Effect of uncertainties on substructure coupling: Modelling and reduction strategies

ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 588–605

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Effect of uncertainties on substructure coupling: Modelling and reduction strategies Walter D’Ambrogio a,, Annalisa Fregolent b a b

` dell’Aquila, Piazza E. Pontieri, 1 - I-67040 Roio Poggio (AQ), Italy Dipartimento di Ingegneria Meccanica, Energetica e Gestionale, Universita ` di Roma ‘‘La Sapienza’’, Via Eudossiana, 18 - I-00184 Roma, Italy Dipartimento di Meccanica e Aeronautica, Universita

a r t i c l e i n f o

abstract

Article history: Received 21 December 2007 Received in revised form 11 July 2008 Accepted 25 July 2008 Available online 15 August 2008

Different strategies for modelling and reducing the effect of uncertainties on the dynamic behaviour of coupled structures are discussed in this paper. In assembled structures, each component satisfies specifications about material properties and dimensional tolerances. Variability of design specifications within the tolerance field may affect the dynamic behaviour of the assembled structure more than that of any individual component substructure. Among modelling strategies, non-predictive techniques such as post-processing of data from a set of randomly assembled structures and analysis of sensitivities to uncertain variables are considered. However, emphasis is placed on a less trivial yet simple strategy such as the use of design of experiments to fit a regression model. Different reduction strategies are considered according to whether small design changes (even revised tolerance allocation) are still possible or not. In the latter case, selective assembly is extended to problems where the performance requirement is not the clearance between two parts, but a prescribed dynamic behaviour. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Uncertainties Substructure coupling Design of experiment

1. Introduction Mass produced industrial structures are often built by assembling together a number of components. Similarly, each component derives from mass production and satisfies some prescribed specifications concerning material properties and dimensional tolerances. Usually, variability of design specifications within the tolerance field does not affect significantly the dynamic behaviour of any individual component substructure. However, the dynamic behaviour of the assembled structure could be much more sensitive to component variability. Uncertainties in structural systems and their effects on dynamics properties have been widely addressed in technical literature. Basically, two kinds of approaches are proposed: stochastic approaches [1] such as Monte Carlo simulation and non-probabilistic approaches [2] such as interval finite element (FE) analysis. Interval FE analysis can be extended to yield fuzzy FE analysis and interval sensitivities [3]. In the implementation of the fuzzy FE analysis, interval arithmetic can be replaced by the transformation method [4] or by computationally more efficient variants such as the short transformation method [5]. In the framework of design of experiments (DoE) [6], transformation methods could be considered as examples of factorial design. Most of the cited literature emphasises how to obtain very accurate and sophisticated models of the combined effect of uncertainties on a given structural system, i.e. a direct problem involving uncertainties. On the contrary, in this paper the  Corresponding author. Tel.: +39 0862 434 352; fax: +39 0862 434 303.

E-mail addresses: [email protected] (W. D’Ambrogio), [email protected] (A. Fregolent). 0888-3270/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.07.018

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emphasis is again on structural systems obtained by assembling together several substructures [7] which in turn bear their own uncertainties. Strategies for modelling and reducing the effect of given substructure uncertainties on the dynamics of the assembled structure are sought, i.e. an inverse problem involving uncertainties arises. Hence the need of very simple models to evaluate the effect of uncertainties, because the solution to inverse of optimisation problems often requires a large number of function evaluations. Simple but not predictive models can be obtained by post-processing of data from a set of randomly assembled structures, or by analysing sensitivities to uncertain variables. Predictive yet simple models can be obtained by using DoE, see for instance [8] where a two level factorial design is used. A regression model that accounts for quadratic and high order interaction effects can be fitted through central composite design (CCD). Different strategies for reducing the effect of uncertainties on the dynamic behaviour of coupled structures can be considered according to whether small design changes are still possible or not. In the former case, either robust design practices or revised tolerance allocation can be considered. In the latter case, selective assembly [9] can be extended to problems where the performance requirement is not the clearance between two parts, but a prescribed dynamic behaviour. Summarising, the aim of the paper is to discuss, by treating an academic simple case, techniques for modelling and reducing variability in the dynamic response of assembled structures. Modelling strategies include non-predictive techniques such as post-processing of data from a set of randomly assembled structures and analysis of sensitivities to uncertain variables. More emphasis is placed on a less trivial yet simple strategy such as the use of DoE to fit a regression model. Different reduction strategies are considered according to whether small design changes (even revised tolerance allocation) are still possible or not. The paper is organised as follows: each section deals both with theoretical and practical issues, by presenting the analysed case immediately after a more general discussion. Section 2 analyses the effect of uncertainties on the dynamics of coupled structures. Section 3 discusses how to model the effect of uncertainties and leads to introduce techniques derived from DoE. Section 4 deals with how to reduce the effects of uncertainties on the dynamics of a coupled structure, either before or after the substructures to be assembled have been produced. 2. Analysis of the effect of uncertainties A procedure to analyse the effect of uncertainties on the dynamics of coupled structures should start from  a model of the (sub)-structures to be coupled;  a definition of uncertainties on structural parameters;  a process to generate the random set of (sub)-structures to be coupled. The model of the substructures can be either numerical (e.g. FEs) or analytical (in the simplest cases). Of course, large scale FE models will be used in most practical cases. Not all strategies for modelling the effect of the uncertainties are compatible with large scale models: this will be discussed later in the paper. The definition of structural uncertainties consists in choosing which parameters are to be considered uncertain and in defining the probability distribution of uncertainties. In this respect, normal distribution is generally a good choice [10]. In some cases, e.g. when dealing with geometric dimensions, the tails of the distribution should be cut away to avoid out of tolerance items. The process to generate the random set of (sub)-structures to be coupled should preliminarily define whether the random variables, associated with the uncertain parameters, are independent or correlated. In the latter case, it should be established whether correlation exists between random variables of the same structure (e.g. length and thickness) or between homologous random variables of different (sub)-structures (e.g. thicknesses). Furthermore, the type of correlation should be established. 2.1. Analysed case A simply supported structure, assembled from three aluminium beams, is considered (Fig. 1). The middle beam behaves as an elastic joint between the two end beams. The mechanical properties of the structures are E ¼ 7  1010 N=m2 and r ¼ 2700 kg=m3 . The nominal dimensions of the items are presented in Table 1.

y

h1

h2 l1

l2

Fig. 1. Sketch of the test structure.

h3 l3

x

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Table 1 Nominal dimensions Item

Length l (mm)

Width w (mm)

Height h (mm)

1 Left beam 2 Middle beam 3 Right beam

400 50 200

40 40 40

8 3 8

2.1.1. Model of structure and uncertainties It is assumed that lengths and heights of the beams are independent random variables, distributed within prescribed dimensional tolerances. Specifically, the probability density functions of random distributions are normal with mean values equal to the nominal dimensions and standard deviations of 1 mm for lengths l1 ; l2 ; l3 and of 0.25 mm for heights h1 ; h2 ; h3 . The tolerance fields for lengths and heights are 2 and 0:5 mm, respectively, and correspond to a very coarse general tolerance class, according to ISO 2768-1. Values outside such fields are discarded. Each substructure is modelled analytically as an Euler beam, instead of using FEs. Boundary and coupling conditions are listed below, where wi represents the displacement of beam i along y direction and ki is the flexural wavenumber of beam i:  simple support of beam 1 at x ¼ 0 w1 ð0Þ ¼ 0

(1)

w001 ð0Þ ¼ 0

(2)

 coupling between beam 1 and beam 2 at x ¼ l1 w1 ðl1 Þ ¼ w2 ðl1 Þ

(3)

w01 ðl1 Þ ¼ w02 ðl1 Þ

(4)

EI1 w001 ðl1 Þ ¼ EI2 w002 ðl1 Þ

(5)

000 EI1 w000 1 ðl1 Þ ¼ EI2 w2 ðl1 Þ

(6)

 coupling between beam 2 and beam 3 at x ¼ l1 þ l2 w2 ðl1 þ l2 Þ ¼ w3 ðl1 þ l2 Þ

(7)

w02 ðl1 þ l2 Þ ¼ w03 ðl1 þ l2 Þ

(8)

EI2 w002 ðl1 þ l2 Þ ¼ EI3 w003 ðl1 þ l2 Þ

(9)

000 EI2 w000 2 ðl1 þ l2 Þ ¼ EI3 w3 ðl1 þ l2 Þ

(10)

 simple support of beam 3 at x ¼ l1 þ l2 þ l3 w3 ðl1 þ l2 þ l3 Þ ¼ 0

(11)

w003 ðl1 þ l2 þ l3 Þ ¼ 0

(12) Pi1 j¼1 lj .

Displacement wi can be expressed as wi ðxÞ ¼ Ai sinðki xi Þ þ Bi cosðki xi Þ þ C i sinhðki xi Þ þ Di coshðki xi Þ, where xi ¼ x  The system of equations, obtained after substituting wi ðxÞ into Eqs. (1)–(12), is shown in Appendix A. To find the characteristic equation in terms of o; ki li can be expressed as sffiffiffiffiffiffiffi 4 rSi pffiffiffiffiffi oli ki li ¼ EIi

where Si ; Ii are the cross-section area and area moment, respectively. The characteristic equation f ðoÞ is found by means of symbolic computation [7]. 2.1.2. Random assembly The effect of component variability can be evaluated using Monte Carlo simulation, that provides a set of results which will serve as reference for subsequent comparisons. The random assembly procedure is defined as follows. For each sample structure, lengths and heights of the component beams are generated according to the normal distribution defined

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previously and by discarding out of tolerance items. Natural frequencies and eigenfunctions of each sample structure are computed from the characteristic equation and stored. The mean values and the standard deviations of the six lowest natural frequencies are estimated at the end of each step. The Monte Carlo simulation is stopped after 1000 realisations, having observed the stabilisation plots of mean values m and standard deviations s (Fig. 2). The percentage minimum, maximum and standard deviations on the first six natural frequencies, computed for the set of random combinations, are presented in Table 2. It can be noticed that the scatter is quite high. 2.1.3. Effect of uncertainties The probability density function of each natural frequency is estimated by dividing the range (encompassed between the minimum and maximum values of each natural frequency) into a number NI of intervals, by counting how many realisations fall inside each interval and by normalising the result according to the meaning of pdf.

50

1400 1200

f4

600

f6

30

σ [Hz]

μ [Hz]

800

40

f6

f5

1000

f5 f4

20 f3

400

10

f2

200

f1

f1

0 0

200

f3

f2

0 400 600 800 Number of realisations

1000

4

0

200

400 600 Number of realisations

800

1000

0

200

400 600 Number of realisations

800

1000

30

(σ − σstab) / σstab [%]

(μ − μstab) / μstab [%]

20 2

0

−2

10 0 −10 −20 −30

−4 0

200

400 600 800 Number of realisations

1000

Fig. 2. Stabilisation plots of mean values m and standard deviations s and of their percentage deviations with respect to stabilised values.

Table 2 Percentage deviations for the 1000 random combinations of control factors Minimum, maximum and standard deviation (%) on natural frequencies

Minimum deviation Maximum deviation Standard deviation

1

2

3

4

5

6

25.36 23.12 10.39

11.19 11.76 3.94

12.97 11.39 5.11

6.89 5.21 2.07

9.48 9.78 3.78

8.14 7.81 2.65

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The range of each natural frequency is relatively large with respect to the nominal natural frequency, i.e. that computed for the nominal dimensions. It goes from a maximum of about 24% for the first natural frequency to a minimum of about 6% for the fourth natural frequency.

0.1

0.2

0.08

0.15

0.06 p2(f)

p1(f)

0.25

0.1

0.04

0.05

0.02

0

0 15

20 25 Frequency [Hz]

30

120

130 140 Frequency [Hz]

150

0.04

0.03

p4(f)

p3(f)

0.03 0.02

0.02

0.01 0.01

0 260

0 280

300 320 Frequency [Hz]

340

360

580

600

620 640 Frequency [Hz]

660

0.015 0.015

0.01

p6(f)

p5(f)

0.01

0.005 0.005

0 800

850 900 Frequency [Hz]

950

0 1150

1200

1250 1300 Frequency [Hz]

1350

1400

Fig. 3. Estimated probability density functions p1 ðf Þ    p6 ðf Þ of the six lowest natural frequencies after Monte Carlo simulations (1000 realisations) versus mean values (m), nominal values (.) and fitted normal distributions (—).

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Fig. 3 shows the estimated (with NI ¼ 50) probability density function of the six lowest natural frequencies, together with the mean value and the nominal value. It can be noticed that the mean value is generally different from the nominal value, i.e. a bias is produced due to the non-linear dependence of natural frequencies from geometrical dimensions. The estimated probability density function looks very similar to the fitted normal distribution, apart from some significant deviation in the third natural frequency and from some scatter in the other pdfs. In Fig. 4 the eigenfunctions of all sample structures are overlayed: a distinct plot is used for each of the first six eigenfunctions. The variability of the global shapes is small as confirmed by computing MAC between nominal and current eigenfunctions: the correlation is always greater than 95% [7]. Local variations are more significant although they are small compared with the typical errors that can be expected in experimental identification of mode shapes. 3. Modelling the effect of uncertainties The results of the previous section show that uncertainties affect natural frequencies and mode shapes, and this can be considered as a general result. However, Monte Carlo simulation is unpractical when dealing with complex structures assembled together that require large scale numerical models for the analysis. Complexity may be related to the number of DoFs and/or the number of uncertain variables. A high number of uncertain variables requires an increasing number of Monte Carlo realisations to stabilise the statistical properties of the solution, while a high number of DoFs increases the time required to evaluate the dynamic properties of each realisation. To understand the main reasons of the variability of the dynamic behaviour, i.e. how each uncertainty affects natural frequencies and mode shapes, several possibilities could be explored:  to post-process the results of Monte Carlo simulation by extracting single variable slices obtained by extracting those realisations where that variable falls within a small neighbourhood around the nominal value, as shown in Fig. 5;  to perform a sensitivity analysis with respect to each uncertain variable;  to identify an approximate model of the effect of variability, e.g. obtained using DoE [6] as recalled in the following section.

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Fig. 4. All realisations (1000) of the first six eigenfunctions.

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155 150 145

f2

140 135 130 125 120 2.5

2.6

2.7

2.8

2.9

3 h2

3.1

3.2

3.3

3.4

3.5

Fig. 5. Natural frequency f 2 versus h2 at the end of Monte Carlo simulation: full set of results () with slice h2 2 ½2:9; 3:1 mm () superimposed.

3.1. Design of experiments An experiment is a test or a series of tests in which the values of the variables that affect an output response are appropriately modified to identify the reasons for changes in the response. Therefore, the objective of an experiment may be to find which variables are most significant in determining the response f , and to find how to set the significant variables so that either the response f is near the desired value or the variability in the response f is small. This definition does not prevent from performing numerical experiments whenever this may be convenient for a better understanding of the numerical problem under investigation. 3.1.1. Factorial design and CCD Many experiments involve the study of the effects of two or more variables or factors. In general, factorial designs investigate all possible combinations of the levels of the factors and are very efficient for this task. A regression model representation of a factorial experiment with two factors A and B at two levels could be written as f ¼ a0 þ a1 x1 þ a2 x2 þ a12 x1 x2 þ e

(13)

where the a’s are parameters whose values are to be determined, the variables x1 and x2 are defined on a coded scale from 1 to þ1 (the low and high levels of A and B) and e is an error term. Specifically, if p factors at two levels are considered, a complete series of experiments requires 2p observations and is called a two-level 2p full factorial design. Usually, each series of experiments should be replicated several times using the same value of the factors to average out the effects of noise. Of course, this is unnecessary if experiments are numerical. A potential concern in the use of two-level factorial design is the assumption of linearity in the factor effects. Actually, the interaction term in Eq. (13) introduces a bilinear effect, but each section of f ðx1 ; x2 Þ with a plane with either x1 ¼ const or x2 ¼ const would be a straight line. To account for possible non-linear effects, a logical extension is to consider quadratic terms, as in the following expression: f ¼ a0 þ a1 x1 þ a2 x2 þ a12 x1 x2 þ a11 x21 þ a22 x22 þ e

(14)

Of course, a three level (low level 1, intermediate level 0, high level þ1) factorial design, involving 3p observations, is a possible choice if quadratic terms are important. However, a more efficient alternative is the CCD that starts from the 2p design augmented with:  the centre point: a single observation with all factors at intermediate level;  axial runs: each factor is considered at two levels (the low level 1 and the high level þ1) while the remaining factors are at the intermediate level, for a total of 2p observations. Overall, a CCD for p factors requires n ¼ 2p þ 2p þ 1 observations instead of 3p observations required by the three level factorial design, with advantages for pX3. The n observations can be conducted by varying the values of the factors as shown in Appendix B.

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3.1.2. Response surface model for CCD For p control factors, the experimental response can be expressed as f ¼ a0 þ

p X

a i xi þ

p X i1 X

aji xj xi þ    þ

i¼1 j¼1

i¼1

p X i1 X



i¼1 j¼1

m 1 X

anmji xn xm    xj xi þ

n¼1

p X

aii x2i þ e

(15)

i¼1

where a regression model representation of a 2p full factorial experiment (involving 2p terms), augmented with p quadratic terms, is used. Overall, the expression contains 2p þ p parameters a. Each parameter provides an estimate of the effect of a single factor (linear or quadratic) or of a combination of factors. Note that Eq. (15) is linear in the parameters a, and it can be rewritten as 9 8 a0 > > > > > > > > > = < a1 > 2 þe (16) f ¼ ½1 x1    xp  .. > . > > > > > > > > > : app ; having arranged the parameters in a vector a. A different equation can be written for each observation by varying the factors ðx1 ; . . . ; xp Þ. In a CCD, a set of equations can be written by choosing the factors as established in Table B.1. By arranging the experimental responses in a vector ff g, a linear relationship between ff g and fag can be expressed in matrix notation as ff g ¼ ½Xfag þ feg

(17)

where ½X is a ð2p þ 2p þ 1Þ  ð2p þ pÞ matrix. The least square estimate of fag is given by fa^ g ¼ ð½XT ½XÞ1 ½XT ff g T

1

T

(18)

þ

where ð½X ½XÞ ½X ¼ ½X is the pseudo-inverse of ½X. The fitted regression model is ff^ g ¼ ½Xfa^ g

(19)

The difference between the actual observation f and the corresponding fitted value f^ is the residual e ¼ f  f^ . The residual accounts both for the modeling error e and for the fitting error due to the least square estimation. A vector of residuals can be defined as feg ¼ ff g  ff^ g

(20)

3.1.3. Error sum of squares P If the squared deviations of each response f i from its average value f¯ ¼ ni¼1 f i =n are considered, their sum is the so-called total sum of squares SST that can be computed as SST ¼

n X

ðf i  f¯ Þ2 ¼

X

2

2

f i  nf¯ ¼ ff gT ff g  ff¯ gT ff¯ g

(21)

i¼1

In fact: X X 2 X 2 X 2 2 ðf i  f¯ Þ2 ¼ ðf i  2f i f¯ þ f¯ Þ ¼ f i  2f¯ f i þ nf¯ X 2 X 2 2 2 2 ¼ f i  2nf¯ þ nf¯ ¼ f i  nf¯ Furthermore, the total sum of squares can be partitioned into a sum of squares due to the model (or regression) and a sum of squares due to residual (or error): SST ¼ SSR þ SSE The sum of squares of the residuals can be computed as n X ðf i  f^ i Þ2 ¼ ðff g  ff^ gÞT ðff g  ff^ gÞ SSE ¼

(22)

(23)

i¼1

Substituting ff^ g ¼ ½Xfa^ g it is SSE ¼ ðff g  ½Xfa^ gÞT ðff g  ½Xfa^ gÞ ¼ ff gT ff g  2fa^ gT ½XT ff g þ fa^ gT ½XT ½Xfa^ g Since fa^ g ¼ ð½XT ½XÞ1 ½XT ff g (Eq. (18)), the last equation becomes SSE ¼ ff gT ff g  fa^ gT ½XT ff g

(24)

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A low value of the ratio SSE =SST between the error sum of squares and the total sum of squares indicates that the chosen regression variables provide a good fit. 3.2. Analysed case Post-processing of Monte Carlo simulation, sensitivity analysis and DoE are used to model the effect of dimensional uncertainties of the component substructures on the dynamic behaviour of the assembled structure defined in Section 2.1. 3.2.1. Post-processing of Monte Carlo simulation The output of Monte Carlo simulation is post-processed as outlined at the beginning of Section 3. This is done to observe whether narrowing the range of a given dimensional variability has some significant effect on the distribution of natural frequencies. For each dimensional uncertainty, a slice around the nominal value is considered: a smaller neighbourhood around the nominal value is defined, and those realisations where the considered length or height falls outside the defined neighbourhood are discarded. New probability density functions can be estimated by considering the realisations within each slice: by comparing the new probability density functions with those estimated using the full set of Monte Carlo simulations, the effect of each dimensional uncertainty can be assessed. Note that, depending on the neighbourhood with, the retained number of realisations can become too small to provide an accurate estimation of new probability density functions. For each considered natural frequency, six new pdf’s can be estimated, because there are six uncertain variables (lengths and heights of the three beams). Fig. 6 compares two typical results: the estimated probability density functions of the first natural frequency after retaining realisations where heights h1 and h2 are within 0:1 mm of their nominal values. By comparing the two new probability density functions with the corresponding p1 ðf Þ of the complete Monte Carlo simulation (see Fig. 3), it can be observed that the effect of narrowing the tolerance field on h1 is almost negligible, while narrowing the tolerance field on h2 yields a much sharper pdf. Furthermore, it can be observed that the scatter has increased with respect to the complete Monte Carlo simulation due to the decrease of retained realisations (332 for h1 and 341 for h2 instead of 1000). 3.2.2. Sensitivity with respect to uncertain variables Sensitivity analysis is numerically performed using the characteristic equation mentioned in Section 2.1.1 to evaluate the natural frequency changes that occur due to small variations of an uncertain variable at a time. Results are presented in Table 3 where the highest sensitivity is underlined for any of the first six natural frequency (column). It can be noticed that the highest effects are due either to h2 or to h1 , i.e. the heights of middle and left beams. This agrees with quite obvious physical reasoning, since natural frequencies of beams are more sensitive to beam heights than to beam lengths. 3.2.3. Approximate model through DoE Among the possible techniques for DoE, CCD is used to model the effect of dimensional uncertainties. Lengths and heights of the beams are considered as control factors. Accordingly, low and high levels of the control factors (Table 4) are chosen as the lower and upper bounds of lengths and heights within the tolerance fields defined in Section 2.1.

0.8

0.25

0.6

0.15

p1(f)

p1(f)

0.2

0.1

0.4 0.2

0.05 0

0 15

20 25 Frequency [Hz]

30

15

20 25 Frequency [Hz]

30

Fig. 6. Estimated probability density function p1 ðf Þ of the first natural frequency with 7:9 mmph1 p8:1 mm (left) and with 2:9 mmph2 p3:1 mm (right) versus mean values (m), nominal values (.) and fitted normal distributions (—).

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Table 3 First order sensitivity of natural frequencies to 1 mm change of uncertain variables

h1 l1 h2 l2 h3 l3

f1

f2

f3

f4

f5

f6

1.1168 0.0464 10:1282 0.1938 0.0325 0.0749

16:1683 0.4776 15.4476 0.6878 5.2575 0.2978

6.9787 0.8070 67:9611 3.0039 6.1865 0.6892

50:0436 2.3669 19.7167 1.0443 20.2201 1.1613

18.7768 0.8554 139:2602 5.4761 37.5467 5.6203

125:4358 5.7247 80.5690 3.3331 3.9421 0.5021

Table 4 Control factors

Low value (mm) High value (mm)

A ¼ h1

B ¼ l1

C ¼ h2

D ¼ l2

E ¼ h3

F ¼ l3

7.5 8.5

398 402

2.5 3.5

48 52

7.5 8.5

198 202

Table 5 Error relative sum of squares for the first six natural frequencies

SSE =SST

f1

f2

f3

f4

f5

f6

6:46  106

7:34  105

2:76  105

3:64  105

8:23  105

1:05  105

A 2p þ 2p þ 1 CCD with p ¼ 6 is implemented, requiring 77 experiments. Since noise is not present in numerical experiments, a single replicate is sufficient. Using the characteristic equation, the eigenvalues of the assembled structure are evaluated as functions of control factors. An estimate a^ of the regression coefficients can be obtained for each natural frequency according to Eq. (18). The ratios SSE =SST for each natural frequency are almost negligible as presented in Table 5. The estimated regression coefficients are presented in Table 6 for the first six natural frequencies. After having estimated the regression coefficients a^ , approximate values of the first six natural frequencies can be computed for any given combination of control factors using the following equation (fitted model): f^ ¼ a^ 0 þ

6 X

a^ i xi þ

6 X i1 X

i¼1

i¼1 j¼1

a^ ji xj xi þ    þ a^ 123456 x1 x2 x3 x4 x5 x6 þ

6 X

a^ ii x2i

(25)

i¼1

Natural frequencies provided by the fitted model (Eq. (25)) can be compared with those provided by Monte Carlo simulation. Natural frequency percentage errors between the fitted model and the numerical experiments, are computed according to the following expression: Error ¼

f^  f  100 f

where f is the natural frequency provided by the characteristic equation. Mean value, minimum and maximum values, median value, lower and upper quartile values of natural frequency errors between fitted model and numerical experiments conducted using the characteristic equation are shown in the box and whisker plot in Fig. 7. It can be noticed that errors are quite small, especially if compared with those obtained by using 26 factorial design [8], shown in Fig. 8. When using plain factorial design, mean error values are quite far from zero and a large scatter occurred, especially on the third natural frequency. (Note the different scales of the two plots.) This demonstrates the benefits of augmenting the factorial design with centre point and axial runs, allowing to estimate quadratic terms. In Table 6, the most important effects are given in bold italic for each natural frequency (column). Of course, the mean a^ 0 , appearing on the first row, can never be neglected. Furthermore, the a^ ’s given in bold are those for which ja^ =a^ 0 j exceeds a cut-off value of 0.5%.

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Table 6 Estimated regression coefficients a^ Model term

Regression coefficients f1

Mean ¼ a0 A ¼ a1 B ¼ a2 C ¼ a3 D ¼ a4 E ¼ a5 F ¼ a6 AA ¼ a11 AB ¼ a12 AC ¼ a13 AD ¼ a14 AE ¼ a15 AF ¼ a16 BB ¼ a22 BC ¼ a23 BD ¼ a24 BE ¼ a25 BF ¼ a26 CC ¼ a33 CD ¼ a34 CE ¼ a35 CF ¼ a36 DD ¼ a44 DE ¼ a45 DF ¼ a46 EE ¼ a55 EF ¼ a56 FF ¼ a66 ABC ¼ a123 ABD ¼ a124 ABE ¼ a125 ABF ¼ a126 ACD ¼ a134 ACE ¼ a135 ACF ¼ a136 ADE ¼ a145 ADF ¼ a146 AEF ¼ a156 BCD ¼ a234 BCE ¼ a235 BCF ¼ a236 BDE ¼ a245 BDF ¼ a246 BEF ¼ a256 CDE ¼ a345 CDF ¼ a346 CEF ¼ a356 DEF ¼ a456 ABCD ¼ a1234 ABCE ¼ a1235 ABCF ¼ a1236 ABDE ¼ a1245 ABDF ¼ a1246 ABEF ¼ a1256 ACDE ¼ a1345 ACDF ¼ a1346 ACEF ¼ a1356 ADEF ¼ a1456 BCDE ¼ a2345 BCDF ¼ a2346 BCEF ¼ a2356 BDEF ¼ a2456 CDEF ¼ a3456 ABCDE ¼ a12345 ABCDF ¼ a12346 ABCEF ¼ a12356 ABDEF ¼ a12456 ACDEF ¼ a13456 BCDEF ¼ a23456 ABCDEF ¼ a123456

21.695 0.49472 0.09456 4.9674 0.372 0.031047 0.14831 0.027289 0.0022488 0.052253 0.0026485 0.033726 0.0060236 0.00059662 0.027067 0.0018276 0.0019811 0.00079334 0.13506 0.051234 0.055783 0.032399 0.0082554 0.00372 0.0032074 0.028314 0.0036622 7.36E05 0.00089229 5.98E05 0.0002944 6.48E05 0.0076034 0.004837 0.00092973 0.00086391 2.95E05 0.00013026 0.00054096 6.59E05 0.00025787 1.69E05 1.86E05 1.74E05 0.0028521 0.00036455 0.00053517 5.29E05 1.64E05 2.53E05 1.98E05 4.90E06 1.38E06 7.09E07 0.00020631 0.00010406 5.48E06 3.64E06 1.90E05 7.00E06 5.24E06 7.97E08 4.62E06 3.11E07 2.75E07 4.83E07 2.82E08 1.14E06 1.26E07 1.89E08

f2 135.32 7.5821 0.86203 8.4477 1.5196 2.6099 0.57252 0.014667 0.023906 0.65984 0.19304 0.19434 0.055294 0.0034106 0.11676 0.026737 0.01611 0.0066275 1.4899 0.34321 0.51605 0.021214 0.012518 0.040365 0.0045135 0.014072 0.030056 0.0047858 0.028203 0.0046896 0.0012362 0.0011939 0.12119 0.092916 0.013009 0.0082172 0.0015897 0.0034523 0.0092944 0.0056068 0.00020188 3.00E05 5.38E05 0.00096074 0.010226 0.0030405 0.0016222 0.00058234 0.0017747 0.00069826 9.98E05 8.31E05 4.19E05 0.00017307 0.0028113 0.00031888 0.001148 0.00022091 0.00023715 8.71E05 4.58E05 1.40E06 0.00049454 0.00011947 5.52E05 2.20E05 2.22E06 9.64E05 1.96E06 4.87E06

f3 308.54 4.0618 1.6671 33.283 5.5279 2.9778 1.315 0.59142 0.025004 2.1647 0.16062 0.36842 0.091469 0.054477 0.14176 0.014562 0.08593 0.005314 6.3853 0.44365 2.2691 0.49192 0.14421 0.25143 0.075143 0.57614 0.17519 0.049968 0.056272 0.010036 0.001366 0.001891 0.24443 0.33335 0.095039 0.041954 0.014024 0.00040487 0.015303 0.03491 0.0073423 0.0057367 0.0010879 0.00012181 0.06437 0.0024296 0.040421 0.0093144 0.0029276 0.00036454 0.00010444 0.00018115 3.88E05 0.00051155 0.036454 0.004882 0.0016243 0.00042638 0.0017467 0.00051846 0.00026365 4.49E05 0.0022255 0.00033667 0.00015014 0.00018977 4.84E05 0.00048284 4.89E05 2.37E05

f4 620.19 25.547 4.7963 10.543 2.2018 9.41 2.2514 0.77928 0.27652 3.0555 0.45132 0.28222 0.070158 0.022949 0.37238 0.033943 0.15953 0.06419 0.66382 0.3695 3.4714 0.86585 0.036225 0.4321 0.11268 0.52414 0.23889 0.049682 0.014671 0.0046461 0.026455 0.015011 0.035094 0.54605 0.098432 0.083865 0.016832 0.037437 0.003313 0.055537 0.0057376 0.0078356 0.00095305 0.012315 0.033004 0.0083247 0.18823 0.020814 0.00089062 0.03253 0.0089951 0.0044491 0.0013133 0.0061535 0.015009 0.0045508 0.057691 0.0078227 0.0070285 0.0013509 0.01536 0.0018023 0.018391 0.00049475 8.67E05 0.0051319 0.00079183 0.0028874 0.00060533 0.00083429

f5 868.69 11.692 1.9155 64.097 9.4191 20.718 11.179 1.4398 0.11782 8.283 1.0956 0.41889 0.19727 0.044297 1.0936 0.13974 0.17741 0.029724 1.338 1.907 7.0331 1.3806 0.3244 1.0093 0.13812 1.5863 1.0224 0.17399 0.15786 0.017079 0.031483 0.025537 0.73586 0.20736 0.29728 0.02761 0.034726 0.047369 0.069557 0.0542 0.032604 0.0064219 0.0039974 0.015662 0.20182 0.034548 0.29859 0.031522 0.003609 0.029154 0.0032791 0.0052649 0.0013264 0.0044049 0.019228 0.01514 0.058874 0.010245 0.0082959 0.0034216 0.014073 0.0018704 0.025096 0.00038997 0.00056692 0.0057817 0.00077199 0.0004482 0.00033293 0.0009667

f6 1277 61.901 11.285 39.93 6.9192 3.0821 1.4118 1.5898 0.88709 7.7868 1.0301 0.15572 0.27926 0.017512 0.5688 0.046412 0.029553 0.028214 7.6337 3.3502 2.5403 1.0988 0.27863 0.34267 0.14246 0.44322 0.082971 0.053512 0.063282 0.0019488 0.006605 0.0099452 0.39291 0.11291 0.18497 9.42E05 0.015193 0.01629 0.037416 0.021849 0.019988 0.003175 0.002469 0.0062396 0.28208 0.10502 0.063668 0.010149 0.0059977 0.0016999 0.0046372 0.0015363 0.00023449 0.001121 0.0072758 0.0027497 0.0075495 0.0015431 0.0028114 0.0011173 0.0038592 0.00013697 0.007915 0.0018259 0.00091162 0.00026287 4.65E05 0.0020565 6.13E05 0.00014464

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599

0.8 0.6 0.4 Error [%]

0.2 0 −0.2 −0.4 −0.6 −0.8 1

2 3 4 5 Natural Frequency Number

6

Fig. 7. Mean value (- - -), median value (thick solid ), minimum and maximum values (thin solid —), lower and upper quartile values (boxes below and above median value –) of natural frequency errors between the model fitted by using central composite design and numerical experiments.

0.5

Error [%]

0 −0.5 −1 −1.5 −2 −2.5 1

2 3 4 5 Natural Frequency Number

6

Fig. 8. Mean value (- - -), median value (thick solid ), minimum and maximum values (thin solid —), lower and upper quartile values (boxes below and above median value –) of natural frequency errors between a model fitted by using 26 factorial design [8] and numerical experiments.

For the sake of further developments, a reduced fitted model including only the regression coefficients that exceed the cut-off value could be considered. For each natural frequency, a different number of model terms should be taken into account. If the same model terms for all natural frequencies are to be used, ten coefficients must be included: a^ 0 , the six linear effects a^ 1 ; . . . ; a^ 6 , the interaction terms a^ 13 and a^ 35 and the quadratic effect a^ 33 . The reduced model becomes f^ red ðx1 ; . . . ; x6 Þ ¼ a^ 0 þ

6 X

a^ i xi þ a^ 13 x1 x3 þ a^ 35 x3 x5 þ a^ 33 x23

(26)

i¼1

The results provided by the reduced fitted model (Eq. (26)) can be compared with those provided by the complete fitted model. To this aim, the reduced fitted model is used to compute the natural frequency errors for the 1000 random combinations of control factors used previously. Mean value, minimum and maximum values, median value, lower and upper quartile values of natural frequency errors between reduced fitted model and numerical experiments conducted using the characteristic equation are shown in the box and whisker plot of Fig. 9. By observing Figs. 7 and 9, it can be noticed that natural frequency errors increase, as expected, when using the reduced fitted model, but they remain acceptably low, especially if compared with natural frequency uncertainties due to beam dimension variability, that are quantified by the deviations in Table 2. Hence, the reduced fitted model could be used instead of the complete fitted model without problems.

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0.8 0.6 0.4 Error [%]

0.2 0 −0.2 −0.4 −0.6 −0.8 1

2 3 4 5 Natural Frequency Number

6

Fig. 9. Mean value (- - -), median value (thick solid ), minimum and maximum values (thin solid —), lower and upper quartile values (boxes below and above median value –) of natural frequency errors between reduced model and numerical experiments.

4. Reducing the effect of uncertainties Two different scenarios can be considered: in one case, it is assumed that the component substructures have not been produced yet so that small changes in the design are possible; in the other case, the sets of component substructures to be assembled are assumed to be available, and they must be matched to minimise the number of assemblies that do not meet some performance requirement and possibly to enhance the average performance requirements. 4.1. Reduction at design stage: tolerance allocation Small design modifications may involve marginal changes in the geometry of the structures or may leave the nominal structure unchanged while modifying tolerance allocation. Obviously, uncertainties due to material properties cannot be reduced by small design modifications. Tolerance allocation can be modified by identifying those tolerances to which the performance requirements are most sensitive and by tightening those tolerances [9]. 4.1.1. Analysed case The analysis of effects provides a way to limit the scatter of natural frequencies of the assembled structure by operating on tolerance allocation. In Section 3.2, it was noticed that the highest effects on the natural frequencies are due either to h2 or to h1 , i.e. the heights of middle and left beams. Tolerances allocation can be modified in subsequent steps: first, tolerance can be tightened on h2 ; secondly, tolerance can be tightened on h1 ; and so on. Each subsequent step has decreasing effects but involves additional costs. Table 7 presents the effect of narrowing the tolerance on beam height h2 from 0:5 to 0:1 mm, and of subsequently reducing of the same amount the tolerance on beam height h1 . It is noticed that the standard deviations of natural frequencies decrease, as obvious. When only the tolerance on h2 is tightened, the decrease is very significant for the first, third and fifth natural frequencies, as it could be expected by observing Table 3 (sensitivity) and Table 6 (regression coefficients). When also the tolerance on h1 is tightened, a further significant decrease is observed for the fourth and sixth natural frequencies, as expected from Tables 3 and 6. The second natural frequency is similarly affected by tightening h1 and h2 . 4.2. Reduction at production stage: selective assembly In order to minimise the number of assemblies that do not meet some performance requirement and possibly to enhance the average performance, a selective assembly procedure can be devised. Selective assembly is typically used in industry for obtaining high-precision assemblies from relatively low-precision components: it consists in measuring each part and choosing one that is the right size to fit in the assembly [9]. In this paper, the same idea is tentatively extended to problems where the performance requirement is not the clearance between two parts, but a different set of quantities such as the natural frequencies of the assembled parts.

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Table 7 Percent standard deviation on the first six natural frequencies before tightening dimensional tolerances (row 1), after reducing dimensional tolerance on h2 from 0:5 to 0:1 mm (row 2), after reducing dimensional tolerances on both h2 and h1 from 0:5 to 0:1 mm (row 3) h2 (mm)

h1 (mm)

3  0:5 3  0:1 3  0:1

8  0:5 8  0:5 8  0:1

Standard deviation (%) f1

f2

f3

f4

f5

f6

10.39 3.00 2.89

3.94 2.92 1.32

5.11 1.69 1.61

2.07 1.96 0.94

3.78 1.58 1.54

2.65 2.25 0.78

Selective assembly is performed (e.g. by automotive industry when assembling engine parts) in two steps: in the first step, a number of dimensional groups or classes is defined for each item to be assembled; in the second step, items are actually assembled by picking them from the appropriate dimensional group in order to satisfy the prescribed tolerance requirements. At the moment, no tentative has been made to extend the concept of dimensional groups [11] but selective assembly is performed on an item by item basis. Two different strategies can be conceived, depending on the quality of information that is available before assembly:  batch processing strategy: information about components to be assembled is assumed to be available before the assembly process starts;  serial processing strategy: information about components is obtained one by one. In many assemblies, it is possible to identify the most critical component, e.g. the one to which the performance requirements are most sensitive. For the most critical component, a sort of fitness can be checked a priori by evaluating (using the regression model fitted through DoE) the effect of assembling it together with the other (non-critical) components. To this aim, a virtual component can be defined as any substructure (even not existing in the set of realisations) with characteristics within the tolerance field. A given critical component is unfit if no other non-critical virtual components can be found so that the assembled structure satisfies some prescribed performance requirements. Unfit components can be discarded and ignored in subsequent steps, both in the batch processing strategy and in the serial processing strategy with advantages for the selective assembly process. Similarly, it would be possible to check fitness of partially assembled substructures instead of single components. 4.2.1. Analysed case In Section 2.1, it was observed that the random assembly of the coupled structure produces quite scattered natural frequencies. With reference to 1000 realisations, the highest deviations are obtained for the first natural frequency (see Table 2). If a tolerance jej ¼ 10% were established on the first natural frequency, 362 over 1000 assembled structures should be discarded, as qualitatively confirmed by Fig. 10. By observing Tables 3 and 6, it can be noted that the uncertain variable that most strongly affects the first natural frequency is the thickness h2 of the intermediate beam (beam 2). Note that the thickness hi of a given beam bi is necessarily associated with length li . Therefore, length and thickness of a given beam cannot be chosen independently. In this case, the most critical component is beam 2. A procedure for fitness check of the intermediate beam can be devised as follows. By using the reduced fitted model (26), the frequency deviation due to any intermediate beam (control factors x3 and x4 ), with beams 1 (left) and 3 (right) at their nominal dimensions, (control factors x1 ; x2 ; x5 and x6 equal to zero) is

Df b2 ¼

f^ 1;red ðx3 ; x4 Þ  f 1 a^ 0 þ a^ 3 x3 þ a^ 4 x4 þ a^ 33 x23  f 1 ¼ f1 f1

(27)

If, for a given intermediate beam, jDf b2 jXe, this effect should be compensated by the contributions of virtual beams 1 and 3. By considering the maximum possible compensation, the condition to satisfy is: jDf b2 j 



 ja^ 1 þ a^ 13 x3 j ja^ 2 j ja^ 5 þ a^ 35 x3 j ja^ 6 j þ þ þ oe f1 f1 f1 f1

(28)

Note that the use of the complete fitted model would lead to much more complex expressions. Batch processing strategy: By using Eq. (28), a number of intermediate beams can be definitely discarded. In the considered case (e ¼ 10%), 210 over 1000 intermediate beams can be discarded immediately.

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30

Deviation [%]

20 10 0 −10 −20 −30 0

200

400 600 Realisations

800

1000

Fig. 10. Percent deviation on the first natural frequency after selective assembly: (-  -) random assembly; (- - -) batch processing; (——) serial processing.

Similar to Eq. (27), the frequency deviation due to any left beam (control factors x1 and x2 ), with beams 2 and 3 at their nominal dimensions, (control factors x3 ; x4 ; x5 and x6 equal to zero) can be defined as

Df b1 ¼

f^ 1;red ðx1 ; x2 Þ  f 1 a^ 0 þ a^ 1 x1 þ a^ 2 x2  f 1 ¼ f1 f1

(29)

Finally, the frequency deviation due to any right beam (control factors x5 and x6 ), with beams 1 and 2 at their nominal dimensions, (control factors x1 ; x2 ; x3 and x4 equal to zero) is

Df b3 ¼

f^ 1;red ðx5 ; x6 Þ  f 1 a^ 0 þ a^ 5 x5 þ a^ 6 x6  f 1 ¼ f1 f1

(30)

The frequency deviations due to any of the three beams can be added up (Df b1 þ Df b2 þ Df b3 ) to provide an approximate estimation of the frequency deviation accounting for the variability of the assembled structure. The result is only approximate because interactions between beam 2 and beams 1 and 3 are neglected. When Df b2 Xe, it should be compensated by negative Df b1 and Df b3 . If for a given intermediate beam, the following relation holds:

Df b1;min þ Df b2 þ Df b3;min Xe

(31)

the corresponding beam must be discarded. Similarly, when Df b2 p  e, it should be compensated by positive Df b1 and Df b3 . If for a given intermediate beam, the following relation holds:

Df b1;max þ Df b2 þ Df b3;max p  e

(32)

the corresponding beam must be discarded too. In the considered case, 18 intermediate beams must be additionally discarded at the end of this stage. The remaining intermediate beams (772) are checked to verify if they can be actually assembled with existing left and right beams to satisfy the constraint on frequency deviation. In the considered case, it turns out that all 772 intermediate beams can be actually assembled. Since only 772 left and right beams would be necessary, those to be effectively used are selected so as to minimise the frequency deviation. For the assembled structures, the percentage deviation on the first natural frequencies is shown in Fig. 10. It can be noticed that more structures fall within the 10% constraint limit and the error distribution is more favourable with respect to the random assembly. Serial processing strategy: In this case, it will be necessary to evaluate the fitness of partial assemblies (namely the assembly of beams 1 and 2). A given partial assembly is unfit if no right virtual beam can be found so that the assembled structure satisfies the prescribed specifications on the first natural frequency deviation. The frequency deviation due to given left and intermediate beams is obtained as Df b1 þ Df b2 (see Eqs. (27) and (29)). This effect should be compensated by the contributions of virtual beam 3. By considering the maximum possible compensation, the condition to satisfy is   ja^ 5 þ a^ 35 x3 j ja^ 6 j jDf b1 þ Df b2 j  þ (33) oe f1 f1

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START

Pick a beam from stack 2

Is beam 2 unfit ?

Discard beam 2

YES

NO Pick a beam from stack 1

Is the assembly of beam1 and beam 2 unfit?

YES

Put beam 1 in stack 1

NO Pick a beam from stack 3

Is the assembly of beam 1 and beam 2 and beam 3 unfit?

YES

Put beam 3 in stack 3

NO Beam 1, beam 2 and beam 3 are ready for assembly

NO

Is stack 2 empty? YES END

Fig. 11. Flowchart summarising the serial processing strategy for selective assembly.

The beams are assumed to be arranged in three stacks: stack 1 containing all the left beams, stack 2 containing all the intermediate beams and stack 3 containing all the right beams. The procedure is summarised in Fig. 11. At the end of the procedure, 772 structures can be assembled that satisfy the constraint on the first natural frequency deviation as shown in Fig. 10. In this case, the error distribution is quite similar to that obtained through random assembly, except for values of deviation close to 10%. It can be noticed that, whenever more information is supplied as in the batch processing strategy, a better error distribution is achieved.

5. Conclusion Different strategies for modelling and reducing the effect of uncertainties on substructure coupling have been discussed in this paper. Modelling techniques include: post-processing of data from a set of randomly assembled structures (Monte Carlo simulation), analysis of sensitivities to uncertain variables, and use of design of experiments (DoE) to fit a regression

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model. The techniques have been applied to a simple academic structure made by three beams coupled together. In principle, all techniques can be extended to large scale numerical models although in this case the use of Monte Carlo simulations would be very time consuming. On the contrary, sensitivity analysis and DoE can be easily applied. Although data post-processing and sensitivity analysis are able to provide useful information about the effect of a given uncertainty, the only way to account for interactions between different uncertainties and to obtain a simple mathematical model that can be used in what if analysis is to consider DoE. More specifically, central composite design has been used that provides much better results with respect to two-level factorial design, without requiring a much larger number of experiments. Strategies for reducing the effect of uncertainties on substructure coupling can be adopted either at design stage or at production stage. In the former case, a careful position is assumed that leads either to robust design practices or to considerate tolerance allocation; in the latter case, no care is taken at design stage and solutions such as selective assembly can be adopted at production stage, using either a batch strategy or a serial strategy. Any of the considered modelling techniques is able to provide hints in order to effectively modify tolerance allocation at design stage. However, only a regression model fitted through DoE can be used if, for instance, a worst case analysis is needed. Also, only a regression model obtained from DoE can be used to implement selective assembly procedures at production stage. In comparison with the considered alternatives, DoE can be used much more effectively within strategies aimed at modelling and reducing the effect of uncertainties. Finally, it should be noted that not only the regression model fitted from central composite design appears to be better compared to relatively simple alternatives, but the results provided by this kind of model seem to be very accurate in absolute terms too, as demonstrated by the low errors obtained in the analysed case.

Acknowledgement This research is supported by MIUR grants. Appendix A. Homogeneous system leading to characteristic equation 8 B1 þ D1 ¼ 0 > > > > > B1  D1 ¼ 0 > > > > > A1 sinðk1 l1 Þ þ C 1 sinhðk1 l1 Þ  B2  D2 ¼ 0 > > > > > A > 1 cosðk1 l1 Þ þ C 1 coshðk1 l1 Þ  A2  C 2 ¼ 0 > > > > A > 1 I1 sinðk1 l1 Þ þ C 1 I1 sinhðk1 l1 Þ þ B2 I2  D2 I2 ¼ 0 > > > < A1 I1 cosðk1 l1 Þ þ C 1 I1 coshðk1 l1 Þ þ A2 I2  C 2 I2 ¼ 0 A2 sinðk2 l2 Þ þ B2 cosðk2 l2 Þ þ C 2 sinhðk2 l2 Þ þ D2 coshðk2 l2 Þ  B3  D3 ¼ 0 > > > > > > A2 cosðk2 l2 Þ  B2 sinðk2 l2 Þ þ C 2 coshðk2 l2 Þ þ D2 sinhðk2 l2 Þ  A3  C 3 ¼ 0 > > > > > A 2 I2 sinðk2 l2 Þ  B2 I2 cosðk2 l2 Þ þ C 2 I2 sinhðk2 l2 Þ þ D2 I2 coshðk2 l2 Þ þ B3 I3  D3 I 3 ¼ 0 > > > > > A2 I2 cosðk2 l2 Þ þ B2 I2 sinðk2 l2 Þ þ C 2 I2 coshðk2 l2 Þ þ D2 I2 sinhðk2 l2 Þ þ A3 I3  C 3 I3 ¼ 0 > > > > > A3 sinðk3 l3 Þ þ B3 cosðk3 l3 Þ þ C 3 sinhðk3 l3 Þ þ D3 coshðk3 l3 Þ ¼ 0 > > > : A sinðk l Þ  B cosðk l Þ þ C sinhðk l Þ þ D coshðk l Þ ¼ 0 3

3 3

3

3 3

3

3 3

3

3 3

Appendix B. Run order in CCD Each series of experiments (runs) for CCD with p factors can be split in two blocks: the first block corresponds to the 2p factorial design; the second block includes the additional centre point and the axial runs. The 2p factorial design is conducted by using the so-called standard order. A recursive definition of the standard order can be given as follows: (1) for a single factor A, the first run considers the factor at the low () level and the second run is conducted with factor at the high (þ) level; (2) whenever a new factor is introduced, all previous runs are repeated once with the new factor at the low level and once with the additional factor at the high level. In the second block, the centre point run is conducted first. For two or more control factors, axial runs are conducted first for factor A, then for factor B and so on. Table B.1 presents how to build the CCD with p ¼ 1 (control factor A; runs 1, 2, 2p þ 1), p ¼ 2 (control factors A; B; runs 1–4, 2p þ 1 to 2p þ 5) or p ¼ 3 (control factors A–C; runs 1–8, 2p þ 1 to 2p þ 7).

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Table B.1 Standard order of runs in CCD

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