LECTURE 17-18
L-17Summary: Purpose and scope cam mechanisms, main advantages and disadvantages. Classification of cam mechanisms. Basic parameters of cam mechanisms. Structure of the cam mechanism. Cyclogram of the cam mechanism operation.
L-18 Summary: Typical laws of pusher motion. Criteria for the performance of the mechanism and the pressure angle during the 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.
Control questions.
Cam mechanisms:
Kulachkov called a three-link mechanism with a higher kinematic pair, the input link is called a cam, and the output link is called a pusher (or rocker arm). Often, to replace sliding friction in the higher pair with rolling friction and reduce wear of both the cam and the pusher, an additional link is included in the mechanism design - a roller and a rotational kinematic pair. The mobility in this kinematic pair does not change the transfer functions of the mechanism and is local mobility.
Purpose and scope:
Cam mechanisms are designed to convert the rotary or translational motion of a cam into the reciprocating or reciprocating motion of a follower. At the same time, in a mechanism with two moving links, it is possible to realize the transformation of motion according to a complex law. An important advantage cam mechanisms is the ability to ensure precise alignment of the output link. This advantage determined their widespread use in the simplest cyclic automation devices (camshaft) and in mechanical computing 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 movement 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 standard laws of motion depending on operating conditions and manufacturing technology.
Classification of cam mechanisms:
Cam mechanisms are classified according to the following criteria:
- by location of links in space
- spatial
- flat
- by cam movement
- rotational
- progressive
- by the movement of the output link
- reciprocating (with pusher)
- reciprocating rotation (with rocker arm)
- according to video availability
- with roller
- without roller
- by the type of cam
- disk (flat)
- cylindrical
- according to the shape of the working surface of the output link
- flat
- pointed
- cylindrical
- spherical
- by the method of closing the elements of the higher pair
- power
- geometric
During a force closure, the pusher is removed 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 the driving link. With geometric closure, the movement of the pusher when moving away is carried out by the action of the outer working surface of the cam on the pusher, and when approaching - by the action of the inner working surface of the cam on the pusher. In both phases of movement, the cam is the leading link, the pusher is the driven link.
Cyclogram of the cam mechanism operation
Rice. 2
Most cam mechanisms are cyclic mechanisms with a cycle period equal to 2p. In the pusher movement cycle, in general, four phases can be distinguished (Fig. 2): removal from the closest (relative to the center of rotation of the cam) to the farthest position, farthest position (or standing in the farthest position), return from the farthest position in the closest and closest standing (standing in the closest position). According to this, cam rotation angles or phase angles are divided into:
- offset angle jy
- far standing angle j d
- return angle j in
- near standing angle j b .
Amount φ y + φ d + φ v is called the working angle and is designated φ r. Therefore,
φ y + φ d + φ c = φ r.
Main parameters of the cam mechanism
The mechanism cam is characterized by two profiles: center (or theoretical) and constructive. Under constructive refers to the outer working profile of the cam. Theoretical or center is a profile that, in the cam coordinate system, describes the center of the roller (or the rounding of the working profile of the pusher) when the roller moves along the structural profile of the cam. The phase angle is called the angle of rotation of the cam. Profile angle di is the angular coordinate of the current operating 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.
In Fig. Figure 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 cam profile is usually represented in polar coordinates by the relationship 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 movements of different functional purpose: W 0 = 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 replace sliding friction in the higher pair with rolling friction.
Kinematic analysis of the cam mechanism
Kinematic analysis of the cam mechanism can be carried out by any of the methods described above. When studying 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, it is necessary to define one of the kinematic diagrams. Since the cam mechanism is specified during kinematic analysis, its kinematic diagram and the shape of the structural profile of the cam are known. The displacement diagram is constructed in the following sequence (for a mechanism with an off-axis translationally moving pusher):
- a family of circles with a radius equal to the radius of the roller is constructed, tangent to the structural profile of the cam; 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 fit into the resulting center profile r0 and r0 +hAmax ,the magnitude of the eccentricity is determined e
- by the size of areas that do not coincide with the arcs of circles of radii r0 and r0 +hAmax , the phase angles jwork, jу, jдв and jс are determined
- arc of a circle r , corresponding to the operating phase angle, is divided into several discrete sections; through the splitting points, straight lines are drawn tangentially 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 r 0
; these segments correspond to the movements of the center of the pusher roller SВi
based on received movements SВi a diagram of the position function of the center of the pusher roller is constructed SВi= f(j1)
In Fig. Figure 17.4 shows a diagram of 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 standard 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, as they require more precise and complex equipment, and therefore are significantly more expensive to manufacture. Laws with hard impacts have very limited application and are used in non-critical mechanisms at low speeds and low durability. It is advisable to use cams with shockless laws in mechanisms with high speeds of movement with strict 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 ensure a rational combination of manufacturing costs and performance characteristics mechanism.
After choosing the type of law of motion, usually using the method of kinematic diagrams, a geometric-kinematic study of the mechanism is carried out and the law of movement of the pusher and the law of change per cycle of the first transfer function are determined (see. lecture 3- method of kinematic diagrams).
Table 17.1
For the exam
Performance criteria and pressure angle during motion transmission V higher kinematic pair.
Pressure angle defines 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 (relative to the cam) speeds of that point of the pusher, which is located at the point of contact A(Fig. 3, a, b):
If we neglect the frictional force between the cam and the pusher, then the force driving the pusher (driving force) is pressure Q cam applied to the pusher at the point A and directed along the common normal p-p to the cam and follower profiles. Let's break down the power Q into mutually perpendicular components Q 1 And Q 2, of which the first is directed in the direction of speed υ 2. Force Q 1 moves the pusher, while overcoming all useful (related to the performance of technological tasks) and harmful (friction forces) resistance applied to the pusher. Force Q 2 increases the friction forces in the kinematic pair formed by the pusher and the stand.
Obviously, with decreasing angle γ force Q 1 decreases and strength Q 2 increases. At a certain angle γ it may turn out that the force Q 1 will not be able to overcome all the resistance applied to the pusher, and the mechanism will not work. This phenomenon is called jamming mechanism, and the angle γ , at which it occurs is called the wedging angle γ seal
When designing a cam mechanism, the permissible value of the pressure angle is specified extra, ensuring the fulfillment of the condition γ ≥ γ min > γ close , i.e. current angle γ at no point in the cam mechanism shall the minimum transmission angle be less than γ m in and significantly exceed the jamming angle γ close .
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 top pair.
Condition that must be satisfied by the position of the center of rotation of the cam ABOUT 1 : pressure angles during the removal phase at all points of the profile must be less than the permissible value. Therefore, graphically the area of the point location ABOUT 1 can be determined by a family of straight lines drawn at an allowable pressure angle to the vector of the possible speed of the center profile point belonging to the pusher. A graphical interpretation of the above for the pusher and rocker arm is given in Fig. 17.5. During the removal phase, a dependency diagram is constructed S B = f(j1). Since at the rocker the point IN moves along an arc of a circle of radius lBC, 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 .
When synthesizing a cam mechanism, as in the synthesis of any mechanism, a number of problems are solved, two of which are considered in the TMM course:
choice block diagram and determination of the main dimensions of the mechanism links (including the cam profile).
Synthesis stages
The first stage of synthesis is structural. The block diagram determines the number of links of the mechanism; number, type and mobility of kinematic pairs; number of redundant connections and local mobility. During structural synthesis, it is necessary to justify the introduction of each redundant connection and local mobility into the mechanism diagram. The determining conditions when choosing a structural diagram are: the specified type of motion transformation, the location of the axes of the input and output links. The input movement in the mechanism is converted into output, for example, rotational into rotational, rotational into translational, etc. If the axes are parallel, then a flat mechanism diagram is selected. When intersecting or intersecting axes, it is necessary to use a spatial diagram. 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 contacting surfaces of the highest pair is increased.
The second stage of synthesis is metric. At this stage, the main dimensions of the mechanism links are determined, which provide the given law of transformation of motion in the mechanism or the 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 speeds. The main criteria that guide the designer when solving problems of metric synthesis are: minimizing dimensions, and, consequently, mass; minimizing the pressure angle in the upper steam; obtaining a technologically advanced cam profile shape.
Statement of the problem of metric synthesis
Given:
Block diagram of the mechanism; law of motion of the output link S
B =
f(j1)
or its parameters - h
B, jwork = jу + jdv + jс, permissible pressure angle -
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Additional Information: Roller Radius r p, cam shaft diameter d c, eccentricity e(for a mechanism with a pusher moving progressively) ,
center distance a wi and rocker length l BC (for a mechanism with reciprocating rotation of the output link).
Define:
radius of the initial cam washer r
0
; roller radius r
0
; coordinates of the center and structural profile of the cam r i = f(di)
and, if not specified, then eccentricity e and center distance a
w.
Algorithm for designing a cam mechanism based on the permissible pressure angle
Center selection is possible in shaded areas. Moreover, you need to choose in such a way as to ensure the minimum dimensions of the mechanism. Minimum radius r 1 * we get, if we connect the vertex of the resulting region, the point About 1* , with the origin. With this choice of radius, at any point in the profile during the removal phase, the pressure angle will be less than or equal to the permissible one. However, the cam must be made with eccentricity e* . At zero eccentricity, the radius of the initial washer will be determined by the point O e0 . The radius is equal to r e 0 , that is, significantly more than the minimum. With the output link - a rocker arm, the minimum radius is determined similarly. Cam Starter Radius r 1aw at a given center distance aw , determined by the point About 1aw , the intersection of an arc of radius aw with the corresponding boundary of the region. Usually the cam rotates in only one direction, but when carrying out repair work, it is desirable to be able to rotate the cam in the opposite direction, that is, to ensure the possibility of reverse movement of the cam shaft. When changing the direction of movement, the phases of removal and approach change places. Therefore, to select the radius of a cam moving reversely, it is necessary to take into account two possible phases of removal, that is, construct two diagrams S B= f(j1) for each of the possible directions of movement. The choice of radius and associated dimensions of the reversible cam mechanism is illustrated by the diagrams in Fig. 17.6.
In this picture:
r 1- minimum radius of the initial cam washer;
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.
Selecting the roller radius
Advantages of cam mechanisms
All mechanisms with VKP are small-linked, therefore, they make it possible to reduce the dimensions of the machine as a whole.
Ease of synthesis and design.
Mechanisms with VCP more accurately reproduce the transfer function.
Provide a wide variety of laws of motion of the output link.
Mechanisms with VKP must have a force or geometric closure.
The contact forces in the VCP are much higher than in the NCP, which leads to wear, i.e. 2 profiles lose their shape and, as a result, their main advantage.
Difficulty in processing the cam profile.
Inability to operate at high speeds and transmit large powers.
Main parameters of the cam mechanism
The cam profile can be composed of arcs of two concentric circles and curves that transition from one circle to another.
Most cam mechanisms are cyclic mechanisms with an equal cycle period. When the cam rotates, the pusher makes a reciprocating or reciprocating rotational movement with a stop in the upper and lower positions. Thus, in the pusher movement cycle, in general, four phases can be distinguished: moving away, far standing (or standing), approaching and close standing. According to this, cam rotation angles or phase angles are divided into:
Removal (ascent) angle
Angle of far (upper) stand
Angle of approach (descent)
Angle of near (lower) stand.
The sum of three angles forms an angle called the working angle
In particular cases, the angles of the upper and lower height may be missing, then.
The mechanism cam is characterized by two profiles:
Center (or theoretical)
Constructive (or working).
Under constructive refers to the outer working profile of the cam.
Theoretical or center is a profile that, in the cam coordinate system, describes the center of the roller (or the rounding of the working profile of the pusher) when the roller moves along the structural profile of the cam.
Phase called the angle of rotation of the cam.
Profile angle is called the angular coordinate of the current operating point of the theoretical profile, corresponding to the current phase angle. In general, the phase angle is not equal to the profile angle.
The movement of the pusher and the angle of rotation of the cam are counted from the beginning of the lifting phase, i.e. from the lowest position of the roller center, located at a distance from the center of rotation of the cam. This distance is called - initial radius or the radius of the zero initial washer and coincides with the minimum radius vector of the cam center profile.
The maximum displacement of the output link is called pusher stroke.
Off-axis of the pusher - eccentricity - for cams with a translationally moving pusher.
Center distance - the distance between the center of rotation of the cam and the fixed point of the rocker arm - for cams with a rocker pusher.
The pressure angle is the angle between the velocity at the point of contact and the normal to the profile (i.e. the direction of the force). Usually this angle is designated or. And at one point of contact, the two profiles have a different pressure angle.
Without taking into account friction, the force is directed along the common normal at the point of contact of the profiles. Thus, in a cam mechanism, the pressure angle is the angle between the normal to the center profile of the cam and the speed of the center of the roller.
The dimensions of the cam mechanism are determined from kinematic, dynamic and structural conditions.
- Kinematic conditions – ensuring the reproduction of the given law of movement of the pusher.
- Dynamic – ensuring high efficiency and no jamming.
- Structural – ensuring the minimum dimensions of the mechanism, strength and wear resistance.
Geometric interpretation of the pusher speed analogue
The cam and the pusher form the VCP. The pusher moves translationally, therefore, its speed is parallel to the guide. The cam performs a rotational movement, so its speed is directed perpendicular to the radius of rotation at the current point and the relative sliding speed of the profiles is directed along a common tangent to them.
where, a is the engagement pole in the VCP, which is located at the intersection of the normal to the profiles at the point of contact with the line of centers. Because The pusher moves translationally, then its center of rotation lies at infinity, and the line of centers runs perpendicular to the speed through the center of the cam.
The velocity triangle and are similar as triangles with mutually perpendicular sides, i.e. the ratio of their corresponding sides is constant and equal to the similarity coefficient: , whence.
Those. The analogue of the pusher speed is depicted by a segment perpendicular to the pusher speed, which is cut off by a straight line parallel to the contact normal and passing through the center of the cam.
Synthesis formulation: If, in the continuation of the ray drawn from the center of the roller perpendicular to the speed of the pusher, a segment of length is set aside from the point and a straight line parallel to the contact normal is drawn through the end of this segment, then this straight line will pass through the center of rotation of the driving link (cam) point.
Thus, in order to obtain a segment depicting an analogue of the pusher speed, the pusher speed vector must be rotated in the direction of rotation of the cam.
The influence of pressure angle on the operation of the cam mechanism
A decrease in the initial radius of the cam, other things being equal, leads to an increase in pressure angles. With increasing pressure angles, the forces acting on the mechanism links increase, the efficiency of the mechanism decreases, and the possibility of self-braking (jamming of the mechanism) arises, i.e. no force from the driving link (cam) can move the driven link (pusher) from its place. Therefore, to ensure reliable operation of the cam mechanism, it is necessary to select its main dimensions so that the pressure angle in any position does not exceed a certain permissible value.
When determining the main dimensions of a cam mechanism with a rocker pusher, it is enough that the pressure angle in any of the positions of the mechanism does not exceed; for a cam mechanism with a progressively moving roller pusher, it is enough that the pressure angle in any of the positions of the mechanism does not exceed.
Synthesis of the cam mechanism. Synthesis stages
When synthesizing a cam mechanism, as in the synthesis of any mechanism, a number of problems are solved, of which two are considered in the TMM course: choosing a structural diagram and determining the main dimensions of the mechanism links (including the cam profile).
The first stage of synthesis is structural. The block diagram determines the number of links of the mechanism; number, type and mobility of kinematic pairs; number of redundant connections and local mobility. During structural synthesis, it is necessary to justify the introduction of each redundant connection and local mobility into the mechanism diagram. The determining conditions when choosing a structural diagram are: the specified type of motion transformation, the location of the axes of the input and output links. The input movement in the mechanism is converted into output, for example, rotational into rotational, rotational into translational, etc. If the axes are parallel, then a flat mechanism diagram is selected. When intersecting or intersecting axes, it is necessary to use a spatial diagram. 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 contacting surfaces of the highest pair is increased.
The second stage of synthesis is metric. At this stage, the main dimensions of the mechanism links are determined, which provide the given law of transformation of motion in the mechanism or the 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 speeds. The main criteria that guide the designer when solving problems of metric synthesis are: minimizing dimensions, and, consequently, mass; minimizing the pressure angle in the upper steam; obtaining a technologically advanced cam profile shape.
Selecting the roller radius (rounding the working area of the pusher)
When choosing the roller radius, the following considerations are used:
The roller is a simple part, the processing of which is simple (it is turned, then heat treated and ground). Therefore, high contact strength can be ensured on its surface. In a cam, due to the complex configuration of the working surface, this is more difficult to ensure. Therefore, usually the radius of the roller is less than the radius of the initial washer of the structural profile and satisfies the relation where is the radius of the initial washer of the theoretical cam profile. Compliance with this ratio ensures approximately equal contact strength for both the cam and the roller. The roller has greater contact strength, but since its radius is smaller, it rotates at a higher speed and the working points of its surface are involved in a greater number of contacts.
The structural profile of the cam should not be pointed or cut off. Therefore, a restriction is imposed on the choice of roller radius, where is the minimum radius of curvature of the theoretical cam profile.
It is recommended to select a roller radius from a standard range of diameters in the range. It must be taken into account that an increase in the radius of the roller increases the dimensions and weight of the pusher, worsens the dynamic characteristics of the mechanism (reduces its natural frequency). Reducing the roller radius increases the dimensions of the cam and its weight; The roller rotation speed increases, its durability decreases.
3.3. Phases of operation of cam mechanisms. Phase and design angles
Cam mechanisms can implement 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 the output link (tappet or rocker arm) as the point of contact between the cam and the pushrod moves away from the center of rotation of the cam.
2. Return (approach) phase. 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. Standing phases. A situation where, with a rotating cam, the point of contact between the cam and the pusher is stationary. At the same time, they distinguish close phase– when the contact point is in the closest position to the center of the cam, long-dwelling phase– when the point of contact is at the farthest position from the center of the cam and intermediate phases. Dwell phases occur when the point of contact moves along a portion of the cam profile that is shaped like a circular arc drawn from the center of rotation of the cam.
The above classification of phases primarily relates to positional mechanisms.
Each phase of operation has its own phase angle of operation of the mechanism and the design angle of the cam.
The phase angle is the angle through which the cam must rotate in order for the corresponding phase of operation to complete. These angles are designated by the letter with an index indicating the type of phase, for example, U – removal phase angle, D – far phase angle, B – return phase angle, B – near phase angle.
The design angles of the cam determine its profile. They are designated by the letter with the same indices. In Fig. Figure 3.2a shows these angles. They are limited by rays drawn from the center of rotation of the cam to points on its center profile at which the profile of the cam 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 eccentricity, is installed in the position corresponding to the beginning of the removal phase; To– 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. From the construction it is clear that in order for the point To took the position To’ the cam must rotate through an angle Y, not equal to Y, but different by an angle e, called the eccentricity angle. For mechanisms with a pusher, we can write the following relationships:
U = U + e, B = B – e,
D = D, B = B
3.4. Selection 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 “throws” the output link from one extreme position to another and back.
In Fig. Figure 3.3 shows the law of motion - a graph of the movement of the pusher of such a mechanism, when the entire work process is represented by a combination of four vases: removal, long stay, return and near stay. Here is the angle of rotation of the cam, and the corresponding phase angles are designated: y, d, c, b. The movement 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 movement of the pusher. 
In this case, the choice of the law of motion consists in determining the nature of the movement of the output link during the removal and return phases. In Fig. 3.3 some kind of curve is depicted for these sections, but it is precisely this that needs to be determined. What criteria form the basis for solving this problem?
Let's go from the opposite. Let's try to do it “simple”. Let us define a linear law of displacement in the removal and return sections. In Fig. 3.4 shows what this will lead to. Differentiating the function () or S() twice, we obtain that theoretically infinite, i.e., will appear at the phase boundaries. unpredictable accelerations, and, consequently, inertial loads. This unacceptable phenomenon is called hard phase shock.
To avoid this, the choice of the law of motion is made based on the acceleration graph of the output link. In Fig. 3.5 shows an example. The desired shape of the acceleration graph is specified and the velocity and displacement functions are found by integrating it. 
The dependence of the acceleration of the output link in the removal and return phases 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 dimensions, the phenomenon is allowed soft blow, when the acceleration graph shows jumps, but by a finite, predictable amount.
In Fig. 3.6 presents examples of the most commonly used types of laws of acceleration change. The functions are shown for the delete phase; for the return phase they are similar, but mirrored. In Fig. 3.6 shows symmetric 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, during the removal (return) phase, a section with a constant speed of the output link is needed.
As a rule, the functions of the laws of acceleration have analytical expressions, in particular 3.6, a, d - sinusoid segments, 3.6, b, c, g - straight segments, 3.6, f - cosine, therefore their integration in order to obtain speed and displacement is not difficult . However, the amplitude values of the acceleration are not known in advance, but the displacement value of the output link in the removal and return phases is known. Let's consider how to find both the acceleration amplitude and all the functions that characterize the movement of the output link.
At a constant angular velocity of rotation of the cam, when the angle of rotation and time are related by the expression = t functions can be considered both from time and from the angle of rotation. We will consider them in time and in relation to a mechanism with a rocker arm.
At the initial stage, we will set the shape of the acceleration graph in the form of a normalized, that is, with unit amplitude, function *( t). For the dependence in Fig. 3.6a it will be *( t) = sin(2 t/T), where T is the time the mechanism goes through the removal or return phase. Actual acceleration of the output link:
2 (t) = m *(t), (3.1) 
where m is the amplitude still unknown.
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 function value 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, related to the dynamics of their operation. The cam can also be designed shockless, in the sense in which we meant this concept above. But at high speeds, in mechanisms with force closure, the pusher (rocker arm) can separate from the cam. After some time, the closing force restores contact, but this restoration occurs with the impact. Such phenomena can occur, for example, when the return phase is set too small. The profile of the cam then at this phase turns out to be steep and at the end of the long-dwell phase the closing force does not have time to ensure contact and the pusher seems to break off from the cam profile at the far-dwell and can even immediately hit some point of the cam at the near-dwell. For mechanisms with positive locking, the roller moves along a groove in the cam. Since there is always a gap between the roller and the walls of the groove, during operation the roller hits the walls, the intensity of these impacts also increases with increasing speed of rotation of the cam. To study these phenomena, it is necessary to create a mathematical model of the operation of the entire mechanism, but these questions are beyond the scope of this course.
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Design of cam mechanisms
Summary: Cam mechanisms. Purpose and scope. Selection of the law of motion of the cam pusher. Classification of cam mechanisms. Main parameters. Geometric interpretation of the speed analogue. The influence of pressure angle on the operation of the cam mechanism. Synthesis of the cam mechanism. Stages of synthesis. Selecting the roller radius (rounding of the pusher working area).
Cam mechanisms
The working process of many machines makes it necessary to have mechanisms in their composition, the movement of the output links of which must be carried out strictly according to a given law and coordinated with the movement of other mechanisms. The simplest, most reliable and compact ones for performing this task are cam mechanisms.
It's called kulachkov three-link mechanism with a higher kinematic pair, the input link of which is called fist, and the day off is pusher(or rocker).
With your fist called the link to which the element of the higher kinematic pair, made in the form of a surface of variable curvature, belongs.
A rectilinearly moving output link is called pusher, and the rotating (swinging) – rocker.
Often, to replace sliding friction in the higher pair with rolling friction and reduce wear of both the cam and the pusher, an additional link is included in the mechanism design - a roller and a rotational kinematic pair. The mobility in this kinematic pair does not change the transfer functions of the mechanism and is local mobility.
They reproduce the movement of the output link - the pusher - theoretically accurately. The law of movement of the pusher, specified by the transfer function, is determined by the cam profile and is the main characteristic of the cam mechanism, on which its functional properties, as well as dynamic and vibration qualities, depend. The design of a cam mechanism is divided into a number of stages: assigning the law of motion of the pusher, choosing a structural diagram, determining the main and overall dimensions, calculating the coordinates of the cam profile.
Purpose and scope
Cam mechanisms are designed to convert the rotary or translational motion of a cam into the reciprocating or reciprocating motion of a follower. An important advantage of cam mechanisms is the ability to ensure precise alignment of the output link. This advantage determined their widespread use in the simplest cyclic automation devices and in mechanical computing 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 movement 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 standard laws of motion depending on operating conditions and manufacturing technology.
Selection of the law of motion of the cam pusher
Law of motion of the pusher called the function of movement (linear or angular) of the pusher, as well as one of its derivatives, taken with respect to time or a generalized coordinate - the movement of the leading link - the cam. When designing a cam mechanism from a dynamic point of view, it is advisable to proceed from the law of change in the acceleration of the pusher, since it is the accelerations that determine the inertial forces that arise during operation of the mechanism.
There are three groups of laws of motion, characterized by the following features:
1. the movement of the pusher is accompanied by hard impacts,
2. the movement of the pusher is accompanied by soft blows,
3. The pusher moves without impact.
Very often, production conditions require the pusher to move at a constant speed. When applying such a law of movement of the pusher in the place of an abrupt change in speed, the acceleration theoretically reaches infinity, and the dynamic loads should also be infinitely large. In practice, due to the elasticity of the links, an infinitely large dynamic load is not obtained, but its magnitude still turns out to be very large. Such impacts are called “hard” and are permissible only in low-speed mechanisms and with low pusher weights.
Soft impacts accompany the operation of the cam mechanism if the speed function does not have a discontinuity, but the acceleration function (or an analogue of acceleration) of the pusher undergoes a discontinuity. An instantaneous change in acceleration by a finite value causes a sharp change in dynamic forces, which also manifests itself in the form of an impact. However, these strikes are less dangerous.
The cam mechanism operates smoothly, without shocks, if the speed and acceleration functions of the pusher do not undergo a break, change smoothly and provided that the speeds and accelerations at the beginning and end of the movement are equal to zero.
The law of motion of the pusher can be specified both in analytical form - in the form of an equation, and in graphical form - in the form of a diagram. In assignments for the course project, the following laws of change in analogues of acceleration of the center of the pusher roller are encountered, given in the form of diagrams:
Uniformly accelerated law of change in the analogue of the acceleration of the pusher; with a uniformly accelerated law of movement of the pusher, the designed cam mechanism will experience soft impacts at the beginning and end of each of the intervals.
The triangular law of changing the analogue of acceleration ensures shockless operation of the cam mechanism.
The trapezoidal law of change in the acceleration analogue also ensures shock-free operation of the mechanism.
Sinusoidal law of change of acceleration analogue. Provides the greatest smoothness of movement (characteristic is that not only speed and acceleration, but also higher order derivatives change smoothly). However, for this law of motion, the maximum acceleration at the same phase angles and stroke of the pusher turns out to be greater than in the case of the uniformly accelerated and trapezoidal laws of change of acceleration analogues. The disadvantage of this law of motion is that the increase in speed at the beginning of the ascent, and, consequently, the ascent itself occurs slowly.
The cosine law of change in the analogue of acceleration causes soft impacts at the beginning and end of the pusher stroke. However, with the cosine law, there is a rapid increase in speed at the beginning of the stroke and a rapid decrease at the end, which is desirable when operating many cam mechanisms.
From the point of view of dynamic loads, shockless laws are desirable. However, cams with such laws of motion are technologically more complex, as they require more precise and complex equipment, so their production is significantly more expensive. Laws with hard impacts have very limited application and are used in non-critical mechanisms at low speeds and low durability. It is advisable to use cams with shockless laws in mechanisms with high speeds of movement with strict 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 ensure a rational combination of manufacturing costs and operational characteristics of the mechanism.
The main dimensions of the cam mechanisms are determined from kinematic, dynamic and structural conditions. Kinematic the conditions are determined by the fact that the mechanism must reproduce the given law of motion. Dynamic The conditions are very varied, but the main thing is that the mechanism has high efficiency. Constructive the requirements are determined from the condition of sufficient strength of individual parts of the mechanism - resistance to wear of contacting kinematic pairs. The designed mechanism must have the smallest dimensions.
Fig.6.4. On the force analysis of a cam mechanism with a translational-moving pusher.
Fig.6.5. To study the pressure angle in the cam mechanism
In Fig. 6.4 shows a cam mechanism with a pusher 2, ending with a point. If we neglect 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 equilibrium of pusher 2 (Fig. 10.5) and bring all forces to point , then the pusher will be under the action of the driving force, reduced resistance force T, taking into account useful resistance, spring force, inertia force, and 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;
Pusher overhang.
Then from the force equilibrium equation we obtain that the friction force is equal to
The instantaneous efficiency of the mechanism without taking into account friction in the higher pair and the cam shaft bearing can be determined by the formula
The extension k of the pusher is equal to (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
Moving the pusher 2.
From Fig. 6.5 we get
From equation (6.7) we obtain
Then the efficiency will be equal to
From equality (6.9) it follows that the efficiency decreases with increasing pressure angle. The cam mechanism may 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 jamming of the mechanism occurs, and is the analogue of speed corresponding to this angle.
Then for the critical pressure angle we will have:
From equality (6.10) it follows that the critical pressure angle decreases with increasing distance, i.e. with increasing dimensions of the mechanism. We can approximately assume that the value of the speed analog 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, the value of the critical pressure angle can be determined. It must be borne in mind that jamming of the mechanism usually occurs only during the lifting phase, which corresponds to overcoming the useful resistance, the inertia force of the pusher and the spring force, i.e. when a certain reduced resistance force T is overcome (Fig. 6.5). During the lowering phase, the phenomenon of jamming does not occur.
To eliminate the possibility of jamming of the mechanism during design, a condition is set that the pressure angle in all positions of the mechanism is less than the critical angle. If the maximum permissible 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 cam mechanisms with a rotating rocker arm, in which jamming is less likely, the maximum pressure angle
When designing cams, you can take into account not the pressure angle, but the transmission angle in the calculations. This angle must satisfy the conditions
6.4. Determination of the pressure angle through the main parameters of the cam mechanism
The pressure angle can be expressed through the basic parameters of the cam mechanism. To do this, consider a cam mechanism (Fig. 6.4) with a progressively moving pusher 2. We draw a normal line and find the instantaneous center of rotation in the relative motion of links 1 and 2. From this we have:
From equality (6.13) it follows that with the chosen law of motion and size, the dimensions of the cam are determined by the radius, we obtain smaller pressure angles, but larger dimensions of the cam mechanism.
And vice versa, if you 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) will take the form