Analysis and synthesis of mechanisms. Structural synthesis and analysis of mechanisms

PRACTICAL WORK No. 1

Subject: Structural synthesis of mechanisms

Purpose of the lesson: familiarization with the elements of the structure of the mechanism, calculation of mobility, elimination of redundant connections.

Equipment: guidelines for performing practical work.

The work is designed for 4 academic hours.

1. General theoretical information.

To study the structure of the mechanism, its structural diagram is used. Often this mechanism diagram is combined with its kinematic diagram. Since the main structural components of the mechanism are links and the kinematic pairs they form, structural analysis means the analysis of the links themselves, the nature of their connection into kinematic pairs, the possibility of rotation, and analysis of pressure angles. Therefore, the work provides definitions of the mechanism, links, and kinematic pairs. In connection with the choice of method for studying the mechanism, the question of its classification is considered. The classification proposed is given. When performing laboratory work, models of flat lever mechanisms available at the department are used.

A mechanism is a system of interconnected rigid bodies with certain relative movements. In the theory of mechanisms, the mentioned rigid bodies are called links.

A link is something that moves in a mechanism as one whole. It may consist of one part, but it may also include several parts that are rigidly connected to each other.

The main links of the mechanism are the crank, the slider, the rocker arm, the connecting rod, the rocker, and the stone. These moving parts are mounted on a fixed stand.

A kinematic pair is a movable connection of two links. Kinematic pairs are classified according to a number of characteristics - the nature of the contact of the links, the type of their relative motion, the relative mobility of the links, and the location of the trajectories of movement of the points of the links in space.

To study the mechanism (kinematic, power), its kinematic diagram is constructed. For a specific mechanism - on a standard engineering scale. The elements of the kinematic diagram are the following links: input, output, intermediate, and also a generalized coordinate. The number of generalized coordinates and, therefore, input links is equal to the mobility of the mechanism relative to the rack –W3.

The mobility of a flat mechanism is determined by Chebyshev’s structural formula (1):

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In a mechanism without redundant connections, q ≤ 0. Their elimination is achieved by changing the mobility of individual kinematic pairs.

Attaching Assur structural groups to the leading link is the most convenient method for constructing a mechanism diagram. The Assur group is a kinematic chain that, when connecting external pairs to a rack, receives a zero degree of mobility. The simplest Assur group is formed by two links connected by a kinematic pair. The stand is not included in the group. A group has class and order. The order is determined by the number of elements of external kinematic pairs with which the group is attached to the mechanism diagram. The class is determined by the number K, which must satisfy the relation:

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Figure 1 - Types of mechanisms

Taking into account the possibility of conditionally transforming almost any mechanism with higher pairs into a lever mechanism, in the following we will consider these mechanisms in more detail.

2. Report preparation

The report must contain:

1. Title of the work.

2. Purpose of the work.

3. Basic formulas.

4. Solving the problem.

5. Conclusion on the solved problem.

Example of structural analysis of a mechanism

Perform a structural analysis of the linkage mechanism.

The kinematic diagram of the lever mechanism is specified in a standard engineering scale at a position determined by the angle α (Fig. 2).

Determine the number of links and kinematic pairs, classify links and kinematic pairs, determine the degree of mobility of the mechanism using the Chebyshev formula, establish the class and order of the mechanism. Identify and eliminate redundant connections.

Sequencing:

1. Classify the links: 1- crank, 2- connecting rod, 3- rocker arm, 4- strut. Only 4 links.

Figure 2 - Kinematic diagram of the mechanism

2. Classify kinematic pairs: O, A, B, C – single-moving, flat, rotational, inferior; 4-kinematic pairs.

3. Determine the mobility of the mechanism using the formula:

W3=3(n-1)-(2P1+1P2)=3(4-1)-(2*4+1*0)=1 (4)

4. Establish the class and order of the mechanism according to Assur:

Outline and mentally select from the diagram the leading part - a class 1 mechanism (M 1K - links 1.4, connection of the crank to the stand, Fig. 3). Their number is equal to the mobility of the mechanism (defined in paragraph 3).

Figure 3 – Mechanism diagram

Decompose the remaining (driven) part of the mechanism diagram into Assur groups. (In the example under consideration, the remaining part is represented by only two links 2,3.)

The first to be identified is the group that is furthest from the mechanism of class 1, the simplest (links 2,3, Fig. 3). In this group, the number of links is n’=2, and the number of whole kinematic pairs and elements of kinematic pairs in total is P =3 (B is a kinematic pair, A, C are elements of kinematic pairs). When selecting each successive group, the mobility of the remaining part should not change. The degree of mobility of the Assur 2-3 group is

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The entire mechanism is assigned the highest class and order, i.e. - M1K 2P.

5. Identify and eliminate redundant connections.

The number of redundant connections in the mechanism is determined by the expression:

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Eliminate redundant connections. We replace the single-moving pair A, for example, with a rotational double-moving one (Fig. 1), and the single-moving pair B with a three-moving one (spherical Fig. 1). Then the number of redundant connections will be determined as follows:

Structural synthesis and analysis of mechanisms

Main types of mechanisms

Based on kinematic, structural and functional properties, mechanisms are divided into:

1. Lever(Fig. 2 a, b) - designed to convert the rotational motion of the input link into the reciprocating movement of the output link. Can transmit great forces and powers.

2. Cam(Fig. 2 c, d) - designed to convert the rotational or reciprocating motion of the input link into the reciprocating or reciprocating motion of the output link. By giving the profiles of the cam and the pusher the corresponding shapes, it is always possible to implement any desired law of movement of the pusher.

3. Toothed(Fig. 2 f) - formed with the help of gears. Serve to transmit rotation between fixed and moving axes. Gear drives with parallel axes are realized using cylindrical gears, with intersecting axes using bevel gears, and with crossing axes using a worm and worm wheel.

4. Friction(Fig. 2 d) - movement from the driving link to the driven link is transmitted due to friction forces arising as a result of the contact of these links.

Structural synthesis of a mechanism is usually called the design of a structural diagram of a mechanism, which consists of fixed and moving links and kinematic pairs. It is the initial stage of drawing up a diagram of a mechanism that satisfies the given conditions. The initial data are usually the types of motion of the driving and working links of the mechanism, the relative position of the axes of rotation and the direction of translational movement of the links, their angular and linear movements, speeds and accelerations. The most convenient method for finding a structural diagram is the method of attaching Assur structural groups to the leading link or main mechanism.

The structural analysis of a mechanism is usually understood as determining the number of links and kinematic pairs, determining the degree of mobility of the mechanism, as well as establishing the class and order of the mechanism.

The degree of mobility of the spatial mechanism is determined by the Somov-Malyshev formula:

W = 6n-(5P 1 +4P 2 + 3P 3 + 2P 4 + P 5) (1)

where P 1, P 2, P 3, P 4, P 5 - the number of one-, two-, three-, four- and five-movable kinematic pairs; n is the number of moving parts.

The degree of mobility of a flat mechanism is determined by the Chebyshev formula:

W=3n-2P H - P B (2)

where рН is the number of lower ones, and Рв is the number of higher kinematic pairs.

As an example, consider a four-link autopilot steering mechanism (Fig. 3.3): links 1 and 2 form a cylindrical pair of the fourth class, having two degrees of freedom; links 2-3 and 4-1 form fifth-class rotational pairs having one degree of freedom; links 3-4 form a ball pair of the third class, having three degrees of freedom; the number of moving links is three, then

W = 6 3-2 5-1 4-1 3 = 1

The degree of mobility of this mechanism is 1.

A kinematic chain, the number of degrees of freedom of which relative to the elements of its external kinematic pairs is zero, is called the Assur structural group, named after L.V. Assur, who for the first time fundamentally researched and proposed a structural classification of flat rod mechanisms. An example of the formation of a flat six-bar mechanism is given in Fig. 4.

Structural groups are divided by class and order. The class of a group is determined by the maximum number of kinematic pairs included in one link (Fig. 5).

The order of the group is determined by the number of elements by which the group is attached to the main mechanism (Fig. 6).

The class and order of the mechanism depend on which link is the leading one.

Mechanisms with an open kinematic chain are assembled without interference, so they are statically definable, without redundant connections ( q=0).

Structural group– a kinematic chain, the attachment of which to a mechanism does not change the number of its degrees of freedom and which does not break up into simpler kinematic chains with zero degree of freedom.

Primary mechanism(according to I. I. Artobolevsky - class I mechanism, initial mechanism), is the simplest two-link mechanism, consisting of a moving link and a stand. These links form either a rotational kinematic pair (crank - strut) or a translational pair (slider - guides). The initial mechanism has one degree of mobility. The number of primary mechanisms is equal to the number of degrees of freedom of the mechanism.

For Assur structural groups, according to the definition and Chebyshev formula (with R vg =0, n= n pg and q n =0), the equality is true:

W pg =3 n pg –2 R ng =0,

(1.5)

Where W pg is the number of degrees of freedom of the structural (leader) group relative to the links to which it is attached; n pg, R ng – the number of links and lower pairs of the Assur structural group.

Figure 1.5 – Division of the crank-slider mechanism into the primary mechanism (4, A, 1) and structural group (B, 2, C, 3, C")

The first group is attached to the primary mechanism, each subsequent group is attached to the resulting mechanism, but a group cannot be attached to one link. Order a structural group is determined by the number of link elements with which it is attached to the existing mechanism (i.e., the number of its external kinematic pairs).

The class of a structural group (according to I. I. Artobolevsky) is determined by the number of kinematic pairs that form the most complex closed contour of the group.

The class of the mechanism is determined by the highest class of the structural group included in it; in the structural analysis of a given mechanism, its class also depends on the choice of primary mechanisms.

Structural analysis of a given mechanism should be carried out by dividing it into structural groups and primary mechanisms in the reverse order of formation of the mechanism. After the separation of each group, the degree of mobility of the mechanism must remain unchanged, and each link and kinematic pair can be included in only one structural group.

Structural synthesis of flat mechanisms should be carried out using the Assur method, which provides a statically definable flat mechanism diagram ( q n =0), and Malyshev’s formula, since due to manufacturing inaccuracies, the flat mechanism to some extent turns out to be spatial.

For a crank-slider mechanism, considered as a spatial one (Figure 1.6), according to Malyshev’s formula (1.2):

q=W+5p 5 +4R 4 +3R 3 +2R 2 +R 1 -6n=1+5×4-6×3=3

Figure 1.6 – Crank-slider mechanism with lower pairs

For a crank-slider mechanism, considered as a spatial one, in which one rotational pair was replaced with a cylindrical two-moving pair, and the other with a spherical three-moving pair (Figure 1.7), according to Malyshev’s formula (1.2):

q=W+5p 5 +4R 4 +3R 3 +2R 2 +R 1 -6n=1+5×2+4×1+3×1-6×3=0

Figure 1.7 – Crank-slider mechanism without redundant connections (statically determinable)

We get the same result by swapping the cylindrical and spherical pairs (Figure 1.8):

q=W+5p 5 +4R 4 +3R 3 +2R 2 +R 1 -6n=1+5×2+4×1+3×1-6×3=0

Figure 1.8 – Option for designing a crank-slider mechanism without redundant connections (statically determinable)

If we install two spherical pairs in this mechanism instead of rotational ones, we get a mechanism without redundant connections, but with local mobility (W m = 1) - rotation of the connecting rod around its axis (Figure 1.9):

q=W+5p 5 +4R 4 +3R 3 +2R 2 +R 1 -6n=1+5×2+3×2-6×3= -1

q=W+5p 5 +4R 4 +3R 3 +2R 2 +R 1 -6n+W m =1+5×2+3×2-6×3+1=0

Figure 1.9 – Crank-slider mechanism with local mobility

Section 4. Machine parts

Features of product design

Product classification

Detail– a product made of a homogeneous material, without the use of assembly operations, for example: a roller made of one piece of metal; cast body; bimetallic sheet plate, etc.

Assembly unit– a product whose components are subject to interconnection by assembly operations (screwing, joining, soldering, crimping, etc.)

Knot- an assembly unit that can be assembled separately from other components of the product or the product as a whole, performing a specific function in products for one purpose only together with other components. A typical example of units are shaft supports - bearing units.

PRACTICAL WORK No. 1

Subject: Structural synthesis of mechanisms

Purpose of the lesson: familiarization with the elements of the structure of the mechanism, calculation of mobility, elimination of redundant connections.

Equipment: guidelines for performing practical work.

The work is designed for 4 academic hours.

general theoretical information.

To study the structure of the mechanism, its structural diagram is used. Often this mechanism diagram is combined with its kinematic diagram. Since the main structural components of the mechanism are links and the kinematic pairs they form, structural analysis means the analysis of the links themselves, the nature of their connection into kinematic pairs, the possibility of rotation, and analysis of pressure angles. Therefore, the work provides definitions of the mechanism, links, and kinematic pairs. In connection with the choice of method for studying the mechanism, the question of its classification is considered. The classification proposed by L.V. Assur is given. When performing laboratory work, models of flat lever mechanisms available at the department are used.

A mechanism is a system of interconnected rigid bodies with certain relative movements. In the theory of mechanisms, the mentioned rigid bodies are called links.

A link is something that moves in a mechanism as one whole. It may consist of one part, but it may also include several parts that are rigidly connected to each other.

The main links of the mechanism are the crank, the slider, the rocker arm, the connecting rod, the rocker, and the stone. These moving parts are mounted on a fixed stand.

The mobility of a flat mechanism is determined by Chebyshev’s structural formula (1):

where n is the number of all links of the mechanism;

P 1, P 2 - the number of one and two movable kinematic pairs in the mechanism.

Due to errors in the manufacture of mechanisms, harmful passive connections q - (excessive) arise, which lead to additional deformations and energy losses due to these deformations. During design, they must be identified and eliminated. Their number is determined using the Somov–Malyshev structural formula (2):

In a mechanism without redundant connections, q ≤ 0. Their elimination is achieved by changing the mobility of individual kinematic pairs.

(3)

where P is the number of kinematic pairs, including elements of pairs, Q 1 is the number of links in the Assur group.

The class and order of this mechanism corresponds to the class and order of the Assur senior group in this mechanism. The purpose of classification is to select a method for studying the mechanism.

Among the variety of mechanism designs, there are: rod (lever), cam, friction, gear mechanisms, mechanisms with flexible links (for example, belt drives) and other types (Fig. 1).

Less common classifications imply the presence of mechanisms with lower or higher pairs in a flat or spatial design, etc.

Figure 1 - Types of mechanisms

Taking into account the possibility of conditionally transforming almost any mechanism with higher pairs into a lever mechanism, in the following we will consider these mechanisms in more detail.

report preparation

The report must contain:

1. Title of the work.

2. Purpose of the work.

3. Basic formulas.

4. Solving the problem.

5. Conclusion on the solved problem.

Example of structural analysis of a mechanism

Perform a structural analysis of the linkage mechanism.

The kinematic diagram of the lever mechanism is specified in a standard engineering scale at a position determined by the angle α (Fig. 1d).

Sequencing:

1. Classify the links: 1- crank, 2- connecting rod, 3- rocker arm, 4- strut. Only 4 links

2. Classify kinematic pairs: O, A, B, C – single-moving, flat, rotational, inferior; 4-kinematic pairs.

3. Determine the mobility of the mechanism using the formula:

W3=3(n-1)-(2P1+1P2)=3(4-1)-(2*4+1*0)=1 (4)

4. Establish the class and order of the mechanism according to Assur:

Outline and mentally select from the diagram the leading part - a class 1 mechanism (M 1K - links 1.4, connection of the crank to the stand, Fig. 2). Their number is equal to the mobility of the mechanism (defined in paragraph 3).

Figure 2. Mechanism diagram

Decompose the remaining (driven) part of the mechanism diagram into Assur groups. (In the example under consideration, the remaining part is represented by only two links 2,3.)

The first to be identified is the group that is furthest from the mechanism of class 1, the simplest (links 2,3, Fig. 3). In this group, the number of links is n’=2, and the number of whole kinematic pairs and elements of kinematic pairs in total is P =3 (B – kinematic pair, A, C – elements of kinematic pairs). When selecting each successive group, the mobility of the remaining part should not change. The degree of mobility of the Assur 2-3 group is

The class of the group is determined from the simplest system of two equations:

whence the Class of the group is 1.

The order of the group is 2, since the group is attached to the main mechanism by two elements of kinematic pairs A, C.

Therefore, the Assur group under consideration is a Class 1, Order 2 group.

Mechanism structure formula:

(7)

The entire mechanism is assigned the highest class and order, i.e. - M1K 2P.

5. Identify and eliminate redundant connections.

The number of redundant connections in the mechanism is determined by the expression:

In the mechanism, all pairs are single-moving P 1 = 4 and the number of links n is 4. The number of redundant links:

They have the same research methods regardless of their area of application or functional purpose.

It is necessary to know what a structural group (Assur group) is, how its class, order, and type are determined. It is advisable to remember the table showing the combination of links and kinematic pairs of the fifth class in the group:

n groups	2	4	6	8	…
P 5 groups	3	6	9	12	…

Solving the problem begins with determining the number of degrees of freedom of the kinematic chain underlying this mechanism. In accordance with the number of degrees of freedom, the number of initial links (or input links) is assigned, after which the chain becomes a mechanism.

After adding each Assur group, an intermediate mechanism should be obtained, with the same number of degrees of freedom as the given one. After adding the last group, the initially specified mechanism should be obtained.

Please note that the class of the mechanism (and therefore the methods for solving it) are determined not only by the mechanism diagram, but also by which link is accepted as the input. With the same scheme, but with different input links, mechanisms of different classes can be obtained, and, therefore, the methods for studying them will be different.

It should also be noted that the presence of closed loops in the circuit diagram of a mechanism does not determine the class of the mechanism, since when divided into Assur groups, these contours can disintegrate. But if some circuit is preserved in the Assur group, then it determines the class of this group, and through the class of the group - the class of the mechanism.

Mechanisms may contain double and more complex hinges, so you need to be careful when determining the number of degrees of freedom, as well as when dividing the mechanism into Assur groups.

The following must be kept in mind:

with the same scheme, it is possible to obtain different mechanisms from the point of view of research methods, if different links are specified as input;
from the same Assur groups you can create different mechanisms with different functional purposes;
structural group (Assur group) has the same properties and research methods regardless of the mechanism in which it is located. This very important property allows us to develop research methods only for Assur groups, and not for each mechanism from their huge number;
The structural classification under consideration is applicable not only for the analysis of existing mechanisms, but also for the targeted synthesis of mechanisms with predictable properties (by attaching Assur groups to the initial ones or to the initial mechanisms and their further layering).

If the mechanism has two degrees of freedom, it is necessary to specify two initial links.

If the mechanism has higher kinematic pairs of class IV, then before dividing the mechanism into structural groups, it is necessary to replace the higher pairs with chains with lower pairs, because Assur groups include only V class couples.

For subsequent analysis, it is advisable to compare the number of degrees of freedom of the given mechanism and the mechanism obtained after replacing the higher pairs.

There may be extra degrees of freedom in the mechanism. The formula for determining the number of degrees of freedom gives the correct result for the general case, but in a particular case, for certain sizes of links, the actual number of degrees of freedom may differ from that determined by the formula.

Usually, the presence of a round roller provides an extra degree of freedom (its rotation around its own axis gives the mechanism an additional degree of freedom, but this movement does not affect the nature of the operation of the remaining links and the entire mechanism as a whole). Therefore, the number of initial mechanisms must be specified according to the effective number of degrees of freedom (W actual = W calculated – W extra).

When replacing the highest pair, the excess degree of freedom automatically disappears (therefore, after replacing the highest pair, the new calculated value of the number of degrees of freedom will be equal to the current number of degrees of freedom). This is convenient for monitoring the correctness of establishing the presence or absence of extra degrees of freedom.

In some cases, it is difficult to determine the class of Assur groups, and, accordingly, the mechanism according to the kinematic scheme, because some triangles degenerate into straight lines, the sides of contours can be represented by sliders, etc. As a result, it is quite difficult to determine the presence of a closed contour in a group and the number of its sides. In this case, it is convenient to use the construction of a block diagram of a mechanism (or a separate group).

The structural diagram is drawn without scale, all links included in three kinematic pairs are depicted in the form of rigid triangles, links included in four kinematic pairs are depicted in the form of rigid quadrangles, etc., all sliders are conventionally replaced by hinges. Thus, another mechanism is formed with the same structure, but with a more visual diagram for solving this problem. Naturally, further research considers the initially specified mechanism.