[机械模具数控自动化专业毕业设计外文文献及翻译]【期刊】区间参数系统的机械优化设计-外文文献

uncertainwere made on the reliability-based analysis and optimum designof structures and mechanical systems 1-8#, Usually normal dis-tribution is assumed for describing the probabilistic behavior ofthe uncertain parameters. The reliability-based optimum designrequires a quantitative assessment of all engineering uncertainty.When past data are available, Bayesian statistical decision theorycan be used to include professional information and past data inachieving an optimal design 9,10#. In the Bayesian approach toreliability-based optimal design, the a priori information about thestatistical parameters is assumed to be known. When exact infor-mation is not known, t- or chi- square distribution is to be used toestablish the confidence levels. In the interval parameters-basedoptimal design, on the other hand, only the known or expectedranges of the parameters are required without any need to estab-lish the confidence levels.Fuzzy theories can be used when the parameters of the systemare described using linguistic and vague terms such as about,nearly, almost and substantially larger. If the parametersof a system are denoted by simple ranges, interval analysis meth-system. In such cases, the influence of an independent intervalvariable, x5(x2Dx,x1Dx), on the dependent variable f can berepresented as an interval as f 5( f 2D f1, f 1D f2), where 2D f1and 1D f2denote the variations in f caused by the changes 2Dxand 1Dx in x.4. Approximations based on Taylors series are used in manyengineering analysis and design problems. For example, if f (x)isapproximated using terms up to the kth derivative as f (x) f (x0)1 f 8(x0)Dx1(1/2!)f 9(x0)(Dx)21fl11k!f(k)(x0)(Dx)kwhere the values of f and its derivatives are known at the refer-ence point x0with x2x05Dx!, the error in f (x) is given bye51k11!f(k11)j!Dx!k11; x0,j,x01DxThus the actual value of f (x), subject to the error e, lies in theinterval, ( f (x)2e, f (x)1e).5. The performance characteristics of most engineering systemsvary degrade! during their lifetimes because of aging, creep,wear, corrosion and changes in operating conditions. Any avail-able data can be used to establish the ranges of variation of per-formance characteristics such as material properties. The analysis*Corresponding authorContributed by the Design Automation Committee for publication in the JOUR-NAL OF MECHANICAL DESIGN. Manuscript received January 2000. Associate Editor.A. Diaz.Copyright 2002 by ASMEJournal of Mechanical Design SEPTEMBER 2002, Vol. 124 465S. S. Rao*Professor, ChairmanLingtao CaoGraduate StudentDepartment of Mechanical Engineering,University of Miami,Coral Gables, FL 33124-0624OptimumSystemsParametersThe imprecision or uncertaintyusing probabilistic, fuzzyuncertain mechanicalsponse. Each of theinterval ranges of responsenumber and/or rangesoperations, a truncationresponse of the system.accurate. The optimumaspects of the methods.used for the design oftribution functions orDOI: 10.1115/1.14796911 IntroductionMuch of the decision making in the real world takes place in anenvironment in which the criteria, the limitations and the conse-quences of possible actions are not known precisely. The mainreason is that many practical systems are too complex to be de-scribed by precise models and in exact terms. The level and extentof imprecision or uncertainty present varies from one system toanother. Different models need to be developed/used for differenttypes of imprecision. As an example, consider a link in a robotmanipulator whose length many be stated in different ways asfollows: lies between 0.501 and 0.503 m, or about 0.502 m,or has a mean value of 0.502 m and a standard deviation of0.0003 m and follows normal distribution. When the parametersof a system are described as random variables with known statis-tics such as means and standard deviations, probabilistic and reli-ability approaches can be used for the analysis and design of thesystem.By using probabilistic models for the uncertainties related togeometry, loads and material properties, several investigationsDesign of MechanicalInvolving Intervalpresent in many engineering systems can be modeledor interval methods. This work presents the optimum design ofsystems using interval analysis for the prediction of system re-parameters is defined by a range of values. Since theparameters is found to increase with an increase in theof input interval parameters with the use of interval arithmeticprocedure is used to obtain approximate but reasonably accurateThis procedure is found to be simple, economical and fairlydesign of a brake is considered to illustrate the computationalThe procedures outlined in this work are quite general and can beany uncertain mechanical system when either the probability dis-the preference information of uncertain parameters are unknown.#ods can be used for the analysis of the system. This work presentsthe optimal design of uncertain mechanical systems in which theuncertain parameters are described by intervals. Some situationsin which an uncertain parameter can be modeled as an intervalnumber are indicated below:1. When a machine part is to be produced with dimension x,itis often specified using a tolerance as x6Dx for convenience ofmanufacturing. In this case, the actual dimension of the part is tobe treated as an interval number as (x2Dx,x1Dx).2. In many engineering applications, external actions such asthe wind and snow loads acting on a tall pressure vessel (P)maybe known to vary over a range P1to P2. If sufficient data areavailable for these external actions, they can be modeled usingextremal distributions. When the exact probability distribution ofthe load P is not known, it can be treated as an interval number asP1,P2! based on the range of values observed.3. During the design stages, designers often conduct sensitivitystudies by changing one or more parameters over specifiedranges! and finding its influence on the response/design of theand design of such systems should take care of the intervals ofdeviation of parameters from their nominal initial! values.3 Interval AnalysisWhen the imprecise design parameters are expressed as intervalnumbers, the design equations can be converted to the form ofinterval expressions. The required computations can then be car-ried using the rules of interval arithmetic. When optimizationtechniques are used in conjunction with interval analysis. certaincomputational problems need to be resolved. With the use of the-oretical interval arithmetic operations, the widths intervals! of theresponse parameters have been observed to be wider than the truewidths with an increase in the number and/or ranges of intervalparameters. To limit the unnecessary growth of the intervals of theresponse parameters, an approximation technique, termed thetruncation method, has been used. This method limits the relativerange of variation of a response quantity on the order of the rela-tive ranges of variation of the input parameters 7#.The optimal design of brakes is considered to illustrate the ef-fectiveness and the computational aspects of the present method-ology. This work represents one of the first attempts at presentingan optimization strategy for the design of mechanical systemsinvolving interval parameters. It is to be noted that nonlinear pro-gramming routines can not be used directly to solve the currentproblem. The procedure has to be modified to overcome the in-herent difficulties associated with interval numbers. Not only theinterval of a response parameter grows with the ranges of inputparameters and the numbers of arithmetic operations involved, butalso the ranges of the response parameters depend on the order orsequence in which the various interval input variables are usedduring the computations. In order to overcome this difficulty, theranges of the response parameters obtained from interval arith-metic are checked and corrected if necessary! with those given bythe combinatorial approach at certain stages of optimization par-ticularly during gradient computations!.2 Brief Literature SurveyThe present day competitive manufacturing requires designersto pursue effective approaches for a more precise and economicalproduct design. The economical product design requires the appli-cation of optimization. However, the results of optimization willbe useful only when accurate response quantities are predicted bythe analysis procedure used during optimization. Since the designparameters associated with most engineering systems are uncer-tain and imprecise, different analysis models have been developedfor predicting the response of different types of uncertain systems.For example, the fuzzy set theory was initiated in 1965 11# andhas been applied to the analysis and design of mechanical systemsin recent years 12-14#. In 1987, Rao 12# developed a fuzzy setmethodology for the description of uncertain mechanical systemsand illustrated its application for the optimum design of four barmechanisms. In 1995, Rao and Sawyer 13# presented a fuzzyfinite element method for the static analysis of imprecisely definedstructural and mechanical system. The vibration analysis of uncer-tain systems was conducted using fuzzy finite elements by Li andRao 14# in 1997. Apart from the methods based on fuzzy theory,probabilistic approaches were also applied for the analysis anddesign of uncertain engineering systems. For example, in 1987,Nakagiri 15# presented a stochastic finite element method to per-form the uncertain eigenvalue analysis of composite plates. Thefuzzy finite element method presented in 13# permitted the appli-cation of interval methods in the analysis of fuzzy systems natu-rally. The arithmetic of interval analysis was established in 1966and was used for the purpose of solving systems of equations 16#.In 1997, Rao and Berke 17# applied interval arithmetic for struc-tural analysis, and compared the accuracy obtained with combina-torial method, direct interval method and interval-truncationmethod. Currently, besides probabilistic and fuzzy methods, inter-val method is an alternative method that can be used to analyzethe imprecise problems that exist in the engineering design field.466 Vol. 124, SEPTEMBER 2002An interval parameter represents the range of uncertainty of adesign parameter. For each imprecise parameter, there are twonumbers that represent the lower and upper bounds of the param-eter. For example, an interval number can be denoted as x5(xI ,x)(x1,x2) where the lower and upper bound values aregiven by xI 5x15x02Dx and x5x25x01Dx where x0denotesthe nominal value and Dx represents the tolerance on x. If thesystem is linear, the exact response can be computed using thecombinatorial approach. If the response parameter is representedas f (x1,x2,fl,xn) where x1,x2,fl,xndenote the input intervalparameters withxi5xi(1),xi(2)#xIi,xi#; i51,2,fl,n (1)then all possible values of f are given by:fr5 fx1(i),x2( j),fl,xn(k)!; i51,2, j51,2, k51,2;r51,2,fl,2n(2)Here frdenotes the value of the response parameter, f , at a par-ticular combination of the end points of the intervals ofx1,x2,fl,xn. The response parameter can be denoted as an inter-val number as:f 5 fI, f#minrfr,maxrfr# (3)Although this method, known as the combinatorial method, ap-pears to be simple, it requires 2nanalyses and becomes tediousand prohibitively expensive for most practical problems that con-tain large number of interval parameters.The interval analysis can be conducted using interval arithmeticoperations. For example, if A5aI ,a# and B5bI ,b# denote twointerval numbers, a general arithmetic operation can be expressedas 16#:aI ,a#*bI ,b#5$x*yuaI 0 j5me11,fl,mXlXXu(16)where X is the set of design variables, f (X) is the objective func-tion, hi(X)isith equality constraint, and gj(X) is the jth inequal-ity constraint, Xlis the lower bound vector, Xuis the upper boundvector, meis the number of equality constraints and m is the totalJournal of Mechanical Design16!18,19#. The gradients of objective function and constraintsare computed by the finite difference method. The iterativescheme is:Xk115Xk1akdk(17)where Xkis the current design vector, dkis the search directionand akis the step length. When the optimum point is not reached,the modified BFGS method is used to generate a new search di-rection dk18,19#.6 Optimization Using Interval ParametersAll the preassigned and design parameters are treated as crispor deterministic quantities in few cases. In some cases, only thedesign variables are assumed to be interval numbers. In othercases the preassigned parameters or design data! as well as thedesign variables or unknown parameters! are considered as inter-val parameters. To demonstrate the interval approach, the rangesof interval numbers are assumed to be 20% (610%) of the re-spective nominal values with triangular memberships. In order todeal with interval numbers, the interval arithmetic rules are ap-plied to every step of calculations. Since a specific interval vari-able may appear several times in different terms of the same equa-tion, the order in which computations are carried willunnecessarily increase the interval ranges of the result to varyingdegrees 17,20#. Thus, during actual programming, we need toadjust the order in which different interval parameters are consid-ered in any specific equation. The purpose of changing the orderof parameters is obviouswhen the program executes the equa-tion using interval parameters, the new order will not only mini-mize the computational time but also leads to a more accurateinterval range for the result. In addition, the truncation approach isused based on a comparison between the input range and theoutput range of the parameters. The purpose of truncation is tomake reasonable modifications to the output range before con-ducting the next interval operation. The response parameters ofthe system, computed using interval arithmetic and truncationmethod, are compared with those given by the combinatorial ap-proach at regular stages during optimization for example, duringgradient computations! to ensure that the ranges do not propagateto unreasonable levels.In some computational steps, the use of interval arithmetic maylead to a result that is in conflict with the physics of the problem.If these invalid operations are used in the computations, the finalsolution is not expected to be correct. For example, let a directioncosine be computed from the coordinates of a point (x,y) as:cos u5xAx21y2(18)For simplicity, let the x value is considered as the only intervalvalue as x5(xI ,x). If we apply the interval operation, we canobtain four possible values:cos u!15xIAxI21y2, cos u!25xAx21y2; (19)cos u!35xIAx21y2, cos u!45xAxI21y2; (20)Here only the first two values indicated by Eq, 19! are correct,the latter two given by Eq. 20! are incorrect, because accordingto the original equation, the values of x used at the two locationsmust be the same. The problem is that we dont know whichgroup would be used to define the interval boundary of the resultduring automatic computations, because the sign of x may beeither negative or positive. After we apply the truncation ap-proach, if the range of cos u need to be truncated, we would loseSEPTEMBER 2002, Vol. 124 467the correct value. Even if truncation method is not used at thisstage, since some incorrect values are included in the intervalpreassigned parameters, namely the coefficient of friction betweenthe friction material and the brake drum ( f ), angle correspondingrange, they may lead to more inaccurate results in the followingcomputational steps. Finally, it is difficult to judge where the cor-rect solution lies in the final interval range.In such cases, it is safe to apply the combinatorial method in-stead of the interval operation in order to comply with the physi-cal logic. Thus it is necessary to understand the physical meaningof each equation not viewing it just as a combination of symbolsand numbers! before implementing the interval analysis. It is to benoted that the optimization method is found to converge to thecorrect solution as long as the ranges of the input interval param-eters are small and the ranges of the response parameters arerestricted not to grow unnecessarily.7 Illustrative Example: Brake DesignThe design of an internal long-shoe drum brake, which iswidely used in automobiles, is considered to illustrate the optimi-zation methodology based on interval analysis. A drum brake hasfriction material applied to the circumference of a cylinder, andcan be simplified as a system composed of only three main parts,namely the brake shoe, the part bonded with the friction material,the brake drum, the part rubbed by the friction material and thehinge Fig. 1!. The shoes are forced against the rim to create afriction torque. In an internal-shoe brake, the shoes expand againstthe inside surface of the drum, and the drum is attached directly tothe wheel. One shoe is self-energizing in the forward direction ofdrum rotation, and the other is self-energizing in the other direc-tion. Energizing means that once the brake is initially engaged, thefriction force tends to increase the normal force, thus nonlinearlyincreasing the friction torque in a positive feedback fashion. If theshoe of a brake contacts over a large angular portion of the druman angle no less than 45!, then it is viewed as a long-shoe brake22#. For a long-shoe brake, the pressure p at any point on theshoe is assumed proportional to the vertical distance from thehinge pin:p5pasin usin ua(21)where pais the permissible maximum pressure acting at an angleuafrom the hinge pin. If the toe angle u2is larger than 90, thenp will be maximum (pa) when ua590. If it is less than 90,then p will be maximum at the toe. In a practical brake, theFig. 1 Brake Shigley 21 468 Vol. 124, SEPTEMBER 2002to the beginning of friction