Research in Rigid Body Dynamics

This web page is an evolving progress report on a collaborative research on a specific problem in rigid body dynamics. The collaborators are:

The Goal

The goal of this research, lead by the interests of the US Department of Agriculture, is to develop a fast, reliable and inexpensive method to orient large quantities of apples so that their stem-calyx axes point in a prescribed direction.

At an apple packaging or processing plant, the properly oriented apples will scanned by optical instruments to detect contamination.

The Experiment

Experiments conducted at UMBC and USDA have shown that apples that roll down an inclined track tend to orient themselves so that they spin about their stem-calyx axes, regardless of their initial orientations. The following sequence of still images captured with a high-speed video camera, illustrates this.

View top to bottom, column 1 first.

The approximate length and inclination of the track are 8 feet and 30 degrees. Note that this apple has oriented itself by the time it has reached about the half-way point down the track.

The Mathematical Model

To understand and explain the mechanism behind the rolling apple's behavior, we have built a mathematical model that serves as a metaphor for the actual apple.

The “apple” in our mathematical model is a perfect sphere however its mass is not distributed uniformly; the center of mass coincides with the sphere's geometric center however the moment of inertia is not a multiple identity as it would be in the case of a homogeneous sphere.

The sphere rides on an inclined track consisting of two straight and parallel rails, as shown in the figure below:

There is Coulomb friction between the sphere at its contact points with the rails. As the sphere picks up speed, its inhomogeneous mass distribution results in dynamical imbalance which causes it to slip against the rails or even lift off from one (or both?) rail.

While the sphere is in contact with both rails of the track, the the motion constraints are holonomic. When the sphere loses contact with one rail, the motion constraints are non-holonomic. The orientation takes place during this non-holonomic phase.

The rigid-body dynamics of this inhomogeneous sphere is described in terms of a rather complicated set of differential-algebraic equations (DAEs). Currently we are investigating techniques for solving such DAEs and generating a series of simulations for statistical analysis of the sphere's behavior.

Publications

1. Priya Narayanan, Alan Marc Lefcourt, Uri Tasch, Rouben Rostamian, Abraham Grinblat, Moon S. Kim, and Yud Ren Chen, Development of technology for orienting apples for automated on-line inspection, Transactions of the ASABE, submitted for publication: June 2006, currently under revision. Early draft.
2. Priya Narayanan, Alan M. Lefcourt, Uri Tasch, Rouben Rostamian, Abraham Grinblat, Moon S. Kim, Theoretical aspects of orienting fruit using stability properties during rotation, Proceedings of the 2006 ASAE Annual Meeting, Portland, Oregon, 9–12 July 2006. Published article (requires login).
3. Priya Narayanan, Alan M. Lefcourt, Uri Tasch, Abraham Grinblat, Rouben Rostamian, and Moon S. Kim, Theoretical Aspects of Orienting Fruit Using Stability Properties During Rotation, Slides from the Portland meeting.
4. Alan M. Lefcourt, Priya Narayanan, Uri Tasch, Rouben Rostamian, Moon S. Kim, and Yud-Ren Chen, Development of video technology to analyze dynamics of inertia-based apple orientation, in Y. R. Chen, G. E. Meyer, and S. I. Tu, editors, Proceedings of Optics East, October 1–4, 2006, Boston, volume 6381 of Optics for Natural Resources, Agriculture, and Foods, SPIE, 2006. Draft manuscript. Published article (requires login).
5. Alan M. Lefcourt, Priya Narayanan, Uri Tasch, Rouben Rostamian, Moon S. Kim, and Yud-Ren Chen, Algorithms for parameterization of dynamics of inertia-based apple orientation, Applied Engineering in Agriculture, submitted for publication: December 2006. Draft manuscript.
6. Priya Narayanan, Alan M. Lefcourt, Uri Tasch, Rouben Rostamian, Moon S. Kim, Tests of the ability to orient apples using their inertial properties, Proceedings of the 2007 ASAE Annual Meeting, Minneapolis, Minnesota, 17–20 June 2007. Draft manuscript. Published article (requires login).
7. Priya Narayanan, Alan M. Lefcourt, Uri Tasch, Rouben Rostamian, Moon S. Kim, Orientation of apples using their inertial properties, Transactions of the ASABE, submitted for publication: Dec 2007. Draft manuscript.

This web page was created on 2007–03–29 and was revised last on 2007–12–26.