One pair of gears
Figure 35
When teeth are re-matched, the next tooth of the second wheel must fall into the next cavity of the first, i.e. the pitches on the pitch circles of the engaged wheels must be the same:
Thus, for one pair of wheels, the gear ratio is directly proportional to the ratio of angular velocities and inversely proportional to the ratio of the number of teeth of the wheels that make up the pair:
The sign of the gear ratio shows the direction of rotation of the wheel at the output in relation to the direction of rotation at the input:
- (+) - the directions of rotation at the input and output are the same. For a pair of wheels, the direction of rotation coincides with internal gearing (Figure 35b);
- (-) - the wheels rotate in opposite directions. This occurs with external engagement (Figure 35a).
Figure 35 shows a frontal projection of the gears, as well as their conditional image on the kinematic diagrams when viewed from the side (or in section).
Multi-stage transmission
To increase the kinematic effect, several gear pairs can be connected in series into a single mechanism. Such a mechanism is called a multi-stage gear mechanism or multistage transmission. A diagram of one of these mechanisms is shown in Figure 36.
Figure 36
Let's write down the gear ratios for each pair of wheels of this mechanism:
It can be seen from the diagram that wheels 2 and 3 are on the same shaft and rotate at the same angular velocity (ω 2 = ω 3 ), similarly ω 4 = ω 5 . Therefore, in the above equation, these terms canceled out.
Thus, the total gear ratio of a multi-stage mechanism is equal to the product of the private gear ratios of the steps that make up this mechanism:
In this formula, “m” is the number of external gears (if the number of external gears is even, then the “+” sign, i.e. the input and output wheels rotate in the same direction; if odd, then the “–” sign. Quantity internal gears are not taken into account, because internal gear does not change the direction of rotation).
In the given example, m=2 (pairs Z 1* Z 2 and Z 3* Z 4; pair Z 5* Z 6 is a pair of internal gearing) and, thus, the wheels "1" and "6" rotate in the same direction.
Planetary and differential gears
In practice, gear mechanisms are used that have wheels with movable geometric axes ( satellites). Such mechanisms are called planetary(if they have one degree of freedom) or differential(if the degree of freedom is two).
Planetary and differential gears allow you to get a higher kinematic effect, higher efficiency, more convenient layout. Differential mechanisms also allow you to decompose one movement into two or add two movements into one.
Figure 37
Figure 37 shows an example of differential (Figure 37 a) and planetary mechanisms (Figure 37 b). In these mechanisms, the wheel "2" has a movable geometric axis - this is the satellite.
The fixed geometrical axis around which the axis of the satellite moves is called central axis. Wheels whose geometric axes coincide with the central one are also called central(in figure 37 wheels "1" and "3" - sometimes such wheels are called solar). The link connecting the pinion axle to the central axle is called the carrier (the carrier is usually designated "H").
We write down the equation of the gear ratio between the central wheels of this multi-stage mechanism (in order to distinguish the gear ratio of the mechanism with the carrier stopped from the originally specified one, the carrier symbol H is put in the superscript. For this example, it reads - the gear ratio from the first to the third with the carrier stopped):
A formula of this type, obtained on the basis of the motion reversal method, is called the Willis formula. In this particular mechanism (Figure 38) there is one more feature - wheel 2 enters successively into two engagements (with the first and third wheels), being driven for the first wheel and driving for the second.
The resulting formula is universal for both mechanisms shown in Figure 37. The differential mechanism shown in Figure 37a has two degrees of freedom, and therefore, for the definiteness of motion, it is necessary to set the laws of motion for two links. In this case, the following options are possible:
- ω 1 and ω 3 are given; from the written formula, ω H is determined (the variant shown in Figure 37 a);
- ω 1 and ω H are given; ω 3 is determined from the written formula;
- ω H and ω 3 are given; ω 1 is determined from the written formula.
Since the links can be assigned any laws of motion, then how special case, we set the angular velocity equal to zero to one of the central wheels. For example, in the considered mechanism we will set ω 3 =0, in other words, we will brake the third wheel. With this technique, one of the two degrees of freedom is taken away, and the mechanism turns from a differential into a planetary one (Figure 37 b).
Thus, the planetary mechanism is a special case of the differential, when one of the central wheels is stationary (braked).
Therefore, these mechanisms are solved in exactly the same way, according to the same equations, only in the planetary mechanism for a stationary wheel, the value of the angular velocity equal to zero is substituted into the equation. For the planetary mechanism shown in Figure 37b.
Lab #26
Kinematic analysis of planetary and differential mechanisms
Objective:familiarization with the kinematics of planetary and differential mechanisms and the determination of their gear ratios by practical and theoretical methods.
Object of study: models of planetary and differential mechanisms.
In the previous laboratory work, gear mechanisms with fixed axes of rotation were studied. A distinctive feature of planetary and differential mechanisms is the presence of gears with a movable axis of rotation. Figure 1 shows the planetary mechanism. It has wheel 4 fixed, the common axis of wheels 2 and 2¢ rotates with the carrier H around wheels 1 and 4, called sun wheels. Wheels 2 and 2¢ are called satellites, and the mechanism is planetary by analogy with the solar system, in which the planets, making a revolution around the Sun, also rotate around their own axis.
The planetary mechanism has a degree of mobility equal to one. If wheel 4 is released, then we get a differential mechanism with two degrees of freedom.
The inversion method is used to determine the gear ratio of planetary mechanisms. In this case, this method equivalent to pinning the carrier and freeing the stationary wheel.
Rice. one
In this case, we get a gear train with fixed axes, the gear ratio of which can be determined by the method described in the previous laboratory work. On fig. 2 shows a diagram of the mechanism in reversed motion. The gear ratio of the planetary mechanism is indicated by the letter U, where the superscript indicates the fixed link, and the lower index indicates the numbers of the input and output links. For the mechanism in Fig. 1, having wheel 1 as an input link, carrier 1 as an output H, with fixed wheel 4. The gear ratio is indicated, and for the inverted mechanism -.
Rice. 2
The gear ratio of the considered planetary mechanism is determined by the Willis formula
where
In general, the gear ratio i-th wheel of the planetary mechanism to the carrier when stationary j-th wheel is determined by the formula
The gear ratio of the differential mechanism (Fig. 3) is determined from the formula for the gear ratio of the inverted mechanism
from which implies that the differential mechanism does not have a certain gear ratio if one input link has a certain angular velocity. Only at a given angular velocity of two input links (for example, 1 and H) the gear ratio becomes defined.
Rice. 3
Determination of the gear ratio empirically.
In the planetary mechanism (Fig. 1) we turn the input link (carrier H) at the cornerφ H =360 ° , determine the angleφ 1 rotation of the output link (wheel 1), then the gear ratio of the mechanism under study is equal to
The sign of the gear ratio is determined visually.
Work order
1. Familiarize yourself with the structure of the studied mechanisms.
2. Fill in the tables below. 1, having drawn diagrams of the investigated and reversed mechanisms.
Table 1
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planetary gear |
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Wheel tooth numbers |
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Formula and result of determining the gear ratio of the planetary mechanism |
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Formula and result of determining the gear ratio of the inverted mechanism |
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Angle of rotation of the output link |
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Empirically obtained gear ratio |
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reversed mechanism |
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Wheel tooth numbers |
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The value and formula of the gear |
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Formula and value of gear |
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Given: Z1=26, Z3=74, Z4=78, Z5=26, m=2
Find:,Z6 ,Z2
We select two contours on the kinematic diagram:
I k \u003d wheels 1,2,3 and carrier H.
II k \u003d wheels 4,5,6.
To determine the unknown values of the number of teeth of the wheels, we make the condition of alignment 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.
We choose an arbitrary vector of the speed 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, serial connection.
W 6 \u003d W H, W 3 \u003d W 4
Based on the constructed picture of instantaneous velocities, we construct a plan of angular velocities.
According to the constructed plan of angular velocities, we determine the gear ratio:
Conclusion
gear mechanism kinetostatic speed
During the implementation of the course project, a kinematic analysis of the mechanism was carried out and plans for speeds and accelerations for the working and idling of the mechanism (3 and 9 positions) were built.
As a result of the kinetostatic calculation, the values of the reactions in kinematic pairs and the balancing force for the working and idling 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 built, 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.