Experimental analysis of nonlinear oscillations in the undergraduate physics laboratory

Abstract

In this paper, we present a simple experiment to introduce the nonlinear behaviour of oscillating systems in the undergraduate physics laboratory. The transverse oscillations of a spring allow reproduction of three totally different scenarios: linear oscillations, nonlinear oscillations reducible to linear for small displacements, and intrinsically nonlinear oscillations. The chosen approach consists of measuring the displacements using video photogrammetry and computing the velocities and the accelerations by means of a numerical differentiation algorithm. In this way, one can directly check the differential equation of the motion without having to integrate it, or perform an experimental study of the potential energy in each of the analysed scenarios. This experiment allows first year students to reflect on the consequences and the limits of the linearity assumption for small displacements that is so often made in technical studies.

Experimental study of viscous friction in undergraduate physics laboratory: introduction of phase diagrams to analyse dynamic equilibrium

“…As it has been proved in [11], important changes in the differential equations can result in small modifications in the solution ( ). For example, in our case, the effect of Coulomb friction could be explained by a slight variation of the slope in the fitting results.…”

“…This is especially important in the viscous friction case, where, in many instances, the model is not clear or even can vary along the motion [10]. Furthermore, as shown in [11], very different differential equations can give as result similar movement equations.…”

“…This strategy has been used in former works by using photogrammetry and efficient techniques of numerical differentiation for estimating the values of the position and its derivatives [11,15].…”

“…For the experimental estimation of the relevant variables of the process, we propose a video photogrammetry system and advanced techniques of numerical differentiation [15]. This measurement system is adequate for the level of the undergraduate physics laboratory and has been utilized in former works for analyzing complex movements, obtaining satisfactory results [11].…”

Exploring viscous damping in undergraduate Physics laboratory using electromagnetically coupled oscillators

“…We also suggest that students explore the possibility of adding a phenomenological term in the denominator of Equation (3) to model the oscillations in this case and follow the method of non-linear regression outlined earlier to determine the parameters. Recently, a method of directly checking the differential equation of the motion without integrating it has been suggested [14] using video photogrammetry. This can be very useful in analyzing such non-linear oscillations.…”