Summary: Complex gear mechanisms. Multistage and planetary mechanisms. Kinematics of in-line and stepped gear mechanisms. Willis formula for differential mechanisms. Kinematic study of typical planetary mechanisms using graphical and analytical methods. Statement of the problem of synthesis of planetary mechanisms. Conditions for selecting the number of teeth. Conditions of coaxiality, proximity and assembly. Examples of solving problems on selecting the number of teeth.
When designing gear mechanisms of many machines and devices, it becomes necessary to ensure transmission of rotation with a large gear ratio or at significant interaxial distances. In such cases, multi-link gear mechanisms are used - either gearboxes that reduce the speed of rotation of the output shaft compared to the speed of the input link, or multipliers that increase it.
Multi-link gear mechanisms are divided into two types: 1) mechanisms with fixed axes of all wheels (ordinary and stepped gear mechanisms); 2) mechanisms in which the axes of individual wheels move relative to the stand (planetary and differential mechanisms).
Mechanisms with fixed axes Gear wheels have a number of degrees of freedom equal to one, due to which the gear ratio is constant.
The overall gear ratio of a multi-link gear mechanism is equal to the product of the gear ratios of the individual stages:
Ordinary gear mechanisms They are a series connection of several pairs of gears (Fig. 14).
The overall gear ratio of an ordinary gear mechanism is constant and equal to the inverse ratio of the numbers of teeth or radii of the outer wheels:
.
Stepped gear mechanisms(Fig. 15) are a series connection of block (paired wheels 1 and 2; 2 ”and 3) or single gears. In general, when j wheels and t external gears full gear ratio of stepped transmission
,
those. is equal to the ratio of the product of the numbers of teeth of the driven wheels to the product of the driving wheels.
Gear mechanisms with movable axes have wheels with moving geometric axes, which are called satellites. In Fig. 16 shows the diagram planetary mechanism: moving link –h, in which the axes of the satellites are placed, is called carrier; rotating fixed axis wheel – 1 , along which the satellites are rolled, is called central; fixed central wheel – 3 is called supporting. As a rule, planetary mechanisms are made coaxial, which means that the wheel axles 1, 3 and drove h are on the same straight line.
Typically, a real mechanism has several symmetrically located satellites. They are introduced in order to reduce the meshing forces, unload the bearings of the central wheels, and improve the balancing of the carrier. But in kinematic calculations, only one satellite is taken into account, since the rest are passive in kinematic terms.
Analytical method The study of planetary mechanisms is based on the method of reversing motion. All links of the mechanism are imparted an angular velocity equal in magnitude and opposite in direction to the angular velocity of the carrier. Then the carrier becomes stationary, and the mechanism turns from a planetary one into a gear mechanism, consisting of several pairs of gears connected in series ( 1,2 And 2`3 ). The gear ratios of the planetary mechanism and the inverted mechanism are related by the condition:
This formula is valid for any planetary gearbox design with a fixed central wheel. This means that the gear ratio from any satellite k to the carrier with the support wheel stationary j is equal to one minus the gear ratio from the same wheel to the supporting one in the reversed mechanism:
.
If in the planetary mechanism (Fig. 16) the support wheel is released from its fastening 3 and impart rotational motion to it, then the mechanism will turn into differential with degree of freedom W=2(Fig. 17).
For the kinematic study of differential mechanisms, the Willis formula is used, also obtained on the basis of the motion reversal method:
,
Where is the gear ratio in reverse motion ().
Graphic definition of gear ratio Multi-link gear mechanisms can be implemented using the method of speed plans (speed triangles). Velocity triangles can be constructed if the linear velocities of at least two points on the link are known (in magnitude and direction). Using this method and constructing speed triangles (Fig. 18), you can get a visual representation of the nature of the change in speeds from one shaft to another, and you can determine graphically the angular speed of any link (wheel).
Input data: m – engagement module, z i- number of wheel teeth, .
Define gear ratio of the mechanism.
Solution. Let's construct a kinematic diagram of the mechanism on a scale, determining the radii of the pitch circles of the gears
Let's find the linear speed t. A in mesh of links 1 And 2
In the coordinate system r0V Let's construct triangles for the distribution of linear velocities of the links. To do this from the point A with ordinate r 1 on a selected arbitrary scale 
put aside the segment aa". We draw a straight line through the end of this segment and the origin of coordinates, which will determine the distribution of velocities for the points of the link 1
, lying on the axis r 1. This straight line forms with the axis r 1 corner . Since at the point C link speeds 2
And 3
are equal to each other and equal to zero, then by connecting the point C straight line with a point a", we obtain the speed distribution line for the link 2
. Since the point B belongs to the links 2
And h, then its speed is determined by the ray ca" for a radius equal r B = (r 1 + r 2), which on a scale corresponds to the segment bb". Connecting the dot b" with the origin of the straight line, we find the speed distribution line for the carrier. This line forms with the axis r corner . The gear ratio of the planetary mechanism, determined from these graphical constructions, can be written as follows:
.
Statement of the problem of synthesis of planetary mechanisms.
When designing planetary mechanisms, it is necessary, in addition to the requirements of the technical specifications (given gear ratio), to fulfill a number of conditions related to the features of planetary and multi-thread mechanisms. The design task in this case can also be divided into structural and metric synthesis of the mechanism. With structural synthesis, the structural diagram of the mechanism is determined, with metric synthesis, the number of gear teeth is determined, since the radii of gears are directly proportional to the number of teeth
r i = m × z i / 2 .
For standard mechanisms first task comes down to choosing a scheme from a set of standard schemes. In this case, they are guided by the range of gear ratios recommended for the circuit and approximate estimates of its efficiency. After selecting the mechanism’s design, it is necessary to determine the combination of numbers of teeth of its wheels, which will ensure the fulfillment of the terms of the technical specifications - for the gearbox, this is the gear ratio and the magnitude of the moment of resistance on the output shaft. The gear ratio sets the conditions for choosing the relative sizes of gears - the numbers of gear teeth; the torque sets the conditions for choosing absolute sizes - gear modules. Since to determine the module it is necessary to select the material of the gear pair and the type of heat treatment, then at the first stages of design the module of the gears is taken equal to one, that is, they solve the problem of kinematic synthesis of the mechanism in relative quantities.
With kinematic synthesis(selection of numbers of wheel teeth), the problem is formulated as follows: for the selected planetary mechanism design with the number of satellites and a given gear ratio, it is necessary to select the number of wheel teeth that will ensure the fulfillment of a number of conditions.
Given: Z1=26, Z3=74, Z4=78, Z5=26, m=2
Find:,Z6 ,Z2
Let us highlight two circuits in the kinematic diagram:
I k = wheels 1,2,3 and carrier N.
II k = wheels 4,5,6.
To determine the unknown values of the numbers of wheel teeth, we create an alignment condition for each contour.
Z2= (Z3- Z2)/2 =(74-26)/2 =24
Z6= Z4-2* Z5=78-2*26=26
Since m=2, then r=z.
To build a picture of the speeds of a closed differential gearbox, consider a closed stage: wheels 6,5,4.
Let us choose an arbitrary speed vector of wheel 5 at point C.
I to =W=3n-2P 5 -P 4 ; W=3*4-2*4-2=2 ,
differential mechanism.
II k, closed stage, series connection.
W 6 =W H, W 3 =W 4
Based on the constructed picture of instantaneous velocities, we will construct a plan of angular velocities.
Using the constructed angular velocity plan, we determine the gear ratio:
Conclusion
gear mechanism kinetostatic speed
During the course project, a kinematic analysis of the mechanism was carried out and plans of speeds and accelerations were constructed for the working and idle speed of the mechanism (3 and 9 positions).
As a result of the kinetostatic calculation, the values of the reactions of kinematic pairs and the balancing force for the working and idle speed of the mechanism (3 and 9 positions) were obtained.
As a result of the kinematic analysis of the gear mechanism, a picture of instantaneous velocities and a plan of angular velocities were constructed, and the gear ratio was also determined.
List of used literature
1. Artobolevsky I. I. Theory of mechanisms - M.: Nauka, 1965 - 520 p.
2. Dynamics of lever mechanisms. Part 1. Kinematic calculation of mechanisms: Guidelines / Comp.: L.E. Belov, L.S. Stolyarova - Omsk: SibADI, 1996, 40 p.
3. Dynamics of lever mechanisms. Part 2. Kinetostatics: Guidelines / Comp.: L.E. Belov, L.S. Stolyarova - Omsk: SibADI, 1996, 24 p.
4. Dynamics of lever mechanisms. Part 3. Examples of kinetostatic calculation: Guidelines / Comp.: L.E. Belov, L.S. Stolyarova - Omsk: SibADI, 1996, 44 p.
Rules for performing structural analysis of a mechanism:1. Eliminate passive connections and extra degrees of freedom (W) from the kinematic diagram of the mechanism.
2. Replace flat kinematic pairs of class 4 with kinematic pairs of class 5, while the replacement mechanism must have the number of degrees of freedom of the previous mechanism and perform all its movements.
3. Start disconnecting the structural group furthest from the leading link of the mechanism.
4. Disconnect first the class II structural group (if it is not possible to disconnect the class II structural group, disconnect the class III structural group, etc.).
5. Make sure that when a structural group is disconnected, the remaining mechanism retains its functionality, i.e. didn't fall apart.
Replacing a class 4 kinematic pair with a class 5 kinematic pair.
Any flat kinematic pair of class 4 is replaced by two kinematic pairs of class 5 (rotational and translational), interconnected by fictitious links.
Examples: Gear mechanism is given. It is required to replace kinematic pairs of class 4 with kinematic pairs of class 5 (Fig.):
Solution :
Here n=2, P 5 =2, P 4 =1(t.B),
Then W=3·2-2·2-1=1
Through t. IN draw a tangent t-t to link 2. Through t. IN at an angle to t-t carry out N-N. From points A And WITH draw perpendiculars to N-N. At the points of their intersection with N-N install rotational kinematic pairs of class 5: TO And L K-L.
The angle of engagement of link 1 and link 2 with each other.
(W).
Here n=3, P 5 =4, P 4 =0, Then W=3·3-2·4=1
Friction mechanism provided, rice.
Here: n=2, P 5 =2, P 4 =1(t.V)
Then: W=3·2-22-1=1
| Rice. eleven |
Draw up a kinematic diagram of the replacement mechanism and determine the number of degrees of freedom W,
Here: n=3, P 5 =4, P 4 =0. Then W=3·3-2·4=1
Given cam mechanism, rice.
Solution:
Here n=2, P 5 =2, P 4 =1
Then W=3·2-2·2-1=1
Through t. IN draw a tangent t-t To
link 1 and link 2. Through t. IN perpendicular to t-t carry out N-N. On N-N find the centers of curvature of link 1 and link 2, install rotational kinematic pairs of class 5 in them: TO And L, which are connected by fictitious links K-L, rice.
Draw up a kinematic diagram of the replacement mechanism and determine the number of degrees of freedom W, rice.
Here n=3, P 5 =4, P 4 =0, Then W=3·3-2·4=1
Examples of performing structural analysis of a mechanism.
Given: Kinematic diagram of the mechanism.
It is required to perform a structural analysis of the mechanism.
Solution:
a) Movable links: 1,2,3,4,5 . Kinematic pairs: A, A", B, C, D, E, E"
b) W=3n-2P 5 - P 4, Here n=5, P 5 =7, P 4 =0 → W=3·5-2·7=1
Consider the remaining mechanism 0,1,2,4,0
The mechanism has fallen apart, because when link 1 rotates, link 4 will be stationary.
Therefore, it was done incorrectly.
In this case, the class III structural group is disconnected
Structural group III class 3rd order.
3. Links 0.1 remain with the kinematic pair A.
W=3·1-2·1=1
Therefore, the leading link is a class I mechanism.
Structure formula I (0,1) → III 3 (2,3,4,5).
Basic mechanism of class III.
1) Disconnect the links 1,2 with kinematic pairs A,B,C
n=2, P 5 =3, W=3·2-2·3=0.
2) disconnect the links 3,4
with kinematic pairs A", D, E,
n=2, P 5 =3, W=3 2-2 3=0
Structural group II class 2nd order
3) links remain 0,5 with kinematic pair E",
n=1, P 5 =1, W=3 1-2 1=1
The leading link is a class I mechanism.
Basic mechanism of class II.A kinematic diagram of a class 5 mechanism is given. It is required to perform a structural analysis of the mechanism.
Links: 0, 1, 2, 3, 4, 5, 0, 6, 0
Kinematic pairs: A, B, C, D, D", E, F, K
W=3n-2P 5 -P 4, Here n=6, P 5 =8, P 4 =0 → W=3 6-2 8=2
1) disconnect the links 4,5 with kinematic pairs D,D",E.
n=2, P 5 =3, W=3·2-2·3=0.
| Rice. 41 |
The main mechanism with links is considered 0,1,2,3,6,0.
The mechanism did not fall apart, because when the link rotates 1 and 6 will be mobile.
The detachment of the structural group was completed correctly.
2) Disconnect links 2 and 3 with kinematic pairs from the main mechanism B,C,F, rice.
n=2, P 5 =3, W=3 2-2 3=0
Structural group II class 2nd order.
3) leading links remain 0,1 with kinematic pair A and links 0,6 with kinematic pair TO.
| Rice. 44 |
Class I mechanism Class I mechanism
4) write down the formula for the structure of the mechanism:
II 2 (2.3) → II 2 (4.5)
I (0.6) Class II mechanism
Kinematic analysis of gear mechanisms.
The task of kinematic analysis of gear mechanisms is to determine their gear ratios.
A gear mechanism is a mechanism consisting of gears designed to transmit rotation from one shaft of a machine to another shaft with a change in the magnitude of the transmitted torque (Mcr).
Torque depends on the gear ratio; the larger the gear ratio, the greater the torque (Mcr). The gear mechanism is installed between the engine and the working mechanism.
A gear mechanism that serves to reduce the rotation speed or number of revolutions of the engine shaft is called a gearbox; to increase - multiplier; Moreover, the gearbox increases the torque (Mcr), and the multiplier reduces it.
There are simple, planetary (satellite), stepped, differential and closed differential gear mechanisms.
Planetary gear mechanisms, gear ratio.
Particular gear ratios of planetary gear mechanisms.
A planetary gear mechanism is a mechanism in which at least one axis with a group of gears (satellites) is movable in space.
Planetary mechanisms are used to obtain large gear ratios with smaller dimensions and weight, compared to simple gear mechanisms. The planetary gear mechanism consists of a central wheel, satellites (the number of satellites from 2 to 12), a fixed wheel and a carrier (the central moving axis of the satellites). They have W=1 and come in the following types: 1) James gearbox (Fig. 8)
Here: 1 – central (solar) wheel; 2 – satellite; 0 – fixed wheel; N– carrier (moving kinematic link).
W = 3n - 2P 5 - P 4
Here: n = 3 (1,2,H), P 5 = 3 (A, B, C), P 4 = 2 (D, E).
Then: W=3·3-2·3-2=1
The gear ratio of the planetary gear mechanism is determined by the Willis formula:
(1)
Ordinary cylindrical planetary gear mechanism 1-0 (Fig. 9).
Then: 
(2)
Substitute (2) into (1): 
Determine: a) reverse gear ratio
c) gear ratio from the central gear to any moving wheel (for example, a xatallite)
.
2) David gearbox with external gearing (Fig. 10).
Two or more gear wheels rigidly fixed on one axis constitute one wheel and are designated by the same numbers; and the second, third gear will have one, two, etc. strokes. In Fig. 10: 2 - 2". 
, (1)
Where – gear ratio of a stepped planetary mechanism.
Then: 
(2)
Substitute (2) into (1): 
.
Gear mechanisms serve to transmit rotational motion from one shaft to another, to change the magnitude and direction of angular velocity and torque.
Based on the relative position of the shafts, flat and spatial gears are distinguished. In flat mechanisms, the axes of rotation of the links are parallel, and all links rotate in parallel planes. In this case, rotation is transmitted with a constant gear ratio using round cylindrical wheels (Fig. 1).
|
|
|
In spatial gears, the rotation axes of the links intersect (bevel gears) or cross (worm, screw, spiroid and hypoid gears).
There are external (Fig. 1.a), internal (Fig. 1.b) and rack and pinion gears.
The ratio of the angular velocity of the drive shaft j to the angular velocity of the driven shaft k is called the gear ratio and is denoted by the letter “u” with the corresponding indices:
The plus sign refers to internal gearing, and the minus sign refers to external gearing. To obtain large gear ratios, more complex multi-stage gear mechanisms are used.
A gear stage is a transmission between two links located on the nearest fixed axes. The number of steps in gear mechanisms is equal to the number of fixed axes minus one.
The steps are simple and planetary. In Fig. 2. A and C - simple, B - planetary stages. If the rotation speed of the driven shaft is less than the rotation speed of the drive shaft, then such a mechanism is called a gearbox.
Gear mechanisms with wheel axles that are motionless relative to the stand are divided into ordinary and stepped. In ordinary mechanisms (Fig. 3), each axis has one wheel. In step mechanisms, each axle, except the drive and driven, has two wheels. In Fig. 4. shows a diagram of a three-stage mechanism. For him
|
|
When transmitting rotation through bevel wheels, the sign of the gear ratio is determined by the rule of arrows (Fig. 2.5). If the arrows on the drive and driven wheels, located on parallel shafts, are directed in the same direction, then the gear ratio will be with a plus sign, if in opposite directions, then with a minus sign.
|
|
|
For the mechanism shown in Fig. 5.
Gear mechanisms that have wheels whose axes move in space are called satellites (Fig. 2.6a). Wheels 1 and 3, rotating around a fixed central axis, are called central, and wheel 2, whose axis moves in space, is called a satellite. Link H, in which the axis of the satellite 2 is fixed, is called the carrier.
|
|
|
Satellite mechanisms with two or more degrees of freedom are called differential, and those with one degree of freedom are called planetary.
The relationship between the angular velocities of the links can be determined using the motion reversal method. Its essence lies in the fact that all links of the mechanism are given additional rotation with an angular velocity equal in magnitude to the angular velocity of rotation of the carrier, but opposite in direction (-ω n). At the same time, the carrier mentally stops and the differential mechanism turns into a reverse mechanism, in which the axes of all wheels are motionless. The new angular velocities of the links in reverse motion are equal
The gear ratio from the first link to the third for the reversed mechanism has the form
Formula (4) is called the Willis formula, where for a specific mechanism according to Fig. 6,a
Given two speeds, formula (4) can be used to determine the third speed.
Note that the Willis formula can be written for any two links. For example, according to formula (5)
Since ω3=0, then
In some cases, it is advisable to use combined gear mechanisms made up of gears of different types. For example, the mechanism shown in Fig. 2.2, has two simple stages and one planetary stage. Gear ratio of the entire mechanism
In technology, satellite mechanisms are used, consisting of a differential, between the leading links of which an intermediate gear is installed. This transmission imposes an additional coupling condition, and the differential mechanism turns into a complex planetary mechanism with one degree of mobility. Such a mechanism is called closed differential.