Classical and Modern Approaches in the Theory of Mechanisms

By Nicolae Pandrea, Dinel Popa and Nicolae-Doru Stanescu

Classical and Modern Approaches in the Theory of Mechanisms is a study of mechanisms in the broadest sense, covering the theoretical background of mechanisms, their structures and components, the planar and spatial analysis of mechanisms, motion transmission, and technical approaches to kinematics, mechanical systems, and machine dynamics. In addition to classical approaches, the book presents two new methods: the analytic-assisted method using Turbo Pascal calculation programs, and the graphic-assisted method, outlining the steps required for the development of graphic constructions using AutoCAD; the applications of these methods are illustrated with examples. Aimed at students of mechanical engineering, and engineers designing and developing mechanisms in their own fields, this book provides a useful overview of classical theories, and modern approaches to the practical and creative application of mechanisms, in seeking solutions to increasingly complex problems.

• Language: English
• Publisher: Wiley
• Release date: Mar 24, 2017
• ISBN: 9781119221760

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Chapter 1

The Structure of Mechanisms

The structure of a mechanism is its composition (elements, kinematic pairs), its classification, and the conditions in which motion is transmitted in a predeterminate way (the conditions of desmotomy). The variety and the complexity of technical systems gives a requirement for conventional representations of the elements of mechanisms and the linkages between them (kinematic pairs), and the establishment of certain rules for their representation, formation and decomposition.

1.1 Kinematic Elements

The component parts of mechanisms that have motions relative to one another are called thekinematic elements of the mechanisms. The elements may consist of one or more component parts. Figure 1.1a,b shows two cranks, the first consisting of one piece, and the second one consisting of more (the body, the cap, the screws and so on). The conventional representation of these elements is given in Figure 1.1c,d.

Schematic for Two elements and their conventional representations: (a) single-piece crank and (c) its conventional representation; (b) multi-piece crank and (d) its conventional representation.

Figure 1.1 Two elements and their conventional representations: (a) single-piece crank and (c) its conventional representation; (b) multi-piece crank and (d) its conventional representation.

The rank j of an element is the number of kinematic pairs of that element. The conventional representation of the elements as a function of their rank is shown in Table 1.1.

Table 1.1 Rank of elements

1.2 Kinematic Pairs

The direct and permanent link between two elements is called a kinematic pair. The class of the kinematic pair represents the number of restrictions (the non-permitted motions) of an element in relative motion with respect to another one. For example, if one considers elements 1 and 2, which are supported by one another (Figure 1.2), then it can be seen that element 2 has only one restriction on its relative displacement with respect to element 1 ( c01-math-0001 ).

Schematic Representation of two elements.

Figure 1.2 Representation of two elements.

Denoting the number of restrictions by m and the number of the degrees of freedom in relative motion by l, it can be seen, in the case considered, that c01-math-0002 , c01-math-0003 , and the kinematic pair has a class equal to one. In general:

1.1 equation

The classification of the kinematic pairs is according to the criteria in Table 1.2 (see also Table 1.3).

Table 1.2 The classification of kinematic pairs

Table 1.3 Classes of kinematic pairs

1.3 Kinematic Chains

The system of elements jointed by kinematic links is called a kinematic chain. Figure 1.3a,b shows two planar kinematic chains, while Figure 1.3c shows a spatial kinematic chain. Kinematic chains are classified by taking into account the rank of the elements, the shape of the chain, and the motion of the elements. This classification, together with some examples, is presented in Table 1.4.

Schematic for Kinematic chains: (a) planar kinematic chain with three elements; (b) planar kinematic with six elements; (c) spatial kinematic chain with four elements.

Figure 1.3 Kinematic chains: (a) planar kinematic chain with three elements; (b) planar kinematic with six elements; (c) spatial kinematic chain with four elements.

Table 1.4 Classification of kinematic chains

The degree of freedom of the kinematic chain is denoted L. Consider a kinematic chain having c elements and c01-math-0009 kinematic links of class m. In the absence of any common restriction, the number of restrictions introduced by the kinematic links is equal to c01-math-0010 .

If the elements are not jointed to one another, they will have a total number of c01-math-0011 degrees of freedom. The degree of freedom L of the kinematic chain is therefore given by:

1.2 equation

The degree of mobility of the kinematic chain M is defined as the degree of freedom of a kinematic chain having a fixed element. It has the relation:

1.3 equation

or, if we denote by c01-math-0040 the number of the elements considered mobile, we get, for the degree of mobility, the expression

1.4 equation

1.4 Mobility of Mechanisms

1.4.1 Definitions

The mechanism is the kinematic chain having one fixed element (the base, the frame) and in which all the elements have predeterminate (desmotomic) motions. According to their functional role, the component elements of the mechanism can be:

driving (motor, input); the elements that receive the motion from outside the mechanism

driven (commanded, output); the elements whose motion depends on the motion of the driving elements.

Representations of the mechanisms and the kinematic chains are called the representation schema. We define:

constructive schemata, in which the representations of the kinematic links and elements contain some structural details

kinematic schemata, in which the elements and kinematic links have conventional representations

structural schemata, which give the fundamental structural mechanisms of the mechanism.

1.4.2 Mobility Degree of Mechanisms without Common Constraints

The mobility degree of a mechanism is the degree of mobility of the respective kinematic chain; that is,

1.5 equation

where n is the number of the mobile elements of the mechanism.

1.4.3 Mobility Degree of Mechanisms with Common Constraints

If the elements of the mechanism have f common restrictions, we say that the mechanism is of family f. For instance, a planar mechanism, where each element has only three degrees of freedom (two translation and one rotation), is of family c01-math-0045 , because all the elements have three common restrictions (two rotations about the axes contained in the plane and one translation along the axis perpendicular to the plane).

Returning to the general case, if the elements have f restrictions, then the number of the degrees of freedom is equal to c01-math-0046 for each element, while a kinematic joint of class m has only c01-math-0047 restrictions, where c01-math-0048 .

To determine the mobility, 6 has to be replaced with c01-math-0049 , m with c01-math-0050 , and M with c01-math-0051 in (1.5). This gives the Dobrovolski formula:

1.6

equation

that is,

1.7

equation

1.8

equation

1.9

equation

1.10 equation

1.11 equation

As already described, planar mechanisms have family c01-math-0058 . Their degree of mobility is given by relation (1.10).

1.4.4 Mobility of a Mechanism Written with the Aid of the Number of Loops

To determine the mobility of a mechanism, we take into account the fact that a mechanism with j loops is obtained from a mechanism with c01-math-0059 loops, to which we add an open kinematic chain with c01-math-0060 elements and c01-math-0061 kinematic pairs.

In Figure 1.4 we have shown how a mechanism with three loops can be obtained by adding an open kinematic chain with three elements and four kinematic pairs to a mechanism with two loops. For the first loop, taking into account that it has a fixed element, the expression c01-math-0105 is obtained. This then results in

1.12

equation

where N is the number of loops.

Schematic for Obtaining a mechanism with three loops from a mechanism with two loops.

Figure 1.4 Obtaining a mechanism with three loops from a mechanism with two loops.

Denoting by c the total number of kinematic links and by n the number of elements, summing (1.12) gives:

1.13 equation

Developing expression (1.6) gives:

1.14

equation

Taking into account (1.13) gives:

1.15 equation

1.4.5 Families of Mechanisms

We defined the family f of a mechanism as the number of the restrictions (constraints) common to all elements. If a mechanism is formed with loops of different families, then, conventionally, we define the apparent family by the relation

1.16 equation

where c01-math-0111 marks the family of the mechanism i.

For a series of mechanisms, the family may be determined based on the analysis of the common constraints for all the elements in regard to a dextrosum reference system. Table 1.5 shows how the family can be established for five types of simple mechanism by analyzing the common restrictions.

Table 1.5 The determination of the families of the kinematic chains

1.4.6 Actuation of Mechanisms

To assure desmotomy, it is necessary that the number of the acted (motor, driving) elements is equal to the degree of mobility M of the mechanism. If the driving elements are defined, the mechanism is called a driving (motor) mechanism. The driving elements, in most cases, are linked to a base or frame, but there are also driving elements that are not linked to a base or frame. Figure 1.5 shows different types of driving mechanism and how they are actuated. In Figure 1.5a,b, the driving elements, denoted by 1, are adjacent to the base, while in Figure 1.5c,d the driving elements, denoted by 2, are not adjacent to the frame.

Schematic for Actuation of mechanisms: in (a) and (b) the driving element is adjacent to the base.

Figure 1.5 Actuation of mechanisms: in (a) and (b) the driving element is adjacent to the base; in (b) and (d) the driving element is not adjacent to the base.

1.4.7 Passive Elements

There are situations in which, due to the geometric particularities of certain mechanisms, some elements can be eliminated, or others added, but the real mobility of the mechanism remains unchanged. So, for instance, in the example in Figure 1.6, in which c01-math-0112 , c01-math-0113 , and c01-math-0114 , the loops ODEC, OABC are parallelograms and the element denoted by 4 can be eliminated without changing the mobility of the mechanism. Such an element is called a passive element. For the calculation of the mobility, the passive elements must be eliminated.

Schematic for Passive element.

Figure 1.6 Passive element.

1.4.8 Passive Kinematic Pairs

In some cases, certain kinematic pairs have a passive role: they can be eliminated and the mobility of the mechanism remains unchanged. These links are called passive kinematic pairs. Figure 1.7 shows a spherical mechanism (with a Cardan link) of three elements, of family equal to three. The axes being concurrent at O (Figure 1.7b), the elements can have only rotational motions. The rotational kinematic pairs c01-math-0115 and c01-math-0116 are passive kinematic pairs. If they are eliminated, the structural variant in Figure 1.7c and the kinematic variant in Figure 1.7d, respectively, are obtained. For calculation of the mobility the kinematic schema in Figure 1.7d has to be used.

Schematic for Passive kinematic links.

Figure 1.7 Passive kinematic links: (a) constructive schema of Cardan linkage; (b) kinematic schema of spherical mechanism; (c) kinematic schema of Cardan linkage; (d) kinematic schema of spherical mechanism without passive linkages.

1.4.9 Redundant Degree of Mobility

Figure 1.8a shows the constructive schema of a spatial RSSR mechanism of family c01-math-0117 , while Figure 1.8b is its kinematic schema.

Illustration for Redundant degree of mobility: (a) constructive schema of the RSSR mechanism and (b) kinematic schema of the same mechanism.

Figure 1.8 Redundant degree of mobility: (a) constructive schema of the RSSR mechanism and (b) kinematic schema of the same mechanism.

In this mechanism, the motion is transmitted in a determined way from the element denoted by 1 to the element denoted by 3. In this case, the degree of mobility is given by the expression c01-math-0118 ; since c01-math-0119 (the pairs at the points O and C) and c01-math-0120 (the pairs at the points A and B), this gives c01-math-0121 .

One of the two degrees of mobility is redundant; that is, the rotation of crank 2 around axis AB. In the calculation of the real degree of mobility, any redundant degrees of mobility have to be eliminated.

1.4.10 Multiple Kinematic Pairs

Consider the planar mechanism in Figure 1.9. For the calculation of the mobility of this mechanism, one has to take into account that at point B there are two fifth-class kinematic pairs. Generally, the number of kinematic pairs is one less than the number of elements concurrent at the same point.

Schematic for Multiple kinematic pair (at point B).

Figure 1.9 Multiple kinematic pair (at point B).

A pair such as that presented in Figure 1.9 is called a multiple kinematic pair, and it has to be taken into account when calculating the mobility. In the present case, the number of elements of the mechanism is c01-math-0122 and the number of class 5 kinematic pairs is c01-math-0123 , so the mobility c01-math-0124 .

1.5 Fundamental Kinematic Chains

By definition, general kinematic chains contain kinematic pairs of different classes, while fundamental kinematic chains contain only kinematic pairs of the fifth class; that is, R, T, H. For the equivalence of a general kinematic chain with a fundamental kinematic chain the theorem of the equivalence of the kinematic pair developed by Gruebler and Harisberger must be used. This states that a kinematic pair of class k in a kinematic chain of any family can be replaced with a simple open kinematic chain consisting of c01-math-0125 elements jointed to one another by c01-math-0126 fundamental kinematic pairs (of class 5).

To demonstrate this theorem, we equate the degree of mobility of the kinematic chain consisting of zero elements and one kinematic pair of class k with the degree of mobility of the replacing kinematic chain consisting of n elements and c kinematic pairs of class 5. Applying the Dobrovolski formula gives:

equation

Knowing that for a kinematic chain with one loop c01-math-0128 , this gives:

equation

or

1.17 equation

In particular, a kinematic pair of the fourth class is equivalent to a kinematic chain consisting of an element and two kinematic pairs of the fifth class. Such a situation is presented by the planar mechanism in Figure 1.10, in which the fourth-class kinematic pair at point A is replaced by the element CD and the fifth-class kinematic pairs at the points C and D, finally obtaining the mechanism OCDB; from the structural point of view, this mechanism is equivalent to the mechanism OAB. If the kinematic equivalence is also required, then the points D and C are situated at the curvature centres of the contact curves.

Schematic for The equivalence of a class 4 kinematic pair.

Figure 1.10 The equivalence of a class 4 kinematic pair.

For the fundamental kinematic chains, the degree of mobility is given by

equation

replacing c01-math-0132 gives:

1.18 equation

In the case of a spatial fundamental chain with one loop and with the degree of mobility c01-math-0134 , c01-math-0135 and the fundamental kinematic chains in Table 1.6 are obtained.

Table 1.6 Fundamental kinematic chains ( c01-math-0136 , c01-math-0137 )

Illustration for The 7R mechanism.

Figure 1.11 The 7R mechanism.

Illustration for The Franke mechanism.

Figure 1.12 The Franke mechanism.

Illustration for The Bricard mechanism.

Figure 1.13 The Bricard mechanism.

Illustration for The Goldberg mechanism.

Figure 1.14 The Goldberg mechanism.

Illustration for The Bennett mechanism.

Figure 1.15 The Bennett mechanism.

The planar fundamental kinematic chains ( c01-math-0140 ) have the degree of mobility given by:

1.19 equation

Planar fundamental mechanisms with mobility c01-math-0142 have the numerical solutions:

1.20

equation

This gives for c01-math-0144 the fundamental four-bar (quadrilateral) kinematic chain (Figure 1.16a; c01-math-0145 , c01-math-0146 ) and for c01-math-0147 the fundamental kinematic chains of Watt (Figure 1.16b) and Stephenson (Figure 1.16c).

Illustration for Planar fundamental kinematic chains (M = 1): (a) fundamental four-bar kinematic chain; (b) fundamental Watt kinematic chain; (c) fundamental Stephenson kinematic chain.

Figure 1.16 Planar fundamental kinematic chains ( c01-math-0138 ): (a) fundamental four-bar kinematic chain; (b) fundamental Watt kinematic chain; (c) fundamental Stephenson kinematic chain.

For c01-math-0148 :

1.21

equation

resulting in a fundamental kinematic chain with five elements (Figure 1.17a) and three fundamental kinematic chains with seven elements (Figure 1.17b–d).

Illustration for Planar fundamental kinematic chains (M = 2): (a) fundamental kinematic chain with five elements; (b), (c) and (d) fundamental kinematic chains with seven elements.

Figure 1.17 Planar fundamental kinematic chains ( c01-math-0139 ): (a) fundamental kinematic chain with five elements; (b), (c) and (d) fundamental kinematic chains with seven elements.

1.6 Multi-pairs (Poly-pairs)

The open, simple kinematic chain, that is equivalent, from the point of view of mobility, to a kinematic pair of class k is called a multi-pair or a poly-pair of class k. Multi-pairs are used in system design for constructive and technological reasons, because, on the one hand, not all independent motions can be realized by direct (simple or complex) contacts, and, on the other hand, because of the tendency to replace superior kinematic pairs with kinematic chains formed from inferior kinematic pairs (lower pairs) of increased mobility. Since c01-math-0150 , by equating the mobilities c01-math-0151 or

1.22

equation

Based on this relation, in Table 1.7 we give a few examples of simple superior kinematic pairs replaced by multi-pairs. Important categories of multi-pairs are couplings and especially homokinetic couplings.

Table 1.7 The equivalence between the simple kinematic links and the multi-links

1.7 Modular Groups

The modular groups are obtained from the fundamental kinematic chains with mobility M by eliminating an element. For c01-math-0161 this gives the passive (kinematic) modular groups, while for c01-math-0162 it gives the driving (motor) modular groups. For planar mechanisms, the numerical relations used to obtain the modular groups are: c01-math-0163 and c01-math-0164 .

The important passive modular groups are:

1.Dyad ( c01-math-0165 , c01-math-0166 , c01-math-0167 ; Figure 1.18b), which is obtained from the fundamental zero-mobile kinematic chain with three elements (see Figure 1.18a). The dyad has five aspects: RRR, RTR, RRT, TRT, RTT (Figure 1.19).

2.Triad ( c01-math-0168 , c01-math-0169 ; Figure 1.20b), which is obtained from the zero-mobile kinematic chain (Figure 1.20a) by eliminating an element of rank c01-math-0170 .

3.Tetrad ( c01-math-0171 , c01-math-0172 ; Figure 1.20c), which is obtained from the same kinematic chain (Figure 1.20a) by eliminating an element of rank c01-math-0173 .

The important driving (motor) modular groups are:

1. the driving element: c01-math-0174 , c01-math-0175 , c01-math-0176 , c01-math-0177 ; see Figure 1.21a

2. the driving dyad: c01-math-0179 , c01-math-0180 , c01-math-0181 , c01-math-0182 ; see Figure 1.21b.

The kinematic pairs of the modular groups can be

exterior: A, B, C (Figure 1.20b); D, C (Figure 1.20c)

interior: D, E, F (Figure 1.20b); A, B, E, F (Figure 1.20c)

and

active: C (Figure 1.21b)

passive: A, B, C, D, E, F (Figure 1.20a,b)

respectively.

Scheme of (a) fundamental kinematic chain with mobility; (b) the dyad.

Figure 1.18 Obtaining a dyad from a fundamental kinematic chain with three elements: (a) fundamental kinematic chain with mobility; (b) the dyad.

Scheme of Aspects of the dyad: (a) the RRR dyad; (b) the RTR dyad; (c) the RRT dyad; (d) the TRT dyad; (e) the RTT dyad.

Figure 1.19 Aspects of the dyad: (a) the RRR dyad; (b) the RTR dyad; (c) the RRT dyad; (d) the TRT dyad; (e) the RTT dyad.

Scheme of a triad and tetrad from a fundamental zero-mobile chain with five elements: (a) fundamental kinematic chain with five elements and mobility; (b) triad; (c) tetrad.

Figure 1.20 Obtaining a triad and tetrad from a fundamental zero-mobile chain with five elements: (a) fundamental kinematic chain with five elements and mobility; (b) triad; (c) tetrad.

Scheme of Driving modular groups (M = 1): (a) driving element; (b) driving dyad.

Figure 1.21 Driving modular groups ( c01-math-0178 ): (a) driving element; (b) driving dyad.

1.8 Formation and Decomposition of Planar Mechanisms

The general principle for the formation of planar mechanisms is the successive connection of the modular groups; the mobility degree of the resulting mechanism is equal to the sum of the mobility degrees of the component mechanisms. The following rules have to be observed:

the exterior pairs of the passive groups are not all connected with the same element

at least one driving modular group is linked with all its exterior pairs to the frame.

Figure 1.22d shows the mode of formation of the mechanism of a shaper with mobility c01-math-0183 from a driving element (Figure 1.22a) with c01-math-0184 and two dyads (Figure 1.22b,c) with c01-math-0185 . Figure 1.22e shows the same mechanism in another variant in which the lengths AB and EF are equal to zero.

Scheme of Formation of a shaper mechanism: (a) driving element; (b) RTR dyad; (c) RTR dyad; (d) and (e) the shaper mechanism.

Figure 1.22 Formation of a shaper mechanism: (a) driving element; (b) RTR dyad; (c) RTR dyad; (d) and (e) the shaper mechanism.

The decomposition of a mechanism into modular groups, which is necessary in reaching geometric, kinematic, and kinetostatic solutions, involves the following steps:

1. Create the kinematic schema of the mechanism.

2. Create the structural schema of the fundamental mechanism.

3. Separate the driving elements linked to the frame.

4. Identify the other modular groups.

An example of decomposition is shown in Figures 1.23 and 1.24. These show the constructive schema of an intake mechanism (Figure 1.23a), then the kinematic schema (Figure 1.23b), the structural schema (Figure 1.24a), and the modular groups (Figure 1.24b).

Illustration of Intake mechanism schemata: (a) structural; (b) kinematic.

Figure 1.23 Intake mechanism schemata: (a) structural; (b) kinematic.

Scheme of The decomposition of an intake mechanism into modular groups: (a) structural schema; (b) modular groups.

Figure 1.24 The decomposition of an intake mechanism into modular groups: (a) structural schema; (b) modular groups.

1.9 Multi-poles and Multi-polar Schemata

Recalling elementary notions of the theory of systems, modular groups are subsystems, called multi-poles, which transmit, in a unique determined way, the information of motion (positions, velocities, accelerations), the mechanism being formed by interconnected multi-poles. The multi-poles are referred to as active if c01-math-0186 , and passive if c01-math-0187 .

A multi-pole can have

exterior input poles, through which it receives information about the motion from the exterior

exterior output poles, through which it transmits in exterior the information of motion

interior poles, through which information is transmitted and manipulated inside the multi-pole.

The exterior input poles can be active (driving pairs) or passive. Table 1.8 shows the correspondence of representation between modular groups and multi-poles.

Table 1.8 The representation of modular groups

The multi-polar schema is the representation of the component multi-poles of the mechanism and the interconnections between them. Figure 1.25 shows the multi-polar schema of the intake mechanism drawn in Figure 1.23.

Illustration of The multi-polar schema of the intake mechanism.

Figure 1.25 The multi-polar schema of the intake mechanism in Figure 1.23.

The structural relation is the sum of the component multi-poles, in their connection order.

equation

The cover sense of the schema is from left to right for the kinematic analysis, and from right to left for the kinetostatic analysis.

1.10 Classification of Mechanisms

The classification of mechanisms according to different criteria is given in Table 1.9.

Table 1.9 Classification of mechanisms

Chapter 2

Kinematic Analysis of Planar Mechanisms with Bars

In this chapter we will study the positions, velocities, and accelerations of the elements of planar mechanisms with bars. The kinematic parameters (positions, velocities, accelerations) are necessary for the calculation of the forces that act upon the elements, while the forces are necessary in the strength of materials calculation (checkout and sizing).

2.1 General Aspects

In the kinematic calculation we will use two methods: the grapho-analytic method and the analytic method. Graphical methods were designed in the 1970s and used until the use of computers became widespread. They facilitated the development of the analytic methods, the graphical ones being retained only for the approximate determination of positions, which is necessary for initiating numerical analytic calculations. The use of the personal computers enabled the assisted graphical approaches, which use computer-aided design (CAD) products. These allow geometric constructions to be created and the coordinates of any desired point to be determined.

From these ideas it may be concluded that graphical methods, with their characteristic simplicity, nevertheless become applicable and also ‘analytic’ through the use of CAD products. For these reasons we will present graphical methods starting from this premise. We will discuss the planar mechanisms that are the most used in practice; from the structural point of view, the majority can be created using dyads.

We will perform the kinematic analysis of the five types of dyad (RRR, RRT, RTR, TRT, RTT). We will also study the 6R triad, and in the final section we will present examples based on programs that use algorithms derived from graphical methods. At the end of the chapter we perform a kinematic analysis for the most used design: planar mechanisms with bars.

2.2 Kinematic Relations

2.2.1 Plane-parallel Motion

Distribution of velocities

The distribution of velocities for an element in plane-parallel motion is given by the Euler relation,

2.1 equation

where A and B are two arbitrary points, while c02-math-0002 is the angular velocity vector, which is perpendicular to the plane of motion. Using the