Solutions Of Complex Differential Equations Research Articles

A closer look at tumbling toast

The study of the mechanics of tumbling toast provides an informative and entertaining project for undergraduates. The relatively recent introduction of software packages to facilitate the analysis of video recordings, and the numerical solution of complex differential equations, makes such a study an attractive candidate for inclusion in an experimental physics course at the undergraduate level. In the study reported here it is found that the experimentally determined free fall angular velocity of a board, tumbling off the edge of a table, can only be predicted at all accurately if slipping is taken into account. The size and shape of the board used in the calculations and in the experiments were roughly the same as that of a piece of toast. In addition, it is found that the board, tumbling from a standard table of height 76 cm, will land butter-side down (neglecting any bounce) for two ranges of overhang (δ0). δ0 is defined as the initial distance from the table edge to a vertical line drawn through the center of mass when the board is horizontal. For our board (length 10.2 cm) the approximate ranges of overhang are 0–0.8 and 2.7–5.1 cm. The importance of the 0–0.8 cm (only 2% of all possible overhangs for which tumbling is possible) favoring a butter-side down landing should not be underestimated when pondering the widely held belief that toast, tumbling from a table, usually falls butter-side down.

Linear and non-linear deflection analysis of thick rectangular plates—II. Numerical applications

Variational methods are widely used for the solution of complex differential equations in mechanics for which exact solutions are not possible. The finite difference method, although well known as an efficient numerical method, was applied in the past only for the analysis of linear and non-linear thin plates. In this paper the suitability of the method for the analysis of non-linear deflection of thick plates is studied for the first time. While there are major differences between small deflection and large deflection plate theories, the former can be treated as a particular case of the latter, when the centre deflection of the plate is less than or equal to 0.2–0.25 of the thickness of the plate. The finite difference method as applied here is a modified finite difference approach to the ordinary finite difference method generally used for the solution of thin plate problems. In this analysis thin plates are treated as a particular case of the corresponding thick plate when the boundary conditions of the plates are taken into account. The method is first applied to investigate the deflection behaviour of clamped and simply supported square isotropic thick plates. After the validity of the method is established, it is then extended to the solution of rectangular thick plates of various aspect ratios and thicknesses. Generally, beginning with the use of a limited number of mesh sizes for a given plate aspect ratio and boundary conditions, a general solution of the problem including the investigation of accuracy and convergence was extended to rectangular thick plates by providing more detailed functions satisfying the rectangular mesh sizes generated automatically by the program. Whenever possible results obtained by the present method are compared with existing solutions in the technical literature obtained by much more laborious methods and close agreements are found. The significant number of results presented here are not currently available in the technical literature. The submatrices involved in the formation of the finite difference equations from the governing differential equations are generated directly by the computer program. The subroutine SOLINV using the change of variable technique illustrated elsewhere takes care of the solution of the general system. Simplicity in formulation and quick convergence are the obvious advantages of the finite difference formulation developed to compute small and large deflection analysis of thick plates in comparison with other numerical methods requiring extensive computer facilities.

Bounds for solutions of 2nd order complex differential equations

1. It is well known [1], [2] that upper and lower bounds for the norm of a solution of ordinary differential systems can be given in terms of solutions of related first order scalar equations. However, the independent variable t is taken to be real there. In [3] upper bounds for solutions of a class of 2nd order complex differential equations were obtained. In this paper we derive upper as well as lower bounds for solutions of the complex differential equation

Two Applications of a Digital Computer to Machine-Design Problems

Abstract Two examples are presented that illustrate two types of problems wherein a high-speed computer can provide important aid to the design engineer: (a) Tedious table-making computations; (b) solution of complex differential equations. The two applications represented are, respectively, a cycloidal crank-drive mechanism and a swing-type hammer mill.