Investigation of dynamics of elastic element of vibration device

ва подхода к решению аэрогидродинамической части задачи, основанные на методах теории функций комплексного переменного и методе Фурье. В результате применения каждого подхода решение исходной задачи сведено к исследованию дифференциального уравнения с частными производными для деформации элемента, позволяющего изучать его динамику. На основе метода Галеркина произведены численные эксперименты для конкретных примеров механической системы, подтверждающие идентичность решений, найденных для каждого дифференциального уравнения с частными производными. The dynamics of an elastic element of a vibration device, simulated by a channel, inside which a stream of a liquid flows, is investigated. Two approaches to solving the aerohydrodynamic part of the problem, based on the methods of the theory of functions of a complex variable and the Fourier method, are given. As a result of applying each approach, the solution to the original problem is reduced to the study of a partial differential equation for the deformation of an element, which makes it possible to study its dynamics. Based on the Galerkin method, the numerical experiments were carried out for specific examples of mechanical system, confirming the identity of the solutions found for each partial differential equation.

Direct expansion of Fourier extrapolator for one-way wave equation using Chebyshev polynomials of the second kind

The Fourier method for one-way wave propagation is efficient, but potentially inaccurate in complex media. The implicit finite-difference method can handle arbitrarily complex media, but can be inefficient in 3D and has limited dip bandwidth. We proposed a new Fourier method based on Chebyshev expansion of the second kind. Both theoretical analyses and numerical experiments show that the proposed method is comprehensively superior to a similar method based on Chebyshev expansion of the first kind in terms of balanced amplitude and error tolerance. Within the dip bandwidth from 0 to 65°, the fourth-order form of our method has an error tolerance of 2%, which is about one-third that of Chebyshev expansion of the first kind. Our method is also superior to the implicit finite-difference method in several important aspects: effective bandwidth, computational efficiency, numerical dispersion and two-way splitting error. It can be easily extended from 2D to 3D compared with the finite-difference method and from low orders to high orders compared with the optimized Chebyshev-Fourier method. The proposed method shows better imaging results of the SEG/EAGE model by providing a well-focused salt dome, flank and bottom as well as the detailed structures beneath the salt body, compared with the implicit finite-difference method and Chebyshev expansion of the first kind; meanwhile, our method has less imaging artifacts since it can better position the reflectors.

Floquet Spectral Almost-Periodic Modulation of Massive Finite and Infinite Strongly Coupled Arrays: Dense-Massive-MIMO, Intelligent-Surfaces, 5G, and 6G Applications

In this study, we introduce a new formulation based on Floquet (Fourier) spectral analysis combined with a spectral modulation technique (and its spatial form) to study strongly coupled sublattices predefined in the infinite and large finite extent of almost-periodic antenna arrays (e.g., metasurfaces). This analysis is very relevant for dense-massive-MIMO, intelligent-surfaces, 5G, and 6G applications (used for very small areas with a large number of elements such as millimeter and terahertz waves applications). The numerical method that is adopted to model the structure is the method of moments simplified by equivalent circuits MoM GEC. Other numerical methods (such as the ASM-array scanning method and the windowing Fourier method) used this analysis in their kernel to treat periodic and pseudo-periodic (or quasi-periodic) arrays.

Spatio-spectral dynamics of soliton pulsation with breathing behavior in the anomalous dispersion fiber laser

We have numerically and experimentally observed the soliton pulsation with obvious breathing behavior in the anomalous fiber laser mode-locked by nonlinear polarization rotation technique. The numerical study of the soliton pulsation with breathing behavior was analyzed through the split-step Fourier method at first, and it was found that the phase difference caused by the polarization controller would affect the breathing characteristics. Then, taking advantage of the dispersive Fourier transform technique, we confirmed the breathing characteristic of soliton pulsation in the same fiber laser as the simulation model experimentally. These results complement the research on the breathing characteristic of soliton pulsation.

Discrete-continuous three-element model of impact device

There have been examined the mathematic model of the impact device provided for geological materials destruction. Basic elements of the impact device are variable cross-section tool, striker and impact device body. The interaction of these elements is described as a movement of two discrete mass and the rod in the presence of rigid and dissipative connections. One equation in partial derivatives and two ordinary differential equations associated by initial and boundary conditions represent the initial-boundary problem. The numerical method parameters of which are determined at tests problems solution by Fourier method is used for looking for solutions of mixed initial-boundary problem. Researches are made, and parameters determining the damping efficiency of tool, striker and impact device body oscillations are evaluated.

Floquet Spectral Almost Periodic Modulation of Massive Finite and Infinite Strongly Coupled Arrays: Dense Massive MIMO, Intelligent Surfaces, 5G and 6G Applications

In this paper, we introduce a new formulation based on Floquet (Fourier) spectral analysis combined with a spectral modulation technique (and its spatial form) to study strongly coupled sublattices predefined in the infinite and large finite extent of almost periodic antenna arrays (e.g metasurfaces). This analysis is very relevant for dense massive MIMO, intelligent surfaces, 5G, and 6G applications (used for very small areas with a large number of elements such as millimeter and terahertz waves applications). The numerical method that is adopted to model the structure is the method of moments simplified by equivalent circuits MoM GEC. Other numerical methods (as the ASM array scanning method and windowing Fourier method) used this analysis in their kernel that to treat periodic and pseudo-periodic (or quasi-periodic) arrays.

Disconnected stationary solutions for 3D Kolmogorov flow problem: preliminary results

The extension of the classical A.N. Kolmogorov’s flow problem for the stationary 3D Navier-Stokes equations on a stretched torus for velocity vector function is considered. A spectral Fourier method with the Leray projection is used to solve the problem numerically. The resulting system of nonlinear equations is used to perform numerical bifurcation analysis. The problem is analyzed by constructing solution curves in the parameter-phase space using previously developed deflated pseudo arc-length continuation method. Disconnected solutions from the main solution branch are found. These results are preliminary and shall be generalized elsewhere.

Application of partial areas method in the problem of sound radiation by a sphere in a waveguide with soft acoustically boundaries

The paper considers the features of the formation of an acoustic field by a spherical source with complicated properties in a regular plane-parallel waveguide, which is of practical importance in marine instrumentation and oceanographic research. The calculation algorithm is based on the use of the Helmholtz equation and the Fourier method for each partial region and the conjugation conditions on their boundaries. The presented calculation allows one to get rid of the idealized boundary conditions on the source surface, with the subsequent determination of the excitation coefficients of the waveguide modes within the framework of the Sturm-Liouville problem. In this case, the attraction of the boundary conditions on the surface and the bottom of the sea, as well as the Sommerfeld conditions, makes it possible to obtain the real distribution of the field in the vertical sections of the waveguide. The obtained frequency dependences of the pressure and vibrational velocity components show their amplitude-phase differences, which reach 90 degrees, which partially explains the appearance of singular points in the intensity field in a regular waveguide. It has been determined that multiple reflections of sound waves from the boundaries of the working space and the space of the waveguide cause oscillations of the pressure components with a change in the amplitude level up to 6 dB. It was found that with an increase in the size of the source, a kind of resonance is formed in the working space, the frequency of which depends on the depth of the sea and corresponds to the region kr=x=5.8. It was found that when the acoustic field is formed in the working space, the frequency response of the impedance components is represented as a multiresonant dependence formed on the basis of the frequency characteristics of the lower modes and their combinations. Experimental studies have shown that the results of calculations of the mode composition of the acoustic field of the emitter, obtained in the conditions of the pool, correspond to the spatial characteristics of the mode components of the acoustic field with an error of up to 3 dB

Initial-boundary value problem for a time-fractional subdiffusion equation on the torus

An initial-boundary value problem for a time-fractional subdiffusion equation with the Riemann-Liouville derivatives on N-dimensional torus is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. It should be noted, that the condition on the initial function found in this paper is less restrictive than the analogous condition in the case of an equation with derivatives in the sense of Caputo.