3.3. Phases of work cam mechanisms. Phase and design angles
Cam mechanisms can implement the laws of motion of almost any complexity at the output link. But any law of motion can be represented by a combination of the following phases:
1. Removal phase. The process of moving an output link (follower or rocker) when the contact point of the cam and the follower moves away from the center of rotation of the cam.
2. Phase of return (approximation). The process of moving the output link as the point of contact between the cam and the follower approaches the center of rotation of the cam.
3. Phases of exposure. The situation when, with a rotating cam, the contact point of the cam and the pusher is stationary. At the same time, they distinguish near dwell phase– when the contact point is at the closest position to the center of the cam, long-range phase– when the point of contact is at the farthest position from the center of the cam and intermediate dwell phases. The dwell phases take place when the point of contact moves along the section of the profile of the Cam, which has the form of an arc of a circle drawn from the center of rotation of the Cam.
The above classification of phases primarily refers to positional mechanisms.
Each phase of work corresponds to its own phase angle of the mechanism and the design angle of the cam.
The phase angle is the angle through which the cam must turn in order to complete the corresponding phase of operation. These angles are denoted by the letter with an index indicating the type of phase, for example, Y is the phase angle of removal, D is the phase angle of far dwell, B is the phase angle of return, B is the phase angle of near dwell.
The design angles of the cam determine its profile. They are denoted by the letter with the same indices. On fig. 3.2a shows these angles. They are limited by rays drawn from the center of rotation of the cam to the points on its center profile, where the cam profile changes during the transition from one phase to another.
At first glance, it may seem that the phase and design angles are equal. Let us show that this is not always the case. To do this, we perform the construction shown in Fig. 3.2b. Here, the mechanism with the pusher, if it has an eccentricity, is set to the position corresponding to the beginning of the removal phase; to- the point of contact between the cam and the pusher. Dot to' is the position of the point to, corresponding to the end of the removal phase. It can be seen from the construction that in order for the point to took a position to’ the cam must rotate through an angle Y, not equal to Y, but different by an angle e, called the angle of eccentricity. For mechanisms with a pusher, the following relations can be written:
Y \u003d Y + e, B \u003d B - e,
D = D, B = B
3.4. Choice of the law of motion of the output link
The method for choosing the law of motion of the output link depends on the purpose of the mechanism. As already noted, according to their purpose, cam mechanisms are divided into two categories: positional and functional.
3.4.1. Positional mechanisms
For clarity, let's consider the simplest case of a two-position mechanism, which simply “transfers” the output link from one extreme position to another and back.
On fig. 3.3 shows the law of motion - a graph of the movement of the pusher of such a mechanism, when the entire process of work is represented by a combination of four vases: removal, far rest, return and near rest. Here is the angle of rotation of the cam, and the corresponding phase angles are denoted: y, d, c, b. The displacement of the output link is plotted along the ordinate axis: for mechanisms with a rocker arm, this is - the angle of its rotation, for mechanisms with a pusher S - the displacement of the pusher. 
In this case, the choice of the law of motion consists in determining the nature of the motion of the output link in the phases of removal and return. On fig. 3.3 for these sections some kind of curve is shown, but it is precisely this curve that must be determined. What criteria are laid down as the basis for solving this problem?
Let's go from the opposite. Let's try to do it "simple". Let us set a linear law of displacement in the areas of removal and return. On fig. 3.4 shows what this will lead to. Differentiating the function () or S() twice, we get that theoretically infinite, i.e. unpredictable accelerations and, consequently, inertial loads. This unacceptable phenomenon is called a hard phase shock.
To avoid this, the choice of the law of motion is made on the basis of the acceleration graph of the output link. On fig. 3.5 is an example. Given the desired shape of the acceleration graph and its integration, the functions of speed and displacement are found. 
The dependence of the acceleration of the output link in the phases of removal and return is usually chosen to be shockless, i.e. as a continuous function without acceleration jumps. But sometimes for low-speed mechanisms, in order to reduce the dimensions, the phenomenon is allowed soft hit, when jumps are observed on the acceleration graph, but by a finite, predictable amount.
On fig. 3.6 presents examples of the most commonly used types of laws of change in acceleration. The functions are shown for the delete phase, they are similar in the return phase, but mirrored. On fig. 3.6 presents symmetrical laws when 1 = 2 and the nature of the curves in these sections is the same. If necessary, asymmetric laws are also applied, when 1 2 or the nature of the curves in these sections is different, or both.
The choice of a specific type depends on the operating conditions of the mechanism, for example, law 3.6d is used when a section with a constant speed of the output link is needed in the removal (return) phase.
As a rule, the functions of the laws of acceleration have analytical expressions, in particular, 3.6, a, e - segments of a sinusoid, 3.6, b, c, g - segments of straight lines, 3.6, e - a cosine wave, so their integration in order to obtain speed and movement is not difficult . However, the amplitude values of the acceleration are not known in advance, but the value of the displacement of the output link during the removal and return phases is known. Let us consider how to find both the acceleration amplitude and all the functions that characterize the motion of the output link.
At a constant angular velocity of rotation of the cam, when the angle of its rotation and time are related by the expression = t functions can be considered both on time and on the angle of rotation. We will consider them in time and in relation to the mechanism with a rocker arm.
At the initial stage, we set the form of the acceleration graph in the form of a normalized, that is, with a unit amplitude, function *( t). For the dependence in Fig. 3.6a it will be *( t) = sin(2 t/T), where Т is the time for the mechanism to pass through the removal or return phase. Real acceleration of the output link:
2 (t) = m *(t), (3.1) 
where m is the currently unknown amplitude.
Integrating expression (3.1) twice, we obtain:
Integration is performed with initial conditions: for the removal phase 2 ( t) = 0, 2 ( t) = 0; for the return phase 2 ( t) = 0, 2 ( t) = m . The required maximum displacement of the output link m is known, therefore, the acceleration amplitude
Each value of functions 2 ( t), 2 ( t), 2 (t) can be assigned to the values 2 (), 2 (), 2 (), which are used to design the mechanism, as described below.
It should be noted that there is another reason for the occurrence of shocks in cam mechanisms, associated with the dynamics of their work. The cam can also be designed to be shockless, in the sense that we put into this concept above. But at high speeds, for mechanisms with a power circuit, the pusher (rocker arm) can be separated from the cam. After some time, the closing force restores contact, but this restoration occurs with a blow. Such phenomena can occur, for example, when the return phase is set too small. The cam profile then turns out to be steep in this phase and at the end of the long-range dwell phase, the closing force does not have time to provide contact and the pusher, as it were, breaks off the cam profile at the far end and can even immediately hit some point of the cam at the near end. For positive locking mechanisms, the roller moves along a groove in the cam. Since there is necessarily a gap between the roller and the walls of the groove, the roller hits the walls during operation, the intensity of these impacts also increases with the speed of rotation of the cam. To study these phenomena, it is necessary to make a mathematical model of the entire mechanism, but these issues are beyond the scope of this course.
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Cam mechanism- this is a mechanism with a higher kinematic pair, which has the ability to provide outage of the output link, and the structure contains at least one link with a working surface of variable curvature.
Cam mechanisms are designed to convert the movement of the leading link into the required type of movement of the output link according to a given law.
The diagram of a typical cam mechanism has a structure containing a rack and two moving links ( fig. 9.1). At the same time, in a cam mechanism with two moving links, it is possible to implement the transformation of movement and force factors according to the law of any complexity.
Rice. 9.1. Kinematic diagrams of cam mechanisms
In typical diagrams of cam mechanisms, the drive link is called cam, and the pusher acts as the output link (Fig. 9.1, but)
or rocker (Fig. 9.1, b).
The cam is a link of the cam mechanism with a working surface of variable curvature.
The pusher is the output link of the cam mechanism that performs translational movements.
The rocker arm is the output link of the cam mechanism, which performs only rotational movements and does not have the ability to rotate through an angle of more than 360 °.
In cam mechanisms, the transformation of movement and force factors is carried out by direct contact of the working surface of the cam with the surface of the output link. In this case, due to the difference in the speeds of movement of the contacting links in the zone of their contact, sliding friction takes place, which leads to intensive wear of these surfaces, as well as to an increase in losses, a decrease in the efficiency and service life of the cam mechanism. To replace sliding friction with rolling friction in the higher kinematic pair, an additional link is introduced into the cam mechanism circuit, which is called a roller. The roller forms a single-moving kinematic pair of the 5th class with the output link (Fig. 9.2). The mobility of this
9. CAM MECHANISMS
kinematic pair does not affect the transfer function of the cam mechanism and is a local mobility.
Rice. 9.2. Kinematic diagrams of cam mechanisms with a roller
When an additional link is introduced into the circuit - a roller - the transformation of movement and force factors is carried out by means of contact of the working surface of the cam with the surface of the roller, which interacts with the output link. In this case, the cam has two types of profiles ( fig. 9.3): constructive and theoretical.
Rice. 9.3. Types of cam profiles in cam mechanisms
Structural (working) profile is the outer profile of the cam. Theoretical ( center) profile is a profile that describes
There is no center of the roller when it rolls without sliding along the structural profile of the cam.
9.1. CLASSIFICATION OF CAM MECHANISMS
Cam mechanisms are classified: 1) according to their official purpose:
cam mechanisms that ensure the movement of the output link according to a given law of motion;
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9. CAM MECHANISMS
9.1.
cam mechanisms that provide only the specified maximum displacement of the output link (stroke of the pusher or swing angle of the rocker arm);
2) according to the location of the links in space: flat cam mechanisms ( rice. 9.1, fig. 9.2);
spatial cam mechanisms ( fig. 9.4);
Rice. 9.4. Schemes of spatial cam mechanisms
3) according to the type of cam movement:
cam mechanisms with rotational movement of the cam (fig. 9.2); cam mechanisms with translational movement of the cam (Fig. 9.5); cam mechanisms with helical cam movement;
Rice. 9.5. Schemes of cam mechanisms with translational movement of the cam
4) according to the type of movement of the output link:
cam mechanisms with translational movement of the output
link (Fig. 9.1, but, Fig. 9.2, but, Fig. 9.4, but, Fig. 9.5, but);
cam mechanisms with rotational movement of the output link
(Fig. 9.1, b, Fig. 9.2, b, Fig. 9.4, b, Fig. 9.5, b);
5) by the presence of a video in the scheme:
cam mechanisms with a roller (fig. 9.2, fig. 9.4, fig. 9.5); cam mechanisms c without roller (fig. 9.1);
6) by type of cam:
cam mechanisms with a flat cam (fig. 9.1, fig. 9.2, fig.
9.5 );
cam mechanisms with a cylindrical cam (fig. 9.4); cam mechanisms with a globoid cam (Fig. 9.6, but); cam mechanisms with a spherical cam (Fig. 9.6, b);
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9. CAM MECHANISMS
9.1. Classification of cam mechanisms
Rice. 9.6. Schemes of cam mechanisms with globoid and spherical cams
Rice. 9.7. Diagrams of de-axial cam mechanisms
7) according to the shape of the working surface of the output link:
cam mechanisms with a pointed working surface
leg link (Fig. 9.1, a, Fig. 9.7, b, Fig. 9.8, b);
cam mechanisms with a flat working surface of the output link (Fig. 9.7, but, Fig. 9.8, but);
cam mechanisms with a cylindrical working surface of the output link (fig. 9.2);
cam mechanisms with a spherical working surface of the output link (Fig. 9.7, c, d, Fig. 9.8, c, d);
8) by the presence of displacement:
deaxial cam mechanisms ( fig. 9.7); axial cam mechanisms (fig. 9.8).
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9. CAM MECHANISMS
9.1. Classification of cam mechanisms
Rice. 9.8. Diagrams of axial cam mechanisms
Deaxial cam mechanism is a cam mechanism, in which
the axis of the path of the output link is shifted relative to the center of rotation of the cam by a certain amount (Fig. 9.7). The amount of displacement is called eccentricity, or deaxial, and is denoted e.
Axial cam mechanism- this is a cam mechanism in which the axis of the path of the output link passes through the center of rotation of the cam ( fig. 9.8).
9.2. METHODS FOR CLOSING ELEMENTS OF THE HIGHEST KINEMATIC PAIR
AT during the movement of the cam mechanisms, a situation is possible leading to the loss of contact of the moving links, which leads to the opening of the elements of the higher kinematic pair. The opening of the elements of the higher kinematic pair leads to the termination of its existence, which is reflected in the law of motion of the links in the form of breaks and is unacceptable for the normal operation of the cam mechanisms. To ensure the constancy of contact of the links that form the highest kinematic pair, the following closing methods are used in cam mechanisms:
Power circuit- this is a way to ensure the constancy of contact of the links of the higher kinematic pair by using the gravity forces of the links or the elastic forces of the springs (Fig. 9.9).
AT In cam mechanisms with power closing of the links forming the higher pair, the movement of the output link in the removal phase is carried out due to the action of the contact surface of the cam on the contact surface of the output link, i.e. the cam is the leading link, and the output link is the driven link: pusher or rocker. In the approach phase, the output link moves due to the action of the elastic force of the spring or the gravity force of the output link, i.e. the leading link is the output link: pusher or rocker, and the driven link is the cam.
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9. CAM MECHANISMS
9.2. Methods for closing elements of a higher kinematic pair
Rice. 9.9. Schemes of cam mechanisms with force closure
Geometric closure- this is a way to ensure the constancy of contact of the links of the higher kinematic pair by means of the configuration of the working surfaces of the cam (Fig. 9.10).
Rice. 9.10. Diagrams of positive cam mechanisms
In cam mechanisms with geometrically locking the links that form the higher pair, the movement of the output link in the removal phase is carried out due to the impact of the outer working surface of the cam on the contact surface of the output link. The movement of the output link in the approach phase is a consequence of the impact of the inner working surface of the Cam on the contact surface of the output link. In both phases, the cam acts as the leading link, and the output link is the driven link: pusher or rocker.
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9. CAM MECHANISMS
9.3. MAIN PARAMETERS OF THE CAM MECHANISM
Cam mechanisms formed on the basis of typical schemes belong to cycloidal mechanisms with a period of operation equal to 2π, and are characterized by the presence of several phases of movement of the output link (Fig. 9.11):
the removal phase is the phase of movement of the cam links by moving the output link from the lower position to the upper one;
upper standing or resting phase
oval mechanisms, accompanied by standing or stand output link in the upper position;
approach phase - this is the phase of movement of the links of the cam mechanisms, accompanied by the movement of the output link from the upper position to the lower;
lower standing or resting phase is the phase of movement of the cam links
oval mechanisms, accompanied by standing or stand output link in the down position.
ϕу |
ϕ c.c. |
ϕс |
ϕ n.v |
|
ϕ r.x |
ϕ x.x |
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Rice. 9.11. Phases of movement of the output link of the cam mechanisms |
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Each phase of the movement of the links of the cam mechanisms is characterized by the corresponding two types of angles (Fig. 9.12):
phase angle ϕ is the angle of rotation of the cam during the action of a certain phase of the movement of the output link;
profile angle δ is the angular coordinate of the operating point of the theoretical cam profile corresponding to the current phase angle.
In accordance with the classification of phases, phase angles are divided into four types ( fig. 9.11):
phase angle of removal ϕ y (Fig. 9.12); phase angle of the upper standing or standing ϕ in. in (Fig. 9.12);
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9. CAM MECHANISMS
9.3. The main parameters of the cam mechanism
phase angle of approach ϕ with (Fig. 9.12); phase angle of the lower standing or standing ϕ n.v (Fig. 9.12).
Rice. 9.12. Phase and profile angles of cam mechanisms
The sum of all four phase angles forms the cyclic phase angle:
ϕ = ϕу + ϕv.v + ϕс + ϕн.v = 2 π.
The sum of the first three phase angles is the phase angle of the working stroke of the cam mechanism (Fig. 9.11):
ϕ r.x = ϕ y + ϕ v.v + ϕ s.
The idle phase angle of the cam mechanism is equal to the phase angle of the lower dwell (Fig. 9.11), i.e.
ϕ x.x = ϕ n.v.
Each phase of the movement of the links of the cam mechanisms has its own profile angle, the angles are also divided into four types ( fig. 9.12):
removal angle δ y ; the angle of the upper standing or standing δ in. in; approach angle δ with ;
angle of lower standing or standing δ n.v.
In the general case, the phase and profile angles of the corresponding phases of movement of the links of typical cam mechanisms are not equal to each other:
ϕ ≠ δ.
The equality of the phase and profile angles of the corresponding phases of the movement of the links is characteristic only in the phase of the lower dwell (Fig. 9.12), and for the remaining phases of the movement of the links, it takes place only for typical cam mechanisms without a roller.
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9. CAM MECHANISMS
9.4. STRUCTURAL ANALYSIS OF FLAT CAM MECHANISMS
The links of typical cam mechanisms move in parallel planes, therefore, these mechanisms are flat, the mobility of which is calculated by the Chebyshev formula.
Cam mechanisms without a roller (Fig. 9.1 ). The structure of both types of ty-
new cam mechanisms consists of three links, of which cam 1 and pusher or rocker arm 2 are movable links, and rack 0 is a fixed link, therefore, n = 2. The rack is represented in the scheme of the mechanism with a pusher with one hinged-fixed support and a fixed slider, and in the scheme of mechanisms with a rocker arm - two hinged-fixed supports. The moving links and the rack form two rotational kinematic pairs with a mobility equal to one: 0 - 1, 2 - 0 and one higher kinematic sail mobility equal to two: 1 - 2, therefore, p 1 = 2, p 2 = 1.
W = 3 2 - 2 2 - 1 = 6 - 4 - 1 = 1.
The result means that one generalized coordinate is enough to unambiguously determine the relative position of the links of mechanisms of this type.
Cam mechanisms with a roller (Fig. 9.2 ). The schemes of both cam mechanisms consist of four links, of which cam 1, pusher or rocker 2 and roller 3 are movable links, and rack 0 is a fixed link, therefore, n = 3. The rack is presented in the scheme of the mechanism with a pusher of onehinged-fixedsupport and a fixed slider, and in the scheme of mechanisms with a rocker arm - twohinged-fixedsupports. The moving links and the rack form three rotational kinematic pairs with a mobility equal to one: 0 - 1, 2 - 3, 3 - 0 and one higher kinematic pair with a mobility equal to two: 1 - 3, therefore, p1 = 2, p2 = 1.
Substituting the obtained data into the structural formula, we obtain
W = 3 3 - 2 3 - 1 = 9 - 6 - 1 = 2 .
Calculation according to the Chebyshev formula for typical cam mechanisms with a roller shows that the mobility is equal to two. The result indicates the presence of structural defects in the schemes of typical cam mechanisms with a roller, which indicates the presence of two types of mobility for different functional purposes. The mobility of a typical flat cam mechanism with one drive link, forming a primary mechanism with a mobility equal to one, is equal to one, therefore, the second unit of mobility is accounted for by the local mobility formed by the roller with the output link:
W = 2 =W 0 +W ì =1 +1,
where W 0 , W m - respectively, the main (calculated) and local mobility of the cam mechanism.
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9. CAM MECHANISMS
9.5. KINEMATIC ANALYSIS OF FLAT CAM MECHANISMS
To carry out a kinematic analysis of typical cam mechanisms, it is necessary to know the main dimensions of all its links or the law of motion of the output link.
In the general case, the purpose of the kinematic analysis of typical cam mechanisms with a given mechanism scheme is to determine the law of motion of the output link, and with known basic dimensions of all links, to determine the law of motion of the output link.
The law of motion of the output link is determined as a function of the angle of rotation of the cam based on the structural features of the cam mechanism and the specified parameters:
S = f(ϕ),
where ϕ is the angle of rotation of the cam.
This functional dependence can be obtained by an analytical or graph-analytical method. The analytical method, as in the analysis of mechanisms of other types, allows obtaining more accurate data, however, the graphical-analytical method is simpler and gives a clear result, which led to its wide use in engineering calculations to obtain a primary idea of the values and patterns of change in the kinematic parameters of cam mechanisms based on given conditions.
Graph-analytical method kinematic analysis can be carried out by two methods: the method of kinematic diagrams or the method of kinematic plans. The method of plans as applied to the analysis of typical cam mechanisms is based on the use of replacement mechanisms.
Replacement mechanism- this is a mechanism, the structure of which contains only lower kinematic pairs, which, at certain positions of the leading link, have the same displacements, velocities and accelerations for the output link as the corresponding mechanism with the higher pair.
When choosing a replacement mechanism scheme, the main attention is paid to the preservation of the laws of motion of the driving and output links of the cam mechanisms and the mutual arrangement of the axes of these links. Each higher kinematic pair is replaced by two lower pairs, which leads to the appearance of a fictitious link 3 in the structure of the replacement mechanism. Based on the foregoing, taking into account the type of movement performed by the output link, the cam mechanisms diagrams are replaced with the corresponding diagram of a typical lever mechanism.
The kinematic analysis of typical lever mechanisms has been discussed above (see Chapter 2).
In most cases, the law of motion of the output link of a typical cam mechanism is given by means of the second derivative of the path with respect to the angle of rotation or with respect to time (acceleration tax). In this case, to directly obtain the law of motion of the output link, the method of kinematic diagrams is used (Fig. 9.13).
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9. CAM MECHANISMS
9.5. Kinematic analysis of planar cam mechanisms
d 2 S |
F(ϕ) |
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dϕ 2 |
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dϕ 2 |
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F(ϕ) |
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S = f(ϕ)
2 π ϕ
Rice. 9.13. Kinematic analysis of cam mechanisms by the method of diagrams
The process of determining the law of motion is carried out in the following sequence.
First, based on the given conditions, a diagram of the analogue of the
integrating the diagram of the acceleration analog, first form the diagram
mu analog speed |
(ϕ) (Fig. 9.14, b), then, using the graphic |
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diagram integration |
speed analog, get a path diagram |
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s \u003d f (ϕ) (Fig. 9.13, c). |
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Kinematic analysis allows obtaining the necessary data for the transition to the stage of metric synthesis of cam mechanisms.
9.6. SYNTHESIS OF FLAT CAM MECHANISMS
The main criteria that are guided in solving the problems of synthesis of cam mechanisms are: minimization of overall and mass characteristics and values of pressure angles, as well as ensuring the manufacturability of the structural profile of the cam.
The synthesis of any cam mechanism is carried out in two stages: structural synthesis and metric synthesis.
At the stage of structural synthesis, the formation of a structural diagram of the cam mechanism is carried out, i.e., the number of links is substantiated
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9. CAM MECHANISMS
9.6. Synthesis of flat cam mechanisms
mobile links and types of movement performed by them; number and type of rack elements; number, class and mobility of kinematic pairs, number and type of kinematic chains. In addition, the introduction of each excess connection and local mobility into the structure of the cam mechanism is substantiated. The determining conditions when choosing a block diagram are: the given laws of transformation of the motion of the input and output links and the relative position of the axes of these links. If the axes of the input and output links are parallel, then a flat scheme of the mechanism is selected. With intersecting or crossing axes, a spatial scheme must be used. In cam mechanisms operating under the influence of small force factors, an output link with a pointed working surface is used. In cam mechanisms operating under the action of large force factors, in order to increase durability and reduce wear, a roller is introduced into the structure or the reduced radius of curvature of the contacting surfaces of the links is increased.
At the stage of metric synthesis, the main dimensions of the links of the cam mechanism and the configuration of the working surfaces of the cam profiles are determined, which ensures the implementation of the specified laws of motion and the transfer function or the maximum displacement of the output link.
9.7. LAWS OF MOTION OF THE OUTPUT LINK
If the law of motion of the output link is not specified in the terms of reference for the metric synthesis of the cam mechanism, then it must be independently selected from a set of typical laws of motion, which are divided into three groups:
unstressed laws (Fig. 9.14); laws with hard hits (Fig. 9.15); laws with soft impacts (Fig. 9.16).
The main representatives of the shockless laws of motion of the output links are: sinusoidal (Fig. 9.14, a) and trapezoidal laws of motion (Fig. 9.14, b). Both laws ensure smooth operation of the mechanism, however, they have a significant drawback, which is expressed in a slow increase in the displacement of the output link, accompanied by large values accelerations.
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dϕ 2 |
|
d 2 S |
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dϕ 2 |
|
Rice. 9.14. Unstressed laws of motion of the output link of the cam mechanism
Unstressed laws of motion of the output links are preferable from the point of view of the perception of force factors by the links of the cam mechanisms. The cams, implemented according to the shockless laws of motion, have structural profiles of a more complex configuration, the manufacture of which is technologically difficult, since it requires the use of high-precision equipment, therefore their manufacture is much more expensive. Cam mechanisms with shockless laws of output links should be used at high speeds and stringent requirements for accuracy and durability.
dϕ 2 |
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d 2 S |
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dϕ 2 |
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Rice. 9.15. Laws of motion of the output link of the cam mechanism with hard impacts
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9.7. Laws of motion of the output link |
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dϕ 2 |
dϕ 2 |
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d 2 S |
d 2 S |
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dϕ 2 |
dϕ 2 |
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Rice. 9.16. Laws of motion of the output link of the cam mechanism |
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with soft strokes |
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The main representatives of the laws of motion of output links with hard impacts are: linear (Fig. 9.15, a) and linear with transition curves (Fig. 9.15, b). The laws with hard impacts are characterized by the presence at the beginning and end of the phases of removal and approach of points having acceleration values theoretically equal to infinity, which causes the appearance of inertia forces in the contact zone of the cam mechanism links, also equal to infinity. This phenomenon indicates the occurrence of collision of the working surfaces of the contacting links. Hard impact laws have limited application and are used in non-critical mechanisms operating at low speeds and low durability.
To ensure the quality indicators of the cam mechanism, the laws of motion of the output links with soft impacts are the most preferable. Similar laws include: uniformly accelerated (Fig. 9.16, a), cosine (Fig. 9.16, b), linearly decreasing (Fig. 9.16, c) and linearly increasing (Fig. 9.16, d).
The laws with soft impacts allow for the presence of collision of the working surfaces of the contacting links of the cam mechanism, which occur when the acceleration values of the contact points change momentarily to the final
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9.7. Laws of motion of the output link
size. Soft hits are less dangerous. The implementation of these laws is carried out in mechanisms operating at low speeds with high durability.
In fact, combined laws are most widespread, i.e., laws of motion formed by functions of the same type or functions of different groups.
9.8. DETERMINING THE RADIUS OF THE ORIGINAL CAM CONTOUR
The overall dimensions of the cam mechanism are determined by the parameters of the original cam contour. The position of the center of rotation of the cam is aligned with the geometric center of the original contour and must satisfy the following condition: the current value of the pressure angle at any point of the structural profile of the cam must not exceed the allowable value. If the cam is flat and rotates, then its initial contour is a circle. In this case, the process of searching for the original contour is reduced to determining its radius.
In most cases, the cam only rotates in one direction, however, when carrying out repairs, it is necessary to be able to reverse the movement of the cam. When the direction of movement changes, the phases of removal and approach are reversed. To determine the area of admissible solutions, i.e., the area of \u200b\u200bthe possible location of the center of rotation
cam, a diagram is constructed S = f d dS ϕ . Graphically, the range of valid
solutions is determined by a family of tangents drawn to the resulting curve at angles of inclination with the corresponding values of the allowable pressure angle (Fig. 9.17, Fig. 9.18).
The choice of the center of rotation of the cam is made only within the region of feasible solutions. In this case, the smallest overall dimensions of the cam mechanism must be ensured. The minimum radius of the original contour R min is obtained by connecting the vertex of the region of feasible solutions of point O with the origin of the coordinate system point 0, i.e. R 0 = R min
(fig. 9.17, fig. 9.18).
The radius of the initial contour of the axial cam mechanisms with a pusher, when the phase angles of removal and approach are equal (Fig. 9.17, but) corresponds to the minimum radius, i.e. R 0 \u003d R min. Determination of the radius of the initial contour of axial cam mechanisms with a pusher with an inequality of phase angles of removal and approach (Fig. 9.17, b) is carried out by connecting the origin of the coordinate system of point 0 with point O 1located in the area of admissible solutions and which is the point of intersection of the path axis with one of tangents, i.e. R 0 = R 1 .
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9. CAM MECHANISMS
9.8.
Rmin |
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Rmin |
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Rice. 9.17. Schemes for determining the radius of the initial contour of cam mechanisms with a pusher
To determine the radius of the initial contour of deaxial cam mechanisms with a pusher, it is necessary to draw two straight lines parallel to the path axis S, offset relative to the path axis by an amount proportional to the eccentricity value (Fig. 9.17). At the intersection of the tangents, limiting the area of feasible solutions, with these straight lines, we find the points O 2 and O 3 . We connect the points O 2 and O 3 with the center of the origin of the coordinate system at point 0. The resulting radii R 2 and R 3 will be slightly larger than the minimum radius of the original contour R min .
For deaxial cam mechanisms with a pusher, if the phase angles of removal and approach are equal (Fig. 9.17, a), the radii R 2 and R 3 will be equal in magnitude. In this case, the radius corresponding to the specified location of the eccentricity (right or left) is taken as the radius of the initial contour. For deaxial cam mechanisms with a pusher, if the phase angles of removal and approach are not equal (Fig. 9.17, b), the radii R 2 and R 3 will not be equal in magnitude. In this case, the radius that has a smaller value is taken as the radius of the original contour. AT
in particular, R 2 > R 3 , i.e. R 0 = R 3 .
In cam mechanisms with a rocker arm for a given center distance a w, we find the positions of the points O 4 and O 5 at the intersection of an arc with a radius R \u003d a w drawn from point E with tangents (Fig. 9.18, a). Connecting the points O 4 and O 5 with the origin point 0, we get the radii R 4 and R 5 . The radius with the smaller value is taken as the radius of the original contour. In particular, R 4 > R 5 , i.e. R 0 = R 4 .
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9. CAM MECHANISMS
9.8. Determining the radius of the initial contour of the cam
Rmin |
Rmin |
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Rice. 9.18. Schemes for determining the radius of the original contour |
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cam mechanisms with rocker arm |
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To determine the radius of the initial contour of the cam mechanisms with a rocker arm at a given angle ϕ 0, we find the positions of the points O 6 and O 7 at the intersection of a straight line drawn through the point E at an angle ϕ 0 plotted from
axes of the analog of speed d dS ϕ with tangents (Fig. 9.18, b). By connecting the points O 6 and
O 7 with the origin point 0, we get the radii R 6 and R 7 . The radius with the smaller value is taken as the radius of the original contour. In particular, R 6 > R 7 , i.e. R 0 = R 7 .
9.9. SELECT RADIUS ROLL
When choosing the radius of the roller, the following provisions are guided -
1. The roller is a simple part, the manufacturing process of which is not complicated. Therefore, high contact strength can be ensured on its working surface. For the cam, due to the complex configuration of the working surface, ensuring high contact strength is very difficult. In order to ensure a sufficient ratio of contact strengths of the working surfaces of the cam and the roller, when choosing the radius of the roller rroll, the following condition is taken into account:
r roll \u003d 0.4 R 0,
where R 0 is the radius of the original cam contour.
The fulfillment of this ratio provides an approximate equality of the contact strengths of the working surfaces of the cam and roller. Radius ro-
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9. CAM MECHANISMS
9.9. Roller Radius Selection
the face is much smaller than the radius of the initial contour of the cam, therefore, the roller rotates with a greater angular velocity, and the points of its working surface come into much more contacts, which leads to uneven wear of the contact surfaces of the cam and roller. To ensure uniform wear of the working surfaces of the cam and roller, the roller surface must have a greater contact strength.
2. The constructive (working) profile of the cam should not be pointed or cut off (Fig. 9.19, but). Therefore, a restriction is imposed on the choice of the radius of the roller:
r roll = 0.7 ρ min,
where ρ min is the minimum radius of curvature of the theoretical cam profile.
A pointed or cut cam profile (Fig. 9.19, b) will not allow the roller to roll over its top, which leads to damage to the working surfaces of both links and to loss of cam mechanism performance.
3. The value of the radius of the roller is selected from a standard range of natural integers in the following range:
r roll \u003d (0.35 - 0.45) R 0.
When choosing the roller radius, the following points must be additionally taken into account: an increase in the roller radius value leads to an increase in the dimensions and mass of the output link, which worsens the dynamic characteristics of the cam mechanism and reduces the angular velocity of the roller. A decrease in the roller radius value leads to an increase in the dimensions of the cam and its mass, which causes an increase in the angular velocity of the roller and a decrease in the load capacity and service life of the cam mechanism.
ρmin
Rice. 9.19. Scheme of forming the top of the constructive profile of the Cam
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9. CAM MECHANISMS
9.9. Roller Radius Selection
AT In some cases, the introduction of an additional link (roller) into the structure of the cam mechanism is impossible for a number of reasons. In this case, there is no local mobility that replaces sliding friction with rolling friction, and a very small working area with a curved surface is provided on the output link. The points of the curved section slide along the working surface of the cam, i.e., the wear of the output link surface is more intense. To reduce wear, the working section of the output link is rounded. An increase in the rounding radius does not cause an increase in the dimensions and mass of the output link, however, it leads to a decrease in the dimensions of the structural profile of the cam. Based on this, the radius of curvature of the working surface of the output link can be taken quite large in value.
9.10. SYNTHESIS OF PROFILES OF FLAT ROTARY MOVEMENT cams
De-axial cam mechanisms with a pusher . Building a pro
cam lei is carried out in the following sequence (Fig. 9.20):
1. μ l .
3. From the selected point About in the scale factor of lengths, concentric circles with radii R 0 and e are drawn.
4. To a circle with a radius e a tangent is drawn to the intersection with
circle R 0 , the resulting intersection point is the origin of the path axis S .
7. From each division point, tangents are drawn to a circle with a radius e.
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9. CAM MECHANISMS
9.10.
Rice. 9.20. Synthesis of a deaxial cam mechanism with a pusher
8. From point O, which is the center of a circle with radius R 0 , we draw circles with radii equal to the sum of R 0 and the corresponding displacement of the pusher until it intersects with the tangents to the circle with radius e.
For the synthesis of deaxial cam mechanisms with a pusher and a roller, it is necessary to additionally perform the following:
10. r roll.
Axial cam mechanisms with a pusher . Building a pro
cam lei is carried out in the following sequence ( fig. 9.21):
1. The length scale factor is determinedμ l .
2. An arbitrary point is chosen in the free space O, which is the center of the original cam contour.
3. From the selected point About in the scale factor of lengths, a circle with radius R 0 is drawn.
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9. CAM MECHANISMS
9.10. Synthesis of profiles of flat rotary motion cams
work profile
Theoretical Profile
Rice. 9.21. Synthesis of an axial cam mechanism with a pusher
4. The axis of the path S is aligned with the vertical axis of symmetry of the circumferential
sti with radius R 0 . At the intersection of the path axis S with a circle of radius R 0, we get the origin point 0.
5. From the origin on a circle with a radius R 0 in the direction of rotation of the crank phase angles are plotted, and on the axis of the path in scale
coefficient μ l − displacement of the pusher.
6. Arcs of the original contour corresponding to the phase angles of removal
and approach, we divide into equal parts, the number of which is equal to the number of points included in the phases of removal and approach. We connect the obtained points with a point About, which is the center of rotation of the cam.
7. From point O, which is the center of a circle with radius R 0 , we draw circles with radii equal to the sum of R 0 and the corresponding re-
displacement of the pusher to the intersection of the straight lines connecting the point O
With division points.
8. The obtained points are connected by a smooth curve, forming a theoretical cam profile, which at this stage coincides with the working profile.
For the synthesis of axial cam mechanisms with a pusher and a roller, it is necessary to additionally perform the following:
9. Based on the given conditions, the radius of the roller is determined r roll .
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9. CAM MECHANISMS
9.10. Synthesis of profiles of flat rotary motion cams
10. From arbitrarily chosen points of the theoretical cam profile
we draw circles with radii r rol, simulating the position of the roller as part of the cam mechanism circuit.
11. Having drawn an envelope curve with respect to all positions of the roller, we obtain the working profile of the cam.
Cam mechanisms with rocker arm. The construction of cam profiles is carried out in the following sequence ( rice. 9.22):
1. The length scale factor is determinedμ l .
2. An arbitrary point is chosen in the free space O, which is the center of the original cam contour.
3. From the scheme for determining the radius of the original contour depending on
from the given conditions we transfer triangles 0EO 4 (Fig. 9.18, but) or 0EO 7
(Fig. 9.18, b).
4. From point E with radius R = 0E we draw an arc corresponding to the axis
path S.
5. From the origin on a circle with a radius R 0 in the direction of rotation of the crank phase angles are plotted, and on the axis of the path in scale
coefficient μ l - displacement of the rocker.
6. Arcs of the original contour corresponding to the phase angles of removal
and approach, we divide into equal parts, the number of which is equal to the number of points included in the phases of removal and approach. We connect the obtained points with a point About, which is the center of rotation of the cam.
7. From point O, which is the center of a circle with radius R 0, we draw circles with radii equal to the sum of R 0 and the corresponding displacement of the pusher until they intersect with straight lines connecting point O with division points.
8. The obtained points are connected by a smooth curve, forming a theoretical cam profile, which at this stage coincides with the working profile.
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9. CAM MECHANISMS
9.10. Synthesis of profiles of flat rotary motion cams
Rice. 9.22. Synthesis of a cam mechanism with a rocker arm
For the synthesis of cam mechanisms with a rocker arm and a roller, it is necessary to additionally perform the following:
9. Based on the given conditions, the radius of the roller is determined r roll .
10. From arbitrarily chosen points of the theoretical cam profile
we draw circles with radii r rol, simulating the position of the roller as part of the cam mechanism circuit.
11. Having drawn an envelope curve with respect to all positions of the roller, we obtain the working profile of the cam.
Theory of mechanisms and machines. Proc. allowance |
The main dimensions of the cam mechanisms are determined from kinematic, dynamic and constructive conditions. Kinematic the conditions are determined by the fact that the mechanism must reproduce the given law of motion. Dynamic conditions are very diverse, but the main one is that the mechanism has a high efficiency. constructive requirements are determined from the condition of sufficient strength of individual parts of the mechanism - wear resistance of contacting kinematic pairs. The designed mechanism should have the smallest dimensions.
Fig.6.4. On the power analysis of a cam mechanism with a progressively moving pusher.
Fig.6.5. To the study of the pressure angle in the cam mechanism
On fig. 6.4 shows a cam mechanism with a pusher 2, ending in a tip. If we neglect the friction in the higher kinematic pair, then the force acting on the pusher 2 from the side of the cam 1. The angle formed by the normal n-n to the profile of the cam 1. The angle formed by the normal n-n and the direction of movement of the pusher 2 is pressure angle and the angle equal to is transmission angle. If we consider the balance of the pusher 2 (Fig. 10.5) and bring all the forces to the point, then the pusher will be under the action of the driving force, the reduced resistance force T, taking into account the useful resistance, the spring force, the inertia force, and the reduced friction force F. From the equilibrium equation forces acting on pusher 2, we have
The reduced friction force T is equal to
Where is the coefficient of friction in the guides;
Guide length;
Departure of the pusher.
Then from the force balance equation we obtain that the friction force is equal to
The instantaneous efficiency of the mechanism without taking into account friction in the upper pair and the cam shaft bearing can be determined by the formula
The value of departure k of the pusher is (Fig. 6.5)
Where b is the constant distance from the point N of the support of the pusher 2 to the axis A of rotation of the cam;
Smallest radius vector of cam 1
Pusher movement 2.
From fig. 6.5 we get
From equation (6.7) we get
Then the efficiency will be equal to
From equality (6.9) it follows that the efficiency decreases with increasing pressure angle. The cam mechanism can jam if the force (Fig. 6.5) is . Jamming will occur if the efficiency is zero. Then from equality (6.9) we obtain
The critical angle at which the mechanism jams, and is the analog of the speed corresponding to this angle.
Then for the critical pressure angle we will have:
It follows from equality (6.10) that the critical pressure angle decreases with increasing distance, i.e. with an increase in the dimensions of the mechanism. Approximately, we can assume that the value of the analog of velocities corresponding to the critical angle is equal to the maximum value of this analog, i.e.
Then, if the dimensions of the mechanism and the law of motion of the pusher are given, it is possible to determine the value of the critical pressure angle . It must be borne in mind that the jamming of the mechanism usually takes place only in the lifting phase, which corresponds to overcoming the useful resistances, the inertial force of the pusher and the spring force, i.e. when some reduced resistance force T is overcome (Fig. 6.5). During the lowering phase, the jamming phenomenon does not occur.
To eliminate the possibility of jamming of the mechanism during the design, the condition is set that the pressure angle in all positions of the mechanism is less than the critical angle . If the maximum allowable pressure angle is denoted by , then this angle must always satisfy the condition
in practice, the pressure angle for cam mechanisms with a progressively moving pusher is taken
For rotary rocker cams where jamming is less likely, the maximum pressure angle is
When designing cam gears, it is possible to take into account not the pressure angle, but the transmission angle. This angle must satisfy the conditions
6.4. Determination of the pressure angle through the basic parameters of the cam mechanism
The pressure angle can be expressed in terms of the basic parameters of the cam mechanism. To do this, consider a cam mechanism (Fig. 6.4) with a progressively moving pusher 2. Draw a normal in t. and find the instantaneous center of rotation in relative motion links 1 and 2. From we have:
It follows from equality (6.13) that for the chosen law of motion and size, the dimensions of the cam are determined by the radius , we get smaller pressure angles , but larger dimensions of the cam mechanism.
Conversely, if we decrease , then the pressure angles increase and the efficiency of the mechanism decreases. If in the mechanism (Fig. 6.5) the axis of movement of the pusher passes through the axis of rotation of the cam and , then equality (6.13) takes the form
LECTURE 17-18
L-17Summary: Purpose and scope of cam mechanisms, main advantages and disadvantages. Classification of cam mechanisms. Basic parameters of cam mechanisms. The structure of the cam mechanism. Cyclogram of the cam mechanism.
L-18 Summary: Typical laws of motion of the pusher. Mechanism operability criteria and pressure angle during transmission of motion in the higher kinematic pair. Statement of the problem of metric synthesis. Stages of synthesis. Metric synthesis of a cam mechanism with a progressively moving pusher.
Test questions.
Cam mechanisms:
Kulachkov a three-link mechanism with a higher kinematic pair is called the input link, which is called the cam, and the output link is called the pusher (or rocker arm). Often, in order to replace sliding friction with rolling friction in the highest pair and reduce wear, both the cam and the pusher, an additional link is included in the mechanism diagram - a roller and a rotational kinematic pair. The mobility in this kinematic pair does not change the transfer functions of the mechanism and is a local mobility.
Purpose and scope:
Cam mechanisms are designed to convert the rotational or translational motion of the cam into a reciprocating rotational or reciprocating motion of the pusher. At the same time, in a mechanism with two moving links, it is possible to implement the transformation of movement according to a complex law. An important advantage cam mechanisms is the ability to provide accurate dwells of the output link. This advantage determined their wide application in the simplest cyclic automatic devices (camshaft) and in mechanical calculating devices (arithmometers, calendar mechanisms). Cam mechanisms can be divided into two groups. The mechanisms of the first ensure the movement of the pusher according to a given law of motion. The mechanisms of the second group provide only the specified maximum displacement of the output link - the stroke of the pusher. In this case, the law by which this movement is carried out is selected from a set of typical laws of motion, depending on the operating conditions and manufacturing technology.
Classification of cam mechanisms:
Cam mechanisms are classified according to the following criteria:
- according to the arrangement of links in space
- spatial
- flat
- according to the movement of the cam
- rotational
- progressive
- according to the movement of the output link
- reciprocating (with pusher)
- reciprocating rotational (with rocker arm)
- by video availability
- with roller
- without roller
- by type of cam
- disk (flat)
- cylindrical
- according to the shape of the working surface of the output link
- flat
- pointed
- cylindrical
- spherical
- according to the method of closing the elements of the higher pair
- power
- geometric
In case of force closing, the removal of the pusher is carried out by the action of the contact surface of the cam on the pusher (the driving link is the cam, the driven link is the pusher). The movement of the pusher when approaching is carried out due to the elastic force of the spring or the force of the weight of the pusher, while the cam is not a leading link. In case of positive lock, the movement of the pusher during removal is carried out by the action of the outer working surface of the cam on the pusher, while approaching - by the action of the inner working surface of the cam on the pusher. In both phases of movement, the cam is the driving link, the pusher is the driven link.
Cyclogram of the cam mechanism
Rice. 2
Most cam mechanisms are cyclic mechanisms with a cycle period of 2p. In the cycle of movement of the pusher, in the general case, four phases can be distinguished (Fig. 2): removal from the closest (in relation to the center of rotation of the cam) to the farthest position, far standing (or standing in the farthest position), return from the farthest position in the closest and closest standing (standing in the closest position). Accordingly, the cam angles or phase angles are divided into:
- removal angle jy
- distance angle j d
- return angle j in
- near standing angle j b .
Amount φ y + φ d + φ in called the working angle and denote φ r. Therefore,
φ y + φ d + φ in = φ r.
The main parameters of the cam mechanism
The cam of the mechanism is characterized by two profiles: center (or theoretical) and constructive. Under constructive refers to the outer working profile of the cam. Theoretical or center a profile is called, which in the cam coordinate system describes the center of the roller (or rounding of the working profile of the pusher) when the roller moves along the constructive profile of the cam. The phase angle is called the angle of rotation of the cam. profile angle di is called the angular coordinate of the current working point of the theoretical profile, corresponding to the current phase angle ji.
In general, the phase angle is not equal to the profile angle ji¹di.
On fig. 17.2 shows a diagram of a flat cam mechanism with two types of output link: off-axis with translational motion and swinging (with reciprocating rotational motion). This diagram shows the main parameters of flat cam mechanisms.
In figure 17.2:
The theoretical profile of the cam is usually represented in polar coordinates by the dependence ri = f(di),
where ri is the radius vector of the current point of the theoretical or center profile of the cam.
Structure of cam mechanisms
In the cam mechanism with a roller, there are two mobilities for different functional purposes: W 0 \u003d 1 - the main mobility of the mechanism by which the transformation of movement is carried out according to a given law, W m = 1 - local mobility, which is introduced into the mechanism to be replaced in the highest pair of sliding friction by rolling friction.
Kinematic analysis of the cam mechanism
The kinematic analysis of the cam mechanism can be carried out by any of the methods described above. In the study of cam mechanisms with a typical law of motion of the output link, the method of kinematic diagrams is most often used. To apply this method, one of the kinematic diagrams must be defined. Since the cam mechanism is given in the kinematic analysis, its kinematic scheme and the shape of the cam constructive profile are known. The construction of a displacement diagram is carried out in the following sequence (for a mechanism with an off-axis translationally moving pusher):
- tangent to the constructive profile of the cam, a family of circles with a radius, equal to the radius roller; the centers of the circles of this family are connected by a smooth curve and the center or theoretical profile of the cam is obtained
- circles of radii are inscribed in the resulting center profile r0 and r0 +hAmax , the value of the eccentricity is determined e
- by the size of the sections that do not coincide with the arcs of circles of radii r0 and r0 +hAmax , the phase angles jwork, jу, jeng and jс
- circular arc r , corresponding to the working phase angle, is divided into several discrete sections; straight lines are drawn through the split points tangent to the circle of the eccentricity radius (these lines correspond to the positions of the axis of the pusher in its movement relative to the cam)
- on these straight lines, the segments located between the center profile and the circle of radius are measured r0
; these segments correspond to the displacements of the center of the pusher roller SVi
according to received movements SVi a diagram of the function of the position of the center of the pusher roller is constructed SВi= f(j1)
On fig. 17.4 shows a scheme for constructing a position function for a cam mechanism with a central (e = 0) translationally moving roller follower.
Typical laws of pusher motion .
When designing cam mechanisms, the law of motion of the pusher is selected from a set of typical ones.
Typical laws of motion are divided into laws with hard and soft impacts and laws without impact. From the point of view of dynamic loads, shockless laws are desirable. However, cams with such laws of motion are technologically more complex, since they require more accurate and sophisticated equipment, so their manufacture is much more expensive. The laws with hard impacts have a very limited application and are used in non-critical mechanisms at low speeds and low durability. Cams with shockless laws are advisable to use in mechanisms with high speeds of movement with stringent requirements for accuracy and durability. The most widespread are the laws of motion with soft impacts, with the help of which it is possible to provide a rational combination of the cost of manufacture and the operational characteristics of the mechanism.
After choosing the type of the law of motion, usually by the method of kinematic diagrams, a geometric-kinematic study of the mechanism is carried out and the law of displacement of the pusher and the law of change per cycle of the first transfer function are determined (see Fig. lecture 3- method of kinematic diagrams).
Table 17.1
For the exam
Performance criteria and pressure angle when transmitting motion in higher kinematic pair.
pressure angle determines the position of the normal p-p in the highest gearbox relative to the velocity vector and the contact point of the driven link (Fig. 3, a, b). Its value is determined by the dimensions of the mechanism, the transfer function and the movement of the pusher S .
Motion transmission angle γ- angle between vectors υ 2 and υ rel absolute and relative (with respect to the cam) velocities of the point of the pusher, which is located at the point of contact BUT(Fig. 3, a, b):
If we neglect the friction force between the cam and the pusher, then the force that sets the pusher in motion (driving force) is pressure Q cam attached to the pusher at the point BUT and directed along the common normal p-p to the profiles of the cam and pusher. Let's decompose the force Q into mutually perpendicular components Q1 and Q 2 , of which the first is directed in the direction of velocity υ 2 . Strength Q1 moves the pusher, while overcoming all useful (associated with the implementation of technological tasks) and harmful (friction forces) resistance applied to the pusher. Strength Q2 increases the friction forces in the kinematic pair formed by the pusher and the rack.
Obviously, as the angle decreases γ strength Q1 decreases and strength Q 2 increases. For some value of the angle γ it may turn out that the power Q1 will not be able to overcome all the resistances applied to the pusher, and the mechanism will not work. Such a phenomenon is called jamming mechanism, and the angle γ , at which it takes place, is called the wedging angle γ cont.
When designing a cam mechanism, the permissible value of the pressure angle is set additional, ensuring the fulfillment of the condition γ ≥ γ min > γ con , i.e. current angle γ in no position of the cam mechanism must be less than the minimum transmission angle γm in and significantly exceed the jamming angle γ con .
For cam mechanisms with a progressively moving pusher, it is recommended γ min = 60°(Fig. 3, a) and γ min = 45°- mechanisms with a rotating pusher (Fig. 3, b).
Determination of the main dimensions of the cam mechanism.
The dimensions of the cam mechanism are determined taking into account the permissible pressure angle in the upper pair.
Condition that the position of the center of rotation of the cam must satisfy O 1 : pressure angles in the retraction phase at all points of the profile must be less than the allowable value. Therefore, graphically, the area of the location of the point O 1 can be determined by a family of straight lines drawn at an admissible pressure angle to the vector of possible velocity of the point of the center profile belonging to the pusher. A graphical interpretation of the above for the pusher and rocker arm is given in fig. 17.5. At the removal phase, a dependency diagram is built S B = f(j1). Since with a rocker a point AT moves along a circular arc of radius l BC , then for a mechanism with a rocker arm, the diagram is constructed in curvilinear coordinates. All constructions on the diagram are carried out on the same scale, that is m l = m Vq = m S .
In the synthesis of a cam mechanism, as in the synthesis of any mechanism, a number of tasks are solved, of which two are considered in the TMM course:
selection of a block diagram and determination of the main dimensions of the mechanism links (including the cam profile).
Stages of synthesis
The first stage of synthesis is structural. The block diagram determines the number of links in the mechanism; number, type and mobility of kinematic pairs; the number of redundant connections and local mobility. In structural synthesis, it is necessary to justify the introduction of the mechanism of each excess bond and local mobility into the scheme. The determining conditions for choosing a block diagram are: a given type of motion transformation, the location of the axes of the input and output links. The input motion in the mechanism is converted into output, for example, rotational to rotational, rotational to translational, etc. If the axes are parallel, then a flat mechanism scheme is selected. With intersecting or crossing axes, a spatial scheme must be used. In kinematic mechanisms, the loads are small, so pushers with a pointed tip can be used. In power mechanisms, to increase durability and reduce wear, a roller is introduced into the mechanism circuit or the reduced radius of curvature of the contact surfaces of the upper pair is increased.
The second stage of synthesis is metric. At this stage, the main dimensions of the links of the mechanism are determined, which provide a given law for the transformation of movement in the mechanism or a given transfer function. As noted above, the transfer function is a purely geometric characteristic of the mechanism, and, therefore, the problem of metric synthesis is a purely geometric problem, independent of time or speed. The main criteria that the designer is guided by when solving problems of metric synthesis are: minimization of dimensions, and, consequently, mass; minimizing the pressure angle in your pair; obtaining a manufacturable form of the cam profile.
Statement of the problem of metric synthesis
Given:
Block diagram of the mechanism; output link motion law S
B =
f(j1)
or its parameters - h
B, jwork = jу + jeng + jс, admissible pressure angle -
|J|
Additional Information: roller radius r p, camshaft diameter d in, eccentricity e(for a mechanism with a pusher moving forward) ,
center distance a w and rocker arm length l BC (for a mechanism with a reciprocating rotational movement of the output link).
Define:
cam starter radius r
0
; roller radius r
0
; coordinates of the center and structural profile of the cam ri = f(di)
and, if not specified, then the eccentricity e and center distance a
w.
Algorithm for designing a cam mechanism according to the allowable pressure angle
Center selection is possible in the shaded areas. Moreover, you need to choose so as to ensure the minimum dimensions of the mechanism. Minimum Radius r 1 * we get, if we connect the vertex of the obtained area, the point About 1* , with the origin. With this choice of radius at any point of the profile in the removal phase, the pressure angle will be less than or equal to the allowable one. However, the cam must be made with an eccentricity e* . With zero eccentricity, the radius of the initial washer is determined by the point About e0 . The value of the radius in this case is equal to r e 0 , which is much larger than the minimum. When the output link is a rocker arm, the minimum radius is determined similarly. Cam Starter Radius r 1aw at a given center distance aw , is determined by the point Oh 1aw , the intersection of an arc of radius aw with the corresponding boundary of the region. Normally the cam only rotates in one direction, but for repair work it is desirable to be able to rotate the cam in the opposite direction, i.e. to allow the camshaft to reverse. When changing the direction of movement, the phases of removal and approach are reversed. Therefore, to select the radius of a cam moving in reverse, it is necessary to take into account two possible removal phases, that is, to build two diagrams S B= f(j1) for each of the possible directions of movement. The choice of radius and related dimensions of the reversible cam mechanism is illustrated by the diagrams in fig. 17.6.
In this figure:
r1- the minimum radius of the initial washer of the Cam;
r 1е- radius of the initial washer at a given eccentricity;
r 1aw- radius of the initial washer at a given center distance;
aw 0- center distance at minimum radius.
Roller Radius Selection
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