Motion Space Analysis of Smooth Objects in Point Contacts

The present work studies instantaneous motion of smooth planar and spatial objects in unilateral point contacts. The traditional first-order instantaneous kinematic analysis is found insufficient to explain many common physical scenarios. The present work looks beyond the velocity state of motion for a comprehensive understanding through higher-order kinematic analysis of the above system. The methodology proposed herein is a Euclidean space approach to second-order motion space analysis of objects in point contacts. The geometries of the objects are approximated up to second-order in the differential vicinity of the point of contact; meaning, up to curvature at the point of contact. The instantaneous motion is approximated up to second-order kinematics, i.e., up to acceleration state. The basic approach consists of impressing an instantaneous motion upon one object while holding the other fixed which is in a single point contact initially, and observing for one of the following three states: penetration, separation, and persistence of contact between the two objects. These three states are characterized by the interference between the geometries of the objects. Penetration and separation of two curves for rotation about points on the plane is geometrically studied based on the relative configuration of the osculating circles at the point of contact. It is shown that the plane is partitioned into four regions of rotation centers. Partitioning of the plane into motion space regions at a contact provided a geometrical framework compose the motion space for multiple contacts. The applications include second-order form-closure (SFC) and synthesis of kinematic pairs. To explore the consequence of a generic motion, an analytical scheme is formulated using the screw theoretic concepts of twist and twist-derivative. It is shown that the characteristics of second-order motions at a single contact depends only upon the geometric kinematic properties of the motion; meaning, the motion characteristics are time-independent. The geometric conditions for the second-order motion that will be admissible or restrained at a contact are not available in the existing literature on \second-order mobility. The classical Euler-Savary equation for enveloping curves is found to represent the condition which is both necessary and sufficient for the second-order roll-slide motion. An elegant generalized geometric characterization of second-order motions is derived. This is made use for deriving condition of immobilization of, planar mechanisms with up to 2-degrees-of-freedom (d.o.f.), with a single point contact. Illustrative examples of four-bar and 2R-mechanisms are presented. Rapid prototyped model of the four-bar mechanism is fabricated and the SFC theory is verified satisfactorily. Through a novel use of Meusnier's theorem, rotational motion characteristics of planar curves in a point contact is used to determine the patterns and distribution of admissible axes of rotation in space for two surfaces in a single point contact. In the…