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Home , Optimal design, Robust parameter design, Taguchi methods

Computers and Structures 83 (2005) 315–326 www.elsevier.com/locate/compstruc

A review of robust and its application in dynamics

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

a Department of Aerospace , 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

Abstract

The objective of robust design is to optimise the 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; ; Taguchi; Vibration absorber

1. Introduction on product design and performance are of paramount concern to researchers and practitioners. Successful manufactured products rely on the best The earliest approach to reducing the output varia- possible design and performance. They are usually pro- tion was to use the strategy [1,2] so duced using the tools of engineering design optimisation that ±6 standard deviations lie between the mean and in order to meet design targets. However, conventional the nearest specification limit. Six Sigma as a measure- design optimisation may not always satisfy the desired ment standard in product variation can be traced back targets due to the significant uncertainty that exists in to the 1920s when Walter Shewhart showed that three material and geometrical such as modulus, sigma from the mean is the point where a process re- thickness, density and residual strain, as well as in joints quires correction. In the early and mid-1980s, Motorola and component assembly. Crucial problems in noise and developed this new standard and documented more than vibration, caused by the in the structural $16 billion in savings as a result of the Six Sigma efforts. dynamics, are rarely considered in design optimisation. In the last twenty years, various non-deterministic Therefore, ways to minimize the effect of uncertainty 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 * Corresponding author. Fax: +44 117 927 2771. design based methods. The reliability methods estimate E-mail address: [email protected] (M.I. Friswell). the of the systemÕs response

0045-7949/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.10.007 316 C. Zang et al. / Computers and Structures 83 (2005) 315–326 based on the known probability distributions of the ran- tions industry and since then the Taguchi robust design dom parameters, and is predominantly used for risk method has been successfully applied to various indus- analysis by computing the probability of failure of a sys- trial fields such as electronics, automotive products, tem. However, the variation is not minimized in the reli- photography, and telecommunications [6–8]. ability approaches [3], which concentrate on the rare The fundamental definition of robust design is de- events at the tails of the probability distribution [4]. Ro- scribed as A product or process is said to be robust when bust design improves the quality of a product by mini- it is insensitive to the effects of sources of variability, even mizing the effect of the causes of variation without through the sources themselves have not been eliminated eliminating these causes. The objective is different from [9]. In the design process, a number of parameters can the reliability approach, and is to optimise the mean per- affect the quality characteristic or performance of the formance and minimise its variation, while maintaining product. Parameters within the system may be classified feasibility with probabilistic constraints. This is achieved as signal factors, noise factors and control factors. Sig- by optimising the product and process design to make nal factors are parameters that determine the of the performance minimally sensitive to the various configurations to be considered by the robust design. causes of variation. Hence robust design concentrates Noise factors are parameters that cannot be controlled on the probability distribution near to the mean values. by the designer, or are difficult and expensive to control, In this paper, robust optimal design methods are re- and constitute the source of variability in the system. viewed extensively and an example of the robust design Control factors are the specified parameters that the de- of a passive vibration absorber is used to demonstrate signer has to optimise to give the least sensitivity of the the application of these techniques to vibration analysis. 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 2. The concept of robust design geometry is then optimised through control factors describing the panel shape, for the different configura- The robust design method is essential to improving tions to be analysed determined by the signal factors, engineering productivity, and early work can be traced such as different applied loads or the response at differ- back to the early 1920s when Fisher and Yates [5] devel- ent frequencies. oped the statistical design of (DOE) ap- A P-diagram [6] may be used to represent different proach to improve the yield of agricultural crops in types of parameters and their relationships. Fig. 1 shows England. In the 1950s and early 1960s, Taguchi devel- the different types of performance variations, where the oped the foundations of robust design to meet the chal- large circles denote the target and the response distribu- lenge of producing high-quality products. In 1980, he tion is indicated by the dots and the associated probabil- applied his methods in the American telecommunica- ity density function. The aim of robust design is to make

Fig. 1. Different types of performance variations. C. Zang et al. / Computers and Structures 83 (2005) 315–326 317 the system response close to the target with low varia- tion is developed. design, sometimes called tions, without eliminating the noise factors in the sys- robust design, identifies factors that reduce the system tem, as illustrated in Fig. 1(d). sensitivity to noise, thereby enhancing the systemÕs Suppose that y = f(s,z,x) denotes the vector of re- robustness. Tolerance design specifies the allowable sponses for a particular set of factors, where s, z and x deviations in the parameter values, loosening tolerances are vectors of the signal, noise and control factors. if possible and tightening tolerances if necessary [9]. The noise factors are uncertain and generally specified TaguchiÕs objective functions for robust design arise probabilistically. Thus z, and hence f, are random vari- from quality measures using quadratic loss functions. ables. The range of configurations are specified by a In the extension of this definition to design optimisation, set of signal factors, V, and thus s 2 V. One possible Taguchi suggested the signal-to-noise ratio (SNR), mathematical description of robust design is then 10log10(MSD), as a measure of the mean squared devi- hi ation (MSD) in the performance. The use of SNR in sys- 2 x^ ¼ arg min max Ez kfðs; z; xÞtk ð1Þ tem analysis provides a quantitative value for response x s2V variation comparison. Maximizing the SNR results in subject to the constraints, the minimization of the response variation and more ro- g ðs; z; xÞ 6 0 for j ¼ 1; ...; m; ð2Þ bust system performance is obtained. Suppose we have j only one response variable, y, and only one configura- where E denotes the (over the uncertainty tion of the system (so the signal factor may be ne- due to z) and t is the vector of target responses, which glected). Then for any set of control factors, x, the may depend on the signal factors, s. Note that x is the noise factors are represented by n sets of parameters, parameter vector that is adjusted to obtain the optimal leading to the n responses, yi. Although there are many solution, given by Eq. (1) as the solution that gives the possible SNR ratios, only two will be considered here. smallest worse case response over the range of configu- The target is best SNR. This SNR quantifies the devi- rations, or signal factors, in the set V. The key step in ation of the response from the target, t, and is the robust design problem is the specification of the objective function, and once this has been done the tools SNR ¼10log ðMSDÞ 10 ! of (such as the ) and the de- 1 X sign of experiments (such as orthogonal arrays) may be ¼10log ðy tÞ2 10 n i used to calculate the solution. i For convenience, the robust design approaches will ¼10log ðS2 þðy tÞ2Þð3Þ be classified into three categories, namely Taguchi, opti- 10 misation and stochastic optimisation methods. Further where S is the population . Eq. (3) is details are given in the following sections. The stochastic essentially a sampled version of the general optimisation optimisation methods include those that propagate the criteria given in Eq. (1). Note that the second form indi- uncertainty through the model and hence use accurate cates that the MSD is the summation of population var- statistics in the optimisation. Also included are optimi- iance and the deviation of the population mean from the sation methods that are inherently stochastic, such as target. If the control parameters are chosen such that the genetic algorithm. These methods are generally time y ¼ t (the population mean is the target value), then consuming, and the statistical representation may be the MSD is just the population variance. If the popula- simplified at the expense of accuracy, for example by tion standard deviation is related to the mean, then the considering only a linear perturbation from the nominal MSD may also be scaled by the mean to give values of the parameters. The take this a step further, using only a limited number of discrete S2 SNR ¼10log ðMSDÞ¼10log values for the parameters and using the design of exper- 10 10 y2 iments methodology to limit the number of times the y2 model has to be run. The response then requires ¼ 10log : ð4Þ 10 S2 analysis of variance techniques to determine the sensitiv- ity of the response to the noise and control factors. The smaller the better SNR. This SNR considers the deviation from zero and, as the name suggests, penalises large responses. Thus ! 3. The Taguchi based methods X 1 2 SNR ¼10log10ðMSDÞ¼10log10 yi ð5Þ TaguchiÕs approach to the product design process n i may be divided into three stages: system design, para- meter design, and tolerance design [6]. System design is This is equivalent to the Target-is-Best SNR, with the conceptual design stage where the system configura- t =0. 318 C. Zang et al. / Computers and Structures 83 (2005) 315–326

The most important task in TaguchiÕs robust design expansion of the objective function about the mean val- method is to test the effect of the variability in different ues of the design variables. Other significant publica- experimental factors using statistical tools. The require- tions in this area include those by Mohandas and ment to test multiple factors that a full factorial Sandgren [20], Sandgren [21], and Belegundu and Zhang experimental design that describes all possible condi- [22]. Chang et al. [23] extended TaguchiÕs parameter de- tions would result in a large number of experiments. sign to the notion of conceptual robustness. Lee et al. Taguchi solved this difficulty by using orthogonal arrays [24] developed robust design in discrete design space (OA) to represent the range of possible experimental using the Taguchi method. The based conditions. After conducting the experiments, the data on the Taguchi concept was utilized to arrange the dis- from all experiments are evaluated using the analysis crete variables, and robust solutions for unconstrained of variance (ANOVA) and the analysis of mean optimization problems were found. (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 4. Optimisation methods the sensitivity to the effects of noise, and adjusting the mean response to the target response. The optimisation procedures aim to minimise the TaguchiÕs techniques were based on direct experimen- objective functions, such as Eq. (1), directly. The uncer- tation. However, designers often use a computer to sim- tainty in the noise factors means that the system perfor- ulate the performance of a system instead of actual mance is a . One option for the robust experiments. Ragsdell and dÕEntremont [10] developed optimisation is to minimize both the deviation in the a non-linear code that applied TaguchiÕs concepts to de- 2 mean value, jlf tj, and the variance, rf , of the perfor- sign optimisation. In some cases, the optimal design is mance function, subject to the constraints, where the least robust, and designers have to make a trade- off between target performance and robustness. Ideally, l ðx; sÞ¼E ½fðs; z; xÞ; f z hi one should optimise the expected performance over a r2ðx; sÞ¼E ðfðs; z; xÞl ðx; sÞÞ2 : ð6Þ range of variations and uncertainties in the noise factors. f z f Although TaguchiÕs contributions to the philosophy of robust design are almost unanimously considered to The quantities of mean and the standard deviation of be of fundamental importance, there are certain limita- system performance, for given signal and control fac- tions and inefficiencies associated with his methods. tors, may be calculated if the joint probability density Box and Fung [11] pointed out that the orthogonal array function (PDF) of the noise factors is known. For most method does not always yield the optimal solution and practical applications these PDFs are unknown, but of- suggested that non-linear optimisation techniques ten it is assumed that all variables have independent nor- should be employed when a computer model of the de- mal distributions. In this case the joint PDF becomes a sign exists. Better results were achieved on a Wheatstone product of the individual PDFs. However, evaluating Bridge circuit design problem used by Taguchi. Mont- Eq. (6) is extremely time consuming and computation- gomery [12] demonstrated that the inner array used for ally expensive and approximations using TaylorÕs series the control factors in the TaguchiÕs approach and the expansions about the mean noise and control factors, z outer array used for noise factors, is often unnecessary and x, may be used. If only the linear terms are retained and results in a large number of experiments. Tsui [13] in the expansion then the mean and variance of the re- showed that the Taguchi method does not necessarily sponse are readily computed in terms of the mean and find an accurate solution for design problems with variance of the noise factors. highly non-linear behaviour. An excellent survey of The constraints must also be satisfied. For a worst these controversies was the panel discussion edited by case analysis the constraints must be satisfied for all val- Nair [14]. ues of control and noise factors, and the constraint in Eq. (2) may be approximated [25] as TaguchiÕs approach has been extended in a number of ways. DÕErrico and Zaino [15] implemented a modifi- X og X og j j 6 cation of the Taguchi method using Gaussian–Hermite gjðz; xÞþ Dzi þ Dxl 0 ð7Þ ozi oxl quadrature integration. Yu and Ishii [16] used the frac- i l tional quadrature factorial method for systems with sig- where the dependence on the signal factors has been nificant nonlinear effects. Otto and Antonsson [17] made implicit. The derivatives are evaluated at z and addressed robust design optimisation with constraints, x, and Dzi and Dxl represent the deviations of the ele- using constrained optimisation methods. Ramakrishnan ments of z and x from these means. Because of the abso- and Rao [18,19] formulated the robust design problem lute values this approximation is likely to be very as a non-linear optimization problem with TaguchiÕs loss conservative. For a statistical analysis the constraint is function as the objective. They used a 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- A robust design is one that attempts to optimise both straints become, the mean and variance of the performance, and is there- fore a multi-objective and non-deterministic problem. g ðz; xÞþkrgjðz; xÞ 6 0 ð8Þ j Optimisation of the mean often conflicts with minimiz- where rgj is an approximation to the standard deviation ing the variance, and a trade-off decision between them of the j th constraint and k is a constant that reflects the is needed to choose the best design. The conventional probability that the constraint will be satisfied. For weighted sum (WS) methods to determine this trade- example, k = 3 means that a constraint is satisfied off have serious drawbacks for the Pareto set generation 99.865% of the time, if the constraint distributions are in multi-objective optimisation problems [34]. The Pare- normal. The constraint variance for a linear Taylor ser- to set [35] is the set of designs for which there is no other ies is design that performs better on all objectives. Using X og 2 X og 2 weighted sum methods, it may be impossible to achieve r2 z; x j r2 j Dx 2: 9 gjð Þ¼ zi þ ð lÞ ð Þ some Pareto solutions and there is no assurance that the ozi oxl i l best one is selected, even if all of the Pareto points The noise factors are assumed to be stochastic, whereas are available. Chen et al. [36] used a combination of the control factors are used to optimise the system and multi-objective mathematical programming methods therefore must remain as parameters in the constraints. and the principles of decision analysis to address the Parkinson et al. [25] proposed a general approach for multi-objective optimisation in robust design. The com- robust optimal design and addressed two main issues. promise programming (CP) approach, that is the Tche- The first issue was design feasibility, where the proce- bycheff or min–max method, replaced the conventional dures are developed to account for tolerances during de- WS method. The advantages of the CP method over sign optimisation such that the final design will remain the WS approach in locating the efficient multi-objective feasible despite variations in parameters or variables. robust design solution (Pareto points) were illustrated The second issue was the control of the transmitted var- both theoretically and through example problems. Chen iation by minimizing sensitivities or by trading off con- et al. [37] made the bi-objective robust design optimisa- trollable and uncontrolled tolerances. The calculation tion perspective more powerful by using a physical pro- of the transmitted variation was based on well-known gramming approach [38–40], where each objective was results developed for the analysis of tolerances [26,27]. controlled with more flexibility than by using CP. Re- Parkinson [28] also discussed the feasibility robustness cently, Messac and Ismail-Yahaya [41] formulated the and sensitivity robustness for robust mechanical design robust design optimization problem from a fully multi- using engineering models. Variability was defined in objective perspective, again using the physical program- terms of tolerances. For a worst-case analysis, a toler- ming method. This method allows the designer to ance band was defined, whereas for a statistical analysis express independent preferences for each design metric, the ± 3r limits for the variable or parameter were used. design metric variation, design variable, design variable A number of engineering examples showed that the variation, and parameter variation. method works well if the tolerances are small. For larger An alternative to CP is the preference aggregation tolerances or for strongly non-linear problems, higher method, and Dai and Mourelatos [42] discussed the fun- order Taylor series expansions must be used. A second damental differences of these approaches. The compro- order worst-case model was described by Emch and Par- mise programming approach is a technique for efficient kinson [29] and a second order model using statistical optimisation to recover an entire Pareto frontier, and analysis was presented by Lewis and Parkinson [30]. does not attempt to select one point on that frontier. The disadvantage of higher order models is that the for- Preference aggregation, on the other hand, can select mulae to evaluate the response become quite compli- the proper trade-off between nominal performance and cated and computationally expensive. variability before calculating a Pareto frontier. CP relies Sundaresan et al. [31] applied a Sensitivity Index on an efficient algorithm to generate an entire Pareto optimisation approach to determine a robust optimum, frontier of robust designs, while preference aggregation but the drawback was the difficulty in choosing the selects the best trade-off before performing any weighting factors for the mean performance and calculation. variance. Mulvery et al. [32] treated robust design as a Mattson and Messac [43] gave a brief literature sur- bi-objective non-linear programming problem and em- vey on how various robust design optimisation methods ployed a multi-criteria optimisation approach to gener- handle constraint satisfaction. The focus was the varia- ate a complete and efficient solution set to support tion in constraints caused by variations in the controlled decision-making. Chen et al. [33] developed a robust de- and uncontrolled parameters. The constraints are con- sign methodology to minimize variations caused by the sidered as two types: equality and inequality constraints. noise and control factors, by setting the factors to zero Discussions on inequality constraint satisfaction were in turn. 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- to demonstrate the accuracy and efficiency of the pro- yam [47]. Research on equality constraint problems are posed method. Chen [65] used Monte Carlo simulation limited to three approaches; to relax the equality con- and quadratic programming. Schueller [66] gave a recent straint [48,49,41], to satisfy the equality constraint in a review on structural stochastic analysis. probabilistic sense [50,51], or to remove the equality Optimisation approaches that are inherently stochas- constraint through substitution [52]. tic include techniques such as simulated annealing, neu- ral networks and evolutionary algorithms (EA) (genetic algorithms, evolutionary programming and evolution 5. Stochastic optimization strategies (ES)), and these have been applied to multi- objective optimisation problems [67–69]. These tech- A slightly different strategy for robust design optimi- niques do not require the computation of gradients, sation is based on stochastic optimisation. The stochas- which is important if the objective function relies on esti- tic nature of the optimisation arises from incorporating mating moments of the response random variables. uncertainty into the procedure, either as the parameter Gupta and Li [70] applied mathematical programming uncertainty through the noise factors, or because of and neural networks to robust design optimisation, and the stochastic nature of the optimisation procedure. the numerical examples showed that the approach is able The earliest work on stochastic optimization can be to solve highly non-linear design optimisation problems traced back to the 1950s [53] and detailed information in mechanical and structural design. Sandgren and Cam- may be obtained from recent books [54,55]. The objective eron [71] used a hybrid combination of a genetic algo- of stochastic optimization is to minimize the expectation rithm and non-linear programming for robust design of the sample performance as a function of the design optimization of structures with variations in loading, parameters and the in the system. Eggert geometry and material properties. Parkinson [72] em- and Mayne [56] gave an introduction to probabilistic ployed a genetic algorithm for robust design to directly optimisation using successive surrogate probability den- obtain a global minimum for the variability of a design sity functions. Some authors [57–59] employed the con- function by varying the nominal design parameter val- cept exploration method for robust design optimisation ues. The method proved effective and more efficient than by creating surrogate global response surface models of conventional optimisation algorithms. Additional stud- computationally expensive simulations. The response ies are required before these methods are suitable for surface methodology (RSM) is a set of statistical tech- application to large-scale optimisation problems. 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 6. Applications in dynamics [60] and Myers et al. [61] considered the RSM as an alter- native to TaguchiÕs robust design method. Monte Carlo The most successful applications of robust design are simulation generates instances of random variables found in the fields of mechanical design engineering (sta- according to their specified distribution types and char- tic performance) and process systems, and there have acteristics, and although accurate response statistics been few applications to the robustness of dynamic per- may be obtained, the computation is expensive and time formance. Seki and Ishii [73] applied the robust design consuming. Mavris and Bandte [62] combined the re- concept to the dynamic design of an optical pick-up sponse model with a Monte Carlo simulation to con- actuator focusing on shape synthesis using computer struct cumulative distribution functions (CDFs) and models and . The response in the probability density functions (PDFs) for the objective first bending and torsion modes were selected as mea- function and constraints. All these methods depend on sures of undesirable vibration energy. The objective the of the statistics, whereby the probabilistic functions were defined as the signal-to-noise ratios of re- distributions of the stochastic input sets are required. sponse frequencies and the sensitivities were derived One concern is that the response surface approximations from the design of experiments using an orthogonal ar- might not generate the accurate sensitivities required for ray. Hwang et al. [74] optimised the vibration displace- robust design [48]. The stochastic finite element method ments of an automobile rear view mirror system for [63] provides a powerful tool for the analysis of struc- robustness, defined by the Taguchi concept. tures with parameter uncertainty. Chakraborty and In this paper the robust design approach is applied to Dey [64] proposed a stochastic finite element method in the dynamics of a tuned vibration absorber due to the domain for analysis of structural dynamic parameter uncertainty, using the optimisation approach problems involving uncertain parameters. The uncertain through non-linear programming. The objective is to structural parameters are modelled as homogeneous determine the stiffness, mass and damping parameters Gaussian stochastic fields and discretised by the local of the absorber, to minimize the displacements of the averaging method. Numerical examples were presented main system over a large range of excitation frequencies, C. Zang et al. / Computers and Structures 83 (2005) 315–326 321 despite uncertainty in the mass and stiffness properties Ω f0 sin( t) of the main system.

The principle of the vibration absorber was attrib- k2 uted to Frahm [75] who found that a natural frequency k of a structure could be split into two frequencies by 1 m m attaching a small spring–mass system tuned to the same 1 c2 2 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. q q Brock [77] and Den Hartog [78] gave criteria for the effi- 1 2 cient optimum operation of a tuned vibration absorber, such as the relationship of the frequency and mass ratios Fig. 2. Forced vibration of a two degree of freedom system. between the absorber and main system and the relation- ship between the damping and mass ratio. Jones et al. q1max [79] successfully designed prototype vibration absorbers F ðX;m1;k1Þ¼ for two bridges using these criteria. However, the effect q1st . 2 2 2 2 2 2 2 2 2 of parameter uncertainty and variations in the dynamic ¼ k1 c2X þðk2 m2X Þ c2X ðk1 m1X m2X Þ performance have not been considered. 1=2 2 2 2 2 We consider the forced vibration of the two degree of þ k2m2X ðk1 m1X Þðk2 m2X Þ ð13Þ freedom system shown in Fig. 2. The original one degree of freedom system, referred to as the main system, con- Suppose a steel box girder footbridge may be repre- sists of the mass m1 and the spring k1, and the added the sented by a single degree of freedom system with mass system, referred to as the absorber, consists of the mass m1 = 17500 kg and stiffness k1 = 3.0 MN/m. The worst m2, the spring k2 and the damper c2. case of dynamic loading is considered to be equivalent The equations of motion are to a sinusoidal loading with a constant amplitude of 0.48 kN at the natural frequency of the bridge. To sim- m1q€1 þ c2ðq_ 1 q_ 2Þþk1q1 þ k2ðq1 q2Þ¼f0 sinðXtÞ ulate the environmental changes, the stiffness k1 and the m q€ c q_ q_ k q q 0 2 2 þ 2ð 2 1Þþ 2ð 2 1Þ¼ mass m1 are allowed to undergo 10% variations ð10Þ (k1 ± Dk1, m1 ± Dqmffiffiffiffi1), and a wide excitation frequency band ð1 6 X 6 3 k1 Þ is considered. This ensures that Inman [80] and Smith [81] should be consulted for fur- m1 ther details of the modelling and analysis of vibration the absorber works well for a wide range of possible absorbers. Solving these equations for the steady-state excitations. The frequency X is the signal factor, and solution, gives the amplitude of vibration of the two the range of this frequency gives the set V. Using the masses as worst-case formulation, the standard deviations of the

! 1=2 2 2 2 2 c2X þðk2 m2X Þ q ¼ f0 1 max 2 2 2 2 2 2 2 2 2 c X ðk1 m1X m2X Þ þðk2m2X ðk1 m1X Þðk2 m2X ÞÞ 2 !11 1=2 ð Þ 2 2 2 c2X þ k2 q ¼ f0 2 max 2 2 2 2 2 2 2 2 2 c2X ðk1 m1X m2X Þ þðk2m2X ðk1 m1X Þðk2 m2X ÞÞ

1 The equations may be non-dimensionalised using a mass and stiffness in the main system are rk1 ¼ 3 Dk1 1 ÔstaticÕ deflection of main system, defined by and rm1 ¼ 3 Dm1. This standard deviation is calculated f for a uniform distribution over the 10% parameter vari- q ¼ 0 ð12Þ ations and is used for the robust design optimisation. 1st k 1 However the original intervals are used in the simula- to give the non-dimensional displacement of mass m1 tions to demonstrate the effectiveness of the solutions. as The mean response function and the standard deviation 322 C. Zang et al. / Computers and Structures 83 (2005) 315–326 of the maximum non-dimensional displacement of mass m may be calculated using the first-order Taylor 1 1 approximation2 as 3 2 2 3 6 7 6 7 lf ¼E4 maxqffiffiffiffiðF ðX;m1;k1ÞÞ5 6 6 k1

1 X 3 * f m1 4 2 3 σ

/ 1.5 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 f 5

σ 6 oF ðX;m ;k Þ 2 oF ðX;m ;k Þ 2 7 6 ¼E6 max @ 1 1 r2 þ 1 1 r2 A7 f 4 qffiffiffiffi k1 m1 5 ok1 om1 6 6 k1 7 1 X 3 m 1 8 9 10 11 ð14Þ 1 Here the expected value is evaluated for the uncertain mass and stiffness parameters m and k , and F is defined 1 1.5 2 2.5 1 1 µ / µ* f f in Eq. (13). The design variables for the absorber are m2, k2 and c2, with lower and upper bounds given by Fig. 3. Efficient solutions of a robust multi-objective 6 m2 2 [10, 1750], c2 2 [10, 2000], k2 2 [100, 10 ]. There- optimization. fore, the robust design of a suitable tuned vibration ab- sorber may be formulated as follows:

Minimize: ½lf ðm2; c2; k2Þ; rf ðm2; c2; k2Þ Table 1 Subject to: j Dk1 j6 0 1k1; j Dm1 j6 0:1m1 6 6 6 6 ð15Þ The robust optimization solutions for the absorber design, 10 m2 1750; 10 c2 2000; together with the textbook solution 6 100 6 k2 6 10 Name a m2 (kg) k2 (N/m) c2 (N s/m) m2m1 The first step of robust design is to seek the ideal design RD1 1 203.72 34509 2000 0.011641 (Utopia point) by a minimisation of lf and rf individu- RD2 0.9 204.05 34576 2000 0.01166 ally as single objective functions. The ideal design ob- RD3 0.8 209.79 35532 2000 0.011988 RD4 0.7 239.12 40421 2000 0.013664 tained is denoted by ½lf ; rf . Since there are only two objective functions, l and r , the two functions are com- RD5 0.6 251.27 42421 2000 0.014358 f f RD6 0.5 259.71 43817 2000 0.01484 bined into a single objective function, G, by the conven- RD7 0.4 310.37 52188 2000 0.017736 tional weighted sum method discussed earlier. The RD8 0.3 348.83 58483 2000 0.019933 objective function is then RD9 0.2 410.44 68483 1999.1 0.023454 l r RD10 0.1 421.27 70238 2000 0.024072 G ¼ a f þð1 aÞ f ð16Þ RD11 0 510.93 84592 2000 0.029196 lf rf BS (book 175 29393 272.2 0.01 where the weighting factor a [0,1] represents the rela- 2 solution) 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- RD1 served. Points 1 and 11 denote the two utopia points, RD2 RD3 showing the minimized mean (a = 1) and variance 10 RD4 (a = 0) response optimization, respectively. The other RD5 8 RD6 points are the optimized results from different weighting RD7 of the mean and variance objection functions. This 1st 6 RD8 / q RD9 weighted sum approach has similarities to the L-curve 4 RD10

1max method used in regularization [82]. The 11 optimised q RD11 2 TD solutions for the absorber parameters (mass, stiffness and damping) and the recommended solution from 0 15 vibration textbooks [80,81], are denoted RD1–RD11 10 40 and BS, and are listed in Table 1. To visualise all of these Design Num 30 5 20 responses over the excitation frequency band of interest, ber 10 0 0 Ω (rad/s) the non-dimensional displacements of mass m1 for these different parameter sets are plotted in Fig. 4, where TD Fig. 4. Robust design results compared with the traditional denotes the text book solution. solution. C. Zang et al. / Computers and Structures 83 (2005) 315–326 323

Most of the damping values for the robust design 25 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, 20 although in practice the amount of damping that can be

1st added is limited. The optimum mass ratio (m2/m1) is be- 15 tween 1% and 3%, and the value of the mass for the RD1 / q case, where only the mean response was minimized, is 1max q close to that of the BS case. With the decreased weight- 10 ing on the mean response and correspondingly increas- ing weighting on the response variance, the absorber 5 mass is gradually increased, and the response shown in Fig. 4 becomes flatter. Although the peak response is de- 0 creased in these cases, the response near the main system 0 10 20 30 40 natural frequency increases. Ω (rad/s) To demonstrate the effectiveness of the robust design Fig. 6. Monte Carlo simulation of the response variation due to approach, Monte Carlo simulation [83] is used to evalu- the uncertainty (RD1). 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- 25 ples of the noise factors (in this case m1 and k1), which are then used to calculate the response over the fre- 20 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. 1st 15 / q The response variations then give some indication of the robustness of the design given by those particular set of 1max 10 q 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 5 Figs. 5–8, together with the corresponding mean re- sponse, shown by the solid line. The amplitudes of the 0 response for the three robust design cases (RD1, RD6, 0 10 20 30 40 Ω (rad/s) RD11) are much smaller than those from the traditional textbook design (BS). The RD1 case has the lowest min- Fig. 7. Monte Carlo simulation of the response variation due to imized mean response and the most sensitive variations the uncertainty (RD11).

25 25

20 20

1st 1st 15 15 / q

/ q

1max

1max q

q 10 10

5 5

0 0 0 10 20 30 40 0 10 20 30 40 Ω (rad/s) Ω (rad/s)

Fig. 5. Monte Carlo simulation of the response variation due to Fig. 8. Monte Carlo simulation of the response variation due to the uncertainty (BS). 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 Acknowledgments the least sensitive variations. The RD6 case is a compro- mise between the RD1 and RD11 cases, with a reason- The authors acknowledge the support of the EPSRC able mean response and insensitive variations due to (UK) through grants GR/R34936 and GR/R26818. the uncertainty. The textbook solution, shown in Fig. Prof. Friswell acknowledges the support of a Royal 5, is most sensitive to the mass and stiffness variations Society-Wolfson Research Merit Award. in the main system, although increasing the damping in the absorber would improve this performance. References 7. Concluding remarks [1] Harry MJ. The nature of Six Sigma Quality. Shaumburg, IL: Motorola University Press; 1997. 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