The Eigenfunction Method for Determining Displacement Time History in Structural Dynamic Analysis

In condition monitoring of structures, acceleration time histories are usually recorded due to ease of instrumentation. In cases where the information about a displacement time history is required, the acceleration data needs to be integrated to obtain the velocity and then the velocity needs to be integrated to obtain the displacements. However, the numerical integration of the acceleration data usually introduces an unrealistic drift component to the velocity as well as displacement. This paper presents an eigenfunction method to derive velocity and displacement time histories from a given acceleration time history. The paper analyzes displacements in two case studies using the numerical integration as well as the proposed eigenfunction method. It is concluded that the eigenfunction method is a viable approach to derive the displacement information from the acceleration data.

Generating Drift‐Free, Consistent, and Perfectly Spectrum‐Compatible Time Histories

The focus of this article is on generating spectrum‐compatible acceleration, velocity, and displacement time histories for seismic analysis and design of engineering structures. If a generated acceleration time history is integrated to obtain the corresponding velocity and displacement time histories, it has been found that there are usually drifts in the resulting processes. Such drifts are due to overdeterminacy in the constants of integration. Baseline correction, although widely used, is not a suitable remedial measure to remove drift because it distorts the frequency content and renders the corrected processes no longer mutually consistent. The objective of this article is to develop an efficient and accurate method for generating drift‐free, consistent, and spectrum‐compatible time histories, which are essential properties for these time histories to be used as seismic input in time history analysis. To ensure drift‐free and consistent behavior, the eigenfunction method is applied to expand the time histories in eigenfunctions of a sixth‐order ordinary differential eigenvalue problem. The influence matrix method considering the influence of one frequency component on all others is capable of achieving perfect spectrum compatibility which has never been accomplished.

Approximation of Current on a Thin Tubular Vibration Antenna Using the Eigenfunction Method

The article considers the method of eigenfunctions for getting the approximated current solution to the internal problem of electrodynamics. The paper considers a polynomial frequency dependence approximation of the eigenvalues and eigenfunctions of the operator for conducting bodies. The current solutions obtained by the straightforward method and with the help of method using the comparison of two approximation types.

Singular treatment of viscous flow near the corner by using matched eigenfunctions

In this paper, the local singular behavior of Stokes flow is solved near the salient and re-entrant corners by the matching eigenfunction method. The flow in a rectangular and an L-shaped cavity are considered as a model for the flow generated by the motion of the upper lid. The solutions of the Stokes equation in polar coordinates are matched with a velocity vector components obtained by analytic or numerical solution for the streamfunction developed for any values of the heights of the rectangular and an L-shaped cavity. Streamline patterns near the corner are simulated for a different aspect ratio A. The techniques are tested on a flow problem undergoing Stokes or Navier–Stokes equations in a square cavity. It is seen that the method appears to be cheaper and more accurate than the numerical and analytical methods. It is expected that the study will lead to useful insights into the understanding of the flow topology near a re-entrant corner from a combined analytical-numerical method. Attention is then focused on the topological behavior near the re-entrant corner of the L-shaped cavity. Careful analysis of the streamlines of streamfunction near the re-entrant corner by using wall shear stress allows us to give a possible flow bifurcation of dividing streamline.

The singularity expansion method and near-trapping of linear water waves

The problem of near-trapping of linear water waves in the time domain for rigid bodies or variations in bathymetry is considered. The singularity expansion method (SEM) is used to give an approximation of the solution as a projection onto a basis of modes. This requires a modification of the method so that the modes, which grow towards infinity, can be correctly normalized. A time-dependent solution, which allows for possible trapped modes, is introduced through the generalized eigenfunction method. The expression for the trapped mode and the expression for the near-trapped mode given by the SEM are shown to be closely connected. A numerical method that allows the SEM to be implemented is also presented. This method combines the boundary element method with an eigenfunction expansion, which allows the solution to be extended analytically to complex frequencies. The technique is illustrated by numerical simulations for geometries that support near-trapping.

Modeling of Random Fatigue Crack Propagation

A statistical model is proposed for the analysis of fatigue crack propagation, based on the theory of fracture mechanics and stochastic process. The fatigue growth process is approximated as a diffusive Markov process. The associated backward Fokker-Plank equation and boundary conditions are written, and the distribution of crack propagation time under a given crack size is obtained by using an Eigenfunction method. The sought distribution is expressed in the form of a convergent infinite series. An examples is presented to illustrate the application of the method. The predicted results seem to agree with the experimental data.