Analytical Solution of the Dynamics Equations for a Wave Solid-State Gyroscope Using the Angular Rate Linear Approximation

The exact solution is provided of the dynamics equation for an elastic inextensible ring being the basic model of a wave solid-state gyroscope with the linear law of the base angular rotation rate alteration. This solution is presented in terms of the parabolic cylinder functions (Weber function). Asymptotic approximations are used in the device certain operating modes. On the basis of the solution obtained, the analytical solution to the equation of the ring dynamics in case of piecewise linear approximation of the angular rate arbitrary profile on a time grid is derived. This significantly expands the class of angular rate dependences, for which the solution could be written down analytically. Earlier, in addition to the simplest case of constant angular rate, solutions were obtained for angular rate varying according to the square root law with time (Airy function), as well as according to the harmonic law (Mathieu function). Error dependence of such approximation on the discretization step in time is estimated numerically. Results obtained make it possible to reduce the number of operations, when it is necessary to study long-term evolutions of the dynamic system oscillations, as well as to quantitatively and qualitatively control convergence of finite-difference schemes in solving dynamics equations for a wave solid-state gyroscope with the ring resonator

Surface Motion of a Half-Space Containing an Elliptical-Arc Canyon under Incident SH Waves

The existence of local terrain has a great influence on the scattering and diffraction of seismic waves. The wave function expansion method is a commonly used method for studying terrain effects, because it can reveal the physical process of wave scattering and verify the accuracy of numerical methods. An exact, analytical solution of two-dimensional scattering of plane SH (shear-horizontal) waves by an elliptical-arc canyon on the surface of the elastic half-space is proposed by using the wave function expansion method. The problem of transforming wave functions in multi-ellipse coordinate systems was solved by using the extra-domain Mathieu function addition theorem, and the steady-state solution of the SH wave scattering problem of elliptical-arc depression terrain was reduced to the solution of simple infinite algebra equations. The numerical results of the solution are obtained by truncating the infinite equation. The accuracy of the proposed solution is verified by comparing the results obtained when the elliptical arc-shaped depression is degraded into a semi-ellipsoidal depression or even a semi-circular depression with previous results. Complicated effects of the canyon depth-to-span ratio, elliptical axis ratio, and incident angle on ground motion are shown by the numerical results for typical cases.

A Mathieu function boundary spectral method for scattering by multiple variable poro-elastic plates, with applications to metamaterials and acoustics

Many problems in fluid mechanics and acoustics can be modelled by Helmholtz scattering off poro-elastic plates. We develop a boundary spectral method, based on collocation of local Mathieu function expansions, for Helmholtz scattering off multiple variable poro-elastic plates in two dimensions. Such boundary conditions, namely the varying physical parameters and coupled thin-plate equation, present a considerable challenge to current methods. The new method is fast, accurate and flexible, with the ability to compute expansions in thousands (and even tens of thousands) of Mathieu functions, thus making it a favourable method for the considered geometries. Comparisons are made with elastic boundary element methods, where the new method is found to be faster and more accurate. Our solution representation directly provides a sine series approximation of the far-field directivity and can be evaluated near or on the scatterers, meaning that the near field can be computed stably and efficiently. The new method also allows us to examine the effects of varying stiffness along a plate, which is poorly studied due to limitations of other available techniques. We show that a power-law decrease to zero in stiffness parameters gives rise to unexpected scattering and aeroacoustic effects similar to an acoustic black hole metamaterial.

Application of Fractional Order Theory of Thermoelasticity in an Elliptical Disk and Associated Thermal Stresses

In this article, a time fractional-order theory of thermoelasticity is applied to an isotropic homogeneous elliptical disk. The lower and upper surfaces of the disk are maintained at zero temperature, whereas the sectional heat supply is applied on the outer curved surface. Thermal deflection and associated thermal stresses are obtained in terms of Mathieu function of the first kind of order 2n. Numerical evaluation is carried out for the temperature distribution, Thermal deflection and thermal stresses and results of the resulting quantities are depicted graphically.

Fast and spectrally accurate numerical methods for perforated screens (with applications to Robin boundary conditions)

This paper considers the use of compliant boundary conditions to provide a homogenized model of a finite array of collinear plates, modelling a perforated screen or grating. While the perforated screen formally has a mix of Dirichlet and Neumann boundary conditions, the homogenized model has Robin boundary conditions. Perforated screens form a canonical model in scattering theory, with applications ranging from electromagnetism to aeroacoustics. Interest in perforated media incorporated within larger structures motivates interrogating the appropriateness of homogenized boundary conditions in this case, especially as the homogenized model changes the junction behaviour considered at the extreme edges of the screen. To facilitate effective investigation we consider three numerical methods solving the Helmholtz equation: the unified transform and an iterative Wiener–Hopf approach for the exact problem of a set of collinear rigid plates (the difficult geometry of the problem means that such methods, which converge exponentially, are crucial) and a novel Mathieu function collocation approach to consider a variable compliance applied along the length of a single plate. We detail the relative performance and practical considerations for each method. By comparing solutions obtained using homogenized boundary conditions to the problem of collinear plates, we verify that the constant compliance given in previous theoretical research is appropriate to gain a good estimate of the solution even for a modest number of plates, provided we are sufficiently far into the asymptotic regime. We further investigate tapering the compliance near the extreme endpoints of the screen and find that tapering with $\tanh $ functions reduces the error in the approximation of the far field (if we are sufficiently far into the asymptotic regime). We also find that the number of plates and wavenumber has significant effects, even far into the asymptotic regime. These last two points indicate the importance of modelling end effects to achieve highly accurate results.

Errors in the numerical solution of the torsion Schr¨odinger equation with Mathieu functions basis set

Рассмотрена погрешность алгоритма аппроксимации функций Матье рядами Фурье, когда коэффициенты ряда Фурье представлены сходящимися цепными дробями. На основании проведенного анализа получены рекуррентные соотношения для абсолютной и относительной погрешностей удерживаемых звеньев цепной дроби и коэффициентов фурье-разложения. Предложен метод оценки точности расчета элементов матрицы гамильтониана торсионного уравнения Шрёдингера в базисе функций Матье. Эффективность предложенного алгоритма подтверждена численными примерами The dependence for the Hamiltonian matrix elements of the Schrodinger torsion equation on the calculation errors of the Mathieu basis set is considered. The Mathieu functions are represented with continued fractions in this study. The analysis of the Mathieu function approximation algorithm using Fourier series expansion is carried out when the coefficients of the Fourier series are represented by convergent continued fractions. It is shown that the major contribution to the errors at the Fourier coefficient calculation is made by the error accumulating in the corresponding elements of the continued fraction. Recurrence relations for the absolute and relative errors of the kept elements of the continued fraction and the Fourier expansion coefficients are obtained. It is shown and illustrated by a numerical example that the absolute and relative errors of the Fourier expansion coefficients in the proposed algorithm are negligible. It is noted that the maximum relative errors of continued fraction are in the highest elements of the kept part. The results of our work are used to estimate the calculation error in the integrals containing Mathieu functions. These integrals constitute the Hamiltonian matrix elements of the Schr¨odinger torsion equation. We developed an algorithm to estimate of the calculation accuracy of the Hamiltonian matrix elements of the Schr¨odinger torsion equation in the basis set of Mathieu functions. We provide the example of this algorithm. The results of the work indicate the adequacy and effectiveness at the application of the Mathieu function basis set to the solution of the Schrdinger torsion equation.¨

Free Vibration Analysis of Fluid-Filled Elliptical Cylindrical Shells

A method is presented for the free vibration analysis of finite fluid-filled elliptical cylindrical shells with variable curvature. Based on the Goldenveizer thin-shell theory, the vibration equations have been expressed as a matrix differential equation by using the transfer matrix and the fluid-loading term is represented as the form of Mathieu function, the transfer matrix is determined by use of the mid-Magnus series method. The natural frequencies are calculated numerically in terms of the matrix elements with a combination of appropriate initial guess and Lagrange interpolation method. The results of the degradation model obtained by the present method are compared to those of existing literatures. It is shown that the present method is highly accurate and the results are reliable. The sensitivity of the frequency parameter to the ellipticity parameter and the length of the shell are investigated respectively.