1,721,060 research outputs found
Forced response of two-dimensional homogeneous media using the wave and finite element method
No full textVeering and Strong Coupling Effects in Structural Dynamics
No full textMode veering is the phenomenon associated with the eigenvalue loci for a system with a variable parameter: Two branches approach each other and then rapidly veer away and diverge instead of crossing. The veering is accompanied by rapid variations in the eigenvectors. In this paper, veering in structural dynamics is analyzed in general terms. First, a discrete conservative model with stiffness, mass, and/or gyroscopic coupling is considered. Rapid veering requires weak coupling: if there is instead strong coupling then there is a slow evolution of the eigenvalue loci rather than rapid veering. The uncoupled-blocked system is defined to be that where all degrees-of-freedom (DOFs) but one are blocked. The skeleton of the system is the loci of the eigenvalues of the uncoupled-blocked system as the variable parameter changes. These loci intersect at certain critical points in the parameter space. Following a perturbation analysis, veering is seen to comprise rapid changes of the eigenvalues in small regions of the parameter space around the critical points: for coupling terms of order u veering occurs in a region of order u around the critical points, with the rate of change of eigenvalues being of order iuâ '1. This is accompanied by rapid rotations in the eigenvectors. The choice of coordinates in the model and application to continuous systems is discussed. For nonconservative systems, it is seen that veering also occurs under certain circumstances. Examples of 2DOFs, multi-DOFs (MDOFs), and continuous systems are presented to illustrate the results
Dynamic and acoustic properties of viscoelastic laminated structures
No full textNumerical computation of wave characteristics in composite laminated structures is obtained using a Wave Finite Element method. The method involves conventional finite element (FE) analysis of a small segment of the structure. Typically, this consists of a stack of solid elements meshed through the cross-section. Each layer can be discretised using either one solid element or more solid elements. The analyst must select element type and number appropriately in order to accurately model the three-dimensional stress-strain state of the structure (in the context of FE analysis). The advantages of the proposed method are that it involves post-processing of a small FE model, commercial FE packages can be used to generate the model, no new elements need to be derived and implemented, and the computational cost is extremely small. Axisymmetric properties of the structure can be exploited, and the method can take advantage of the capability of commercial FE packages to model acoustic fluids and fluid-structure interaction. The method is described and illustrated by application to a laminated viscoelastic sandwich panel in vacuo and a laminated viscoelastic sandwich cylinder filled with fluid. Complex dispersion curves and wave modes are given. The global loss factor as a function of the frequency is also predicted, and the effect of the initial pre-stress is discussed.<br/
Waves in a three-dimensional model of the cochlea
No full textThe conventional travelling wave theory of the cochlea assumes that only a single “slow” wave, which determines the overall response in the cochlea, can propagate. Various different mechanisms, such as longitudinal coupling in the fluid or the basilar membrane, BM, may give rise to other types of wave. In this paper the wave finite element method is used to predict all possible waves in a three-dimensional model of the passive cochlea using an orthotropic plate model for the BM, in terms of wave mode shape and wavenumber as a function of position along the cochlea. Mode conversion in waves can then be explored by decomposing results from a full finite element model. It is found that only one wave, the slow wave, is dominant basal to the characteristic place and then a higher order fluid mode starts to make a significant contribution to the overall response when system damping is small
Structural vibration analysis with random fields using the hierarchical finite element method
No full textElement-based techniques, like the finite element method, are the standard approach in industry for low-frequency applications in structural dynamics. However, mesh requirements can significantly increase the computational cost for increasing frequencies. In addition, randomness in system properties starts to play a significant role and its inclusion in the model further increases the computational cost. In this paper, a hierarchical finite element formulation is presented which incorporates spatially random properties. Polynomial and trigonometric hierarchical functions are used in the element formulation. Material and geometrical spatially correlated randomness are represented by the Karhunen–Loève expansion, a series representation for random fields. It allows the element integration to be performed only once for each term of the series which has benefits for a sampling scheme and can be used for non-Gaussian distributions. Free vibration and forced response statistics are calculated using the proposed approach. Compared to the standard h-version, the hierarchical finite element approach produces smaller mass and stiffness matrices, without changing the number of nodes of the element, and tends to be computationally more efficient. These are key factors not only when considering solutions for higher frequencies but also in the calculation of response statistics using a sampling method such as Monte Carlo simulation
Wave propagation in slowly varying waveguides using a finite element approach
No full textThis work investigates structural wave propagation in one dimensional waveguides with randomly varying properties along the axis of propagation, specifically when the properties vary slowly enough such that there is negligible backscattering, even if the net change is large. Wave-based methods are typically applied to homogeneous waveguides but the WKB (after Wentzel, Kramers and Brillouin) approximation can be used to find a suitable generalisation of the wave solution in terms of the change of phase and amplitude but is restricted to analytical solutions. A wave and finite element (WFE) approach is proposed to extend the applicability of the WKB method to cases where no analytical solution of the equations of motion is available. The wavenumber is expressed as a function of the position along the waveguide. A Gauss-Legendre quadrature scheme is subsequently used to obtain the phase change, while the wave amplitude is calculated using conservation of power. The WFE method is used to evaluate the wavenumbers at each integration point. Moreover, spatially correlated randomness can be included in the formulation by random field properties and in this paper is expressed by a Karhunen-Loève expansion. Numerical examples are compared to a standard FE approach and to available analytical solutions. They show good agreement when compared to either a full FE or analytical solution and require only a few WFE evaluations, providing a suitable framework for efficient stochastic analysis in waveguides
Calculating the forced response of two-dimensional homogeneous media using the wave and finite element method
No full textThe forced response of two-dimensional, infinite, homogenous media subjected to time harmonic loading is treated. The approach starts with the wave and the finite element (WFE) method where a small segment of a homogeneous medium is modelled using commercial or in-house finite element (FE) packages. The approach is equally applicable to periodic structures with a periodic cell being modelled. This relatively small model is then used, along with periodicity conditions, to formulate an eigenvalue problem whose solution yields the wave characteristics of the whole medium. The eigenvalue problem involves the excitation frequency and the wavenumbers (or propagation constants) in the two directions. The wave characteristics of the medium are then used to obtain the response of the medium to a convected harmonic pressure (CHP). Since the Fourier transform of a general two-dimensional excitation is a linear combination of CHPs, the response to a general excitation is a linear combination of the responses to CHPs. Thus, the response of a two-dimensional medium to a general excitation can be obtained by evaluating an inverse Fourier transform. This is a double integral, one of which is evaluated analytically using contour integration and the residue theorem. The other integral can be evaluated numerically. Hence, the approach presented herein enables the response of an infinite two-dimensional or periodic medium to an arbitrary load to be computed via (a) modelling a small segment of the medium using standard FE methods and post-processing its model to obtain the wave characteristics, (b) formulating the Fourier transform of the response to a general loading, and (c) computing the inverse of the Fourier transform semi-analytically via contour integration and the residue theorem, followed by a numerical integration to find the response at any point in the medium. Numerical examples are presented to illustrate the approac
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