Asee peer logo

Prediction Comparisons Between Non Linear And Linear Models For Dynamics Enhanced Education

Download Paper |


2010 Annual Conference & Exposition


Louisville, Kentucky

Publication Date

June 20, 2010

Start Date

June 20, 2010

End Date

June 23, 2010



Conference Session

Computational Tools and Simulation I

Tagged Division

Computers in Education

Page Count


Page Numbers

15.970.1 - 15.970.20

Permanent URL

Download Count


Request a correction

Paper Authors


Arnaldo Mazzei Kettering University

visit author page

ARNALDO MAZZEI is an Associate Professor of Mechanical Engineering at Kettering University. He received his Ph.D. in Mechanical Engineering from the University of Michigan in 1998. He specializes in dynamics and vibrations of mechanical systems and stability of drivetrains with universal joints. His current work relates to modal analysis, stability of drivetrains, finite element analysis and CAE. He is a member of ASME, ASEE and SEM.

visit author page


Richard Scott University of Michigan

visit author page

RICHARD A. SCOTT received his Ph.D. in Engineering Science from The California Institute of Technology. He is a Professor of Mechanical Engineering at the University of Michigan, Ann Arbor. He has obtained a teaching award from the College of Engineering and was selected as professor of the semester four times by the local chapter of Pi-Tau-Sigma.

visit author page

Download Paper |

NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

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®.

Physical Examples

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.

ASEE holds the copyright on this document. It may be read by the public free of charge. Authors may archive their work on personal websites or in institutional repositories with the following citation: © 2010 American Society for Engineering Education. Other scholars may excerpt or quote from these materials with the same citation. When excerpting or quoting from Conference Proceedings, authors should, in addition to noting the ASEE copyright, list all the original authors and their institutions and name the host city of the conference. - Last updated April 1, 2015