Dynamics of Quantum State and Effective Hamiltonian with Vector Differential Form of Motion Method

Effective Hamiltonians in periodically driven systems have received widespread attention for realization of novel quantum phases, non-equilibrium phase transition, and Majorana mode. Recently, the study of effective Hamiltonian using various methods has gained great interest. We consider a vector differential equation of motion to derive the effective Hamiltonian for any periodically driven two-level system, and the dynamics of the spin vector are an evolution under the Bloch sphere. Here, we investigate the properties of this equation and show that a sudden change of the effective Hamiltonian is expected. Furthermore, we present several exact relations, whose expressions are independent of the different starting points. Moreover, we deduce the effective Hamiltonian from the high-frequency limit, which approximately equals the results in previous studies. Our results show that the vector differential equation of motion is not affected by a convergence problem, and thus, can be used to numerically investigate the effective models in any periodic modulating system. Finally, we anticipate that the proposed method can be applied to experimental platforms that require time-periodic modulation, such as ultracold atoms and optical lattices.

THE EQUATIONS OF MOTION OF A MECHANICAL SYSTEM IN MATRIX FORM

This work recommends methods of construction of equations of motion of mechanical systems in matrix form. The use of a matrix form allows one to write an equation of dynamics in compact form, convenient for the in vestigation of multidimensional mechanical systems with the help of computers. Use is made of different methods of constructing equations of motion, based on the basic laws of dynamics as well as on the principles of D Alambert-Le range, Hamilton-Ostrogradski and Gauss.

Precision Design for Machine Tool Based on Error Prediction

Digitization precision analysis is an important tool to ensure the design precision of machine tool currently. The correlative research about precision modeling and analysis mainly focuses on the geometry precision and motion precision of machine tool, and the forming motion precision of workpiece surface. For the machine tool with complex forming motion, there is not accurate corresponding relationship between the existing criterion on precision design and the machining precision of workpiece. Therefore, a design scheme on machine tool precision based on error prediction is proposed, which is divided into two-stage digitization precision analysis crucially. The first stage aims at the technology system to complete the precision distribution and inspection from the workpiece to various component parts of technology system and achieve the total output precision of machine tool under the specified machining precision; the second stage aims at the machine tool system to complete the precision distribution and inspection from the output precision of machine tool to the machine tool components. This article serves YK3610 gear hobber as the example to describe the error model of two systems and basic application method, and the practical cutting precision of this machine tool achieves to 5-4-4 grade. The proposed method can provide reliable guidance to the precision design of machine tool with complex forming motion.