The image source method for calculating the vibrations of simply supported convex polygonal plates

Abstract

To cite this version:Jacques Cuenca, François Gautier, Laurent Simon. The image source method for calculating the vibrations of simply supported convex polygonal plates. AbstractThe aim of this paper is to show that the Image Source Method (ISM) can be used both for analytically calculating Green's functions of particular simply supported convex polygonal plates and for predicting medium and high frequency vibrations of arbitrarily shaped simply supported convex polygonal plates with controllable precision. In the first part of the paper, the method for obtaining the Green's function of a polygonal plate by ISM is developed. Examples for plates of different geometries (rectangle, isosceles right triangle, half-equilateral triangle and equilateral triangle) are given. In the second part of the paper, the pertinence of ISM for predicting medium and high frequency vibrations of arbitrarily shaped simply supported convex polygonal plates is investigated. An approximation based on the exclusion of image sources beyond a certain distance from the receiver is used in order to take advantage of the dissipation of vibrational energy through wave propagation. We investigate the influence of structural damping and truncation distance on the accuracy of such approximation. The computed responses are in good agreement with reference solutions, which are analytically known or obtained by the finite element method.

Measurement of thickness or plate velocity using ambient vibrations

“…For an observation point located on the plate, the result observed corresponds to interference between the direct wave and the multiple reflected waves. According to this reasoning, also defended by other authors [6][7][8] the free space Green function is still valid in a plate of finite dimension but an additional correction term must be added in order to both satisfy the inhomogeneous plate equation and the boundary conditions. Therefore, in the absence of any external source to the plate, it is reasonable to write the Green function of the plate as follows:…”

Estimation of plate material properties by means of a complex wavenumber fit using Hankel's functions and the image source method

“…For a plate with infinite lateral dimensions, the solution of Eq. (1) is given by [5] where G ∞ is the Green's function of the infinite plate, H is the cylindrical Hankel's function of the first kind of order 0, and r = (x − x 0 , y − y 0 ) is the source-to-receiver distance.…”

“…The approach of Cuenca et al [5] combines the use of the image source method and Hankel's functions to construct the Green's function of a finite plate with simply supported boundary conditions. Such a model is much better suited to represent the vibrational field of a point excited plate, especially in the higher frequency range, as the accuracy of the image source method is known to increase with frequency and damping [6].…”

Thickness or phase velocity measurements using the Green's function comparison method

“…For an observation point located on the plate, the result observed consists of wave interferences between the direct wave and multiple reflected waves. as shown in [18]- [20], the free space green's function is still valid in a plate of finite dimension but an additional correction term must be added. Thus, the sum of the two terms is a solution of the inhomogeneous plate equation and satisfies the boundary conditions.…”

“…c D h f = 11 4 /r w (19) in the stripes direction and c D h f = 22 4 /r w (20) in the transverse direction. Finally, the product V P h is linked to flexural rigidities by…”