The division of gears into ordinary and stepped gears makes sense when they contain more than two moving links. On fig. 17, A,b examples of ordinary and stepped gears are shown, respectively.
The wheels of an ordinary transmission are located in the same plane or, otherwise, in one row, hence the name. In a stepped transmission, each pair of meshing wheels is located in its own plane, forming its own stage. in sight b):1 ,2 - first stage; 3, 4 - the second.
In an ordinary transmission, each link contains only one gear. The link number and wheel number are the same. In stepped transmission wheels 2 And 3 located on the same stage (one shaft), so sometimes they are designated by the number of the shaft on which they are located, and differ in index, for example 2', 2'' or 2a, 2b. The link number is not affixed, as it is contained in the designations of the wheels.
Ordinary and stepped transmissions form gear class with fixed axles wheels. For any such transmission, the gear ratio from the first link to the last, n-mu, is equal to the product of intermediate gear ratios. This rule is expressed by the formula
u1,n= u1,2u2,3...un-1,n, (12)
where the indices indicate the numbers of links. To prove the validity of the formula, we present each u as a ratio of speeds:
ω1= ω1ω2 L ωn−1 .
ωn ω2ω3ωn
After reductions, the equation becomes an identity, which proves the validity of formula (12). The formula is valid for any sequential chain of mechanisms. Wherein uij(hereinafter we omit the comma) is the gear ratio of a separate mechanism.
Let's apply formula (12) to the gears shown above. For regular transmission
| u=u u u | z | 2 | − | z | 3 | − | z | 4 | =− | z | 4 | . | ||||||
| = | − | |||||||||||||||||
| z | z | z | ||||||||||||||||
| 14 12 23 | 34 | 2 | 3 | z | ||||||||||||||
| 1 | 1 |
For step transmission
| z2′ | z3 | z2′ z3 | |||||||||
| u=u u= | − | − | = | . | |||||||
| z | z | ||||||||||
| 13 12 23 | z z | ||||||||||
| 1 | 2 ′′ | 1 2 ′′ |
The results show that the gear ratio of the inline gear depends on the number of teeth of the outer wheels only. A stepped transmission does not have this property. Minus sign in gear ratio u14 shows that the wheels 1 ,4 rotate in different directions.
11. Differential and planetary gears. Willis formula.
Complex gear mechanisms in which the axle of at least one wheel is movable are called planetary mechanisms.
Planetary mechanisms are divided into planetary gearboxes and multipliers, which have one degree of freedom and necessarily have a reference link, and gear differential mechanisms, the number of degrees of freedom of which is two or more, and which usually do not have a reference link.
Typical planetary gears include:
single-row planetary gear with mixed gearing (James gear);
· double-row planetary mechanism with mixed gearing;
· two-row planetary mechanism with two external gearings;
· two-row planetary mechanism with two internal gearings.
The elements of the planetary mechanism have special names:
gear wheel with external teeth located in the center of the mechanism is called "solar";
wheel with internal teeth called "crown" or "epicycle";
Wheels, the axles of which are movable, are called "satellites";
· the movable link, on which the satellites are installed, is called the "carrier". This link is usually denoted not by a number, but by Latin letter h.
When the sun wheel rotates, the satellites turn like a lever relative to the instantaneous center of rotation (the support wheel is stationary) and make the planet carrier rotate. In this case, the planetary wheels (satellites) perform a complex movement: they rotate around their own axis (relative to the carrier) with angular velocity and, together with the carrier, roll around its axis (portable movement). The number of degrees of freedom of this mechanism is equal to one. Therefore, the gearbox has a constant gear ratio.
Usually, a real mechanism has several symmetrically located satellite blocks. They are introduced in order to reduce the dimensions of the mechanism, reduce the engagement force, unload the bearings of the central wheels, improve the balance of the carrier, although the mechanism in this case has redundant connections, i.e. is statically indeterminate. In kinematic calculations, one satellite is taken into account, since the rest are passive in kinematic terms.
If, in the considered mechanism, the support wheel (gear housing) is released from fastening and rotation is imparted to it, then all the central wheels will become movable and the mechanism will turn into a differential one, since the number of degrees of freedom will be equal to two.
Thus, differential mechanism is a planetary mechanism with the number of degrees of freedom.
The number of degrees of freedom (mobility) of the mechanism shows how many links of the differential must be given independent movements in order to obtain the certainty of the movement of all other links. Here, depending on the direction of rotation of the outer shafts, either the decomposition of the movement (one leading into two slaves) or the addition of the movement can occur. A shaft is considered to be the leading one, in which the direction of the speed of rotation and the moment coincide. Hence, planetary reductor(or multiplier), which has a fixed wheel, can be turned into a differential if the fixed (support) wheel is released and rotated. On the contrary, any differential can be turned into a planetary gear by fixing one (w = 2) or several of its central wheels. This is the so-called reversibility property of planetary gears, which allows you to apply the same research and design methods for gearboxes and differentials. In this case, each elementary differential will correspond to two planetary gearboxes
The table shows the block diagrams of typical planetary mechanisms, as well as the ranges of recommended gear ratios and approximate efficiency values for these gear ratios.
Typical planetary gears
| № | Block diagram of the mechanism | Ed | efficiency |
| 3....10 | 0.97....0.99 | ||
| 7....16 | 0.96....0.98 | ||
| 25....30 | 0.9....0.3 | ||
 | 30....300 | 0.9....0.3 |
Willys formula
The Willis formula is derived on the basis of the basic linking theorem and establishes the relationship between the angular velocities gear wheels in the planetary gear. Consider the simplest planetary mechanism with one external and one internal gearing. The entire mechanism is given an angular velocity equal in magnitude and opposite in direction to the angular velocity of the carrier, while the carrier will stop and the support wheel will begin to turn. Thus, the planetary mechanism will turn into a mechanism with fixed axes, consisting of several gears connected in series. Such a mechanism is called an inverted mechanism.
The angular velocities of the links in each of the considered movements are given in the table
In the movement of the links relative to the carrier, the angular velocities of the links are equal to the angular velocities in motion relative to the rack minus the angular velocity of the carrier. If in motion relative to the rack the axis of the satellite is movable, then in motion relative to the carrier the axes of both gears are stationary. Therefore, the basic linking theorem can be applied to motion relative to the carrier.
Reversed gear ratio 
, the final gear ratio of the planetary gearbox can be determined by the Willis formula:
The gear ratio of the planetary gear from any wheel to the carrier is equal to one minus the gear ratio of the inverted mechanism from this wheel to the reference one.
Kinematic study of spatial planetary mechanisms by the method of plans of angular velocities.
Let's consider this method of research on the example of the planetary mechanism of the conical differential rear axle car. On fig. 15.8 shows a diagram of the mechanism and plans for angular velocities.
Angular velocity plans are built in accordance with vector equations:
|
w2=w1+w21; w4=w3+w43 |
w3=w2+w32; w5=w3+w53 |
The vectors of relative angular velocities are directed along the axes of instantaneous relative rotation:
w
21
- along the line of contact of the initial cones of the links 2
And 1
;
w
32
- along the hinge axis WITH
;
w
43
4
And 3
;
w
53
- along the line of contact of the initial cones of the links 5
And 3
.
The absolute angular velocity vectors are directed along the axes of kinematic pairs that form links with a rack:
w
2
- along the axis of the pair IN
;w
1
- along the axis of the pair A
;
w
4
- along the axis of the pair E
; w
5
- along the axis of the pair D
.
Satellite angular velocity direction 3 determined by the ratio of the values of the angular velocities w 2 And w 32 .
Consider three driving modes:
- rectilinear motion w 4 = w5(vector diagram in Fig.15.8a). In this driving mode, the differential housing 2 and axle shafts 4 And 5 rotate at the same angular speed w 4 = w 5 = w 2, and the relative angular velocity of the satellite w 32=0 .
- car turning right w 4w5(vector diagram in Fig. 15.8b). When turning to the right, the angular velocities of the semiaxes are not equal and are related by the inequality w 4w5, so the satellite will rotate with such an angular velocity w 32 , which ensures the constancy of the angular velocity of the differential housing w 2.
- left wheel slip w 4 = 0 (vector diagram in Fig. 15.8c). When the left wheel slips, the right wheel stops w 4 = 0 , and the left one will rotate with an angular velocity w5 = 2h w 2 .
In order to reduce the risk of wheel slippage in the differentials of high-traffic vehicles, friction or blocking elements are included in conditions of low adhesion of wheels to the ground.
Test questions for lecture 15
1. Which gear mechanism is called complex? (p. 1)
2. What mechanism is called planetary? (page 1)
3. How to determine the gear ratio of one of the planetary gear schemes in an analytical way? (p. 2-4)
4. How are graphical and analytical methods used to determine the angular velocities of the links of planetary gear mechanisms? (pp. 6-9)
5. How are kinematic relationships established in a planetary bevel gear? (p.1-11)
6. How to use graphic way to determine the angular velocities of the differential links? (p. 10-11)
7. What is the purpose of applying the motion reversal method in the kinematic analysis of planetary mechanisms? (pp. 4-6)
The central wheel 1 is called solar, and the fixed 3 is called crown or crown. Gear 2 having a movable axle is called satellite. Link H is called a carrier or a leash. Mechanisms that include gear wheels with movable axes are called planetary or differential.
Planetary (Fig. 14 a) are called mechanisms that have one degree of freedom. Differential (Fig. 14 b) mechanisms have two or more degrees of freedom.
These mechanisms must necessarily be coaxial, that is, the axes of the sun wheels must be located on the same straight line.
Consider the differential mechanism (Fig. 15).
where: n=4; ; .
Thus, there will be certainty in the movement of the links of this mechanism if the laws of motion of its two leading links are known.
Since satellites have movable axles, it is not possible to use formulas for calculating the gear ratio of mechanisms with fixed axles. In this case, resort to the inversion method (reversed motion method).
We will consider the movement of all wheels relative to the carrier. We will ask all the links rotary motion with the angular velocity of the carrier, but in the opposite direction, and we will find the speeds of all links of the mechanism. To do this, we subtract the angular velocity of the carrier from all the angular velocities of the wheels.
|
|
Table 2.
| No. of Links | Link speed in actual motion (before inversion) | Link speed in inverted motion (after inversion) |
|
|
||
|
|
||
| Wheel 2' |
|
|
|
|
||
|
|
The mechanism obtained as a result of inversion (carrier stop) is called inverted (Fig. 16). As a result, we got the usual gear train with fixed axles.
|
|
This dependence (1) is called the Willis formula for differential mechanisms.
If there were n - wheels, then:
where s is the sun wheel.
The differential mechanism has no definite gear ratio if one of the links (wheel or carrier) is the leading one, and becomes definite if there are two driving wheels.
The graphical determination of the gear ratio of such mechanisms is carried out by the method of speed plans (velocity triangles). The velocity triangle can be constructed if the link has linear velocities of at least two points of the link (in magnitude and direction). Using this method, you can get a visual representation of the nature of the change in speed from one wheel shaft to another, and you can also determine graphically the angular velocity of any wheel.
,
Then 
, where the sign of the ratio is determined by the sign of the tangent.
The gear ratio of ordinary mechanisms is usually not large, since it is limited by the limiting dimensions of the outer wheels (1.4), and the number of teeth of the intermediate wheels (2.3) does not affect the overall gear ratio. Such mechanisms are used where it is necessary to change the rotation of the driven shaft, without changing the direction of movement of the drive - gearbox, or when transferring movement over considerable distances, if it is not possible to increase the size of the drive and driven wheels.
Kinematics of the stepped gear mechanism
Consider the kinematics of a stepped mechanism composed of three gears: two external gears (1-2) and (3-4) and one internal gear (5-6). The scheme of the mechanism is shown in fig.
Analytical determination of the gear ratio
The analytical definition of the gear ratio is based on the formula:
since the wheels 2-3 and 4-5 are on the same shaft, respectively, rotate with the same angular speeds.
Using the main theorem of Willis, for a given mechanism, we obtain:
On the basis of which you can get a general formula for determining the gear ratio of a stepped gearbox:
.
The total gear ratio of a stepped gear mechanism is equal to the ratio of the product of the numbers of teeth of the driven wheels to the product of the numbers of teeth of the drive wheels. The sign of the gear ratio is determined by the multiplier 
, Where 
- number of external gears.
Graphic definition of gear ratio
The graphical determination of the gear ratio is also carried out by the method of speed plans (velocity triangles).
,
where the sign of the ratio is determined by the sign of the tangent or by the rule of arrows.
By selecting the number of teeth in a stepped mechanism, large gear ratios can be obtained with the same dimensions as for an ordinary one.
Stepped gear mechanisms are often used in gearboxes where the gear ratio changes abruptly. This allows, at a constant angular velocity on the leading link, to inform the output link of the mechanism of speeds different in magnitude and direction, and to reproduce any series of gear ratios with a given pattern.
planetary gears
Complex gear mechanisms in which the axis of at least one wheel is movable are called planetary mechanisms.
Planetary mechanisms are divided into planetary gearboxes and multipliers, which have one degree of freedom and necessarily have a reference link, and gear differential mechanisms, the number of degrees of freedom of which is two or more, and which usually do not have a reference link.
Typical planetary gears include:
single-row planetary mechanism with mixed gearing (James mechanism);
double-row planetary gear with mixed gearing;
two-row planetary gear with two external gears;
two-row planetary gear with two internal gears.
The elements of the planetary mechanism have special names:
a gear wheel with external teeth located in the center of the mechanism is called "solar";
a wheel with internal teeth is called a "crown" or "epicycle";
wheels whose axles are movable are called "satellites";
the movable link on which the satellites are installed is called the "carrier". This link is usually denoted not by a number, but by the Latin letter h.
When the sun wheel rotates, the satellites turn like a lever relative to the instantaneous center of rotation (the support wheel is stationary) and make the planet carrier rotate. In this case, the planetary wheels (satellites) perform a complex movement: they rotate around their own axis (relative to the carrier) with an angular velocity 
and together with the carrier roll around its axis (portable movement). The number of degrees of freedom of this mechanism is equal to one. Therefore, the gearbox has a constant gear ratio.
Usually, a real mechanism has several symmetrically located satellite blocks. . They are introduced in order to reduce the dimensions of the mechanism, reduce the engagement force, unload the bearings of the central wheels, improve the balance of the carrier, although the mechanism in this case has redundant connections, i.e. is statically indeterminate. In kinematic calculations, one satellite is taken into account, since the rest are passive in kinematic terms.
If, in the considered mechanism, the support wheel (gear housing) is released from fastening and rotation is imparted to it, then all the central wheels will become movable and the mechanism will turn into a differential one, since the number of degrees of freedom will be equal to two.
Thus, differential mechanism
is a planetary mechanism with the number of degrees of freedom 
.
The number of degrees of freedom (mobility) of the mechanism shows how many links of the differential must be given independent movements in order to obtain the certainty of the movement of all other links. Here, depending on the direction of rotation of the outer shafts, either the decomposition of the movement (one leading into two slaves) or the addition of the movement can occur. A shaft is considered to be the leading one, in which the direction of the speed of rotation and the moment coincide. Therefore, a planetary gearbox (or multiplier) with a fixed wheel can be turned into a differential if the fixed (reference) wheel is released and rotated. On the contrary, any differential can be turned into a planetary gear by fixing one (w = 2) or several of its central wheels. This is the so-called reversibility property of planetary gears, which allows you to apply the same research and design methods for gearboxes and differentials. In this case, each elementary differential will correspond to two planetary gearboxes
The table shows the block diagrams of typical planetary mechanisms, as well as the ranges of recommended gear ratios and approximate efficiency values for these gear ratios.
