Vibration analysis by dynamic partitioning.

Abstract

process of iteration, thus contributing to the uniformity of the programing and facilitating treatment of the problem by modern digital computing machines. Hansen's original lunar theory requires the use of the derivatives of the disturbing function with respect to the eccentricity and the parameters determining the orbit plane's position. This means that a literal development of the disturbing function must be obtained before the values of the elements can be substituted. The present modification permits a numerical development of the disturbing function. In order to reproduce some of Hansen's series, the author developed methods for the computer to perform both numerical and algebraic operations. The trigonometric series employed in the lunar problem are slowly convergent. The original program allowed 50 terms to a series. Of the two completed versions, the program written for the IBM7094 Mod II allows for 200 terms to a series and the Univac 1107 version allows 3333 terms to a series. The IBM 7094 program utilizes the double core storage of 65,000 words. The Univac 1107/8 program uses 500,000 words of drum storage and allows 10,000 words of storage for each series. Almost the entire 65,000-word core storage of the Univac 1107/8 is used as accumulators to bring in the series to be operated on and to accumulate the results. Core storage of 20,000 words is used as an accumulator to store the resultant series. A complete package of routines to multiply, add, subtract, scalar multiply, differentiate, integrate, and evaluate trigonometric series and to extract the numerical coefficient of any argument was written. By the uses of these tools, series operations are programed almost as easily as numerical calculations. The foregoing is a technical description of the program employed. I also would like to discuss the history of this continuing effort. Analytic solutions to orbit problems were pursued by P. Musen at the Goddard Space Flight Center in 1959 and 1960. It was suggested that if we could multiply A sm(IX) -fB sin(JY), where A, B, I, and J are numbers and X and Y are literals, analytic theories could be calculated using digital computers. These original efforts have led to the present programs. Some of the mistakes in the program were extremely difficult to correct. As an example, a discrepancy in the calculation of n$y (the motion of perigee of the moon) was not corrected until we returned to the problem after suspending it for a period of six to eight months. Further efforts will be in the direction of additional improvement of the Jupiter X orbit through differential correction and the calculation of the orbits or other Jupiter satellites which are also strongly perturbed. Some of the results of the differential correction process as applied to Jupiter X are included as Tables 1 and 2. The difference between observation and prediction is of the order of 2 sec of arc.