And the most costly cam method. Design of cam mechanisms. The choice of the law of motion of the pusher of the cam mechanism
The force acting on the pusher from the side of the cam and causing it to move is directed along the normal to the cam at the point of contact with the pusher. Therefore, in the general case, it is directed at an angle to the direction of movement of the pusher (Figure 46).
Figure 46
The angle between the force acting on the pusher and the direction of its movement is called pressure angle(denoted α), and the angle between the acting force and the direction perpendicular to the direction of movement of the pusher is called transmission angle(denoted γ). In sum, these angles make up an angle equal to 90 0 , therefore, when considering the operability of the mechanism, taking into account the direction of force transfer, you can operate with any of them.
With a decrease in the angle of transmission of motion, the driving component of the acting force decreases (the component coinciding with the direction of movement of the pusher). At the same time, the component that presses the pusher against the guides increases, increasing the friction force between the pusher and the support, which prevents the pusher from moving.
V T = S’∙ω cool
However, an increase in the minimum radius circumference leads to an increase in the dimensions, weight, and material consumption of the entire structure. Therefore, the task of dynamic synthesis is to determine such a value of r min , at which the angle of motion transmission would be not less than that allowed in all positions of the mechanism, and the dimensions would be minimal.
The solution of the problem of dynamic synthesis is carried out graphically. The following technique is used (see Figure 46b): if the segment OW is moved parallel to itself, aligning point W with point A, and draw a straight line at an angle
γ to it through the second point O, then it will pass through the center of rotation of the cam (i.e., an O-O line is formed, parallel to the normal N-N and passing through the center of rotation of the cam).
To determine r min, a diagram is built, plotting along the y-axis the values of the displacements of the pusher ( Si) for “p” positions of the mechanism in accordance with the given law of motion. From each marked point, parallel to the abscissa axis, the value of the analogue of velocities corresponding to this position is plotted ( Si'). Displacements and analogues of speeds should be plotted on the same scale (Figure 47).
Figure 47
The ends of the segments of speed analogues are connected by a smooth curve and tangents to it are drawn to the right and left at an angle γ min to the x-axis ( γ min- the minimum allowable angle of motion transmission from the condition of the absence of jamming). These two straight lines separate the allowed zone for choosing the center of rotation of the cam (below these lines) from the forbidden one.
The choice of the center of rotation of the cam at any point in the allowed zone ensures that there is no jamming in all positions of the mechanism. To ensure minimum dimensions, it is necessary to choose the center of rotation of the cam at the boundaries of the allowed zone (or with a slight deviation from the boundaries, providing some margin for the transmission angle). This method also allows the most rational choice of eccentricity.
When designing a mechanism with a rocker pusher, the approaches to solving the problem of dynamic synthesis are similar. However, in this case, the angle of transmission of motion is measured from the corresponding position of the rocker arm. Therefore, when determining the allowed zone, to select the center of rotation of the cam, rays are drawn at an angle
γ min in each rocker position. As a result, the allowed zone is determined by the intersection of several beams (Figure 48).
Figure 48
When designing a mechanism with a rocker pusher, the law of rotational motion of the rocker is set. Therefore, the parameters of the angular motion will be known (the angle of rotation of the rocker arm, the analog of the angular velocity, the analog of the angular acceleration). To determine the analogue of the speeds, which is deposited from the end of the rocker in each of its positions, it is necessary to multiply the analogue of the angular velocity by the length of the rocker:
In mechanisms with a flat pusher, the angle of motion transmission is determined by the angle between the pusher plate and the pusher itself (the axis of its translational movement). Therefore, from the point of view of the transfer of motion, the most advantageous value of this angle is 90 0 .
From the point of view of the technology of manufacturing the pusher and assembling the mechanism, the angle between the pusher and its plate, equal to 90 0, is also the most advantageous. Therefore, this case is usually used in practice. In this case, the entire force acting from the side of the cam on the pusher, in all positions of the mechanism, is the driving force (there is no component pressing the pusher to the guides).
Thus, the phenomenon of jamming for this type of mechanism is not relevant. However, the cam must have a convex profile at all points (because a flat poppet cannot work with concave areas). It turns out that the greater the value of the circle of the minimum radius, the less the probability of formation of concave sections on the profile. Therefore, in this case, a problem similar to the problem of dynamic synthesis is solved - choose r min so that there are no concave sections on the profile, and the dimensions would be minimal (in other words, r min is selected from the condition of the convexity of the cam).
The objectives of the work are:
- performing a kinematic analysis of the cam mechanism, which consists in determining the position, speed and acceleration of the pusher depending on the position of the cam;
– performing a kinematic synthesis of this mechanism, which consists in constructing a cam profile based on the known minimum radius of the latter and the diagram of the movement of the pusher.
5.1. Basic information from the theory
The cam is a link of the cam mechanism, which has a variable curvature of the profile and informs the pusher of the required law of motion. The concepts of profile and phase angles of the cam, as well as the angles of transmission of motion and pressure are given earlier in section 4.1 of the laboratory work "Synthesis of cam mechanisms".
In a kinematic study (analysis), a specific cam mechanism is considered. The study is aimed at determining the kinematic characteristics of the pusher at various positions of the cam.
The simplest and most illustrative way of kinematic research in the case of a cam mechanism with a progressively moving pusher and in the case of the same mechanism with a swinging pusher is a method based on constructing in the first indicated case an experimental diagram "displacement - time" () for the driven link, followed by its graphic integration to obtain diagrams "velocity - time" () and "acceleration - time" (), and in the second case - the experimental diagram "angle of rotation - time" (ψ = ψ( t)) for a similar link with its subsequent integration to find the diagrams "angular velocity - time" (ω = ω( t)) and "angular acceleration - time" (ε = ε( t)). On fig. 5.1. as an example, these diagrams are presented for a progressively moving pusher.
In the laboratory work, a cam mechanism is used, implemented in the form of a model, the main elements of which are the base and the pusher and cam installed on it, on which the disk is fixed. To ensure the possibility of constructing an experimental diagram (or ψ = ψ( t)) on the disk there is a scale graduated from 0 O to 360 O, and on the pusher or on the plate attached to the base, there is a scale with divisions in millimeters or degrees.
Usually in a cam mechanism, the cam moves evenly. In this case the time t the movement of the cam is proportional to the angle of its rotation φ. Therefore, diagrams and ψ = ψ( t) are both diagrams (φ) and ψ = ψ(φ).
The time scale on the charts is determined based on the following.
1) The working angle of the cam corresponds to the length of the cut l on the diagram (Fig. 5.1). Consequently,
where L is the length of the segment of the diagram corresponding to one revolution of the cam.
2) Time of one revolution
where P- number of cam revolutions per minute.
Then the time scale is
In the case of a cam mechanism with a progressively moving pusher, the scales of the displacement diagram , speed and acceleration are calculated using the known formulas:
where H 1 and H 2 – pole distances, mm; s– true displacement, m; s diagr – size on the diagram, mm.
In the case of a cam mechanism with a rocking pusher, the scales of the diagrams of the angle of rotation ψ = ψ( t), angular velocity and angular acceleration ε = ε( t) of the pusher are determined by the formulas:
In formula (5.7) ψ is the true angle of rotation, rad., ψ diagr is the size on the diagram, mm.
Kinematic diagrams, built in accordance with the above, are the basis for performing the kinematic synthesis of the cam mechanism. Features of the implementation of this synthesis are set out in the lecture course on the discipline.
5.2. Work order
1. Slowly turning the cam, fix the moment when the pusher starts to rise and the moment when it ends. On the scale on the disk, rigidly connected to the cam, determine the angle of rotation φ y. Similarly, determine the angle φ c. Each of the angles φ y and φ in divided into several ( n) equal parts (for example, six).
2. Turning the cam through angles φ i, measure the displacement of the pusher s i in millimeters or ψ i in degrees from the scale on the driven link or based on the model of the cam mechanism, first in the removal section, and then in the return section. Collect the obtained data in a table.
3. According to the table, plot a graph (or ), which is also a graph (or ).
4. Using the method of graphical differentiation, construct graphs and (or and )
5. Determine the scales of time, path, speed and acceleration using formulas (5.3) ... (5.9).
6. Perform the synthesis of the mechanism. Construct a kinematic diagram of the cam mechanism according to the dimensions obtained during its study. Required for construction of the minimum radius of the Cam r 0 , eccentricity e, distance between axles O and AT rotation of the cam and pusher, respectively, as well as the length AB the rocker arms of the pusher are measured on the model of the mechanism.
7. Show all phase and profile angles of the cam.
8. In one of the intermediate positions of the cam, show the pusher in reversed motion, and for this position determine the angle of motion transfer γ and the pressure angle α of the cam mechanism.
9. Prepare a report.
5.3. Questions for self-control
1. Which cam angles are called profile, and which are phase? What is their difference?
2. How is graphical differentiation performed?
3. How to calculate chart scales?
4. What is the essence of the motion reversal method?
5. How to build a cam profile in cam mechanisms with progressively moving and oscillating pushers?
6. What is called the angle of pressure and the angle of transmission of motion?
7. How does the pressure angle affect the operation of the cam mechanism?
8. Show pressure and transmission angles at any point on the cam profile.
3.3. Phases of the 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.
| " |
Cam design
Summary: Cam mechanisms. Purpose and scope. The choice of the law of motion of the pusher of the cam mechanism. Classification of cam mechanisms. Main parameters. Geometric interpretation of the analogue of velocity. Influence of the pressure angle on the operation of the cam mechanism. Synthesis of the cam mechanism. Stages of synthesis. Selection of the radius of the roller (rounding of the working section of the pusher).
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 most simple, reliable and compact to perform such a task are cam mechanisms.
Kulachkov is called a three-link mechanism with a higher kinematic pair whose input link is called cam, and the output pusher(or rocker).
cam is called the link to which the element of the higher kinematic pair belongs, made in the form of a surface of variable curvature.
A rectilinearly moving output link is called pusher, and rotating (swinging) - rocker.
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.
They reproduce the movement of the output link - pusher theoretically exactly. The law of movement of the pusher, given by the transfer function, is determined by the profile of the cam and is the main characteristic of the cam mechanism, on which its functional properties, as well as dynamic and vibrational qualities, depend. The design of the cam mechanism is divided into a number of stages: the assignment of the law of motion of the pusher, the choice of a block diagram, the determination of the main and overall dimensions, the calculation of the coordinates of the cam profile.
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. An important advantage of 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 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.
The choice of the law of motion of the pusher of the cam mechanism
The law of motion of the pusher called the displacement function (linear or angular) of the pusher, as well as one of its derivatives, taken in time or a generalized coordinate - the displacement 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 inertia forces that arise during the 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 blows,
2. the movement of the pusher is accompanied by soft impacts,
3. The movement of the pusher occurs without shock.
Very often, according to the conditions of production, it is necessary to move the pusher at a constant speed. When applying such a law of motion of the pusher in the place of an abrupt change in speed, the acceleration theoretically reaches infinity, and the dynamic loads must also be infinitely large. In practice, due to the elasticity of the links, an infinitely large dynamic load is not obtained, but its magnitude is still very large. Such impacts are called "hard" and are permissible only in low-speed mechanisms and with small weights of the pusher.
Soft impacts are accompanied by the operation of the cam mechanism if the speed function does not have a discontinuity, but the acceleration function (or an analog of acceleration) of the pusher undergoes a discontinuity. An instantaneous change in acceleration by a finite amount causes a sharp change in dynamic forces, which also manifests itself in the form of a shock. However, these attacks are less dangerous.
The cam mechanism operates smoothly, without shocks, if the functions of the speed and acceleration of the pusher do not undergo a break, change smoothly and provided that the speeds and accelerations at the beginning and at the end of the movement are equal to zero.
The law of motion of the pusher can be given both in analytical form - in the form of an equation, and in graphical form - in the form of a diagram. In assignments for a course project, the following laws of change in analogs of accelerations of the center of the pusher roller are found, given in the form of diagrams:
The uniformly accelerated law of change of the analog of the pusher acceleration, with the uniformly accelerated law of the pusher movement, the designed cam mechanism will experience soft shocks at the beginning and at the end of each of the intervals.
The triangular law of change of the analogue of acceleration ensures the shockless operation of the cam mechanism.
The trapezoidal law of change of the analogue of acceleration also ensures the shock-free operation of the mechanism.
Sinusoidal law of change of analog of acceleration. Provides the greatest smoothness of movement (characteristic is that not only speed and acceleration, but also derivatives of a higher order change smoothly). However, for this law of motion, the maximum acceleration at the same phase angles and the stroke of the pusher is greater than in the case of uniformly accelerated and trapezoidal laws of change in analogues of accelerations. The disadvantage of this law of motion is that the increase in speed at the beginning of the rise, and, consequently, the rise itself is slow.
The cosinusoidal law of change of the analogue of acceleration causes soft shocks at the beginning and at the end of the pusher stroke. However, with the cosine law, there is a rapid increase in speed at the beginning of the stroke and its rapid decrease at the end, which is desirable when many cam mechanisms are operating.
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 complex 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.
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):
- a family of circles with a radius equal to the radius of the roller is built, 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 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|
Further 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 picture:
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
