Comments on “A direct solution for the transverse vibration of Euler-Bernoulli wedge and cone beams”
✍ Scribed by Ji Wang
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 164 KB
- Volume
- 185
- Category
- Article
- ISSN
- 0022-460X
No coin nor oath required. For personal study only.
✦ Synopsis
In earlier papers [1,2], Naguleswaran studied the vibrations of beams with non-uniform thickness by solving the differential equation with the Frobenius method, which gives the solutions in infinite power series. The author observed the fact that, for some cases, even the quadruple precision could not provide a satisfactory solution, as was found in a previous study [3].
It should be emphasized that the high precision requirement of the computation is caused by the precise evaluation of the functions which are given by power series, not the determinant. The problem is, as we can observe from the solutions, that the frequency is always appearing in the coefficients of the polynomials, and the recursive relations of the coefficients make the coefficients increase with the power of the each terms. It is even clear that these functions converge for parameters such as the frequency and truncation factor, that they do so very slowly, and that the intermediate terms could go beyond the limits of many machines. This is the phenomenon observed by Soni et al. [4,5], who investigated plates with non-uniform thickness. Furthermore, as in the beam case, the high frequencies which are important in some applications could not be found.
The difficulty in computing for high frequency is overcome in a series of papers by Lee and Wang [6,7], dealing with the vibrations of bevelled quartz crystal plates with the aid of a multiple precision FORTRAN program [8], which was developed by D. H. Bailey of Ames Research Center, NASA. Since this software guarantees the computing precision to almost any degree that the users request, the evaluation of functions is precise enough to yield exact solutions. We found that the CPU time to perform such a high precision computation is not greatly increased.