|Volume 21 Issue 07 - Publication Date: 1 July 2002
|Persistent Passive Hopping and
Juggling is Possible Even With Plastic Collisions
|A. Chatterjee and R.
Pratap Mechanical Engineering Indian Institute of Science Bangalore,
India , C. K. Reddy Engineering Science and Mechanics Virginia
Polytechnic Institute and State University Blacksburg, VA, USA and A.
Ruina Theoretical and Applied Mechanics and Mechanical and Aerospace
Engineering Cornell University, Ithaca, NY, USA
|We describe simple one-dimensional
models of passive (no energy input, no control), generally dissipative,
vertical hopping and one-ball juggling. The central observation is that
internal passive system motions can conspire to eliminate collisions in
these systems. For hopping, two point masses are connected by a spring and
the lower mass has inelastic collisions with the ground. For juggling, a
lower point-mass hand is connected by a spring to the ground and an upper
point-mass ball is caught with an inelastic collision and then re-thrown
into gravitational free flight. The two systems have identical dynamics.
Despite inelastic collisions between non-zero masses, these systems have
special symmetric energy-conserving periodic motions where the collision
is at zero relative velocity. Additionally, these special periodic motions
have a non-zero sized, one-sided region of attraction on the higher-energy
side. For either very large or very small mass ratios, the one-sided region
of attraction is large. These results persist for mildly non-linear springs
and non-constant gravity. Although non-collisional damping destroys the
periodic motions, small energy injection makes the periodic motions stable,
with a two-sided region of attraction. The existence of such special energy
conserving solutions for hopping and juggling points to possibly useful
strategies for both animals and robots. The lossless motions are demonstrated
with a table-top experiment.