Mathematical Model of Worm-like Motion Systems with Finite and Infinite Degree of Freedom
This paper presents some theoretical and practical investigations of worm-like motion systems that have the earthworm as live prototype. In the first part of the paper these systems are modeled in form of straight chains of n ≥ 1 interconnected mass point
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1 Faculty of Mechanical Engineering, Technische Universitlit llmenau, Germany Faculty of Mathematics and Natural Sciences, Technische Universitlit Ilmenau, Germany
Abstract This paper presents some theoretical and practical investigations of worm-like motion systems that have the earthworm as live prototype. In the first part of the paper these systems are modeled in form of straight chains of n ~ 1 interconnected mass points. The ground contact can be described either by non-symmetric dry friction or by unilateral differential constraints. The second part the paper deals with the peristaltic movement of a body due to a wavelike disturbance of the boundary surface. The investigations concentrate on motion in a tube or channel, and on motion on a horizontal plane as well. In both cases the body is modeled as a viscous Newton fluid. The dependence of the massflow through a cross section on disturbance and material data (viscosity, dimensions) is discussed. The paper presents first prototypes of technically implemented artificial worms. 1
Introduction
Locomotion systems following biologically inspired ideas are currently dominated by walking machines, i.e., pedal locomotion. All biological models from bipedal to octopedal constructions were transferred by engineers in different forms of realization, i.e., from commercial systems in series up to a small number of prototypes for research. Non-pedal forms oflocomotion show their advantages in inspection techniques or application in medical technology for diagnostic systems and minimally invasive surgery. Observing the locomotion of worms one recognizes a conversion of (mostly periodic) internal and internally driven motions into change of external position (undulatory locomotion). Realization of this type of locomotion requires non-symmetry in the external friction forces.
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Systems with Finite Degree of Freedom
2.1 Motion under the Action of Nonlinear Friction and Periodic External Force The motion of a chain of equal masses m placed on a rough straight line and connected consecutively by equal linear viscoelastic elements under the action of a non-symmetric Coulomb dry frictional force is described in the article of Zimmermann et al., 2001. Now consider we the motion of three masses in a common straight line. We suppose two of them to be equipped with scales contacting the ground. One mass is under the action of a harmonic external force - M
Cf> 0 sin .Q t (Figure 1). The equations of motion then are:
G. Bianchi et al. (eds.), Romansy 14 © Springer-Verlag Wien 2002
K. Zimmermann, I. Zeidis and J. Steigenberger
508
(1)
Here OJ
2
= mc ,
2
OJ0
= Mc ,
0 > 0 . The function F ( X5 ) , describes a non-symmetric dry
friction, i.e., the frictional force is taken to be different in magnitude depending on the direction of body motion. F( X5 ) may be specified as follows:
F+, x>O F(x) ={
F0 ,
- F_, where
F_ > F+
~
x=O,
x
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