Robust Design, Sensitivity Analysis, and Tolerance Setting

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).

The reliability and safety of engineering objects are mostly formed during the design. Every design process has three stages:

1. Proposal of conception,

2. Determination of parameters,

3. 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 design point. Its position should ensure the low sensitivity of the output parameters to the deviations of input quantities from nominal values. This is called robust design[1]. Figure 1 illustrates its principle on an example with one input variable x: the design point 1 is with high sensitivity, whereas point 2 is with low sensitivity. One can see that the changes of the output quantity yaround point 2 are much smaller than around point 1, in both cases for the same changes of x. This also means that acceptable scatter of ycan sometimes be achieved with lower demands on the accuracy of input parameters. The reliability is influenced not only by the scatter of input quantities, but also by the position of design point. The ideal position, with the lowest sensitivity to the parameter variations, corresponds to an extreme of the response function y= f(x₁, x₂, ..., x_n). Various optimization methods exist for finding this position, analytical or with computer modeling. Universal is the ”simplex method“, where the input variables approach the optimum step-by-step according to a simple algorithm [2, 3]. The graphical representation of the response is very informative. Also, the procedures of design of experiments(DOE) are suitable; see books by G. Taguchi and other authors [4 - 7]. The determination of optimum parameters should go hand in hand with the sensitivity analysis.

Figure 1.

Principle of robust design. Note the influence of the position of the design point on the sensitivity of the output (∆y) to variations of input variablex.

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 y,

$$y=f(x_1,x_2,\mathrm{...},x_n),$$E1

is known exactly only in simple cases (e.g. resonant frequency of an oscillator or deflection of a beam). Often, the response functionmust be found by numerical solution (e.g. using the finite element method). Then, an approximate expression is obtained by regression fitting the response computed for several combinations of input variables (Fig. 3 in Chapter 15).

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 (x_i) is changed, whereas the others keep their nominal values x_1,0, x_2,0, ..., x_n,0, corresponding to the design point. Then, the yvalues for the individual groups are fitted by a suitable regression function (e.g. a polynomial),

$$y_i=a_0+a_{i1}{x}_i+a_{i2}{x}_i{}^2+a_{i3}{x}_i{}^3+\mathrm{....},$$E2

or

$$y_i=y_0+a_i(x_i-x_{i,0})+b_i{(x_i-x_{i,0})}^2+\mathrm{....},$$E3

the latter expression characterizes the changes of yas a function of deviations of the i-th input variable from the design point. These regression functions correspond to the cuts through the response surface (Fig. 3 in Chapter 15). The sensitivity analysis will depend on whether the deviations are small or large.

Small changes of the input and output quantities

In this case, linear approximation of the response function may be used, which yields simple expressions. The sensitivity of the responseto the variations of individual variables is obtained from partial derivatives at the pertinent point,

$$c_i=\partial \text{}y/\partial \text{}{x}_i\approx \mathrm{\Delta }y/\mathrm{\Delta }{x}_i.$$E4

For linear approximation, the sensitivity coefficients c_icorrespond to the constants a_i,1in (2) and a_iin (3). Further information is obtained from relative sensitivities,

$$c_{ri}=\frac{\partial y}{\partial x_i}\frac{x_{i,\text{}0}}{y_0}\approx \frac{\mathrm{\Delta }y}{y_0}/\frac{\mathrm{\Delta }{x}_i}{x_{i,0}},$$E5

where y₀and x_i,0are the values corresponding to the design point. Coefficient c_riexpresses the change of y(in %, for example) caused by 1% deviation of x_ifrom the nominal value x_i,0. For linear approximation, c_r,i= a_i(x_i,0/y₀).

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 Cis given by the formula:

$$C=y/F=4L^3/\left(Ew{t}^3\right);$$E6

yis deflection, Fis load, Lis length, Eis elastic modulus, wis spring width, and tis spring thickness.

Figure 2.

Spring for a measuring device (a schematic).

Deterministic analysis for small deviations

The increments of yare calculated via the first derivatives. The response surface is replaced by a tangent plane at the investigated point. For y= f(x₁, x₂,..., x_n), the infinitesimal increment of ycan be expressed generally as

$$\text{d}y=(\partial y/\partial x_1)\text{d}{x}_1+(\partial y/\partial x_2)\text{d}{x}_2+\mathrm{...}+(\partial y/\partial x_n)\text{d}{x}_n,$$E7

where ∂y/∂x₁ expresses partial derivatives. For practical reasons, the differentials are replaced by small finite increments ∆,

$$\mathrm{\Delta }y=(\partial y/\partial x_1)\mathrm{\Delta }{x}_1+(\partial y/\partial x_2)\mathrm{\Delta }{x}_2+\mathrm{...}+(\partial y/\partial x_n)\mathrm{\Delta }{x}_n.$$E8

In our example with the spring, the partial derivative of Equation (6) with respect to the first variable (x₁ = L) is

$$\partial C/\partial L=3L^2\times 4/\left(Ew{t}^3\right)=\left[4L^3/\left(Ew{t}^3\right)\right]\times 3/L=\left(3/L\right)\times C,$$E9

and the increment of compliance due to a small increment of the beam length ∆Lis thus

$$\mathrm{\Delta }C=3C\left(\mathrm{\Delta }L/L\right).$$E10

The formulas for other variables are obtained in a similar way. The resultant expression, involving the changes of all variables, is

$$\mathrm{\Delta }C=C(3\mathrm{\Delta }L/L–\mathrm{\Delta }E/E–\mathrm{\Delta }w/w–3\mathrm{\Delta }t/t),$$E11

and the relative sensitivity of the stiffness is

$$\mathrm{\Delta }C/C=3\mathrm{\Delta }L/L–\mathrm{\Delta }E/E–\mathrm{\Delta }w/w–3\mathrm{\Delta }t/t.$$E12

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 Ewill be higher by 1%, the compliance will be lower by 1%, etc. The constants at individual terms correspond to their exponents in Equation (6), and the signs depend on whether the quantity was in the numerator or denominator.

This preliminary analysis reveals which input quantities have very small influence on the variability of the output quantity yand may thus be considered as constants in the following analysis of simultaneous random variance of the input quantities. However, one must always keep in mind that the variance of the output depends on both the sensitivity c_i and the variance of the pertinent input quantity x_i.

Deterministic analysis for large deviations

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 j-th variable can be studied from Equation (1), in which only x_j varies, whereas the others keep their values corresponding to the design point.

Figure 3.

Error caused by linear approximation of response functiony=f(x).

Influence of random variability – small scatter

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,

$$s_y{}^2={\left(\frac{\partial \text{}y}{\partial x_1}\right)}^2{s}_{x1}{}^2+{\left(\frac{\partial \text{}y}{\partial x_2}\right)}^2{s}_{x2}{}^2+\mathrm{...}+2\left(\frac{\partial \text{}y}{\partial x_1}\right)\left(\frac{\partial \text{}y}{\partial x_2}\right)\mathrm{cov}(x_1,x_2)+\mathrm{...},$$E13

where s_xj² is the scatter of the j-th variable (quadrate of standard deviation). The far right-hand term is nonzero if the variables are correlated; often, it can be omitted. For linear approximation of y,

$$y=a_0+a_1{x}_1+a_2{x}_2+\mathrm{...}+a_n{x}_n,$$E14

the scatter is

$$s_y{}^2=a_1{}^2{s}_{x1}{}^2+a_2{}^2{s}_{x2}{}^2+\mathrm{...}+a_n{}^2{s}_{xn}{}^2+\mathrm{...}$$E15

The individual components, s_yj² = a_j²s_xj², give the scatter of ycaused by random variations of j-th variable. Similarly to deterministic analysis, the contribution of a certain variable x_jto the total scatter is larger for large scatter of this variable (s_xj²) and for large sensitivity (a_j) of the output yto its changes.

The expression obtained by dividing Equation (10) or (12) by the total scatter s_y² gives the relative proportions of individual factors in the total scatter,

$$1=a_1{}^2\frac{s_{x1}{}^2}{s_y{}^2}+a_2{}^2\frac{s_{x2}{}^2}{s_y{}^2}+\mathrm{...}+a_n{}^2\frac{s_{xn}{}^2}{s_y{}^2}+\mathrm{...}$$E16

The square root of scatter (10) is the standard deviation s_y. If the input quantities have normal distribution, the confidence interval for the output quantity ycan be calculated as

$$y_{\text{lower,upper}}=y_0\pm u_{\alpha }{s}_y;$$E17

the + or – sign corresponds to the upper (or lower) confidence limit and u_αis the α-critical value of standard normal distribution. The probability that ywill lie out of these limits is 2α.

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 L:

$$s_{CL}=3C(s_L/L);$$E18

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:

$$s_C={[{\left(3C/L\right)}^2{s}_L{}^2+{\left(C/E\right)}^2{s}_E{}^2+{\left(C/w\right)}^2{s}_w{}^2+{\left(3C/t\right)}^2{s}_t{}^2]}^{1/2}.$$E19

The ratio of the standard deviation of a quantity and its mean is the variation coefficient,

$$v=\sigma /\mu ,$$E20

so that the combination of Equations (16) and (17) gives the variation coefficient of the compliance,

$$v_C=s_C/C={[9v_L{}^2+v_E{}^2+v_w{}^2+9v_t{}^2]}^{1/2}.$$E21

Stochastic analysis for large scatter

The above approach, based on the linearization of the response function, is suitable for small values of variance coefficients of input quantities, say v_j ≤ 10%. If their scatter is large, it is better to study the influence of variability or deviations of input quantities by the Monte Carlo simulation method. A preliminary assessment consists of making msimulation experiments with random variable only x_j, for j= 1, 2,... n, and then calculating partial scatter s_yj² of the obtained values y. Using the characteristics s_xi, x_i,0, and y₀,one can determine the variation coefficients v_j or the sensitivity coefficients a_j(= s_y/s_xj).

The approximate value of the total scatter is obtained by summing up the partial scatters,

$$s_y{}^2=s_{y1}{}^2+s_{y2}{}^2+\mathrm{...}+s_{yn}{}^2+\mathrm{...}$$E22

More accurate value is obtained if all input variables, x₁, x₂, ..., x_nare considered as random in the Monte Carlo simulations, and the scatter is calculated from all values y_j. Dividing Equation (19) by the total scatter s_y²gives the relative influence of individual factors, like in Equation (13).

3. Determination of tolerances of input quantities

If the variability or deviation of the output quantity yfrom the nominal value is larger than allowed, it must be reduced. The procedure depends on whether the variability is random or deterministic.

Deterministic deviations

If the deviation of yis caused by the deviation of one or more input quantities, Equation (12) or (12), showing the contribution of individual factors to the total deviation of y, can be used to decide which factor should be aimed at. Let us assume that the deviation of yin Equation (12) is caused only by the deviation of x_j. The allowable magnitude of ∆x_j, ensuring that the deviation of ydoes not exceed ∆y, is

$$\mathrm{\Delta }{x}_j\le \mathrm{\Delta }y/\left(\partial y/\partial x_j\right).$$E23

For example, the allowable length tolerance of the above spring, ensuring the compliance tolerance ∆C, is

$$\mathrm{\Delta }L=\mathrm{\Delta }C\left(L/3C\right),or\mathrm{\Delta }L/L=\left(1/3\right)\left(\mathrm{\Delta }C/C\right).$$E24

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 ydepend on the deviations of input quantities and also on the sensitivity of yto the changes of x_j. The reduction of the tolerance of ycan thus be accomplished by tightening the tolerances of individual input quantities or by changing the position of the design point towards lower sensitivity. The decision will also depend on the costs related to the individual adjustments.

Random variability of input quantities

The following analysis assumes that the range of probable occurrence of y(i.e. the half-width ∆y_αof the α-confidence interval for y) is directly proportional to the standard deviation s_y, equal to the square root of the scatter. In production, the allowable limits of a quantity xare also often determined as x_nom± ks_x, where kis a constant (e.g. a suitable quantile of standard normal distribution). With this assumption, the tolerance of ycan be reduced from ∆y_αto ∆y_α'by reducing the standard deviation of yfrom the original value s_yto s_y'. This may be accomplished by the reduction of the variance or influence of input factors.

Often, the influence of one factor prevails (e.g. x_k). In such case, most of the scatter of ycan be reduced by reducing its component due to this factor. As it follows from Equation (12), the scatter of ycan be reduced by reducing the standard deviation s_x,kor the sensitivity of yto the changes of x_k(coefficient a_k). The reduction of variance of y_kcan be attained by more accurate manufacturing or by better control and sorting out the components that are out of the tolerance limits. The reduction of sensitivity of yto the changes of x_kcan be accomplished by changing the parameters of the design point (Fig. 1). An example is a prestressed flange connection in steam turbines: the use of long bolts increases the compliance of the joint and reduces the sensitivity of the prestress to the variations of pressure in the pipe and thermal dilatations of the flanges. Sometimes, both ways, the reduction of s_x,kand a_kis combined.

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 s_xi', the lower (L) and upper (U) allowable limit for the input quantity x_jcan be determined as

$$x_{i,L,U}=x_{i,0}\pm k_{L,U}{s}_{xi},$$E25

x_j,0is the nominal (design) value and kis a constant (e.g. 5% quantile of standard normal distribution). k_Lcorresponds to the lower limit, whereas k_Ucorresponds to the upper limit.

The above optimization can be performed even if the scatter s_y² from the preliminary design is smaller than the allowable value. The optimization assigns such tolerances that the total costs are minimal. Sometimes, the tolerances may even be made wider, with lower costs.

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].

Example 1

A cantilever microbeam from Figure 2, with length L= 10 mm, width w= 1.0 mm and thickness t= 50 μm, made of a material with elastic modulus E= 200 GPa, has compliance C= 4L³/(Ewt³) = 0.16 mm/mN. Each input quantity has coefficient of variation v_l= v_w= v_t= v_E= v= 0.01 = 1%. The variation coefficient of the compliance, Equation (17), is v_C= (9v_L² + v_E² + v_w² + 9v_t²)^1/2 = 0.0447, and the standard deviation s_C= Cv_C= 0.00716 mm/mN. Such variation of compliance is unacceptably high and must be reduced to s_C' = 0.004 mm/mN.

Solution. The corresponding reduced variation coefficient is v_C' = s_C'/C= 0.004/0.16 = 0.025. It is possible to reduce the scatter of L, w, and t; the material (E) remains unchanged. The easiest way is to reduce the scatter of L. However, even if this scatter were zero, the variation coefficient of compliance would be v_C' = 0.033, which is much more than demanded. Therefore, the variance of all three quantities (L, w, and t) must be reduced by more accurate manufacturing. If the new variation coefficients of L, w, and twould have the same value, v_L' = v_w' = v_t' = v', this value v' can be calculated from the modified Equation (18):

$$v_C\text{'}=[9v{\text{'}}^2+v_E{}^2+v{\text{'}}^2+9v{\text{'}}^2){]}^{1/2}.$$E26

With the variation coefficient of elastic modulus unchanged, v_E = 0.01, the new coefficients of variation of L, w, and tmust be reduced to v' ≤ 0.005256, which is approximately v' = 0.005. The corresponding allowable standard deviations, obtained by multiplying the variation coefficient v' by the nominal values of L,w, and t, are s_l' ≤ 0.05 mm, s_w' ≤ 0.005 mm, and s_t' ≤ 0.25 μm. In the limit case, v_C' = 0.024 and s_C' = 0.0038 m/N. However, the tolerances of individual dimensions could be adjusted with respect to the manufacturing possibilities, the principal condition being s_C' ≤ 0.004 m/N.

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]

$$t_{\min }=\frac{2{\sigma }_{pt}{}^{N-2}}{(N-2)A{Y}^2{K}_{IC}{}^{N-2}{\sigma }_0{}^N},$$E27

where K_IC is the fracture toughness of the material, Nand Aare the parameters of subcritical crack growth, Yis the geometrical factor of typical crack, responsible for fatigue failure, σ₀ is the characteristic operational stress (assumed constant), and σ_pt is the proof-test stress. A rearrangement of Equation (24) gives the formula for the proof stress needed to guarantee the minimum lifetime:

$${\sigma }_{pt}={\sigma }_0{}^{N/(N-2)}{K}_{IC}{\left[\frac{N-2}{2}A{Y}^2{t}_{\min }\right]}^{1/(N-2)}.$$E28

However, K_IC, 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.