THE RISE OF THE PLANETARY GEARS

Manual Vs Automatic transmissions!! Google this out and a myriad of information pertaining to the same will come flooding before your eyes. What is a manual transmission, what’s an automatic transmission, advantages and disadvantages of both and eventually the heated discussion usually concludes by saying that ‘’Manual or Automatic, The choice is yours!’’. Well, safety, economy and time are of crucial value that a car suffers and besides the purchase, it’s the maintenance cost of a vehicle that is also looked upon. Fuel economy is one major factor that differentiates the two of these transmission types, manual having an advantage on this over its automatic counterpart. The rest you can check out for yourselves on the internet. What’s intriguing is the ingenious mechanism used in automatic transmissions that asks for the least amount of inputs from the driver while the rest is handled by the car itself. One such fascinating thing about it is the mechanism involved in the shifting of gears without the use of clutch, unlike a manual transmission. We will delve more into this design feature further here.

You may have heard or read about a standard gearbox used in cars that incorporates three shafts with various gear combinations on them that are controlled by the driver via a gear stick in order to change gears. This is the typical gearbox in a manual transmission. But when it comes to automatic, there’s a whole distinct mechanism of shifting gears. The key to the entire automatic system is the ”Planetary Gearset” that has a bunch of advantages to outperform a standard gearbox. BUT! Planetary gearsets are also used in Formula Student cars, particularly in Electric Vehicles to transfer the motor power to the wheels. EVs using both in-wheel as well as chassis mounted motors can incorporate this gearset.

First of all, let us know what these two terms mean: Epicycloid and Hypocycloid. Their definitions can’t be simpler than that in a dictionary. Epicycloid is a curve traced by a point on the circumference of a circle rolling on the exterior of another circle whereas Hypocycloid is a  curve traced by a point on the circumference of a circle rolling on the interior of another circle. The figures below depict the two(epicycloid and hypocycloid respectively).

It all starts with an epicyclic gearing design. An epicyclic gear train consists of two gears mounted so that the center of one gear revolves around the center of the other. A carrier connects the centers of the two gears and rotates to carry one gear, called the planet gear, around the other, called the sun gear. The planet and sun gears mesh so that their pitch circles roll without slip. A point on the pitch circle of the planet gear traces an epicycloid curve.

 

sun-planet mesh
Sun-Planet mesh

 

An epicyclic gear train can also be assembled so that the planet gear rolls on the inside of the pitch circle of an outer ring gear. In this case, the curve traced by a point on the pitch circle of the planet is a hypocycloid.

 

planet-ring mesh
Ring-Planet mesh

 

Now if we combine the two methods of meshing the gears as discussed above, i.e., the sun gear concentric with the ring gear and a planet gear engaging both of them, all three gears being in the same plane, we will get a variety of different combinations of gearsets by putting constraints on the relative motions of the three gears. The sun and planet gears are connected via a carrier or arm.

 

planetary
Ring-Planet-Sun mesh

 

The gear ratio in an epicyclic gearing system is somewhat non-intuitive, particularly because there are several ways in which an input rotation can be converted into an output rotation. The three basic components of the epicyclic gear are:

  1. Sun gear : the central gear.
  2. Planet carrier : holds one or more peripheral planet gears of the same size meshed with the sun and ring gears.
  3. Ring gear : An outer ring with inward-facing teeth that mesh with the planet gears.

In these epicyclic gear systems, usually one of the three components is held stationary, one of the two remaining components is an input and the last one is the output. On the basis of this, three basic epicyclic gearbox designs are possible, namely: Planetary, Star and Solar. We will discuss more about these types later in this blog. For the time being, let us perform some calculations to derive an interesting relationship between the angular speeds and the number of teeth of the three gears. This result will be used to calculate the gear ratios of the different types of epicyclic gearbox designs and finally decide the one suitable for a Formula Student electric car.

This is basically a superposition technique to simplify the calculations. The planetary gearset is divided into two arrangements as shown below.

total=part1+part2

We will now carry out calculations for the individual arrangements and sum up the results.

_20171210_174042sun-planet calci

 

_20171210_173650

planet-ring calci

Hence, adding up (i) and (ii), we get:

planetary formula

This will serve as the fundamental formula to calculate the gear ratios for planetary, star and solar gearsets. Let us see how. (Note: the fixed gears have their angular velocities= 0)

gr1

gr2

gr3

gr4

Hence, on the basis of the various gear ratios obtained above for different gearsets, the planetary set is the optimum one that is used in almost all of the automatic transmissions and also in Electric Vehicles owing to its highest achievable gear ratio suitable for attaining the maximum acceleration due to the torque multiplication all the way from the motor to the wheels. In Formula Student EVs, just a single gear ratio suffices the needs of the transmission. However, in cars with automatic transmission, compound epicyclic gearsets are used that help to achieve various gear ratios with the help of multiple planetary gearsets connected to each other.

Also, there is another important formula to take into consideration:

teeth relationship

As in Case 1, the gear ratio can range from 3:1 to about 11:1. But what’s the optimum ratio to choose? A 3:1 ratio has the sun gear twice as large as the planet gear. A 4:1 is a well-balanced ratio with the sun and planet gears of the same size. 5:1 and 6:1 too are quite balanced ratios. But as the ratios get large, the sun gear becomes smaller and smaller and hence its load-bearing capacity decreases. Also, at higher gear ratios, the planet gears become larger creating chances for them to interfere with each other. Hence, 3:1 to 6:1 are fair choices to go for, keeping in mind the desired gear ratio.

Well, the constraints do not end here. The planet gears just can’t be placed anywhere to mesh with the sun and ring gears! The choice of the number of teeth and the number of planet gears needs to satisfy the following criteria: (Z(R) and Z(S) are the no. of teeth in ring and sun gears respectively).

  1. If the planet gears are evenly spaced and all engage the next tooth at the same time, then both Z(R)/N and Z(S)/N must be even, N being the number of planet gears.
  2. If the planet gears are evenly spaced but all may not engage the next tooth at the same time, then ( Z(R)+Z(S) )/N must be even.
  3. If the planet gears need to unevenly spaced, then the minimum angle between any two planet gears about the sun gear must be a whole number multiple of (360°/(Z(R)+Z(S))).

Until now, we have discussed about almost all the key points that must be kept in mind while deciding the size, the number of teeth, gear ratios and the relative positions of the planet gears in planetary gearsets. But these were just the quantitative aspects of planetary gearsets involving calculations and derived facts that are used as thumb rules to design them. What significant values do we extract out of all this? On what grounds are these gearsets better than their standard gearbox counterparts? Let us look into the advantages that this ingenious gearset has to offer.

The principal advantage of epicyclic gears over the parallel shaft gears is the compact and light package. Compact meaning a high torque to volume ratio whereas light meaning a high torque to weight ratio. This stems from the fact that the use of multiple planets allows the load to be distributed among them. If you use a set of three planet gears, each sun-planet contact will carry one-third of the total load and hence, the sun gear size would be approximately one-third of a parallel shaft gear designed to transmit an equivalent torque. Increase the number of planets and you get more sun-planet contact points and hence, the torque density increases. When it comes to the inertia of rotating components; compared to a same torque rating standard gearbox, it is a fair approximation that the planetary gearbox inertia is smaller by the square of the number of planets. Also, in a parallel shaft gearbox, the lubricant is slung away at high speeds of the shaft as a result of which pressurized forced lubrication is used. In contrast, lubricant can’t escape through a planetary gearset as it is continuously redistributed among the components, mixed into the gear contacts and hence, these can be grease lubricated for life. Besides, the smaller gears can be made more accurately than the larger ones of the parallel shaft gearing.

The main losses concerning planetary gearsets are the tooth friction loss and bearing loss. While the tooth loss depends on the tooth load and pitch-line velocities of the gears, these are less than parallel shaft gears running with the same rotational speed with the same load. Bearing losses depend on the bearing size which is smaller on epicyclic gears. The planetary gears outperform on these grounds too.

One major design aspect to look out for is the optimum number of planet gears to be used in a planetary gearset. There is a difference between the effective number of planets sharing the load and the number of planets actually used in the set. D. L. Seager did a theoretical analysis of loading in a simple planetary gear system in 1970 wherein he developed a set of equations to determine the load sharing among the planet gears. He found out that if the number of planets is greater than three, the load was no longer divided equally among the planets. This analysis took into consideration the type of support that these gears have, either fixed or floating. In a fixed support, the gears are supported in bearings and the centers of sun, planet and ring will not be coincident due to manufacturing errors. Thus, fewer planets are simultaneously in mesh, resulting in a lower ”effective” number of planets sharing the load. In a floating support, one or two members are allowed a small amount of radial freedom/float which could be as little as 0.002 inches. This allows the members to adjust their positions to provide equalizing action so that their centers are all coincident and hence, three planets will always be in mesh resulting in a higher ‘’effective’’ number of planets sharing the load. The vibrations that may occur due to the clearances are dampened by the thick film of oil used in them. The gears are therefore, not rigidly fixed with respect to each other in a planetary gearset. Interesting! Isn’t it?

When it comes to the design of transmissions in formula student cars, especially the EVs, planetary gearset is the key feature to look out for. Whether the motor is mounted on the chassis or in the wheel, these gearsets are the connecting links between the motor and wheels. Based on the traction that the road can offer, motor specifications, etc. these gears can be analyzed and designed. It is the best choice to opt for! But there are a lot of design constraints associated with them and also various do’s and don’ts that must be kept in mind while deciding on the relative sizes and number of teeth in the gears. An insight into the various research experiments done in the past pertaining to epicyclic gears would be recommended to get a more detailed explanation of how these gears actually work because just formulae and facts are not sufficient to end up with a planetary gearset design, though they are crucial to be kept in mind all the time.

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