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A Review of Robust Optimal Design and Its Application in Dynamics

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A Review of Robust Optimal Design and Its Application in Dynamics Computers and Structures 83 (2005) 315–326 www.elsevier.com/locate/compstruc 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 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. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Robust design; Stochastic optimization; 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 Six Sigma Quality 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 parameters 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 variance in the structural $16 billion in savings as a result of the Six Sigma efforts. dynamics, are rarely considered in design optimisation. In the last twenty years, various non-deterministic Therefore, ways to minimize the effect of uncertainty methods have been developed to deal with design uncer- tainties. These methods can be classified into two ap- proaches, namely reliability-based methods and robust * Corresponding author. Fax: +44 117 927 2771. design based methods. The reliability methods estimate E-mail address: [email protected] (M.I. Friswell). the probability distribution of the systemÕs response 0045-7949/$ - 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 range 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 experiments (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. Parameter 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 À10log10(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 expected value (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 statistics (such as the analysis of variance) and the de- 1 X sign of experiments (such as orthogonal arrays) may be ¼10log ðy À tÞ2 10 n i used to calculate the solution. i For convenience, the robust design approaches will ¼10log ðS2 þðy À tÞ2Þð3Þ be classified into three categories, namely Taguchi, opti- 10 misation and stochastic optimisation methods. Further where S is the population standard deviation. 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.
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