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 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.

Theory of water waves derived from a Lagrangian. Part 1. Standing waves

2000 ◽  
Vol 423 ◽  
pp. 275-291 ◽  
Author(s):  
MICHAEL S. LONGUET-HIGGINS
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Fourier Coefficients ◽  
Equations Of Motion ◽  
Standing Waves ◽  
System Of Equations ◽  
Practical Applications ◽  
Low Degree ◽  
Sharp Corners ◽  
New System

A new system of equations for calculating time-dependent motions of deep-water gravity waves (Balk 1996) is here developed analytically and set in a form suitable for practical applications. The method is fully nonlinear, and has the advantage of essential simplicity. Both the potential and the kinetic energy involve polynomial expressions of low degree in the Fourier coefficients Yn(t). This leads to equations of motion of correspondingly low degree. Moreover the constants in the equations are very simple. In this paper the equations of motion are specialized to standing waves, where the coefficients Yn are all real. Truncation of the series at low values of [mid ]n[mid ], say n < N, leads to ‘partial waves’ with solutions apparently periodic in the time t. For physical applications N must however be large. The method will be applied to the breaking of standing waves by the forming of sharp corners at the crests, and the generation of vertical jets rising from the wave troughs.

Theoretical and Experimental Studies on Reduction for Multi-Modal Seismic Responses of High-Rise Structures by Tuned Liquid Dampers

2004 ◽  
Vol 10 (7) ◽  
pp. 1041-1056 ◽  
Author(s):  
Hong-Nan Li ◽  
Ying Jia ◽  
Su-Yan Wang
Keyword(s):  
Passive Control ◽  
Equations Of Motion ◽  
Experimental Studies ◽  
Tall Buildings ◽  
Shaking Table ◽  
Earthquake Ground Motion ◽  
Seismic Responses ◽  
High Rise ◽  
Tuned Liquid Dampers ◽  
Liquid Dampers

This paper focuses on the use of multiple tuned liquid dampers (TLDs) as passive control devices to reduce the multi-modal responses of tall buildings and high-rise structures to earthquake ground motion excitation. A model of a three-story building with one and two TLDs was installed on a shaking-table. The system was subjected to three earthquake time histories. Then, the mechanical models and the equations of motion for the systems of tall buildings and high-rise structures with TLDs are established. Here, the solution of the dynamic liquid pressure is based on the method of the volume of fluid and the seismic responses are obtained by use of the state equation. The comparisons show that theoretical results are generally in good agreement with experiments. It is observed that the approach presented in this paper has proved to be quite effective both in the numerical example and in the seismic simulating tests.

scholarly journals A non-conventional discontinuous Lagrangian for viscous flow

2017 ◽  
Vol 4 (2) ◽  
pp. 160447 ◽  
Author(s):  
M. Scholle ◽  
F. Marner
Keyword(s):  
Variational Formulation ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Local Equilibrium ◽  
Navier Stokes ◽  
Mathematical Treatment ◽  
Navier Stokes Equations ◽  
New Formalism ◽  
Limiting Case ◽  
Physical Phenomena

Drawing an analogy with quantum mechanics, a new Lagrangian is proposed for a variational formulation of the Navier–Stokes equations which to-date has remained elusive. A key feature is that the resulting Lagrangian is discontinuous in nature, posing additional challenges apropos the mathematical treatment of the related variational problem, all of which are resolvable. In addition to extending Lagrange's formalism to problems involving discontinuous behaviour, it is demonstrated that the associated equations of motion can self-consistently be interpreted within the framework of thermodynamics beyond local equilibrium, with the limiting case recovering the classical Navier–Stokes equations. Perspectives for applying the new formalism to discontinuous physical phenomena such as phase and grain boundaries, shock waves and flame fronts are provided.

scholarly journals Instability of holographic superfluids in optical lattice

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Peng Yang ◽  
Xin Li ◽  
Yu Tian
Keyword(s):  
Optical Lattice ◽  
Equations Of Motion ◽  
Stable State ◽  
Bloch Wave ◽  
Linear Perturbation ◽  
Holographic Model ◽  
Chaotic State ◽  
Final State ◽  
Bloch Waves ◽  
Soliton Generation

Abstract The instability of superfluids in optical lattice has been investigated using the holographic model. The static and steady flow solutions are numerically obtained from the static equations of motion and the solutions are described as Bloch waves with different Bloch wave vector k. Based on these Bloch waves, the instability is investigated at two levels. At the linear perturbation level, we show that there is a critical kc above which the superflow is unstable. At the fully nonlinear level, the intermediate state and final state of unstable superflow are identified through numerical simulation of the full equations of motion. The results show that during the time evolution, the unstable superflow will undergo a chaotic state with soliton generation. The system will settle down to a stable state with k < kc eventually, with a smaller current and a larger condensate.

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.

scholarly journals Maggi’s Equations Used in the Finite Element Analysis of the Multibody Systems with Elastic Elements

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 399 ◽  
Author(s):  
Sorin Vlase ◽  
Marin Marin ◽  
Maria Luminița Scutaru
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
The Finite Element Method ◽  
Element Analysis ◽  
Main Method ◽  
The Finite Element Analysis ◽  
Elastic Systems ◽  
Lagrange’S Multipliers ◽  
Elastic Elements

The main method used to determine the equations of motion of a multibody system (MBS) with elastic elements is the method of Lagrange’s multipliers. The assembly of equations for the whole system represents an important step in the elastodynamic analysis of such a system. This paper presents a new method of approaching this stage, by applying Maggi’s equations. In this way, the links that exist between the finite elements and the connections that exist between different bodies of the MBS system are conveniently taken into account, each body having a distinct velocity and acceleration field. Although Maggi’s equations have been used, sporadically, in some applications so far, we are not aware that they have been used in the study of elastic systems using the finite element method. Finally, an algorithm is presented that uses the Maggi formalism to obtain the equations of motion for an MBS system.

Experimental Verifications on Seismic Response Control of Tall Structures by Multiple Tuned Liquid Dampers

2004 ◽  
Author(s):  
Hong-Nan Li ◽  
Ying Jia ◽  
Su-Yan Wang
Keyword(s):  
Passive Control ◽  
Equations Of Motion ◽  
Tall Buildings ◽  
State Equation ◽  
Shaking Table ◽  
Earthquake Ground Motion ◽  
High Rise ◽  
Tuned Liquid Dampers ◽  
Tall Structures ◽  
Liquid Dampers

The focus of this paper is on the use of multiple Tuned Liquid Dampers (TLDs) as passive control devices to reduce the multi-modal responses of tall buildings and high-rise structures to earthquake ground motion excitation. A model of a 3-story building with one and two TLDs was installed on a shaking-table. The system was subjected to three earthquake time histories. Then the mechanical models and the equations of motion for the systems of tall buildings and high-rise structures with TLDs are established. Here, the solution of the dynamic liquid pressure is based on the method of the Volume of Fluid and the seismic responses are obtained by use of the state equation. The comparisons show that theoretical results are generally in good agreement with experiments. It is observed that the approach presented in this paper proved to be quite effective both in the numerical example and in the seismic simulating tests.

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