- The degree of precision obtained for the dependent variable depends on the accuracy of the predominant term in the particular equation, as related to algebraic operations. - In what follows, we will assume that the accuracy of the computational device is better than the num- ber of significant figures in a determined value.. - The independent com- ponents are separated by introduction of the operator j = V^l. - In numerical applications, generally the value of a dependent variable is determined for a set of values of the independent variables using an appropriate functional expression. - However, for many other applications requir- ing automated calculations, it may be appropriate to transform at least some of the tabular data into equations by curve-fitting techniques. - In general, curves present a valuable pic- ture of how a dependent variable changes as a function of the independent variables.. - For example, for the stepped shaft previously mentioned, stress concentration factor K ts would be expressed by an equation as a function of r, d, and D derived from the curves of the given graph. - For the special case of n = 1, the equation y = b + C 1 X is linear in x. - For the special case of n = 2, the equation y = b + C 1 X + C 2 x 2 is known as a quadratic equation. - For the special case of n = 1 with C 1 = 1, the equation y = bx is a simple straight line.. - As a specific example of the more general case expressed by Eq. - (4.3), resulting in the following equation:. - log Jc n , respectively.Thus the equa- tion y = bx c will plot as a straight line on log-log graph paper, regardless of the val- ues for constants b and c.. - However, any of the more complicated equations found in machine design may be considered as special combinations of the basic equations, with the terms related by algebraic operations.. - As a specific example of a combined equation, a polynomial equation is merely the sum of positive simple exponential terms, each of which has the general form of the right side of Eq. - The criterion for determining how many terms of the sequence are necessary is based on a consideration of convergence. - The number of terms used must be sufficient for convergence of the determined value to an acceptable level of accuracy when compared with the entire series evaluation. - For the Fourier series expansion of Eq. - The magnitude of the error is \e. - This will be illustrated specifically by application of the theory of variance, as presented later under relative change.. - In some equations of analysis we have a term of the form (1 + x) in the denominator.. - By dropping all but the first two terms of the series, 1/(1 + x) may be approximated by 1 - jc, expressed as follows:. - Hence a denominator term of the form 1 + x could be replaced in an equation with a numerator term 1 - x, providing the error is acceptably small over the antici- pated range of variation for x. - Similarly, a denominator term of the form 1 - x could be replaced with a numerator term 1 + x if the error is likewise acceptably small. - For the summa- rized equations, angle x must be in radians. - (4.7) as follows:. - (4.8) as follows:. - Its first-order Taylor's series approximation about x = a is obtained by using only the first two terms of the Eq.. - (4.9) series, resulting in the following equation:. - (4.30) will be applied, and a good choice for a would be the midpoint of the x range, with a = 55°(7i/180. - For that value of*, the error by Eq. - For any differentiable /(;c), a more accurate approximation can be obtained by using the first three terms of the Eq. - They determine the significant harmonic content of the periodic function. - (4.10) series for the approximation derived. - X n can be substituted respectively for the differentials dxi, dx 2. - (4.3) results in the following simple approximation ([4.5], pp.. - (4.35) as follows:. - (4.34), accuracy estimates can quickly be made for simple exponential equations of the Eq.. - Suppose that relatively small errors are anticipated on the theoretical values of the independent variables JCi, J C 2. - Designate the standard deviation of the normal distribution for each variable respectively by C^ 1 , C^ 2. - Suppose each of the independent variables Jc 1 , J c 2. - If the tolerance band Ax t corresponds to three standard deviations, 99.73 percent of the total population for x t values would be within the range jc. - AjC n would correspond to three standard deviations and would encompass 99.73 percent of the total popula- tion for each variable.. - (4.39) as follows:. - To check the accuracy of the approximations, for y - sin jc we know by calculus that dy/dx = cos jc and d 2 y/dx 2 = -sin x. - (4.41) and (4.42) may be applied to calcu- late partial derivatives for the case of a differentiable function of several variables.. - Hence, for the equation y = /(XI, Jt 2. - Often it is necessary to evaluate a definite integral of the following form, where y = f(x) is a general integrand function:. - The values of y are then calculated at each of the net points so determined, giving ^ 0 , 3>i»3>2. - For the test example, a value of n = 20 is arbitrarily chosen. - gram ML-09 was used for the test example [4.11]. - To check the accuracy of the approximation, from elementary calculus we know that Jsin x dx is -cos x. - Validity of the equation over the range of interest could then be established.. - In such cases, a simple exponential equation of the following form can readily be derived for passing through two precision points (it is assumed that both x and y are positive):. - Either one of the two precision points can then be used to calculate coefficient b as follows, as derived from Eq. - (4.13) for all known data points to determine the validity of the derived equation over the range of interest.. - A polynomial equation of the following form can be derived to pass through (n + 1) given precision points:. - 4, we will obtain a polynomial equation of the third degree, since n = 3.. - (4.2), we obtain the following:. - 4.5, which also shows the curve of the equation y =f(x) to be derived. - 2 terms for k = 1 to M of the given data set is known as the least-squares fit. - c = -0.1305 b = 1.2138 r = 0.9929 Therefore, the derived equation for the least-squares fit is as follows:. - Jc n ) for the case where we have n independent variables. - The final equation is derived using /i(JC 1 ) and/ 2 (jc 2 ) satisfying the y p , (jci) p , and (jc 2 ) p values of the given data.. - For Jc 2 = 4.5 For jci = 3.0 XI y X2 J^. - Thus the first and last data points are used for both parts of the table to calculate exponents c\ and C 2 using Eq. - (4.50) as follows:. - Coefficient b for the equation is then calculated using the common precision-point values as follows:. - b = 3.190 Therefore, the derived equation for the curve fit is as follows:. - (4.68) as follows:. - (4.69) as follows:. - For practical cases, to start, it is generally desired to determine the general charac- teristics of the complicated function y = f(x), and this can be accomplished by exe- cution of an exploratory search. - it is advantageous to locate the approximate neighborhoods of the roots Xj before a more accurate determination is made for each. - This is accomplished by an ex- ploratory search stage, which calculates the y values at successive step points of the range of interest jt min <. - The increment AJC is chosen rela- tively large to save on computation time, but it must be small enough to identify the neighborhoods of the roots.. - These are recognized by an algebraic sign change in y for successive step points of the search. - specific example for the exploratory. - Consider the problem of finding the roots of the following equation:. - 3.8, and we choose AJC = 0.2 for the exploratory search to be made in accordance with Fig. - Thus a new interval of uncertainty is determined based on the sign of the calculated y, as shown in the figure. - Xj + J of the figure. - fore, for the input of Fig. - 4.9, we would use x k = 2.0 and x k + i = 2.2, and we choose an accuracy specification of e = 10~ 6 for the root to be determined. - 4.9 calcula- tion process was programmed on a TI-59 calculator, resulting in the following root value at the conclusion of the search (which took approximately 60 seconds for the execution time):. - If in the iterative search process of the Newton-Raphson method we are at some point. - 4.10, we determine an improved estimate Xj +1 for the root by extrapo- lation as follows:. - There- fore, for the input in Fig. - Also, we choose the accuracy specification for the root as. - 4.11 calculation process was programmed on a TI-59 calcu- lator, resulting in the following root value at the conclusion of the search (which took approximately 15 seconds for the execution time):. - A comparison of the root findings for Eq. - Depending on the general characteristics of the functions, we could devise an iterative scheme for converging to the neighborhood of the solution point as schematically shown in Fig. - (4.73), from Table 4.1 the root is as follows:. - Thus we have found for the solution point the following value for Jc 1. - For the situation where Eqs. - (4.82) and (4.83) cannot readily be combined by eliminating either Jc 1 or X 2 , curve-fitting techniques can generally be applied to either one of the equations, giving an explicit equation for either Jc 1 or X 2 expressed in terms of the other variable. - The equation-combination procedure is then reversed to find the values of the other variables. - Based on the most critical aspects of the particular problem, an appropriate optimization objective must be chosen and mathematically formulated. - Hence the engineer must simultaneously address a com- plicated equation system of the following general form for arriving at decisions of opti- mal design:. - The gen- eral symbol ^ means that the required relation is one of the following, for any of the variables: >, >. - and [4.12]).
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