June 24, 2007
June 24, 2007
June 27, 2007
12.327.1 - 12.327.14
Broadening student knowledge of dynamics by means of simulation software
Some examples are given with the aim of broadening undergraduate student knowledge and understanding of dynamics. Typically the examples involve non-linear equations and numerical methods must be employed. Here, because of its wide availability and of its increasing use in undergraduate mathematics courses, MAPLE is employed. The central goal of the work is to introduce new phenomena, and the examples treated are: (i) The effect of viscous damping on the stability of an inverted pendulum. It is shown that with a linear model viscous damping does not stabilize an unstable state, whereas damping plays an important role when a non-linear model is considered. (ii) Forced harmonic motion of a non-linear hardening spring-mass system. The numerical simulation of the response illustrates the “jump phenomena” in which the steady state amplitude undergoes a jump in passing through frequencies close to the linear resonance frequency. (iii) A simple pendulum with an oscillating support, illustrating parametric resonance. Depending on the system parameter values instabilities can occur (parametric resonance). This is shown numerically and confirmed with an available analytic expression. The associated MAPLE files are given in an appendix.
The availability of commercial codes such as MAPLE has made it possible to numerically treat problems in dynamics which are analytically intractable. Of course other codes such as MATHCAD® and MATLAB®, for example, are also available. However the thrust here is not to debate the relative merits of finite difference schemes in various software packages. The students should be aware of the nature of finite difference schemes (a simple illustrative example is given in a previous work) then, at least in dynamics classes, the software may be treated as a “black box”. Several examples that broaden student physical knowledge and understanding were given previously, namely: a non-linear pendulum subjected to various initial conditions, showing how the period depends on the amplitude; a non-linear softening spring showing the existence of instabilities; an undamped inverted pendulum restrained by a spiral spring, illustrating the existence of multiple equilibrium states and their stability; a simulation of a sweep test (forced motion of a single-degree-of-freedom system in which the forcing frequency varies with time), showing that if the sweep rate is too fast, no resonances will be observed. Here several new examples are presented (for convenience both sets are included in TABLE 1 in an appendix, which has MAPLE worksheet objects included). The examples are: (i) the effect of viscous damping on the stability of an inverted pendulum; (ii) forced harmonic motion of a non-linear hardening spring-mass system; and (iii) a simple pendulum with an oscillating support, illustrating parametric resonance. Examples (i) and (iii) can readily be handled in a beginning course, whereas example (ii) may be more suitable for an intermediate course.
Mazzei, A., & Scott, R. (2007, June), Broadening Student Knowledge Of Dynamics By Means Of Simulation Software Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. 10.18260/1-2--1570
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