Stiffness matrix method with improved efficiency for elastic wave propagation in layered anisotropic media

Dynamic stiffness of three-dimensional anisotropic multi-layered media based on the continued-fraction method

The propagation of elastic transient waves in a multi–layered medium subjected to in–plane loadings is investigated in this study. One of the objectives of this study is to develop an effective analytical method for determining transient full–field solutions in the layered medium. A matrix method is developed by expanding the matrix solution obtained directly from the boundary–value problem in the integral transform domain into a power series of the phase–related reflection and transmission matrix which characterizes the multiple reflections and transmissions of all waves in every layer. The transient response of the multi–layered medium is decomposed into infinite wave groups in which the waves are either reflected by, or transmitted through, the interfaces. The connection between the proposed matrix method and the generalized ray method is established for the layered medium in the transform domain. The matrix representation of the solution enables us to calculate the transient response of the layered medium without tracing the ray path manually. The obtained analytical solution can easily be applied to numerical calculations. The double inverse transform is performed based on Cagniard9s method and the theoretical transient solution for a layered half–space subjected to an in–plane dynamic force will be presented in part II. An experimental set–up that simulates the plane stress condition for a layered half–space was established to obtain the dynamic displacement response. The experimental result agrees well with the theoretical solution. The proposed methodology in this study can be extended to solve more complicated problems such as waves propagating in three–dimensional space.

Static response of a layered magneto-electro-elastic half-space structure under circular surface loading

SUMMARYThis paper presents a complete derivation of the combined flexural-joint stiffness matrix and the elastic deformation field of flexible manipulator arms treated in a three-dimensional fashion. The stiffness properties are derived directly from the differential equations used in the engineering beam theory. The expressions developed here can readily be used in the modeling, control and design of light weight flexible robot manipulators. A two-link arm is used to formulate these expressions and the results can be generalized to n–link manipulators. The stiffness matrix for a robotic link element in 3-D is of the order of 12 X 12, and for an n–link robotic arm the total elemental and system stiffness matrices will be of the order of the (12n X 12n) and 6(n + 1) X 6(n + 1), respectively.

This study focuses on advancements in three broad aspects of active surface wave methods: modeling, testing, and inversion. Transfer matrix, global matrix, and stiffness matrix methods were employed to comprehensively model layered structures with half-space boundary conditions for soil structures with increasing/anomalous stiffness profiles, or pavement structures with decreasing stiffness profiles and leaky waves. All three methods were programmed in MATLAB as forward algorithms. The finite element method was adopted to simulate surface wave testing for various half-space site structures with absorbing layers using increasing damping technique. An improved experimental dispersion analysis scheme was developed by scanning phase-velocity and intercept-time followed by a power-spectrum analysis to minimize side lobes and increase sharpness of dispersion images. The multichannel simulation with one receiver (MSOR) method was applied to capture the dispersion characteristics of soil sites. The reciprocity principle for surface Rayleigh waves was verified by comparing dispersion images from MSOR and multichannel analysis of surface waves (MASW) testing at the same site with the same testing system. A multichannel surface wave acquisition system was developed to improve the accuracy of measuring high-frequency and high-velocity dispersion data on pavement sites. A minimally-invasive multimodal surface wave (MMSW) method was proposed to measure multi-mode dispersion data of Rayleigh waves by using either embedded receivers at various depths to record surface wave motions generated from moving impacts on the ground surface or using a multichannel seismograph with an array of geophones on the soil surface for measuring surface wave motions caused by Standard Penetration Test hammer blows at various depths in a borehole. Stiffness matrix and finite element simulations of the MMSW method were employed to identify the critical geophone depths for optimum measurement of higher-mode motions. A hybrid genetic-simulated-annealing (GSA) algorithm was applied to solve multiple minimization and non-linear optimization problems to match the theoretical dispersion curves against their experimental counterparts. Results from simulation and real-world studies demonstrate that the advancements made in the three aspects of surface wave methods can improve the accuracy of surface wave testing results with higher resolution of experimental dispersion data, more complete multi-modal dispersion data, and higher certainty of inversion.

Of the two main finite element approaches, the displacement method has progressed more rapidly than the force method in the area of element representation. Many authors have contributed to the displacement method with both static and dynamic stiffness matrices for beam elements, plane stress elements and plate bending elements. The plane stress and bending elements being rectangular, triangular or quadrilateral in form. A dynamic stiffness matrix consists of a static stiffness matrix and a mass matrix. In the force method, .static flexibility matrices have been developed for beam elements and plane stress elements. However, static flexibility matrices for plate bending elements have only recently been published (Kaufman and Hall, Morley, Robinson, Przemieniecki). For the past few years, the author has been investigating the rank force method for structural vibration analysis. In ref. 6 a dynamic flexibility matrix is presented for a beam element. This matrix consists of a static flexibility matrix and an inverse mass matrix. Ref. 5 contains the derivation of a dynamic flexibility matrix for a rectangular plate element in bending, twisting and shear. The author, in collaboration with Petyt, demonstrated how a dynamic stiffness and flexibility matrix can be extended to include material damping.

In this paper, wave propagation in two-layered anisotropic media when uniaxially anisotropic medium has its optic axis oriented vertically and tilted around z-axis has been investigated. The Fresnel coefficients are derived for horizontally and vertically polarized incident waves with arbitrary incident angles. The contribution of the cross polarized components of the reflected and transmitted waves due to anisotropic- isotropic or isotropic-anisotropic interfaces have been analyzed. The numerical results for the Fresnel coefficients have been presented. The results of this work can be used in calculation of the electric and magnetic fields in radiation and scattering problems.

The electromagnetic field formation at kHz frequencies when passing through a layered anisotropic medium with tensor values of electrical conductivity was considered. In one series of numerical experiments, the field is excited by an electromagnetic field source distributed along with one of the boundaries of the computational domain. In another, a field source is an inclined loop. This work allows one to estimate the behavior of the components of the electric field E at the interface between two anisotropic media, as well as an isotropic and anisotropic medium in the frequency range.

Rayleigh wave on the half-space with a gradient piezoelectric layer and imperfect interface

P105 CONVERTED-WAVE TRAVELTIME EQUATIONS IN LAYERED ANISOTROPIC MEDIA – AN OVERVIEW Abstract 1 There are various types of converted-wave (C-wave) traveltime equations: for example two-term or three-term Taylor series expansions and double-squared-root (DSR) equations. These different types of equations often have different kinds of parameterization and applicability. In this paper we examine the accuracy and limitations of these equations and present improved equations under a unified system for multi-layered anisotropic media. These new forms of equations are accurate for far offsets up to twice the reflector depth and can be used for velocity analysis parameter estimation and prestack imaging in

Stiffness and transfer matrices of a non-naturally curved 3D-beam element

Preliminary Considerations Nodal Variables - Stiffness and Flexibility Matrices of a Structure Element Variables - Total Stiffness Matrix of an Element Direct Stiffness Method Supplementary Procedures to the Stiffness Method Basic Element Variables The Equations of Equilibrium and the Equations of Compatibility of a Structure Computation of the Displacements of Statically determinate Framed Structures The Flexibility Method The Displacement or Stiffness Method Appendix A. The International System of Units Appendix B. End Actions in Fixed-At-Both-Ends Elements Appendix C. Vector Quantities Appendix D. Computation of the Total Stiffness Matrix of an Element and the Equivalent Actions to be Placed at the Nodes of a Structure Using Shape Functions.

The direct stiffness method is used to analyze the structural response of the cross section of an arteriole which is both heterogeneous and nonlinear in terms of structural response. A typical segment of the cross section is represented by an assembly of triangular elements. The stiffness matrix and strain-displacement relationships are then derived in terms of polar coordinates for a triangular plate element. Viscoelasticity is included in this structural analysis by using a piecewise linear approach. Stress-time and displacement-time data are presented for two typical cross sections of the arteriole. These results reveal special properties of a heterogeneous structure which is undergoing viscoelastic deformations.

This paper introduces the direct global stiffness matrix (DGSM) method for analyzing the dynamics of cylindrical shells with axial symmetry. The dynamic stiffness matrix of a straight, circular cross-section cylindrical element is formulated in terms of ring forces and displacements at each end on a per circumferential mode basis. The stiffness matrix is based on an exact wave-type solution of the thin shell Donnell equations. More complicated shell systems are analyzed by assembling any number of cylindrical elements. For example, a cylinder with a gradual variation in radius may be modeled by a series of shell elements welded together at their ends. Total system dynamics are found at any given frequency by inverting the global system matrix formed from the individual stiffness element matrices and summing over circumferential mode number. Results for a test case are presented to demonstrate and verify the implementation. [Research sponsored by ONR.]

Modified scaled boundary finite element analysis of 3D steady-state heat conduction in anisotropic layered media