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Using Logarithms to Test the Solution of a Differential Equation in the Lab

Abstract

The mathematical modeling of the damping force on a spring-mass system oscillating in a fluid as a linear function of velocity (linear damping) is a simplifying assumption that leads to an ordinary and linear differential equation with constant coefficients. This model provides a simple means to account for the experimental fact that energy is dissipated during oscillations as the moving mass pushes against and displaces the surrounding fluid. The analytical solution to this differential equation is compared with experimental data collected from testing a spring-mass system in the open air of a laboratory. Collected data are analyzed using the concept of the logarithms. It is shown that the model is reliable under special conditions.

Introduction

One way to bring excitement in the use of mathematics in the engineering classroom is to show that it can be applied 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 courses1-3.

We consider the ordinary differential equation given by

2 x 2 x n x 0, (1)

with the following initial conditions x (t 0) x0 (2) x (t 0) v0 2 where x is a function of time, the dots indicate derivatives of x with respect to time, and n and are constants that characterize the system.

Picture A. Photo of a smooth sphere on a linear spring oscillating in the open air of a laboratory.

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Njock-Libii, J. (2009, June), Using Logarithms To Test The Solution Of A Differential Equation In The Lab Paper presented at 2009 Annual Conference & Exposition, Austin, Texas. 10.18260/1-2--4843