Isomorphism of Mathematical Descriptions of the Rotation Operation

Currently the practice of constructing various algorithms of inertial navigation, including strapdown navigation system (SINS), indicates that two mathematical methods for describing rotation are mainly used: using quaternions or direction cosines. Moreover, in SINS algorithms it is often convenient to use two of these parameter systems in parallel: quaternions for the angular motion algorithms (the orientation problem), and the direction cosines for the problem of calculating speed and position. In this case, parallel calculation of these two groups of parameters is carried out under the assumption of an exact isomorphic accordance between them. However, if the formalism of quaternions is single-valued, then the apparatus of matrix operations using direction cosines does not possess such a feature and admits several interpretations, which should be borne in mind. From this point of view, the question posed in this article makes sense not only as the question of the existence of an isomorphic accordance (there is no doubt about it), but in what form it exists for the matrix formalism of direction cosines.     

基于误差四元数的捷联惯导全姿态导航与控制

在舵偏控制回路中使用欧拉角姿态信息,是现有飞行器捷联惯导系统普遍采用的方法。为避免垂直状态时欧拉角姿态表示法的算法奇异问题,引入了四元数姿态表示法,但是最终形成偏差输入到舵机的仍旧是欧拉角姿态信息。对此问题,文中分析讨论了捷联惯导系统中,以欧拉角信号控制飞行器舵偏输出(推力矢量输出)的传统方法的两点不足之处:垂直状态时欧拉角姿态算法的物理奇异问题;控制路径过长问题。提出由误差四元数直接控制舵偏输出(推力矢量输出)的新的捷联导航方法。此方法控制路径最短,且克服了物理奇异问题,不局限于飞行器所处的姿态,因此,将其称之为全姿态导航方法。仿真结果表明了此方法的可行性与有效性。     

Convergence and exact solutions of spline finite strip method using unitary transformation approach

The spline finite strip method(PSM) is one of the most popular numerical methods for analyzing prismatic structures.Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems.To date,no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such,in this paper,the mathematical exact solutions of spline finite strips in the plat...     

An exact solution for a multilayered two-dimensional decagonal quasicrystal plate

By extending the pseudo-Stroh formalism to two-dimensional decagonal quasicrystals, an exact closed-form solution for a simply supported and multilayered two-dimensional decagonal quasicrystal plate is derived in this paper. Based on the different relations between the periodic direction and the coordinate system of the plate, three internal structure cases for the two-dimensional quasicrystal layer are considered. The propagator matrix method is also introduced in order to treat efficiently and accurately the multilayered cases. The obtained exact closed-form solution has a concise and elegant expression. Two homogeneous quasicrystal plates and a sandwich plate made of a two-dimensional quasicrystal and a crystal with two stacking sequences are investigated using the derived solution. Numerical results show that the differences of the periodic direction have strong influences on the stress and displacement components in the phonon and phason fields; different coupling constants between the phonon and phason fields will also cause differences in physical quantities; the stacking sequences of the multilayer plates can substantially influence all physical quantities. The exact closed-form solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and laminated plates. The numerical results can also be employed to verify the accuracy of the solution by numerical methods, such as the finite element and difference methods, when analyzing laminated composites made of quasicrystals.     

Nonlinear wave propagation and reflection — Comparing the numerics with the analytics

The accuracy of numerical methods needs always a special attention. In this paper, analytical and numerical methods have been compared to describe the initial stage of nonlinear propagation and reflection of longitudinal ultrasonic waves. The perturbation method has been used to derive the analytical solution and the finite difference scheme to find the numerical solution for multiple free-boundary reflections of a harmonic burst at ultrasonic frequencies. The comparison of results at relatively small nonlinearities reveals a good qualitative and quantitative agreement between the analytical and numerical solutions. The method for determining analytically the exact region of interaction for counter-propagating waves is outlined in detail. At higher frequencies and larger nonlinear effects some quantitative differences between analytical and numerical results appear. The results are applicable in modelling nonlinear wave motion, including NDT and nonlinear one-dimensional vibrations.     

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.     

An Exact Analysis for Free Vibration of a Composite Shell Structure-Hermetic Capsule

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Limit cycle oscillations of rectangular cantilever wings containing cubic nonlinearity in an incompressible flow

Limit cycle oscillations (LCO) as well as nonlinear aeroelastic analysis of rectangular cantilever wings with a cubic nonlinearity are investigated. Aeroelastic equations of a rectangular cantilever wing with two degrees of freedom in an incompressible potential flow are presented in the time domain. The harmonic balance method is modified to calculate the LCO frequency and amplitude for rectangular wings. In order to verify the derived formulation, flutter boundaries are obtained via a linear analysis of the derived system of equations for five different cases and compared with experimental data. Satisfactory results are gained through this comparison. The problem of finding the LCO frequency and amplitude is solved via applying the two methods discussed for two different cases with hardening cubic nonlinearities. The results from first-, third- and fifth-order harmonic balance methods are compared with the results of an exact numerical solution. A close agreement is obtained between these harmonic balance methods and the exact numerical solution of the governing aeroelastic equations. Finally, the nonlinear aeroelastic analysis of a rectangular cantilever wing with a softening nonlinearity is studied.     

Exact dynamic solutions to piezoelectric smart beams including peel stresses Part II: Numerical results,comparison and approximate solution

This work presents numerical results for the exact dynamic solution of piezoelectric (PZT) smart beams including peel stresses, which was developed in Part I. Numerical results are presented in details for frequency spectra, natural frequencies, normal mode shapes, harmonic responses of the shear and peel stresses, and sensing electric charges for a cantilever beam with a bonded PZT patch to the clamped end. The exact dynamic solution can provide useful data for benchmarking other methods. The numerical results of the present model including peel stresses (PSM) are also compared with those obtained using the shear lag beam model and the shear lag rod model. On the basis of the equivalent forces derived in the static analysis, simple approximate dynamic solutions are obtained and compared with the exact solutions, and then the application and limitation of the simple approximate solutions are investigated. By comparing numerical results predicted by the present PSM model with the shear lag models and the approximate solutions based on the static equivalent forces, effects of the dynamic shear and peel stresses on natural frequencies and dynamic responses of the smart structures are examined.     

Green’s functions of an exponentially graded transversely isotropic half-space

By virtue of a complete set of displacement potential functions and Hankel transform, the analytical expressions of Green’s function of an exponentially graded elastic transversely isotropic half-space is presented. The given solution is analytically in exact agreement with the existing solution for a homogeneous transversely isotropic half-space. Employing a robust asymptotic decomposition technique, the Green’s function is decomposed to the closed-form Green’s function corresponding to the homogeneous transversely isotropic half-space and grading term with strong decaying integrands. This representation is very useful for numerical methods which are based on boundary-integral formulations such as boundary-element method since the numerically evaluated part is not responsible for the singularity. The high accuracy of the proposed numerical scheme is confirmed by some numerical examples.     

关于气动声学数值计算的方法与进展

气动声学数值计算是近年才出现的研究领域。本文介绍了气动声学数值计算的方法和有关的问题、边界条件的处理以及计算非线性声波的数值方法和进展。讨论了计算气动声学(CAA)的特性及其与计算流体力学(CFD)的差异，指出气动声学数值方法的关键是建立能保持色散关系的差分方程和正确处理无反射边界条件。对于非线性声波传播的问题，为了得到正确的解，应注意提高差分格式对短波的分辨能力，同时发展能抑制“伪”振荡(短波)而对长波基本不起作用的数值方法。     

数值模拟在钱塘江涌潮分析中的应用──Ⅱ.计算结果和分析

关于钱塘江涌潮分析中用数值模拟的计算结果和分析,对于一维情况,采用有精确解问题的计算、水跃的数值模拟和溃坝的数值模拟来检验数值方法的正确性,并将该方法应用于钱塘江涌潮的计算．对于二维情况,对存在理论解的直道水激波与有实验数据的湾道溃坝和非线性水波爬坡问题进行了计算,并将该方法应用于钱塘江涌潮的计算,数值计算结果表明了方法的有效性．     

Insights on the continuous representations of piecewise-smooth nonlinear systems: limits of applicability and effectiveness

Nonlinear Dynamics - It is known that three-parameter representations of spatial rotation suffer from kinematic singularities. Euler parameters (unit quaternions) are a four-parameter solution for...     

Exact analytical solution of the MHD Jeffery-Hamel flow problem

This article shows that the well known nonlinear boundary value problem namely MHD Jeffery-Hamel flow problem, investigated in recent years by many numerical and semi-analytical approximative methods, is exactly solvable and furthermore, gives analytical exact solution in the implicit form for further physical interpretation.     

Solution of the laminar duct flow problem by a boundary integral equation method

Summary A boundary integral equation method is proposed for approximate numerical and exact analytical solutions to fully developed incompressible laminar flow in straight ducts of multiply or simply connected cross-section. It is based on a direct reduction of the problem to the solution of a singular integral equation for the vorticity field in the cross section of the duct. For the numerical solution of the singular integral equation, a simple discretization of it along the cross-section boundary is used. It leads to satisfactory rapid convergency and to accurate results. The concept of hydrodynamic moment of inertia is introduced in order to easily calculate the flow rate, the main velocity, and the fRe-factor. As an example, the exact analytical and, comparatively, the approximate numerical solution of the problem of a circular pipe with two circular rods are presented. In the literature, this is the first non-trivial exact analytical solution of the problem for triply connected cross section domains. The solution to the Saint-Venant torsion problem, as a special case of the laminar duct-flow problem, is herein entirely incorporated.     

The Riemann problem for a hyperbolic model of two-phase flow in conservative form

This article is to continue the present author's work (International Journal of Computational Fluid Dynamics (2009) 23 (9), 623–641) on studying the structure of solutions of the Riemann problem for a system of three conservation laws governing two-phase flows. While existing solutions are limited and found quite recently for the Baer and Nunziato equations, this article presents the first instance of an exact solution of the Riemann problem for two-phase flow in gas–liquid mixture. To demonstrate the structure of the solution, we use a hyperbolic conservative model with mechanical equilibrium and without velocity equilibrium. The Riemann problem solution for the model equations comprises a set of elementary waves, rarefaction and discontinuous waves of various types. In particular, such a solution treats both the wave structure and the intermediate states of the two-phase gas–liquid mixture. The resulting exact Riemann solver is fully non-linear, direct and complete. On this basis then, we use locally the exact Riemann solver for the two-phase flow in gas–liquid mixture within the framework of finite volume upwind Godunov methods. In order to demonstrate the effectiveness and accuracy of the proposed solver, we consider a series of test problems selected from the open literature and compare the exact and numerical results by using upwind Godunov methods, showing excellent oscillation-free results in two-phase fluid flow problems.     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Impact of a strip of compressible liquid on a barrier

The exact solution of the plane problem of the impact of a finite liquid strip on a rigid barrier is obtained in the linearized formulation. The velocity components, the pressure and other elements of the flow are determined by means of a velocity potential that satisfies a two-dimensional wave equation. The final expressions for them are given in terms of elementary functions that clearly reflect the wave nature of the motion. The exact solution has been thoroughly analyzed in numerous particular cases. It is shown directly that in the limit the solution of the wave problem tends to the solution of the analogous problem of the impact of an incompressible strip obtained in [1]. A logarithmic singularity of the velocity parallel to the barrier in the corner of the strip is identified. A one-dimensional model of the motion, which describes the behavior of the compressible liquid in a thin layer on impact and makes it possible to obtain a simple solution averaging the exact wave solution, is proposed. Inefficient series solutions are refined and certain numerical data on the impact characteristics for a semi-infinite compressible liquid strip, previously considered in [2–4] in connection with the study of the earthquake resistance of a dam retaining water in a semi-infinite basin, are improved. The solution obtained can be used to estimate the forces involved in the collision of solids and liquids. It would appear to be useful for developing correct and reliable numerical methods of solving the nonlinear problems of fluid impact on solids often examined in the literature [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 138–145, November–December, 1990.The results were obtained by the author under the scientific supervision of B. M. Malyshev (deceased).     

Sensitivity analysis of low Reynolds number channel flow using the finite volume method

An unsteady finite volume‐based fractional step algorithm solved on a staggered grid has been developed for computing design sensitivity parameters in two‐dimensional flows. Verification of the numerical code is performed for the case of low Reynolds number, pressure‐driven flow through a straight channel, which has an exact steady‐state solution to the Navier–Stokes equations. Sensitivity of the flow to the channel height, fluid viscosity, and imposed pressure gradient is considered. Three different numerical techniques for computing the design sensitivity parameters: finite difference, complex‐step differentiation, and sensitivity equation method (SEM), are compared in terms of numerical error (relative to the exact solution), computational expense, and ease of implementation. Results indicate that, of all the three methods, complex step is the most accurate and requires the least computational time. In addition, treatment of the boundary conditions in SEM is addressed, within the framework of the present finite volume approach, with special attention given to parameter dependence in the boundary conditions. Error estimation based on the Grid Convergence Index provides a good indication of the exact error in the SEM solutions. An example of application of the use of sensitivity parameters to estimate the propagation of input uncertainty through the numerical simulation is also provided. Copyright © 2007 John Wiley & Sons, Ltd.