June 20, 2010
June 20, 2010
June 23, 2010
Computers in Education
15.970.1 - 15.970.20
Prediction comparisons between non-linear and linear models for dynamics enhanced education
In previous works 1, 2, 3, 4 examples were given illustrating benefits of introducing modern software, such as MAPLE®, into undergraduate and beginning graduate mechanics courses.
There are many articles on the use of simulation in engineering education. For example, Fraser et al. 5 give a very informative and useful discussion on the use of simulations in fluid mechanics. Student difficulties were assessed using questions from the Fluid Mechanics Concepts Inventory (FMCI). The impact of the simulation was assessed using a second administration of the FMCI. Similar work could probably be done using the Dynamics Concept Inventory of Gray et al. 6 However that is not the goal of the present work, which is to enlarge and augment presentations given in most standard texts. For example consider the amplitude of resonant motions as given by a linear damped model. Are the traditional predictions accurate? Since near resonance the underlying equations are non-linear, substantial differences from the linear model may (and usually are) found.
With modern numerical simulation software (for example, MAPLE®) the non-linear ordinary differential equations of motion can be readily solved and differences assessed.
In the following, three illustrative examples are presented. First a damped one-degree of freedom system with a non-linear spring is investigated. Many texts give expressions for power flow at resonance based on a linear model, however near resonance large amplitudes would be encountered and a non-linear model should be employed. Here numerical solutions to the differential equations are used to obtain the velocity, from which the power flow per cycle can be calculated. It is shown that for reasonable values of damping differences can be substantial.
Other interesting problems involve the effects of non-linearities on frequencies and mode shapes in vibrations. A simple single degree of freedom model is developed here which shows the effect of the non-linearity on the period of vibrations. This is also done for a two degree of freedom example. Another interesting problem involves the linear normal modes of a two-degree of freedom mechanical system. It is well known that if the system is initialized in a normal mode, it remains in that mode. What happens if the underlying physical model is non-linear? Here, for purposes of classroom demonstration, simple models, showing some interesting effects, are developed and analyzed using MAPLE®.
Power Flow in Linear and Non-linear oscillators
The equation of motion for a non-linear (hardening spring) viscously damped system subjected to a harmonic forcing function can be written as:
Mazzei, A., & Scott, R. (2010, June), Prediction Comparisons Between Non Linear And Linear Models For Dynamics Enhanced Education Paper presented at 2010 Annual Conference & Exposition, Louisville, Kentucky. https://peer.asee.org/15739
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