scholarly journals Analysis of the engineering mathematical model of the physical properties of a three-layer hydroacoustic screen with anisotropic components

2021 ◽  
Vol 2094 (2) ◽  
pp. 022011
Author(s):  
S V Storozhev ◽  
V I Storozhev ◽  
V E Bolnokin ◽  
S A Sorokin
Keyword(s):  
Normal Incidence ◽  
Algebraic Equations ◽  
Engineering Model ◽  
Analytical Technique ◽  
Analytical Integration ◽  
Physical Effects ◽  
Physical Fields ◽  
Functional Matrix ◽  
Amplitude Characteristics ◽  
Exponential Inhomogeneity

Abstract A numerical-analytical technique for analyzing the physical effects of the formation of fields of hydroacoustic waves in the area in front of a flat three-layer hydroacoustic screen and in the space behind the screen at normal incidence of a stationary hydroacoustic wave on it is presented. The engineering model of the screen uses the assumption that its components are made of anisotropic functional-gradient materials with exponential inhomogeneity in thickness, and thin, absolutely flexible, inextensible coatings can be applied to the outer and contact surfaces of the layers. The technique is based on the analytical integration of the equations of wave deformation of the screen components and obtaining complex amplitude characteristics for the reflected and generated hydroacoustic waves behind the screen when solving a system of algebraic equations with a functional matrix, which follows from the boundary conditions for the investigated problem. Parametric descriptions for the characteristics of the investigated physical fields are obtained and examples of numerical analysis of the considered engineering model are presented.

scholarly journals The Meshless Analysis of Scale-Dependent Problems for Coupled Fields

Materials ◽  
2020 ◽  
Vol 13 (11) ◽  
pp. 2527
Author(s):  
Jan Sladek ◽  
Vladimir Sladek ◽  
Pihua H. Wen
Keyword(s):  
Weak Form ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Electric Field Gradients ◽  
Test Function ◽  
Field Gradients ◽  
Mlpg Method ◽  
Physical Fields ◽  
Local Weak Form ◽  
Local Integral Equations

The meshless local Petrov–Galerkin (MLPG) method was developed to analyze 2D problems for flexoelectricity and higher-grade thermoelectricity. Both problems were multiphysical and scale-dependent. The size effect was considered by the strain and electric field gradients in the flexoelectricity, and higher-grade heat flux in the thermoelectricity. The variational principle was applied to derive the governing equations within the higher-grade theory of considered continuous media. The order of derivatives in the governing equations was higher than in their counterparts in classical theory. In the numerical treatment, the coupled governing partial differential equations (PDE) were satisfied in a local weak-form on small fictitious subdomains with a simple test function. Physical fields were approximated by the moving least-squares (MLS) scheme. Applying the spatial approximations in local integral equations and to boundary conditions, a system of algebraic equations was obtained for the nodal unknowns.

scholarly journals Analysis of Linearly Elastic Inextensible Frames Undergoing Large Displacement and Rotation

2011 ◽  
Vol 2011 ◽  
pp. 1-37 ◽  
Author(s):  
Jaroon Rungamornrat ◽  
Peerasak Tangnovarad
Keyword(s):  
Linear Approximation ◽  
Stiffness Matrix ◽  
Large Displacement ◽  
Algebraic Equations ◽  
Tangent Stiffness ◽  
Tangent Stiffness Matrix ◽  
Analytical Technique ◽  
Nonlinear Algebraic Equations ◽  
Direct Stiffness ◽  
Assembly Procedure

This paper presents an efficient semi-analytical technique for modeling two-dimensional, linearly elastic, inextensible frames undergoing large displacement and rotation. A system of ordinary differential equations governing an element is first converted into a system of nonlinear algebraic equations via appropriate enforcement of boundary conditions. Taylor's series expansion is then employed along with the co-rotational approach to derive the best linear approximation of such system and the corresponding exact element tangent stiffness matrix. A standard assembly procedure is applied, next, to obtain the best linear approximation of governing nonlinear equations for the structure. This final system is exploited in the solution search by Newton-Ralphson iteration. Key features of the proposed technique include that (i) exact load residuals are evaluated from governing nonlinear algebraic equations, (ii) an exact form of the tangent stiffness matrix is utilized, and (iii) all elements are treated in a systematic way via direct stiffness strategy. The first two features enhance the performance of the technique to yield results comparable to analytical solutions and independent of mesh refinement whereas the last feature allows structures of general geometries and loading conditions be modeled in a straightforward fashion. The implemented algorithm is tested for various structures not only to verify its underlying formulation but also to demonstrate its capability and robustness.

A Direct Numerical Formulation of Galerkin’s Method for Nonlinear Lumped Systems

1971 ◽  
Vol 38 (4) ◽  
pp. 1023-1028
Author(s):  
P. R. Johansen ◽  
R. H. Kohr
Keyword(s):  
Initial Value Problems ◽  
Numerical Procedure ◽  
Point Problem ◽  
Point Boundary ◽  
Algebraic Equations ◽  
Initial Value ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Analytical Integration ◽  
Numerical Formulation

Use of Galerkin’s method for the approximate analysis of nonlinear lumped systems is considered. A numerical procedure for finding Galerkin solutions of initial value problems is developed and illustrated by two examples. The approach employs Galerkin’s method formulated as a two-point boundary-value problem. While solving the two-point problem, a recursion algorithm is obtained; the algorithm is applied to the Galerkin integral directly, and therefore analytical integration of the weighted equation residuals and solution of specific algebraic equations are not required. The numerical formulation renders Galerkin’s method systematic, and it extends the utility of the method.

LOW-FREQUENCY ACOUSTIC SCATTERING OF A PLANE WAVE FROM A SOFT AND HARD TORUS

2007 ◽  
Vol 15 (02) ◽  
pp. 181-197 ◽  
Author(s):  
GEORGE VENKOV
Keyword(s):  
Separation Of Variables ◽  
Near Field ◽  
Scattering Problem ◽  
Acoustic Scattering ◽  
Low Frequency ◽  
Diagonal Form ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Analytical Technique ◽  
Linear Algebraic Equations

A plane acoustic wave is scattered by either a soft or a hard small torus. The incident wave has a wavelength which is much larger than the characteristic dimension of the scatterer and thus the low-frequency approximation method is applicable to the scattering problem. It is shown that there exists exactly one toroidal coordinate system that fits the given geometry. The R-separation of variables is utilized to obtain the series expansion of the fields in terms of toroidal harmonics (half-integer Legendre functions of first and second kind). The scattering problem for the soft torus is solved analytically for the near field, governing the leading two low-frequency coefficients, as well as for the far field, where both the amplitude and the cross-section are evaluated. The scattering problem for the hard torus appears to be much more complicated in calculations. The Neumann boundary condition on the surface of the torus leads to a three-term recurrence relation for the series coefficients corresponding to the scattered fields. Thus, the potential boundary-value problem for the leading low-frequency approximations is reduced to infinite systems of linear algebraic equations with three-diagonal matrices. An analytical technique for solving systems of diagonal form is developed.

Bending of orthotropic rectangular thin plates with two opposite edges clamped

2019 ◽  
Vol 234 (6) ◽  
pp. 1220-1230 ◽  
Author(s):  
Yangye He ◽  
Chen An ◽  
Jian Su
Keyword(s):  
Boundary Conditions ◽  
Integral Transform ◽  
High Order ◽  
Thin Plates ◽  
Constant Thickness ◽  
Order Partial Differential Equation ◽  
Algebraic Equations ◽  
Simply Supported ◽  
Linear Algebraic Equations ◽  
Analytical Integration

This work presents integral transform solutions of the bending problem of orthotropic rectangular thin plates with constant thickness, subject to five sets of boundary conditions: (a) fully clamped; (b) three edges clamped and one edge simply supported; (c) three edges clamped and one edge free; (d) two opposite edges clamped, one edge simply supported, and one edge free; and (e) two opposite edges clamped and two edges free. By adopting eigenfunctions of Euler–Bernoulli beams with corresponding boundary conditions for each direction of the plate, the governing fourth-order partial differential equation is integral transformed into a system of linear algebraic equations. Boundary conditions at the free edges are treated exactly by carrying out integral transform in the boundary formulations, which are incorporated in the transformed governing equations by integration by parts. The numerical difficulties with the high-order beam functions are overcome by using modified exponential forms, thus limiting the eigenfunctions to the range between −2 and 2. Analytical integration forms are used for the integrals of the coefficients of the transformed equations, further avoiding numerical difficulties with large high-order eigenvalues. The accuracy and convergence of the solutions are shown through numerical examples in comparison with available solutions in the literature and with finite element solutions obtained by using Abaqus program.

Period-3 Motions in a Periodically Forced, Damped, Double-Well Duffing Oscillator With Time-Delay

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Analytical Method ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Algebraic Equations ◽  
Implicit Mapping ◽  
Mapping Structures ◽  
Nonlinear Algebraic Equations ◽  
Amplitude Characteristics

In this paper, period-3 motions in a double-well Duffing oscillator with time-delay are predicted by a semi-analytical method. The implicit mapping structures of period-3 motions are constructed through the implicit mappings obtained by discretization of the corresponding differential equation. Complex period-3 motions are predicted through nonlinear algebraic equations of the implicit mappings in the mapping structures and the corresponding stability and bifurcation are carried out through eigenvalue analysis. Numerical and analytical results of complex period-3 motions are obtained and the corresponding frequency-amplitude characteristics are presented.

scholarly journals Free Vibration Analysis Of Isotropic Rectangular Plates On Winkler Foundation Using Differential Transform Method

2013 ◽  
Vol 18 (2) ◽  
pp. 589-597 ◽  
Author(s):  
Y. Kumar
Keyword(s):  
Free Vibration Analysis ◽  
Free Edge ◽  
Differential Transform Method ◽  
Winkler Foundation ◽  
Rectangular Plates ◽  
Algebraic Equations ◽  
Transform Method ◽  
Analytical Technique ◽  
Simply Supported ◽  
Differential Transform

A differential transform method (DTM) is used to analyze free transverse vibrations of isotropic rectangular plates resting on a Winkler foundation. Two opposite edges of the plates are assumed to be simply supported. This semi-numerical-analytical technique converts the governing differential equation and boundary conditions into algebraic equations. Characteristic equations are obtained for three combinations of clamped, simply supported and free edge conditions on the other two edges, keeping one of them to be simply supported. Numerical results show the robustness and fast convergence of the method. Correctness of the results is shown by comparing with those obtained using other methods.

scholarly journals PHYSICAL FIELDS OF CIRCULAR CYLINDRICAL PIEZOCERAMIC RECEIVER IN PRESENCE OF A FLAT ACOUSTIC SOFT SCREEN

2017 ◽  
Vol 8 (2) ◽  
pp. 168-176
Author(s):  
A. V. Derepa ◽  
A. G. Leiko ◽  
O. N. Pozdniakova
Keyword(s):  
Electric Fields ◽  
Mechanical Energy ◽  
Radial Symmetry ◽  
Imaging Method ◽  
Wave System ◽  
Sound Waves ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Physical Fields ◽  
Piezoceramic Transducer

System in the form of a circular cylindrical piezoceramic transducer near a flat acoustic screen was analyzed. The aim of the work was to solve the problem of receiving plane sound waves by «cylindrical piezoceramic transducer – flat acoustically soft screen» system.Considered system was characterized by a violation of the radial symmetry of the radiation load of the transducer while maintaining the radial symmetry of the electric load. At the same time, the energy perceived by the system under consideration is distributed between all modes of oscillation of the transducer, while the conversion of mechanical energy into electric is realized only at zero mole of oscillations.Special attention was paid to the method of coupled fields in multiply connected domains using the imaging method. The design model of the «transducer–creen» system was formulated taking into account the interaction of acoustic, mechanical and electric fields in the process of energy conversion, the interaction of a cylindrical transducer with a flat screen and the interaction of a converter with elastic media outside and inside it. The physical fields of the system under consideration were determined by following solutions: the wave equation; equations of motion of thin piezoceramic cylindrical shells in displacements; equations of stimulated electrostatics for piezoceramics for given boundary conditions, conditions for coupling fields at interfaces and electrical conditions.A general conclusion was made concerning solving of an infinite system of linear algebraic equations with respect to the unknown coefficients of the expansion of the fields. As an example of the application of the obtained relations, a calculation was made and an analysis of the dependences of the electric fields of the system under consideration for various parameters of its construction on the direction of arrival on the plane wave system was conducted.

Trends in Electron Energy Loss Spectroscopy

1977 ◽  
Vol 35 ◽  
pp. 228-229
Author(s):  
C. Colliex ◽  
P. Trebbia
Keyword(s):  
Energy Loss ◽  
Electron Energy ◽  
Electron Energy Loss Spectroscopy ◽  
Materials Science ◽  
Analytical Technique ◽  
Recent Developments ◽  
Quantitative Results ◽  
Electron Microscopes ◽  
Electron Energy Loss ◽  
Primary Electrons

The physical foundations for the use of electron energy loss spectroscopy towards analytical purposes, seem now rather well established and have been extensively discussed through recent publications. In this brief review we intend only to mention most recent developments in this field, which became available to our knowledge. We derive also some lines of discussion to define more clearly the limits of this analytical technique in materials science problems.The spectral information carried in both low ( 0<ΔE<100eV ) and high ( >100eV ) energy regions of the loss spectrum, is capable to provide quantitative results. Spectrometers have therefore been designed to work with all kinds of electron microscopes and to cover large energy ranges for the detection of inelastically scattered electrons (for instance the L-edge of molybdenum at 2500eV has been measured by van Zuylen with primary electrons of 80 kV). It is rather easy to fix a post-specimen magnetic optics on a STEM, but Crewe has recently underlined that great care should be devoted to optimize the collecting power and the energy resolution of the whole system.

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