Longitudinal waves in one dimensional non-uniform waveguides

Wave approach is used to analyze the longitudinal wave motion in one dimensional non-uniform waveguides. With assumptions of constant wave velocity and no wave conversion, there exist four types of non-uniform rods and corresponding traveling wave solutions are investigated. The obtained results indicate that the kinetic energy is preserved as a constant and the wave amplitude is inversely proportional to square root of the cross-sectional area of the rod. Under certain condition, there exists a cut-off frequency for the rod with variation in geometric or material properties, below which waves do not propagate along the non-uniform rod. For the rod with arbitrary variable cross-section, the conclusions are similar if the wave frequency is high enough. And a series solution of the wave motion is presented.     

含电学边界的压电层合梁的非线性弯曲波

非线性科学己成为近代科学发展的一个重要标志, 特别是非线性动力学和非线性波的研究对于解决自然科学各领域中遇到的复杂现象和问题有着极其重要的意义. 本文研究了含电学边界条件的压电层合梁的非线性弯曲波传播特性.首先, 考虑几何非线性效应和压电耦合效应, 利用哈密顿原理建立了一维无限长矩形压电层合梁弯曲波的非线性方程.其次, 采用Jacobi椭圆函数展开法对非线性弯曲波方程进行求解, 得到了非线性弯曲波动方程在近似情况下对应的冲击波解和孤波解.最后, 利用约化摄动法得到了非线性薛定谔方程, 进一步得到了亮孤子和暗孤子解.基于两种方法具体研究了外加电压、压电层厚度等参数对冲击波和孤立波以及亮孤子和暗孤子特性的影响. 研究结果表明, 在波速较小时, 外加电压对冲击波的影响较大, 波速较大时, 外加电压对孤立波影响减弱.通过调整作用在压电层合梁上的电压发现了存在亮孤子和暗孤子, 分析结果表明随着外加电压值的增大, 亮孤子和暗孤子的振幅都增大.      

Solitary waves and chaos in nonlinear visco-elastic rod

The dynamics behavior of a nonlinear visco-elastic rod subjected to axially periodic load is investigated theoretically and numerically. The weak longitudinal periodic load is distributed uniformly along the rod. Firstly, equation of motion of the rod is derived. Utilizing perturbation technique, we acquire Kdv type equation describing strain wave in the rod. By use traveling wave method, the elliptic cosine wave solution and the solitary wave solution in the rod are provided. Then, Melnikov method is applied to analyze the dynamic behaviour of the rod qualitatively. The explicit conditions for the onset of chaotic dynamics are yielded. With the help of the Poincare map method, phase trajectory and time-displacement history diagrams, the theoretical results obtained are checked.     

楔形杆中弹塑性波的一种渐近展开式

据我们所知,楔形杆中弹塑性波尚未有很好的分析方法。对弹性波有文献[1,2]等,其中文献[1]研究了圆锥壳轴向撞击的波动问题,发现楔形杆是其很好的近似,故后者的研究对圆锥壳具有重要意义。文中采用拉氏变换方法求得两种特殊情况下(波阵面和冲击端附近,的渐近解,而一般情形下的解未能得到。也有人用WKB方法讨论了类似问题,但仅限于波长很短的情形,局限性很大。另外,文献[5]用正则摄动法研究了楔形杆的自振问题。 本文针对楔形杆(和圆锥壳)的特点建议了一种渐近展开式,并求解了弹性波和弹塑性波问题,并与其他一些方法及其结果进行了比较。     

非圆截面杆中的非线性扭转波及其行波解

研究了非圆截面杆中非线性扭转波的传播特性.由于非圆截面杆的扭转运动会伴随有横截面的翘曲,这种翘曲运动将引起扭转波的弥散.如果同时考虑有限扭转变形和翘曲弥散的共同作用,将会得到非线性扭转波的方程.在相平面上,对非线性扭转波动方程进行定性分析,结果表明,在一定条件下方程存在同宿轨道或异宿轨道,分别相应于方程的孤波解或冲击波解.本文利用Jacobi椭圆函数展开法,对该非线性方程进行求解,得到了非线性波动方程的三类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.这些条件与定性分析的结果相一致.     

Uniaxial wave propagation in a nonlinear thermoviscoelastic medium with temperature dependent properties

The problem of finite wave propagation in a nonlinearly thermoviscoelastic thin rod whose viscoelastic properties are temperature dependent is considered. The rod is subjected to mechanical or thermal time-dependent loading. The coupled equations of motion and heat conduction are based on a constitutive theory of nonisothermal nonlinear viscoelasticity which is described by single-integral terms only. This theory is reformulated here for the uniaxial motion of a compressible rubbery material. The solution of the field equations is obtained by a numerical procedure which is developed for the present case and is able to handle successfully shock waves whenever they built up in the nonlinear material.     

分层有杆抽油系统诊断模型的反问题计算法

建立了分层有杆抽油系统的诊断模型 ,研究了在实心抽油杆与空心抽油杆的界面处空心泵对应力波的影响 ,给出了抽油杆柱不同杆段的波动方程的显示差分格式 ,提出了一种诊断模型的反问题计算法。用此算法可在缺少一个边界条件的情况下求解诊断模型 ,获得抽油系统各个工况下抽油杆柱任意截面的受力状态 ,从而可绘制出分层有杆抽油系统的空心泵示功图和管式泵示功图。将此算法与模式识别技术相结合就可对分层有杆抽油系统的故障进行诊断。     

Propagation of a low-frequency wave component in a model of a block medium

This paper studies the properties of a model of a block medium consisting of absolutely rigid blocks separated by deformable layers. The model is proposed to describe the low-frequency spectral region of a perturbation wave propagating in the medium of this structure. The model is based on the assumption that the low-frequency part of the wave train provides the least distorted information on the average characteristics of the structure of the medium on the wave pathway. Calculation of waves in a one-dimensional assembly of blocks (rods) and deformable layers show that the model ignoring the deformation of the blocks is applicable only in the case where the stiffness of the layer is low compared to the stiffness of the rod. A correction is applied to eliminate this restriction in the case of a long-wave approximation.     

高频纵向导波在钢杆中传播特性的研究

计算了钢杆中纵向轴对称导波模态的衰减频散曲线和群 速度频散曲线. 分析了1~3MHz范围内高频纵向轴对称超声导波在钢杆中的传播特 性. 理论分析表明，各高阶纵向模态都存在一个衰减最小值，在此衰减最小值所对应频率下 的高阶模态能传播较远距离，可用于钢杆导波检测. 建立实验系统，采用轴对称同端激 励接收的方法，根据第1次端面回波做出群速度和端面回波幅值随频率变化曲线，实验结 果与理论分析基本吻合. 表明考虑材料衰减特性的钢杆频散曲线可以作为实验指导依据.     

Wave propagation in prestrained polyethylene rods

The influence of prestrain on the propagation of mechanical waves along a slender rod of low-density unoriented polyethylene was experimentally investigated. The investigation consisted of two major parts: first, a uniaxial continuous-wave technique was used to determine the dynamic mechanical properties of the polyethylene in the form of the frequency-dependent phase velocity and damping factor for frequencies spanning the audio spectrum and for levels of uniaxial static prestrain up to 10 percent. A linear incremental dynamic viscoelastic behavior about a state of finite-static prestrain was shown to obtain over the range of strains and frequencies used. In the second part, the propagation of an incremental strain pulse along a slender rod of the same material used in the first part was investigated. With the rod in a state of static prestrain, an incremental impact-induced strain pulse was introduced into the polyethylene rod and monitored at two positions along the rod. Assuming a linear incremental dynamic viscoelastic behavior of the material, the equations necessary to describe the resulting uniaxial strain as a function of time and position along the rod are presented and the solution obtained by Fourier transform methods. The resulting Fourier inverse transform was numerically evaluated, using the material properties determined in the first part. The strain measured at the first position was used as the input boundary condition for computing the strain at the second position. Results of the continuous-wave studies indicate that the phase velocity decreases and the damping factor increases with increasing prestrain in the range of prestrains used. The change in the phase velocity with prestrain is relatively uniform over the audio-frequency range. Good correlation of the leading edges of the experimentally measured and numerically synthesized strain pulses supports the high-frequency phase-velocity data of the first part.     

有限变形弹性杆中的非线性波及周期解

利用Ham ilton变分原理,导出了计及有限变形和横向Possion效应的弹性杆中非线性纵向波动方程.利用Jacob i椭圆正弦函数展开和第三类Jacob i椭圆函数展开法,对该方程和截断的非线性方程进行求解,得到了非线性波动方程的两类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

梯形应力脉冲在弹性杆中的传播过程和几何弥散

在应力波传播过程中,几何弥散效应往往难以避免.对应力波在弹性杆中传播的几何弥散效应进行解析分析,对于基础波动问题研究以及材料动态力学行为表征等课题,显得至关重要.本文简单说明了弹性杆中考虑横向惯性修正的一维 Rayleigh-Love应力波理论,概述了其波动控制方程的变分法推导过程;针对 Hopkinson杆实验中常用的梯形应力加载脉冲,建立了相应的偏微分方程初边值问题的求解模型,并运用 Laplace变换方法研究了脉冲在杆中传播的几何弥散现象;根据留数定理进行 Laplace反变换,给出了杆中不同位置和时刻的应力波的级数形式解析解,分析了计算项数对结果收敛性的影响;将解析计算结果与采用三维有限元数值模拟的计算结果进行对比,两者吻合程度良好,从而证明 Rayleigh-Love横向惯性修正理论可以有效地表征典型 Hopkinson杆实验中的几何弥散效应.在此基础上围绕梯形加载脉冲的弥散效应进行参数研究,定量描述了传播距离、泊松比、脉冲斜率等参数的影响.本文给出的 Rayleigh-Love杆在梯形加载条件下的解析解,揭示了几何弥散效应的本质规律,可以用于实际实验的弥散修正过程.      

Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod

Peridynamics is a nonlocal theory of continuum mechanics, which was developed by Silling (2000). Since then peridynamics has been applied to a variety of solid mechanics problems ranging from fracture, damage, failure to wave propagation, buckling, and detonation physics. Since the governing equation of peridynamics is an integro-differential equation, most of the treatment in the literature is often numerical. However, the analytical treatment is very important for the development of the peridynamic theory, which is continually developing at the present time. In this paper, peristatic and peridynamic problems for a 1D infinite rod are analytically investigated. We have developed a method to obtain a valid analytical solution starting from a formal analytical solution, which may be divergent. The primary contribution of the present paper is a systematic analytical treatment of peristatic and peridynamic problems for a 1D infinite rod. Additionally, dispersion curves and group velocities for the materials with three different micromoduli are also studied. It is found from the study that some peridynamic materials can have negative group velocities in certain regions of wavenumber. This indicates that peridynamics can be used for modeling certain types of dispersive media with anomalous dispersion such as the one discussed by Mobley (2007).     

Experimental investigation of longitudinal wave propagation in an elastic rod with coulomb friction

This paper presents the experimental results obtained from the propagation of a compression stress wave generated by the longitudinal impact of two cylindrical elastic rods. One of these rods is subjected to a uniformly distributed coulomb-friction force. In order to determine the stress-wave shape and the decay rate, the rod is subjected to longitudinal impact for different values of coulomb friction. As the stress wave propagates along the rod, it is measured at strain-gage stations located on the stationary rod.In order to correlate the experimental results, the solution of the modified wave equation with coulomb friction is obtained for the longitudinal impact of two semi-infinite rods.     

曲面曲率对Rayleigh波传播特性的影响

对任意形状的均匀各向同性线弹性曲面物体,用 WKB～（1）方法求解了沿曲面传播的Rayleigh表面波的运动微分方程,同时考虑了波传播方向及其垂直方向曲面曲率对波的穿透性的影, 所获波动方程的势函数解答表明,在一般情况下垂直波传播方向的曲面曲率对波的穿透深度的影响是不容忽视的.进而以同种介质平面表面情况下的Rayleigh面波的传播特性为基准,给出了曲面曲率引起波数或波速变化的解析表达式.通过理论分析和数值算例,描述了曲面上Rayleigh面波传播行为的一些基本特征.     

初应力下压电半导体板中SH波的传播特性

针对无限大n-型压电半导体板,论文理论研究了其在初应力作用下水平剪切波的传播特性。基于压电半导体三维宏观理论和边界条件得到色散关系,结合数值算例,系统分析了边界条件、初始载流子浓度、板的厚度和初应力大小对SH波传播特性的影响。此外,讨论了初应力下两种不同材料中的SH波传播。研究显示：较小的初应力对相速度影响很小可忽略,当初应力达到一定值时波速急剧下降;类似地,初应力足够大时衰减才会逐渐加强。计算结果对压电半导体器件设计具有一定的理论指导意义。     

Hybrid finite‐volume finite‐difference scheme for the solution of Boussinesq equations

A hybrid scheme composed of finite‐volume and finite‐difference methods is introduced for the solution of the Boussinesq equations. While the finite‐volume method with a Riemann solver is applied to the conservative part of the equations, the higher‐order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy in space for the finite‐volume solution is achieved using the MUSCL‐TVD scheme. Within this, four limiters have been tested, of which van‐Leer limiter is found to be the most suitable. The Adams–Basforth third‐order predictor and Adams–Moulton fourth‐order corrector methods are used to obtain fourth‐order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ‘HYWAVE’, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi‐chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.     

Acceleration wave propagation in hyperelastic rods of variable cross-section

It is shown that when an acceleration wave propagates in hyperelastic rod with slowly varying cross-section, the transport equation for the wave intensity is a generalized Riccati equation. The three coefficients in the equation all depend on the material properties, but only the coefficient of the quadratic term is independent of the effect of area change. Three theorems are proved, based on the use of comparison equations, which establish that in general the acceleration wave intensity will become infinite (escape) after the wave has propagated only a finite distance along the rod. The existence of thresholds for the initial intensity are also established in certain cases, with their most notable property being that as the intial intensity decreases towards the threshold, so the distance the wave propagates to escape increases without bound.     

Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment

The problem of determining the elastoplastic properties of a prismatic bar from the given experimental relation between the torsional moment M and the angle of twist per unit length of the rod’s length θ is investigated as an inverse problem. The proposed method to solve the inverse problem is based on the solution of some sequences of the direct problem by applying the Levenberg-Marquardt iteration method. In the direct problem, these properties are known, and the torsional moment is calculated as a function of the angle of twist from the solution of a non-linear boundary value problem. This non-linear problem results from the Saint-Venant displacement assumption, the Ramberg–Osgood constitutive equation, and the deformation theory of plasticity for the stress–strain relation. To solve the direct problem in each iteration step, the Kansa method is used for the circular cross section of the rod, or the method of fundamental solutions (MFS) and the method of particular solutions (MPS) are used for the prismatic cross section of the rod. The non-linear torsion problem in the plastic region is solved using the Picard iteration.