Obliczanie Sił Krytycznych dla Sprężystych Prętów Niepryzmatycznych Metoda Interpolacji Częściowej - Życzkowski, M. (Engineering Transactions, 1956)
The existing tables of critical forces for the most common perfectly elastic non-prismatic bars, prepared chiefly by Dinnik, are not accurate. The subject of this paper is to present a method for obtaining 5 or 6 figure tables with small intervals of the argument (the taper k). The values of the critical forces can be obtained in an accurate manner by solving transcendental equations of the type (3.20) involving Bessel’s functions. Such an equation determines a composite function expressing the dependency of the stability coefficient on- the taper. Equations of the type (3.20) appearing in various domains of theoretical physics have already been considered by several investigators, J. Mac Mahon developed a method based on the expansion of a function in a power series convergent in a certain neighbourhood of the point k = 1 (a prismatic bar), thus computing the values of the first several derivatives of the sought function, at the point k = l. A. Kalähne proposed a method based on the knowledge of the zeros of B e s s e l functions Jp(x) and Yp(x). This method enables us to solve the equation considered for certain values of k non-uniformly scattered over the interval 0<k<1. For other values of the taper k the result can be obtained by interpolation. A new interpolation method called the method of partial interpolation is proposed in this paper. This method enables us to use the results of Kalähne and Mac Mahon, and to calculate with required accuracy the roots of equations of the type (3.20): The amount of calculation work is relatively small. The ordinary partial interpolation is illustrated in Fig. 1. First, we construct the basic approximation w(z) using some of the conditions available only. These are usually conditions of the Hermitian type, for the ends of the interval (the agreement of the function itself and its first few derivatives). Next, using the remaining conditions we calculate the error at the interpolation in nodal points. This error, b/(x), is interpolated in a suitable manner in each particular sub-interval (in the most simple case the interpolation is linear). The sum of the basic approximation and the interpolated error is believed to be the approximation of the function required. The accelerated partial interpolation method differs from the ordinary method by the fact that in the second stage we interpolate the ratio of the error bi(x) to a certain reference function U(x) instead of interpolating the error itself. This enables a considerable reduction of the final interpolation error ba(x). The paper gives an example of the evaluation of the error for the ordinary and the accelerated partial interpolations Eqs. (8.17) and (8.21), respectively and presents formulae for the coefficients of basic approximations assumed in the form of a polynomial, (9.3), (9.7), and (9.9). The method proposed is compared to known interpolation methods using the example of the well known function y = sin (7/2) x. Finally, the accelerated partial interpolation method is used to calculate the roots of the corresponding transcendental equations and the required values of stability coefficients (Tabs. 2-6).