Universal approach to derivation of quaternion rotation formulas

This paper introduces and defines the quaternion with a brief insight into its properties and algebra. The main part of this paper is devoted to the derivation of basic equations of the vector rotation around each rotational x, y, z axis. Then, the equations of generalized quaternion rotation and express the general rotation operator is derived. Finally the utilization of equations is demonstrated on a simple example. For purposes of simplicity the quaternions theory is demonstrated around the z-axis by γ angle. For the purpose of this paper, the fact that the subspace of vector quaternions may be regarded as being equivalent to the ordinary is used.

Effects of Surface Tension and General Rotation on the Rayleigh-Taylor Instability of Three-Layer Flow

The stability of two interfaces separating three fluids, where the fluids are assumed to be incompressible, inviscid, and of constant density, has been investigated for a system that is acted upon by a general rotation. The effect of surface tension at the two interfaces is taken into account. A general dispersion relation for the system is obtained analytically by formulating the problem in terms of complex variables. Numerical calculations were performed for a hexane-NaCl-CCl4 system to investigate stable case, and special cases that isolate the effect of various parameters on the growth rate of the Rayleigh-Taylor instability are discussed. It is found that the two cutoff wave numbers for the system with surface tension are unchanged by the addition of a general rotation, and that for the system considered, all growth rates are reduced in the presence of a general rotation.

LOCAL WAVELET TRANSFORM ON THE SMOOTH SURFACE OF ROTATION CLASS

For the smooth surface of rotation class M, the explicit construction of two local flattening maps p are given. By virtue of the inverse projection p-1, where the bijection projection p is the local flattening map from a two-dimensional smooth surface M of rotation class to a plane, the local wavelet transform on M can be generated from wavelet transform on a plane. For the two examples of the quadratic parameter rotation surface and parabolic general rotation surface, by using the two local flattening maps p respectively, the construction of the local dilation is systematically studied, and local wavelet transform of them is offered. We also show the local wavelet graphical representation on the quadratic parameter rotation surface and parabolic general rotation surface. Finally, the inverse transform formula of the local wavelet transform, that is, the reconstruction formula is also offered.

General Rotation Union Design Experiment for Seed Dispensing Performance of Seed Metering Disc

Via the single-factor experiment for the seed dispensing performance of the metering disc, the main factors that affect the seeding performance are determined. Then, the important order, the optimization level, the optimal combination, and the priority level of the test factor are confirmed by taking the orthogonal test for the seed dispensing performance of seed metering disc. Following the single factor experiments and the orthogonal tests, the general rotation union design experiments are taken, and the mathematical models between the dispensing performance indices and the main factors are developed, which establish a foundation for further parameters optimization.