Exact solution of Qian equation of slender toroidal shells

In 1979 Qian Weichang studied the slender toroidal shell systematically and derived a called Qian’s equation, then obtained a series solution with the expression of continued fractions. But Qian did not mention if the series solution can be converted to a well-known special functions. In this paper, a linear transformation has been introduced, which will transfer the equation into a Mathieu equation, whose solution can be expressed in terms of Mathieu functions. This study has revealed a intrinsic relationship between the Qian’s solution and the Mathieu solutions. The present work will make the calculation much easier on the shells due to the fact of the Mathieu functions have been well studied in the literature. 

Citation: Sun Bohua. Exact solution of Qian's equation of slender toroidal shells. Mechanics in Engineering, 2016, 38(5): 567-569 (in chinese)