The curl of angular velocity and its mechanical significance

The curl of the vector field is widely used in modern field theory, fluid mechanics, mathematics, electromagnetic field, and other fields. In this paper, by introducing an auxiliary vector parameter (We called 𝑷𝑼⃗⃗⃗⃗⃗⃗ ) whose direction satisfies the right-hand thread rule the mathematical expression of angular velocity vector curl (𝛁×𝝎⃗⃗⃗ ) was obtained by analogy with the method of defining velocity vector curl (𝛁×𝒗⃗⃗ ). We also pointed out that the laminar flow of viscous fluid in a circular pipe (Hogen-Poiseuille flow) in nature is a typical real example of angular velocity vector curl (𝛁×𝝎⃗⃗⃗ ). Moreover, a concise mathematical equation (Equation(11)) was given, which could be used to describe some motion characteristics of vortex ring theoretically; Therefore, the motion of a single vortex ring has the dual characteristics of the velocity curl(𝛁×𝒗⃗⃗ ) and the angular velocity curl(𝛁×𝝎⃗⃗⃗ ) at the same time.

Application of the auxiliary performance index for automatic optimality control of discrete Kalman filter

Предложен новый метод автоматического контроля оптимальности дискретного фильтра Калмана, основанный на равенстве нулю градиента вспомогательного функционала качества (ВФК) по параметрам адаптивного дискретного фильтра. Для вычисления градиента ВФК применяется численно устойчивый к ошибкам машинного округления алгоритм модифицированной взвешенной ортогонализации Грама-Шмидта (MWGS-ортогонализации). Алгоритм реализован на языке Matlab. Результаты проведенных численных экспериментов подтверждают эффективность предложенного метода The paper proposes a new method for automatic control of the nominal operating mode of a dynamic stochastic system, based on a combination of two previously developed methods: the auxiliary performance index (API) method and the LD modification of an adaptive filter numerically robust to roundoff errors. The API method was previously developed to solve the problems of identification, adaptation, and control of stochastic systems with control and filtering. We suggest using the API not only as a tool for identifying the parameters of the stochastic system model from the measurement data but also for automatically monitoring the optimality of the adaptive filter, namely, the condition that the API gradient is close to zero should be satisfied (with the necessity and sufficiency) at the point corresponding to the optimal value of the vector parameter in the adaptive Kalman filter. The main result is the new eLD-KF-AC algorithm (extended LD Kalman-like adaptive filtering algorithm with automatic optimality control). The advantages of the obtained solution are as follows: 1) the choice of the adaptive filter structure in the form of an extended LD algorithm can significantly reduce the effect of machine roundoff errors on the calculation results when supplemented by the ability to calculate the sensitivity functions by the system vector parameter of the adaptive filter; 2) the application of the API method allows controlling the optimality of the adaptive filter by the condition that the API gradient is zero at the minimum point, which corresponds to the optimal value of the parameter in the adaptive filter; 3) the calculation of the API gradient in the adaptive extended LD filter does not require significant computational costs and such a control method can be carried out in real-time. The results of the work will be applied to solving problems of joint control and identification of parameters in the class of discrete-time linear stochastic systems represented by equations in the state-space form.

Asymptotic method for solution of identification problem of the nonlinear dynamic systems

A dynamic system, when the motion of the object is described by the system of nonlinear ordinary differential equations, is considered. The right part of the system involves the phase coordinates as a unknown constant vector-parameter and a small number. The statistical data are taken from practice: the initial and final values of the object coordinates. Using the method of quasilinearization the given equation is reduced to the system of linear differential equations, where the coefficients of the coordinate and unknown parameter, also of the perturbations depend on a small parameter linearly. Then, by using the least-squares method the unknown constant vector-parameter is searched in the form of power series on a small parameter and for the coefficients of zero and the first orders the analytical formulas are given. The fundamental matrices both in a zero and in the first approach are constructed approximately, by means of the ordinary Euler method. On an example the determination of the coefficient of hydraulic resistance (CHR) in the lift in the oil extraction by gas lift method is illustrated, as the obtained results in the first approaching coincides with well-known results on order of 10-2.

From the Kinematics of Precession Motion to Generalized Rabi Cycles

We use both vector-parameter and quaternion techniques to provide a thorough description of several classes of rotations, starting with coaxial angular velocity Ω of varying magnitude. Then, we fix the magnitude and let Ω precess at constant rate about the z-axis, which yields a particular solution to the free Euler dynamical equations in the case of axially symmetric inertial ellipsoid. The latter appears also in the description of spin precessions in NMR and quantum computing. As we show below, this problem has analytic solutions for a much larger class of motions determined by a simple condition relating the polar angle and z-projection of Ω (expressed in cylindrical coordinates), which are both time-dependent in the generic case. Relevant physical examples are also provided.