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

Michael J Panza

doi:10.1155/2010/817194

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.

Highlights

The broadband high-frequency analysis of distributed parameter systems governed by partial differential equations usually requires a very large multimodal model, a many-node discrete grid finite difference or finite element model, or a statistical energy analysis SEA approach
Most work to date has been for disturbance inputs such as impulse type or a low-frequency random type
The finite state space model is derived from the dynamics of an elastic string fixed at each end

Summary

Introduction

The broadband high-frequency analysis of distributed parameter systems governed by partial differential equations usually requires a very large multimodal model, a many-node discrete grid finite difference or finite element model, or a statistical energy analysis SEA approach. Disturbance force identification problems for structures such as beams and plates using a multimodal model of vibration have been the subject of studies 1, 2 where several lower-order modes are used to represent the structure subjected to either an impulse-type input or a low-frequency random-type input. These reduced order models are usually sufficient for frequencies less than about 200 Hz. A novel discrete high-frequency forced vibration method using a large number of plate modes and based on the high-frequency plate modal density was applied for a harmonic force varied. The Euler-Maclaurin finite string model may provide a reasonable alternative to a multimodal plate model with the same fundamental frequency

Approximate Green’s Function for String Wave Equation

Finite State Space Model from Approximate Green’s Function

Comparison with Multimodal Plate Model

Inverse Dynamics Application for Estimating a Disturbance Input

Conclusions