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Spatial mechanisms, 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 which contains a screw pair, for example, is a spatial mechanism, since the relative motion within a screw pair is helical.
Thus, the overwhelming 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, which 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 represented graphically in a single view. Thus, graphical techniques are well suited to their solution. Since spatial mechanisms do not all have this fortunate geometry, visualization becomes more difficult and more powerful techniques must be developed for their analysis.
Since the vast majority of mechanisms in use today are planar, one might question the need of the more complicated mathematical techniques used for spatial mechanisms. There are a number of reasons why more powerful methods are of value even though the simpler graphical techniques have been mastered.
1. They provide new, alternative methods, which will solve the problems in a different way. Thus they provide a means of checking results. Certain problems by their nature may also be more amenable to one method than another.
2. Methods which are analytical in nature are better suited to solution by calculator or digital computer than graphical techniques.
3. Even though the majority of useful mechanisms are planar and well suited to graphical solution, the few remaining must also be analyzed, and techniques should be known for analyzing 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 quite 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 only-be approximately parallel. If they are far out of parallel, there will be binding in no uncertain terms, and the mechanism will only move 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 no parallelism is to connect the links with self-aligning bearings, actually spherical joints allowing three-dimensional rotation. Such a "planar" linkage is thus a low-grade spatial linkage.
Degrees of Freedom
A three-bar linkage (containing three bars linked together) is obviously a rigid frame; no relative motion between the links is possible. To describe the relative positions of the links in a four-bar linkage it is necessary only to know the angle between any two of the links. This linkage is said to have one degree of freedom. Two angles are required to specify the relative positions of the links in a five-bar linkage; it has two degrees of freedom. Linkages with one degree of freedom have constrained motion; i. e., all points on all of the links have paths on the other links that are fixed and determinate. The paths are most easily obtained or visualized by assuming that, the link on which the paths are required is fixed, and then moving the other links in a manner compatible with the constraints.