On the motion of lumped‐mass and distributed‐mass self‐propelling systems in a linear resistive environment

The motion of self-propelling limbless locomotion systems in a linear viscous environment is considered. The resistance (friction) force acting on an element of the systems is assumed to be proportional to the velocity of this element relative to the environment, the coefficient of proportionality (coefficient of friction) being constant. Two models of interaction of the locomotor with the environment are distinguished. In the first model, the coefficient of friction is constant for a mass element, whereas in the second model, this coefficient is constant for a length element. It is shown that progressive locomotion is impossible for the first model and is possible for the second model. This is explained by the fact that in the second model, the coefficient of friction for a mass element is in fact controlled by changing the length of this element due to deformation of the locomotor's body. The first model applies for lumped mass systems, while the second model is adequate for distributed mass limbless locomotors, like worms. For both models, the equations of motion of the system's center of mass are derived and analyzed.