ANALYSIS AND EXPERIMENT ON TRANSIENT DYNAMIC RESPONSE IN FINITE MINDLIN PLATE

The transient wave propagation in the finite rectangular Mindlin plate is investigated by the analytical and experimental methods. The generalized ray method (GRM) which has been successfully applied to study the transient responses of beams, planar trusses, space frames and infinite layered media is extended to investigate the transient wave propagation and early short time transient response in finite Mindlin plate. Combining the wave solution, the shock source and the boundary conditions, the ray groups transmitted in the finite rectangular plate can be determined. Numerical simulations and experiments are performed and compared with each other. The results show that the transient wave propagation and early short time transient responses in the finite plate can be studied using the GRM. The early short time transient accelerations are very large for the finite plate subjected to the unit impulse, while the early short time transient displacements are very small. The early short time transient accelerations under the unit impulse are much larger than those under the unit step impulse. The thickness and material characteristics have remarkable effects on the early short time transient responses.     

Transient impact responses of laminated composite cylindrical shells

The generalized ray method (GRM) has been successfully used to study the transient elastic wave transmitting in the beams, planar trusses, space frames and infinite layered media. In this letter, the GRM is extended to investigate the early short time transient responses of laminated composite cylindrical shells under impact load. By using the Laplace transformation and referring to the boundary conditions, the ray groups transmitting in the finite laminated cylindrical shells under the shock load are obtained and the transient response related to each ray group can be derived via FFT algorithm. From the numerical results, it is shown that the early short time transient accelerations of the laminated composite cylindrical shell under impact loads are very large. But the short time transient shear strain and displacement are very small.     

The wave and vibratory power transmission in a finite L-shaped Mindlin plate with two simply supported opposite edges

In this paper,wave and vibratory power transmission in a finite L-shaped Mindlin plate with two simply supported opposite edges are investigated using the wave approach.The dynamic responses,active and reactive power flow in the finite plate are calculated by the Mindlin plate theory (MPT) and classic plate theory (CPT).To satisfy the boundary conditions and continuous conditions at the coupled junction of the finite L-shaped plate,the near-field and far-field waves are entirely contained in the wave approach.The in-plane longitudinal and shear waves are also considered.The results indicate that the vibratory power flow based on the MPT is different from that based on the CPT not only at high frequencies but also at low and medium frequencies.The influence of the plate thickness on the vibrational power flow is investigated.From the results it is seen that the shear and rotary inertia correction of the MPT can influence the active and reactive power at the junction of the L-shaped plate not only at high frequencies but also at low and medium frequencies.Furthermore,the effects of structural damping on the active and reactive power flow at the junction are also analyzed.     

A procedure of the method of reverberation ray matrix for the buckling analysis of a thin multi-span plate

A procedure of the method of reverberation ray matrix(MRRM) is developed to perform the buckling analysis of thin multi-span rectangular plates having internal line supports or stiffeners. A computation algorithm for the reverberation ray matrix in the MRRM is derived to determine the buckling loading. Specifically, the analytical solutions are presented for the buckling of the structure having two opposite simply-supported or clamped-supported edges with spans, while the constraint condition of two remaining edges may be in any combination of free, simply-supported, and clamped boundary conditions. Furthermore, based on the analysis of matrices relating to the unknown coefficients in the solution form for the deflection in terms of buckling modal functions, some recursive equations(REs) for the MRRM are introduced to generate a reduced reverberation ray matrix with unchanged dimension when the number of spans increases, which promotes the computation efficiency. Several numerical examples are given, and the present results are compared with the known solutions to illustrate the validity and accurateness of the MRRM for the buckling analysis.     

Theoretical analysis,numerical calculation and experimental measurement of transient elastic waves in strips subjected to dynamic loadings

In this study, the transient response of an elastic strip subjected to dynamic in-plane loadings on the surface is investigated in detail. One of the objectives of this study is to develop an effective analytical method for determining transient solutions in a strip. By applying Laplace transform, the analytical solution in the transformed domain is derived and expressed in matrix form. The solution is then decomposed into infinite wave groups in which the multiple reflected waves with the same reflection are involved. Each multi-reflected wave can be identified by a coding method and be verified by the theory of generalized ray. The inverse transform is performed by using the well-known Cagniard method. The transient solutions in time domain for stresses and displacements are expressed in a closed form and are discussed in detail by an example. The experimental results show that the early time transient responses of displacements on the surface agree very well with the numerical calculations based on the theoretical solutions.     

Method of reverberation ray matrix for static analysis of planar framed structures composed of anisotropic Timoshenko beam members

Based on the method of reverberation ray matrix(MRRM), a reverberation matrix for planar framed structures composed of anisotropic Timoshenko(T) beam members containing completely hinged joints is developed for static analysis of such structures.In the MRRM for dynamic analysis, amplitudes of arriving and departing waves for joints are chosen as unknown quantities. However, for the present case of static analysis, displacements and rotational angles at the ends of each beam member are directly considered as unknown quantities. The expressions for stiffness matrices for anisotropic beam members are developed. A corresponding reverberation matrix is derived analytically for exact and unified determination on the displacements and internal forces at both ends of each member and arbitrary cross sectional locations in the structure. Numerical examples are given and compared with the finite element method(FEM) results to validate the present model. The characteristic parameter analysis is performed to demonstrate accuracy of the present model with the T beam theory in contrast with errors in the usual model based on the Euler-Bernoulli(EB) beam theory. The resulting reverberation matrix can be used for exact calculation of anisotropic framed structures as well as for parameter analysis of geometrical and material properties of the framed structures.     

Vibration band gap properties of periodic Mindlin plate structure using the spectral element method

In this paper, vibration band gap properties of periodic Mindlin plate structure with two simply supported opposite edges are studied using the spectral element method (SEM). Based on the Mindlin plate theory (MPT), the spectral Mindlin plate stiffness matrix is deduced. The spectral equation of the whole periodic structure is further established. Frequency responses are calculated by the spectral equations to illustrate band gap properties. Compared with the results based on the classic plate theory (CPT), the effects of the shear and rotary inertia on the band gap properties are analyzed. The results are also compared with those calculated by the finite element method (FEM). It can be observed that the SEM is more accurate in high frequency ranges. Furthermore, the influences of the material properties and structure dimension on the band gap behaviors are investigated.     

基于厚板理论分析深水域中弹性浮板的水波响应

基于线性水波理论和Mindlin厚板动力学理论,采用Wiener-Hopf 方法,研究了不同水深水面上弹性浮板在不同入射波数水波作用下的动力学响应问题。首先推导了无限深水域中弹性浮板水波响应的解析解,并将本文分析计算结果与采用其他方法(经典薄板理论)得到的计算结果进行了对比和分析;其次,采用本文方法研究了大型浮板在三种入射波数的水波作用下动弯矩幅值的分布情况;最后,根据其他文献的方法计算了不同水深(有限水深)情况下浮板的动响应,并与本文的计算结果进行了对比分析。     

Computation of the stress resultants of a floating Mindlin plate in response to linear wave forces

In this paper, we focus on the computation of stress resultants of a floating elastic plate using the Mindlin plate theory. The proposed method makes use of the linear wave theory and shallow-draft assumption. However, the usual Kirchhoff theory is replaced by the Mindlin theory for the plate. For a single frequency, the coupled water-plate problem is solved using a higher-order-coupled finite element–boundary element method. The solutions for the stress-resultants computed using the proposed method are more satisfactory than these based on the Kirchhoff plate theory.     

用回传波射矩阵法分析框架的地震效应

结构在地震载荷作用下的瞬态响应分析对抗震设计很重要. 大部分地震效应 研究采用时程积分等数值方法. 回传波射矩阵法是桁架和框架结构瞬态响应的一种新的频域 矩阵分析法. 基于Bernulli梁理论的回传波射矩阵法扩展到地面加速度载荷的情形, 对一榀钢制框架在地面El Centro波加速度作用下的瞬态应力计算, 结果表明该方法 能准确有效地进行结构的地震效应分析.     

Application of fractional-derivative standard linear solid model to impact response of human frontal bone

The impact of a rigid body on a thin plate with a buffer is investigated in this paper. A buffer is assumed as a linear spring fractional derivative dashpot which exhibits the viscoelastic features. The fractional-derivative standard linear solid model is suggested for describing the shock interaction of the impactor with a circular elastic plate. We assume that a transient wave of transverse shear is generated in the plate and the reflected wave does not have sufficient time to interact with the plate before the impact process is completed. The ray method is used outside the contact spot, but the Laplace transform method is applied within the contact region. The time-dependence of the contact force is determined. A numerical example is carried out by considering crash scenarios in frontal impacts of the human head which could estimate brain injury risks.     

Theoretical analysis of transient waves in a simply-supported Timoshenko beam by ray and normal mode methods

The dynamic transient responses of a simply-supported Timoshenko beam subjected to an impact force are investigated by two theoretical approaches – ray and normal mode methods. The mathematical methodology proposed in this study for the ray method enable us to construct the solution for the interior source problem and to extend to solve the complicated problem for the multi span of the Timoshenko beam. Numerical results based on these two approaches are compared. The comparison in this study indicates that the normal mode method is more computationally efficient than the ray method except for very short time after the impact. The long-time transient responses are easily calculated using the normal mode method. It is shown that the average long-time transient response converges to the corresponding static value. The Timoshenko beam theory is more accurate than the Bernoulli–Euler beam theory because it includes shear and rotary inertia. This study also provides the slender ratio for which the Bernoulli–Euler beam can be used for the transient-response analysis of the displacement. Moreover, the resonant frequencies obtained from finite element calculation based on the three-dimensional model are compared with the results calculated using the Timoshenko beam and Bernoulli–Euler beam theories. It is noted in this study that the resonant frequency can be accurately determined by the Timoshenko beam theory if the slender ratio is larger than 100, and by the Bernoulli–Euler beam theory if the slender ratio is larger than 400.     

水中冲击波与弹性薄板耦合作用的研究

对水中冲击波与弹性薄板的流固耦合作用进行了研究,建立了实验装置。利用全息干涉法定量测量了水中动态流场,并测出受水中冲击波作用的弹性薄板在不同时刻的变形,得出相应的全息干涉条纹图。同时利用有限元法对水中冲击波与弹性薄板的流固耦合问题进行了数值计算。研究结果表明,用实验方法能够较好地模拟水中冲击波与弹性结构的耦合作用,用有限元法计算水中冲击波传播问题是可行的。     

PARALLEL FINITE ELEMENT ANALYSIS OF HIGH FREQUENCY VIBRATIONS OF QUARTZ CRYSTAL RESONATORS ON LINUX CLUSTER

Quartz crystal resonators are typical piezoelectric acoustic wave devices for frequency control applications with mechanical vibration frequency at the radio-frequency （RF） range. Precise analyses of the vibration and deformation are generally required in the resonator design and improvement process. The considerations include the presence of electrodes, mountings, bias fields such as temperature, initial stresses, and acceleration. Naturally, the finite element method is the only effective tool for such a coupled problem with multi-physics nature. The main challenge is the extremely large size of resulted linear equations. For this reason, we have been employing the Mindlin plate equations to reduce the computational difficulty. In addition, we have to utilize the parallel computing techniques on Linux clusters, which are widely available for academic and industrial applications nowadays, to improve the computing efficiency. The general principle of our research is to use open source software components and public domain technology to reduce cost for developers and users on a Linux cluster. We start with a mesh generator specifically for quartz crystal resonators of rectangular and circular types, and the Mindlin plate equations are implemented for the finite element analysis. Computing techniques like parallel processing, sparse matrix handling, and the latest eigenvalue extraction package are integrated into the program. It is clear from our computation that the combination of these algorithms and methods on a cluster can meet the memory requirement and reduce computing time significantly.     

基于Mindlin板理论的偏移损伤成像数值仿真研究

提出了一种应用散射Lamb波的偏移技术对板结构中多部位损伤进行实时识别. 基于Mindlin板理论,推导了板结构中弥散性弯曲波频率-波数域的快速偏移方法. 首先对由线性传感器阵列激励和接收到的入射和散射波场在波数-频率域分别进行延拓,然后根据Huygens原理,结合波场延拓的时间一致性原理施加成像条件,对损伤进行成像识别. 数值仿真研究采用基于Mindlin板理论的有限差分法模拟结构中含不同形状及尺寸损伤时的散射波场. 对模拟散射波场进行偏移成像的结果表明该方法不仅能够识别多部位损伤的位置,还具有识别损伤程度的能力,其快速计算的优点满足在线结构健康监测系统对实时性的要求.      

复合材料层合加肋板中的瞬态波传播

研究复合材料加肋层合板结构受冲击载荷作用下的瞬态波传播.加肋层合板模型被处理成层合板的弯曲运动与层合梁的扭转与弯曲运动的耦合连续模型.通过Laplace变换,并采用回传射线矩阵法(MRRM),基于一阶剪切变形层合板理论,确定复合材料加肋层合板结构中传播的各条射线群.采用FFT变换,计算各种冲击载荷作用下加肋层合板结构的短时瞬态响应,并分析讨论脉冲类型对加肋层合板结构短时瞬态响应的影响.     

ON FREE WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA

The method of reverberation-ray matrix （MRRM） is extended and modified for the analysis of free wave propagation in anisotropic layered elastic media. A general, numerically stable formulation is established within the state space framework. The compatibility of physical variables in local dual coordinates gives the phase relation, from which exponentially growing functions are excluded. The interface and boundary conditions lead to the scattering relation, which avoids matrix inversion operation. Numerical examples are given to show the high accuracy of the present MRRM.     

一弹性层覆盖于半无限弹性介质对表面点冲击载荷的瞬态响应

利用广义射线理论,研究了一弹性层覆盖于半无限弹性介质表面受法向动集中荷载作用情形表面处的位移响应.给出了一种计算瞬态响应的有效方法,首先根据反射次数的不同将射线自动分组,然后将同一组中具有相同积分路径的积分式自动合并并一次性完成积分.运用该技术,计算量大大减少,使得广义射线法应用于较长时间响应计算的工作得到推进.数值结果中给出了两种情形位移响应的特点,一种情形是弹性层比半空间层硬,另一种情形是弹性层比半空间层软.     

Nonlinear transient response of strain rate dependent composite laminated plates using multiscale simulation

The effects of strain rate dependency and inelasticity on the transient responses of composite laminated plates are investigated. A micromechanics model which accounts for the transverse shear stress effect, the effect of strain rate dependency and the effect of inelasticity is used for analyzing the mechanical responses of the fiber and matrix constituents. The accuracy of the micromechanics model under transverse shear loading is verified by comparing the results with those obtained using a general purpose finite element code. A higher order laminated plate theory is extended to capture the inelastic deformations of the composite plate and is implemented using the finite element technique. A complete micro–macro numerical procedure is developed to model the strain rate dependent behavior of inelastic composite laminates by implementing the micromechanics model into the finite element model. Parametric studies of the transient responses of composite plates are conduced. The effects of geometry, ply stacking sequence, material models, boundary conditions and loadings are investigated. The results show that the strain rate dependency and inelasticity influence the transient responses of composite plates via two significantly different mechanisms.     

梯度饱和土瞬态响应分析

基于Biot多孔介质理论,建立了饱和土体在动载荷作用下的一维回传射线矩阵法的计算列式,其中考虑了土体的非均匀性、惯性、黏滞以及固体颗粒和流体的可压缩性.利用计算结果与已有结果相比较,二者相吻合,验证了算法的正确性.作为数值算例,考虑饱和土的物理力学性质沿深度方向按幂函数连续变化,利用数值Laplace逆变换求解了在冲击性载荷作用下的位移,应力和孔隙压力等物理量的瞬态响应,重点分析讨论了材料非均匀性对饱和土介质动力特性的影响.