Dynamic Response of Lattice-Type Structures

Number of terms required in the fourier expansion of the reflection function for optically thick atmospheres

We prove a fundamental representation formula for all solutions of the matrix Riccati differential equation and of the corresponding algebraic Riccati equation. This formula contains the complete information on the phase portrait of the matrix equation and on the structure of the set F of all solutions of the corresponding algebraic equation. In particular we describe all constant, periodic and almost-periodic solutions of the matrix Riccati differential equation. Further we give an application of the fundamental representation formula to the investigation of non-autonomous Riccati equations.

The main objective of this paper is to describe a class of functional series expansions, known as Fliess operators, which admit inputs from a ball in an L p space as well as Poisson random processes. Conditions are given under which these functional series expansions converge absolutely in the mean. Then, it is shown that a continuous-time switched input-affine nonlinear system with a Poisson switching signal can be represented as a Fliess operator and that for certain cases a closed form solution can be obtained in terms of Poisson integrals.

In this paper, the large-amplitude oscillation of a triple-well non-natural system, covering both qualitative and quantitative analysis, is investigated. The nonlinear system is governed by a quadratic velocity term and an odd-parity restoring force having cubic and quintic nonlinearities. Many mathematical models in mechanical and structural engineering applications can give rise to this nonlinear problem. In terms of qualitative analysis, the equilibrium points and its trajectories due to the change of the governing parameters are studied. It is interesting that there exist heteroclinic and homoclinic orbits under different equilibrium states. By adjusting the parameter values, the dynamic behavior of this conservative system is shifted accordingly. As exact solutions for this problem expressed in terms of an integral form must be solved numerically, an analytical approximation method can be used to construct accurate solutions to the oscillation around the stable equilibrium points of this system. This method is based on the harmonic balance method incorporated with Newton's method, in which a series of linear algebraic equations can be derived to replace coupled and complicated nonlinear algebraic equations. According to this harmonic balance-based approach, only the use of Fourier series expansions of known functions is required. Accurate analytical approximate solutions can be derived using lower order harmonic balance procedures. The proposed analytical method can offer good agreement with the corresponding numerical results for the whole range of oscillation amplitudes.

The article addresses second-order algebraic differential equations that have a separated linear part and admit a finite-order integer function as a solution. All possible integer solutions of such equations are described. It is shown that all solutions are the solutions of certain second-order linear differential equations the coefficients of which are represented by rational functions. It has been demonstrated that any such integer function y = f(z) is either a solution of the algebraic equation R(z, exp{Q(z)}, y) ≡ 0 (where R is a polynomial of three variables, and Q(z) is a polynomial of one variable), or a solution of a differential equation with separable variables y′ = a(z)y (for some rational function a(z)).

Fundamental frequency evolution in slender beams subjected to imposed axial displacements

In this paper, noise estimation based on series expansion of orthogonal functions is proposed. The proposed method searches speechless frequency regions and estimates the noise spectrum from the searched speechless frequency regions. The proposed method can adapt to changing of noise power without estimation delay. Experimental results show that the proposed method provides a good performance against time-varying noise.

The singularity expansion method as applied to the elastodynamic scattering problem

Theory of generalized hermite polynomials

Application of two preconditioned generalizedconjugate gradient methods to three-dimensional neutron and photon transport equations

Heterophase interfaces are boundaries, which join two material types with different physical and chemical nature. Therefore, heterophase interfaces can exhibit a large variety of geometric morphologies ranging from atomically sharp boundaries to gradient materials, in which an interface-specific phase is formed, which provides a continuous change of the structural parameters and thus reduces elastic strains and deformations. In addition, also the electronic properties of the two materials may be different, e.g. at boundaries between an electronically conducting metal and a semiconductor or an insulating material. Due to the deviations in the electronic structure, various bonding mechanisms are observed, which span the range from weakly interacting systems to boundaries with strong, directed bonding and further to reactively bonding systems which exhibit a new phase at the interface. Thus, both elastic and electronic factors may contribute to the formation of a new, often amorphous phase at the interface. Numerical simulations based on electronic structure theory are an efficient tool to distinguish and quantify these different influence factors, and massively parallel computers nowadays provide the required numerical power to tackle structurally more demanding systems. Here, this power has been exploited by the parallelisation over an optimised set of integration points, which split the solution of the Kohn-Sham equations into a set of matrix equations with equal matrix sizes. In this way, the analysis and prediction of material properties at the nanoscale has become feasible.

In this paper we have presented a general scheme whereby the method of Padé approximants may be applied to the power series expansion of time dependent correlation functions. It has been demonstrated by actual numerical calculations, using the computer simulated results of Berne and Harp [5], that even a few terms in the power series expansion are capable of reproducing the correlation curves to a fairly large extent. It is pointed out that the method can be extended to higher orders without the introduction of fitting parameters, and without violating the time parity requirements of the Liouville equation. In the final analysis the method is a truly microscopic method since it relies on the microscopic hamiltonian, and the underlying principles are very straightforward. The basic requirements are the spectral moments, but we do not need many of these as the Padé approximants are highly convergent.

Modal parameters, i.e., modal frequencies, damping ratios, and mode shapes of a two disk-shaft system are estimated in Multiple Input/Multiple Output (MIMO) scheme. The response at the output degrees of freedom (dof) of the considered structure due to the excitation at the input dofs is estimated from theoretical analysis. The generated theoretical data is used to estimate the Frequency Response Function (FRF) matrix, which relates the output (vibration) at response points with the input at the excitation points in the frequency domain. The corresponding Impulse Response Function (IRF) matrix, which relates response and excitation in the time domain, is obtained by Inverse Fast Fourier Transform (IFFT) of the FRF matrix. The unified Matrix Polynomial Approach (UMPA) is employed in the frequency and time domains with the estimated FRF and IRF matrices, respectively, to estimate the desired modal parameters. The obtained results are compared with results from Theoretical Modal Analysis (TMA).

We prove that if an analytic function f with an isolated singular point at ∞ is a solution of the differential equation P(zlnz, f, f′) = 0, where P is a polynomial in all variables, then f has finite order. We study the asymptotic properties of a meromorphic solution with logarithmic singularity.

In the history of Mathematics, frequently happens that certain topics (especially the numeric calculation) has perfectly defined its theoretical solution, but the heap of operations to find a conclusive result, in multiple occasions, don’t allow to arrive to the final results. The solution of differential equations represents a demonstrative example. At the end of the XIX century, the electricity was a fundamental theme in the society and every time new situations appeared that complicates the solution of technical problems, especially with the theory of the electric circuits, where was necessary solving differential equations with classic methods, making intense use of the techniques of integration and derivation, which constituted, in multiple occasions, a remarkable obstacle from the engendering point of view, and it was at the end of the century when an English electrical engineer established a group of practical rules to arrive to this solutions without necessity of using the basics of calculus. Although these rules propitiated the solutions, was necessary to do complex algebraic operations, in occasions, long and tedious, so, comparatively, it was not clear which procedure would be the best. At the present time, with the support of mathematical software, especially Mathcad, we can arrive to this solution in a quick way and with a high grade of precision. In the present text is described an algorithmic procedure to find the solution of differential equations with initial conditions, applying the transformed of Laplace. DOI: http://dx.doi.org/10.21017/rimci.2019.v6.n12.a64