Experimental design theory and structural optimization : design of a major-third church bell

Experimental design theory and structural optimization :

design of a major-third church bell

Citation for published version (APA):

Schoofs, A. J. G., van Asperen, F., Maas, P. J. J., & Lehr, A. (1985). Experimental design theory and structural optimization : design of a major-third church bell. (DCT rapporten; Vol. 1985.043). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1985 Document Version:

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WFW 85.043 -1-

EXPERIMENTAL DESIGN THEORY AND STXUCTURAL OPTIMIZATION; DESIGN O F A MAJOR-THIRD CHURCH BELL

THEORIE DER VERSUCHSPLANUNG UND OPTIMIERUNG VON KON- CTRUKTIONEN; ENTWURF EINET? GROSSE-TERZ GLOCKE

R. Schoof s

,

F. van Asperen, P. Maas, Eindhoven Universi-

t y of Technology.

A . Lelir, Koninklijke Ej.-jsbouts, Bell Foundry, Asten, Netherlands.

ABSTRACT

After a short introduction, Section 2 describes the

i t e r a t i v e procedure f o r coniputer aided structural op-

tj mization. The general op%imj.zation problem i s defined

and the algorithm t o solve i t i s presented. The use of the f i n i t e element method, together w i t h calculation of s e n s i t i v i t i e s , i s discussed.

I n Section 3 the extended experimental design theory i s

treated. The extension concerns the application o f sen-

s i t i v i t i e s . The mentioned theory i s used t o derive re-

duced mathemtical niodels e f a structure.

Section 4 contains some ideas f o r the application o f

such matheniatical models and argunients are given for the

integrati on of the experimental desi gn theory and %he

i t e r a t i v e optimization procedure.

I n Section 5 t h e proposed method i s applied i n order t o

design the geometry o f a so-called niajor-third church

b e l l .

F i n a l l y some conclusions are given.

I. INTRODUCTION

A widely used method for automated structural optimiza-

tion consists o f the combination o f the f ini.te element method and non-linear prograiiiming. The f i n i t e element method is used because o f i t s modelling f a c i l i t i e s and i t s analysi s accuracy. Non-linear programmi n y is used t o

-2-

the objective function (to be minimized) and of the con-

straints with respect to the design variables and the structural behaviour. Generally optimization problems

are solved iteratively, oft.en using gradient search nie-

Experimental design techniques are well suit.ed to esta-

blish eff kient program f o r physical experiments and to

derive from the results of the experiments a mathemati-

cal description cif the physical phenomenon under consi- derakj on.

Experimental design proves to be also applicable to nu-

merical experimentation t especially in the case o f

structural optimization, where the "measurements" con-

sist of finite element analyses.

thodfi.

2 . THE ITERATIVE PROCEDURE IN STRUCTURAL OPTIMIZATION We first present the general optimization problem and the algorithm to solve it. Then we discuss the re-

commendation of the finite element method as an analy-

sing module and the relation between a finite element

model and the design variables.

Finally some relations are given for the calculation of

sensitivities o f the objective function and the con-

straints.

GENERAL OPTIMIZATION PROBLEM

Assumhg that the structure under consideration can be described uniquelly by a column matrix 5:

XT

= (xl,..-,x,f

( 1 1

xiIi.=lI,*.,n: so-callecl design variables

w e can write the constrained optimization problem as follows.

Minimize: F($ objective function (23

-3- g . . ( x )

i

O j = l 1 . . . , m inequality constraints ( 3 ) J h k ( x ) = O k = l , . . . I l equality constraints ( 4 ) 1 x i

i

x i xy i = l

,

.

.

. ,

n side constraints ( 5 )

I n the optimization problem the design variables are va- ried i n order t o obtain the best possible structure by

minimizing the objective function. Those variables may

consist o f :

- rnechanj.cal and physical properties of the material.

- the topology, i . e . the pattern of connections i n the

-

the gecmetry (the shape 1 and cross-sectional dimen-

The objective functi.oii P i s related t o the most impor-

tant property of the structure (the structural weight is

often used), or a weighted sum of several properties.

The inequality constraints g- express the limitations

J

w i t h respect t o the behaviour of the structure, such iis allowable stress, minimum b u c k l i n g load, lowest eigen- frequency, e t c .

The equality constraints hk c m consist of a s e t o f ex- p l i c i t relations between the desi yn variables (and they can be used then t o reduce the number of the design va- r i a b l e s j ; inostiy they consist of a s e t of iinplicit ana- l y s i s equations which are used t o calculate the beha- viour of the structure

the range of variation of tiie design variables. They can

be derived from various considerations such as functio-

n a l i t y * fabricitifin o r aesthetics ~

The objective function F and the constraints c j j and hk niay be linear or nonlinear functions of the desiyn va- ri ables .

Most optimization algorithms require that a n i n i t k t 1 s e t o € design variables,

zo,

i s specified. Starting from

t h i s i n i t i a l s e t , the design I s updated i t e r a t i v e l y . Probably the most common form of t h i s i t e r a t i v e proce- dure i.s given by

structure.

::ions o€ the structure.

-4-

where q i s the iteration number and C i s a vector search

direction i n the design space. The scalar quantity Q

defines the distance that we wish t o move i.n the direc- t i o n C . So the iteration s t e p ( 6 ) consists of two parts:

1. F i n d i n g a convenient search direction S. For that purpose the most e f f e c t i v e optimization algorithms

require gradients t o be calculated of the objective

function and of the constraints w i t h respect t o the

design variables.

*

2.

Findi.ng the scalar Q 50 that, moving i n the direc-

t i o n S I the objective function i s miniitii.zed. A great

number o f matheniatical programming techniques i linear

and nonlinear, can be used for the execution of

it.erati.on step ( 6 )

.

Several numerical software Libra- r i e s provide many appropriate subroutines as b u i l d i n g blocks for optj.nii.zation methods. A wj-dly used method f o r the solution of structural optimization problems

i s the solution of a sequence of linear programming

problem [ S1,P-nieZ:hod) . I n this method the optimiza-

tion problem i s 1j.nearj.zed about the current solution

5"' w i t h i n a restricted area, specified by so-called "move limits" f o r the design variables.

The f i n i t e element method has the following d i s t i n c t ad- vantages

f o r numerical optimization.

*

as an analysing module i.n computer progranis

1.

2 .

A great v a r i e t y of structures can be modelled i n a

general and f l e x i b l e way due t o the great number o f

element types available for different theories i n ap-

plied mechanics; s o very r e a l i s t i c engineering models

The numerical accuracy of the analysis can be con-

t r o l l e d e a s i l y by an appropriate modification of the f i n i t e element model. I n optimization problems one is

often interested i n only s l i g h t improvements of a structure, c o a r e a l i s t i c model and accurate analysis

results are of great. importance.

C.ZQ he formulated and an;.i?ye- clJr,d.

A major problem i n the optimization procedure i s the l i n k between the s e t of design variables and the f i n i t e element model. The desj.gn variables are user-oriented whereas the f i n i t e element model i s computer oriented.

I n our opt.imj.zation program DYNOPT [ 1 , 2 ] for: insi:ance, t h i s link i s realized €or two-dimensional problems. On the other hand, i n general purpose f i n i t e element

packages the incorporation o€ the design variable con-

cept has just s t a r t e d with the most simple types of va- r i a b l e s , such as cross-sectional dimensions.

I n the f i e l d o f computer graphics the rapid development o f s o l i d modelling programs can become of great iinpor-

tance for the solution of this shortcoming, becau:ie the

i n p u t needed f o r an advanced s o l i d modelling prograiii i s much more user oriented t h a n a f i n i t e element model, and the l i n k between a s o l i d model and a f i n i t e element mo- del h a s already been made.

By s e n s i t i v i t i e s of a c e r t a i n function we understand i n

this paper the gradj-ents o f the function w i t h respect t o the design variables.

The s e n s i t i v i t i e s o f the objective function can usually

be calculated straight-forwardly by perturbation o f each

o f the design variables. W i t h

zo

as the current design variables .it follows t h a t :

where &xi i s a smaii perturbation of

xp.

The s e n s i t i v i t i e s o f the c o n s t r a h t s usually require the intermediate calculation of the s e n s i t i v i t i e s o f d i s -

placements since most commonly constraints (i. e .

,

s t r e s s limits) are d i r e c t l y related t o the displace- ment s

.

We f i r s t consider a linear f i n i t e element problem w i t h sti3t.i.c loading. T h i s problem is described by the matrix equation

Kg = p ( 8 )

where K i s the s t i f f n e s s matri.x of the structure g i s a coluitin matrix o f unknown displacements

-6-

We d i f f e r e n t i a t e ( 8 ) w i t h respect t o design vari.able xi

Rearranging and multi.plyiizg by K-' g.i.ves

The s e n s i t i v i t i e s on the right-hand-side of ( I O ) can e a s i l y be calculated by the perturbation method (as i n

( 7 ) ) . Assuming that the matrix K was decomposed during a pievious analysis

,

K-' i s e € € e c t j vely available

,

so the

calculation o f

h/axi.

is

For a linear e l a s t i c dynamically loaded struckure the s e n s i t i v i t i e s o f the eigenfrequencies w.. can be derived

J

i n f a c t , straightforward.

:i

where K is the stiffness matrix, M is the mass matrix

and y . ₇ i s the eigenvector coupled w i t h the ei.genfre- qüeiicy w Again the sensj.ti.vi.ti.es of K and M can be calculated by a perturbation method. I f we assuine that the eigenfrequencies and eigenvectors are available from a previous analysis, then the evaluation of ( î l ) is again straightforward.

I f t h e l i n k between design variables ( i n a broad sense] and the f i n i t e e l f t n ~ n t n!odel i s incorpoiated ia f i i i i t e

element packages, it w i l l n o t be too diffj.cult t o calcu- l a t e the mentioned s e n s i t i v i t i e s as a standard o u t p u t option of the package. T h i s w i l l briny structural o p t i - mization by mans o f s e n s i t i v i t y analysis much closer t o

a great many of the engineering people.

j .

3. EXPERIMEMTAL DESIGN THEORY

The development of the experimental design theory (EDT)

- 7 -

evaluation of physical experiments and for the control

of production q u a l i t y [3j. We are using techniques based

on this theory i n together w i t h the i t e r a t i v e structural optimization procedure. T h i s application can be seen as.

the p l a n n i n g and evaluation of numerical experiments,

i,. e . f i n i t e element analyses. I n contravention of the

expeririiental design l i t e r a t u r e , we describe EDT i n ternis of structural optimization according t o the notation i n section

2.

Together w i t h t.he description o f the common theory, we w i l l discuss the extension we have made, namely the use of s e n s i t i v i t i e s i n EDT. Also some impllcatj.ons w i l l be discussed of the f a c t t h a t the numerical process i s de- terministic.

When a structure can be described by n design variables

3 according t o ( I ) , we may search f o r functions

y j = y j ( g ) j = l r . . . i m i n a certain limited area

( 1 2 )

The functions y : have t o do w i t h several properties of f

the structure such as weight

,

displacements o r stresses

a t certain points. I n other words, they represent the

objective function and the constraints i n a structural

optiinization problem. I n the following we w i l l comider only OE5 prsperty ?' y and f û ï bxevity we ömit t h e index

j .

LINEAR NODEL

To f i n d the r e l a t i o n

Y = Y($ ( 1 4 3

- 8 -

i n which :

Bi, i = l c . . . f k ; the unknown parameters we are looking

f i , i = ? ,

. . . ,

k ; t h e known functions of the design varia-

for, the model i s l h e a r i n the B i s .

bles; we can choose both linear and nonlinear func-

t i o n s f o r them.

e ; a variable which i s a stochastic error at physical

experiments, or a deterministic model error a t de-

terministi c f i n i t e eleinent analyses. EXPERIMENTAL DESIGN

The formulation of an experimental design (ED) implies

the iiiakiny of a programme of measurements. I n a nurtieri- c a l experimental design a "measurement" may consist of a f i n i t e element analy:.; is. More concrete the formulation

of an ED implj-es:

1 .

The choice of discrete values ( l e v e l s ) f o r a l i design

2.

The choice of certain combinatims o f l e v e l s «€ clif- variables x i .

f erent

xi

s

experiment o r Each one f i n i t e element analysis. combination leads

t o une physical

ESTIMATION OF THE PARAMETERS B,

I f we have a number of p combinations of levels i n the

ED, speci.fied by the p column matrices

Y "p

_- X' _,"I i _{" L f " '} Y -

and we analyse the structure a t these p s e t s of design

variable values, then according t o ( 1 5 ) we write

I n matrix notation

2 = X @ 3-

-9-

where: X i s a p * k inatrix consisting of the components: y is a column matxix w i t h p observations o f y

e

is a column matrix w i t h p errors

@ i s a column inatri.x w i t h k unknown parameters.

Xt,i = f i ( x t ) , t=l, . . . , p ; i=I,...,k

Generally, the number o f observations p exceeds the num-

ber of parameters k and a least-syu+res technique should

be applied.. I n t h a t case estimates @ f o r the parameters @ are calculated i n such a way t h a t the residual sum of square.; E?, i s winimuin, w i t h

. L

3KS

Setting 3 = O for .i,-1 I . .

.

, k

39-i

results i n a s e t o f k linear eyuatj-ons

from which the estimates p can be calculated i f t.he na-

T T

t r i x (X X) i s not singular. A regular matrix (X X ) can

be obtclined by a n appropriate formulation o f the ED. Assuniing (XTX) i s regular it follows from

(20)

that

S u b s t i t u t i n g ( 1 8 1 i n ( 2 1 1 gives :

The estimates

i

are unbiassed f o r a stochastic experi-

-10-

and consequently the expected value of $ j.5:

On the other hand the estimates @ are biassed i n the

case of deterininistic observations, such as t.he results

o f f i n i t e element analy:jes, since equation (23) does not; hold for model errors, see ( I ? ) . Nevertheless i t proves

t o be successful t o estimate @ in t h i s way, and t o achieve a linear model t o predict values f o r y _{y ( ? )} which areA su f f i c i e n t l y a c p r a t e

.

U:.;ing the estimated pa- rameters

&

the estimate y f o r the function y cas be calculated from :

THE USE O F SENSITIVITIES TO ESTIMATE @

Differentiation of the natiiematical model ( I 5 ) w i t h ïes- peck t o the design variable xpi gives

i n Section 2 we saw that for several s i t u a t i o n s sensiti- v i t i e s can be calculated r e l a t i v e l y easy, because the matrices needed f o r the calculation of the result: y can be used, see (10) and ( 1 1 ) . I n our experience the coinpu- tational e f f o r t for the calculation of the n s e n s i t i v i - t i e s of y i s only a sinall fraction of the e f f o r t for the calculation of the function value of y i t s e l f . Conside- r i n g t h i s , ( 2 6 ) can be used advantageously, together w i t h ( 1 5 ) , t o estimate the parameters fl. For t h a t pur- pose we extend the s e t of equations (17) w i t h t:he f o l -

-1 1- I n niatrix notation C o n b i n a t i o n of ( 1 8 ) and ( 2 8 ) gives De f i n i n g and

x*T

= [X X'] ii leas-i-squares solution OF

*

f ( 3 0 ) y

= x

@

can be used t o e s t h a t e the parameters [3 as, see ( 2 1 3

i

= (X*T x * ) - l X f T

_Y

*

_{( 3 1 )}

The f u n c t i o n s f i , ( x ) t h a t appear i n ( 1 5 ) are mostly o f a form l i k e

where exp may have any value i n the r a n g e j

expj = O,

1 , .

.

.

, m

-12-

( 3 3 ) provided a l l functj.ons f i ( x ) are different.

Model ( 1 5 ) then be(:oIae:j a polynomial of the order m. Ge- nerally the highest exponent i s EOS: the came for a l l de- sign variables i n the inodel. We t r y t o attune these ex- ponents t o the expected behaviour o f the structure By

means of explorative calculations, The collection o f

terms i n the model w i t h only one and the same design va- r i a b l e i s c a l l e d a main e f f e c t . For instance

can be the main e f f e c t of x j up t o the order 3 . General- l y the model a l s o contains so-called i.n.teractions i n the form o f products o f two or more different design varia-

bles. The highest exponent of a certain design variable

i n the interaction terms is derived from the order: o f .the concerning main e f f e c t ; niostly we r e s t r i c t the model

t;o f i r s t and second order interaction terms, such as

The formu1at:i.m of the ED implies the choice of sets o f

di.screte values for the design variables w i t h which %he

structure w i l l be ana1y:ied. We proceed as follows. An estimation o f the range of interest i s made for the

individual design variables; experience and explorative

calculations a r e used for t h i s purpose. With each design variable the range i s divided i n one or more intervals,

m o s t l y of the same length, resulting i n a certain nuniber

o f l e v e l s of the design variable values. T h a t number has

%o be matched t o the order o f the concerning main ef- f e c t ; it: M i l l be clear that it i s important whether or not s e n s i t i v i t i e s s h a l l be used in the Ell. Appropriate numbers o f l e v e l s are a requirement t o obtain a regular

niatrix (X X

1

~

I n io so-called complete ED a l l possible combinations o f

levels appear. Generally this results i.n a very large

-13-

nuìtiber of analyses t o be carried out. A so-called frac-

ber of analyses needed for a complete ER. A fractional

ED can be formulated I n such a way that it gives a mini- mum loss of accuracy w i t h respect t o t h e complete ED. For the case no sensj.tivj.ti.es are used i n the ED, tables for fractional ED'S can be found i n the literature [ 6 ] . tional ED only needs a 2 , 1 3, 1 q , . . . 1 fraction of the nurn-

REDUCTION OF TEE MODEL

A

After the parameters @ have been estimated by ( 3 1 ) we t r y t o reduce the number of paraineters i n the matheinati- cal model by considering the r e l a t i v e influence of the individual terms i n the inodel. For that purpose the va-

riance-covariance matrix Y(@) i s calculated, where

V j . j , i # j : co-variance between B i and p j

It can be shown that

: variance of 8 .

'ii 3.

where KS,: d u f :

residual sum of squares, see ( 1 9 )

Y;l;mb2ï û2 degrees cif freedom, which i s tile number of rows i n the 1natrj.x X rni.nus the i i a ~ b e r k of the estiïiiates

@ .

*

To reduce the mathematical model we range i t s terms w i t h respect t o the value of

I i = l t . . . s k ( 3 5 )

Pi

i. i

The term j.n the model a t whj.ch ( 3 5 ) proves t o be the *i-

niliium, may be expected as the least important: one. Sub- sequently we omit one o r inore of least important terms and make new estimates f o r the reduced number of pasaine-

ters [s. The reduction of the model i s controlled by a graph 02 the residual sun of squares RC, against the number k of parameters i n the model, see Figure 1.

- 1 4 -

I

I

I

Figuïe 1. Residual sum of squares vexsus number of

parameters

e .

The procedure mentioned above for reduckion of the model

i s repeated until a rapid growth o f KS, occurs. Just be- fore this point the model has the lowest nuznber of

terms, giving the highest predictional, c a p a b i l i t y . I n terms of the theory o f experiniental design this reduced niodel is the b e s t one.

4. APCSICATICNS GF REDuLED Î;IOUELs : INTEGRATION WITH THE

ITERATIVE OPTIMIZATION PROCEDURE

The procedure described i n Section 3 i s suitable for the derivation of reduced inathemati cal models f o r the beha- viour o f structures. The rnodels, based on a sei: of f j

-

n i t e element analyses! can be used in a number o f ways:

1. d i r e c t l y , as an accurate and easy t o evaluate niathe-

aiiatical model o f the structure. These iiiodels can be implemented on a microcomputer a t a design o f f i c e

.

For that purpose a l i b r a r y o f models can be b u i l t up.

The models can also be considered as systematic para-

meter studies.

2.

as a f a s t analysing module i n an optimization pro- gramme, operating according t o the i t e r a t i v e proce-

dure as descrjbed i n Section

2.

The resulting optimum

-15-

3 .

or can serve as a h i g h q u a l i t y starting point: for the i t e r a t i v e procedure i n which a f i n i t e element package i s the analysing module. Because a d i r e c t f i n i t e ele-

ment analysis will be more accurate t h a n reduced mo-

del evaluations, an improvement of the optinia1 struc-

tural design i s t o be expected.

as a fast analysing module i n a computer programme

for simulation of the behaviour of a structure I

The integration o f the i t e r a t i v e optiiflj.zation procedure w i t . h a numerical experiinental design proves t o be very

successful. The reasons are that both procedures have

much i n common. I n each procedure

- a considerable number o f f i n i t e element analyses o f s l i g h t l y modified structures has t o be carried out. - the concept o f design variables i s used; a l i n k be-

tween these variables and the f i n i t e element model. i s needed.

- the s e n s i t i v i t i e s of the structural behaviour have t o be calculated and can be used w i t h great advantage.

On the other hand both procedures complement each other

i n a rather nice way due t o :

- the explorative parameter s t u d y capability of the re- duced model by which a good starting point f o r the i t e r a t i v e procedure can be found.

- the p o s s i b i l i t y t o obtain h i g h q u a l i t y optimal solu- tions by means o f the i t e r a t i v e procedure based on d i - r e c t f i n i t e element: analyses.

-

the p o s s i b i l i t y t o carry out a fast optinlization run, i f a reduced model i.s used as analysing module i n tiie i t e r a t i v e procedure.

5. SOME APPLICATIONS: DESIGN O F A MAJOR-THIRD BELL

I

We have derived several reduced niathematical models and

have applied them i n structural optimization problems

such as:

- the behaviour o f penhole j o i n t f o r the connection of a

-16-

- shape opt.intization o f a l i n k of a chain for a conti- - op t i m i zat i on o f the c L' o s s - s e c ti ona

1

d j. inen s j. ons o f a

1

u -

nuous variable transmission. minum beains .

recent and very nice application of numerical experi-

mental design concerns the design o f a :m-called major-

t h i r d b e l l , and w i l l be discussed i n d e t a i l [ 1 , 2 ] .

PROBLEM DEFINITION

Figure

2.

Church b e l l and i t s cross section. These b e l l s a r e characterized by a range o f higher liar- inoni c s forming a ininor-third chord. Especially the b e l l s

of a c a r i l l o n a r e tuned very precisely i n order t o achieve a consonant sound; the t u n i n g i s carried out by s l i g h t modifications o f the inner contour of the b e l l by means of inetal cxt.ting processes. Froin a musical point o f view it would be interesting t o design bells which are characterized by a major-third chord i n t h e i r range

of higher harmonics. Figure 3 shows a qualitatAve pic-

ture o f the vibration modes of the b e l l i n cross-sec-

tions through, respectively perpendicular t o the rota- tional a x i s .

-17- 2 5 P 3 6 4

Figure 3 . Vibration riiodes of a b e l l .

Table

1

shows t h e r a t i o s of the most iniportant eigen- frequencies €or both a minor- and a major-third b e l l .

The code indicates the relevant vibration mode as a com-

bination of an a x i a l and a transversal mode according t o Figure 3 . F o r p r a c t i c a l reasons the frequencies are nor- rtialized w i t h respect t o the octave. I n the practice of the b e l l foundry the frequency r a t i o are measured i n so- called cents, a measure which i s defined as

-18-

1

200

*LOC ( 4 f / f 0 )

freq. rati.0 i n cents = LOC(2) ( 3 6 1 where f o : frequency of the octave

f : considered frequency 1 t F r e qu en cy rat i o 3cocf;ave / f -

1

[cents] 4.0000 2.0000 1 .E818 1 . 5 8 7 4 1.3348 1 .o000 O. 6674 0 1200 1500 1600 1900 2 400 3100 0.5000

I

3 6 0 0

Table I . Most important frequency ratias o f a b e l l . The reader is referred t o [ 4 , 5 ] for iiiore inforinatj.on

about the musi.cal aspects of b e l l s .

A minor-third and a major-third b e l l only d i f f e r i n t h e t h i r d toner as can be seen i n Table 1; the other fre-

quency r a t i o s have t o be just t h e same.

Experienced b e l l founders have tried t o design major- t h i r d b e l l s by means o f a t r i a l and error approach. I n s p i t e of considerable empirical e f f o r t needed for shape variations, they d j d not succeed.

We solved the problem as a structural shape optimization

problem by mearis of a mathematical inodel as described i n Section 3 . For that purpose the b e l l is modelled as an

axisymmetric structure subjected t o a non-axisymmetric

-19-

GEOMETRIC MODET, OF THE BELL

The geometry of the b e l l can be described i n a flexi.ble

way be nieans of so-called cubical B-splines

[2]

and i s shown i n Figure 4 .

Figure 4 . Geometric definition of the b e l l .

The r, z-coordinates a r e specified for a certain number

of points P i . Subsequently a cubical B-spline function

is used t o f i t . a smooth curve, t h e so-called p r o f i l e

curve!, going through these points. An other coordinate

s, along the p r o f i l e curve, i s defined. Next the wall thickness f:. of the b e l l i s defined i n a number of

points on t i e p r o f i l e curve; these points need not be

pers6 t h e same point.; Pi f o r which r and z were speci- f i e d . The wall thickness i s taken perpendicular t o the p r o f i l e curve. In intermediate points of the curve the wall. thickness i s again defined by means of a cubical B-

s n l i n o ;= *I-.,. - , . - . - : 1 : - - -

coordinates of the points P i can be taken on constant values withoui: l o s s of generality of the geometric mo- d e l . The specified r-coordinates r i together w i t h the specified wall thickness t _j

r i a b l e s i n a shape optimization paoblein.

LUll\rIrLwll 1 1 1 L . J i c L . U L V J . L L L I ~ ~ ~ L coordinate S . The Z-

b u i l d up a s e t of design vc1-

FINITE ELEMENT MESH : EXPLORATIVE CALCULATIONS

isTe analysed the b e l l using the optiuiization program DYNOPT [l]. I n DYNOPT the f i n i t e element method i s used

as an analysing inodule . W i t h this programme eiyenfre-

quencies and vibration modes can be calculated o f an

axisymnietric structure w i t h non-axisy1timetri.c deforma- ti on. For t.hi s purpose two appropriate f i n i t e elements

-20-

are implemented, naniely a 6-node and a 8-node iaopara-

metric ring element w i t h curved element edges.

I n DYNOPT the SLP-niethod, mentioned i n Section

2

i s used as an optimization iiiodule.

The optimization problem can be specified i n the input: f j.le of DYNOPT by the design variables

and the objective function. I n addition i n the i n p u t f j2.e a combinati on of s e t s of design variable values

,

according t o (161, can be specified. T h i s i n p u t defines

the f i n i t e element analyses t o be carried out i n a nu-

merical experimental design as described i n Section 3 .

For shape o p t i mization purposes DSTNOPT incorporates the p o s s i b i l i k y t:o specify an user supplied subroutine as a l i n k between the design varj.afsles and the f i n i t e element model. For the b e l l S U C h a user supplied subroutine i s

based on the geometric ;Rodel described i n the previous

section [ 2 ] .

F i n a l l y i n DYNOPT sensitivltj.ec of the eigenfrequencies can be calculated.

We made some explorative calculations f o r the b e l l . It

has been proved t h a t the eiyenfreyuencies can be calcu- lated s u f f i c i e n t l y accurate brhen 15 8-node elements are

the constraints

Figure 5 . a .

Figure 5 . (a) Nesh and ( b ) f i r s t estimate f o r the geomtry of a major t h i r d b e l l .

With the explorative calculations we a l s o considered the s e n s i t i v i t i e s of the eigenfrequencies. Based on those s e n s i t i v i t i e s we made a f i r s t estimate a t the geoiiietry of a major-third b e l l , see Figure 5 . b . Notable i s the s l i g h t bump a t h a l f the height of the b e l l . The fre- quency r a t i o s of this b e l l are not close enough t o the

-21 -

desired values such that the major-third b e l l cannot yet be realized by tuning the b e l l .

NUMERICAL EXPERIMENTAL DESIGN

I n order t o proceed w i t h the search for the major t h i r d b e l l we formulated an experimental design about the geo- metry according t o Figure 5 . b . With t h a t design 5 radii

and

2

thicknesses were chosen as the design variables;

they a r e indicated i n Figure 5 . b.

We varied a l l design variables on two l e v e l s , namely 5% higher and 5% lower than the nominal values. S e n s i t i v i t y values were a l s o used t o estimate the parameters, so a

mathematical model of order 3 could be chosen.

For all frequency r a t i o s , except the octave which i s the reference (see Table I ) , the same model containing 48 unknown parameters @ i s postulated. O f course the e s t i - mated parameters @ €or t.he individual r a t i o s are diffe- rent.

We used a complete experimental design f o r the 7 design variables varied on

2

l e v e l s , so

2

exp7 = 128 f i n i t e

element analyses had t o be carried out. Subsequently the

parameters $ have been estiinated.

The resulting mathematical models proved t o be very ac-

curate and they are suitable t o predict the results o€

d i r e c t f i n i t e element analyses very c l o s e l y .

OPTJMIZATION USING THE DERIVED ~ ~ ~ H ~ M A T I C A L MODELS I n order t o f i n d the geometry o€ a major-third b e l l , a

zero-order optimization was carried out, using the de-

rived mathematical nioGeis. we proceeäeä as follows. For

each frequency r a t i o a s u f f i c i e n t l y narrow band about the desired value i s defined

<

ri (ideal)

i

r!f i-1,

...,

7

“i.

-

( 3 7 )

Tiie range of variation o€ each design varj.able i.:i de- vj.ded i n a rather large number of equl d i stant intervals. Thus a narrow spaced 7--diinensional g r i d i s specified on the design variable space. Subsequently the following

-23-

w i t h , cast a bel1 according the proposed geometry. T h i s b e l l , w i t h a greatest: diazrieter o f about

1

m.

,

could be

tuned exactly t o the frequency rati os characterizing t h e major-third b e l l and it can be considered as the first; r e a l major-third bell i n the world.

6 . CONCLUSIONS

The experimental design theory, extetidec2 w j t h the use of

s e n s i t i v i t i e s t o estimate the model parameters

,

proves to be a successful t o o l far t h e planning and evaluation

of numerical experiments. Especially the integrati.on, w i t h the i t e r a t i v e opti rnl zation procedure

,

provides a useful optimization method for ;il a r g e class o f structu-

ral optimization problems.

The f i r s t r e a l major-third b e l l has been realized.

LITERATURE

[ I ] F. van Asperen ( 1 9 8 4 ) , Het optiinaliseren van de

eigenfrequenties van axiaalsynimetrische construc- t i e s toegepast op een luid- of carrilionklok, Re-

port WFW-84. O 12, Techn

.

Univ. Eindhoven ( i n Dutch)

.

[2]

P. Maas (19851, Onderzoek naar de geometrie van een

grote-terts klok, Report WFW-85.027, Techn. Univ.

Eindhoven C j.n Dutch)

.

[ 3 ] U . A . Cox (19581, Planning of Experiments, John Wiley 6r Sons.

[4] And16 Lehr 119761, Leerboek der campanologie, Na-

tj onaal Beiaardmuseum, Asten, Holland ( i n Dutch)

.

[SI R. Perrin, T. Charnley, J. De Pont, Noma1 modes of

the inodern English church bell, Journal o f Sound

[6] W.G. Cochran and G.M. Cox, Experimental designs,

and ?ihLUtj.G:: ( 1983 1 3û 1 ;

,

29-49. John Wiley 6r Sons, 1957.

-22-

Take a gridpoint: and evaluate the first frequency r a t i o . I f the side constraints ( 3 7 ) a r e not violated then pro- ceed w i t h the next frequency ratio; otherwise pass t o

the next g r i d point. I f all frequency r a t i o s for a cer-

tain g r i d point satisfy ( 3 7 ) , then note this point as a possible solution f o r the geometry of A major t h i r d b e l l .

Executing t h i s procedure several possible geometries

were found. Verified by means of d i r e c t f i n i t e element analyses, they proved t o be indeed very close t o the

ideal. Figure 6 shows some of the resulting geometries.

Note t h a t a i l the geometries show a inore o r l e s s pro-

nounced bump a t h a l f the height o f the b e l l .

.4

. -675

Figure 6 . Same possible geometries f o r a major- .

t h i r d b e l l .

,

After some s l i g h t modifications of the most promising geometry the b e l l foundry, that we axe co-operating