Computers and Structures 83 (2005) 315–326

www.elsevier.com/locate/compstruc

A review of robust optimal designand its application in dynamics

C. Zang a, M.I. Friswell a,*, J.E. Mottershead b

a Department of Aerospace Engineering, University of Bristol, Queen’s Building, Bristol BS8 1TR, United Kingdomb Department of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, United Kingdom

Received 13 November 2003; accepted 9 October 2004

Abstract

The objective of robust design is to optimise the mean and minimize the variability that results from uncertainty

represented by noise factors. The various objective functions and analysis techniques used for the Taguchi based

approaches and optimisation methods are reviewed. Most applications of robust design have been concerned with static

performance in mechanical engineering and process systems, and applications in structural dynamics are rare. The

robust design of a vibration absorber with mass and stiffness uncertainty in the main system is used to demonstrate

the robust design approach in dynamics. The results show a significant improvement in performance compared with

the conventional solution.

2004 Elsevier Ltd. All rights reserved.

Keywords: Robust design; Stochastic optimization; Taguchi; Vibration absorber

1. Introduction

Successful manufactured products rely on the best

possible design and performance. They are usually pro-

duced using the tools of engineering design optimisation

in order to meet design targets. However, conventional

design optimisation may not always satisfy the desired

targets due to the significant uncertainty that exists in

material and geometrical parameters such as modulus,

thickness, density and residual strain, as well as in joints

and component assembly. Crucial problems in noise and

vibration, caused by the variance in the structural

dynamics, are rarely considered in design optimisation.

Therefore, ways to minimize the effect of uncertainty

0045-7949/$ - see front matter 2004 Elsevier Ltd. All rights reserv

doi:10.1016/j.compstruc.2004.10.007

* Corresponding author. Fax: +44 117 927 2771.

E-mail address: m.i.friswell@bristol.ac.uk (M.I. Friswell).

on product design and performance are of paramount

concern to researchers and practitioners.

The earliest approach to reducing the output varia-

tion was to use the Six Sigma Quality strategy [1,2] so

that ±6 standard deviations lie between the mean and

the nearest specification limit. Six Sigma as a measure-

ment standard in product variation can be traced back

to the 1920s when Walter Shewhart showed that three

sigma from the mean is the point where a process re-

quires correction. In the early and mid-1980s, Motorola

developed this new standard and documented more than

$16 billion in savings as a result of the Six Sigma efforts.

In the last twenty years, various non-deterministic

methods have been developed to deal with design uncer-

tainties. These methods can be classified into two ap-

proaches, namely reliability-based methods and robust

design based methods. The reliability methods estimate

the probability distribution of the systems response

ed.

316 C. Zang et al. / Computers and Structures 83 (2005) 315–326

based on the known probability distributions of the ran-

dom parameters, and is predominantly used for risk

analysis by computing the probability of failure of a sys-

tem. However, the variation is not minimized in the reli-

ability approaches [3], which concentrate on the rare

events at the tails of the probability distribution [4]. Ro-

bust design improves the quality of a product by mini-

mizing the effect of the causes of variation without

eliminating these causes. The objective is different from

the reliability approach, and is to optimise the mean per-

formance and minimise its variation, while maintaining

feasibility with probabilistic constraints. This is achieved

by optimising the product and process design to make

the performance minimally sensitive to the various

causes of variation. Hence robust design concentrates

on the probability distribution near to the mean values.

In this paper, robust optimal design methods are re-

viewed extensively and an example of the robust design

of a passive vibration absorber is used to demonstrate

the application of these techniques to vibration analysis.

2. The concept of robust design

The robust design method is essential to improving

engineering productivity, and early work can be traced

back to the early 1920s when Fisher and Yates [5] devel-

oped the statistical design of experiments (DOE) ap-

proach to improve the yield of agricultural crops in

England. In the 1950s and early 1960s, Taguchi devel-

oped the foundations of robust design to meet the chal-

lenge of producing high-quality products. In 1980, he

applied his methods in the American telecommunica-

Fig. 1. Different types of pe

tions industry and since then the Taguchi robust design

method has been successfully applied to various indus-

trial fields such as electronics, automotive products,

photography, and telecommunications [6–8].

The fundamental definition of robust design is de-

scribed as A product or process is said to be robust when

it is insensitive to the effects of sources of variability, even

through the sources themselves have not been eliminated

[9]. In the design process, a number of parameters can

affect the quality characteristic or performance of the

product. Parameters within the system may be classified

as signal factors, noise factors and control factors. Sig-

nal factors are parameters that determine the range of

configurations to be considered by the robust design.

Noise factors are parameters that cannot be controlled

by the designer, or are difficult and expensive to control,

and constitute the source of variability in the system.

Control factors are the specified parameters that the de-

signer has to optimise to give the least sensitivity of the

response to the effect of the noise factors. For example,

in an automotive body in white, uncertainty may arise in

the panel thicknesses given as the noise factors. The

geometry is then optimised through control factors

describing the panel shape, for the different configura-

tions to be analysed determined by the signal factors,

such as different applied loads or the response at differ-

ent frequencies.

A P-diagram [6] may be used to represent different

types of parameters and their relationships. Fig. 1 shows

the different types of performance variations, where the

large circles denote the target and the response distribu-

tion is indicated by the dots and the associated probabil-

ity density function. The aim of robust design is to make

rformance variations.

C. Zang et al. / Computers and Structures 83 (2005) 315–326 317

the system response close to the target with low varia-

tions, without eliminating the noise factors in the sys-

tem, as illustrated in Fig. 1(d).

Suppose that y = f(s,z,x) denotes the vector of re-

sponses for a particular set of factors, where s, z and x

are vectors of the signal, noise and control factors.

The noise factors are uncertain and generally specified

probabilistically. Thus z, and hence f, are random vari-

ables. The range of configurations are specified by a

set of signal factors, V, and thus s 2 V. One possible

mathematical description of robust design is then

x ¼ argminx

maxs2V

Ez kfðs; z; xÞ tk2h i

ð1Þ

subject to the constraints,

gjðs; z; xÞ 6 0 for j ¼ 1; . . . ;m; ð2Þ

where E denotes the expected value (over the uncertainty

due to z) and t is the vector of target responses, which

may depend on the signal factors, s. Note that x is the

parameter vector that is adjusted to obtain the optimal

solution, given by Eq. (1) as the solution that gives the

smallest worse case response over the range of configu-

rations, or signal factors, in the set V. The key step in

the robust design problem is the specification of the

objective function, and once this has been done the tools

of statistics (such as the analysis of variance) and the de-

sign of experiments (such as orthogonal arrays) may be

used to calculate the solution.

For convenience, the robust design approaches will

be classified into three categories, namely Taguchi, opti-

misation and stochastic optimisation methods. Further

details are given in the following sections. The stochastic

optimisation methods include those that propagate the

uncertainty through the model and hence use accurate

statistics in the optimisation. Also included are optimi-

sation methods that are inherently stochastic, such as

the genetic algorithm. These methods are generally time

consuming, and the statistical representation may be

simplified at the expense of accuracy, for example by

considering only a linear perturbation from the nominal

values of the parameters. The Taguchi methods take this

a step further, using only a limited number of discrete

values for the parameters and using the design of exper-

iments methodology to limit the number of times the

model has to be run. The response data then requires

analysis of variance techniques to determine the sensitiv-

ity of the response to the noise and control factors.

3. The Taguchi based methods

Taguchis approach to the product design process

may be divided into three stages: system design, para-

meter design, and tolerance design [6]. System design is

the conceptual design stage where the system configura-

tion is developed. Parameter design, sometimes called

robust design, identifies factors that reduce the system

sensitivity to noise, thereby enhancing the systemsrobustness. Tolerance design specifies the allowable

deviations in the parameter values, loosening tolerances

if possible and tightening tolerances if necessary [9].

Taguchis objective functions for robust design arise

from quality measures using quadratic loss functions.

In the extension of this definition to design optimisation,

Taguchi suggested the signal-to-noise ratio (SNR),

10 log10(MSD), as a measure of the mean squared devi-

ation (MSD) in the performance. The use of SNR in sys-

tem analysis provides a quantitative value for response

variation comparison. Maximizing the SNR results in

the minimization of the response variation and more ro-

bust system performance is obtained. Suppose we have

only one response variable, y, and only one configura-

tion of the system (so the signal factor may be ne-

glected). Then for any set of control factors, x, the

noise factors are represented by n sets of parameters,

leading to the n responses, yi. Although there are many

possible SNR ratios, only two will be considered here.

The target is best SNR. This SNR quantifies the devi-

ation of the response from the target, t, and is

SNR ¼ 10 log10ðMSDÞ

¼ 10 log101

n

Xi

ðyi tÞ2 !

¼ 10 log10ðS2 þ ðy tÞ2Þ ð3Þ

where S is the population standard deviation. Eq. (3) is

essentially a sampled version of the general optimisation

criteria given in Eq. (1). Note that the second form indi-

cates that the MSD is the summation of population var-

iance and the deviation of the population mean from the

target. If the control parameters are chosen such that

y ¼ t (the population mean is the target value), then

the MSD is just the population variance. If the popula-

tion standard deviation is related to the mean, then the

MSD may also be scaled by the mean to give

SNR ¼ 10 log10ðMSDÞ ¼ 10 log10S2

y2

¼ 10 log10y2

S2

: ð4Þ

The smaller the better SNR. This SNR considers the

deviation from zero and, as the name suggests, penalises

large responses. Thus

SNR ¼ 10 log10ðMSDÞ ¼ 10 log101

n

Xi

y2i

!ð5Þ

This is equivalent to the Target-is-Best SNR, with

t = 0.

318 C. Zang et al. / Computers and Structures 83 (2005) 315–326

The most important task in Taguchis robust designmethod is to test the effect of the variability in different

experimental factors using statistical tools. The require-

ment to test multiple factors means that a full factorial

experimental design that describes all possible condi-

tions would result in a large number of experiments.

Taguchi solved this difficulty by using orthogonal arrays

(OA) to represent the range of possible experimental

conditions. After conducting the experiments, the data

from all experiments are evaluated using the analysis

of variance (ANOVA) and the analysis of mean

(ANOM) of the SNR, to determine the optimum levels

of the design variables. The optimisation process con-

sists of two steps; maximizing the SNR to minimize

the sensitivity to the effects of noise, and adjusting the

mean response to the target response.

Taguchis techniques were based on direct experimen-

tation. However, designers often use a computer to sim-

ulate the performance of a system instead of actual

experiments. Ragsdell and dEntremont [10] developed

a non-linear code that applied Taguchis concepts to de-

sign optimisation. In some cases, the optimal design is

the least robust, and designers have to make a trade-

off between target performance and robustness. Ideally,

one should optimise the expected performance over a

range of variations and uncertainties in the noise factors.

Although Taguchis contributions to the philosophy

of robust design are almost unanimously considered to

be of fundamental importance, there are certain limita-

tions and inefficiencies associated with his methods.

Box and Fung [11] pointed out that the orthogonal array

method does not always yield the optimal solution and

suggested that non-linear optimisation techniques

should be employed when a computer model of the de-

sign exists. Better results were achieved on a Wheatstone

Bridge circuit design problem used by Taguchi. Mont-

gomery [12] demonstrated that the inner array used for

the control factors in the Taguchis approach and the

outer array used for noise factors, is often unnecessary

and results in a large number of experiments. Tsui [13]

showed that the Taguchi method does not necessarily

find an accurate solution for design problems with

highly non-linear behaviour. An excellent survey of

these controversies was the panel discussion edited by

Nair [14].

Taguchis approach has been extended in a number

of ways. DErrico and Zaino [15] implemented a modifi-

cation of the Taguchi method using Gaussian–Hermite

quadrature integration. Yu and Ishii [16] used the frac-

tional quadrature factorial method for systems with sig-

nificant nonlinear effects. Otto and Antonsson [17]

addressed robust design optimisation with constraints,

using constrained optimisation methods. Ramakrishnan

and Rao [18,19] formulated the robust design problem

as a non-linear optimization problem with Taguchis lossfunction as the objective. They used a Taylor series

expansion of the objective function about the mean val-

ues of the design variables. Other significant publica-

tions in this area include those by Mohandas and

Sandgren [20], Sandgren [21], and Belegundu and Zhang

[22]. Chang et al. [23] extended Taguchis parameter de-

sign to the notion of conceptual robustness. Lee et al.

[24] developed robust design in discrete design space

using the Taguchi method. The orthogonal array based

on the Taguchi concept was utilized to arrange the dis-

crete variables, and robust solutions for unconstrained

optimization problems were found.

4. Optimisation methods

The optimisation procedures aim to minimise the

objective functions, such as Eq. (1), directly. The uncer-

tainty in the noise factors means that the system perfor-

mance is a random variable. One option for the robust

optimisation is to minimize both the deviation in the

mean value, jlf tj, and the variance, r2f , of the perfor-

mance function, subject to the constraints, where

lfðx; sÞ ¼ Ez½fðs; z; xÞ;

r2f ðx; sÞ ¼ Ez ðfðs; z; xÞ lfðx; sÞÞ

2h i

: ð6Þ

The quantities of mean and the standard deviation of

system performance, for given signal and control fac-

tors, may be calculated if the joint probability density

function (PDF) of the noise factors is known. For most

practical applications these PDFs are unknown, but of-

ten it is assumed that all variables have independent nor-

mal distributions. In this case the joint PDF becomes a

product of the individual PDFs. However, evaluating

Eq. (6) is extremely time consuming and computation-

ally expensive and approximations using Taylors seriesexpansions about the mean noise and control factors, zand x, may be used. If only the linear terms are retained

in the expansion then the mean and variance of the re-

sponse are readily computed in terms of the mean and

variance of the noise factors.

The constraints must also be satisfied. For a worst

case analysis the constraints must be satisfied for all val-

ues of control and noise factors, and the constraint in

Eq. (2) may be approximated [25] as

gjðz; xÞ þXi

ogjozi

Dzi

þX

l

ogjoxl

Dxl

6 0 ð7Þ

where the dependence on the signal factors has been

made implicit. The derivatives are evaluated at z andx, and Dzi and Dxl represent the deviations of the ele-

ments of z and x from these means. Because of the abso-

lute values this approximation is likely to be very

conservative. For a statistical analysis the constraint is

not always satisfied, and the probability that the con-

C. Zang et al. / Computers and Structures 83 (2005) 315–326 319

straints are satisfied must be chosen a priori. The con-

straints become,

gjðz; xÞ þ krgjðz; xÞ 6 0 ð8Þ

where rgj is an approximation to the standard deviation

of the j th constraint and k is a constant that reflects the

probability that the constraint will be satisfied. For

example, k = 3 means that a constraint is satisfied

99.865% of the time, if the constraint distributions are

normal. The constraint variance for a linear Taylor ser-

ies is

r2gjðz; xÞ ¼

Xi

ogjozi

2

r2ziþXl

ogjoxl

2

ðDxlÞ2: ð9Þ

The noise factors are assumed to be stochastic, whereas

the control factors are used to optimise the system and

therefore must remain as parameters in the constraints.

Parkinson et al. [25] proposed a general approach for

robust optimal design and addressed two main issues.

The first issue was design feasibility, where the proce-

dures are developed to account for tolerances during de-

sign optimisation such that the final design will remain

feasible despite variations in parameters or variables.

The second issue was the control of the transmitted var-

iation by minimizing sensitivities or by trading off con-

trollable and uncontrolled tolerances. The calculation

of the transmitted variation was based on well-known

results developed for the analysis of tolerances [26,27].

Parkinson [28] also discussed the feasibility robustness

and sensitivity robustness for robust mechanical design

using engineering models. Variability was defined in

terms of tolerances. For a worst-case analysis, a toler-

ance band was defined, whereas for a statistical analysis

the ± 3r limits for the variable or parameter were used.

A number of engineering examples showed that the

method works well if the tolerances are small. For larger

tolerances or for strongly non-linear problems, higher

order Taylor series expansions must be used. A second

order worst-case model was described by Emch and Par-

kinson [29] and a second order model using statistical

analysis was presented by Lewis and Parkinson [30].

The disadvantage of higher order models is that the for-

mulae to evaluate the response become quite compli-

cated and computationally expensive.

Sundaresan et al. [31] applied a Sensitivity Index

optimisation approach to determine a robust optimum,

but the drawback was the difficulty in choosing the

weighting factors for the mean performance and

variance. Mulvery et al. [32] treated robust design as a

bi-objective non-linear programming problem and em-

ployed a multi-criteria optimisation approach to gener-

ate a complete and efficient solution set to support

decision-making. Chen et al. [33] developed a robust de-

sign methodology to minimize variations caused by the

noise and control factors, by setting the factors to zero

in turn.

A robust design is one that attempts to optimise both

the mean and variance of the performance, and is there-

fore a multi-objective and non-deterministic problem.

Optimisation of the mean often conflicts with minimiz-

ing the variance, and a trade-off decision between them

is needed to choose the best design. The conventional

weighted sum (WS) methods to determine this trade-

off have serious drawbacks for the Pareto set generation

in multi-objective optimisation problems [34]. The Pare-

to set [35] is the set of designs for which there is no other

design that performs better on all objectives. Using

weighted sum methods, it may be impossible to achieve

some Pareto solutions and there is no assurance that the

best one is selected, even if all of the Pareto points

are available. Chen et al. [36] used a combination of

multi-objective mathematical programming methods

and the principles of decision analysis to address the

multi-objective optimisation in robust design. The com-

promise programming (CP) approach, that is the Tche-

bycheff or min–max method, replaced the conventional

WS method. The advantages of the CP method over

the WS approach in locating the efficient multi-objective

robust design solution (Pareto points) were illustrated

both theoretically and through example problems. Chen

et al. [37] made the bi-objective robust design optimisa-

tion perspective more powerful by using a physical pro-

gramming approach [38–40], where each objective was

controlled with more flexibility than by using CP. Re-

cently, Messac and Ismail-Yahaya [41] formulated the

robust design optimization problem from a fully multi-

objective perspective, again using the physical program-

ming method. This method allows the designer to

express independent preferences for each design metric,

design metric variation, design variable, design variable

variation, and parameter variation.

An alternative to CP is the preference aggregation

method, and Dai and Mourelatos [42] discussed the fun-

damental differences of these approaches. The compro-

mise programming approach is a technique for efficient

optimisation to recover an entire Pareto frontier, and

does not attempt to select one point on that frontier.

Preference aggregation, on the other hand, can select

the proper trade-off between nominal performance and

variability before calculating a Pareto frontier. CP relies

on an efficient algorithm to generate an entire Pareto

frontier of robust designs, while preference aggregation

selects the best trade-off before performing any

calculation.

Mattson and Messac [43] gave a brief literature sur-

vey on how various robust design optimisation methods

handle constraint satisfaction. The focus was the varia-

tion in constraints caused by variations in the controlled

and uncontrolled parameters. The constraints are con-

sidered as two types: equality and inequality constraints.

Discussions on inequality constraint satisfaction were

given by Balling et al. [44], Parkinson et al. [25], Du

320 C. Zang et al. / Computers and Structures 83 (2005) 315–326

and Chen [45], Lee and Park [46] and Wang and Kodial-

yam [47]. Research on equality constraint problems are

limited to three approaches; to relax the equality con-

straint [48,49,41], to satisfy the equality constraint in a

probabilistic sense [50,51], or to remove the equality

constraint through substitution [52].

5. Stochastic optimization

A slightly different strategy for robust design optimi-

sation is based on stochastic optimisation. The stochas-

tic nature of the optimisation arises from incorporating

uncertainty into the procedure, either as the parameter

uncertainty through the noise factors, or because of

the stochastic nature of the optimisation procedure.

The earliest work on stochastic optimization can be

traced back to the 1950s [53] and detailed information

may be obtained from recent books [54,55]. The objective

of stochastic optimization is to minimize the expectation

of the sample performance as a function of the design

parameters and the randomness in the system. Eggert

and Mayne [56] gave an introduction to probabilistic

optimisation using successive surrogate probability den-

sity functions. Some authors [57–59] employed the con-

cept exploration method for robust design optimisation

by creating surrogate global response surface models of

computationally expensive simulations. The response

surface methodology (RSM) is a set of statistical tech-

niques used to construct an empirical model of the rela-

tionship between a response and the levels of some

input variables, and to find the optimal responses. Lucas

[60] andMyers et al. [61] considered the RSM as an alter-

native to Taguchis robust design method. Monte Carlo

simulation generates instances of random variables

according to their specified distribution types and char-

acteristics, and although accurate response statistics

may be obtained, the computation is expensive and time

consuming. Mavris and Bandte [62] combined the re-

sponse model with a Monte Carlo simulation to con-

struct cumulative distribution functions (CDFs) and

probability density functions (PDFs) for the objective

function and constraints. All these methods depend on

the sampling of the statistics, whereby the probabilistic

distributions of the stochastic input sets are required.

One concern is that the response surface approximations

might not generate the accurate sensitivities required for

robust design [48]. The stochastic finite element method

[63] provides a powerful tool for the analysis of struc-

tures with parameter uncertainty. Chakraborty and

Dey [64] proposed a stochastic finite element method in

the frequency domain for analysis of structural dynamic

problems involving uncertain parameters. The uncertain

structural parameters are modelled as homogeneous

Gaussian stochastic fields and discretised by the local

averaging method. Numerical examples were presented

to demonstrate the accuracy and efficiency of the pro-

posed method. Chen [65] used Monte Carlo simulation

and quadratic programming. Schueller [66] gave a recent

review on structural stochastic analysis.

Optimisation approaches that are inherently stochas-

tic include techniques such as simulated annealing, neu-

ral networks and evolutionary algorithms (EA) (genetic

algorithms, evolutionary programming and evolution

strategies (ES)), and these have been applied to multi-

objective optimisation problems [67–69]. These tech-

niques do not require the computation of gradients,

which is important if the objective function relies on esti-

mating moments of the response random variables.

Gupta and Li [70] applied mathematical programming

and neural networks to robust design optimisation, and

the numerical examples showed that the approach is able

to solve highly non-linear design optimisation problems

in mechanical and structural design. Sandgren and Cam-

eron [71] used a hybrid combination of a genetic algo-

rithm and non-linear programming for robust design

optimization of structures with variations in loading,

geometry and material properties. Parkinson [72] em-

ployed a genetic algorithm for robust design to directly

obtain a global minimum for the variability of a design

function by varying the nominal design parameter val-

ues. The method proved effective and more efficient than

conventional optimisation algorithms. Additional stud-

ies are required before these methods are suitable for

application to large-scale optimisation problems.

6. Applications in dynamics

The most successful applications of robust design are

found in the fields of mechanical design engineering (sta-

tic performance) and process systems, and there have

been few applications to the robustness of dynamic per-

formance. Seki and Ishii [73] applied the robust design

concept to the dynamic design of an optical pick-up

actuator focusing on shape synthesis using computer

models and design of experiments. The response in the

first bending and torsion modes were selected as mea-

sures of undesirable vibration energy. The objective

functions were defined as the signal-to-noise ratios of re-

sponse frequencies and the sensitivities were derived

from the design of experiments using an orthogonal ar-

ray. Hwang et al. [74] optimised the vibration displace-

ments of an automobile rear view mirror system for

robustness, defined by the Taguchi concept.

In this paper the robust design approach is applied to

the dynamics of a tuned vibration absorber due to

parameter uncertainty, using the optimisation approach

through non-linear programming. The objective is to

determine the stiffness, mass and damping parameters

of the absorber, to minimize the displacements of the

main system over a large range of excitation frequencies,

m1

k1

f sin(Ωt)0

q1 q2

m2

k2

c2

Fig. 2. Forced vibration of a two degree of freedom system.

C. Zang et al. / Computers and Structures 83 (2005) 315–326 321

despite uncertainty in the mass and stiffness properties

of the main system.

The principle of the vibration absorber was attrib-

uted to Frahm [75] who found that a natural frequency

of a structure could be split into two frequencies by

attaching a small spring–mass system tuned to the same

frequency as the structure. Den Hartog and Ormonroyd

[76] developed a theoretical analysis of the vibration ab-

sorber and showed that a damped vibration absorber

could control the vibration over a wide frequency range.

Brock [77] and Den Hartog [78] gave criteria for the effi-

cient optimum operation of a tuned vibration absorber,

such as the relationship of the frequency and mass ratios

between the absorber and main system and the relation-

ship between the damping and mass ratio. Jones et al.

[79] successfully designed prototype vibration absorbers

for two bridges using these criteria. However, the effect

of parameter uncertainty and variations in the dynamic

performance have not been considered.

We consider the forced vibration of the two degree of

freedom system shown in Fig. 2. The original one degree

of freedom system, referred to as the main system, con-

sists of the mass m1 and the spring k1, and the added the

system, referred to as the absorber, consists of the mass

m2, the spring k2 and the damper c2.

The equations of motion are

m1€q1 þ c2ð _q1 _q2Þ þ k1q1 þ k2ðq1 q2Þ ¼ f0 sinðXtÞm2€q2 þ c2ð _q2 _q1Þ þ k2ðq2 q1Þ ¼ 0

ð10Þ

Inman [80] and Smith [81] should be consulted for fur-

ther details of the modelling and analysis of vibration

absorbers. Solving these equations for the steady-state

solution, gives the amplitude of vibration of the two

masses as

q1max ¼ f0c22X

2 þ ðk2 m2X2Þ2

c22X2ðk1 m1X

2 m2X2Þ2 þ ðk2m2X

2 ðk1 m1X2Þðk2 m2X

2ÞÞ2

!1=2

q2max ¼ f0c22X

2 þ k22c22X

2ðk1 m1X2 m2X

2Þ2 þ ðk2m2X2 ðk1 m1X

2Þðk2 m2X2ÞÞ2

!1=2ð11Þ

The equations may be non-dimensionalised using a

static deflection of main system, defined by

q1st ¼f0k1

ð12Þ

to give the non-dimensional displacement of mass m1

as

F ðX;m1;k1Þ¼q1max

q1st

¼ k1 c22X2þðk2m2X

2Þ2

c22X2ðk1m1X

2m2X2Þ2

.þ k2m2X

2ðk1m1X2Þðk2m2X

2Þ 21=2 ð13Þ

Suppose a steel box girder footbridge may be repre-

sented by a single degree of freedom system with mass

m1 = 17500 kg and stiffness k1 = 3.0 MN/m. The worst

case of dynamic loading is considered to be equivalent

to a sinusoidal loading with a constant amplitude of

0.48 kN at the natural frequency of the bridge. To sim-

ulate the environmental changes, the stiffness k1 and the

mass m1 are allowed to undergo 10% variations

(k1 ± Dk1, m1 ± Dm1), and a wide excitation frequency

band ð1 6 X 6 3ffiffiffiffik1m1

qÞ is considered. This ensures that

the absorber works well for a wide range of possible

excitations. The frequency X is the signal factor, and

the range of this frequency gives the set V. Using the

worst-case formulation, the standard deviations of the

mass and stiffness in the main system are rk1 ¼ 13Dk1

and rm1¼ 1

3Dm1. This standard deviation is calculated

for a uniform distribution over the 10% parameter vari-

ations and is used for the robust design optimisation.

However the original intervals are used in the simula-

tions to demonstrate the effectiveness of the solutions.

The mean response function and the standard deviation

1 1.5 2 2.5

1

1.5

2

µf / µ

f*

σ f / σ f*

123

45

6

78 9 10 11

Fig. 3. Efficient solutions of a robust multi-objective

optimization.

Table 1

The robust optimization solutions for the absorber design,

together with the textbook solution

Name a m2 (kg) k2 (N/m) c2 (N s/m) m2m1

RD1 1 203.72 34509 2000 0.011641

RD2 0.9 204.05 34576 2000 0.01166

RD3 0.8 209.79 35532 2000 0.011988

RD4 0.7 239.12 40421 2000 0.013664

RD5 0.6 251.27 42421 2000 0.014358

RD6 0.5 259.71 43817 2000 0.01484

RD7 0.4 310.37 52188 2000 0.017736

RD8 0.3 348.83 58483 2000 0.019933

RD9 0.2 410.44 68483 1999.1 0.023454

RD10 0.1 421.27 70238 2000 0.024072

RD11 0 510.93 84592 2000 0.029196

BS (book

solution)

175 29393 272.2 0.01

010

2030

40

0

5

10

150

2

4

6

8

10

Ω (rad/s)

Design Number

q 1max

/ q 1s

t

RD1RD2RD3RD4RD5RD6RD7RD8RD9RD10RD11TD

Fig. 4. Robust design results compared with the traditional

solution.

322 C. Zang et al. / Computers and Structures 83 (2005) 315–326

of the maximum non-dimensional displacement of mass

m1 may be calculated using the first-order Taylor

approximation as

lf ¼E max

16X63

ffiffiffiffik1m1

q ðF ðX;m1;k1ÞÞ

2664

3775

rf ¼E max

16X63

ffiffiffiffik1m1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffioF ðX;m1;k1Þ

ok1

2

r2k1þ oF ðX;m1;k1Þ

om1

2

r2m1

s0@

1A

2664

3775

ð14ÞHere the expected value is evaluated for the uncertain

mass and stiffness parameters m1 and k1, and F is defined

in Eq. (13). The design variables for the absorber are m2,

k2 and c2, with lower and upper bounds given by

m2 2 [10, 1750], c2 2 [10, 2000], k2 2 [100, 106]. There-

fore, the robust design of a suitable tuned vibration ab-

sorber may be formulated as follows:

Minimize : ½lfðm2; c2; k2Þ; rfðm2; c2; k2ÞSubject to : j Dk1 j6 0 1k1; j Dm1 j6 0:1m1

10 6 m2 6 1750; 10 6 c2 6 2000;

100 6 k2 6 106

ð15Þ

The first step of robust design is to seek the ideal design

(Utopia point) by a minimisation of lf and rf individu-ally as single objective functions. The ideal design ob-

tained is denoted by ½lf ; r

f . Since there are only two

objective functions, lf and rf, the two functions are com-

bined into a single objective function, G, by the conven-

tional weighted sum method discussed earlier. The

objective function is then

G ¼ alf

lf

þ ð1 aÞ rf

rf

ð16Þ

where the weighting factor a 2 [0,1] represents the rela-

tive importance of the two objectives. G is minimised

for a between 0 and 1 in steps of 0.1, and Fig. 3 shows

the results in the objective space, formed by the normal-

ized values of lf versus rf. The trade-off between the

mean and the standard deviation can clearly be ob-

served. Points 1 and 11 denote the two utopia points,

showing the minimized mean (a = 1) and variance

(a = 0) response optimization, respectively. The other

points are the optimized results from different weighting

of the mean and variance objection functions. This

weighted sum approach has similarities to the L-curve

method used in regularization [82]. The 11 optimised

solutions for the absorber parameters (mass, stiffness

and damping) and the recommended solution from

vibration textbooks [80,81], are denoted RD1–RD11

and BS, and are listed in Table 1. To visualise all of these

responses over the excitation frequency band of interest,

the non-dimensional displacements of mass m1 for these

different parameter sets are plotted in Fig. 4, where TD

denotes the text book solution.

0 10 20 30 400

5

10

15

20

25

Ω (rad/s)

q1m

ax /

q 1st

Fig. 6. Monte Carlo simulation of the response variation due to

the uncertainty (RD1).

0 10 20 30 400

5

10

15

20

25

Ω (rad/s)

q1m

ax /

q 1st

Fig. 7. Monte Carlo simulation of the response variation due to

the uncertainty (RD11).

C. Zang et al. / Computers and Structures 83 (2005) 315–326 323

Most of the damping values for the robust design

cases (except for the RD9 case) are selected as the upper

limit (2000 Ns/m). Larger damping values are able to re-

duce the amplitude of the peak response very effectively,

although in practice the amount of damping that can be

added is limited. The optimum mass ratio (m2/m1) is be-

tween 1% and 3%, and the value of the mass for the RD1

case, where only the mean response was minimized, is

close to that of the BS case. With the decreased weight-

ing on the mean response and correspondingly increas-

ing weighting on the response variance, the absorber

mass is gradually increased, and the response shown in

Fig. 4 becomes flatter. Although the peak response is de-

creased in these cases, the response near the main system

natural frequency increases.

To demonstrate the effectiveness of the robust design

approach, Monte Carlo simulation [83] is used to evalu-

ate the possible response variations of the main system

due to the mass and stiffness uncertainty in the main sys-

tem, at the different optimum points. In Monte Carlo

simulation a random number generator produces sam-

ples of the noise factors (in this case m1 and k1), which

are then used to calculate the response over the fre-

quency range of interest for a given set of control fac-

tors. The random number generator approximates the

PDF of the noise factors for a large number of samples.

The response variations then give some indication of the

robustness of the design given by those particular set of

control factors. Four representative cases, namely RD1

(a = 1), RD6 (a = 0.5), RD11 (a = 0) and BS, are inves-

tigated, and frequency response variations are plotted in

Figs. 5–8, together with the corresponding mean re-

sponse, shown by the solid line. The amplitudes of the

response for the three robust design cases (RD1, RD6,

RD11) are much smaller than those from the traditional

textbook design (BS). The RD1 case has the lowest min-

imized mean response and the most sensitive variations

0 10 20 30 400

5

10

15

20

25

Ω (rad/s)

q1m

ax /

q 1st

Fig. 5. Monte Carlo simulation of the response variation due to

the uncertainty (BS).

0 10 20 30 400

5

10

15

20

25

Ω (rad/s)

q1m

ax /

q 1st

Fig. 8. Monte Carlo simulation of the response variation due to

the uncertainty (RD6).

324 C. Zang et al. / Computers and Structures 83 (2005) 315–326

while the RD11 case has the highest mean response and

the least sensitive variations. The RD6 case is a compro-

mise between the RD1 and RD11 cases, with a reason-

able mean response and insensitive variations due to

the uncertainty. The textbook solution, shown in Fig.

5, is most sensitive to the mass and stiffness variations

in the main system, although increasing the damping

in the absorber would improve this performance.

7. Concluding remarks

The state-of-art approaches to robust design optimi-

sation have been extensively reviewed. It is noted that

robust design is a multi-objective and non-deterministic

problem. The objective is to optimise the mean and min-

imize the variability in the performance response that re-

sults from uncertainty represented through noise

variables. The robust design approaches can generally

be classified as statistical-based methods and optimisa-

tion methods. Most of the Taguchi based methods use

direct experimentation and the objective functions for

the optimisation are expressed as the signal to noise

ratio (SNR) using the Taguchi method. Using the

orthogonal array technique, the analysis of variance

and analysis of mean of the SNR are used to evaluate

the optimum design variables to ensure that the system

performance is insensitive to the effects of noise, and

to tune the mean response to the target. The optimisa-

tion approaches for robust design are based on non-lin-

ear programming methods. The objective functions

simultaneously optimise both the mean performance

and the variance in performance. A trade-off decision

must be made, to choose the best design with the maxi-

mum robustness. Recently, novel techniques such as

simulated annealing, neural networks and the field of

evolutionary algorithms have been applied to solving

the resulting multi-objective optimisation problem.

The application of robust design optimisation in

structural dynamics is very rare. This paper considers

the forced vibration of a two degree of freedom system

as an example to illustrate the robust design of a vibra-

tion absorber. The objective is to minimize the displace-

ment response of the main system within a wide band

of excitation frequencies. The robustness of the re-

sponse due to uncertainty in the mass and stiffness of

the main system was also considered, and the maxi-

mum mean displacement response and the variations

caused by the mass and stiffness uncertainty were min-

imized simultaneously. Monte Carlo simulation demon-

strated significant improvement in the mean response

and variation compared with the traditional solution

recommended from vibration textbooks. The results

show that robust design methods have great potential

for application in structural dynamics to deal with

uncertain structures.

Acknowledgments

The authors acknowledge the support of the EPSRC

(UK) through grants GR/R34936 and GR/R26818.

Prof. Friswell acknowledges the support of a Royal

Society-Wolfson Research Merit Award.

References

[1] Harry MJ. The nature of Six Sigma Quality. Shaumburg,

IL: Motorola University Press; 1997.

[2] Pande PS, Neuman RP, Cavanagh RR. The Six Sigma

way: how GE, Motorola, and other top companies are

honing their performance. New York: McGraw-Hill;

2000.

[3] Siddall JN. A new approach to probability in engineering

design and optimisation. ASME J Mech, Transmiss,

Autom Des 1984;106:5–10.

[4] Doltsinis I, Kang Z. Robust design of structures using

optimization methods. Comput Methods Appl Mech Eng

2004;193:2221–37.

[5] Fisher RA. Design of experiments. Edinburgh: Oliver &

Boyd; 1951.

[6] Phadke MS. Quality engineering using robust

design. Englewood Clifffs, NJ: Prentice-Hall; 1989.

[7] Taguchi G. Quality engineering through design optimiza-

tion. White Plains, NY: Krauss International Publications;

1986.

[8] Taguchi G. System of experimental design. IEEE J

1987;33:1106–13.

[9] Fowlkes WY, Creveling CM. Engineering methods for

robust product design: using Taguchi methods in technol-

ogy and product development. Reading, MA: Addison-

Wesley Publishing Company; 1995.

[10] Ragsdell KM, dEntremont KL. Design for latitude using

TOPT. ASME Adv Des Autom 1988;DE-14:265–72.

[11] Box G, Fung C. Studies in quality improvement: minimiz-

ing transmitted variation by parameter design. In: Report

No. 8, Center for Quality Improvement, University of

Wisconsin; 1986.

[12] Montgomery DC. Design and analysis of experiments. 3rd

ed. New York: John Wiley & Sons; 1991.

[13] Tsui KL. An overview of Taguchi method and newly

developed statistical methods for robust design. IIE Trans

1992;24:44–57.

[14] Nair VN. Taguchis parameter design: a panel discussion.

Technometrics 1992;34:127–61.

[15] DErrico JR, Zaino Jr NA. Statistical tolerancing using a

modification of Taguchis method. Technometrics

1988;30(4):397–405.

[16] Yu J, Ishii K. A robust optimization procedure for systems

with significant non-linear effects. In: ASME Design

Automation Conference, Albuquerque, NM 1993; DE-

65-1:371–8.

[17] Otto JN, Antonsson EK. Extensions to the Taguchi

method of product design. ASME J Mech Des

1993;115:5–13.

[18] Ramakrishnan B, Rao SS. A robust optimization approach

using Taguchis loss function for solving nonlinear opti-

C. Zang et al. / Computers and Structures 83 (2005) 315–326 325

mization problems. ASME Adv Des Autom 1991;DE-

32(1):241–8.

[19] Ramakrishnan B, Rao SS. Efficient strategies for the

robust optimization of large-scale nonlinear design prob-

lems. ASME Adv Des Autom 1994;DE-69(2):25–35.

[20] Mohandas SU, Sandgren E. Multi-objective optimization

dealing with uncertainty. In: Proceedings of the 15th

ASME Design Automation Conference Montreal, vol. DE

19-2; 1989.

[21] Sandgren E. A multi-objective design tree approach for

optimization under uncertainty. In: Proceedings of the 15th

ASME Design Automation Conference Montreal, vol. 2;

1989.

[22] Belegundu AD, Zhang S. Robustness of design through

minimum sensitivity. J Mech Des 1992;114:213–7.

[23] Chang T, Ward AC, Lee J, Jacox EH. Distributed design

with conceptual robustness: a procedure based on Tagu-

chis parameter design. Concurrent Product Des ASME

1994;DE 74:19–29.

[24] Lee KH, Eom IS, Park GJ, Lee WI. Robust design for

unconstrained optimization problems using the Taguchi

method. AIAA J 1996;34(5):1059–63.

[25] Parkinson A, Sorensen C, Pourhassan N. A general

approach for robust optimal design. J Mech Des ASME

Trans 1993;115:74–80.

[26] Cox NE. In: Cornell J, Shapiro S. editors. Volume II: how

to perform statistical tolerance analysis, the ASQC basic

reference in quality control: statistical techniques; 1986.

[27] Bjorke O. Computer-aided tolerancing. New York:

ASME Press; 1989.

[28] Parkinson A. Robust mechanical design using engineering

models. J Mech Des ASME Trans 1995;117:48–54.

[29] Emch G, Parkinson A. Robust optimal design for worst-

case tolerances. ASME J Mech Des 1994;116:1019–25.

[30] Lewis L, Parkinson A. Robust optimal design with a

second-order tolerance model. Res Eng Des 1994;6:25–37.

[31] Sunderasan S, Ishii K, Houser DR. A robust optimization

procedure with variation on design variables and con-

straints. ASME Adv Des Autom 1993;DE 65(1):379–86.

[32] Mulvey J, Vanderber R, Zenios S. Robust optimization of

large-scale systems. Operat Res 1995;43:264–81.

[33] Chen W, Allen JK, Mistree F, Tsui K-L. A procedure for

robust design: minimizing variations caused by noise factors

and control factors. ASME J Mech Des 1996;118:478–85.

[34] Das I, Dennis JE. A closer look at drawbacks of

minimizing weighted sums of objectives for Pareto set

generation in multi-criteria optimization problems. Struct

Optim 1997;14:63–9.

[35] Steuer RE. Multiple criteria optimisation: theory, compu-

tation and application. New York: John Wiley; 1986.

[36] Chen W, Wiecek MM, Zhang J. Quality utility: a

compromise programming approach to robust design.

ASME J Mech Des 1999;121(2):179–87.

[37] Chen W, Sahai A, Messac A, Sundararaj GJ. Exploration

of the effectiveness of physical programming in robust

design. J Mech Des 2000;122:155–63.

[38] Messac A. Physical programming: effective optimization

for computational design. AIAA J 1996;34:149–58.

[39] Messac A. Control-structure integrated design with closed-

form design metrics using physical programming. AIAA J

1998;36:855–64.

[40] Messac A. From the dubious construction of objective

functions to the application of physical programming.

AIAA J 2000;38:155–63.

[41] Messac A, Ismail-Yahaya A. Multi-objective robust design

using physical programming. Struct Multidisciplin Optimiz

2002;23(5):357–71.

[42] Dai Z, Mourelatos ZP. Robust design using preference

aggregation methods. In: ASME 2003 Design Engineering

Technical Conferences and Computer and Information in

Engineering Conference, Chicago, IL, USA, DETC2003/

DAC-48715.

[43] Mattson C, Messac A. Handing equality constraints in

robust design optimization. In: 44th AIAA/ASME/ASCE/

AHS Structures, Structure Dynamics, and Materials Con-

ference. 2003, Paper No. AIAA-2003-1780.

[44] Balling RJ, Free JC, Parkinson AR. Consideration of

worst-case manufacturing tolerances in design optimiza-

tion. ASME J Mech, Trans, Autom Des 1986;108:438–41.

[45] Du X, Chen W. Towards a better understanding of

modeling feasibility robustness in engineering design.

ASME J Mech Des 2001;122:385–94.

[46] Lee KH, Park GJ. Robust optimization considering

tolerances of design variables. Comput Struct 2001;79:

77–86.

[47] Wang L, Kodialyam S. An efficient method for probabi-

listic and robust Design with non-normal distributions. In:

AIAA 43rd AIAA/ASME/ASCE/AHS Structures, Struc-

ture Dynamics, and Materials Conference. 2002, Paper No.

AIAA-2002-1754.

[48] Su J, Renaud JE. Automatic differentiation in robust

optimization. AIAA J 1997;35:1072–9.

[49] Fares B, Noll D, Apkarian P. Robust control via sequen-

tial semidefinite programming. SIAM J Contr Optimiz

2002;23(5):357–71.

[50] Sunderasan S, Ishii K, Houser DR. Design optimization

for robustness using performance simulation programs.

ASME Adv Des Autom 1991;DE 32(1):249–56.

[51] Putko MM, Newman PA, Taylor AC, Green LL.

Approach for uncertainty propagation and robust design

in CFD using sensitivity derivatives. In: AIAA 42nd

AIAA/ASME/ASCE/AHS Structures, Structural Dynam-

ics, and Materials Conference. 2001. Paper No. AIAA-

2001-2528.

[52] Das I. Robust optimization for constrained, nonlinear

programming problems. Eng Optim 2000;32(5):585–

618.

[53] Beale EML. On minimizing a convex function subject

to linear inequalities. J R Statist Soc 1955;17B:173–

84.

[54] Birge JR, Louveaux FV. Introduction to stochastic pro-

gramming. New York, NY: Springer; 1997.

[55] Kall P, Wallace SW. Stochastic programming. New York,

NY: Wiley; 1994.

[56] Eggert RJ, Mayne RW. Probabilistic optimal-design using

successive surrogate probability density-functions. ASME

J Mech Des 1993;115(3):385–91.

[57] Chen W, Allen JK, Mavis D, Mistree F. A concept

exploration method for determining robust top-level spec-

ifications. Eng Optimiz 1996;26:137–58.

[58] Koch PN, Barlow A, Allen JK, Mistree F. Configuring tur-

bine propulsion systems using robust concept exploration.

326 C. Zang et al. / Computers and Structures 83 (2005) 315–326

In: ASME Design Automation Conference. Irvine, CA, 96-

DETC/DAC-1285; 1996.

[59] Simpson TW, Chen W, Allen JK, Mistree F. Conceptual

design of a family of products through the use of the robust

concept exploration method. In AIAA/NASA/USAF/

ISSMO Symposium on Multidisciplinary Analysis and

Optimisation, Bellevue, Washington, vol. 2(2); 1996. p.

1535–45.

[60] Lucas JM. How to achieve a robust process using response

surface methodology. J Quality Technol 1994;26:248–60.

[61] Myers RH, Khuri AI, Vining G. Response surface alter-

native to the Taguchi robust parameter design approach. J

Am Stat 1992;46:131–9.

[62] Mavris DN, Bandte O. A probabilistic approach to

multivariate constrained robust design simulation. 1997.

Paper No. AIAA-97-5508.

[63] Elishakoff I, Ren YJ, Shinozuka M. Improved finite

element method for stochastic structures. Chaos, Solitons

& Fractals 1995;5:833–46.

[64] Chakraborty CS, Dey SS. A stochastic finite element

dynamic analysis of structures with uncertain parameters.

Int J Mech Sci 1998;40:1071–87.

[65] Chen S-P. Robust design with dynamic characteristics

using stochastic sequential quadratic programming. Eng

Optimiz 2003;35(1):79:89.

[66] Schueller GI. Computational stochastic mechanics recent

advances. Comput Struct 2001;79:2225–34.

[67] Hajela P. Nongradient methods in multidisciplinary design

optimisationstatus and potential. J Aircraft 1999;36(1):

255–65.

[68] Kalyanmoy DC. Multi-objective optimization using evo-

lutionary algorithms. West Sussex, England: John Wiley

& Sons; 2001.

[69] Fouskakis D, Draper D. Stochastic optimisation: a review.

Int Statist Rev 2002;70(3):315–49.

[70] Gupta KC, Li J. Robust design optimization with math-

ematical programming neural networks. Comput Struct

2000;76:507–16.

[71] Sandgren E, Cameron TM. Robust design optimization of

structures through consideration of variation. Comput

Struct 2002;80:1605–13.

[72] Parkinson DB. Robust design employing a genetic algo-

rithm. Qual Reliab Eng Int 2000;16:201–8.

[73] Seki K, Ishii K. Robust design for dynamic performance:

optical pick-up example. In: Proceedings of the 1997

ASME Design Engineering Technical Conference and

Design Automation Conference, Sacrament, CA, Paper

No. 97-DETC/DAC3978; 1997.

[74] Hwang KH, Lee KW, Park GJ. Robust optimization of an

automobile rearview mirror for vibration reduction. Struct

Multidisc Optim 2001;21:300–8.

[75] Frahm H. Device for damping vibrations of bodies. US

patent No. 989958. 1911.

[76] Den Hartog JP, Ormondroyd J. Theory of the dynamic

absorber. Trans ASME 1928;APM50-7:11–22.

[77] Brock J. A note on the damped vibration absorber. ASME

J Appl Mech 1946;68(4):A-284.

[78] Den Hartog JP. Mechanical vibration. Boston: McGraw-

Hill; 1947.

[79] Jones RT, Pretlove AJ, Eyre R. Two case studies of the use

of tuned vibration absorbers on footbridges. Struct Eng

1981;59B:27–32.

[80] Inman DJ. Engineering vibration. Englewood Cliffs,

NJ: Prentice-Hall; 1994.

[81] Smith JW. Vibration of structures: applications in civil

engineering design. London: Chapman & Hall; 1988.

[82] Hansen PC. Analysis of discrete ill-posed problems by

means of the L-curve. SIAM Rev 1992;34:561–80.

[83] Law AM, Kelton WD. Simulation modeling and analysis.

3rd ed. Boston: McGraw-Hill; 2000.