Euler‐Maclaurin Closed Form Finite State Space Model for a String Applied to Broadband Plate Vibrations

Abstract

The Euler-Maclaurin sun formula is applied to the infinite series Green's function solution in the space-time Laplace transform domain for the one dimensional wave equation for a string fixed at each end. The resulting approximate closed form solution is used to derive a single third order input-output ordinary differential equation to model the string dynamics. The average modal density of a plate is shown to be comparable to a string. A finite three state-space model is developed for the string and applied to the vibrations of a plate subjected to broadband random and impulse inputs. The applications include the direct problem of determining the response to a disturbance input and the inverse problem of identifying the disturbance input with a finite state observer based on the finite string model. Numerical simulations using many plate modes are obtained in the time and frequency domains and are used to compare the multimodal plate model to the finite string based model and to demonstrate how the finite string based model can be used to represent the multimodal vibrations of the plate.

A Finite State Space Model for Representing the Broadband Infinite Series for Acoustic Reverberation between Parallel Reflecting Planes

“…In view of the Euler-Maclaurin closed form results given in [8][9][10], in both time domain response and the transfer function based control for frequencies up to 1000 Hz, and the successes in [1][2][3][4][5][6][7] for the truncated state space models for various types of low frequency control in acoustic and structural systems, the work in [11] focuses on the development of a low order but full finite state space model that would naturally include a broadband frequency range and essentially all of the terms required for the convergent infinite series in the Green's function representation of bounded structures. Also, in [11], a one-dimensional structural system of a string is used to develop a suitable state space for a two-dimensional plate. The developed state space model in [11] is shown to give an accurate trend for representing of the series transfer function for high frequencies up to 2000 Hz.…”

“…Also, in [11], a one-dimensional structural system of a string is used to develop a suitable state space for a two-dimensional plate. The developed state space model in [11] is shown to give an accurate trend for representing of the series transfer function for high frequencies up to 2000 Hz. and the accurate time response for both a direct problem with an exponential input and for use in an observer based inverse problem of estimating a disturbance input.…”

“…In this paper, the ideas and approach in [11] for structural systems are expanded, modified, and improved to develop a low order but full finite space model that would naturally include a broadband frequency range and essentially of all the terms required for a convergent infinite series in the Green's function representation of the acoustic field between reflecting planes commonly known as reverberation. The distinguishing difference between this paper and the development in [11] is that, in [11], the Euler-Maclaurin sum formula is applied to an infinite series of string modes.…”

“…The distinguishing difference between this paper and the development in [11] is that, in [11], the Euler-Maclaurin sum formula is applied to an infinite series of string modes. In this paper, the Euler-Maclaurin sum formula is applied directly to the Laplace transform form of the Green's function infinite series for acoustic reverberation in [8] providing all polynomial type functions in the Laplace transform domain which are directly combined into a system function.…”

“…In this paper, the Euler-Maclaurin sum formula is applied directly to the Laplace transform form of the Green's function infinite series for acoustic reverberation in [8] providing all polynomial type functions in the Laplace transform domain which are directly combined into a system function. This system transfer function in [11] leads directly to an input/output ordinary differential equation which serves as the basis for defining the state variables and determining the state equations. For the acoustic space of the current paper, the Euler-Maclaurin sum formula must be directly applied to the time domain form of the Green's function infinite series as is done in [8].…”