Classification of kinematic pairs. There are several classifications of kinematic pairs. Kinematic pairs and connections Signs of classification of kinematic pairs
rotational;
progressive;
screw;
spherical.
Symbols of links and kinematic pairs on kinematic diagrams.
The kinematic scheme of the mechanism is a graphic representation on the selected scale of the relative position of the links included in the kinematic pairs, using symbols according to GOST 2770-68. Large letters of the Latin alphabet on the diagrams indicate the centers of the hinges and other characteristic points. The directions of movement of the input links are marked with arrows. The kinematic diagram must have all the parameters necessary for the kinematic study of the mechanism: the dimensions of the links, the number of gear teeth, the profiles of the elements of the higher kinematic pairs. The scale of the circuit is characterized by the length scale factor Kl, which is equal to the ratio of the length AB l of the link in meters to the length of the segment AB depicting this link in the diagram, in millimeters: Kl = l AB / AB
The kinematic scheme, in essence, is a model that is replaced by a real mechanism for solving the problems of its structural and kinematic analysis. We note the main assumptions that are implied in this schematization:
a) the links of the mechanism are absolutely rigid;
b) there are no gaps in the kinematic pairs
Kinematic chains and their classification.
Kinematic chains according to the nature of the relative motion of the links are divided into flat and spatial. A kinematic chain is called flat if the points of its links describe trajectories lying in parallel planes. A kinematic chain is called spatial if the points of its links describe non-planar trajectories or trajectories lying in intersecting planes.
Classification of kinematic chains:
Flat - when one link is fixed, the remaining links make a flat movement, parallel to some fixed plane.
Spatial - when one link is fixed, the remaining links move in different planes.
Simple - each link includes no more than two kinematic pairs.
Complicated - at least one link has more than two kinematic pairs.
Closed - no more than two kinematic pairs are included, and these links form one or more closed loops
Open - links do not form a closed loop.
The number of degrees of freedom of the kinematic chain, the mobility of the mechanism.
The number of input links for the transformation of a kinematic chain into a mechanism must be equal to the number of degrees of freedom of this kinematic chain.
The number of degrees of freedom of the kinematic chain in this case means the number of degrees of freedom of the movable links relative to the rack (the link taken as fixed). However, the rack itself in real space can move.
Let us introduce the following notation:
k is the number of links of the kinematic chain
p1 is the number of kinematic pairs of the first class in a given chain
p2 is the number of pairs of the second class
p3 is the number of pairs of the third class
p4 is the number of pairs of the fourth class
p5 is the number of pairs of the fifth class.
The total number of degrees of freedom k of free links placed in space is 6k. In a kinematic chain, they are connected into kinematic pairs (i.e., connections are superimposed on their relative movement).
In addition, a kinematic chain with a rack (a link taken as a fixed one) is used as a mechanism. Therefore, the number of degrees of freedom of the kinematic chain will be equal to the total number of degrees of freedom of all links minus the constraints imposed on their relative motion:
The number of bonds imposed by all pairs of class I is equal to their number, since each pair of the first class imposes one connection on the relative movement of the links connected in such a pair; the number of bonds imposed by all pairs of class II is equal to their doubled number (each pair of the second class imposes two bonds), etc.
All six degrees of freedom are taken away from the link, taken as fixed (six bonds are superimposed on the rack). In this way:
S1=p1, S2=2p2, S3=3p3, S4=4p4, S5=5p5, Spillars=6,
and the sum of all connections
∑Si=p1+2p2+3p3+4p4+5p5+6.
The result is the following formula for determining the number of degrees of freedom of a spatial kinematic chain:
W=6k–p1–2p2–3p3–4p4–5p5–6.
Grouping the first and last terms of the equation, we get:
W=6(k–1)–p1–2p2–3p3–4p4–5p5,
or finally:
W=6n–p1–2p2–3p3–4p4–5p5,
Thus, the number of degrees of freedom of an open kinematic chain is equal to the sum of the mobilities (degrees of freedom) of the kinematic pairs included in this chain. In addition to degrees of freedom, the quality of work of manipulators and industrial robots is greatly influenced by their maneuverability.
Types of gear mechanisms, their structure and a brief description.
A gear transmission is a three-link mechanism in which two moving links are gears, or a wheel and a rack with teeth that form a rotational or translational pair with a fixed link (body).
The gear train consists of two wheels, through which they interlock with each other. A gear with a smaller number of teeth is called a gear, with a large number of teeth a wheel.
The term "gear" is generic. The gear parameters are assigned index 1, and the wheel parameters 2.
The main advantages of gears are:
The constancy of the gear ratio (no slippage);
Compactness compared to friction and belt drives;
High efficiency (up to 0.97 ... 0.98 in one step);
Great durability and reliability in operation (for example, for general purpose gearboxes, a resource of 30,000 hours is set);
Possibility of application in a wide range of speeds (up to 150 m/s), power (up to tens of thousands of kW).
Flaws:
Noise at high speeds;
The impossibility of a stepless change in the gear ratio;
The need for high precision manufacturing and installation;
Overload protection;
The presence of vibrations that occur as a result of inaccurate manufacturing and inaccurate assembly of gears.
Involute profile gears are widely used in all branches of mechanical engineering and instrument making. They are used in an exceptionally wide range of operating conditions. The power transmitted by gears varies from negligible (instruments, clockwork) to many thousands of kW (aircraft engine gearboxes). Gears with cylindrical wheels are the most widespread, as they are the easiest to manufacture and operate, reliable and small-sized. Bevel, screw and worm gears are used only in cases where it is necessary according to the layout of the machine.
Basic law of engagement.
To ensure the constancy of the gear
relations: it is necessary that the profiles of the mating teeth be outlined by such curves that would satisfy the requirements of the main gearing theorem
The basic law of engagement: the general N-N normal to the profiles, drawn at the point C of their contact, divides the center distance a w into parts inversely proportional to the angular velocities. With a constant gear ratio ( = const) and fixed centers O 1 and O 2, the point W will occupy a constant position on the line of centers. In this case, the velocity projections k 1 and k 2 are not equal. Their difference indicates the relative sliding of the profiles in the direction of the K-K tangent, which causes their wear. The equality of projections of velocities and is possible only in one position, when the contact point C of the profiles coincides with the point W of the intersection of the N-N normal and the line of centers O 1 O 2 . Point W is called the pole of engagement, and circles with diameters d w1 and d w2 that touch at the pole of engagement and roll over each other without slipping are called initial.
To ensure the constancy of the gear ratio, theoretically, one of the profiles can be chosen arbitrarily, but the shape of the profile of the mating tooth must be strictly defined to fulfill the condition (1.82). The most technologically advanced in manufacturing and operation are involute profiles. There are other types of engagement: cycloidal, lantern, Novikov engagement, satisfying this requirement.
Types of kinematic pairs and their brief description.
A kinematic pair is a connection of two contacting links, allowing their relative movement.
The set of surfaces, lines, points of a link, along which it can come into contact with another link, forming a kinematic pair, is called a link element (element of a kinematic pair).
Kinematic pairs (KP) are classified according to the following criteria:
according to the type of contact point (connection point) of the link surfaces:
the lower ones, in which the contact of the links is carried out along a plane or surface (sliding pairs);
higher, in which the contact of the links is carried out along lines or points (pairs that allow sliding with rolling).
according to the relative motion of the links forming a pair:
rotational;
progressive;
screw;
spherical.
according to the method of closing (ensuring the contact of the links of the pair):
power (due to the action of weight forces or the force of elasticity of the spring);
geometric (due to the design of the working surfaces of the pair).
Physical quantities and units of measurement,
Used in mechanics
| Physical quantity | Unit of measurement | ||
| Name | Designation | Name | Designation |
| Length Mass Time Plane angle Displacement of a point Linear speed Angular speed Linear acceleration Angular acceleration Frequency of rotation Material density Moment of inertia Force Moment of force Torque Work Kinetic energy Power | L, l, r m T, t a, b, g, d S u w a e n r J F, P, Q, G M T A E N | Meter Kilogram Second Radian, Degree Meter Meter per second Radian per second Meter per second squared Radian per second squared Revolution per minute Kilogram per cubic meter Kilogram meter squared Newton Newton meter Newton meter Joule Joule Watt | m kg s rad, α 0 m m / s rad / s, 1 / s m / s 2 rad / s 2, 1 / s 2 rpm kg / m 3 kg. m 2 N (kg. m / s 2) Nm Nm J \u003d Nm J W (J / s) |
STRUCTURE AND CLASSIFICATION OF MECHANISMS
Mechanism structure
The mechanisms include solid bodies who are called links. The links may not be solid (for example, a belt). Liquids and gases in hydraulic and pneumomechanisms are not considered links.
The conditional representation of links on the kinematic diagrams of mechanisms is regulated by GOST. Examples of images of some links are shown in fig. 1.1.
Rice. 1.1. Link Image Examples
on kinematic diagrams of mechanisms
Links happen:
– input(leading) - their distinguishing feature is that the elementary work of the forces applied to them is positive (the work of the force is considered positive if the direction of the force coincides with the direction of movement of the point of its application or at an acute angle to it);
– weekends(slave) - the elementary work of the forces applied to them is negative (the work of the force is considered negative if the direction of the force is opposite to the direction of movement of the point of its application);
– mobile;
– motionless(bed, rack).
On the kinematic diagrams, the links are indicated by Arabic numerals: 0, 1, 2, etc. (see fig. 1.1).
The movable connection of two adjoining links is called kinematic pair. It allows the possibility of movement of one link relative to another.
Classification of kinematic pairs
1. By elements of the connection of links kinematic pairs are divided:
- for higher(they are available, for example, in gear and cam mechanisms) - the links are connected to each other along a line or at a point:
– lower- the connection of the links with each other occurs on the surface. In turn, the lower compounds are divided:
for rotational
progressive
cylindrical
|
spherical
2. By the number of superimposed connections. The body, being in space (in the Cartesian coordinate system X, Y, Z) has 6 degrees of freedom. It can move along each of the three axes X, Y and Z, as well as rotate around each axis (Fig. 1.2). If a body (link) forms a kinematic pair with another body (link), then it loses one or more of these 6 degrees of freedom.
According to the number of degrees of freedom lost by the body (link), kinematic pairs are divided into 5 classes. For example, if the bodies (links) that formed a kinematic pair lost 5 degrees of freedom each, this pair is called a kinematic pair of the 5th class. If 4 degrees of freedom are lost - the 4th class, etc. Examples of kinematic pairs of different classes are shown in fig. 1.2.
Rice. 1.2. Examples of kinematic pairs of various classes
On a structural and constructive basis kinematic pairs can be divided into rotational, translational, spherical, cylindrical, etc.
Kinematic chain
Several links interconnected by kinematic pairs form kinematic chain.
Kinematic chains are:
closed
open
To from the kinematic chain get gear, necessary:
- make one link immovable, i.e. form a frame (rack);
- set the law of motion for one or several links (make them leading) in such a way that all other links perform required purposeful movements.
Number of degrees of freedom of the mechanism- this is the number of degrees of freedom of the entire kinematic chain relative to the fixed link (rack).
For spatial kinematic chain in a general form, we conditionally denote:
number of moving parts - n,
the number of degrees of freedom of all these links is 6n,
number of kinematic pairs of the 5th class - P5,
the number of bonds imposed by kinematic pairs of the 5th class on the links included in them, - 5Р 5 ,
number of kinematic pairs of the 4th class - R 4,
the number of bonds imposed by kinematic pairs of the 4th class on the links included in them, - 4P 4 etc.
For flat kinematic chain and, accordingly, for a flat mechanismThis formula is called P.L. Chebyshev (1869). It can be obtained from the Malyshev formula, provided that on the plane the body has not six, but three degrees of freedom:
W \u003d (6 - 3)n - (5 - 3)P 5 - (4 - 3) P 4.
The value of W shows how many driving links the mechanism should have (if W= 1 - one, W= 2 - two leading links, etc.).
Kinematic couple is a movable connection of two contacting links, allowing relative movements
according to the relative movement of the links:
rotational; progressive; screw; planar; spherical;
according to the type of contact of the links:
lower- these are kinematic pairs in which the contact of the links that form them is carried out along a plane or along a surface;
higher- these are kinematic pairs in which the contact of the links that form them is carried out along a line or at a point;
according to the method of ensuring the contact of links forming kinematic pairs: power- these are kinematic pairs in which the constancy of the contact of the links is ensured due to the action of gravity forces or the elastic force of the spring; geometric- these are kinematic pairs in which the constancy of the contact of the links is realized due to the design of the working surfaces of the links;
according to the number of connection conditions imposed on the relative motion of the links that form the kinematic pair (the number of connection conditions determines the class of the kinematic pair);
according to the number of mobilities in the relative motion of the links (the number of mobilities determines the mobility of the kinematic pair).
Connections- these are restrictions imposed on the movements of the links of the mechanism, making them not free and intended for the transfer of energy or information between these links.
For the formation of a kinematic pair, it is necessary to have at least one bond, because if the number of bonds is equal to zero, the links do not interact, i.e., they do not touch, therefore, the kinematic pair does not exist
6.Kinematic chains. Types of kinematic chains
All mechanisms consist of a set of links that form kinematic pairs that make up kinematic chains.
Kinematic chain is a system of links that form kinematic pairs with each other
Kinematic chains are divided into:
by design:
simple- this is a kinematic chain, each link of which is part of no more than two kinematic pairs, that is, it contains only one or two vertex links.
complex- this is a kinematic chain that has links that are part of three or more kinematic pairs, that is, it contains at least one link with three or more vertices
on the interaction of links:
closed or open is a kinematic chain in which at least one link has a free element that does not interact with other links and does not form kinematic pairs with them.
closed- this is a kinematic chain, each link of which is part of at least two kinematic pairs
Kinematic connection is a kinematic pair formed by links of several kinematic chains.
Depending on the complexity of the structure, there may be several kinematic connections in the mechanism.
The nature of the relative motion of the links allowed by the kinematic pair depends on the shape of the links at their contact points.
The set of possible points of contact forms on each of the two links element kinematic pair. An element of a kinematic pair can be dot , line , surface.
Kinematic pairs whose element dot or line , are called higher ; kinematic pairs, the element of which surface , called inferior .
Depending on the geometry of one (or both) of the contacting links, kinematic pairs are distinguished: spherical, conical, cylindrical, planar, helical.
According to the nature of the relative movement of the links allowed by the kinematic pair, rotational (B), translational (P), rotational-translational (B + P) and with screw motion of the VP are distinguished . The difference between pairs of type B + P and VP is that in the first, the relative movements (rotational and translational) are independent, and in the second, one movement cannot be carried out without the other.
Along with pairs of links that are in contact along the same surface, line or point, pairs with multiple contact are used in practice. This is either a repetition of interaction elements (splined, multi-start screw, gear pairs), or the use of simultaneous contact along the surface and line (spherical pair with a pin), along cylindrical and flat surfaces (pair with a sliding key). The repetition of contact between links characterizes the equivalence of pairs of different types. A pair with a three-point contact can be equivalent to a planar or spherical lower pair in terms of the nature of the movement of the links.
For a rigid body moving freely in space, the number of degrees of freedom (the number of possible movements of a mechanical system independent of each other) is six: three translational along the axes X, Y, Z and three rotational around these axes (Fig. 2.1 ).
For links included in a kinematic pair, the number of degrees of freedom is always less than six, since the conditions of contact (bonds) reduce the number of possible movements of one link relative to another: one link cannot penetrate into another and cannot move away from it.
In the general case, each kinematic pair imposes S bonds on the relative movement of the links, allowing H=6 - S relative movements of the links. Depending on the number of superimposed bonds S (the remaining degrees of freedom H), 5 classes of kinematic pairs are distinguished. Such a classification of kinematic pairs was proposed by I.I. Artobolevsky (table 2.1)
Tables 2.2-2.4 show examples of the design of kinematic pairs. The pairs shown in Tables 2.2 and 2.4 are classified based on the assumption that there is no friction and deformation of the links. Friction allows separate pairs to be used in friction gears. Given the deformation, pairs with point contact can be converted into pairs with surface contact.
Table 2.1
Types of kinematic pairs
Basic concepts and definitions in the theory of mechanisms
The theory of mechanisms and machines studies the structure, kinematics and dynamics of mechanisms and machines.
mechanism An artificially created system of bodies is called, designed to convert the movement of one or more bodies into the required movements of other bodies.
The solid bodies that make up the mechanism are called links.
Each movable part or group of parts that forms one rigid movable system of bodies is called moving link mechanism.
All fixed parts form one rigid fixed system of bodies, called a fixed link or rack.
Therefore, any mechanism has one fixed and one or more moving links.
The connection of two contacting links, allowing their relative movement, is called a kinematic pair.
Surfaces, lines, points of a link, along which it can come into contact with another link, forming a kinematic pair, are called link elements.
A connected system of links that form kinematic pairs with each other is called a kinematic chain.
Mechanism- there is a kinematic chain used to carry out the required movement.
The mechanisms that make up the machine are varied. From the point of view of their functional purpose, machine mechanisms are divided into the following types:
a) mechanisms of motors and converters:
engine mechanisms convert various types of energy into mechanical work;
converter mechanisms carry out the transformation of mechanical work into other types of energy;
b) transmission mechanisms, transferring motion from the engine to the technological machine or executive body;
in) executive mechanisms, directly affecting the processed environment or object;
G) governance mechanisms, control and regulation, carrying out process control, control, etc.;
e) automatic counting mechanisms, weighing and packaging used in machines that produce mass piece products.
Kinematic pairs and their classification
The main property of a pair is the number of geometric parameters that can be used to determine the relative position of the connected links. For example, when touching on the surface of revolution, the relative position of the links is completely determined by setting only one parameter - the angle of relative rotation of the links in the plane perpendicular to the axis of rotation.
When touching on a spherical surface, there are already three such parameters - these are the angles of rotation around three mutually perpendicular axes intersecting at the center of the sphere.
Consequently, the elements of the kinematic pair impose some restrictions on the relative movement of the links, linking the coordinates of the points of both links in a certain way.
The constraints imposed by the elements of a kinematic pair on the relative motion of the links forming the pair are called constraints, and the controls expressing these constraints are called constraint equations.
Let us consider what bonds and in what quantity can be imposed on the relative motion of the links of a kinematic pair.
As is known, in the general case, any absolutely rigid body freely moving in space has six degrees of freedom:
three rotations around the X, Y, Z axes and three translational movements along the same axes.
The constraints imposed on the relative motion of a link of a kinematic pair limit the same possible relative motions that the links have in a free state.
As a result of these restrictions, some of the six possible relative motions of a freely moving link become bound for it. The remaining independent possible motions determine the number of degrees of freedom of the links of the kinematic pair in their relative motion.
Kinematic pairs, depending on the number of connection conditions imposed on the relative movement of its links, are divided into five classes:
A pair of class I - (Fig. 1 a) a five-moving pair, has a number of degrees of freedom of links equal to five and a number of connection conditions equal to 1;
A pair of class II - (Fig. 1b) a four-moving pair, the number of degrees of freedom of the link of the kinematic pair is four, the number of connection conditions is 2;
A pair of class III - (Fig. 1 c, i, d) a three-moving pair, the number of degrees of freedom of the link of the kinematic pair is three, the number of connection conditions is 3;
A pair of class IV - (Fig. 1 e, i, f) a two-moving pair, the number of degrees of freedom of the link is 2, the number of connection conditions is 4;
A class V pair is (Fig. 1 g, h. i) a single-moving (rotary pair), the number of degrees of freedom of the link is equal to one, the number of connection conditions is 5.
Kinematic pairs are divided into spatial and flat. A spatial kinematic pair is a pair whose link points in relative motion describe spatial curves. Planar kinematic pairs are called such pairs, the points of the links of which in relative motion move in parallel planes, i.e. their trajectories are plane curves. In modern mechanical engineering, flat mechanisms, the links of which are included in pairs of classes IV and V, are especially widely used.
Kinematic pairs also differ in the nature of the contact of the links. If the elements of a kinematic pair are such that at each relative position of the links they have contact on the surface, then the pair is called the lowest. If the touch occurs at separate points or along lines, then the pair is called the highest.
With the relative motion of the links forming the lower pair, the surfaces of their contact slide over each other. If the links form a higher pair, then their relative movement can occur both with sliding of the elements of the pair, and without it - by rolling.