gives the variation coefficient of the compliance,
The above approach, based on the linearization of the response function, is suitable for small values of variance coefficients of input quantities, say
The approximate value of the total scatter is obtained by summing up the partial scatters,
More accurate value is obtained if all input variables,
3. Determination of tolerances of input quantities
If the variability or deviation of the output quantity
If the deviation of
For example, the allowable length tolerance of the above spring, ensuring the compliance tolerance ∆
The tolerances of other quantities can be determined in similar way. One must respect that the deviations of some input quantities influence the output in one direction, whereas the deviations of other quantities can have the opposite influence. Generally, the deviations of
The following analysis assumes that the range of probable occurrence of
Often, the influence of one factor prevails (e.g.
If several input variables vary, one must decide, which of them should be reduced. As the standard deviation equals the square root of the scatter, it is obvious that the reduction of scatter of a quantity, contributing to the total scatter by only 5% to 10%, will have negligible effect. Also, the costs of the pertinent improving operation must be considered, as they usually increase with tightening the tolerances.
After having obtained the corrected standard deviation
The above optimization can be performed even if the scatter
Often, the scatter of some input quantities cannot be changed continuously. In such cases, the response must be evaluated for each possible value of every discontinuous quantity.
The determination of suitable tolerances will be illustrated on the following example, adapted from [9].
A cantilever microbeam from Figure 2, with length
Solution. The corresponding reduced variation coefficient is
With the variation coefficient of elastic modulus unchanged,
4. Uncertainties in ensuring safety and lifetime using proof testing
If the high reliability of a certain object must be ensured, a proof-test is often used: the component is exposed to some overload, specified so that only sufficiently strong components survive it; the weaker ones are destroyed. In the same way, sufficient lifetime can be ensured for components made of brittle materials suffering by static fatigue. The minimum time to failure of a component that has passed a proof-test is [10 - 12]
where
However, KIC, N, and A were determined by measurement and are known only approximately and Y was estimated. Therefore, it is recommended to perform sensitivity analysis and correct the proof stress appropriately. The pertinent theory, based on probabilistic analysis, is explained in [11, 12] or in [10]. For easier application, strength-probability-time diagrams were developed [13 – 15], in which the necessary proof stress can be found for the demanded time to failure and confidence level.
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