Classification of kinematic pairs, Kinematic chains - Applied Mechanics

Classification of kinematic pairs

One of the main characteristic properties of kinematic pairs is the number of the simplest relative motions, which are deprived of links of the mechanism when they are connected to kinematic pairs. Let us explain this with an example. It is known that a free solid has six degrees of freedom. Arbitrary displacement of it in space can be represented as the result of the addition of six independent motions: three translational axes Ox, Oy, Oz parallel to the axes of coordinates and three rotational ones around axes parallel to these axes (Figure 1.2). Depending on the type of connections of the links of the mechanism, one of them can perform relative to the other one, two, three, four or five movements of the six listed above. Consequently, the kinematic pairs impose certain restrictions on the relative movements of the links, which depend on the way they are connected. These restrictions are called links . The number S of links (geometric) constraining the relative motion of links is determined by the equality S = 6 - IT, where W - the number of degrees of freedom of the links forming kinematic pair.

Academician II Artobolevsky introduced a classification of kinematic pairs, according to which all pairs of de

Fig. 1.2

are divided into five classes depending on the number S. The separation of kinematic pairs by classes is presented in Table. 1.1. The arrows indicate the possible movements of the links, which are preserved after the formation of the pair. For each class, the number of degrees of freedom W and the number of geometric constraints S are given. Conditional images of kinematic pairs of different classes are given.

Table 1.1

Schematic representation of kinematic pairs

Conditional

image

S

W

The kinematic pair class

1

5

1

2

4

2

3

3

3

3

3

3

4

2

4

4

2

4

5

1

5

5

1

5

5

1

5

Kinematic pairs are divided into lower and higher, depending on the type of their constituent elements. The lower kinematic pairs, whose elements are surfaces, include pairs of translational, rotational, spherical, helical and planar (see Table 1.1). Points and lines are elements of higher kinematic pairs. The higher kinematic pairs are the pairs "ball in the plane" and cylinder on the plane (see Table 1.1). The advantages of lower pairs are their ability to transmit considerable effort with less wear compared with higher pairs; for higher pairs - the ability to reproduce with their help rather complex relative movements.

Kinematic chains

The kinematic chain - is the connected system of links forming kinematic pairs. Kinematic chains can be divided into flat and spatial, simple and complex, closed and nonclosed (Fig. 1.3). K simple are chains in which each link enters no more than two kinematic pairs (Figure 1.3, a, b, d), to complex - chains, which have links that enter into three or more kinematic pairs (Figure 1.3, c); to closed - chains, for which each link enters at least two kinematic pairs (Figure 1.3, b- r), to unclosed - chains that have links that enter only one kinematic pair (Figure 1.3, a). All the moving links of a plane kinematic chain make motions parallel to the same fixed plane (see Fig. 1.1). In spatial kinematic chains, the points of the links describe spatial curves or move along planar curves lying in intersecting planes (Figure 1.4).

Introducing the concept of a kinematic chain, we can give a different definition for mechanisms composed only of solids. Mechanism is a kinematic chain in which for one fixed link ( the rack) and the given motion of one or more links ( slave ) perform uniquely defined motions. Mechanisms can be formed by both closed and non-closed kinematic chains. An example of an unclosed kinematic chain can be the mechanism of an elementary manipulator (Figure 1.5).

Fig. 1.3

Fig. 1.4

Fig. 1.5

Most mechanisms are formed by closed kinematic circuits (see Figures 1.1, 1.4).

In the study of mechanisms, their conditional images are used, structural, kinematic and other schemes are compiled. Structural diagrams are executed in the form of a drawing, in which, with the conventional symbols set by GOST, the links, kinematic pairs are shown, indicate the rack and the leading links (see Figure 1.1). Structural diagrams executed on a specific scale are called kinematic schemes .

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