World Academy of Science, Engineering and Technology International Journal of Industrial and Manufacturing Engineering Vol:6, No:2, 2012 Mechanical Structure Design Optimization by Blind Number Theory: Time-dependent Reliability Zakari Yaou, Lirong Cui Abstract—In a product development process, understanding the a kind of unascertained system, which the concern/research functional behavior of the system, the role of components in achiev- subject of unascertained mathematical theory. ing functions and failure modes if components/subsystem fails its Different from other kinds of uncertainties such as random, required function will help develop appropriate design validation and verification program for reliability assessment. The integration fuzzy, grey and so on, the blind number theory which of these three issues will help design and reliability engineers in is discrete and numerical tool for expressing all sorts of identifying weak spots in design and planning future actions and uncertainty variables from the perspective of microcosm will testing program. provide an effective solution to the above stated problem. This case study demonstrate the advantage of unascertained theory Up to now the unascertained mathematical theories have described in the subjective cognition uncertainty, and then applies blind number (BN) theory in describing the uncertainty of the been successfully applied to several research fields [6- mechanical system failure process and the sametime used the 7]. Therefore, the blind number, as the expansion of the same theory in bringing out another mechanical reliability system unascertained mathematics, has more general description on model. The practical calculations shows the BN Model embodied unascertained problems [5]. The statistic values of uncertainty the characters of simply, small account of calculation but better variables used by BN come from fitting special distribution forecasting capability, which had the value of macroscopic discussion to some extent. and data of the actual measurement. The practical calculation shows that the BN model embodied the characters of simply, small account of calculation but better forecasting capability, Keywords—Mechanical structure Design, time-dependent stochas- tic process, unascertained information, blind number theory. which has the value microscopic discussion to some extent. I. INTRODUCTION II. BN DESCRIPTION HE development of mechanical design reliability A. Unascertained Number and Unascertained Rational Num- theory, the reliability of design optimization and the T ber stochastic time-dependent characteristics of mechanical system reliability is gaining more attention. In that theory, When decision makers are faced with information that load process on the mechanical structure was considered to are certain and not to certain degree i.e. unascertained, then Poisson Process while the actual process is much complex this information is called as unascertained information [8]. and unpredictable [1-2]. The machine lifetime and the Unascertained mathematics and grey number prove consistent resistance of mechanical structure degenerate with the course in dealing with this case and the main difference between of time [3-4], which is difficult to predict precisely due to the the two is the certain amount of unascertained number is degradation of component characteristics. A more advanced larger than the grey number. The most basic and simplest stochastic process is employed in describing these mechanical unascertained number is unascertained rational number [9] load and resistance. which is the expansion of real number. The blind number Due to several design, state and constraint variables, the theory and its application were developed on the basis of reliability design optimization is difficult to achieve. While unascertained mathematics [10-11]. It adopts the expression applying also the advanced stochastic process in describing type of grey variable and unascertained variable to express un- time-dependent uncertain information in mechanical structure certainty variables and thus unifying the two expression types. design, this complicates further the problem as it exist The BN describes objectively the unascertained information International Science Index, Industrial and Manufacturing Engineering Vol:6, No:2, 2012 waset.org/Publication/2515 many distribution types of random variables including in details; avoid flaw of information omission and distortion complex operations among them, in every single reliability which is produced by the single real number in expressing optimization design model. The traditional stochastic method these amounts. Liu et al [11] have derived the mean and the for the complex reliability optimization model is also variance of BN to describe the center point and the distribution very cumbersome and inadequate in solving the problem. characteristics of BN. Therefore, it is believed the mechanical design process is Let for arbitrary closed interval [a, b] so that a = x1 <x2 < ···<xn = b, and if function φ(x) satisfy Z. Yaou is with the School of Management and Economics, Beijing ⎧ ⎪ Institute of Technology, Beijing, 100081 P. R. CHINA e-mail: (see za- ⎨⎪αi,x= xi where i =1, 2, ..., n [email protected]). L. Cui is with the School of Management and Economics, Bei- φ(x)=⎪ (1) jing Institute of Technology, Beijing, 100081 P. R. CHINA e-mail: ⎩⎪ (see [email protected]) 0, otherwise. International Scholarly and Scientific Research & Innovation 6(2) 2012 536 scholar.waset.org/1307-6892/2515 World Academy of Science, Engineering and Technology International Journal of Industrial and Manufacturing Engineering Vol:6, No:2, 2012 and variable (e.g random variable or fuzzy variable) whose n distribution type and parameters are known can be expressed αi = α (2) i=1 by BN, the narrower the GNIs, the more approximate to original distribution function will be. ≤ with 0 <α 1[a, b] and φ(x) are called unascertained When the distribution type or parameters of uncertainty { } rational number with n order where [a, b],φ(x) ; variable are not known, the uncertainty variable can also be α, [a, b]; and φ(x) represent the total confidence degree; expressed by BN based on statistical analysis of the known distribution interval and confidence distribution density data, and to reduce the data fitting error experts engineering function respectively. practice experience is used. B. BN Concept III. TIME-DEPENDENT CHARACTERISTIC OF Let R be a set of real number,and R a is the set of MECHANICAL STRUCTURE BASED ON STOCHASTIC unascertained rational number, g(I) is the set of grey interval PROCESS number, and let suppose that xi ∈ g(I), αi ∈ [a, b]; i =1, 2,...,n and if f(x) represents a grey interval number Material properties, manufacturing processes, operation defined on the g(I),we get conditions, physical environment and maintenance practices ⎧ are someofthemain factors determining the lifetimeof ⎪ ⎨⎪αi,x= xi with i =1, 2, ..., n mechanical structures or machines resulting to uncertainties in load on the mechanical structure and its structural strength. f(x)=⎪ (3) ⎩⎪ Through out their lifetime, the load and the strength of 0, otherwise. structure follow a stochastic process. n Let consider a probability space (Ω, F, P) and let {z(t),t ≥ } when i = j, xi = xj and αi = α, 0 <α≤ 1 the function 0 be a standard 1-dimensional Brownian motion defined on i=1 this probability space. Assuming {F,t≥ 0} to be the natural f(x) is said to be blind number, αi is the confidence degree filtration generated by Brownian motion: {z(t),t ≥ 0} , i.e. of f(x) for x to be xi and α represents the overall confidence F = σ{z(t),t≥ 0}. degree of f(x). Let stochastic process S satisfy Itole¯ mma drift-diffusion ≤ 1 Let a, b be real numbers so that a b, and [a, b]= 2 (a+b), process 1 and let assumed that 2 (a + b) is the ”core” value of grey dS = μSdt + ωSdz (5) interval number [a, b]. The mean value of BN is given as follow: And assuming that G =lnS, then ⎧ n ⎪ 1 ∂G ∂G ω2 ∂2G ∂G ⎪αi,x= αixi 2 ⎨ dG = + μ S + 2 S dt + ω Sdz (6) α i=1 ∂t ∂S 2 ∂S ∂S E(f(x)) = (4) ⎪ ⎪ 2 ⎩⎪ dG = μ − ω dt + ωdz (7) 0, otherwise. 2 Where G, (GT − Gt), (μ − ω )(T − t) and ω (T − t) C. Processing unascertained information by BN represent a Generalized Wiener Process, a mean of normal distribution, and a standard deviation respectively. As the random variable represents usually a distribution of probability density on a certain open interval in mathematics, 2 but in engineering practice the BN expression is to take the ln ST ∼ φ ln St + μ − ω (T − t),ω (T − t) (8) aggregate of probabilities on limited numbers of great-narrow interval (GNIs) as random variable’s probability density ST and St are stochastic values of the future period (T )and distribution approximate [12]. Let consider X as a random the present period (t) respectively. μ is drift rate and measure variable in normal distribution for example, its probability of of certainty while ω is a volatility and measure of uncertainty. International Science Index, Industrial and Manufacturing Engineering Vol:6, No:2, 2012 waset.org/Publication/2515 falling within [μ − 6σ, μ +6σ] reach 100% substantially. They are estimated as follow from the historical data [13] By dividing this interval into several GNIs (e.g. 200), n each X distribution can be considered as certain typical 1 Si μ = η = ln (9) distribution (usually equal distribution). Summing up those n Si−1 GNIs’ probabilities can be view as random variable X’s i=1 probability density distribution. The uncertainty of random 1 n 2 variable is strictly limited to a ”defined” range. Transferring 1 2 ω = (ηi − η) (10) strong uncertainty to weak uncertainty on the GNI by the − n 1 i=1 result of dispersing method on the extensive interval, the random variable then can be numerically expressed and where ηi = ln(Si/Si−1) and ηi is the mean value of ηi calculated easily.