- N a Number of active turns N t Total number of turns. - Documentation of the design. - Computer-aided analysis (CAA) involves use of the computer in an "if this then that". - The data base created during computer-aided drafting can be used by computer-aided manu- facturing. - The basis for this programming must be their understanding of the problem. - This capability can be described as decision making.. - This can be called iteration.. - The specification set is not couched in terms of the designer's thinking parameters or concerns, and so the designer recasts it as a decision set. - In the case of the spring, a correspond- ing decision set is. - These five decisions can be made a priori, and are consequently called a priori decisions. - Note that the dimensionality of the task has been identified. - An adequacy assessment consists of the cerebral, empirical, and related steps undertaken to determine if a specification set is satisfactory or not. - In the case of the spring, the adequacy assessment can look as follows:. - In the case where large numbers of satisfactory specification sets are expected, an opti- mization strategy is needed so that the best can be identified without exhaustive examination. - Often the competition is among steels and the weight density y can be omitted.. - An optimization strategy is chosen in the light of the number of design variables present (dimensionality), the number and kinds of constraints (equality, inequality, or mixed. - An example follows to show how simply this can be done, how a hand-held programmable calculator can be the only computational tool needed, and how some tasks can be done manually.. - 1.25 in 0.5 <. - 1.50 in 0.5 <. - DECISION SET. - The above computational steps can be programmed on a computer using a lan- guage such as Fortran, or on a hand-held programmable pocket calculator. - The fol- lowing table comes from a pocket calculator when one inputs wire size and the remaining elements of the column are presented.. - Figure 5.4 shows a plot of the figure of merit vs. - Ponder the structure that identified the dimensionality of the task and guided the component computational arrangement.. - In the discussion in the previous section of the adequacy-assessment task and the conversion to a specification set as illustrated by the static-service spring example, there occurred a number of routine computational chores. - These were simple alge- braic expressions representing mathematical models of the reality we call a spring.. - FIGURE 5.4 The figure of merit as a function of wire diameter in Example 5.1. - However, it is of the same character. - In Fortran such a program can be a function subprogram or a subroutine subprogram. - This simplicity is welcome as the tasks become more complicated, such as finding the stress at the inner fiber of a curved beam of a tee cross section or locating the neutral axis of the cross section.. - Such routine answers to computational chores can be added to a subroutine library to which the computer and the users have access. - A library of analysis subroutines can be created which the designer can manipulate in an executive manner simply by calling appropriate routines. - Such subroutines are called design subroutines because through an inverse-analysis strat- egy they can be made to yield design decisions. - Within them is the essence of the reality of the physical world. - When decisions are made which completely describe a helical compression spring intended for static use, the computer can be used to examine important features. - a large number of attributes can be viewed:. - This too can be assessed, and the computer can assist routinely. - The relative contribution of the various tolerances can be observed. - To make a statistical statement as to the probability of observing a value of r\ s of a particular magnitude, we need an estimate of the variance of r\ s. - The estimate of the variance and standard deviation of r^ is tf . - Recurring tasks can be coded as subprograms which represent convenient building blocks for use in solving larger problems.. - Rewrite in the form x = F(X), thereby defining F(JC).. - The secant equation is of the form nPIA = F(nP!A). - If a function of x is expanded about the point Jt 0 in the neighborhood of the root as a Taylor series (Ref. - O, and if the series is truncated after two terms, solution for x is a better estimate of the root than is Jt 0 . - and the better estimate of the root is Jt 1 = X Q + A*. - Ay, then the preceding equations can be written as. - Jt <— x + Ax y <— y + Ay. - As an example of the use of the Newton-Raphson method, consider a position analysis of the four-bar linkage depicted in Fig. - Such a solution algorithm for simultaneous equations can be generalized to n equations. - The previous kinematic problem can be coded for a hand-held calculator in approximately a hundred steps.. - 78) showed that if an integration is performed in interval a, b with TV 2 panels and then repeated with N 1 = N 2 /2 panels (using every other ordinate), then the value of the integral is given by. - The number of panels N 2. - The approximate relation between the number of panels and the error is. - Estimate the value of /J sin x dx to five significant digits to the right of the decimal point.. - Estimate the number of panels necessary to attain requisite accuracy:. - An improved estimate of the value of the integral might be I 24 + R which rounded to five significant digits to the right of the decimal point is still 2.000 00.. - There exist programs for large computational machines which can be imitated or approximated on smaller machines. - If M 1 is the number of successes, M 2 is the number of failures, and M 1 >. - 10, then the sampling distribution of the number of runs M is approx- imately gaussian, with (Ref. - The null hypothesis that the sample is random can be based on the statistic. - The mean number of runs expected Ji n and the standard deviation expected G n are. - If the differences between y and y are ranked in the order of their corresponding abscissas and placed in the column vector DY, then the number of runs can be detected with the following Fortran coding:. - NUMBER=I DO 100 1=2,N A=DY(I)XABS(DY(I)) B=DY(I-I)/ABS(DY(I-I)) IF(A/B.LT.O.)NUMBER=NUMBER+!. - where the integer NUMBER has a magnitude equal to the number of runs.. - The structure of the design-decision problem and that of the optimization problem are similar. - Many ideas and techniques of the latter are applicable to the former. - The optimization problem can be posed as. - In terms of the ideas in Sec. - The adequacy assessment can be performed by a Fortran subroutine:. - (Establish the adequacy of the decision set.). - (Evaluate the figure of merit if the decision set is adequate.) The choice of d is provided either manually (interactively) or by an appropriate optimization algorithm which makes successive choices of d which have superior merit. - The program ADEQ is durable and once pro- grammed can be used. - The user is solving a problem to which the answer is not known and can- not be sure of having attained the global extreme of the figure-of-merit function.. - This procedure is best made interactive and can be presented to a user without requiring a knowledge of programming.. - For the static-service helical-coil compression spring using one decision variable d and two decision variables d and € 0 , the specification sets are. - Armed with rationales from probability the- ory, statistics developed methods for gathering, analyzing, and summarizing data and formulating inferences to learn of systematic relationships, together with an esti- mate of the chance of being incorrect. - To simulate is to mimic some or all of the behavior of one system with another, with equipment, or with a computer using random numbers.. - Random numbers from the uniform distribution U[O, I] can be selected using a machine-specific subprogram supplied by the computer manufac- turer. - These numbers can be trans- formed into another distribution of interest using software. - Through selec- tion of random numbers and calcula- tions performed with them, data can be gathered and answers to useful ques- tions obtained. - What is the distribution of the radial clearance c? What is its cumulative distribution function F(C)? It takes more than elementary statistics to go straight to the answers.. - A simulation can be run to obtain robust answers without knowing or using the sta- tistical knowledge.. - Example 2 illustrates the power of the computer simulation process.. - The cumulative distribution function can be well approximated for various values of radial clearance c and a polynomial fitted to the data.. - c, in F(C) 0.0010 O . - A least-squares quadratic fit of the form F(c. - For distributions with survival equations that can be explicitly solved for R or F 9. - The question of the accuracy of the reliability estimate is addressed as follows. - The sum of the elements in {x} is np, where n is the number of entries (trials) and p is the probability of success. - The mean of the ele- ments in [x] is. - n n ^ so x is an estimator of the probability of success p. - The column vector of the squares of the elements in [x] is identical to the elements in [x}. - The sum of the squares of the elements in [x] is also np. - The standard deviation of the mean x is. - The number of trials n 2 necessary to attain an error e m ) associated with m significant digits to the right of the decimal point is. - Some of the left-hand digits are correct. - c Initialize counters so simulation proceeds in steps (economically) c under the control of the user.. - 2 print*,'Enter number of trials in similation n' read*,nmore. - if(wl.lt.w2)x=0.
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