A review of robust optimal design and its ... - Michael I Friswell

Abstract
A review of robust optimal design
and 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 Kingdom
b Department of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, United Kingdom
Received 13 November 2003; accepted 9 October 2004
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.
Ó 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 produced
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
* Corresponding author. Fax: +44 117 927 2771.
E-mail address: m.i.friswell@bristol.ac.uk (M.I. Friswell).
Computers and Structures 83 (2005) 315–326
0045-7949/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruc.2004.10.007
www.elsevier.com/locate/compstruc
on product design and performance are of paramount
concern to researchers and practitioners.
The earliest approach to reducing the output variation
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 measurement
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 requires
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 uncertainties.
These methods can be classified into two approaches,
namely reliability-based methods and robust
design based methods. The reliability methods estimate
the probability distribution of the systemÕs response

316 C. Zang et al. / Computers and Structures 83 (2005) 315–326
based on the known probability distributions of the random
parameters, and is predominantly used for risk
analysis by computing the probability of failure of a system.
However, the variation is not minimized in the reliability
approaches [3], which concentrate on the rare
events at the tails of the probability distribution [4]. Robust
design improves the quality of a product by minimizing
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 performance
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 reviewed
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] developed
the statistical design of experiments (DOE) approach
to improve the yield of agricultural crops in
England. In the 1950s and early 1960s, Taguchi developed
the foundations of robust design to meet the challenge
of producing high-quality products. In 1980, he
applied his methods in the American telecommunica-
Fig. 1. Different types of performance variations.
tions industry and since then the Taguchi robust design
method has been successfully applied to various industrial
fields such as electronics, automotive products,
photography, and telecommunications [6–8].
The fundamental definition of robust design is described
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. Signal
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 designer
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 configurations
to be analysed determined by the signal factors,
such as different applied loads or the response at different
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 distribution
is indicated by the dots and the associated probability
density function. The aim of robust design is to make

the system response close to the target with low variations,
without eliminating the noise factors in the system,
as illustrated in Fig. 1(d).
Suppose that y = f(s,z,x) denotes the vector of responses
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 variables.
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 arg min
x
h i
max
s2V Ez kfðs; z; xÞ tk 2
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 configurations,
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 design
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, optimisation
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 optimisation
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 experiments
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 sensitivity
of the response to the noise and control factors.
3. The Taguchi based methods
TaguchiÕs approach to the product design process
may be divided into three stages: system design, parameter
design, and tolerance design [6]. System design is
the conceptual design stage where the system configura-
C. Zang et al. / Computers and Structures 83 (2005) 315–326 317
tion is developed. Parameter design, sometimes called
robust design, identifies factors that reduce the system
sensitivity to noise, thereby enhancing the systemÕs
robustness. Tolerance design specifies the allowable
deviations in the parameter values, loosening tolerances
if possible and tightening tolerances if necessary [9].
TaguchiÕs 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 log 10(MSD), as a measure of the mean squared deviation
(MSD) in the performance. The use of SNR in system
analysis provides a quantitative value for response
variation comparison. Maximizing the SNR results in
the minimization of the response variation and more robust
system performance is obtained. Suppose we have
only one response variable, y, and only one configuration
of the system (so the signal factor may be neglected).
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 deviation
of the response from the target, t, and is
SNR ¼ 10 log 10ðMSDÞ
¼ 10 log 10
1
n
X
ðyi i
tÞ 2
¼ 10 log 10ðS 2 þðy tÞ 2 Þ 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 indicates
that the MSD is the summation of population variance
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 population
standard deviation is related to the mean, then the
MSD may also be scaled by the mean to give
SNR ¼ 10 log 10ðMSDÞ ¼ 10log 10
y
¼ 10log10 2
S 2 : 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 log 10ðMSDÞ ¼ 10log 10
!
S 2
y 2
1
n
X
i
y 2
i
!
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 TaguchiÕs robust design
method is to test the effect of the variability in different
experimental factors using statistical tools. The requirement
to test multiple factors means that a full factorial
experimental design that describes all possible conditions
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 consists
of two steps; maximizing the SNR to minimize
the sensitivity to the effects of noise, and adjusting the
mean response to the target response.
TaguchiÕs techniques were based on direct experimentation.
However, designers often use a computer to simulate
the performance of a system instead of actual
experiments. Ragsdell and dÕEntremont [10] developed
a non-linear code that applied TaguchiÕs concepts to design
optimisation. In some cases, the optimal design is
the least robust, and designers have to make a tradeoff
between target performance and robustness. Ideally,
one should optimise the expected performance over a
range of variations and uncertainties in the noise factors.
Although TaguchiÕs contributions to the philosophy
of robust design are almost unanimously considered to
be of fundamental importance, there are certain limitations
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 design
exists. Better results were achieved on a Wheatstone
Bridge circuit design problem used by Taguchi. Montgomery
[12] demonstrated that the inner array used for
the control factors in the TaguchiÕs 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].
TaguchiÕs approach has been extended in a number
of ways. DÕErrico and Zaino [15] implemented a modification
of the Taguchi method using Gaussian–Hermite
quadrature integration. Yu and Ishii [16] used the fractional
quadrature factorial method for systems with significant
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 TaguchiÕs loss
function as the objective. They used a Taylor series
expansion of the objective function about the mean values
of the design variables. Other significant publications
in this area include those by Mohandas and
Sandgren [20], Sandgren [21], and Belegundu and Zhang
[22]. Chang et al. [23] extended TaguchiÕs parameter design
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 discrete
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 uncertainty
in the noise factors means that the system performance
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, r2 f , of the performance
function, subject to the constraints, where
lfðx; sÞ ¼Ez½fðs; z; xÞŠ;
r 2
f ðx; sÞ ¼Ez ðfðs; z; xÞ lfðx; sÞÞ 2
h i
: ð6Þ
The quantities of mean and the standard deviation of
system performance, for given signal and control factors,
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 often
it is assumed that all variables have independent normal
distributions. In this case the joint PDF becomes a
product of the individual PDFs. However, evaluating
Eq. (6) is extremely time consuming and computationally
expensive and approximations using TaylorÕs series
expansions about the mean noise and control factors, z
and x, may be used. If only the linear terms are retained
in the expansion then the mean and variance of the response
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 values
of control and noise factors, and the constraint in
Eq. (2) may be approximated [25] as
gjðz; xÞþ X ogj Dzi þ
ozi i
X ogj Dxl 6 0 ð7Þ
oxl l
where the dependence on the signal factors has been
made implicit. The derivatives are evaluated at z and
x, and Dzi and Dxl represent the deviations of the elements
of z and x from these means. Because of the absolute
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-

straints are satisfied must be chosen a priori. The constraints
become,
gjðz; xÞþkrgjðz; 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 series
is
r 2
gjðz; xÞ ¼X
i
og j
ozi
2
r 2
X
þ zi
l
og j
oxl
2
ðDxlÞ 2 : 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 procedures
are developed to account for tolerances during design
optimisation such that the final design will remain
feasible despite variations in parameters or variables.
The second issue was the control of the transmitted variation
by minimizing sensitivities or by trading off controllable
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 tolerance
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 Parkinson
[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 formulae
to evaluate the response become quite complicated
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 employed
a multi-criteria optimisation approach to generate
a complete and efficient solution set to support
decision-making. Chen et al. [33] developed a robust design
methodology to minimize variations caused by the
noise and control factors, by setting the factors to zero
in turn.
C. Zang et al. / Computers and Structures 83 (2005) 315–326 319
A robust design is one that attempts to optimise both
the mean and variance of the performance, and is therefore
a multi-objective and non-deterministic problem.
Optimisation of the mean often conflicts with minimizing
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 tradeoff
have serious drawbacks for the Pareto set generation
in multi-objective optimisation problems [34]. The Pareto
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 compromise
programming (CP) approach, that is the Tchebycheff
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 optimisation
perspective more powerful by using a physical programming
approach [38–40], where each objective was
controlled with more flexibility than by using CP. Recently,
Messac and Ismail-Yahaya [41] formulated the
robust design optimization problem from a fully multiobjective
perspective, again using the physical programming
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 fundamental
differences of these approaches. The compromise
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 survey
on how various robust design optimisation methods
handle constraint satisfaction. The focus was the variation
in constraints caused by variations in the controlled
and uncontrolled parameters. The constraints are considered
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 Kodialyam
[47]. Research on equality constraint problems are
limited to three approaches; to relax the equality constraint
[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 optimisation
is based on stochastic optimisation. The stochastic
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 density
functions. Some authors [57–59] employed the concept
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 techniques
used to construct an empirical model of the relationship
between a response and the levels of some
input variables, and to find the optimal responses. Lucas
[60] and Myers et al. [61] considered the RSM as an alternative
to TaguchiÕs robust design method. Monte Carlo
simulation generates instances of random variables
according to their specified distribution types and characteristics,
and although accurate response statistics
may be obtained, the computation is expensive and time
consuming. Mavris and Bandte [62] combined the response
model with a Monte Carlo simulation to construct
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 structures
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 proposed
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 stochastic
include techniques such as simulated annealing, neural
networks and evolutionary algorithms (EA) (genetic
algorithms, evolutionary programming and evolution
strategies (ES)), and these have been applied to multiobjective
optimisation problems [67–69]. These techniques
do not require the computation of gradients,
which is important if the objective function relies on estimating
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 Cameron
[71] used a hybrid combination of a genetic algorithm
and non-linear programming for robust design
optimization of structures with variations in loading,
geometry and material properties. Parkinson [72] employed
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 values.
The method proved effective and more efficient than
conventional optimisation algorithms. Additional studies
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 (static
performance) and process systems, and there have
been few applications to the robustness of dynamic performance.
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 measures
of undesirable vibration energy. The objective
functions were defined as the signal-to-noise ratios of response
frequencies and the sensitivities were derived
from the design of experiments using an orthogonal array.
Hwang et al. [74] optimised the vibration displacements
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,

despite uncertainty in the mass and stiffness properties
of the main system.
The principle of the vibration absorber was attributed
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 absorber
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 efficient
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 relationship
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, consists
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€q 1 þ c2ð_q 1 _q 2Þþk1q1 þ k2ðq1 q2Þ¼f0 sinðXtÞ
m2€q 2 þ c2ð_q 2 _q 1Þþk2ðq2 q1Þ¼0 ð10Þ
Inman [80] and Smith [81] should be consulted for further
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
q 1 max ¼ f0
q 2 max ¼ f0
The equations may be non-dimensionalised using a
ÔstaticÕ deflection of main system, defined by
q1st ¼ f0
k1
c 2 2 X2 þðk2 m2X 2 Þ 2
c 2 2 X2 ðk1 m1X 2 m2X 2 Þ 2 þðk2m2X 2
c 2 2 X2 þ k 2
2
c 2 2 X2 ðk1 m1X 2 m2X 2 Þ 2 þðk2m2X 2
ð12Þ
to give the non-dimensional displacement of mass m1
as
C. Zang et al. / Computers and Structures 83 (2005) 315–326 321
F ðX;m1;k1Þ¼ q 1max
q 1st
¼ k1 c 2
2 X2 þðk2 m2X 2 Þ 2
þ k2m2X 2
k 1
.
ðk1 m1X 2 Þðk2 m2X 2 Þ 2 1=2
c 2
2 X2 ðk1 m1X 2 m2X 2 Þ 2
ð13Þ
Suppose a steel box girder footbridge may be represented
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 simulate
the environmental changes, the stiffness k1 and the
mass m1 are allowed to undergo 10% variations
(k1 ± Dk1, m1 ± Dm
qffiffiffiffi
1), and a wide excitation frequency
k1
band ð1 6 X 6 3 Þ is considered. This ensures that
m1 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
ðk1 m1X 2 Þðk2 m2X 2 ÞÞ 2
ðk1 m1X 2 Þðk2 m2X 2 ÞÞ 2
! 1=2
! 1=2
m 1
mass and stiffness in the main system are rk1
q 1
f sin(Ωt)
0
Fig. 2. Forced vibration of a two degree of freedom system.
ð11Þ
¼ 1
3 Dk1
and rm 1 ¼ 1
3 Dm1. This standard deviation is calculated
for a uniform distribution over the 10% parameter variations
and is used for the robust design optimisation.
However the original intervals are used in the simulations
to demonstrate the effectiveness of the solutions.
The mean response function and the standard deviation
k 2
c 2
m 2
q 2

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
2
3
6
lf ¼E6 4
7
max qffiffiffiðF ðX;m1;k1ÞÞ7
5
16X63
k 1
m 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rf ¼E max qffiffiffi
oF ðX;m1;k1Þ
ok1
2
r2 k þ 1
oF ðX;m1;k1Þ
om1
2
r2 2
3
0s
1
6
4
@
7
A7
m1 5
16X63
k 1
m 1
ð14Þ
Here 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 m 2,
k2 and c2, with lower and upper bounds given by
m2 2 [10, 1750], c2 2 [10, 2000], k2 2 [100, 10 6 ]. Therefore,
the robust design of a suitable tuned vibration absorber
may be formulated as follows:
Minimize: ½lfðm2; c2; k2Þ; rfðm2; c2; k2ÞŠ
Subject 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 10 6
ð15Þ
The first step of robust design is to seek the ideal design
(Utopia point) by a minimisation of lf and rf individually
as single objective functions. The ideal design obtained
is denoted by ½l f ; r f Š. Since there are only two
objective functions, lf and rf, the two functions are combined
into a single objective function, G, by the conventional
weighted sum method discussed earlier. The
objective function is then
G ¼ a lf þð1 aÞ
lf rf
ð16Þ
rf where the weighting factor a 2 [0,1] represents the relative
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 normalized
values of lf versus rf. The trade-off between the
mean and the standard deviation can clearly be observed.
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.
*
σ / σ
f f
2
1.5
1
1
2
3
4
5 6
7
8 9 10 11
1 1.5 2 2.5
*
µ / µ
f f
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
q 1max / q 1st
10
8
6
4
2
15 0
10
Design Number
5
0
0
10
20
Ω (rad/s)
RD1
RD2
RD3
RD4
RD5
RD6
RD7
RD8
RD9
RD10
RD11
TD
Fig. 4. Robust design results compared with the traditional
solution.
30
40

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 reduce
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 between
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 weighting
on the mean response and correspondingly increasing
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 decreased
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 evaluate
the possible response variations of the main system
due to the mass and stiffness uncertainty in the main system,
at the different optimum points. In Monte Carlo
simulation a random number generator produces samples
of the noise factors (in this case m 1 and k 1), which
are then used to calculate the response over the frequency
range of interest for a given set of control factors.
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 investigated,
and frequency response variations are plotted in
Figs. 5–8, together with the corresponding mean response,
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 minimized
mean response and the most sensitive variations
q 1max / q 1st
25
20
15
10
5
0
0 10 20
Ω (rad/s)
30 40
Fig. 5. Monte Carlo simulation of the response variation due to
the uncertainty (BS).
C. Zang et al. / Computers and Structures 83 (2005) 315–326 323
q 1max / q 1st
25
20
15
10
5
0
0 10 20 30 40
Ω (rad/s)
Fig. 6. Monte Carlo simulation of the response variation due to
the uncertainty (RD1).
q 1max / q 1st
25
20
15
10
5
0
0 10 20
Ω (rad/s)
30 40
Fig. 7. Monte Carlo simulation of the response variation due to
the uncertainty (RD11).
q 1max / q 1st
25
20
15
10
5
0
0 10 20 30 40
Ω (rad/s)
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 compromise
between the RD1 and RD11 cases, with a reasonable
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 optimisation
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 minimize
the variability in the performance response that results
from uncertainty represented through noise
variables. The robust design approaches can generally
be classified as statistical-based methods and optimisation
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 optimisation
approaches for robust design are based on non-linear
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 maximum
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 vibration
absorber. The objective is to minimize the displacement
response of the main system within a wide band
of excitation frequencies. The robustness of the response
due to uncertainty in the mass and stiffness of
the main system was also considered, and the maximum
mean displacement response and the variations
caused by the mass and stiffness uncertainty were minimized
simultaneously. Monte Carlo simulation demonstrated
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 optimization.
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 technology
and product development. Reading, MA: Addison-
Wesley Publishing Company; 1995.
[10] Ragsdell KM, dÕEntremont KL. Design for latitude using
TOPT. ASME Adv Des Autom 1988;DE-14:265–72.
[11] Box G, Fung C. Studies in quality improvement: minimizing
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. TaguchiÕs parameter design: a panel discussion.
Technometrics 1992;34:127–61.
[15] DÕErrico JR, Zaino Jr NA. Statistical tolerancing using a
modification of TaguchiÕs 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 TaguchiÕs loss function for solving nonlinear opti-

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 problems.
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 TaguchiÕs
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 worstcase
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 constraints.
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, computation
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 closedform
design metrics using physical programming. AIAA J
1998;36:855–64.
C. Zang et al. / Computers and Structures 83 (2005) 315–326 325
[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 Conference.
2003, Paper No. AIAA-2003-1780.
[44] Balling RJ, Free JC, Parkinson AR. Consideration of
worst-case manufacturing tolerances in design optimization.
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 probabilistic
and robust Design with non-normal distributions. In:
AIAA 43rd AIAA/ASME/ASCE/AHS Structures, Structure
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 sequential
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 Dynamics,
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 programming.
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 specifications.
Eng Optimiz 1996;26:137–58.
[58] Koch PN, Barlow A, Allen JK, Mistree F. Configuring turbine
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 alternative
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 evolutionary
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 mathematical
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 algorithm.
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.