Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 817194, 14 pages
Research Article

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

Mechanical Engineering, Gannon University, 109 University Square, Erie, PA 16541, USA

Received 29 July 2009; Revised 9 February 2010; Accepted 2 March 2010

Academic Editor: Paulo Batista Gonçalves

Copyright © 2010 Michael J. Panza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.