Rigid body nonholonomic systems serve as models for locomotion of several terrestrial animals such as snakes as well as for fish-like swimming motion. Several well known nonholonomic systems have also found applications in the field of mobile robotics in everything from wheeled vehicles to articulated snake like robots. However, one aspect of their dynamics has remained unexplored. This is to do with the effects of increasing the degrees of freedom by adding additional ‘segments’ such as in a chain, with the joints between segments having a nonzero torsional stiffness. Such nonholonomic systems when subjected to periodic actuation or inputs have additional modes of oscillation. The interplay of the nonholonomic constraints, linear elastic potentials and additional degrees of freedom can produce rich frequency-amplitude response in the dynamics of the system and can lead to significantly higher speed and efficiency. In this paper we explore such dynamics with the example of a well known nonholonomic system, the Chaplygin sleigh and a variant of it with an additional degree of freedom. Such models can be expected to better match the dynamics of biological swimmers and have widespread applications for soft and under-actuated robots.