Free vibration of axially loaded,rotating Timoshenko shaft systems by the wave-train closure principle

A systematic approach for the free vibration analysis of a rotating Timoshenko shaft system subjected to axial forces is presented in this paper. The system has multiple point discontinuities such as elastic supports, rotor masses, and cross-sectional changes. Wave reflection and transmission matrices are employed to characterize the wave motions between the sub-spans of the shaft system. These matrices are combined with the field transfer matrices expressed in wave forms to obtain the characteristic equation in a straightforward manner. The solutions are exact since effects of attenuating wave components are included in the formulation. The wave propagation-based matrix algebra leads to recursive algorithms which are suitable for computer coding. Three examples are presented to illustrate the numerical procedure.     

Shock-Induced Borehole Waves and Fracture Effects

We perform wave experiments using a vertical shock tube setup. Shock waves are generated by the rupture of a thin membrane. In the test section, the incident pressure waves generate borehole-guided waves along water-saturated samples. The tube is equipped with side wall gages and a mobile pressure probe, so that the attenuation and reflection of the wave can be measured. The computation for a single horizontal fracture intersecting a vertical borehole gives a quantitative prediction of reflection and transmission of borehole-guided waves. Three different fracture apertures are used for the calculation. Fracture aperture significantly affects both reflection and transmission coefficients. Large fractures increase reflectivity and decrease transmissivity. In the experiment, we found that both pressures above and below the fracture are influenced by the fracture aperture indeed, thus indicating the potential for fracture detection by borehole waves.     

Shock-Induced Borehole Waves and Fracture Effects

We perform wave experiments using a vertical shock tube setup. Shock waves are generated by the rupture of a thin membrane. In the test section the incident pressure waves generate borehole-guided waves along water-saturated samples. The tube is equipped with side wall gages and a mobile pressure probe, so that the attenuation and reflection of the wave can be measured. The computation for a single horizontal fracture intersecting a vertical borehole gives a quantitative prediction of reflection and transmission of borehole-guided waves. Three different fracture apertures are used for the calculation. Fracture aperture significantly affects both reflection and transmission coefficients. Large fractures increase reflectivity and decrease transmissivity. In the experiment, we found that both pressures above and below the fracture are influenced by the fracture aperture indeed, thus indicating the potential for fracture detection by borehole waves.     

阶梯压电层合梁的波动动力学特性

采用行波理论系统地研究了压电阶梯梁的自由振动分析以及强迫响应的分析方法. 基于分布 参数理论研究了压电阶梯梁的波传播特性，忽略柔性梁横向剪切和转动惯量的影响，给出了 梁的轴向和横向的简谐波解. 将压电阶梯梁离散化为单元，考虑压电片的刚度和质量的影响， 建立了节点散射模型. 应用位移连续和力平衡条件，推导了节点的波反射和波传递矩阵，在 此基础上，引入波循环矩阵的概念，给出波循环矩阵、波传递系数矩阵的确定方法. 应用波 循环矩阵可以有效地计算结构的固有频率. 另外，应用波传递系数研究了压电陶瓷作动器位 置对其驱动能力的影响. 得出两个主要结论：1)作动器靠近悬臂梁固定端将有较强的驱动 能力，悬臂梁边界反射行波产生弯曲消失波有利于增大压电波的模态传递系数；2)模态传 递系数与固有频率的灵敏度密切相关，波传递系数越大, 对应该处固有频率变化灵敏度越大. 另外，数值算例表明了行波方法比有限元方法具有更高的计算精度.     

The interaction of surface waves with fixed semi immersed cylinders having symmetric cross sections

The reflection of water waves by a semi immersed cylinder having a symmetric cross section is studied for both Dirichlet and Neumann boundary conditions on the cylinder. The method of conformal transformations as utilized by Ursell and by Tasai for the radiation problem is adapted to the present diffraction problem. The problem is solved by expansions of the reflected wave potential using nonorthogonal functions (wave free potentials). These functions are not complete, and an additional source and a dipole are required. Infinite systems of linear equations are obtained for the unknown expansion coefficients and the unknown strengths of the source and the dipole terms. Numerical results are obtained for the reflection coefficient, transmission coefficient, horizontal force on cylinder, vertical force on cylinder. In the long wave region analytical approximations are obtained for these functions when the cross section is circular. The reflection and transmission coefficients are very different for the two boundary conditions in the long wave region, the Dirichlet reflection coefficient being much larger than the corresponding Neumann coefficient. This behavior is similar to acoustic and electromagnetic diffraction problems in two dimensions. On leave of absence from Itek Corporation, Lexington (Mass.), U.S.A.     

Rational finite element method for elastic bending of reissner plates

IntroductionAsawell-establishedandsophisticatednumericalanalysistechniqueforthedifferentialequahons,thefiniteelementmethodwasoriginatedindependentlybythedomesticandtheforeignmechanicalinvestigatorstheyyearsago.ItfoundforthefirsttimewideapplicationsintheelasticstfUctolproblemsandwiththedevelopmentofcomputerextendedinvaryingdegreestOalmostalltheengineeringdisciplinesandmanyscientificandtechnologicalfields.Thisleadstoadramaticbreakthroughforthecurrentcomputationalmathematics.Themathematicalbasis…     

输液曲管平面内振动的波动方法研究

采用Flügge曲梁模拟弯曲管道,推导了管内流体的加速度,在总体轴线不可伸长假定的基础上建立了曲管平面内振动的动力学方程;采用波动方法,获得了曲管内振动波的传播和反射矩阵,提出了计算曲管平面内振动固有频率的数值方法。算例分析中,通过计算两端固定半圆形曲管的临界流速并与已有文献结果对比,验证了本文方法的正确性。最后,计算了两端固定半圆形曲管在四种不同流速下的前四阶固有频率,结果表明,管内流速的增大会降低管道的固有频率,当流速增大到某一特定值时,管道的一阶固有频率消失。     

Transmission boundary condition for mixed variable finite-difference method

All kinds of numerical methods based on the discretization of first-order velocity-stress hyperbolic system of elastic wave equations are named mixed variable finite-difference method in this paper, and the transmission boundary condition of artificial boundaries is studied here by the mixed variable finite-difference method. The transmission condition of complex geometrical boundaries of a transversely isotropic medium is presented based on the characteristic variables of wavefields propagating in the normal direction of the boundary. The boundary condition proposed in this paper is a local artificial boundary condition, with which the computation cost is very low. Elastic wave propagations in transversely isotropic medium are modelled by the staggered grid finite-difference method incorporated with the boundary condition presented. Numerical results and analysis of reflection coefficients show that the reflections of incoming waves by artificial boundaries are efficiently reduced.Supported by National Natural Science Foundation and Liaoning Province Science Foundation.     

Reflection and transmission in anisotropic dissipative multilayers

A planarly-stratified multilayer is considered with material properties depending on the Cartesian coordinate normal to the layers. Upon the assumption that the time dependence be harmonic, the equations of motion and the constitutive equations (of a viscoelastic solid) are given the form of a first-order system of ordinary differential equations. The propagator of the whole multilayer is determined and hence the reflection and transmission matrices are obtained for different boundary conditions. Next a new algorithm, which avoids some drawbacks of other procedures, is outlined. The reflection and transmission matrices of the multilayer are determined by recursive relations via the matrices, of reflection and transmission, associated with the single layers.     

Propagation and scattering of waves by dense arrays of impenetrable cylinders in a waveguide

A coupled numerical scheme, based on modal expansions and boundary integral representations, is developed for treating propagation and scattering by dense arrays of impenetrable cylinders inside a waveguide. Numerical results are presented and discussed concerning reflection and transmission, as well as the wave details both inside and outside the array. The method is applied to water waves propagating over an array of vertical cylinders in constant depth extended all over the water column, operating as a porous breakwater unit in a periodic arrangement (segmented breakwater). Focusing on the reflection and transmission properties, a simplified model is also derived, based on Foldy–Lax theory. The latter provides an equivalent index of refraction of the medium representing the porous structure, modeled as an inclusion in the waveguide. Results obtained by the present fully coupled and approximate models are compared against experimental measurements, collected in wave tank, showing good agreement. The present analysis permits an efficient calculation of the properties of the examined structure, reducing the computational cost and supporting design and optimization studies.     

SH0导波在粘接结构中传播时的相位变化1）

对粘接结构进行超声导波无损检测与评估是一个有挑战性的前沿性课题.针对此问题,研究了SH₀导波在界面为理想连接的三层板状粘接结构中传播时的相位变化情况.首先基于波传播的控制方程,建立了粘接结构中反射和透射SH₀导波相对于入射SH₀导波的相位差解析模型.然后利用数值模拟计算了铝/环氧树脂/铝粘接结构中反射和透射SH₀导波的相位差曲线.最后分析了入射角度和频厚积的变化对反射和透射SH₀导波相位差的影响.结果表明,对于具体的粘接结构,反射和透射SH₀导波在其中传播时的相位差变化主要依赖于入射角度、频率等参数.在特定的频厚积下,当声波水平入射时,反射和入射SH₀导波同相位.无论入射角度为多大,随着频厚积的增大,反射SH₀导波的相位差曲线均会产生周期性谐振.对于透射SH₀导波,当声波垂直入射时,其相位差曲线的改变无规律可循;但是随着入射角度逐渐增大,透射SH₀导波的相位差曲线逐渐变规则.所得结果可为实验时研究板状粘接结构中SH₀导波的传播特性以及提取SH₀导波回波中的有用信息和定位提供一定的理论指导.     

Interfacial imperfection on reflection and transmission of plane waves in anisotropic micropolar media

This study is concerned with the reflection and transmission of plane waves at an imperfectly bonded interface between two orthotropic micropolar elastic half-spaces with different elastic and micropolar properties. There exist three types of coupled waves in xy-plane. The reflection and transmission coefficients of quasi-longitudinal (QLD) wave, quasi-coupled transverse microrotational (QCTM) wave and quasi-coupled transverse displacement (QCTD) wave have been derived for different incidence waves and deduced for normal force stiffness, transverse force stiffness, transverse couple stiffness and perfect bonding. The numerical values of modules of the reflection and transmission coefficients are presented graphically with the angle of incidence for orthotropic micropolar medium (MOS) and isotropic micrpolar medium (MIS). Some particular cases of interest have been deduced from the present investigation.     

Reflection and transmission of quasi-plane waves at the interface of piezoelectric semiconductors with initial stresses

We examine the reflection and transmission phenomena of quasi-longitudinal plane(QP) waves in an AlN-ZnO laminated composite structure. The structure is designed under the influence of the initial stresses in which one carrier piezoelectric semiconductor(PSC) half-space is in welded contact with another PSC half-space.The secular equations in the transversely isotropic PSC material are derived from the general dynamic equation, taking the initial stresses into consideration. It is shown that the incident quasi-longitudinal wave(QP-mode) at the interface generates four types of reflected and transmitted waves, namely, QP wave, quasi-transverse(QSV) wave,electric-acoustic(EA) wave, and carrier plane(CP) wave. The algebraic equations are obtained by imposing the boundary conditions on the common interface of the laminated structure. Reflection and transmission coefficients of waves are obtained by implementing Cramer's rule. Profound impacts of the initial stresses and exterior electric biasing field on the reflection and transmission coefficients of waves are investigated and presented graphically.     

Acoustic diode: Wave non-reciprocity in nonlinearly coupled waveguides

The paper describes a passive time-independent setting for non-reciprocal wave transmission in mechanical and acoustic systems with strong nonlinearities. In the proposed system vibro-impact elements with pre-defined clearances are used to couple two non-dispersive waveguides. The asymmetry necessary for the non-reciprocal behavior is realized through unequal grounding springs of the vibro-impact elements. We show that, for appropriate selection of the parameters, the proposed system acts as a mechanical diode, allowing the transmission of acoustic waves in one direction and completely preventing reverse transmission. Two different designs of the coupling elements are suggested, with the possibility of single-sided or double-sided impacts. A unique feature of the proposed non-reciprocal acoustic system is that minimal distortion of the harmonic content of the transmitted wave occurs, in contrast to current designs where nonlinear non-reciprocity is achieved at the expense of a rather strong distortion of the transmitted signals. For both designs, we derive exact solutions for propagation and reflection of the harmonic waves, and demonstrate the possibility for strong non-reciprocity. Stability properties of the observed solutions in the space of parameters are also explored.     

Hydroelastic analysis of gravity wave interaction with submerged horizontal flexible porous plate

The present study deals with the surface gravity wave interaction with submerged horizontal flexible porous plate under the assumption of small amplitude water wave theory and structural response. The flexible porous plate is modeled using the thin plate theory and wave past porous structure is based on the generalized porous wavemaker theory. The wave characteristics due to the interaction of gravity waves with submerged flexible porous structure are studied by analyzing the complex dispersion relation using contour plots. Three different problems such as (i) wave scattering by a submerged flexible porous plate, (ii) wave trapping by submerged flexible porous plate placed at a finite distance from a rigid wall and (iii) wave reflection by a rigid wall in the presence of a submerged flexible porous plate are analyzed. The role of flexible porous plate in attenuating wave height and creating a tranquility zone is studied by analyzing the reflection, transmission and dissipation coefficients for various wave and structural parameters such as angle of incidence, depth of submergence, plate length, compression force and structural flexibility. In the case of wave trapping, the optimum distance between the porous plate and rigid wall for wave reflection is analyzed in different cases. In addition, effects of various physical parameters on free surface elevation, plate deflection, wave load on the plate and rigid wall are studied. The present approach can be extended to deal with acoustic wave interaction with flexible porous plates.     

Nonlinear vibration of a rotating cantilever beam in a surrounding magnetic field

The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are obtained. A nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented. The Galerkin method is adopted to discretize the dynamic equations. The proposed method is validated by comparison with the literature. The nonlinear behaviors of the responses are studied. Then simulations for different kinds of magnetic field are conducted. The effects of magnetic field parameters, including the amplitude, plane angle, spatial angle and time-varying frequency, on the dynamic behaviors of the stretching motion and bending motion are investigated in detail. The results illustrate that the interaction effects between the rotating cantilever beam and the magnetic field will increase the vibration amplitude and fluctuation of the beam. In particular, we found that: collinear magnetic fields with equal amplitude lead to the same dynamic responses; the amplitude of magnetic field intensity increases the dynamic responses remarkably; the response amplitude changes nonlinearly with the plane angle and spatial angle of the magnetic field; and the increase of time-varying frequency enhances dynamic responses of the rotating cantilever beam.     

Synthesis of highly convergent 2D and 3D finite elements for short acoustic wave simulation

New higher-order finite elements of enhanced convergence properties for acoustic wave simulation are presented in the paper. The element matrices are obtained by combining modal synthesis and optimization techniques in order to achieve minimum errors of higher modes of the computational domain. As a result, simulation models of propagating wave pulses require a smaller number of finite element divisions per wavelength compared to the conventional element model thus significantly reducing computational costs. Though finite element matrices are obtained in optimization, the resulting patterns of the matrices are versatile and further can be used in any wave propagation model. The mass matrices of the elements are diagonal, so explicit time integration schemes are applicable. The usage of new elements is especially efficient in situations where wavelengths of the simulated signal are much shorter than the dimensions of the computational domain. This is referred to as short wave propagation analysis. The results of wave propagation simulation for ultrasonic measurements are presented as application examples. The B-scans and computed dispersion curves are provided for visual interpretation of the results.     

Flow rate through a long channel with a rotating damper at the end

The experimental data on the flow rate (velocity) of a fluctuating air flow are presented on a wide fluctuation frequency range at a constant pressure difference between the channel entry and exit. The superimposed flow fluctuations were produced by periodic cut-off of the exit section by a rotating damper. A considerable dependence of the mean flow rate (velocity) on the wave structure of the flow is established. A flow rate minimum corresponds to resonance flow modes with a maximum relative amplitude of the flow velocity fluctuations.     

基于电磁声激励Lamb波的裂纹深度检测

本文采用电磁声传感器接收单一S₀模态激励,测量Lamb波在板中不同深度的槽形裂纹处的反射与透射,用于对已知扩展长度的表面裂纹进行深度测量。然后分析了Lamb波模态的频散与波动特性,由超声Lamb波的波结构,近似计算了S₀模态入射到裂纹时的反射系数。计算结果与实验结果一致。结果表明,Lamb波在有限长裂纹处的反射系数可采用二维模型中的反射系数求解公式近似求解。采用电磁超声传感器接收单一S₀模态激励,重复测量稳定性高,适用于Lamb波反射系数的测量。Lamb波的反射系数与裂纹深度有很好的对应关系,可用于板表面已知长度裂纹的深度测量。     

P-version hybrid analytical finite element method for plate bending problems

Two sets of trial functions with different variables are constructed for the admissible space of the finite element analysis. The trial functions satisfy the equilibrium differential equation inside elements, while the deflections and rotations on the edges of the elements are approximated by the Peano hierarchical interpolation functions. Then, a generalized variational principle is applied to set up the p-version hybrid analytical finite element method for plate bending problems. The accuracy of finite element computation can be improved by increasing the order of the interpolation polynomials with fixed mesh. In the finite element formulation, to obtain the stiffness matrices and the load vectors, it is only necessary to perform quadrature over the edges of the elements. These matrices and vectors possess an embedding structure. The conformability between the elements can be controlled automatically.This work is supported by the Natural Science Foundation of China and the Aeronautical Science Foundation of China.