Planar, Spherical, And Spatial Mechanisms

Mechanisms may be categorized in several different ways to emphasize their similarities and differences. One such grouping divides mechanisms into planar, spherical, and spatial categories. All three groups have many things in common; the criterion that distinguishes the groups, however, is to be found in the characteristics of the motions of the links.

A planar mechanism is one in which all particles describe plane curves in space and all these curves lie in parallel planes; that is, the loci of all points are plane curves parallel to a single common plane. This characteristic makes it possible to represent the locus of any chosen point of a planar mechanism in its true size and shape on a single drawing or figure. The motion transformation of any such mechanism is called coplanar. The plane four-bar linkage, the plate cam and follower, and the slider-crank mechanism are familiar examples of planar mechanisms. The vast majority of mechanisms in use today are planar.

Planar mechanisms utilizing only lower pairs are called planar linkages; they include only revolute and prismatic pairs. Although a planar pair might theoretically be included, this would impose no constraint and thus be equivalent to an opening in the kinematic chain. Planar motion also requires that all revolute axes be normal to the plane of motion and that all prismatic pair axes be parallel to the plane.

A ~pherical mechanism is one in which each link has some point that remains stationary as the linkage moves and in which the stationary points of all links lie at a common location; that is, the locus of each point is a curve contained in a spherical surface, and the spherical surfaces defined by several arbitrarily chosen points are all concentric. The motions of all particles can therefore be completely described by their radial projections, or "shadows," on the surface of a sphere with a properly chosen centre. Hooke's universal joint is perhaps the most familiar example of a spherical mechanism.

Spherical linkages are constituted entirely of revolute pairs. A spheric pair would produce no additional constraints and would thus be equivalent to an opening in the chain, while all other lower pairs have non spheric motion. In spheric linkages, the axes of all revolute pairs must intersect at a point. Spatial mechanisms, on the other hand, include no restrictions on the relative motions of the particles. The motion transformation is not necessarily coplanar, nor must it be concentric. A spatial mechanism may have particles with loci of double curvature. Any linkage that contains a screw pair, for example, is a spatial mechanism, because the relative motion within a screw pair is helical.

Thus, the overwhelmingly large category of planar mechanisms and the category of spherical mechanisms are only special cases, or subsets, of the all-inclusive category spatial mechanisms. They occur as a consequence of special geometry in the particular orientations of their pair axes.

If planar and spherical mechanisms are only special cases of spatial mechanisms, why is it desirable to identify them separately? Because of the particular geometric conditions that identify these types, many simplifications are possible in their design and analysis. As pointed out earlier, it is possible to observe the motions of all particles of a planar mechanism in true size and shape from a single direction. In other words, all motions can be rep-. resented graphically in a single view. Thus, graphical techniques are well-suited to their solution.

Because spatial mechanisms do not all have this fortunate geometry, visualization becomes more difficult and more powerful techniques must be developed for their analysis. Because the vast majority of mechanisms in use today are planar, one might question the need for the more complicated mathematical techniques used for spatial mechanisms.

There are a number of reasons why more powerful methods are of value even though tJ;1e simpler graphical techniques have been mastered: "

1.      They provide new, alternative methods that will solve the problems in a differeJllt way. Thus they provide a means of checking results. Certain problems, by their nature, may also be more amenable to one method than to another.

2.      Methods that are analytic in nature are better suited to solution by calculator or digital computer than by graphic techniques.

3.      Even though the majority of useful mechanisms are planar and well-suited to graphical solution, the few remaining must also be analysed, and techniques should be known for analysing them.

4.      One reason that planar linkages are so common is that good methods of analysis for the more general spatial linkages have not been available until relatively recently. Without methods for their analysis, their design and use has not been common, even though they may be inherently better suited in certain applications.

5.      We will discover that spatial linkages are much more common in practice than their formal description indicates.

Consider a four-bar linkage. It has four links connected by four pins whose axes are parallel. This "parallelism" is a mathematical hypothesis; it is not a reality. The axes as produced in a shop-in any shop, no matter how good-will be only approximately parallel. If they are far out of parallel, there will be binding in no uncertain terms, and the mechanism will move only because the "rigid" links flex and twist, producing loads in the bearings. If the axes are nearly parallel, the mechanism operates because of the looseness of the running fits of the bearings or flexibility of the links. A common way of compensating for small nonparallelism is to connect the links with self-aligning bearings, which are actually spherical joints allowing three-dimensional rotation. Such a "planar" linkage is thus a low-grade spatial linkage.