scholarly journals Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems

2012 ◽  
Vol 468 (2142) ◽  
pp. 1629-1651 ◽  
Author(s):  
A. N. Norris ◽  
A. L. Shuvalov ◽  
A. A. Kutsenko
Keyword(s):  
Effective Medium Theory ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Expansion Method ◽  
Bloch Wave ◽  
Dimensional System ◽  
Medium Theory ◽  
Material Parameters ◽  
Elastic Systems ◽  
Arbitrary Frequency

Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.

Phononic Band Gap Systems in Structural Mechanics: Finite Slender Elastic Structures and Infinite Periodic Waveguides

2013 ◽  
Vol 135 (4) ◽  
Author(s):  
Michele Brun ◽  
Alexander B. Movchan ◽  
Ian S. Jones
Keyword(s):  
Equations Of Motion ◽  
Standing Waves ◽  
Tall Buildings ◽  
Bloch Wave ◽  
Elementary Cell ◽  
Mathematical Treatment ◽  
Asymptotic Model ◽  
Spectral Approach ◽  
Elastic Systems ◽  
Waveguide Problem

The paper presents a novel spectral approach, accompanied by an asymptotic model and numerical simulations for slender elastic systems such as long bridges or tall buildings. The focus is on asymptotic approximations of solutions by Bloch waves, which may propagate in a infinite periodic waveguide. Although the notion of passive mass dampers is conventional in the engineering literature, it is not obvious that an infinite waveguide problem is adequate for analysis of long but finite slender elastic systems. The formal mathematical treatment of a Bloch wave would reduce to a spectral analysis of equations of motion on an elementary cell of a periodic structure, with Bloch–Floquet quasi-periodicity conditions imposed on the boundary of the cell. Frequencies of some classes of standing waves can be estimated analytically. One of the applications discussed in the paper is the “dancing bridge” across the river Volga in Volgograd.

scholarly journals Energy of Accelerations Used to Obtain the Motion Equations of a Three- Dimensional Finite Element

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 321 ◽  
Author(s):  
Sorin Vlase ◽  
Iuliu Negrean ◽  
Marin Marin ◽  
Maria Luminița Scutaru
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
The Finite Element Method ◽  
Method Of Calculation ◽  
Elastic Systems ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element ◽  
Multi Body ◽  
Elastic Elements

When analyzing the dynamic behavior of multi-body elastic systems, a commonly used method is the finite element method conjunctively with Lagrange’s equations. The central problem when approaching such a system is determining the equations of motion for a single finite element. The paper presents an alternative method of calculation theses using the Gibbs–Appell (GA) formulation, which requires a smaller number of calculations and, as a result, is easier to apply in practice. For this purpose, the energy of the accelerations for one single finite element is calculated, which will be used then in the GA equations. This method can have advantages in applying to the study of multi-body systems with elastic elements and in the case of robots and manipulators that have in their composition some elastic elements. The number of differentiation required when using the Gibbs–Appell method is smaller than if the Lagrange method is used which leads to a smaller number of operations to obtain the equations of motion.

Some Problems of Polar Missile Control

1960 ◽  
Vol 64 (596) ◽  
pp. 482-488 ◽  
Author(s):  
D. Best
Keyword(s):  
Control Systems ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Dimensional System ◽  
Two Dimensional ◽  
Special Design ◽  
Design Problems ◽  
Yaw Control ◽  
Homing Missiles ◽  
Three Dimensional System

A polar-controlled missile is one in which manoeuvre is carried out by rotations about roll and pitch axes, that is, in the manner of a conventional aeroplane. This paper discusses some problems in the application of this form of control to homing missiles.In comparison with the alternative Cartesian configuration, this method presents some special design problems. In the former case, it is often possible to resolve the motion into two planes and consider the pitch and yaw control systems as independent two-dimensional problems. This simplification is not possible in the case of polar control and it is usually necessary to consider the whole three-dimensional system. The equations of motion which result are, in general, not susceptible to analysis. Because of this, the design of control systems requires extensive use of simulators.

Three-Dimensional Sonic Band Gaps Tunned by Material Parameters

2010 ◽  
Vol 29-32 ◽  
pp. 1797-1802 ◽  
Author(s):  
Xiao Zhou Zhou ◽  
Yue Sheng Wang ◽  
Chuan Zeng Zhang
Keyword(s):  
Bulk Modulus ◽  
Mass Density ◽  
Density Ratio ◽  
Three Dimensional ◽  
Expansion Method ◽  
Small Mass ◽  
Band Gaps ◽  
Face Centered Cubic ◽  
Modulus Ratio ◽  
Material Parameters

In this paper, band gaps tunned by material parameters in three-dimensional fluid-fluid sonic crystals are studied. From the basic wave equation, it is found that the material parameters directly determining the three-dimensional sonic band gaps are the mass density ratio and bulk modulus ratio. The calculation of the sonic band gaps is completed by the plane-wave expansion method. The effects of these parameters on sonic band gaps are discussed in details for the simple-cubic (sc), face-centered cubic (fcc) and body-centered cubic (bcc) lattices. The results show that the first potential sonic band gap easily appears at both small mass density ratio and bulk modulus ratio, and becomes wider with both of these two parameters decreasing. The bulk modulus ratio plays a more important role than the mass density ratio in tuning the sonic band gaps. The present analysis can be applied to artificially design band gaps.

AN INVERSE DESIGN METHOD FOR METAMATERIAL SHRINKING DEVICE

2013 ◽  
Vol 27 (25) ◽  
pp. 1350182 ◽  
Author(s):  
TINGHUA LI ◽  
MING HUANG ◽  
JINGJING YANG ◽  
JIA ZENG ◽  
JIN LU
Keyword(s):  
Inverse Method ◽  
Effective Medium Theory ◽  
Design Method ◽  
Transformation Function ◽  
Inverse Design ◽  
Medium Theory ◽  
Material Parameters ◽  
Full Wave ◽  
Traditional Design ◽  
Inverse Design Method

In this paper, an inverse method to determine the parameters of metamaterial shrinking device is developed. Different from the traditional design method, of which the transformation function must be known in advance, this method allows us to directly obtain material parameters of device without any knowledge of the corresponding transformation function. Moreover, to further remove the inhomogeneity and anisotropy of material parameters, layered device composed of only homogeneous and isotropic materials is presented based on effective medium theory. The validity of such a method and shrinking effect of designed device are confirmed by full-wave simulations.

Creeping flow in two-dimensional networks

1982 ◽  
Vol 119 ◽  
pp. 219-247 ◽  
Author(s):  
Joel Koplik
Keyword(s):  
Effective Medium Theory ◽  
Three Dimensional ◽  
Effective Medium ◽  
Creeping Flow ◽  
Absolute Permeability ◽  
Two Dimensional ◽  
Linear Network ◽  
Medium Theory ◽  
Reynolds Number Flow ◽  
Number Flow

We discuss creeping incompressible fluid flow in two-dimensional networks consisting of regular lattice arrays of variable-sized channels and junctions. The intended application is to low-Reynolds-number flow in models of porous media. The flow problem is reduced to an analogue linear-network problem and is solved by numerical matrix inversion. It is found that ‘effective-medium theory’ provides an excellent approximation to flow in such networks. Various qualitative features of such flows are discussed, and an elegant general form for the absolute permeability is derived. The latter, and the effective-medium approximation, are equally applicable to three-dimensional networks.

Topological spin current

2017 ◽  
Author(s):  
E. Saitoh
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Spin Current ◽  
Edge State ◽  
Dimensional System ◽  
Electron Band ◽  
Electrically Conducting ◽  
Spin Currents ◽  
Surface Spin ◽  
Three Dimensional System

This chapter discusses another type of equilibrium-spin current similar to the exchange-spin current—the topological spin current. Topological spin currents are driven by topological-band structure and classified into bulk and surface topological spin currents. The former is confined onto electron-band manifolds, sometimes affecting their motions. This confinement is addressed through the standard method of combining the equations of motion and the Boltzmann equation for semi-classical electrons in a band. The latter class, on the other hand, is a surface-spin current, which is limited near surfaces of a three-dimensional system and flows along these surfaces. This type is known to appear in topological insulators, where the bulk is insulating but the surface or edge is electrically conducting due to the surface or edge state.

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