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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.
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