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Using the Fourier Theorem to Test the Solution of a Differential Equation in the Laboratory

Abstract

This paper presents work conceived and implemented to test the validity of a second- order differential equation that is commonly used to represent the oscillation of a spring- mass system. We designed a load cell and used it with computer data-acquisition equipment to collect data that give the position of the suspended mass versus time. We analyzed the collected data using the Fourier theorem and compared our experimental results to those obtained by solving the differential equation analytically. Students found it remarkable that the analysis of collected data demonstrated in a convincing manner that the analytical solution represented the motion of a mass suspended from a spring accurately.

Introduction

One way to bring excitement in the use of mathematics in the engineering classroom is to show that it can be used to model physical reality accurately. This paper presents work conceived and implemented to test the extent to which an ordinary differential equation and its solution are valid for use in actual applications. The equation chosen is commonly used in mathematics, physics, and engineering courses.

We consider the ordinary differential equation given by 2 x nx 0, (1) with the following initial conditions x (t 0) x0 (2) x (t 0) v0

where x is a function of time, the dots indicate derivatives of x with respect to time, and 2 n is a quantity that is independent of time.

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Njock-Libii, J. (2008, June), Using The Fourier Theorem To Test The Solution Of A Differential Equation In The Lab Paper presented at 2008 Annual Conference & Exposition, Pittsburgh, Pennsylvania. 10.18260/1-2--3092