Lecture 11
CRANKSHAFT KINEMATICS
11.1. KShM types
11.2.1. Piston movement
11.2.2. Piston speed
11.2.3. Piston acceleration
Crank mechanism ( K W M ) is the main mechanism of the piston internal combustion engine, which perceives and transmits significant loads.Therefore, the strength calculation K W M it's important. In turn calculations of many details engine depend on the kinematics and dynamics of the KShM. Kinematic analysis of the CSM establishes the laws of motion of its links, primarily the piston and connecting rod.
To simplify the study of the KShM, we will assume that the cranks crankshaft rotate uniformly, i.e., with constant angular velocity.
11.1. KShM types
Three types of KShM are used in piston internal combustion engines:
- central (axial);
- mixed (deaxial);
- with a trailed connecting rod.
In the central KShM the axis of the cylinder intersects with the axis of the crankshaft (Fig. 11.1).
Rice. 11.1. Central KShM scheme:φ
- current angle of rotation of the crankshaft; β is the angle of deviation of the connecting rod axis from the cylinder axis (when the connecting rod is deflected in the direction of rotation of the crank, the angle β is considered positive, in the opposite direction - negative); S - piston stroke;
R - radius of the crank; L is the length of the connecting rod; NS - movement of the piston;
ω - angular speed of the crankshaft
The angular velocity is calculated by the formula
An important design parameter of the KShM is the ratio of the radius of the crank to the length of the connecting rod:
It was found that with a decrease in λ (due to an increase in L) there is a decrease in inertial and normal forces. This increases the height of the engine and its mass, therefore, in automobile engines λ is taken from 0.23 to 0.3.
The λ values for some automobile and tractor engines are given in table. 11.1.
Table 11. 1. Values of the parameter λ for р different engines
|
Engine |
|
|
VAZ-2106 |
0,295 |
|
ZIL-130 |
0,257 |
|
D-20 |
0,280 |
|
SMD-14 |
0,28 |
|
YaMZ-240 |
0,264 |
|
KamAZ -740 |
0,2167 |
V disaxial KShM(Fig.11.2) the cylinder axis does not intersect the crankshaft axis and is offset relative to it by a distance a .
Rice. 11.2. Diagram of deaxial KShM
Disaxial KShM have some advantages relative to the central KShM:
- increased distance between the crank and camshafts, as a result of which the space for movement of the lower head of the connecting rod is increased;
- more uniform wear of engine cylinders;
- with the same values R and λ is a longer piston stroke, which helps to reduce the content of toxic substances in the engine exhaust gases;
- increased engine displacement.
In fig. 11.3 showsKShM with a trailed connecting rod.The connecting rod, which is pivotally connected directly to the journal of the crankshaft, is called the main one, and the connecting rod, which is connected to the main one by means of a pin located on its head, is called trailed.Such a KShM scheme is used on engines with a large number of cylinders when they want to reduce the length of the engine.Pistons connected to the main and trailed connecting rod do not have the same stroke, since the crank axis is trailed th The connecting rod during operation describes an ellipse, the semi-major axis of which is greater than the radius of the crank. V V -shaped twelve-cylinder D-12 engine, the difference in the piston stroke is 6.7 mm.
Rice. 11.3. KShM with trailed connecting rod: 1 - piston; 2 - compression ring; 3 - piston pin; 4 - piston plug finger; 5 - bushing of the upper head connecting rod; 6 - the main connecting rod; 7 - trailed connecting rod; 8 - bushing of the bottom head of the trailed connecting rod; 9 - pin for attaching the connecting rod; 10 - locating pin; 11 - inserts; 12 - tapered pin
11.2. Kinematics of the central KShM
In the kinematic analysis of the crankshaft, it is assumed that the angular velocity of the crankshaft is constant.The task of the kinematic calculation is to determine the movement of the piston, the speed of its movement and acceleration.
11.2.1. Piston movement
The movement of the piston depending on the angle of rotation of the crank for an engine with a central control gear is calculated by the formula
(11.1)
Analysis of equation (11.1) shows that the movement of the piston can be represented as the sum of two movements:
x 1 - displacement of the first order, corresponds to the displacement of the piston with an infinitely long connecting rod(L = ∞ for λ = 0):
x 2 - second order displacement, is a correction for the final length of the connecting rod:
The value of x 2 depends on λ. For a given λ extreme values x 2 will take place if
i.e. within one revolution extreme values x 2 will correspond to the angles of rotation (φ) 0; 90; 180 and 270 °.
The displacement will reach its maximum values at φ = 90 ° and φ = 270 °, i.e. when s φ = -1. In these cases, the actual displacement of the piston will be
The value λR / 2, is called the Brix correction and is a correction for the final length of the connecting rod.
In fig. 11.4 shows the dependence of the piston movement on the angle of rotation of the crankshaft. When the crank is turned 90 °, the piston travels more than half its stroke. This is due to the fact that when the crank is turned from TDC to BDC, the piston moves under the action of the connecting rod displacement along the cylinder axis and its deviation from this axis. In the first quarter of the circle (from 0 to 90 °), the connecting rod, simultaneously with its movement to the crankshaft, deviates from the cylinder axis, and both movements of the connecting rod correspond to the movement of the piston in one direction, and the piston travels more than half of its path. When the crank moves in the second quarter of the circle (from 90 to 180 °), the directions of the connecting rod and piston movements do not coincide, the piston travels the shortest path.
Rice. 11.4. Dependence of the displacement of the piston and its components on the angle of rotation of the crankshaft
The movement of the piston for each of the angles of rotation can be determined graphically, which is called the Brix method.To do this, from the center of a circle with a radius R = S / 2 the Brix amendment is postponed towards Brix, a new center is located About 1. From the center O 1 through certain values of φ (for example, every 30 °), the radius vector is drawn until it intersects with the circle. The projections of the points of intersection on the cylinder axis (line TDC — BDC) give the desired piston positions for the given values of the angle φ. The use of modern automated computing tools allows you to quickly get addiction x = f (φ).
11.2.2. Piston speed
The derivative of the piston movement - equation (11.1) with respect to the rotation time gives the piston movement speed:
(11.2)
Likewise displacement of the piston, the piston speed can also be represented in the form of two components:
where V 1 Is the first-order piston velocity component:
V 2 - second-order piston velocity component:
Component V 2 represents the piston speed with an infinitely long connecting rod. Component V 2 is a correction to the piston speed for the final length of the connecting rod. The dependence of the change in the speed of the piston on the angle of rotation of the crankshaft is shown in Fig. 11.5.
Rice. 11.5. The dependence of the piston speed on the angle of rotation of the crankshaft
The speed reaches its maximum values at crankshaft angles of rotation less than 90 and more than 270 °.The exact value of these angles depends on the λ values. For λ from 0.2 to 0.3, the maximum piston speeds correspond to the angles of rotation of the crankshaft from 70 to 80 ° and from 280 to 287 °.
The average piston speed is calculated as follows:
The average piston speed in automobile engines is usually between 8 and 15 m / s.Meaning maximum speed piston with sufficient accuracy can be determined as
11.2.3. Piston acceleration
The acceleration of the piston is defined as the first derivative of the velocity over time or as the second derivative of the displacement of the piston over time:
(11.3)
where and - harmonic components of the first and second order of piston acceleration, respectively j 1 and j 2. In this case, the first component expresses the acceleration of the piston with an infinitely long connecting rod, and the second component expresses the acceleration correction for the final length of the connecting rod.
The dependences of the change in the acceleration of the piston and its components on the angle of rotation of the crankshaft are shown in Fig. 11.6.
Rice. 11.6. Dependences of changes in the acceleration of the piston and its components
from the angle of rotation of the crankshaft
Acceleration reaches maximum values at the position of the piston at TDC, and minimum values at BDC or near BDC.These curve changes j in the area from 180 to ± 45 ° depend on the valueλ. For λ> 0.25, the curve j has a concave shape towards the φ axis (saddle), and the acceleration reaches its minimum values twice. At λ = 0.25 the acceleration curve is convex and the acceleration reaches the largest negative value only once. Maximum acceleration of the piston in an automobile internal combustion engine 10,000 m / s 2. Kinematics of deaxial KShM and KShM with trailer a few connecting rods distinguishes Xia from kinematics central KShM and in the present edition not considered.
11.3. The ratio of piston stroke to cylinder bore
Piston Stroke Ratio S to cylinder diameter D is one of the main parameters that determines the size and weight of the engine. In automobile engines, the values S / D from 0.8 to 1.2. Engines with S / D> 1 are called long-stroke, and with S / D< 1 - short-stroke.This ratio directly affects the piston speed, and hence the engine power.With decreasing value S / D the following advantages are obvious:
- the engine height decreases;
- due to a decrease in the average piston speed, the mechanical losses and wear of parts is reduced;
- the conditions for the placement of valves are improved and the prerequisites are created for increasing their size;
- it becomes possible to increase the diameter of the main and connecting rod journals, which increases the rigidity of the crankshaft.
However, there are also negative points:
- the length of the engine and the length of the crankshaft increase;
- the loads on the parts from the forces of gas pressure and from the forces of inertia increase;
- the height of the combustion chamber decreases and its shape deteriorates, which in carburetor engines leads to an increase in the tendency to detonation, and in diesel engines - to a deterioration in the conditions of mixture formation.
It is considered expedient to decrease the value S / D with an increase in the speed of the engine. This is especially beneficial for V -shaped engines, where an increase in short-stroke makes it possible to obtain optimal mass and overall dimensions.
S / D Values for various engines:
- carburetor engines- 0.7-1;
- diesel engines of medium speed - 1.0-1.4;
- high-speed diesel engines - 0.75-1.05.
When choosing values S / D it should be borne in mind that the forces acting in the crankcase are more dependent on the cylinder diameter and to a lesser extent on the piston stroke.
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The initial value when choosing the size of the KShM links is the value of the full stroke of the slider, specified by the standard or for technical reasons for those types of machines, for which the maximum value of the stroke of the slider is not specified (scissors, etc.).
The following designations have been introduced in the figure: dО, dА, dВ - the diameters of the fingers in the hinges; e - the magnitude of the eccentricity; R is the radius of the crank; L is the length of the connecting rod; ω - angular speed of rotation of the main shaft; α - angle of under-reach of the crank to KNP; β is the angle of deflection of the connecting rod from the vertical axis; S - the value of the full stroke of the slide.
For a given value of the stroke of the slider S (m), the radius of the crank is determined:
For axial crank connecting rod mechanism the functions of the movement of the slider S, the speed V and the acceleration j from the angle of rotation of the crank shaft α are determined by the following expressions:
S = R, (m)
V = ω R, (m / s)
j = ω 2 R, (m / s 2)
For the deaxial crank mechanism, the functions of the slider movement S, the speed V and the acceleration j from the angle of rotation of the crank shaft α, respectively:
S = R, (m)
V = ω R, (m / s)
j = ω 2 R, (m / s 2)
where λ is the coefficient of the connecting rod, the value of which for universal presses is determined in the range of 0.08 ... 0.014;
ω is the angular speed of rotation of the crank, which is estimated based on the number of strokes of the slider per minute (s -1):
ω = (π n) / 30
The nominal force does not express the actual force developed by the drive, but represents the ultimate strength of the press parts, which can be applied to the slider. The nominal force corresponds to a strictly defined angle of rotation of the crank shaft. For single-acting crank presses with one-sided drive, the nominal force is taken as the force corresponding to the angle of rotation α = 15 ... 20 о, counting from the bottom dead center.
Kinematics of the crank mechanism
In autotractor ICEs, two types of crank mechanism (KShM) are mainly used: central(axial) and displaced(disaxial) (fig. 5.1). An offset mechanism can be created if the cylinder axis does not intersect the ICE crankshaft axis or is offset relative to the piston pin axis. A multi-cylinder internal combustion engine is formed on the basis of the indicated KShM schemes in the form of a linear (in-line) or multi-row structure.
Rice. 5.1. Kinematic diagrams of KShM of an autotractor engine: a- central linear; b- offset linear
The laws of motion of the CRS parts are studied using its structure, the main geometric parameters of its links, without taking into account the forces causing its movement and friction forces, as well as in the absence of gaps between the conjugated moving elements and a constant angular velocity of the crank.
The main geometric parameters that determine the laws of motion of the elements of the central CRM are (Fig.5.2, a): g- the radius of the crank of the crankshaft; / w - the length of the connecting rod. Parameter A = g / 1 w is a criterion for the kinematic similarity of the central mechanism. In autotractor internal combustion engines, mechanisms with A = 0.24 ... 0.31 are used. In dexial KShM (Fig.5.2, b) the amount of mixing of the axis of the cylinder (pin) relative to the axis of the crankshaft (a) affects its kinematics. For autotractor ICEs, the relative displacement To = a / g= 0.02 ... 0.1 - additional criterion of kinematic similarity.
Rice. 5.2. Design scheme of KShM: a- central; b- displaced
The kinematics of the KShM elements is described when the piston moves, starting from TDC to BDC, and clockwise rotation of the crank by the laws of time variation (/) of the following parameters:
- ? piston movement - x;
- ? the angle of rotation of the crank - (p;
- ? the angle of deflection of the connecting rod from the cylinder axis - (3.
The analysis of the KShM kinematics is carried out at constancy the angular velocity of the crankshaft crank or the crankshaft rotation frequency (α), interconnected by the relation ω = kp / 30.
At ICE operation the movable elements of the KShM make the following movements:
- ? the rotational motion of the crank of the crankshaft relative to its axis is determined by the dependences of the angle of rotation cp, angular velocity ω and acceleration e on time t. In this case, cp = co /, and at constant co - e = 0;
- ? the reciprocating movement of the piston is described by the dependences of its displacement x, velocity v and acceleration j from the angle of rotation of the crank cf.
Central piston movement KShM when turning the crank at an angle cp is determined as the sum of its displacements from turning the crank at the angle cp (Xj) and from the deflection of the connecting rod at the angle p (xn) (see Fig.5.2):
This dependence, using the relation X = g / 1 w, the relationship between the angles cp and p (Asincp = sinp) can be represented approximately as a sum of harmonics, multiples of the crankshaft speed. For example, for X= 0.3 the first amplitudes of the harmonics are related as 100: 4.5: 0.1: 0.005. Then, with an accuracy sufficient for practice, the description of the piston movement can be limited to the first two harmonics. Then for cp = co /
Piston speed defined as 
and approximately
Piston acceleration calculated by the formula 
and approximately
In modern internal combustion engines v max = 10 ... 28 m / s, y max = 5000 ... 20,000 m / s 2. Friction losses and engine wear increase with increasing piston speed.
For a displaced CRM, the approximate dependences have the form
These dependences, in comparison with their counterparts for the central KShM, differ in an additional term proportional to kk. Since for modern engines its value is kk= 0.01 ... 0.05, then its influence on the kinematics of the mechanism is small and in practice it is usually neglected.
The kinematics of the complex plane-parallel movement of the connecting rod in the plane of its swing consists of the movement of its upper head with the kinematic parameters of the piston and the rotational motion relative to the point of articulation of the connecting rod with the piston.
Kinematic studies and dynamic calculation of the crank mechanism are necessary to find out the forces acting on the parts and elements of engine parts, the main parameters of which can be determined by calculation.
Rice. 1. Central and deaxial
crank mechanisms
Detailed studies of the kinematics and dynamics of the crank mechanism of the engine due to the variable operating mode of the engine are very difficult. When determining the loads on engine parts, simplified formulas are used, obtained for the condition of uniform rotation of the crank, which give sufficient accuracy in the calculation and greatly facilitate the calculation.
The schematic diagrams of the crank mechanism of autotractor-type engines are shown: in Fig. 1, a - the central crank mechanism, in which the cylinder axis intersects the crank axis, and in Fig. 1 , b - deaxial, in which the cylinder axis does not intersect the crankshaft axis. The axis 3 of the cylinder is displaced relative to the axis of the crankshaft by an amount, a. Such a displacement of one of the axes relative to the other allows, somewhat to change the pressure of the piston on the wall of the cylinders, to reduce the speed of the piston y in. m t
The following designations are adopted on the diagrams: - the angle of rotation of the crank, measured from v. m.t. in the direction of rotation of the crank (crankshaft); S = 2R
- piston stroke; R- radius of the crank; L
- the length of the connecting rod;
- the ratio of the radius of the crank to the length of the connecting rod. Modern car engines 
, for tractor engines 
; - angular speed of rotation of the crank; a- displacement of the cylinder axis from the crankshaft axis; - the angle of deflection of the connecting rod from the axis of the cylinder; for modern automotive engines
In modern engines, the relative displacement of the axes is taken 
... With this displacement, the engine with the deaxial mechanism is calculated in the same way as with the central crank mechanism.
In kinematic calculations, the displacement, speed and acceleration of the piston are determined.
The piston movement is calculated using one of the following formulas:
Values in brackets and braces for different values and see appendices.
The displacement of the piston S is the sum of two S 1
and S 2
harmonic components: 
; .
The curve describing the movement of the piston depending on the change is the sum n + 1... harmonic components. These components above the second have a very small effect on the value of S, therefore, they are neglected in the calculations, limiting themselves only to S = S 1 + S 2 .
The time derivative of the expression S is the speed of movement of the piston
here v and 
- respectively, the first and second harmonic components.
The second harmonic component, taking into account the final length of the connecting rod, leads to a displacement to v. m., i.e.
One of the parameters characterizing the design of the engine is the average piston speed (m / s)
where NS - crankshaft rotation frequency per minute.
The average speed of the piston in modern automotive engines fluctuates within m / s. Larger values refer to motors passenger cars, the smaller ones - to the tractor.
Since wear piston group is approximately proportional to the average piston speed, something engines tend to do with to increase durability. lower average piston speed.
For automotive, engines:; at at
at 
Time derivative of piston speed - piston acceleration
When studying the KShM kinematics, it is assumed that the engine crankshaft rotates at a constant angular velocity ω , there are no gaps in the mating parts, and the mechanism is considered with one degree of freedom.
In fact, due to the uneven torque of the engine, the angular velocity is variable. Therefore, when considering special issues of dynamics, in particular torsional vibrations of the crankshaft system, it is necessary to take into account the change in angular velocity.
The independent variable is the angle of rotation of the crankshaft crank φ. In the kinematic analysis, the laws of motion of the KShM links, and first of all the piston and connecting rod, are established.
The initial position of the piston at the top dead center (point IN 1) (Fig. 1.20), and the direction of rotation of the crankshaft is clockwise. At the same time, to identify the laws of motion and analytical dependencies, the most characteristic points are established. For the central mechanism, these points are the axis of the piston pin (point V), performing, together with the piston, a reciprocating motion along the axis of the cylinder, and the axis of the crank pin of the crank (point A) rotating around the crankshaft axis O.
To determine the dependences of the KShM kinematics, we introduce the following designations:
l- the length of the connecting rod;
r- radius of the crank;
λ - the ratio of the radius of the crank to the length of the connecting rod.
For modern automobile and tractor engines the value is λ = 0.25–0.31. For high-speed engines, in order to reduce the inertial forces of reciprocating moving masses, more long connecting rods than for low-speed ones.
β - the angle between the axes of the connecting rod and the cylinder, the value of which is determined by the following relationship:
The largest β angles for modern automobile and tractor engines are 12–18 °.
Move (path) piston will depend on the angle of rotation of the crankshaft and is determined by the segment NS(see fig. 1.20), which is equal to:
Rice. 1.20. Central KShM scheme
Of triangles A 1 AB and OA 1 A follows that
Considering that , we get:
Of right-angled triangles A 1 AB and A 1 OA we establish that
Where
then, substituting the obtained expressions into the formula for the piston movement, we get:
Since then
The resulting equation characterizes the movement of the KShM parts depending on the angle of rotation of the crankshaft and shows that the path of the piston can be conventionally represented as consisting of two harmonic displacements:
where is the first order piston path, which would take place in the presence of a connecting rod of infinite length;
- the path of the second order piston, that is, additional displacement, depending on the final length of the connecting rod.
In fig. 1.21 shows the curves of the piston path along the angle of rotation of the crankshaft. It can be seen from the figure that when the crankshaft is turned through an angle of 90 °, the piston travels more than half its stroke.
Rice. 1.21. Change in the piston path depending on the angle of rotation of the crankshaft
Speed
where is the angular speed of rotation of the shaft.
The piston speed can be represented as the sum of two terms:
where is the harmonically varying speed of the first-order piston, i.e., the speed with which the piston would move in the presence of a connecting rod of infinitely long length;
- harmonically changing speed of the second order piston, i.e. the speed of additional displacement arising from the presence of a connecting rod of a finite length.
In fig. 1.22 shows the curves of the speed of the piston on the angle of rotation of the crankshaft. The angles of rotation of the crankshaft, where the piston reaches the maximum speed, depend on? and its increase are shifted towards the dead points.
For practical assessments of engine parameters, the concept is used average piston speed:
For modern car engines Vav= 8-15 m / s, for tractor - Vav= 5-9 m / s.
Acceleration The piston is defined as the first time derivative of the piston path:
Rice. 1.22. Change in piston speed depending on the angle of rotation of the crankshaft
Piston acceleration can be represented as a sum of two terms:
where is the harmonically varying acceleration of the first order piston;
- harmonically varying second-order piston acceleration.
In fig. 1.23 shows the curves of the piston acceleration in the angle of rotation of the crankshaft. Analysis shows that the maximum acceleration occurs when the piston is at TDC. When the piston is positioned at BDC, the value of acceleration reaches the minimum (maximum negative) value opposite in sign, and its absolute value depends on?.
Figure 1.23. Change in piston acceleration depending on the angle of rotation of the crankshaft
