Model of Snake Locomotion by Frictional Forces
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In this Demonstration, a curve of fixed length represents a snake as it moves in one direction in a plane. As time increases, the curve advances forward, with plotted vectors representing frictional forces exerted by the snake on the surface at various points along the snake's length. It is shown how different selected materials affect a milk snake's locomotion. These force vectors are calculated using the proposed equation from David Hu's study .
Contributed by: Christopher Umeki and Gregor Wettermann (July 2015)
After work by: David L. Hu, Jasmine Nirody, Terri Scott, and Michael J. Shelley
Special thanks to the University of Illinois NetMath Program and the mathematics department at William Fremd High School
Open content licensed under CC BY-NC-SA
Snapshot 1: initial state with the tail of the snake to the left of the origin and the surface set to cloth
Snapshot 2: tail of snake at meters and the surface set to cloth
Snapshot 3: tail of snake at meters and the surface set to smooth fiberboard
The frictional forces that a milk snake exerts on a surface as its center of mass moves at a constant velocity are represented by vectors. It is theorized that the frictional forces at a point are given by the formula:
The variables above are defined as:
: mass per unit length
: gravitational acceleration 9.8
: coefficient of forward kinetic friction, for when the scales are oriented parallel to the direction of sliding (unitless)
: coefficient of tangential kinetic friction, for when the scales are oriented perpendicular to the direction of sliding (unitless)
: coefficient of backward kinetic friction, for when the scales are oriented antiparallel to the direction of sliding (unitless)
: the unit vector of the left-footed normal to the body at a given point along the snake (unitless)
The function is defined by .
The Demonstration is based on this proposed formula. The vectors point in the opposite direction to the snake's overall velocity because the force of friction between a snake's scales and the surface propels that part of the snake in the opposite direction, according to Newton's third law.
The control "surface material" changes the forward, backward, and lateral coefficients for a cloth surface and those for a smooth fiberboard. With cloth, the frictional forces are larger because the coefficient of friction between the snake's skin and cloth is greater than the coefficient of friction between the snake's skin and smooth wood. It can be noted that the lateral components of friction diminish greatly on the smooth surface, because the coefficient of friction on a smooth surface is the same regardless of the snake's orientation, while on a rough surface the lateral frictional coefficient is much greater than the forward and backward coefficients.
 D. L. Hu, J. Nirody, T. Scott, and M. J. Shelley, "The Mechanics of Slithering Locomotion," Proceedings of the National Academy of Sciences, 106(25), 2009 10081–10085. doi:10.1073/pnas.0812533106.
 New York University. "Snakes Use Friction and Redistribution of Their Weight to Slither on Flat Terrain." www.nyu.edu/about/news-publications/news/2009/06/08/snakes_use_friction_and.html.