Optimum transmission of elastic waves through joints

An elastic wave transmitting system (e.g. a nonuniform elastic rod) consisting of an input section with constant impedance, a joint with variable impedance, and an output section with constant impedance is considered. The efficiency of energy transmission (the ratio of transmitted to incident elastic wave energy) is determined and the following optimization problem is solved for two sample cases: Given the properties of the wave transmitting system and a limited duration of the incident wave, determine the shape of this wave such that the efficiency of energy transmission is maximized. In the first case (a joint with constant impedance) the optimization problem leads to a matrix eigenvalue problem. In the second case (a joint with concentrated mass) it leads to an eigenvalue problem for an integral equation. The efficiencies obtained for the optimum incident waves are compared with those obtained for rectagular incident waves an it is found that the differences are generally small. The results, which are discussed with particular reference to the transmission of elastic wave energy through drill rod joints, can also be interpreted for e.g. shallow water waves and electromagnetic waves in a transmission line.     

Some fundamental transmission properties of impedance transitions

Reflection and transmission of waves by impedance transitions from a constant input to a constant output characteristic impedence are considered. Several fundamental properties are explored, primarily for impedance transitions with piece-wise constant characteristic impedance in an arbitrary number N of intervals of equal length. For example, the following properties are shown: (i) The relative momentum transmission depends only on the ratio of output to input characteristic impedance. (ii) For a given impedance transition there are at most, and generally exactly, 2^N different transitions, including the original one, with identical transmission properties. (iii) For monotoneous impedance transitions the efficiency of energy transmission is minimized by one with an abrupt change in characteristic impedance. (iv) There exists an optimal impedance transition, with a certain antisymmetry, which maximizes the efficiency of energy transmission for a given incident wave of finite duration and energy. Several of the results can be extended to more general classes of impedence transitions. Simple illustrative examples are given.     

Rayleigh–Lamb Waves in a Compound Wave Guide. Reflection from and Transmission Through the Vertical Interface between Media with Different Impedances

The method of superposition is used to study the first normal wave reflecting from and transmitting through the interface in a compound waveguide consisting of two rigidly joined elastic half-strips with equal width and different mechanical properties. We study how the impedances of the contacting media influence the transformation of the energy of the incident wave to those of the reflected and transmitted waves. Two cases are considered — propagating waves of higher orders appear in the reflected wavefield earlier than in the transmitted wavefield and propagating waves of higher orders appear in the transmitted wavefield earlier than in the reflected wave field. For both cases, the impedances vary so that the incident wave can propagate in both more rigid and softer media. It is shown that by increasing the impedances of the contacting media, the interface can be made more transparent     

Dynamic Analysis of Large Space Structures Using Transfer Matrices and Joint Coupling Matrices

Abstract

Linear dynamic analysis of lattice structures using transfer matrices and joint coupling matrices is presented. A lattice structure is defined as a network of one-dimensional members that are connected by joints. Two examples are considered to illustrate how transfer matrices and joint coupling matrices may be used to compute natural frequencies of vibration. These two examples indicate that the transfer matrix and joint coupling matrix analysis is numerically accurate over a wide range of frequencies and becomes increasingly efficient, compared to the finite element method, as the frequency increases. Some suggestions for further improvements in computational efficiency and some comments about applicability to numerical analysis of wave propagation problems are given.     

Modified Kolsky Formulas for an Increased Measurement Duration of SHPB Systems

The so-called incident, reflected and transmitted strain histories are typically recorded during standard Split Hopkinson Pressure Bar (SHPB) experiments. Subsequently, the stress-strain curve for the specimen material is determined based on these recordings. Unless wave deconvolution techniques are employed, the reliable measurement of the reflected wave requires an input bar which is at least twice as long as the striker bar (of equal impedance). The present brief technical note elucidates the advantages of a simple alternative configuration which has only been seldom used in the past. Based on the assumption of quasi-static equilibrium at the specimen level, we present a modification of Kolsky’s formulas such that the stress-strain curve for the specimen material can be obtained from the measurement of the incident and transmitted strain histories only. As a result, the measurement of the reflected wave may be omitted and a much shorter input bar can be chosen. Conversely, a much longer striker bar may be used for a given input bar length, thereby increasing the valid duration of standard SHPB experiments by up to 100 % through the use of the modified Kolky formulas. An example experiment is shown where the duration of valid measurements has been increased by more than 70 %.     

SHPB实验中粘弹性试件内部应力波的传播

建立描述SHPB实验中线性粘弹性试件内部应力波传播的控制方程组,根据试件两端与入射杆及透射杆接触的应力波特征关系给出耦合边界条件.对方程组和定解条件进行Laplace变换,求得试件内部应力在变换域像函数的表达式.采用数值反变换技术进行反Laplace变换,获得试件两端的应力时程曲线.对现有的固定Tal-bot反变换算法进行改进:将入射波像函数分解为基本部分和延迟部分,利用固定Talbot算法对基本部分入射波作用下的波动问题求解,其他部分的解通过延迟定理得到,最终解为两部分的叠加.采用这种改进算法得到的不同入射波下粘弹性试件的内部应力解与传统的基于特征线数值模拟方法的结果吻合.在此基础上探讨了粘弹性试件的几何参数和材料本构参数对透射波波形的影响.     

Calculation of the motion of a gas behind the front of a luminous detonation wave taking account of the lateral expansion of the plasma column

Nonsteady-state gasdynamic processes are considered in a plasma column (hot channel), formed behind the front of a shock wave moving toward a laser beam. A quasi-one-dimensional approximation is used-the parameters in the channel are assumed to be compensated with respect to cross section, but depend on the time and distance along the axis. Motion in the cold dense casing surrounding the channel is assumed to be one-dimensional and cylindrically symmetrical. The solutions of the corresponding systems of equations in partial derivatives permit the parameters to be determined approximately both in the case when the mean free path of the radiation is small in comparison with the radius of the beam (luminous detonation) and also in the case when the mean free path is comparable with the radius. Examples are given of the corresponding numerical calculations. It is shown that in the latter case, a cycle of incomplete absorption can be achieved when behind the shock wave front, moving with constant velocity up to the Jouguet plane, only a part of the radiation energy incident on the front is released.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 18–28, May–June, 1976.     

非线性变形节理中纵波传播特性的理论研究

基于波前动量守恒理论和位移不连续方法所提出的时域分析新方法,引入岩石非线性法向本构关系,对弹性纵波在岩石非线性节理中的传播特性进行了理论分析。采用节理变形的双曲线模型(BB模型),获得纵波P波斜入射非线性节理的传播波动方程,并通过参数研究分析了在岩石节理中节理非线性系数、节理初始刚度、应力波入射角和入射波幅值等因素对纵波传播规律的影响。结果表明:所推导的应力波传播方程在考虑多种非线性问题时,通过迭代计算即可方便求出透射波和反射波的数值解,避免了复杂的数学运算;当波斜入射节理面时,产生了波型转换,节理变形的非线性对透射波和反射波有较大影响,透射系数和反射系数并非随着非线性参数的变化而单调变化。时域内所推导的波传播方程更有益于波斜入射时非线性参数的广泛研究,为开展该方面的理论研究工作提供了借鉴。     

SH波对一维六方准晶中直裂纹的散射

分析了SH波对一维六方准晶中直裂纹的散射问题。利用积分变换技术,结合Copson方法,通过求解对偶积分方程,得到声子场和相位子场应力、位移及裂纹尖端动应力强度因子的解析表达式。通过数值算例讨论了裂纹长度、入射角和入射波频率对标准动应力强度因子的影响,此研究在工程材料应用中有一定的参考价值。     

On the T-matrix for water-wave scattering problems

A rigid cylinder of infinite length is floating in the free surface of deep water. The cylinder is held fixed and a given time-harmonic wave of small amplitude is incident upon it. The corresponding linear two-dimensional boundary-value problem for a velocity potential φ is treated using the null-field method, and an expression for the T-matrix is obtained. (The T-matrix connects the diffraction potential away from the cylinder to the given incident potential.) Fundamental properties of the T-matrix are derived from considerations of energy and reciprocity. For regular wavetrains incident from the right or from the left, there are well-known relations between the corresponding reflection and transmission coefficients; these relations are recovered by specialising the equations satisfied by the T-matrix. Two extensions to water of constant finite depth are described: one uses multipole potentials whilst the other uses Havelock wavemaker functions; this second approach also leads to a new method for treating the problem of waves in a semi-infinite channel with an end-wall of arbitrary shape.     

Modeling of fault slip in earthquakes

The total energy release and the length of interfacial crack growth associated with an earthquake are analyzed based on a horizontal one-dimensional semi-infinite elastic model of fault of uniform rectangular cross section between two rigid rocks. Initial interfacial cracks of finite length are assumed to exist on the top and bottom surfaces of the fault. A compressive force is applied at the end of the fault and uniform frictional forces are exerted on the crack surfaces. When the compressive force reaches a critical value, fault slip against the surrounding rock occurs and the frictional force changes from a static value to a dynamic value within a short time interval. The change in frictional force generates longitudinal wave propagation in the fault. As a result, a large amount of energy release takes place. The total amount of energy release and the length of crack growth can be analyzed based on the balance of energy.     

Numerical study on the dynamics of a two-raft wave energy conversion device

This paper presents a dynamic analysis of a two-raft wave energy conversion device based on the three-dimensional wave radiation-diffraction method. The device consists of two hinged cylindrical rafts of elliptical cross section and a power take-off system at the joint. The effect of raft length, linear damping and spring coefficient in the power take off (PTO) system, axis ratio (ratio of minor axis to major axis of raft elliptical cross section) and raft radius of gyration on wave energy capture factor has been investigated in frequency domain, while the effects of a nonlinear Coulomb power take-off, raft radius of gyration and latching control have been studied in time domain. The difference in the performance of a raft-typed device obtained using a linear damping and a Coulomb damping is also illustrated.     

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.     

激波管双爆轰驱动段性能的数值模拟研究

采用频散可控的耗散格式(DCD),求解Euler方程和一种改进的二阶段化学反应模型, 对氢氧反向-正向双爆轰驱动段激波管进行了数值模拟. 计算结果表明：当辅驱动段与主驱动 段初始压力比小于临界值时,Taylor波仍会出现,但波扇夹角较单一前向爆轰驱动段小,入 射激波马赫数衰减率变小;当初始压力比等于临界值时,主驱动段中的Taylor波完全被消除, 入射激波马赫数不再衰减. 当初始压力比大于临界值时,在主驱动段中能产生过驱动爆轰波, 不仅Taylor波被完全消除,而且驱动能力较单一前向爆轰驱动段强.     

Impact friction test method by applying stress wave

To understand the dynamic response of two bodies in contact, kinetic friction during impact presently is focused on. A new testing technique, which provides the normal and the tangential impact force independently, is developed by modifying the split Hopkinson pressure bar method. Normal and torsional stress wave propagation in a one-dimensional framework of an axial impact of an input tube on a rotating output tube is analyzed and is experimentally verified. Kinetic friction of brass was clarified at a high rate of sliding up to 5 m/s and is found to be almost constant independent of normal force and sliding velocity. The present technique provides direct measurement of kinetic friction with simple configuration and data analysis.     

SHPB试验中试件的轴向应力均匀性

针对SHPB试验中试件的轴向应力均匀性问题,采用一维弹性波理论,推导了具有任意形状前沿的入射波加载下,试件内应力的时空分布计算公式。以脉冲前沿的上升时间为参数,将矩形、梯形和坡形3种典型的输入脉冲统一表示为梯形波的形式,计算了不同入射波上升时间和不同试件-压杆波阻抗比情况下试件中的应力传播过程,得到了相应的应力均匀度时程曲线以及应力平衡时间。分析了入射波上升时间和试件-压杆波阻抗比对应力平衡时间的影响,得到了一些有意义的认识,为SHPB试验的设计与分析提供了一定的理论依据。     

Definition and updating of simplified models of joint stiffness

The objective of this work is to define a simple linear model of joints used in aeronautics and to update this model efficiently.Industrial designers usually resort to semi-empirical linear joint models to represent the behavior of the joints of a large aeronautical structure. Here, we propose to develop a one-dimensional linear joint model which is capable of representing the behavior of every joint of a large structure globally while enabling local nonlinear reanalysis of the most highly loaded joints. Work on nonlinear reanalysis is not considered in this paper.In order to solve the numerical difficulties encountered in some of modeling situations, an updating strategy based on the constitutive relation error is proposed. Since the updating efficiency is significantly affected by the ratios of the stiffnesses of the different parts of the model, the strategy consists in rigidifying some parts of the model in order to control the updating accuracy and the rate of convergence. The numerical results of a standard model and a rigidified model illustrate the updating improvements allowed by the strategy.     

Shock wave attenuation by grids and orifice plates

The interaction of weak shock waves with porous barriers of different geometries and porosities is examined. Installing a barrier inside the shock tube test section will cause the development of the following wave pattern upon a head-on collision between the incident shock wave and the barrier: a reflected shock from the barrier and a transmitted shock propagating towards the shock tube end wall. Once the transmitted shock wave reaches the end wall it is reflected back towards the barrier. This is the beginning of multiple reflections between the barrier and the end wall. This full cycle of shock reflections/interactions resulting from the incident shock wave collision with the barrier can be studied in a single shock tube test. A one-dimensional (1D), inviscid flow model was proposed for simulating the flow resulting from the initial collision of the incident shock wave with the barrier. Fairly good agreement is found between experimental findings and simulations based on a 1D flow model. Based on obtained numerical and experimental findings an optimal design procedure for shock wave attenuator is suggested. The suggested attenuator may ensure the safety of the shelter’s ventilation systems.     

超高速撞击下波阻抗梯度防护结构碎片云特性研究

碎片云特性是影响空间碎片防护结构防护性能的重要因素。通过实验对比了相同面密度波阻抗梯度材料、铝合金材料的碎片云特性,并借助数值模拟进行了更深入的研究,结果表明,当弹丸分别撞击波阻抗梯度材料、铝合金材料时,碎片云结构中弹丸的破碎特征明显不同。撞击波阻抗梯度材料时,弹丸头部破碎更加充分,弹丸侧向扩展程度提高;在高速段（6.5 km/s）,受阻抗梯度及材料熔化效应的共同作用,波阻抗梯度材料碎片云头部出现分层现象。研究结果表明,超高速撞击波阻抗梯度材料碎片云特性的变化是其防护性能优于相同面密度铝合金的重要因素之一。     

Simulation of adaptation of blood vessel geometry to flow and pressure: Implications for arterio-venous impedance

Aortic input impedance relates pressure to flow at the aortic entrance distal to the aortic valve. We designed the CircAdapt three-element model of this impedance, consisting of resistive wave impedance, arterial compliance and peripheral resistance. Direct association of the elements with physical properties facilitated incorporation of nonlinear elastic properties of wall material and adaptation of vessel geometry to mechanical load. Use of the CircAdapt impedance model is extended to all arterial and venous connections to the heart. After incorporation in the existing CircAdapt model of whole circulation dynamics, vascular geometry was determined by adaptation to hemodynamic load as generated by the CircAdapt model itself. Model generated vascular geometry and hemodynamics appear realistic. Since the same adaptation rules are used for arteries and veins, all vascular impedances are determined mainly by two parameters only. Thus, large changes in hemodynamic load, like exercise or hypertension, were simulated realistically without the need to change parameter values. Simulation of adaptation enables to predict consequences of chronic change in hemodynamics, e.g. due to pathology or proposed therapy.