Kinematic pair

A kinematic pair is a connection between two physical objects that imposes constraints on their relative movement. In other words: two links or elements of a machine, when in contact with each other, are said to form a pair; if the relative motion between them is completely or successfully constrained (i.e. in a definite direction), the pair is known as kinematic pair. Franz Reuleaux introduced the kinematic pair as a new approach to the study of machines that provided an advance over the motion of elements consisting of simple machines.

From the time of Archimedes through the Renaissance, mechanisms were viewed as constructed from simple machines, such as the lever, pulley, screw, wheel and axle, wedge, and inclined plane. Reuleaux focused on bodies, called links, and the connections between these bodies called kinematic pairs, or joints. To use geometry to study the movement of a mechanism, its links are modeled as rigid bodies. This means that distances between points in a link are assumed to not change as the mechanism moves that is, the link does not flex. Thus, the relative movement between points in two connected links is considered to result from the kinematic pair that joins them.

Types of pairs

Kinematic pairs, or joints, are considered to provide ideal constraints between two links, such as the constraint of a single point for pure rotation, or the constraint of a line for pure sliding, as well as pure rolling without slipping and point contact with slipping. A mechanism is modeled as an assembly of rigid links and kinematic pairs. Kinematic pairs can be classified according to the nature of the contact, nature of mechanical constraint and, the nature of relative motion. The kinematic pairs are often distinguished in shape-closing pairs (or self closing pairs) and force-closing pairs according to whether or not their contact is ensured by the shape of the kinematic elements or through external means such as springs.

• Self closed pair. When the two elements of a pair are connected together mechanically in such a way that only required kind of relative motion occurs, it is then known as self closed pair. The lower pairs are self closed pair.
• Forced closed pair. When the two elements of a pair are not connected mechanically but are kept in contact by the action of external forces, the pair is said to be a force-closed pair. The cam and follower is an example of force closed pair, as it is kept in contact by the forces exerted by spring and gravity.

Reuleaux called the ideal connections between links kinematic pairs. He distinguished between higher pairs with line contact between the two links and lower pairs with area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple model.

Lower pair

A lower pair is an ideal joint that has surface contact between the pair of elements (examples: nut and screw, shaft rotating in bearing, all pairs of slider-crank mechanism, universal joint, etc.), as in the following cases:

• A revolute pair, or hinged joint, requires a line in the moving body to remain co-linear with a line in the fixed body, and a plane perpendicular to this line in the moving body must maintain contact with a similar perpendicular plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom.
• A sliding pair, or prismatic joint, requires that a line in the moving body remain co-linear with a line in the fixed body, and a plane parallel to this line in the moving body must maintain contact with a similar parallel plane in the fixed body (the two elements of a pair are connected in such a way that one can only slide relative to the other). This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom.
• A cylindrical joint requires that a line in the moving body remain co-linear with a line in the fixed body. It combines a revolute joint and a sliding joint. This joint has two degrees of freedom.
• A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body (the two elements of a pair are connected in such a way that one element (with spherical shape) turns or swivels about the other fixed element). This joint has three degrees of freedom.
• A planar joint requires that a plane in the moving body maintain contact with a plane in fixed body. This joint has three degrees of freedom.
• A screw joint, or helical joint, has only one degree of freedom because the sliding and rotational motions are related by the helix angle of the thread (the two elements of a pair are connected in such a way that one element turns or swivels about the other fixed element).

Higher pairs

Generally, a higher pair is a constraint that requires a line or point contact between the elemental surfaces. For example, the contact between a cam and its follower is a higher pair called a cam joint. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.

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