Institute of Mechanical and Automotive Engineering - Chair of Machine Elements

Technical University of Munich

Calculation method of multi-shaft planetary coupling gears

Complete reprint of the thesis published by the Faculty of Mechanical Engineering of the Technical University of Munich for the attainment the academic degree of doctoral engineer approved dissertation.

Chairman: Univ.-Prof. Dr.-Ing. Hans Peter Kau

Dissertation examiner:

Univ.-Prof. Dr. Bernd-Robert Höhn

Univ.-Prof. Peter Tenberge, University of Chemnitz

the dissertation was submitted to the Technical University of Munich on 5 April 2006 and accepted by the Faculty of Mechanical Engineering on 25 September 2007.

0 Preface

This thesis was written while I was working as an assistant at the Chair of for Machine Elements, the Research Centre for Gears and Gear Construction (FZG), at the Technical University of Munich. The basics for this work were during the research project “Mechanical Electric Drive Train”, a joint project of the Forschungsvereinigung Antriebstechnik e.V. (FVA) between the three chairs LVK(Chair of Combustion Engines), LEMG (Chair of Electrical Machines and Devices, now EWT) and the FZG in the comparative investigation of different hybrid concepts. My thanks go to all members of the accompanying working groups for their help, suggestions and interest in my work.

I would like to thank my doctoral supervisor, Prof. Dr.-Ing. Hoehn for his encouragement, his critical comments, which were a valuable help to me, and his constant support.

Special thanks go to my research group leader, Dr.-Ing. Klaus Michaelis for his great patience, his advice and his trust in me.

I would like to thank all colleagues, staff and students for their support, their esteemed advice and assistance, without which I would not have been as successful in multiple areas, such as my teaching position.

Most of all, I thank my parents, who made my education possible, and my siblings, who always helped me to solve many everyday problems. Without them, this work would certainly not have come about.

Munich, 2007

1 Introduction

In all areas of mechanical engineering, gearboxes are used in all areas of mechanical engineering for the conversion of speed and torque. Leather belts, ropes and cogwheels connected to water wheels or animal cogs have been used for a very long time to irrigate fields, lift loads and perform heavy work.

fig:1.1

sketch of epicyclic revolving wheels by Leonardo da Vinci

In the 15th century, the genius architect, painter and engineer Leonardo da Vinci sketched gears with epicyclic revolving elements (figure 1.1). This is currently the first written proof of the idea of gears with multiple degrees of freedom. Also, Anikythera shows it has been well known and used already for a long time. However, the merits of this type lie not only in the easy-to-integrate coaxial position of input and output shafts, but also in their smaller and lighter construction, which is achieved by splitting the power to several planets. With planetary gears you get, compared with conventional spur gear chains, the highest transmission ratios with the smallest required volume. In the following picture (figure 1.2) you see this by the representation of a one to three-stage spur gear chain in comparison with several different planetary gearboxes. In each case, the ratio of the largest required diameter to the diameter of the sprocket above the respective achievable transmission ratio. In the case of a rectangular shape of the gear (e.g. with the spur gear chains), the circle with the same area is used to form the ratio.

fig:1.2

space requirement of different gearbox types, expressed by the comparison of the related diameters with the same input torque as a function of the transmission ratio, source (MUE, 1)

The most important characteristic of the so-called epicyclic gearboxes, however, lies in the possibility of superimposing speeds of two or more shafts, which is made possible by the degrees of freedom. Shifting between two gears under load is also possible. These special properties are used, for example, in the differential of rear axles, in variable valve control of engines, in bicycle gear sticks, wheel hubs of agricultural and construction machinery, in wind turbines or in combination with infinitely variable hydro-stats and hydrodynamic converters in automatic transmissions, power steering systems and in many areas of heavy engineering, shipbuilding and aviation. Despite all these numerous areas of application the computational methods for planetary gears are considered difficult and a field for specialists, in which no more than the most general cases can be calculated or designed. Despite a degree in mechanical engineering, for many the determination of the transmission of gears with more than two degrees of freedom is an unsolvable task. Statements about self-locking are also often only assigned to certain types of gearboxes, such as the Wolfrom set, instead of making this property dependent on the standard ratios of the elementary gears (MUE, 2), in the preface of his book, “The planetary gear transmission” also states that:

“The author believes to be justified in assuming that there are cases in which the planetary gear was not used, only because it was not possible to determine the kinematics of the gearbox. …”

As far as the nomenclature of planetary gearboxes is concerned, one can find in the literature numerous variants for categories of planetary gears (one/two-stage planetary gears, plus/minus gears, elementary/reduced coupling gears, etc.), depending on what was important to the author as a common characteristic of a group. The term ‘multi-shaft’ planetary gearboxes was chosen as the title for this work because most of the methods for the calculation of planetary gearboxes are based on the systems with three active connecting shafts, i.e. there is usually an input and output, one shaft is fixed and all others rotate freely during operation. Gearbox systems with four or more connecting shafts must be composed of these basic cases for the calculation. Systems with multiple in- and outputs become particularly complicated, especially in the determination of their efficiency.

In addition to the derivation of the basic equations for planetary gears, this paper is to show a method which enables every engineer to determine all speeds and torques, i.e. also the transmission ratios of each gear, the spread and the degree of freedom of any combination of gears. It does not matter whether the gearbox is made of known components, such as a simple plus or minus gearbox, a Ravigneaux or Wolfrom set, coupled planetary gearboxes, reduced coupling sets, gears with double and stepped planets, or any other types of gears. Furthermore, the integration of the efficiency based on the data of empirical equations or measurements for gear and bearing losses is shown. In the second part of the paper is presented how this calculation method can be implemented on a computer using graph theory. For this purpose, all the paths the power can take through a gearbox are used to construct a graph, the shape of which then indicates the equations to be used. At the end of the paper there are hints on how the given algorithm can be used together with an automatic gear synthesis for gear analysis and thus for computer-controlled conceptual design. The most suited methods / algorithms for the generation and selection of reasonable solutions are briefly elucidated and their use for this purpose is explained.

2 Theoretical foundations of planetary gears

2.1 Fundamental Equations

planetary gearboxes belong to the group of epicyclic gearboxes. This type of gearbox has a revolving housing in contrast to the normally known stationary gearbox. Gearboxes are used for the mechanical conversion of speed or torque. If you look at an industrial gearbox, one only pays attention to the moving parts and easily overlooks the fact that the housing of the gearbox fulfils an important function. In torque conversion it is transferring the support torque from the bearings to the ground.

Example:

With a torque increase of 1:10 and a driving torque of 100 Nm, 1000 Nm will be taken from the output. However, 900 Nm have to be supported by the gearbox housing (according to the old mechanical theorem according to the old theorem of mechanics that the sum of the torques is always zero, ΣM = 0). Now, if we consider the outer gearbox housing to be a shaft as well and mount it rotatably, one has to transfer this supporting torque from the outside e.g. via a cut splined shaft (see circles as symbolism for ball bearings around the housing in figure 2.1, right). The stationary gearbox has thus become an epicyclic gearbox.

fig:2.1

derivation of an epicyclic gearbox from a stationary gearbox

Compared to stationary gearboxes, this type of gearbox has some special properties due to its degree of freedom of at least F = 2, which are described in more detail by the following equations. These equations describe all possible states of motion, torque ratios between the individual shafts and the degrees of freedom of the entire gearbox. They can therefore be applied independently of their constructional design. Examples of this are for example a hydrostat (when using hydraulic power transmission neglecting its leakage losses), an eccentric gear, a friction gear (neglecting slip), or a harmonic drive.

2.1.1 Speeds

In the case of a simple spur gear stage, as contained in a stationary transmission (figure 2.2), the transmission ratio (i) of a gearbox is determined from the ratios of its shaft speeds (n) as follows:

eq:2.1

\[ i_{12} = \frac{n_{1}}{n_{2}} \]

fig:2.2

simple stationary gear made of two splined shafts and spur gears

This transmission ratio, i.e. the speed ratio of the two shafts, can be very easily calculated from the ratio of the number of teeth or pitch circles of the gearwheels involved. A simple spur gear, input on shaft 1, output on shaft 2, with a transmission ratio of \(|i_{12}| > 1\) means a slow transmission, one with \(0 < |i_{12}| < 1\) a fast transmission. A reversal of the direction of rotation is usually expressed by a change of sign of the speeds. A spur gear stage with two identical gearwheels has a transmission ratio of -1. Non-parallel shafts are therefore handled like parallel shafts. To use the real 3D information of a shafts, vectors would have to be used.

An epicyclic gearbox, however, has not only two, but three outward led shafts:

two central gears (sun or ring gears with index 1 or 2 respectively)

a web shaft with the planets mounted on it and possibly rotating (with index s).

All three shafts can have one rotational speed, i.e. also that at a gear of this kind, with only two rotating shafts in each case, already six ratios (speed ratios) are possible. There are the possibilities of central wheel 1 to the web (1-s, s-1), central wheel 1 to central wheel 2 (1–2, 2–1) and central wheel 2 to the web (2-s, s-2). Willis already made this seemingly very confusing speed behaviour easier to understand around 1840 by describing it as a superposition of two partial movementsMUE:

the rolling case (in the following denoted by ')

the coupling case (in the following denoted by ")

Any state of motion of a planetary gear can be obtained by the addition of these two states with the correct amount and direction of the respective velocity vector. In the left part of figure 2.3 one can see the velocity distribution in the rolling case. The web is stationary ( \(n_{s}' = 0\) ) and the drive is provided by one of the central wheels (sun or ring gear). The right part shows the clutch case is shown: The gearbox rotates as a block, as if it were filled with sand. For this to happen, all three shafts of the planetary gear must rotate at the same speed ( \(n_{1}'' = n_{2}'' = n_{s}''\) ). Any speed state ( \(n_{1}, n_{2}, n_{s}\) ) is now always the sum of these two cases. With these preconditions, we get:

\[ n_{1} = n_{1}' + n_{1}'' = n_{1}' + n_{s}'' \\ n_{2} = n_{2}' + n_{2}'' = n_{2}' + n_{s}'' \\ n_{s} = n_{s}' + n_{s}'' = 0 + n_{s}'' = n_{s}'' \]

then follows from the third line for \(n_{1}\) and \(n_{2}\): \[ n_{1} = n_{1}' + n_{s} \\ n_{2} = n_{2}' + n_{s} \] These equations can be solved for \(n_{1}'\) and \(n_{2}'\). If one now takes the equation of the transmission ratio of a spur gear stage (equation 2.1), which is also valid here for the ‘stand case’, it follows for the transmission ratio of a planetary gear the so called Willis-equation:

eq:2.2

\[ \frac{n_{1}'}{n_{2}'} = i_{12} = \frac{n_{1} - n_{s}}{n_{2} - n_{s}} \]

transformed, this gives

eq:2.3

\[ n_{1} - n_{2} \cdot i_{12} - n_{s} \cdot (1 - i_{12}) = 0 \]

The ratio of the relative speeds \(n_{1}' = n_{1} - n_{s}\) and \(n_{2}' = n_{2} - n_{s}\) is therefore always constant and equal to the transmission ratio of the gearbox in the stationary case. This, in turn, can be easily determined from the number of teeth or the pitch circle diameters. The relative speeds are often also referred to as ‘rolling speeds’. Correspondingly, the web speed is also called the ‘clutch speed’ of each shaft 1, 2 or s.

fig:2.3

The rolling case (') (left) and the coupling case (") (right) of an epicyclic gearbox

KutzbachKZB has developed a very simple graphical method for the visualisation of the speeds according to size and direction from the diameters of the gears used, but this will not be discussed in detail here, as its creation is shown in the chapter sec:Kutbachplan by means of an example.

2.1.2 Transmissions and Stand Transmission

The ‘Standübersetzung’ in german or ‘stationary gear ratio’ of a planetary gearbox is, as the name implies, the ratio between the two shafts of the planetary gearbox when the web is standing still. Since the stationary gear ratio is a central variable for calculating the speeds and torques, the determination of this value from the existing geometry will be demonstrated on two planetary gearboxes. These are a ‘plus’ and a ‘minus’ gearbox. In the case of a Plus gearbox, the input and output shafts rotate in the same direction. With a minus gear, the input and output shafts rotate in opposite directions. There are several types of categorisation of planetary gearboxes (VDI2157, 3), e.g. according to

Number of webs: main categories single, double and multiple web gearboxes, subcategories simple and compound gearboxes.

Number of running connecting shafts: Two-, three-, multi-shaft gearboxes

Mode of use: Gear ratio, superimposed, manual, reverse, summing and differential gearboxes

movement of the web: stationary transmission or recirculating gearboxes (standing or rotating web)

The subdivision of planetary gears into the two categories of ‘plus’ and ‘minus’ gearboxes has proved to be particularly useful for statements on efficiency, self-locking capacity and power flows MUE,STR.

In the figure 2.4 you can see a minus gearbox. The power flows from the sun (1) via the planets (p) (only one shown) to the web (s). The ring gear (2) is fixed in the gearbox housing.

For the determination of the standing transmission of a gearbox you go step by step from one central gear to the other central gear. During the steps you multiply the negative ratio of each pair of teeth involved.

(No matter what the actual kinematics of the construction look like, as the ring gear is fix here, this procedure is strictly followed).

fig:2.4

minus gearbox

The equation for the final drive ratio of this minus gearbox would be: \[ i_{12} = -\frac{z_{p}}{z_{1}} \cdot -\frac{z_{2}}{z_{p}} = \frac{z_{2}}{z_{1}} \] Since one has to use a negative number of teeth for the ring gear, the result is a stationary transmission ratio < 0 (the pairing of planet to ring gear does not produce a reversal of the direction of rotation!)

For the plus gearbox in the following figure 2.5 the result is therefore: \[ i_{12} = -\frac{z_{p1}}{z_{1}} \cdot -\frac{z_{2}}{z_{p2}} = \frac{z_{p1} \cdot z_{2}}{z_{1} \cdot z_{p2}} \] As can be seen immediately, this number is > 0, since no ring gear is involved and there is no negative sign. Here also the power flows from the first sun via the planets to the web. The second central wheel is again fixed in the housing.

fig:2.5

plus gearbox

With the knowledge of the standing ratio, this fixed ratio of a planetary gear, it is now possible to calculate all five other possible ratios from the Willis equation (equation 2.2). In the above example, for the determination of the output speed, the gear ratio from the centre wheel 1 to the web is required. From equation 2.3 we get:

eq:2.6

\[ n_{1} - n_{2} \cdot i_{12} - n_{s} \cdot (1 - i_{12}) = 0 \enspace |:n_{s}\]

eq:2.7

\[\frac{n_{1}}{n_{s}} - \frac{n_{2}}{n_{s}} \cdot i_{12} - (1 - i_{12}) = 0 \enspace |n_{2} = 0\]

eq:2.8

\[i_{1s} = 1 - i_{12}\]

The translation from a sun to the web in both examples is therefore always 1 minus the standing transmission. One can also see that, if not the case of only two rotating shafts (\( n_{2} \ne 0\)), at least one speed ratio between the two shafts would have to be known in order to be able to be determined.

Speed ratios with three rotating shafts are usually designated with a different letter (e.g. k) to distinguish them from the ratio of only two rotating shafts, which can be calculated from the number of teeth. If three shafts rotate at the same time, one would have to calculate the ratio from sun to bridge as follows:

eq:2.9

\[ n_{1} - n_{2} \cdot i_{12} - n_{s} \cdot (1 - i_{12}) = 0 \enspace |:n_{s}

\\]

eq:2.10

\[ \frac{n_{1}}{n_{s}} - \frac{n_{2}}{n_{s}} - i_{12} - (1 - i_{12}) = 0 \]

eq:2.11

\[ k_{1s} = 1 - i_{12} \cdot (1 - k_{2s}) \]

2.1.3 clutch power and rolling power

The two submovements defined by Willis for calculating the speed ratios can also be used to illustrate the internal power flows of an epicyclic gearbox. The consideration of these powers is helpful for the interpretation of the equations for the determination of the torques on the individual shafts and the efficiency. As already described in chapter [2.1.1][txt:Drehzahlen], every possible state of motion can be derived from the superposition of rolling and coupling case.

In the rolling case, the entire power applied from the outside is transmitted via the planets from one central wheel (1, e.g. the sun) to the other (2, e.g. a ring gear). The power is transmitted with losses due to the sliding between the teeth. The resulting rolling power is defined by the circumferential force of the teeth in the rolling circle and the relative speed occurring between the respective wheel and the rolling point. Since the rolling point is always in a constant position relative to the web, the relative speed between the web shaft and the rolling point is always zero. Therefore power can only be transmitted between the two central shafts. (see also figure 2.6), but not between a central shaft and the web shaft. The equations for this are:

eq:2.12

\[ P_{W1} = M_{1} \cdot (\omega_{1} - \omega_{s}) \]

eq:2.13

\[ P_{W2} = M_{2} \cdot (\omega_{2} - \omega_{s}) \] In figure 2.6 the triangular circles symbolise the input shafts. All arrows coming from outside represent external power input, while all arrows within the triangle represent the power triangle denote the power flow inside the gearbox.

fig:2.6

all possible power flows of the rolling power

If the web is now allowed to rotate faster and faster from a standstill until all three shafts have reached the same speed in the same direction of rotation, this is called clutch fall. The total power transmitted by the gearbox is then passed on without rolling off the gears, i.e. it is transmitted without loss. (The thereby occurring plane torque of the web in the oil and the load-independent bearing losses of the outer shafts are neglected). The power then transmitted results from the product of the applied torque and the web speed, which is then applied to all shafts, or formulated mathematically in the following:

eq:2.14

\[ P_{K1} = M_{1} \cdot \omega_{s}\]

eq:2.15

\[ P_{K2} = M_{2} \cdot \omega_{s}\]

eq:2.16

\[ P_{Ks} = M_{s} \cdot \omega_{s}\]

The coupling powers are thus always in the same ratio to each other as the torques of the shafts, which can easily be determined by shortening with \(\omega_{s}\). An interesting point of this relationship is - since the sum of the torques is always 0 and therefore one of the torques must always be as great as the two others, that the coupling powers of two shafts must always flow towards one or away from one to the other two.

Figure 2.7 shows all possible types of coupling powers which can occur within an epicyclic gearbox. In any case, both coupling powers always flow towards or away from a shaft.

fig:2.7

All possible power flows of the coupling power

Now, the external powers of any state of motion result from the addition of these two types of ‘internal powers’:

eq:2.17

\[ P_{1} = P_{W1} + P_{K1} = M_{1} \cdot (\omega_{1} - \omega_{s}) + M_{1} \cdot \omega_{s} = M_{1} \cdot \omega_{1} \]

eq:2.18

\[ P_{2} = P_{W2} + P_{K2} = M_{2} \cdot (\omega_{2} - \omega_{s}) + M_{2} \cdot \omega_{s} = M_{2} \cdot \omega_{2} \]

eq:2.19

\[ P_{s} = 0 + P_{Ks} = M_{s} \cdot \omega_{s} \] From the idea that coupling power is transmitted without loss and that power can only flow between the two central shafts, some phenomena in certain types of planetary gearboxes are easier to explain, e.g. the fact that the recirculating efficiency of minus gearboxes must always be better than their stationary efficiency, since the coupling and rolling capacities can never overlap unfavourably. The total shaft of minus gearboxes is always their web shaft (for explanation see page ErklaerungMinusgetriebe), therefore the coupling power of minus gearboxes always flows from one of the two central shafts to the web shaft or back. The lossy part of the power always flows from central shaft 1 to 2 or vice versa. At standstill the entire external power always flows as rolling power, i.e. it is lossy. In the case of superimposition, part of the power is transmitted as coupling power, i.e. the efficiency of a negative gear improve in any case. Figure 2.8 shows all the possible cases of power flows that can occur in a minus gearbox. It can be seen that coupling power and rolling power never coincide, i.e. run along the same path. With a positive transmission one of the two central shafts will always be the summation shaft, i.e. the power will always flow from/to 1 or 2 and overlap with the rolling power.

fig:2.8

external and internal power flows of a minus gearbox in case of power sharing

2.1.4. Torque and Torque ratios

In the following section, the relationship of the torque ratio between the three shafts of an epicyclic gearbox will be shown. This is based on the fact that all powers acting on a system must result in zero.

eq:2.20

\[ \sum\limits_{i=1}^{n}{P} = \sum\limits_{i=1}^{n_{An}}{P_{An}} + \sum\limits_{i=1}^{n_{Ab}}{P_{Ab}} + \sum {P_{V}} = 0 \]

All powers at the input shafts are equal to the sum of the power losses plus the output power. It follows mathematically that the output and power loss can be interpreted as the supply of negative power. Thus, if the power at a (splined) shaft is positive, it is a drive shaft (\(P_{An} > 0\)), if it is negative, it is a driven shaft (\(P_{Ab} < 0\)).

In the following, one proceeds to determine the relationship between the torques of the input and output shafts and the efficiency of the gearbox again based on a simplified approach:

Any gearbox with exactly one input and one output shaft, shown in figure 2.9 as a black box.

fig:2.9

gearbox as black box. Equation of the efficiency depends on the flow direction of the power

The efficiency of such a gearbox is determined by the general relationship between benefit by effort:

eq:2.21

\[ \eta = \frac{- P_{Ab}}{P_{An}} \]

The negative sign only serves to turn the negative output power into a positive number in order to not having to reckon with negative efficiencies. If the power now flows from shaft 1 to shaft 2 (so \(P_{1} > 0\) and \(P_{2} < 0\)), the result is:

\[ \eta_{12} = \frac{- P_{2}}{P_{1}} = \frac{- M_{2} \cdot n_{2}}{M_{1} \cdot n_{1}} = -\frac{M_{2}}{M_{1}} \cdot \frac{1}{i_{12}} \]

it follows:

eq:2.22

\[ (\frac{M_{1}}{M_{2}})_{P_{1}>0} = \frac{- 1}{i_{12} \cdot \eta_{12}} \]

But if the power flows from shaft 2 to shaft 1 (so \(P_{1} < 0\) and \(P_{2} > 0\)), we get:

\[ \eta_{21} = \frac{- P_{1}}{P_{2}} = \frac{- M_{1} \cdot n_{1}}{M_{2} \cdot n_{2}} = - \frac{M_{1}}{M_{2}} \cdot i_{12} \]

from which follows:

eq:2.23

\[ (\frac{M_{1}}{M_{2}})_{P_{1}<0} = \frac{- \eta_{21}}{i_{12}} \] Both formulas are not identical! (Only if \(\eta = 1\)). However, this means:

-To know what the efficiency of a gearbox is or to calculate the ratio of the torques between the input and output shafts, one must know in which direction the power flows through the gearbox. -loss symmetry of a gearbox, i.e. the same efficiency when power passes through from shaft 1 to 2 or vice versa, is by no means self-evident. (It is \(\eta_{12} \ne \eta_{21}\) until proven otherwise).

In order to prevent the need of giving two formulas each time for the torque ratio of two shafts and the conditions for their use, the equations 2.22 and 2.23 of (BRA, 4) were combined into

eq:2.24

\[ \frac{M_{2}}{M_{1}} = - i_{12} \cdot \eta_{o}^{w1} \]

Whereby applies: \[ w1 =+1 → \eta_{o}^{w1} = \eta_{12}\\ w1 =-1 → \eta_{o}^{w1} = \frac{1}{\eta_{21}} \]

\(w1\) is determined from the direction of flow of the rolling power at shaft 1. Mathematically speaking this means:

\[ w1 = \frac{M_{1} \cdot (n_{1} - n_{s})}{|M_{1} \cdot (n_{1} - n_{s})|} \]

In order to be able to draw conclusions from this simple spur gear stage to the torque ratios of an epicyclic gearbox, the formula of the equilibrium of torques is used

eq:2.25

\[ \sum\limits_{i=1}^{n}{M} = M_{1} + M_{2} + M_{s} = 0 \] From this we can derive the two missing relations for the torque ratios to the web shaft:

eq:2.26

\[ \frac{M_{s}}{M_{1}} = i_{12} \cdot \eta_{o}^{w1} - 1 \]

eq:2.27

\[ \frac{M_{s}}{M_{2}} = \frac{1}{i_{12} \cdot \eta_{o}^{w1}} - 1 \]

With this, we can now calculate all the torques on the shafts of an epicyclic gearbox using the equations 2.24, 2.26 and 2.27, as soon as you have calculated its stall ratio (\(i_{12}\)) and stall efficiency (\(\eta_{12}\) or \(\eta_{21}\)) are known. It is to be noted that the torque ratios are thus only dependent on the number of teeth of the gearbox and not on its speeds. The power flow is determined by the speed ratios, but not the torque ratios.

2.1.5 Stand Efficiency and Circulation Efficiency

In the previous section, the equations for the torque ratios between two of the three shafts of an epicyclic gearbox are being derived, starting from a simple stationary gearbox. These equations can be easily applied to an epicyclic gearbox with a stationary web. The resulting efficiency is also referred to as the stationary efficiency, since the web is stationary. It has also been shown that the direction of the power flow is decisive for the choice of the correct equation. For two rotating shafts, there was the simple approach that the gearbox has to be passed through once from right to left or vice versa by the power. Due to the resulting equations for the torque ratios (equations 2.24, 2.26 or 2.27), the calculation of the efficiency for these cases is relatively simple.

With three shafts rotating at the same time, there are no longer just these two, but six possible cases (see figure 2.10)):

Three cases for power summation and three for power sharing.

fig:2.10

All options for power sharing and power summation on an epicyclic gearbox

In each of the three cases mentioned, one of the three shafts is the sole input or output for the power. In this operating case, it is no longer referred to as the standstill efficiency but as the ‘circulating efficiency’. In the following the equation for one of these cases will be derived: The case of power sharing from central wheel 1 to web and central wheel 2 (figure 2.10, top centre) . From the power approach and the equations for the torque ratios we get:

eq:2.28

\[ \eta_{{1}<^{2}_{s}} = -\frac{P_{2} + P_{s}}{P_{1}} = -\frac{M_{2} \cdot n_{2} + M_{s} \cdot n_{s}}{M_{1} \cdot n_{1}} \]

eq:2.29

\[ = -\frac{- M_{1} \cdot i_{12} \cdot \eta_{o}^{w1} \cdot n_{2} + M_{1} \cdot (i_{12} \cdot \eta_{o}^{w1} - 1) \cdot n_{s}}{M_{1} \cdot n_{1}} \] after shortening with \(M_{1}\) and the shortened notation of k for the speed ratios for three rotating shafts, we obtain:

eq:2.30

\[ \eta_{{1}<^{2}_{s}} = i_{12} \cdot \eta_{o}^{w1} \cdot k_{21} - (i_{12} \cdot \eta_{o}^{w1} - 1) \cdot k_{s1} \] Since it is easier to have to insert only one speed ratio into the equation one makes it only dependent on e.g. \(k_{12}\).

\( k_{21} = \frac{1}{k_{12}} \) and \(k_{s1} = \frac{1 - i_{12}k_{12}}{1 - i_{12}} \)

substituting these two relations, we finally obtain for the efficiency of this case:

eq:2.31

\[ \eta_{{1}<^{2}_{s}} = \frac{k_{12} - i_{12} + i_{12}\eta_{o}^{w1}(1 - k_{12})}{k_{12}(1 - i_{12})} \] The other cases can be determined equivalently. All these cases have been listed by Müller in a very clear table (MUE, 5). In case of minus gears this efficiency is always better than the static efficiency, which results from the consideration of the favourable overlapping of rolling (see fig. 2.4), or the comparison of the result of the equation 2.24 with its behaviour \(n_{s}\) against the extreme values (0/∞).

2.2 Definition and use of Wolf’s internal gear diagrams

When determining the elementary quantities of speed, torques and power of an epicyclic gearbox, one limits oneself to a few input parameters (static transmission ratio and static efficiency), as explained in the previous chapter. Once these parameters have been obtained, the design is no longer relevant for the calculation. For this reason, a form of representation was soon devised that was intended to show only the essential features and properties. In 1958, Wolf introduced symbols that represent a gear by means of a circle with three lines going outwards (the three connecting shafts). These symbols, type and use are described e.g. in the VDI guideline (VDI672, 6). The most important terminology is also explained there more detailed. In the following, therefore, only the most important features of this form of presentation will be briefly discussed in order to provide the basis for the following chapter on the determination of the degree of freedom in the Wolf representation.

The ridge shaft is always specially marked by this line extending somewhat into the circle, while the other two end at the edge. Wolf has also marked the ‘sum shaft’ by a double web, i.e. the shaft with the greatest torque. In this way it is possible to detect at first sight whether it is a plus or minus gearbox.

In the case of a minus gearbox, the web shaft is always the same as the sum shaft, whereas with a Plus gearbox one of the two central shafts must always be the sum shaft. This can be easily understood by the following consideration:

If you have a minus gearbox, then one of the central shafts, W1 (e.g. a sun), must rotate in a different direction than the other one, W2 (e.g. a ring gear), because this is the definition of a minus gearbox. If one now applies a positive drive torque to W1 (the sun), the positive speed results in a drive power. Since the web is stationary, the output must be guided outwards via W2 (the ring gear). But since the speed of this shaft is negative, the torque applied to W2 must also be positive, since otherwise the product of speed and torque would not result in a negative output power. If the sum of the torques at the planetary gear is zero, \(M_{1} > 0\) and \(M_{2} > 0\), the web torque \(M_{s}\) has to be negative and as large as these two together. Consequently, the web shaft of a negative gear must be the sum shaft. (see 2.2). A corresponding consideration can be made for a Plus gearbox. The result is, that with a positive transmission one of the two central shafts must be the sum shaft. Given that, this is the shaft with the gear, that has the greater number of teeth.

fig:2.11

Minus gear - the web is always the sum shaft, torques, speeds and powers

According to Wolf, this representation is particularly interesting for the interlinking or interconnection of several elementary gearboxes (higher coupling gearboxes), in order to keep overview. The torques or speeds can be noted on the individual shafts in order to gradually solve more complex problems. There are also symbols for fixing a shaft in the housing or the connection of a clutch or brake. Together with a shift logic several gears can be generated. (e.g. in a planetary gearbox, shown in figure 2.12).

fig:2.12

Simpsonsatz as an example of a switchable planetary gear, construction and symbolism according to Wolf

table:2.1

simpsonset logic of image 2.12

Depending on the switching state, the individual torques and speeds can be visualised and the interaction of the individual components can be better understood, similar as in hydraulic or electrical circuit diagrams. The wolf symbolism is useful both at the beginning of the design phase, for determining the correct Gratuity regulations of the gear ratios, as well as for recalculating existing gearboxes. When the number of planetary gearboxes used and all their ratios (number of teeth) have been specified, an assumed static gear can be used to determine the efficiency of the overall system in each elementary gearbox. Statements about self-locking can also be derived very well from these plans.

2.3 Gearbox degrees of freedom

“The degree of freedom of a physical system is equal to the number of determinants that can and must be given arbitrary and independently of each another, in order to unambiguously determine its state.”(MUE, 7).

A gearbox has as relevant determinants speeds [\(s^{-1}\)] and torques [Nm], which are determined according to the requirements of the design. At the degree of freedom of a gearbox, a distinction is made between its kinematic (speeds) and its static degree of freedom (torques). The degree of freedom of a gearbox must contain all the necessary determinants for the operation of a system and is often referred to as the ‘degree of operational freedom \(F_{B}\)’. It is defined by the sum of the kinematic and static degrees of freedom

eq:2.32

\[F_{B} = F_{stat} + F_{kin} \]

In order to be able to specify a certain number of operating variables, one needs as many force application points into the gearbox as there are specifications. If one assumes that a shaft end represents exactly one point of force introduction and that there are no losses in the system, the number of outwardly guided shafts (W) must correspond to the degrees of freedom.

eq:2.33

\[ W = F_{stat} + F_{kin} = F_{B}\]

It also follows: If one wants to transmit power with the gear as well, it is to be noted that at least one shaft must exist more than the system has kinematic degrees of freedom, otherwise the static degree of freedom is zero and no torque can be introduced. A transmission with exactly one input shaft and a degree of freedom of 2 is only conceivable if only losses and heat are produced within the gearbox.

In the following, only the abstract forms of planetary gears according to geometry, bearings, clutches and brakes will be considered, as each shifting state and gear of a gearbox can also be described in a separate model according to Wolf. The definition of the degrees of freedom is also derived from these, as the calculation model presented later is based on the real geometry and, in addition to the speeds and torques, also determines the degrees of freedom. Even if clutches and brakes are represented in Wolf’s symbolism, a concrete shifting condition must always be assumed for the calculation of a gear.

2.3.1 Static degree of freedom

If one assumes in the following that the static transmission ratio and the static efficiency are known, one always has three unknown quantities for the torques on an epicyclic gearbox:

The torque

at the web (\(M_{s}\))

at the centre wheel 1 (\(M_{1}\))

and at the centre wheel 2 (\(M_{2}\)) Two equations are available to determine these quantities:

sum of torques (equation 2.25): \[M_{1} + M_{2} + M_{s} = 0\]

the ratio of two torques (equations 2.24 or 2.26 or 2.27). \[ \frac{M_{s}}{M_{1}} = i_{12} \cdot \eta_{o}^{w1} - 1 \frac{M_{s}}{M_{2}} = \frac{1}{i_{12} \cdot \eta_{o}^{w1}} - 1 \frac{M_{2}}{M_{1}} = - i_{12} \cdot \eta_{o}^{w1} \] So there are two equations and three unknowns, consequently the static degree of freedom of a simple epicyclic gearbox = 1, i.e. one torque must be given in order to be able to calculate all the others.

The resulting degree of freedom of a complete gearbox is not always immediately obvious.

fig:2.13

spur gear with static degree of freedom 2

Even with a simple spur gearbox, a few design changes can be made several torques can be applied to one shaft, so that the static degree of the static degree of freedom is increased (see figure 2.13 Spur gear with stat. \(F_{stat} = 2\) ). Only by unambiguous specification of the quantities at each force application point (and if it is \(M_{x} = 0\)) can input and output power or the missing torques be calculated. With the combination of, e.g., two simple elementary planetary gearboxes, depending on the type of combination of the individual shafts, the choice of the bindings can change the degree of static freedom (figure 2.14 Four-shaft transmission).

fig:2.14

Two four-shaft gearboxes with differences in their internal meshing and degree of freedom

Thus the left-hand gearbox has a single, the right-hand one a double static degree of freedom. For the exact determination of the degree of freedom of a certain gear construction, one abstracts this procedure into the representation according to Wolf and then proceeds most simply with the method described in section 2.3.4. Also with application of the alternative calculation method presented in this work, one of course obtains a statement about the degree of freedom of the entire construction. (With this example one can also see that a categorisation of epicyclic gearboxes according to the number of their connecting shafts would not make sense if one has in mind the summary of general statements).

2.3.2 Kinematic degree of freedom

Here also the static ratio \(i_{12}\) and the static efficiency \(\eta_{o}\) are assumed to be known. Three shafts are also involved in determining the rotational speed, i.e. three variables are unknown:

at the web (\(n_{s}\))

at the central wheel 1 (\(n_{1}\))

and at centre wheel 2 (\(n_{2}\)) However, here only the so-called Willis equation (equation 2.2) which was derived on page 8. \[ n_{1} - n_{2} \cdot i_{12} - n_{s} \cdot (1 - i_{12}) = 0\] An equation for three unknowns gives a kinematic degree of freedom of 2 for a simple epicyclic gearbox, i.e. two speeds must be known to calculate the missing third.

The stationary case of an epicyclic gearbox, when the web shaft is stationary, is also a speed specification, namely \(n_{s} = 0\). A change in the bindings and the static degree of freedom automatically means a change in the kinematic degree of freedom.

2.3.3 Constrained and unconstrained gears

Instead of the term ‘degree of freedom’ in the literature on the general gear theory the term ‘degree of running’ is often used. Since there the mobility of a gear is usually examined, the kinematic freedom of the gear usually means its kinematic degree of freedom. Reuleaux (REU, 8) denotes a gear with the running degree, i.e. kinematic degree of freedom, \(F_{kin}\) as constrained. Accordingly, gears with the degree of freedom of 2 are called double-rotation gears, and those with a higher degree of freedom are called multi-rotation gears. This kind of distinction is mainly important due to the different operating behaviour resulting in practice:

Forced-rotation gearboxes are given exactly one speed, but can have several input or output shafts and thus a higher static degree of freedom. Only through the knowledge of each torque on all connecting shafts (these can of course also be 0!) the complete power flow can be determined. However, all shafts of the gearbox always rotate in direct relation to the drive speed (a multiple of the drive speed, depending on their transmission ratio). As an example of a positive transmission Mueller (MUE, 9) cites a distributor gearbox of a drill with several drill spindles (figure 2.15).

fig:2.15

drill as an example of a constrained gear, source (MUE, 10)

In the case of involuntary gearboxes, however, multiple speeds can and must be specified, while the torques on all shafts have an invariable relation to each other, which is given by the number of teeth used. This leads to a different operating behaviour than with positive drive gearboxes:

If there is no torque on one of the output shafts, the complete gearbox can suddenly no longer transmit any power, since all output shafts also receive a torque of 0. However, the output shaft without torque revs up empty until it reaches its highest idling speed. An example of an unconstrained transmission is the rear axle differential (figure 2.16) of a passenger car.

fig:2.16

differential of a rear axle as an example of a casual gear

2.3.4 Determination of the degree of freedom from the Wolf transmission diagrams

On the basis of the sectional drawing of the construction, a statement about the degree of freedom of a complete gearbox is not always possible even for the experienced designer. Especially with automatic transmissions, which consist of several elementary plus and minus gearboxes the interconnection of the individual components is essential for the resulting degree of freedom. If clutches, brakes and freewheels that are closed, open or active depending on the gear come along, it is often very difficult and confusing to see the power flows, speeds and torques of a given gear. Here it is advantageous to note each gear of a gearbox in the symbolism according to Wolf and to represent all possible shifting states of the system. From the circuit diagram according to Wolf, it is now very easy to make a statement about the operating behaviour and the degree of freedom of a transmission system. One takes the plan drawing of the gearbox and abstracts all existing elementary gearboxes from it. Now, the lines of all elementary gears, which in reality lie aside on a shaft as a sun, ring gear or web, are connected to each other via a node. All circles of the epicyclic gears are connected at the end in a way that there is always exactly one node between them and the next circle. Now you can determine the degree of freedom of the respective system:

This is to be set up in such a way that a circle symbol of an epicyclic gearbox has exactly one equation for the speeds (the Willis equation 2.2) and two for the torques (torque ratio is the ratio of the number of teeth equation 2.24 and sum of torques is zero, equation 2.25). Thus the result is for the symbol of an epicyclic gearbox the following figure with the corresponding equations as listed in table 2.2.

table:2.2

necessary equations on an epicyclic gear

A node that connects e.g. n elementary gears (i.e. n lines that converge), always has exactly one equation for the torque equilibrium (sum of torques is zero) and \(n - 1\) equations of speed. For a node with four connections, the following equations for torques and speeds therefore result as given in table 2.3.

table:2.3

necessary equations at a node with four connections

In the following section, two examples will be used to demonstrate the application of these two rules for the determination of the degree of freedom. This is based on a ready-made existing model of the gearbox in the symbolism according to Wolf. In general, the following procedure for determining the degree of freedom of planetary gears is recommended:

Numbering the shafts of each elementary gear from 1 to \(N_{shafts}\).

Create a table with one column each for all speeds and torques following the example of table 2.4

Counting of all revolving gearboxes and entry of this number \(N_{epicyclic gear}\) in the first row of both columns

Counting of all connection nodes and entry into the torque column of the second row

Counting of all connecting nodes with the same number of connections and entry into the speed column of the second row and its following

Determining the number of given equations by multiplying each digit with the corresponding valence:

6.1. the row with the number of revolving gears gets the multiplier 1 in the speed column and 2 in the torque column

6.2. the row of each node gets the multiplier \((N_{connections} - 1)\) in the speed column and 1 in the torque column

Sum the numbers in each row and compare them with the number of numbered shafts. The difference results in the specifications still missing for the calculation, i.e. their degree of freedom (in the left column the static one, in the right column the kinematic one).

table:2.4

table to determine the degree of freedom of a gearbox from the gearbox inner diagrams according to Wolf

Two epicyclic gearboxes connected to a shaft are to serve as a first example. On the left the corresponding Wolf symbolism is shown and next to it one of the possible constructions for this kind of coupling (see fig. figure 2.17)

fig:2.17

four-shaft transmission from the simple combination of two elementary planetary gears, on the left wolf symbolism, on the right a possible construction

All the shafts of the wolf symbolism are numbered. This results in six shafts. Then the table is filled with the number of the respective elements. Here there are two gear symbols and one node with two connections. Subsequently the results of each line are calculated and the sum for the speeds and the torques are determined. The difference between the number of shafts and the sum results in the respective degree of freedom. This gearbox therefore has the static degree of freedom of \(6 - 5 = 1\) and the kinematic degree of freedom of \(6 - 3 = 3\) table 2.5.

table:2.5

degree of freedom of a gearbox consisting of two elementary gears with a coupling

fig:2.18

Four-shaft transmission from the double combination of two elementary planetary gears, on the left wolf symbolism, on the right a possible construction

table:2.6

degree-of-freedom of a gearbox consisting of two elementary gears with two couplings

These considerations are particularly interesting for the production gears or when connecting several motors to one gearbox, e.g. in hybrid concepts. In this way, it is possible to determine whether a connected electric motor is actually working as a motor or generator in the desired sense.

2.4 installation criteria

The installation criteria of recirculating gearboxes are:

conditions for the choice of diameters

conditions for the choice of number of teeth and meshing pitch of sun, ring gears and planets

conditions for diameters: When determining the number of teeth, one should consider the interrelationships between the gears meshing with each other. In the case of a minus gearbox, the sun and planets must fit into the ring gear. Consequently, the geometrical condition is:

eq:2.34

\[ d_{ring gear} = d_{sun} + 2 \cdot d_{planet} \]

If only a hollow wheel with revolving planetary wheels is used without a central sun, the geometrical relationship should be observed that the pitch circles of all planets can be inscribed in the pitch circle of the ring gear without overlapping each other. With three planets this results in an equilateral triangle on which the axes of the planets revolve. The maximum size of a is according to equation 2.37 limited by the diameter of the ring gear (for a planetary radius of 0, see (image 2.19)).

fig:2.19

Relationship between an equilateral triangle (edge length a) and its circumcircle (radius r)

The minimum length of a is limited to the size of the planetary diameter at which all planets within the ring gear just touch each other (fig. figure 2.20). Between the side length a of an equilateral triangle and its inscribing circle results in general:

eq:2.35

\[ \frac{\frac{a}{2}}{r} = cos(30°)\]

eq:2.36

\[ \frac{a}{2} = cos(30°) \cdot r = \frac{1}{2} \sqrt{3} \cdot r \]

eq:2.37

\[ a = \sqrt{3} \cdot r \]

As maximum Quantity for the edge length a of the three-sided planet carrier thus follows:

eq:2.38

\[ a = \sqrt{3} \cdot r_{ring gear}\]

For the minimum size of the edge length a of the planet carrier, is obtained according to figure 2.20:

eq:2.39

\[ d_{web} + d_{planet} = d_{ring gear} \]

eq:2.40

\[ a = d_{planet} \] from equation 2.37 it follows with the help of equation 2.39, resolved by the ridge diameter:

eq:2.41

\[ a = \sqrt{3} r_{web} = \frac{\sqrt{3}}{2} \cdot (d_{ring gear} - d_{planet}) \]

Substituting equation 2.40 into 2.41 and thereby eliminating the diameter of the planet, we get for the minimum edge length a:

eq:2.42

\[ a = (2\sqrt{3} - 3) \cdot d_{ring gear} = 0.46410 \cdot d_{ring gear} \] If you want to determine the diameter ratio between the smallest possible sun and the ring gear, you need the ring gear, one needs the relation in equation 2.43:

eq:2.43

\[ d_{sun} + 2 \cdot d_{planet} = d_{ring gear} \]

fig:2.20

Relationship between web diameter, planet diameter, ring gear diameter and distance of the planet centres at maximum planet size

eq:2.44

\[ \frac{d_{so}}{d_{H}} = 1 - \frac{\sqrt{3}}{2 \cdot (2 + \sqrt{3})} = 0.0718 \] In practice, when determining this minimum, usually the minimum height of \(2 \cdot Modul\) is added to the inscribed radii, because otherwise the teeth of the planets would interpenetrate. The same also applies in use of only one eccentrically rotating planet, as is found in gears with particularly high transmission ratios. (MUE, 11) therefore determines a maximum ratio of about \(11.3^{-1}\) a minus gearbox of this kind.

With five planets this figure forms an equilateral pentagon, with n an n-corner etc… The derivation for all cases with the number of planets > 3 follows the same principle, it is only necessary to use instead of \(cos(30°)\) the expression \(cos(90° - \frac{180°}{n_planets})\) in formula equation 2.37.

condition between number of teeth and number of planets: One of the advantages of planetary gears is power sharing. Therefore it is important to have a web with symmetrically distributed planets over the circumference, as this avoids bending torques on the sun or ring gears. However, this is only guaranteed if when the sun is fixed, through a rotation of the web each planet is moved from its original starting position to exactly the same position into exactly the same engagement position as its predecessor at this point. With, for example, three planets evenly distributed over the circumference this would be the case after a rotation of the web by \(\frac{360°}{3} = 120°\). This is formulated in the following equation:

eq:2.45

\[ \phi_{web} = 2 \cdot \frac{\pi}{n_{planets}} \] For this to be possible, the distance travelled by the planet at the ring gear (wheel 2) and the sun (wheel 1) must be an integer multiple \(Z_{1}\) or \(Z_{2}\) of the division p. Mathematically formulated, this means:

eq:2.46

\[ r_{2} \cdot \phi_{web} = Z_{2} \cdot p\]

eq:2.47

\[ r_{1} \cdot \phi_{web} = Z_{1} \cdot p\]

From gear geometry we know the two relations between module m, pitch p and diameter d:

eq:2.48

\[ p = m \cdot \pi \]

eq:2.49

\[ d = m \cdot z\] If one now adds both equations 2.48 and 2.49, we get \[(r_{1} + r_{2}) \cdot \phi_{web} = (Z_{1} + Z_{2}) \cdot p = Z \cdot p\] Adding two integers \(Z_{1}\) and \(Z_{2}\) again produces an integer Z. With \(\phi\) replaced by the expression of equation 2.45 and p in equation 2.48, this gives: \[(r_{1} + r_{2}) \cdot \frac{2\pi}{n_{planets}} = Z \cdot m \cdot \pi \] Finally, truncating this expression with \(\pi\) and eliminating the radii r by the numbers of teeth z with equation 2.49 gives:

eq:2.50

\[ \frac{z_{2} + z_{1}}{n_{planets}} = Z\] In words, the sum of the numbers of teeth of the two participating gears of a minus gearbox should always be divisible by the number of planetary gears used.

In the case of a simple positive gear consisting of a step planet whose first planet \(p_{1}\) with the left sun \(z_{1}\) and whose second planet \(p_{2}\) with the right sun \(z_{2}\) (see image 2.5), the same approach yields:

eq:2.51

\[ \frac{(z_{p1} \cdot z_{2}) - (z_{1} \cdot z_{p2})}{q \cdot n_{planets}} = Z\] Where q is the greatest common divisor of \(z_{p1}\) and \(z_{p2}\).

3 Previous method of calculating planetary gears

Wolf’s gearbox schematics and the abstract methods derived from them are mainly used for the design of gearboxes in development, i.e. before the geometric parameters such as the number of teeth, tooth width, module, profile shift or even the type of gearing such as bevel or spur gearing. With this technique it is possible to determine the key data of a gearbox such as spread, degrees of freedom, efficiency, and transmission ratio of each gear of one or a combination of epicyclic gears.

The same procedure serves as a basis for even more complex arrangements, as it was developed for the design of a simple epicyclic gearbox. In numerous books and publications, more and more special cases on this subject (e.g. calculation methods for reduced coupling sets or meshing of two elementary gearboxes (SCHTZ1, 12), (SCHTZ2, 13)) and their results were finally laid down in several VDI guidelines. The terms and basic equations are described in (VDI2157, 14). In addition, there’s a 10-point procedure presented, which has turned out to be favourable for the calculation of planetary gears, as it has already been used in a similar form in numerous books, but often in a different order and without naming the individual steps explicitly as such. The applicability of this procedure was described by Prof. Looman in general in (VDI672, 15) and especially for reduced planetary gears in (VDI1460, 16). The recommended operations are:

gear scheme, designations

Kutzbach’s speed diagram

Sign of n, T and P

splitting into subgearboxes

Ratios of stationary gearboxes

Basic speed equations -> Transmission ratio of the gearbox

Torques

Power

Efficiency

Installation criteria

fig:3.1

Wolfrom set as an example for a reduced planetary gearbox

In the following, using an example (a Wolfrom set, see picture 3.1, taken from (VDI1460, 17) these 10 working steps will be explained in the following chapter in order to make the differences to the new proposed method clearer.

3.1 gear diagram, designations

First, the gearbox is drawn in section in its wheel and shaft geometry. All wheels are numbered in ascending order from 1. Afterwards the gear scheme according to Wolf is derived. For this purpose, each planetary gear, characterised by its web, which is always present exactly once, and its planetary group mounted on it, is expressed by a circle. All shafts with the central wheels meshing in this planet or planetary group (hollow or sun wheels) are drawn on this circle by a line. If it is a gearbox with four or more shafts, it can be made up of a combination of its elementary planetary gears with exactly three connections. An epicyclic gearbox with four shafts, i.e. three central shafts and one web shaft contains, for example, three possible elementary gears: 1–2-s, 1–3-s, 2–3-s (picture 3.2). The web is common to all elementary drives. The other two shafts are composed of two of the remaining central wheels. An epicyclic gear consisting of five connecting shafts could therefore be composed of \(\binom{4}{2}\) possible elementary gears (1–2-s, 1–3-s, 1–4-s, 2–3-s, 2–4-s and 3–4-s), which would have to be put together to form an equivalent image with the same effect.

fig:3.2

all elementary gearboxes of a four-shaft gearbox

fig:3.3

all possible combinations of the elementary gearboxes to the equivalent model for the actually real gearbox

3.2 Kutzbach’s speed diagram

sec:Kutzbachplan In order to determine all speeds of the gearbox, one makes use of the speed plan according to Kutzbach, which is based on the similarity relation between the numbers of teeth / diameters of the gears involved and their relative movements (see equation 2.3). It is important that first the known outer speeds (kinematic degree of freedom) must be entered with the correct sign as the remaining speeds of all inner components, e.g. of the planetary components such as the planets can be derived from these.

fig:3.4

coordinate system and construction line for the rotational speed diagram according to Kutzbach

For the drawing of the movement plan itself one starts with the coordinate system, which must be at the height of the central axis. Then draw a straight line parallel to the x-axis (see picture 3.4) through each tooth mesh and the web axis of the gear. On these the absolute speeds of the respective components can be tapped later. For the most accurate comparison of all angular velocities, a straight line is chosen which is as far away as possible from the central axis; it is advisable to choose the one that goes through the wheel with the largest diameter. Now one can apply all known speeds (here \(n_{A}, n_{C}\)) in relation to this diameter on this straight line. Through the connecting line, which goes through the end of the speed vector (e.g. \(n_A\)) with the origin (see e.g. the straight line from A to 0, picture 3.5), the angular velocity of this shaft is fixed. Now the intersections of the lines through the larger planetary diameter with the connecting straight lines C0 and A0 span the velocity field of the planet. At the level of the parallel through the web, the straight line describing the angular velocity of the web 0S is drawn (see picture 3.5, middle). Proceed accordingly with the remaining interventions. In the Kutzbach plan the superimposition of the individual velocities of all shafts involved is shown. Now the absolute values of all speeds can be read off (\(r \cdot \omega\) at the associated radius), as well as a comparison of all angular velocities among each other (length of the vectors at the outermost radius) can be made.

fig:3.5

development of the speed plan

3.3 signs of n, T and P

For the determination of the signs of the rotational speeds one uses the direction vectors of the created Kutzbach plan. Particularly convenient is the box triple used by Loomann (see picture 3.6), into which the signs of speed, torque and power are entered. In the upper line he records these values absolutely, while in the second line he records relative speed, torque and power. The absolute speed of the three central shafts can simply be read from the direction of the speed arrow. When creating the Kutzbach plan, the following was already defined right as positive and left as negative. The direction of the relative speeds can be read off from the length and direction of the diameter. The sign of the applied torques can generally be gathered from the knowledge of the power at the input and output shafts or the type of the gearbox (negative/positive gear, summation shaft).

fig:3.6

sign of speed, torque and power at the connecting shafts, upper line absolute and lower line relative

3.4 division into sub-gears

The speeds created in the Kutzbach plan are now important for the signs of the pitch powers, because from these the really involved elementary gears are identified. As is well known, pitch powers can only occur between two central shafts (see also picture 2.6). By looking at the sign of the pitch power (lower right corner of the sign boxes), one can see that the pitch power flows into A and B, but only comes out in C. Accordingly, there are only power flows from A to C and B to C, i.e. the partial gear from B to A, or more precisely: B-5–4–2–1-A does not participate in the power flow. Accordingly, for the selection of the equivalent model for the gearbox, only the combination of A-C-s with B-C-s can be considered. In picture 3.7 the equal-acting planetary coupling gear ‘pulled apart’ is shown.

fig:3.7

effective replacement model of the four-shaft gearbox

3.5 parking gear ratios

The determination of the static gear ratios of both individual gearboxes is now carried out in the same way as for a normal epicyclic gearbox. You hold the respective web in your mind and move from one central shaft to the other. Meanwhile, multiply the reciprocal negative ratio of the number of teeth of the gears involved in the direction of movement. A result for the static transmission ratio of each elementary gear are the following equations:

eq:3.1

\[i_{13'} =- \frac{z_2}{z_1} - \frac{z_3}{z_2} = \frac{z_3}{z_1}\]

eq:3.2

\[i_{53"} = -\frac{z_4}{z_5} - \frac{z_3}{z_2}= \frac{z_4 \cdot z_3}{z_5 \cdot z_2}\]

3.6 basic speed equations, gear ratio

The speed follows immediately from Willis' equation via the final drive ratio. Basic equations for both gearbox sections:

eq:3.3

\[i_{13'} = \frac{n_1 - n_{s'}}{n_{3'} - n_{s'}} \]

eq:3.4

\[i_{53"}= \frac{n_5 - n_{s"}}{n_{3"} - n_{s"}} \] From the combination of both elementary gears to the equivalent image with the same effect, resolved in each case according to the outer connecting shafts, result in the following relations:

eq:3.5

\[n_A = n_1\]

eq:3.6

\[n_B = n_5\]

eq:3.7

\[n_C = n_{3'} = n_{3"} = 0\]

eq:3.8

\[n_{s'} = n_{s"}= n_s\] The equations 3.3 and 3.4, transformed with 3.5, 3.6 and 3.7, resolved according to the speeds of the connecting shafts:

eq:3.9

\[n_A =i_{13'}(n_{3'} - n_s) + n_s= n_s (1-i_{13'})\]

eq:3.10

\[n_B =i_{53"}(n_{3"} - n_s) + n_s= n_s (1-i_{53"})\] From equations 3.9 and 3.10 it follows for the total transmission ratio of the gearbox:

eq:3.11

\[i_AB = n_An_B = 1-i_{13'}1-i_{53"} = 1-z_3z_11-z_4 z_3z_5 z_2\]

Since gear 3 and 5 are ring gears, negative numbers of teeth must be inserted into the equation for \(z_3\) and \(z_5\). If one assumes that the remaining numbers of teeth will be positive, this results in a positive numerator in any case. If the product of the numbers of teeth in the denominator is less than 1, the result is a plus gear, otherwise a minus gear.

3.7 Torques

For the determination of the torques for \(\eta = 1\) one follows the rules \(\sum M = 0\) at each shaft and \(i_{12} = -\frac{T2}{T1}\) at each planetary gear. Starting from drive A (torque and speed are positive -> drive power), the result is a negative torque for the sun. With a minus gearbox, the sum shaft is always the web, i.e. S' must oppose the torque of the two central shafts 1 and 3', i.e. positive. For the right-hand gearbox, the signs are obtained in the same way, with the difference that one starts from the output B. The equations for the torques on the connecting shafts are obtained by determining the transmission ratio from A to B by transforming the static transmission ratios of both elementary gearboxes:

fig:3.8

torques at the Wolfrom set pulled apart

By deriving equation 2.8 it was shown how any other gear ratio of a simple planetary gearbox can be derived (table according to Müller see appendix [C][TabellenMueller]). \[i_{1s} = 1 - i_{12}\] \[i_{s1} = \frac{1}{1 - i_{12}}\] In terms of the two standing ratios \(i_{1s'}\) and \(i_{s"5}\), this means: \[i_{1s'} = 1 - i_{13'}\] \[i_{s"5} = \frac{1}{1 - i_{53"}}\] If we now determine the torque ratio from shaft A to B, we get for the torque ratio \(\mu = \frac{T_B}{T_A} = -i_{AB} \cdot \eta_{AB}\) of the total system: \[\mu = \frac{T_B}{T_A} = - i_{1s'} \cdot i_{s"5} \eta_{AB} = - (1 - i_{13'}) \cdot \frac{1}{1 - i_{53"}} \cdot \eta_{AB}\] for \(\eta_{AB} = 1\) it follows that: \[\mu = \frac{T_B}{T_A} = - \frac{1 - i_{13'}}{1 - i_{53"}}\]

3.8 power

Since the direction of the pitch power is decisive for the choice of the correct equation for the efficiency (see explanation in picture 2.9), its course must first be determined for the pulled-apart replacement gearbox. After transferring the speeds from the speed diagram, one can determine the absolute and pitch powers and enter the actual power flow of the pitch powers (see grey arrows in image 3.9). The existence of reactive power (if the configuration has one) can be verified this way too. By the direction of the arrows one can see that power flows appear in the left gearbox only from 1 to 3' and in the right gearbox only from 5 to 3".

fig:3.9

speeds, torques and power at the pulled apart Wolfrom set; the flow of the pitch power are shown in the arrows

3.9 efficiency

Now that we know the direction of the pitch power (from sun to ring gear), we can write the correct efficiency equations for each gearbox section as a function of the final drive ratio for each subgear:

eq:3.13

\[i_{13'} \cdot \eta_{13'} = -\frac{T_{3'}}{T_1}\]

eq:3.14

\[T_1 + T_{3'} + T_{S'} = 0\]

eq:3.15

\[i_{53"} \cdot \eta_{53"} = -\frac{T_{3"}}{T_5}\]

eq:3.16

\[T_{3"} + T_5 + T_{S"} = 0\] Create the links of the two nodes: \[T_3 + T_{3'} + T_C = 0\] \[T_{S'} + T_{S"} = 0\] Accordingly, the lossy torque ratio is:

eq:3.17

\[\mu = \frac{T_B}{T_A} = -\frac{1 - i_{13'} \cdot \eta_{13'}}{1 - i_{53"} \cdot \eta_{53"}}\]

3.10 installation criteria

The installation criteria are those already mentioned in the previous chapter for the selection of the diameters

eq:3.18

\[d_1 + 2 \cdot d_2 = d_3\] with three planets apply as limits for their diameters:

eq:3.19

\[d_2 < (2\sqrt{3} - 3) \cdot d_3\]

eq:3.20

\[d_4 < (2\sqrt{3} - 3) \cdot d_5\] and the relationship of the integer Z between numbers of teeth and numbers of planets:

eq:3.21

\[\frac{z_3 + z_1}{n_{planets}} = Z\]

eq:3.22

\[\frac{(z_2 \cdot z_5) - (z_3 \cdot z_4)}{q \cdot n_{planets}} = Z\] Where q is the greatest common divisor of \(z_2\) and \(z_4\). The specific values to satisfy these relationships are usually determined in an iterative process by the designer. It is common to specify a minimum number of teeth that can be implemented (e.g. 13) and a maximum diameter ratio for a stage as a starting point.

4 Weaknesses and limitations of existing calculation methods

Even if the method explained in the previous chapter can in principle be used to calculate any type of epicyclic gears, whose combinations or reduced coupling set can be calculated, consequently also complex gearboxes, which are composed of the basic types of epicyclic gearboxes, these methods nevertheless show some weaknesses in practice. These weaknesses will be addressed in the following sections.

4.1 Complexity and special knowledge

In general, for the calculation of the key data of recirculating gearboxes, a certain understanding of their basic interrelationships is necessary. Without the knowledge of the determination of the stance ratio as a central variable or the special feature of the larger number of degrees of freedom of this type of gearbox, a productive implementation or occupation with this topic is certainly not possible. Thus, a transmission ratio and thus the power flow in a gearbox with three shafts is no longer a fixed variable, but rather a state that arises as a result of the applied speed ratios. These can be understood and applied to a simple epicyclic gearbox within a relatively short period of time, so that at least a statement about the transmission (speed ratio) and torques of an arrangement with an assumed efficiency of 100% can be determined. This knowledge is not referred to when speaking of complexity and the need for special knowledge in the following.

However, it is undisputed that the design of one or more concepts, e.g. of a larger automatic transmission, turns out to be very time-consuming. The designer not only has to contend with the combinatorial explosion that results from the increasing number of all networking possibilities of the elementary transmissions involved. As an example in picture 4.1 the combinatorial possibilities of two planetary gearboxes are shown.

fig:4.1

All 9 possible combinations of two orbital gears. A complete naming of the central shafts of the elementary gears and naming of the main shafts with I II and S is only done in the middle of the diagrams for a better overview

The creation of the gear scheme according to Wolf

The Kutzbach plan

Calculation of the efficiency

The acquisition of these methods and their safe handling requires time and interest in this area. The 10 steps procedure is not a simple scheme that can be carried out without thought and leads safely to the right goal. This kind of complexity and the need for specialised knowledge is to be seen as a weakness, as it means that only a smaller circle of people can deal with it in more detail.

4.1.1 Creation of the gear scheme according to Wolf

The procedure for the creation of the gear schematic according to Wolf has already been described in chapter 2.2. It is the first step of the procedure. If the gearbox is a simple one with two central shafts and a bar, the equivalent image is very easy to determine (minus gearbox with a web as the sum shaft, plus gearbox with one of the two central shafts as the sum shaft). If you want to see the diagram of a 5-, or even 7- or 9-gear automatic gearbox, the equivalent pattern for each gear must be constructed. When using special cases of planetary gearboxes as components, e.g. reduced coupling sets with more than two central shafts, a solution is more difficult. Only through further literature (see the example used in chapter 3) or basic considerations, one can integrate these special cases.

fig:4.2

problem of creating the wolf symbolism of a five-shaft gear

4.1.2 The Kutzbach Plan

The Kutzbach plan is the second step in the 10 steps procedure. The results of the Kutzbach plan are the basis of all following calculations. The signs of speed and torque result in input or output power. For the determination of the rolling powers and relative speeds an arrow with the correct sign is also necessary. The preparation of the Kutzbach diagram is very error-prone, especially with larger gear chains. Non-parallel axles, e.g. in the case of a differential, must also be taken into consideration separately. A further weakness of the graphical method of the Kutzbach plan is the fact that in the event of a reversal of the direction of rotation or a major change of one of the shafts involved, the sign of the rolling power can reverse, so that the gearbox can then suddenly change to a different operating state, e.g. from power sharing to power summation. Therefore, if statements are to be made for a variation of the speeds involved, the designer has to go through all possible speed variants that may occur. Setting up an analytical equation makes such a variation easier, but this only exists in the literature for known combinations. All others have to be derived by theirselves (e.g. from the relations of the ray theorem).

fig:4.3

All possible speed variants of a ring gear-planet pairing in a Kutzbach diagram

4.1.3 Calculation of efficiency

In the 10 steps procedure, the fourth point is the ‘decomposition of larger gearboxes into sub-gearboxes’. This is one of the most important steps for the determination of the efficiency. However, only the partial gear which also carries rolling power may be used. Only when the directions of the power flow is known, the equations for the efficiency can be established. If the wrong partial gears are selected, the efficiency can suddenly result in efficiencies greater than 100 or less than 0. If now, for example, a reduced coupling set is to be represented directly in the symbolism according to Wolf, without having to go through the previous steps of the Kutzbach plan, the determination of the signs, etc. (which is possible on the basis of the form of the symbols), this in turn presupposes a kind of special knowledge. Of the possible variants, the correct combination without reactive power must be used.

fig:4.4

construction and associated wolf symbolism of the variants of the replacement image for a reduced coupling gear

Another problem in the determination of the efficiency according to the methods described in chap. 2.1.5 is that one assumes a fixed standing efficiency, independent of the oil used and the operating temperature or variable bearing resistances over the speed. The stall efficiency of an epicyclic gearbox is obtained by transmitting the total power applied as rolling power from one central gear to the other. It is not specified under what load and at what speed this occurs. This static efficiency is subsequently assumed to be constant for each operating condition. By means of the equations it is merely calculate what proportion of the power is transmitted as clutch power and thus without loss, in order for a ‘circulating efficiency’ for the operating to be achived. However, this does not correspond to reality, where the load-independent losses normally follow a polynomial over the speed.

4.2 Transmissions with four or more rotating shafts

The literature on planetary gearboxes almost always assumes gear combinations or gear arrangements which, in the operating state, have an overall total degree of freedom of two or at most three. Mostly, in the case of two kinematic degrees of freedom and one static degree of freedom are used. In the case of automatic transmissions with 5, 6 or more gears, there are normally always exactly as many clutches or brakes applied that result in exactly one kinematic and one static degree of freedom for each gear. Even with four or more connecting shafts, there is always one free rotating and one connected coupling shaft is defined between the two planetary gearboxes. In this 24 possible transmission ratios are created in this way for a four-shaft gearbox, see (SCHTZ2, 18). But how should one proceed if all four shafts are connected simultaneously to an input and output, i.e. transmitting power to the outside? Especially for marine transmissions and hybrid concepts with at least two engines, rear axles with one / two drives and two rear wheels as downforce or else four-wheel drive concepts, transmissions with a higher degree of freedom are required. Despite this, calculation methods for speeds, torques and efficiency of these combinations are very seldom dealt with in the literature. Such concepts can certainly be solved by breaking them down into several sub-gearboxes and then solving them by conventional methods. However, this entails the problems mentioned in the previous section: determination of the correct substitute model according to Wolf, of the Kutzbach plan and above all the correct efficiency, which requires a known power flow.

4.3 Self-penetration

The creation of the equivalent images according to Wolf form the great advantage of being able to quickly derive basic statements about a gear concept. However, unfortunately, it is also quite possible to design gears with it, which can never be realised in reality, because some components would have to penetrate themselves. This is the case, for example, when as in picture 4.5, one central wheel is connected to the web of the opposite wheel and at the same time an inner central wheel with an outer central wheel. A shaft connection between the right and left sun of the construction is not possible because of the necessary connection of the brake plates to the housing. However, an alternative design can be defined and represented mathematically.

fig:4.5

gear with self-penetration: Wolf symbolism and construction

4.4 Computability of existing methods

Every manual calculation procedure is based on rules, similar to a recipe, the application of which leads to the desired result. If this algorithm can be executed and solved by a human being, it is also transferable to a computer. It is therefore possible to have the procedures presented so far solved by a programme. However, the effort required for this is sometimes not insignificant. If one now creates a programme on the basis of the wolf symbolism for the determination of all torques and the efficiency of an epicyclic gearbox, this is still easily solvable for an elementary gearbox. However, if one has to consider the equations of all possible special cases, all the coupling possibilities between two or more elementary gearboxes, all exceptions or rules for the calculation, this will be a combinatorial explosion and quickly become unfeasible. In any case, it is better to use a generally valid procedure in order to avoid many exceptions or special cases, which would otherwise increase its probability of error.

5 Explanation of an alternative calculation methodology with examples

In chapter 3 an example of a calculation method using a reduced coupling theorem was shown. Certainly there are other possible solutions for certain problems, e.g. for the determination of the torques an equation system from equilibria of forces, as is common in classical mechanics. In this case, however, all angular relations between the wheels for the vectors of the forces in space, which would normally also require the inclusion of the gearing geometry used (spur gear, bevel gear, hypoid gearing). The following procedure is a compromise between the prediction with purely abstract models from symbols according to Wolf and the design of the detailed gear geometry, which is only required towards the end of the design process for the determination of the service life and load-bearing capacity. In addition, it can be used for the preliminary design and concept creation as well as for the control of more detailed methods in the computer. In the following section, the procedure is first applied to a few simple and one more complex example before it is formulated in general terms. The following section explains how the losses obtained from empirical equations or experiments are integrated into the lossless model.

5.1 Examples

5.1.1 Switchable spur stage

Even if the calculation of speeds and torques at simple spur gear stages seems trivial, some basic relationships can be seen very well. The following example in picture 5.1 was chosen because of existing loose gearwheels. It consists of two shafts with one splined shaft and two gearwheels, one of which is designed as a clutchable wheel. In the following, the gearbox is to be used in the first gear, i.e. with the clutch \(K_1\) closed.

fig:5.1

symbolic sectional drawing of a shiftable spur gearbox, with engaged first stage

First of all, all elements of the gearbox that can apply a torque to the shaft are numbered consecutively. Then the equations (speeds and torques) of all stage relationships and then for all elements which sit together on one shaft are set up. Finally, the still missing equations must be supplemented by specifications (according to the existing loose gearwheels and the degree of freedom of the entire wheel arrangement). As a basis for the relationship of the speed ratios of a spur gear stage between wheel 2 and wheel 4 with fixed axles in space, serves the equation 2.1. For the relationship of the torques between wheel 2 and wheel 4 one uses equation 2.22 at \(\eta = 1\). All step relations of this gear can be described by the following four equations:

eq:5.1

\[\frac{n_2}{n_4} = -\frac{z_4}{z_2}\]

eq:5.2

\[\frac{n_3}{n_5} = -\frac{z_5}{z_3}\]

eq:5.3

\[\frac{M_4}{M_2} = \frac{z_4}{z_2}\]

eq:5.4

\[\frac{M_5}{M_3} = \frac{z_5}{z_3}\] Here the equations 5.1 and 5.3 represent the relation for the first, the equations 5.2 and 5.4 represent the second stage. Furthermore, all elements which are together on a shaft are equal in speed and \(\sum M = 0\). From this follows:

eq:5.5

\[n_1 = n_2\]

eq:5.6

\[n_2 = n_3\]

eq:5.7

\[n_4 = n_6\]

eq:5.8

\[M_1 + M_2 + M_3 = 0\]

eq:5.9

\[M_4 + M_6 = 0\] For the upper shaft, the two equations 5.5 and 5.6 describe the speed of the shaft, equation 5.8 for the torques applied to it. The lower shaft is described by the speed equation 5.7 and the torque equilibrium by equation 5.9. Now there are five equations for the rotational speeds and four for the torques of the gearbox. However, since six unknowns could be identified in the numbering of all the elements, one more quantity is needed on the left-hand side of the speeds, but on the right-hand side of the torques two are needed. To calculate a gearbox, at least one input or output power must always be specified, as otherwise no torque and speed conversion can take place. Therefore, at least one torque and one speed not equal to 0 must be specified. The apparently still missing equation on the side of the torques results from the fact that wheel 5 has not been engaged in the currently displayed state, i.e. no power flows from wheel 3 to wheel 5. Consequently, the torque applied to wheel 5 is 0, it is a loose wheel. This results in:

eq:5.10

\[n_x = X\]

eq:5.11

\[M_5 = 0\]

eq:5.12

\[M_y = Y\] The equations of the defaults (equations 5.10 and 5.12), can be given at any of the elements. Equation 5.11 takes into account the power branch of the transmission of the gearbox, as specified by the shift position. The equations listed can also be used as in table 5.1 for the speeds and in table 5.2 for the torques, in a matrix in order to solve them more easily using the Gauss-Jordan method:

table:5.1

equation matrix for the speeds of a shiftable spur gearbox in first gear

Since the defaults can be specified at any element, in this example the star (\(\star\)) has been written into the matrix instead of the number 1. This should make clear that the first column does not necessarily have to be chosen as the default, but rather the element for which the data are available. This gear needs a speed and torque specification, has a kinematic degree of freedom (kin. FG) of 1, a static degree of degree of freedom (stat. FG) of 1 and a total degree of freedom of 2. The torque of the loose wheel results from the current shifting situation and is therefore rather a boundary condition of the switching logic. It is not explicitly implanted into the gearbox from the outside.

table:5.2

Equation matrix for the torques of a shiftable spur gearbox in first gear

5.1.2 Minus gearboxes

The elementary minus gear in image 5.2 was used for the demonstration of the calculation method on an epicyclic gearbox, because it contains two stages on one element: The pairing of planet to sun and planet to ring gear.

fig:5.2

symbolic illustration of a minus gear

fig:5.3

minus gear in which each wheel has exactly one engagement partner

eq:5.13

\[\frac{n_3}{n_2 - n_6} = -\frac{z_2}{z_3}\]

eq:5.14

\[\frac{n_3}{n_4 - n_6} = -\frac{z_4}{z_3}\]

eq:5.15

\[\frac{M_3}{M_2} = \frac{z_3}{z_2}\]

eq:5.16

\[\frac{M_4}{M_3} = \frac{z_4}{z_3}\]

For all components on the shafts of the minus gearbox, again applies speed equality and \(\sum M = 0\):

eq:5.17

\[n_1 = n_2\]

eq:5.18

\[n_3 = n_8\]

eq:5.19

\[n_4 = n_5\]

eq:5.20

\[n_6 = n_7\]

eq:5.21

\[M_1 + M_2 = 0\]

eq:5.22

\[M_3 + M_8 = 0\]

eq:5.23

\[M_4 + M_5 = 0\]

eq:5.24

\[M_6 + M_7 = 0\]

For the solar shaft the first line of the block is valid, i.e. equation 5.17 and 5.21, for the planetary shaft the second line, equation 5.18 and 5.22, finally, for the ring gear shaft, equation 5.19 and 5.23. The ridge shaft is represented by the last line, equation 5.20 and 5.24. The only missing equation on the side of the torques is a specification of the web torque. All torques on an epicyclic gear are determined by the step relations and number of teeth. The web torque is obtained from the fact that all torques on an epicyclic gearbox must add up to 0 (equation 5.27):

eq:5.25

\[n_x = X\]

eq:5.26

\[n_y = Y\]

eq:5.27

\[M_2 + M_4 + M_6 = 0\]

eq:5.28

\[M_x = X\]

The remaining equations are obtained by filling the system of equations with the necessary specifications. Since there are eight unknowns, there must also be eight equations. This results in a kinematic degree of freedom of 2 (equation 5.25 and 5.26) and a static degree of freedom of 1 (equation 5.28). The equations can now be solved very easily by hand using substitution or addition methods by hand. In table 5.3 and table 5.4 these are once again summarised in a matrix.

table:5.3

Equation matrix for the speeds of a simple negative gearbox

table:5.4

equation matrix for the torques of a simple minus gear

5.1.3 Five-shaft transmission: combination of Ravigneaux and Wolfrom gearbox

A Ravigneaux wolfrom set with five connecting shafts (figure 5.4) shall be the last example to show that the presented approach can also be applied to more complex gearboxes and that a generally applicable procedure can be derived. Furthermore, one also recognises from this that the degree of freedom of a system - which here cannot necessarily be determinable at first glance - arises as a result of the procedure. An equivalent circuit diagram according to Wolf no longer needs to be generated. In this way, the difficulty illustrated in fig. 4.2 is eliminated.

fig:5.4

Ravigneaux-Wolfrome set with five connecting shafts (I-V)

fig:5.5

dragged-apart Ravigneaux wolfrome set

eq:5.29

\[\frac{n_3}{n_2 - n_{15}} = -\frac{z_2}{z_3}\]

eq:5.30

\[\frac{n_6}{n_9 - n_{15}} = -\frac{z_9}{z_6}\]

eq:5.31

\[\frac{n_7}{n_{11} - n_{15}} = -\frac{z_{11}}{z_7}\]

eq:5.32

\[\frac{n_8}{n_{13} - n_{15}} = -\frac{z_{13}}{z_8}\]

eq:5.33

\[\frac{n_4}{n_5} = -\frac{z_5}{z_4}\]

eq:5.34

\[\frac{M_2}{M_3} = \frac{z_2}{z_3}\]

eq:5.35

\[\frac{M_9}{M_6} = \frac{z_9}{z_6}\]

eq:5.36

\[\frac{M_{11}}{M_7} = \frac{z_{11}}{z_7}\]

eq:5.37

\[\frac{M_{13}}{M_8} = \frac{z_{13}}{z_8}\]

eq:5.38

\[\frac{M_5}{M_4} = \frac{z_5}{z_4}\]

Two intermeshing planetary gears are considered as two spur gear stages fixed in space (equation 5.33). Since both have to revolve on the same ridge, no superimposed speed can occur between them. The following relations:

eq:5.39

\[n_1 = n_2\]

eq:5.40

\[n_3 = n_4\]

eq:5.41

\[n_5 = n_6\]

eq:5.42

\[n_6 = n_7\]

eq:5.43

\[n_7 = n_8\]

eq:5.44

\[n_9 = n_{10}\]

eq:5.45

\[n_{11} = n_{12}\]

eq:5.46

\[n_{13} = n_{14}\]

eq:5.47

\[n_{15} = n_{16}\]

eq:5.48

\[M_1 + M_2 = 0\]

eq:5.49

\[M_3 + M_4 = 0\]

eq:5.50

\[M_5 + M_6 + M_7 + M_8 = 0\]

eq:5.51

\[M_9 + M_{10} = 0\]

eq:5.52

\[M_{11} + M_{12} = 0\]

eq:5.53

\[M_{13} + M_{14} = 0\]

eq:5.54

\[M_{15} + M_{16} = 0\] The torque at the web (\(M_{15}\)) is obtained from the sum of the torques at all the wheels that mesh with a planet:

eq:5.55

\[M_{15} + M_2 + M_9 + M_{11} + M_{13} = 0\] There are now 14 speed equations and 13 torque equations, i.e. for the solution of the 16 unknowns in each case, 2 speed specifications ( equation 5.56 and 5.57) and 3 torque defaults (equation 5.58, 5.59, 5.60). For the presented construction this means that the torques of two shafts of the five-shaft gearbox are always determined, as soon as a torque is applied to three shafts. The kinematic degree of freedom does not differ from a simple elementary planetary gear, which one would not have suspected at first glance.

eq:5.56

\[n_x = X\]

eq:5.57

\[n_y = Y\]

eq:5.58

\[M_x = X\]

eq:5.59

\[M_y = Y\]

eq:5.60

\[M_z = Z\]

The equation matrices are not shown, as they do not differ in structure from those of the gearboxes.

5.2 General formulation of the determination of speeds and torques

The three examples presented served as a basis for a general procedure for the calculation of the speeds and torques, and thus the power flow, the transmission ratio and the degree of freedom of any arbitrary wheel arrangements. The standardised procedure for the calculation of multi-shaft gears is divided into three sections:

preparation of the geometric specifications

1.1 Disassembling the gear: dividing gears that have more than one meshing partner until each gear is used in exactly one stage

1.2 Determining the number of existing unknowns: Numbering of all gears, splined shafts etc., i.e. all elements on the shafts that introduce a torsional torque.

List of all descriptive equations

2.1 Notation of the equation(s) of the step ratios

2.2 Notation of the equation(s) for each shaft

2.3 Consideration of the special cases in the equations of torques: web sum and loose wheel specification

2.4 Derivation of the kinematic, static and general degree of freedom: Counting the set up equations, comparison with the unknowns and completion with the necessary specifications

determination of the quantities

3.1 Solving the system of equations for speeds and torques

3.2 Determination of all input and output powers, the spread and transmission ratio of the gearbox

Points 1 and 3 can be described in general terms. For setting up the equations in the second section, a distinction is made for speeds and torques respectively.

5.2.1 Preparation of the geometric specifications

Extracting the construction The ‘pulling apart’ of the gears precedes the calculation in order to determine how many unknowns the actual gear really contains.

fig:5.6

preparation of the geometry of the construction: pulling apart the wheels with several interventions

numbering of all involved elements

After the geometric modification, so that each tooth meshing can be addressed uniquely, all elements that can introduce a torque into a shaft must be numbered consecutively. This results in the number of equations that are at least necessary to describe the entire system. This is shown in the right-hand part of the picture 5.6.

5.2.2 Set-up of all descriptive equations

notation of the equation(s) of the step ratios

fig:5.7

spur gear stage with epicyclic revolving wheel (p), central wheel (x) and web (s)

The rotational speeds of a single stage of an epicyclically rotating planetary gear (see picture 5.7) can be calculated by equation 5.61. It results from the speed relations of the Kutzbach plan. The possibility of speed superposition of the central shaft (\(n_x\)) with the web shaft (\(n_s\)) results here as well in a double kinematic degree of freedom for the determination of the planetary speed (\(n_p\)), given numbers of teeth of the central gear (\(z_x\)) and of the planet (\(z_p\)) are assumed.

eq:5.61

\[\frac{n_p}{n_x - n_s} = -\frac{z_x}{z_p}\] If the central wheel is a ring gear, the correct direction of rotation is obtained by inserting a negative number of teeth. The wheel arrangement in picture 5.7 is the minimum configuration only to represent the speed ratios on an epicyclic gearbox. A power transmission is possible with this wheel arrangement (at \(\eta = 1\), i.e. without friction or torque of inertia) is not possible, because the planetary gear has no possibility to transmit the torque. One can therefore impress no torque on one of the spline shafts, since the planetary wheel will always have a torque of 0. The static degree of freedom of the entire arrangement in picture 5.7 is therefore also 0. If it is a step with two axes that are fixed to each other in space, one sets the rotational speed of the web in equation 5.61 to 0. This gives equation 5.62, with index x for the wheel and index y for the opposite wheel of the step.

eq:5.62

\[\frac{n_y}{n_x} = -\frac{z_x}{z_y}\] torques

The equation for the torques at a spur gear stage is dependent on the direction of power flow, as already shown in equation 2.23 and 2.22. This in turn is calculated from the signs of the applied speeds and torques, i.e. in any case a calculation of the gearbox with \(\eta = 1\) before a further statement is possible. The integration of power losses into the system of equations and thus the determination of the efficiency, is described in the following section. For the first calculation, the following applies for the determination of the torques: Equation 5.63 with \(M_x\) and \(M_y\) as torques of the elements X and Y.

eq:5.63

\[\frac{M_x}{M_y} = \frac{z_x}{z_y}\] set-up of equation(s) for each shaft

fig:5.8

a shaft with n elements

For all elements that are firmly connected to each other by a shaft (picture 5.8), there is equality of speed. For a shaft with N force-initiating elements, the following applies in general terms:

eq:5.64

\[n_1 = n_2\\ n_2 = n_3\\ ... \enspace = \enspace ...\\ n_{N-1} = n_N\] torques

For the torques on a shaft with N force introducing elements, \(\sum M = 0\) holds. In general:

eq:5.65

\[M_1 + M_2 + \enspace ... \enspace+ M_{N-1} + M_N = 0\] considerations of special cases in torque equations

As shown in the example of the chapter 5.1.1 on the basis of a gear stage, in the case of loose gearwheels without friction or loads, it is necessary to insert the specification of the torque of 0 into the system of equations.

eq:5.66

\[M_{free wheel} = 0\] This is different for gearboxes with synchronisers, clutches or brakes from gear, depending on which stage is running under load and which loose gearwheels it contains.

The other special case occurs in the presence of a ridge. A ridge is connected directly via a shaft to the other connections of the epicyclic gearbox, but represents the supporting torque of the housing (see also picture 2.1). The web torque is therefore calculated from the sum of all the torques of the central shafts and the web. The corresponding wheels of the central shafts of an epicyclic gearbox can be identified by the fact that they always mesh with a planet which revolves on the searched web.

eq:5.67

\[M_{Z1} + M_{Z2} + \enspace ... \enspace + M_{Zn} + M_S = 0\] deriving the kinematic, static and general degree of freedom

One compares the number of unknowns numbered in the first step with the number of equations obtained. The difference represents the necessary specifications. For the speed equations this is the kinematic one, for the obtained torque equations it is the static degree of freedom. Both degrees of freedom are necessary complete solution of the equation system. To check the obtained, one can check whether the sum of the kinematic and static degrees of freedom is equal to the number of connecting shafts of the gearbox.

5.2.3 Determination of the quantities

Solving the system of equations for speeds and torques

The determination of the rotational speed and the torque of each element is achieved by the respective linear system of equations by the classical methods of addition-, substitution- and equation-procedure. Once the coefficients in a matrix with the corresponding result vector, direct or iterative methods can be used for the solution. A direct method is, for example, the addition-based Gaussian elimination method, which is based on the addition method, or Cramer’s rule, which works with determinants. The iterative methods include, for example, the Gauss-Seidel and the Jacobi method. Since the matrices that are set up are usually very large, sparse matrices, they are particularly suitable for the modern methods such as the preconditioned Krylov subspace method (BRO, 19).

determination of power, spread and transmission ratio

With the help of the obtained speeds, each gear ratio (between all input and output shafts) and from all gears also the maximum spread of the gearbox can be determined. Whether the torques acting on the spline shafts result in input and output powers can be determined from the signs. Negative power is output, positive power is input. Also the necessary power flows in the gearbox can also be easily determined by applying this rule to each element in the gearbox or directly to each shaft.

5.3 Integration of losses, calculation of efficiency

Two types of power loss generally occur in a gearbox:

Losses acting on elements on the shaft (Ex: sealing losses, splash losses or ventilation losses of gears, friction of a multi-plate clutch in the housing).

Losses acting between two elements (bearing losses between inner and outer ring, gearing losses in the stage between pinion and wheel). Losses can be included in the calculation model presented as virtual brakes or through torque specifications on individual elements. The main difficulty is in the specification of the losses between two elements, since for this the flow direction of the power must be known. It is not considered how these losses are determined, i.e. which theoretical or empirical equations behind their absolute values or whether they were obtained from measurements. The losses must always be set in such a way that they counteract their cause. For clarification the integration of losses on individual elements and gear and bearing losses will be shown as an example of losses between elements.

5.3.1 Losses at individual elements

Every element on a shaft that introduces force can also imprint losses on this shaft. A gear wheel can therefore not only be used to transmit power through its transmission by its step, but also generates a loss torque by its splashing in the oil or its ventilation, which has to be passed on to the shaft. The sign of the torque loss depends on the speed of the shaft. A power loss, like an output power, must always be negative, as it is removed from the power flow and is no longer available to the system. Even in the case of a bearing in which a bearing ring is fixed in the housing, its drag torque must be overcome. In this situation it therefore acts like an impressed loss torque of a simple element on the shaft.

fig:5.9

driving and loss torques on a rotating shaft (top) and resulting power flow (bottom)

5.3.2 gearing losses

In picture 5.10 a simple spur gear stage is shown with the magnitudes and torques for \(\eta = 1\), which will serve as a basic model in the following. In the hashs the respective shaft speed is entered. The rectangles contain the torque impressed by the corresponding element on the shaft. The power flow is shown at the bottom and right of the picture.

fig:5.10

speeds, torques (left) and powers (right) at a lossless spur gear stage with the ratio 2:1

fig:5.11

Real behaviour of the power loss: Problem in calculating the torques at a lossy spur gear stage. Solution by diversions via the power

fig:5.12

speeds and torques at a simple spur gear stage with representation of the stage loss as a virtual element, power flow from the gear to the pinion

fig:5.13

speeds and torques at a simple spur gear stage with representation of the stage loss as a virtual element, power flow from the pinion to the wheel

5.3.3 Bearing losses

The bearing losses, like the gearing losses, occur between two elements, the inner and outer ring of the bearing. The direction of rotation of the bearing is of decisive importance for the transmission of the importance. This will be illustrated in the following using a single bearing which is used to support a shaft in a hollow shaft. The power is supplied on one or both sides, i.e. from the left onto the inner ring or from the right via the hollow shaft onto the outer ring. The dimensions of the numbers listed in the picture 5.14 and picture 5.15 are given by way of example only and are arbitrary in amount.

fig:5.14

torques and power flow due to bearing losses when the shafts involved are rotating in the same direction

eq:5.68

\[P_V = 2\pi \cdot M_V \cdot \Delta n \] From the occurring loss torque the drive torques with opposite sign result on the outer and inner ring of the bearing (otherwise the sum of the torques on the bearing would not be 0). From this the torques at the spline shafts are derived. It results that the faster shaft drives the slower one. The power at the the left, faster rotating spline shaft is positive, that of the slower one (right) negative. If both bearing rings rotate in opposite directions (fig. 5.15), it follows from the bearing’s torque of loss that the drive of this constellation must originate from both sides.

fig:5.15

speeds, torques and power flow through the losses in a bearing with counter-rotating shafts

5.4 General formulation for integration of losses

Losses can be integrated into the existing system of equations of the lossless calculated without losses by creating virtual brakes that are positioned at the position in the gearbox. The system of equations with the lossless calculated torques is merely extended and not changed in essence. In the case of losses in individual elements (gears, sliding clutches, seals, bearings with a ring in the housing), a new brake with a new sequential number is created next to the element. The torque loss of this brake is according to the sign of its rotational speed, so that the total power is negative power (i.e. at negative speed as positive torque, at positive speed as negative torque). In the case of losses between two gears, the virtual brake is always applied to the shaft that lies in the direction of power flow. The amount of the torque loss is calculated from the power loss of the gear stage and the speed of the shaft to which the brake is applied.

In order to pass on bearing losses in the case of bearings installed between two shafts, the direction of rotation of the shafts involved must be taken into account. If the shafts move in the same direction, the torque must be applied in such a way that the faster shaft is braked by the bearing loss torque (the left shaft in picture 5.14), the slower shaft is accelerated by the faster one (right shaft in image 5.14). With the two shafts moving in opposite directions, the braking torque with the opposite sign to the rotational speed must be imprinted (picture 5.15).

5.5 Three examples of calculations of gears with losses

In the following section, three examples are used to show the application of the calculation methodology of losses as virtual brakes, as implemented in the FVA programme WTplus2.0. The first example, a two-stage spur gear chain, illustrates the basic calculation principle, how the results of empirical equations contribute to the determination of the true torque distribution in the system of equations described and why iterations are necessary for this. In the second example, it is shown how with the now created possibility of calculating the load-free, co-rotating components, it is possible to make general statements about the power flows in the individual gears. This makes it possible, for example, to demonstrate the existence of circular power flows at low oil levels/flange losses or low gearing losses. In the third example the application to a Ravigneaux set as a more complex gearbox with several degrees of freedom is shown.

fig:5.16

geometry of a two-stage helical gearbox with speeds and torques at \(\eta = 1\)

fig:5.17

mapping between a number and all loss producers in a two-stage spur gear

  .                    -------------------
  .                    - General results -
  .                    -------------------
  +-- Shaft 1:
  Drive power . . . . . . . . . P_AN    37.69           kW
  Drive speed . . . . . . . . . n_AN        -3600.00    1/min
  Drive torque . . . . . . . . .T_AN        -100.00     Nm
  +-- Shaft 3:
  Output power . . . . . . . . . P_AB  -36.26           kW
  Output speed . . . . . . . . . n_AB       -578.05     1/min
  Output torque . . . . . . . . .T_AB        599.04     Nm

  Efficiency . . . . . . . . . . eta    96.19 percent
                                        -------------
  Total power dissipation . . . P_V     1436            W

  Gear power loss . . . . . . . .P_VZ   1227            W
    Load-dependent component . . P_VZP          1185    W
    Load-independent component . P_VZ0            42    W
  Bearing power loss . . . . . . P_VL    208            W
    Load-dependent component . . P_VLP           193    W
    Load-independent component . P_VL0            15    W
  Seal power loss . . . . . . . .P_VD      0            W

table:5.5

table showing all losses [watts] at a two-stage spur gear stage during the iteration. On the left the pictures of the corresponding elements of the gear

fig:5.18

Switchable spur gear stage as an example for determining the power flow in a no-load gear branch

fig:5.19

Possible power flows (white arrows) and losses (black arrows) at a co-rotating stage when the shaft and idler wheel rotate in the same direction. Final destination of the circular power is marked in grey

table:5.6

high oil level; examples of reactive power occurring in an idling gear stage (transmission from slow to fast), synchronisation on input shaft (top) and output shaft (bottom), occurring torques (bottom), occurring torques (box) and speeds of the idler / shafts (rhombus)

low oil level; examples of reactive power occurring in an idling gear stage (transmission from slow to fast), synchronisation on input shaft (top) and output shaft (bottom), occurring torques (bottom), occurring torques (box) and speeds of the idler / shafts (rhombus)

table:5.7

high oil level; examples of reactive power occurring in a rotating loose gearwheel stage (transmission from fast to slow), synchronisation on input shaft (top) and output shaft (bottom), occurring torques (bottom), occurring torques (box) and speeds of the loose gearwheel / the shafts (hash)

low oil level; examples of reactive power occurring in a rotating loose gearwheel stage (transmission from fast to slow), synchronisation on input shaft (top) and output shaft (bottom), occurring torques (bottom), occurring torques (box) and speeds of the loose gearwheel / the shafts (hash)

As a third example, a Ravigneaux set with four shafts connected, shown in fig. 5.20 was used in order to demonstrate the applicability of several rotating shafts, i.e. higher degrees of freedom and different inputs and outputs. This calculation was carried out first with three and then with five planets. The output torque of -200 Nm on shaft 4 and the input torque of 10 Nm on shaft 2 were fixed. In addition, the speeds of 1800 1/min at the sun on shaft 1 and the standstill of the sun of shaft 3 were fixed. This results in the following two result blocks. As can easily be seen, the main changes in this operating range are the of the load-dependent and load-independent losses. The load-dependent losses become smaller, since the normal force in the gearing is distributed evenly over all planets. This reduces the value for the friction coefficient in the gear meshing and thus also the load-dependent losses. On the other hand, the load-independent losses increase, since not only three but five planets have to run along and produce squeezing and impulse losses. The same phenomenon can be observed with the bearing losses.

fig:5.20

geometry of the third example, a Ravigneaux set with two kinematic and two static degrees of freedom

  .# three planets:    ---------------------
  .                    -- General results --
  .                    ---------------------
  +-- Shaft 1:
  Drive power . . . . . . . . . P_AN    23.11        kW
  Drive speed . . . . . . . . . n_AN        1800.00  1/min
  Drive torque . . . . . . . . .T_AN         122.62  Nm
  +-- Shaft 2:
  Drive power . . . . . . . . . P_AN     0.78        kW
  Drive speed . . . . . . . . . n_AN         741.17  1/min
  Drive torque . . . . . . . . .T_AN          10.00  Nm
  +-- Shaft 3:
  Output power . . . . . . . . .P_AB     0.00        kW
  Output speed . . . . . . . . .n_AB           0.00  1/min
  Output torque . . . . . . . . T_AB          68.43  Nm
  +-- Shaft 4:
  Output power . . . . . . . . .P_AB   -23.44        kW
  Output speed . . . . . . . . .n_AB        1119.32  1/min
  Output torque . . . . . . . . T_AB        -200.00  Nm

  Efficiency . . . . . . . . . . eta    98.12 percent
                                       --------------
  Total power dissipation . . . P_V    539           W

  Gear power loss . . . . . . . P_VZ   499           W
    Load-dependent component . .P_VZP        496     W
    Load-independent component .P_VZ0          3     W
  Bearing power loss . . . . . .P_VL    40           W
    Load-dependent component . .P_VLP         16     W
    Load-independent component .P_VL0         24     W

With five planets correspondingly higher load-independent bearing and gearing losses, with lower load-dependent losses:

  .# five planets:     ---------------------
  .                    -- General results --
  .                    ---------------------
  +-- Shaft 1:
  Drive power . . . . . . . . . P_AN   23.13        kW
  Drive speed . . . . . . . . . n_AN        1800.00 1/min
  Drive torque . . . . . . . . .T_AN         122.71 Nm
  +-- Shaft 2:
  Drive power . . . . . . . . . P_AN    0.78        kW
  Drive speed . . . . . . . . . n_AN         741.17 1/min
  Drive torque . . . . . . . . .T_AN          10.00 Nm
  +-- Shaft 3:
  Output power . . . . . . . . .P_AB    0.00        kW
  Output speed . . . . . . . . .n_AB           0.00 1/min
  Output torque . . . . . . . . T_AB          68.37 Nm
  +-- Shaft 4:
  Output power . . . . . . . . .P_AB  -23.44        kW
  Output speed . . . . . . . . .n_AB        1119.32 1/min
  Output torque . . . . . . . . T_AB        -200.00 Nm

  Efficiency . . . . . . . . . . eta   98.06 percent
                                      --------------
  Total power dissipation . . . P_V   545           W

  Gear power loss . . . . . . . P_VZ  492           W
    Load-dependent component . .P_VZP        486    W
    Load-independent component .P_VZ0          6    W
  Bearing power loss . . . . . .P_VL   53           W
    Load-dependent component . .P_VLP         13    W
    Load-independent component .P_VL0         40    W

6 Implementation of the calculation method in the EDP

For the realisation of the described calculation methology (chapter 5) on an EDP system, the necessary equation system to describe the entire gearbox must be created by a programme. the matrices presented for the speed and torque relationships must be set up and solved on the basis of the given geometric shafts, stages and gears, it must be possible to set up and solve them without the user having to enter the abstract models of the wolf symbolism or having to enter any other additional information. The trick is to use only the relevant relationships without redundant information so that the equation system is not over- or under determined. This is realised with the help of a graph. Normally, the relationships of these abstract constructs are used directly for decisions on logistic or assignment problems. The innovation of the method presented here is that the shape of the graph is used to set up a correct system of equations. The graph is obtained from the geometric relationships of the gearbox. The method is therefore similar to the ‘finite element method’, in which a system of equations is also set up on the basis of geometry. However, this is usually solved numerically and iteratively for many elements and nodes, whereas in the following presented below, an analytical equation for the entire gearbox system is created and solved under programme control. In a slightly different form, this method has already been used by DOM for the determination of manual transmissions with multi-group design.

Since the method presented here is based on a graph, we will first discuss the basic concepts of graph theory and an application case from information theory, so-called ‘backtracking’, is explained, as this is required for the equations. Subsequently, the methods described in chapter 5 are then used to visualise the computer-aided algorithm and the relationships between the gear geometry and the graph, as well as the graph and the equation system are explained.

6.1 Graph theory

Graph theory is a field of mathematics that has found its way into many areas of life nowadays, especially because of computer science. Frank Harary HAR writes in the foreword to his book, “that graph theory is used in some areas of physics, chemistry, in transport and communications, in computer and civil engineering, operations research, genetics, sociology, economics, anthropology and linguistics. The theory is also closely related to many branches of mathematics, such as group theory, topology and combinatorics.”

6.1.1 What is a graph?

Leonhard Euler (1707–1782) is regarded as the father of graph theory as well as topology since he solved the so-called ‘Koenigsberg bridge problem’ in 1736. The ‘Koenigsberg bridge problem’ is a famous unsolved problem of its time. Euler was concerned with whether there was a circular route that crosses all seven of the city’s bridges over the Pregel River exactly once and return to the starting point HAR (see picture 6.1, left).

fig:6.1

The park in Koenigsberg and its bridges

The corresponding graph to the ‘Koenigsberg bridge problem’

If you abstract the shore and the island of the sketch into points of the drawing plane and the bridges into lines between the corresponding points, we obtain the graph shown in fig. 6.1. A graph is therefore a construction of points that are connected by lines. This graph can only be traversed in the desired way if it is connected and each node has an even number of connections. The figure created here from the bridges and islands illustrates that a path searched for by Euler in this way (‘Euler’s path’) does not exist. Euler’s achievement in this field lay in the proof of in which case such an undirected graph visualised that way allows such a circular path (‘Euler’s circle’). This question is particularly relevant to everyday life for professions with logistical problems, such as that of the postman ('postman’s problem NOL, p 165) and for haulage companies.

6.1.2 Basic concepts of graph theory

There are no precisely defined conventions regarding the nomenclature of graphs and so there are also differences in terminology form author to author. For this reason, a few definitions are given below so that they can be used unambigously in this work. They are takes from the books (NOL, 20).

definition:

A graph consists of a set of nodes and a set of edges. The set of edges also provides a connection between the nodes, because an edge is the connection between two nodes.

directed and undirected:

For undirected graphs, the order of the nodes is not relevant. In contrast, directed graphs are orders pairs of nodes

fig:6.2

Undirected (left) and directed graph (right)

multigraph and pseudograph:

Several edges can also exist between two nodes in a graph (multigraph). However, the definition of the graph does not allow a loop, a connection from a node to itself. If both multiple edges and loops are permitted, it is a pseudograph.

fig:6.3

Pseudograph with loop and double edge

indexing:

A graph is indexed if the nodes are distinguished from each other by names such as \(e_1, ..., e_i\).

valency:

The valency of the node \(e_i\) of a graph is equal to the number of indexing edges of \(e_i\), e.g. if three lines go to a node, it has a valency of 3. As each edge is always connected to two nodes, it contributes the value 2 to the sum of the valencies of the nodes. This is a result found by Euler, which was the first theorem of graph theory.

“The sum of the valencies of the nodes of the graph G is equal to twice the number of edges:”

eq:6.1

\[valency \enspace e_i = 2q\] whereas \(e_i\) equals the nodes of the graph from 1 to i and q the number of edges.

‘Teilgraph’ and ‘Untergraph’:

A graph \(G_1\) is called ‘Teilgraph’ of G if it has fewer nodes or edges than the graph G. A ‘Untergraph’ \(G_2\) has fewer nodes and fewer edges than graph G.

planarity:

In graph theory, the term planarity refers to a graph, that can be represented in the drawing plane without intersections. This question of planarity plays an important role in the production of electrical circuits, for example.

fig:6.4

Planar graph \(K_{2,3}\) and non-planar graph \(K_{3,3}\)

completeness:

A complete graph \(K_n\) is an undirected graph without multiple edges with n nodes and exactly \(\binom{n}{2} = \frac{n \cdot (n - 1)}{2}\) edges for \(n > 1\). In a complete graph, every node is connected to every other node by an edge.

fig:6.5

Complete graphs from \(K_1\) to \(K_5\)

6.1.3 Mathematical representation

In information theory, graphs are written down for the computer in a mathematically more tangible form. This is normally done in the form of matrices, which can be stored in arrays, lists or dictionaries. Basically, two forms of matrices are used: The adjacency matrix and the incidence matrix. Circle matrices and concrete matrices are special forms that were not used in this work.

The adjacency matrix (lat. adiacere - neighbouring) is also called a neighbourhood or node-node matrix. This is obtained by numbering the nodes and writing a 1 in the i-th row of the j-th column if you can get from node i to node j via a line. As a result, undirected graphs are represented by symmetrical and directed graphs by asymmetrical matrices. The graph shown in picture 6.3 has the adjacency matrix shown in table 6.1. In it, the relationships between the nodes are noted by the number of connections. If you now want to know to which neighbouring node number 4 is connected by a line, read the fourth row of the adjacency matrix. The nodes to which there is exactly one connection then contain the number 1.

table:6.1

Adjacency matrix of the pseudograph in image 6.3

table:6.2

Incidence matrix of the pseudograph in image 6.3

6.1.4 Examples of applications of graph theory in practice

general transport problem:

This problem relates to the distribution of products from individual production sites to locations where the produced goods are in demand. The problem was first formulated by Sir William Hamilton in 1859. He invented a game whose board was constructed in the shape of a regular dodecahedron fig. 6.6. At its 20 corners/nodes were the names of famous cities that the players had to visit by travelling once around the world, which means that he must find a closed path along the edges that passes through each node exactly once (Hamilton’s path).

fig:6.6

Game by Sir William Hamilton

fig:6.7

Graph of a transport system: P production sites, T transfer point, S sites with demand for NOL

The solution to the general transport problem contains the answers to questions that play an important role in the relationship between supply and demand. The most important ones are mentioned here:

Is there a transport plan in place that respects transport and production capacities and satisfies demand?

What is the maximum amount of demand that can be satisfied?

Where and by how much must production and transport capacity be increased if there is a change in minimum demand?

How do you obtain a cost-optimised production and transport plan that satisfies the required demand at the sites when production and transport costs are included? Figure 6.7 shows a simple graph of such a problem. The nodes with the name P are production sites, T are transfer points and S are sinks with demand for the produced goods. At this point, a few procedures for solving this problem are briefly mentioned. For example, the ‘north-west corner method’, the ‘ranking method’, the ‘matrix minimum method’ and the ‘Vogel’s approximation method’ are suitable, whereby the latter variant in most cases represents a relatively optimal solution for the problem NOL. However, the north-west corner method is often favoured in data processing because it is easier to programme and the number of iterations required is not significant.

electrical networks:

In 1847, the physicist Kirchhoff developed a theory that was intended to solve the linear system of equations that determines the current in each branch and circuit of an electrical network. This became known as the so-called ‘theory of trees’.

fig:6.8

Electrical network, its underlying graph and a spanning tree

In this way, Kirchhoff proved that it is not necessary to consider each circle in the graph of an electrical network separately in order to solve the system of equations and showed that the independent circles of a graph are sufficient, which are determined by any of its spanning trees (fig. 6.8, right).

6.1.5 Backtracking algorithm

The term ‘backtracking’ refers to a problem-solving method within algorithmics. Backtracking works according to the Trial-and-Error principle, i.e. a partial solution is gradually developed into a complete solution.

fig:6.9

Backtracking, explained using the graph of a labyrinth: Labyrinth and associated graph (top), path from entrance to exit in the graph (bottom)

Figure 6.9 illustrates this procedure using a labyrinth. The node of a graph is drawn at each intersection and end point. It represents the decision to change direction (forwards, right, left, etc.). If you now want to go through the labyrinth from the upper entrance to the lower exit and always chooses to go ‘right’ as the strategy in the graph, the backtracking method results in the following sequence of steps along the graph. Each dead end that was taken and then discarded is shown in grey. The path is always continued at the last unvisited path of the fork passed and the path to the exit is found.

6.2 Examples for visualising the computer-aided algorithm

The following section uses the gear examples already shown in chapter 5 to visualise the structure of a graph. This graph is a description of all possible power paths of the gearbox. The relationship between its shape and the corresponding equations are listed in a table at the end.

As you can already see from the formulae in chapter 5.2, the descriptive equations consist of step and shaft relationships. There are also the two special cases in the torque equations for the treatment of idler gears and the web of epicyclic gears. These components must therefore also all occur in the generated graph or can be derived from it. The general formulation of the procedure is derived by applying the algorithm from the simple to the complex example.

6.2.1 Switchable spur gear stage

For the shiftable spur gearbox with its coupled first stage (gear 2 is firmly connected to the shaft) in figure 6.10 the graph to the right applies to the power flow.

fig:6.10

Switchable spur gear stage with closed clutch K1 in the first stage and associated graph with all possible power flow paths

To create the graph, you draw a node instead of each element of the gearbox. Two wheels that form a step are connected by solid lines and all elements on a shaft with zigzag lines. It is important to ensure that each element of a shaft is connected to all other elements on it (the ‘Untergraph’ of the shaft is a complete graph).

You could draw all lines with simple lines without differentiating by connection type. The resulting graph, like a city map, would be a network with intersections (grey) and roads (white) on which the power can flow (fig. 6.12). If you start at node 1, the power can either continue via node 2 or via node 3. If you come from node 3, you can decide in favour of node 2 or 1. The node clusters are therefore an intersection. The lines are the connecting roads.

fig:6.11

Development of an easier-to-read version of the graph of the switchable spur gear stage

However, it is then no longer so easy to assign the correct corresponding equations. It is therefore better to retain the cause of a line through its shape (solid, dashed, zigzag line). A simple undirected, unweighted graph like the one on the right in fig. 6.10, only represents which element is connected to which neighbouring node. You can move the positions of the nodes without changing the graph. In fig. 6.11, the descriptive graph of the shiftable spur gear is shown again with shifted node positions. It is now easier to recognise which paths the power can take. Assuming that the power can only fed into or extracted from the system by elements 1 and 6 (the splined shafts), you can immediately recognise that there is actually only one power path. This is spanned by the elements 1–2–4–6 (or in reverse order), if one assumes that the power does not make any detours.

fig:6.12

The flow of performance as a road map. Roads (white) connected by intersections (grey)

All elements that lie together on a shaft are in the node clusters, which are connected by zigzag lines: Nodes 1–2–3 and 4–6

All ends and edges of the graph consist of single nodes, i.e. nodes that are are connected to other nodes via exactly one level relationship (straight line) or which are not connected to other nodes in any way. These are: Nodes 1, 5 and 6. The specifications are always impressed into the gearbox at at least one of these nodes. They represent the input and output of the gearbox or idler gears. Here, node 1 is the drive and node 6 is the output. Consequently, node 5 as an edge node must represent an idler gear with torque 0 which it is, as can be seen by looking at fig. 6.10. These results are shown once again in table 6.3.

table:6.3

Relationship between graph and gearbox geometry for a spur gearbox

6.2.2 Minus gearbox

The special features of the negative gearbox are the double engagement of the planetary gear and the web. Now all nodes with elements on a shaft are again connected by zigzag lines and all nodes of the gear stages by straight lines.

fig:6.13

Geometry and first graph of the negative gearbox

fig:6.14

Geometry of the graph of the minus gear after the reorganisation of node 3 with two-step bindings: New node 8

The only remaining special feature of this graph is the web, which is included in the equations as a torque sum over it and all centre gears of the associated planetary gear. Since power can also flow to the web via the planet, it is necessary to connect the planets to the web. However, this cannot be one of the two lines used so far, as there is neither a step relationship between the planet and the web, nor do they both lie on a shaft. A new type of line is therefore required, shown in fig. 6.15.

fig:6.15

Final graph of the negative gear with connection of the web (6) to the planets (8,3)

The elements together on a shaft are again in the node clusters: Nodes 1–2, 3–8, 6–7 and 4–5. The torque sum of the bar runs over it and all the central wheels, which are neighbours of the planets: Nodes 6, 4 and 2.

All the ends and edges of the graph are drives and outputs: nodes 1,5 and 7. These results are summarised once again in table 6.4.

table:6.4

Relationship between graph and gearbox geometry for a negative gearbox

6.2.3 Five-shaft gearbox: Combination Ravigneaux and Wolfrom set

The procedure for determining the graph of a five-shaft gearbox, consisting of the combination of a Ravigneaux and a Wolfrom set, follows the same steps as for the minus gearbox. The starting point is again the creation of nodes based on the geometry of the shafts and stages. Each element that can apply a torque to the shaft is given a number.

fig:6.16

Geometry and first graph of the five-shaft gearbox

fig:6.17

Graph of the five-shaft gearbox after conversion of the double stages: Completed nodes 13 - 15 from number 4 and node 16 from number 3 (left). Graph of the involute shaft gearbox with the node positions pushed together (right)

table:6.5

Relationships between graph and gearbox geometry in the five-shaft gearbox

6.3 General formulation of the computer-aided algorithm

The computer-aided algorithm works in the same way as the procedure described in chapter 5. It is again divided into three areas:

Structure and drawing of the graph

1.1 Structure of a node for each element of the gearbox

1.2 Creation of stage relationships

1.3 Creation of shaft relationships

1.4 Creation of the web relationships

1.5 Pulling apart the nodes with more than one level relationship

List of all descriptive equations

2.1 Setting up the equation(s) of the step ratios

2.2 Set up the equation(s) for each shaft

2.3 Special cases in the torque equations: Webs and idler wheels

2.4 specifications of the gearbox: According to the kinematic, static and general degree of freedom

Determination of sizes

3.1 Solving the system of equations for speeds and torques

3.2 Determination of all input and output powers, the spread and transmission ratio of the gearbox

The first paragraph is always used to determine the unknowns in the equations and to identify all the elements actually involved. This is done in a computer-aided process using graph theory. The equations for setting up the descriptive matrix for speeds and torques are the same as in chapter 5.2.2. Reference is therefore made to this in the sections on ‘Setting up all descriptive equations’ and ‘Determining the variables’.

6.3.1 Structure and drawing of the graph

structure of the nodes

A unique number is assigned to each element of the gearbox that transmits power to a shaft. The same number is also written in the node of a graph that is to represent this element (fig. 6.18). This relationship can be noted in an assignment table, for example (table 6.6).

fig:6.18

Assignment of the nodes of the graph from the section through the geometry of the gearbox

table:6.6

Mapping table between elements of the gearbox and the graph

creation of all level relationships

Each step relationship between two gears is represented by a straight line between their two nodes (fig. 6.19).

fig:6.19

Creating the level relationships with a solid line between two nodes

creation of shaft relationships

All elements that are firmly attached to a shaft are connected to each other. This is represented by a zigzag line to make it easier to distinguish them from the step relationships.

fig:6.20

Notation of the togetherness of all elements on a shaft by a zigzag line

creation of the web relationships

As the web of a planetary gearbox is not located directly on the same shaft as the planets, but is also not connected via a step, the connection between the web and planet is shown by a dashed line (fig. 6.21). This connection is used to determine all neighbouring elements of the planets as soon as you want to calculate the sum of the torques on all connecting shafts of a planet set.

fig:6.21

Notation of the relationship between the bridge and planets with a dashed line

pulling the gearbox apart

As soon as a gear meshes with not just one, but two or more gears, there are several forces/torques on this element.

fig:6.22

Pulling apart the gearbox on the graph

fig:6.23

Expanding the graph at each node with more than one level relationship

6.3.2 List of all descriptive equations

The corresponding equations are derived from the shape of the graph: In table 6.7, the necessary equations are noted next to the shape of the respective graph. For the derivation of the assignment of the correct associated equation, please refer to chapter 5.2.2.

table:6.7

Relationships between graph and equation for n and M

6.3.3 Determination of sizes

This procedure has already been described in chapter 5.2.3.

7 Computer-aided synthesis of gearboxes

The subject of this work is a calculation method for determining speeds and torques on any planetary gearbox. This is based on the geometry and obtains a system of equations for the complete description of all variables (in EDP via the diversions of a graph). During the work, the idea arose that one could also go the other way round, i.e. from the graph to the actual gear geometry. The thoughts on this will be explained in the following chapter on the synthesis of gears.

7.1 Current methods of gear synthesis

The work of (XIY, 21) on gear synthesis also uses graph theory. (XIY, 22) on gearbox structures in a layshaft design. In contrast to this work, in which a graph is used to set up a descriptive system of equations for speeds and torques, the graphs of the aforementioned works are used to avoid collisions/self-penetration or to construct meaningful gear combinations and geometric relationships.

(XIY, 23) uses line graphs as the starting point for its synthesis of planetary gears. By correlating the line graph with the associated speed equations, the author is able to generate any automatic gearbox. He generates the line graph with the desired distribution and spread of the gearbox. He then uses the gradients of three lines to determine the possible stationary ratios of planetary gearboxes that must be used to achieve these ratios. The combination of the simple epicyclic gears with the found stall ratios then results in the possible overall gearbox. Gearboxes with two (kinematic) degrees of freedom can therefore also be represented very elegantly in the drawing plane as a function of \(\omega_1\) and \(\omega_2\). With three degrees of freedom, however, the possible combinations are already surfaces in three-dimensional space, so that only their intersection lines with the main coordinate surfaces (\(\omega_1 - \omega_2\), \(\omega_1 - \omega_3\), \(\omega_2 - \omega_3\)) can be represented. The author does not deal with gears with a higher kinematic degree of freedom than three, as ‘’…the realisation of a gear in planetary gears with more than four degrees of freedom is extremely complex‘’ and not suitable for motor vehicle construction. Unfortunately, his method also does not deal with gearboxes with higher static degrees of freedom than one, which would be essential for hybrid concepts or gearboxes with several input and output shafts. His explanations on the feasibility of a transmission concept without geometric contradictions, based on Kuratowski’s theorem and the flatness of a graph, make a very practicable method for generating gearbox sections possible. The filter criteria he proposes for selecting good concepts (by excluding excessively high stall ratios, relative speeds and shift torques for brakes/clutches) and selecting the right specifications for a synthesis are also trend-setting.

(DOM, 24)’s work is based on so-called gear train structures, which, when several are put together, form so-called gear base structures (GBS). The GBS is also described using graphs and their adjacency matrices. This basic structure of a countershaft gearbox contains the information about which gear is located on which shaft, which are neighbouring and which gears mesh with each other. The graph structure is used to prevent collisions and avoid isomorphisms, so that there is really only one variant at a time. Normally, these isomorphisms are transformed into each other by congruent transformations. However, as this is too computationally intensive in practice, this is achieved by comparing key data (number of gears and shafts) and characteristic values (determinant and dimension of the matrix) of the gear base structures found.

The so-called gearbox trivial structure (GTS) is then generated from the GBS by adding coupling elements on all wheels, the load-bearing cylinder sequence matrix, wheel shaft coupling matrix or coupling vector to the GBS. The GTS in turn becomes the gearbox primitive structure (GPS) by reducing the GTS by unfavourable coupling combinations. The so-called final gearbox structure is finally achieved by reducing the GPS to the necessary minimum of shiftable couplings, e.g. by turning couplable gears that never need to be coupled into fixed gears. With this approach, it is possible to generate all possible variants of a gearbox in a pre-coupling design under computer control, with multiple use of the wheels for different gears and thus also a statement about the minimum necessary complexity of a gearbox with n gears in this design. To select the generated structures, (DOM, 25) uses an evaluation and importance matrix, which is mainly based on the gearbox complexity (number of gear trains, shaft trains, gears, shafts, idler gears, coupling and actuation units) and the expected gear efficiency.

7.2 Necessary specifications for gearbox synthesis

For the automated generation of gear concepts, several boundary conditions must be specified by the user in order to limit the number of possibilities during synthesis. Otherwise, chains of gears would simply be generated in space, which would have nothing to do with the solution of the desired problem other than by chance. A meaningful application of such a programme is only possible if some of the following specifications are used:

Desired degree of freedom of the overall system

Number and position of input and output shafts

Number of possible gears

Maximum number of shafts and elements involved (wheels, etc.)

Complexity (type and maximum number of basic elements to be used: fixed steps, epicyclic steps with bridge, clutches, brakes, etc.)

Spreading of the gearbox

Grading of the gears (progressive/geometric) and permitted deviation in each case

Design (coaxial input/output), elongated/tall/flat

Maximum/minimum diameter and number of teeth ratio

Different usage scenarios (alternating input and output on different shafts, straight ahead or cornering with a differential, etc.), directions of rotation and power flows (+/-) on input and output shafts

(DOM, 26) refers to these specifications as synthesis parameters. In the case of planetary gearboxes, the parameters he mentions must be supplemented by a number of application scenarios compared to gearboxes of the layshaft type, which the design must fulfil, as several input and output shafts can exist and the internal degree of freedom between these is normally also relevant (e.g. differential functionality between input shafts A and B, but drive always only via shaft C). As evaluation parameters that are easy to determine after synthesis, (DOM, 27) states:

Number and arrangement of coupling options (clutches, brakes)

Number of elements required for the solution (gears, shafts) and their manufacturing effort (ring gear, spur gear, number of gears with the same number of teeth)

Speed analysis (maximum occurring speed/relative speed)

These figures are used as a guideline for the main requirements of weight, installation space and costs. Design and optimisation with regard to installation space (e.g. fewer elements by combining two wheels) or load capacity (variation of e.g. width, profile displacement, material) can also only take place after the actual concept has been found in a separate optimisation cycle. These figures can then be included in the further evaluation.

7.3 Problems that arise when creating computer-generated solutions

Several authors ((DOM, 28), (XIY, 29) have already proposed and programmed computer-aided syntheses of gear concepts. However, these However, these usually only covered a few sub-areas of gearboxes (e.g. manual gearboxes with several groups, only negative gearboxes as components, only a structure consisting of combinations of wolf symbols, only consideration of load capacity, only calculation of speeds, etc.). No arbitrary gear chains in space were considered, as the combinatorial explosion due to different networking of elementary gears and the optimisation possibilities of the generated solutions alone are already very computationally intensive.

It was also often assumed that there were certain larger basic components (e.g. a complete plus or minus gearbox) and their corresponding equations, instead of choosing the smallest element, a step between two wheels. If this was implemented in this way, then only for the determination of the speeds (KAH, 30). For forces and torques in space, the concept again had to be translated into many 2D sections of the pitch circles in order to calculate the forces in each stage. This is quite time-consuming for a concept determination, as specific pitch circle diameters, centre distances and angles between the centres of these circles must already be found.

The gear ratio alone is usually relatively easy to create. However, the creation of a complete transmission concept with several gears, including the relationships between them, or a statement about reactive power, raises several problems that cannot be solved without graph theory or a similar consideration of networking. If you want to create variants of concepts that have already been found, the influences of newly installed steps on all other gears and the entire degree of freedom must also be taken into account.

Once the program finally works, you have the problem that a lot of possible combinations are often generated, all of which should normally be viewed by designers, since the avoidance of nonsensical concepts cannot always be automatically prevented during synthesis. For example, if you only use wolf symbols for generation, concepts with self-penetration must first be recognized and removed (see image 4.5). If you do this reduction of solutions programmatically, you have to be particularly careful when choosing the exclusion criteria in order to discard unsuitable concepts. Here you would also have the problem that, due to the recognized logic, a designer always first has to translate the abstract concept into an actual geometry, i.e. steps and wheels with actual numbers of teeth.

If the basis of a synthesis of transmission concepts is simply the creation of wheels/cylinders in space, coupled via stages and shafts, special emphasis must be placed on the collision of the individual wheel cylinders with one another. Every time a new wheel is created, it is recommended to check all the wheels that have already been created by recalculating them for collisions. This also occurs if all the pitch circles of three meshing wheels touch each other (picture 7.3).

fig:7.1

Simple collision of two gears (left), collision caused by three meshing pitch circles (right), source (DOM, 31)

Complexity: Restriction of the synthesis to gears made of certain components, e.g. according to Wolf symbolism, negative gears, spur gear stages

Variation: Changing a found concept by adding/removing wheels/stages/couplings to achieve the desired degree of freedom

Logic: Automatically determines the correct shift logic for the new transmission

Combinatorial explosion: Exponentially longer runtimes with increasing number of components

Constructive implementation: Any post-processing, review and constructive implementation of each solution is necessary

Impossibility of solution: Check for self-penetration of the shafts. Collision of the wheels with each other during or after generation is necessary

7.4 Proposed solution to the problems that arise

For all the problems described in the previous chapter 7.3, a proposed solution will be given in the following section, so that in the end the implementation of automatic transmission optimization is possible. However, the main problem with such a program was not mentioned in the previous chapter because it is a problem from computer science: This would lie in the general procedure according to which the best possible gearbox suggestion is selected. The best approach to program-controlled gear synthesis would be to proceed like the human design process. During the creation of a design, the following phases are usually passed through for the created concepts:

synthesis

analysis

detailed optimization

evaluation

selection and rejection

mutation and recombination

The order in which these points are processed can vary. For example, depending on the time available, the solutions found can be detailed before or after a preliminary evaluation. For the first few cycles this may not be necessary at all. Often several loops are formed in which concept development and rejection alternate.

However, if possible, one tries to obtain a wide range of solutions from which a few are then selected that go in the desired direction. The methods used in each phase may differ depending on the problem at hand and priority. In one case, the focus may be on the simplicity of a solution, represented for example by the number of parts required, while in the other case the accuracy is more relevant, for example if the desired gear ratio should be achieved as precisely as possible.

Since there are always references and mutual dependencies when making design changes in gears, especially planetary gears, standard methods such as the gradient method cannot be used for optimization, since discontinuities and constant changes between maxima and minima are the rule. The need for whole numbers of teeth alone is an example of this. The optimization would get stuck at local extremes and only reach the areas of the global optimum by chance. It would also be very difficult to deal with actually contradictory goals, such as load-bearing capacity and installation space. Furthermore, one has a multi-parametric solution space:

For example, if you want to change the gearing parameters of several stages, you could do this at each stage in terms of width, module, number of teeth, profile shift, helix angle, etc.. Many possible solutions are conceivable, but how should one sensibly evaluate this quantity and determine the relevant quantities?

For all of these reasons, less deterministic and more stochastic or heuristic methods are suitable for optimizing the concepts, such as the Monte Carlo method, simulated annealing, evolutionary strategies (ES) or genetic algorithms (GA). Evolutionary strategies (see appendix) are best suited for this, as this approach includes all the steps described above. GA is quite difficult to implement because it requires finding a binary representation of a constructive solution. Using one or more penalty functions, it is possible to determine the fitness of an individual (a number for the quality of a solution found) and to weight this based on the user’s specifications. For example, a penalty function returns a number between 0 (very bad) and 100 (best of the group) as a return value, whereby the entire pool of solution objects is normally used for evaluation. The only condition for this is that it can be evaluated mathematically based on a clearly defined criterion.

The problem of combinatorial explosion explained in chapter 7.3 cannot simply be eliminated or even proven to be ‘difficult’. However, because it is divided into individual steps, it can be carried out in parallel on several computers or a cluster.

However, by using the method of calculating gears presented in this work, the other problems mentioned can be eliminated. If you use the method presented in the opposite direction, i.e. by inferring a real gear from a graph, you can also use it to generate gear concepts. If you only create planar graphs, you prevent the self-penetration and impossibility of the generated solutions. The constructive implementation is also not that difficult, as there is a clear table between elements of the graph and the true construction. The problems of coupling and variation are reduced to attempts at coupling and releasing individual wheels or shafts, which can be carried out on the abstract model until the desired degree of freedom for the entire network is achieved from the nodes. Since a complete transmission can be depicted as a graph, there is no need to limit the components used, meaning that any level of complexity can be achieved.

When generating a graph as the basis for a gear concept, a so-called suitable Chomsky grammar or a Lindenmayer system with different axioms can be used. A simple random generator can take on the problem of varying solutions. The rules of grammar implicitly guarantee a usable solution. Each solution found is then available in three forms due to its origins: as a sequence of words and character strings in the L-system, as a flat graph and finally as a geometric representation of gears connected via steps and shafts.

fig:7.2

Example of creating a new spur gear from a character string, without the intermediate graph (k: spline, z: gear, *: shaft connection, -: step connection

fig:7.3

Example of the further development of a planetary gear from a character string

table:7.1

Illustration of the development of a manual transmission, starting from a simple splined shaft using an L-system. (The 1st time step is represented by the first row, the 2nd time step by the second row, etc.) The growth of a gear from the axiom and the generation rules is shown in the 1st column. The character strings of the L-system are visualized in the 2nd column, and the graph created from them and a possible construction are visualized in the 3rd and 4th columns

The two-dimensional representation of the graph can then be assembled from the character strings of the L-system.
This can be used to calculate the degrees of freedom, speeds and torques using the method presented in this work if you assign numbers of teeth to the individual elements.
If you supplement the step relationships in the graph with axis angles, modules, etc., the three-dimensional information of a gear can be derived from this.
In this way, one can move from a simple basic structure to an increasingly concrete constructive solution, whereby those that cannot be used can be excluded or modified to meet the desired requirements.
So the end result in the last line is the basic structure for a classic manual transmission with two gears and an intermediate gear.
In order to achieve the desired kinematic degree of freedom of 1, the circle Z2, Z4, Z5, Z6 must be resolved by one of the wheels mentioned in each of the two stages becoming an idler wheel.
The character strings and regulations for converting into idler gears, for planetary gears, clutches and housings were no longer explained in order to keep the example simple and to focus on the functional principle.

The advantage of this process is that no nonsensical solutions are generated if the production instructions are formulated correctly. In addition, you can easily change concepts that have already been created, as you know the influence on the (kin./stat.) degree of freedom and the complexity of the gearbox for each regulation. The concept development and variant production are reduced to random text substitutions, so to speak.

In summary one can say:

Heuristic methods such as evolutionary strategies are suitable for monitoring and controlling the optimization process. To synthesize concepts, a so-called L-system must be chosen, which can generate a flat graph. To calculate the degree of freedom, to analyze speed and torque ratios and to generate switching logic, use the graph method described in this work.

8 Summary and outlook

This work deals with an alternative calculation method for more complex and composite planetary gears with several degrees of freedom and many input and output shafts, such as those used for hybrid systems or automatic transmissions. At the beginning of this work, the theoretical basics for calculating the most important variables of a planetary gear are explained. The methodology proposed today in the VDI guidelines for treating a more complex planetary gear is then presented using a Wolfrom theorem. Since the procedure recommended there can only be implemented with the help of more specific knowledge and also has some weaknesses when it comes to implementation in a computer system, an alternative calculation methodology will be developed in the next chapter. This enables every engineer to very easily determine the degrees of freedom, speeds and moments of all conceivable wheel combinations. It is developed and explained using a spur gear stage, a simple negative gear and a more complex Ravigneaux set. The algorithm required for the computational implementation of the method and its information theoretical foundations are explained in the following chapter. To demonstrate this procedure, the same examples as in the previous visualization chapter are used. The method is also shown for including any loss torque on a gearbox in the calculation. The calculation method based on graph theory was implemented in the FVA program WTplus, a tool for determining the efficiency and heat balance of transmissions. The calculation examples at the end of chapter 5 were also created with this.

In the last section of the work, a possible application of the method shown is presented: computer-aided concept development of gearboxes. This process makes it possible to have different transmission concepts automatically created and compared by the computer without a greater level of detail. The weaknesses of previous computer-supported methods (concentration on a few basic elements such as only minus gears or spur gear stages, use of abstract symbols without concrete constructive implementation, coverage of sub-areas such as only calculation of the speeds, etc.) can be avoided and complete concepts with power flows, spread and associated Switching logic can be determined. Two possible methods for solving the problems mentioned are proposed: Lindenmayer systems for geometric concept generation and genetic algorithms for concept selection. Lindenmayer systems are methods for generating two- or three-dimensional geometries from character strings and assignment rules. Genetic algorithms are characterized by determining an optimum of solution objects in multi-parametric solution spaces. The combination of the methods mentioned will hopefully be used in the future to make it easier to find solutions for gear concepts in industry.

A Evolutionary Algorithms

The first approaches and ideas of evolutionary algorithms (abbreviation: EA, English: evolutionary computation) already go back to work in the late 1950s and 1960s (e.g. Friedman, 1956; Box, 1957; Fraser, 1957; Friedberg, 1958; Friedberg, Dunham North, 1959; Rechenberg, 1964; Fogel, Owens Walsh, 1965, 1966; Bremermann, Rogson Salaff, 1966; Crosby, 1967; Bossert, 1967; Kaufman, 1967; Fraser, 1968; Holland, 1969).(WEI, 32).

fig:A.1

classes of evolutionary algorithms

In the early years of research in this field, the tasks were mostly of academic interest. Only in recent years have these approaches found their way into practical application in industry and business on a broad front.

In biological evolution, a process of adaptation of individual species to environmental conditions takes place. In the process, the genetic code, the hereditary substance of the individuals in a population, changes from one generation to the next. This genetic material contains all the essential information about the structure, organisation, functionality and appearance of the individual. The following factors play an important role: Mutation, recombination and selection. This biological process serves as a model for the evolutionary algorithms, and so the evolutionary cycle can also be found in the evolutionary algorithms (figure A.2). However, the focus is no longer on adaptation, but on a concrete optimisation problem. Here, the goodness of an individual can be measured concretely and is referred to as fitness. Such a fitness value is then used in the selection process to ensure that the better individuals are more likely to be adopted into the next generation or to raise them more frequently as parents. In most evolutionary algorithms, however, only one of the two selection mechanism is used to guide the search.

fig:A.2

optimizing with genetic algorithms

Genetic algorithms (Holland, 1969, 1973, 1992) in the classic sense work on a binary representation of the problem space. The selection pressure is generated via a stochastic, fitness-proportional selection of the parents from the predecessor population. The operators are a mutation operator, which inverts individual bits in the individual, and a recombination operator, which mixes the chromosomes of the parents (a so-called crossover operator). In the theory of genetic algorithms, the crossover operator is given particular importance by the schema theorem, which postulates the widespread distribution and combination of advantageous information in the population. A subtype of genetic algorithms are classifier systems (Holland & Reitman, 1978), in which rules for classifying or controlling a system are learned in the binary representation.

Evolutionary strategies (Rechenberg, 1973, 1994; Pelz, 1981) work on a real-valued representation of the problem space. The parents are chosen at random, and the selection pressure arises exclusively from environmental selection, which deterministically adopts only the best individuals. In the classic evolution strategy, only mutation was used, where additive changes are made to the real-valued elements of an individual according to a normal distribution. However, the use of recombination operators and the use of self-adaptive control of the mutation parameters also became apparent very soon. The process arose from the desire to obtain the best possible solution for constructive and engineering problems modeled on nature. A milestone is Rechenberg’s lecture at the annual meeting of the WGLR and DGRR on September 16, 1964 in the Berlin Congress Hall with the title: Cybernetic solution control of an experimental research task (REC, 33). Here he demonstrated the now famous Darwin in the wind tunnel experiment, in which a joint plate folded into a zigzag shape evolves into a flat shape with the least resistance. Figure A.4 with the development of a two-phase supersonic nozzle is given as an example of a practical application of this process.

Evolutionary programming (Fogel et al., 1966) is not limited to a specific representation, but rather works on the natural representation of the problem. Evolutionary programming was initially used for the development of finite automata for time series prediction. Characteristic features of this process are the absence of the recombination operator and a special tournament selection to produce the next generation. Independent of the evolution strategies, self-adaptive controls of the mutation parameters were also developed in evolutionary programming.

Genetic programming (Koza, 1989, 1992) works on a dynamic representation, such as tree structures, and is therefore primarily used for the evolutionary development of computer programs, etc.. In addition to these strictly separated classic forms, a large number of mixed forms have now emerged. In recent years, evolutionary algorithms have found their way into solving numerous practical problems. However, especially for industrial applications, in addition to the sobering result discussed in the previous section that the algorithm must be adapted to each problem, there are often additional requirements that make optimization more difficult. These can go far beyond the benchmark problems considered in research. Various such requirements are briefly outlined below.

Explosion in dimensions: A large increase in the search space dimensions not only leads to a significantly larger number of local optima, but the convergence speed also decreases disproportionately. This can partly be countered by splitting it into sub-problems, as long as there are no strong mutual dependencies in the search space dimensions (e.g. Potter, 1997; Valenzuela & Jones, 1994).

Boundary conditions: A large number of boundary conditions can have the effect of making the search more complicated, as it is difficult to balance the fulfillment of the boundary conditions and the optimization of the fitness function during the optimization process. A standard solution approach is penalty terms in the fitness function when boundary conditions are violated. However, this approach often only delivers inadequate results and cannot be used at all if the fitness function can no longer be calculated if the boundary conditions are violated (Michalewicz, 1995; Michalewicz & Schoenauer, 1996).

Multi-objective optimization: In industrial problems, not just one but several target attributes are usually used to evaluate a solution, which can often contradict each other. An a priori weighting of the attributes in the fitness function only leads to a sensible solution for good-natured problems, namely the compromise between the different target attributes. Other methods that calculate various alternative solutions are often only efficient for no more than two target attributes (Fonseca & Fleming, 1995, 1997; Horn, 1997).

Noisy fitness functions: Noisy target values can occur due to measurement errors or stochastic simulations. This can be countered by multiple evaluations of a solution and subsequent averaging. However, this leads to significantly higher costs and is not feasible, especially for expensive experimental evaluations. In recent years, initial strategies have been developed to minimize the necessary multiple assessments. The same problem can also arise if the input variables of a technical system have to be adjusted and can only be recorded using measuring sensors. Then a solution to the problem should be stable against small variations in the input variables (compare figure A.3) (Aizawa & Wah, 1994; Nissen & Propach, 1998; Branke, 1998).

fig:A.3

noise in the evaluation of a solution

Dynamic fitness functions: Problems can change dynamically during optimization. Finding an optimum in such an environment is much more difficult than with static problems. Likewise, moving optima must be tracked by the search algorithm and new, better optima must be discovered. Research into these problems has focused in particular on methods for maintaining diversity in populations (and thus adaptivity), reactive methods that respond to changes in the environment and procedures with memory for oscillating problems (Grefenstette 1992; Lewis, Hart & Ritchie, 1998; Branke 1999; Weicker & Weicker 1999).

Although research results are already available for all problem areas, there are still no suitable standard techniques. A sharp increase in work in these areas at specialist conferences in recent years shows the pressure with which work is being done to solve these tasks ((NIS, 34). It should not be forgotten that these methods do not represent a panacea, but are not a good method for some types of problems: If two weaknesses of an individual, if they occur at the same time in the appropriate environment, suddenly represent a strength, then a heuristic procedure would only come up with this solution by chance. However, result objects, each of which has a major weakness, would not survive the cycles during the selection process. In such problems, this method could only rely on chance, as it always combines strong objects with the strengths of other objects. As always, you should choose the right tool for the problem in question.

fig:A.4

Rechenberg, Schwefel, Bienert 1966: Development of a two-phase supersonic nozzle from initial shape 0 to optimum shape 45

B Lindenmayer Systems

For centuries, many biologists have tried to capture the beauty of flowers and trees in illustrated books, herbaria or textbooks. Mathematicians have always been fascinated by the bilateral symmetry of the leaves and the rotationally symmetrical flowers or fruits. Why are they the way they are? And how can their formation be described in simple terms? How does the self-similarity arise in plants, which Mandelbrot was also able to find in the number sets of his fractals? In 1968, Aristid Lindenmayer (1925 - 1989) succeeded in summarizing these properties of the growth of a multicellular organism in a mathematical formalism. This is now called the Lindenmayer system, or ‘L system’ for short, after him. Lindenmayer subsequently also dealt with higher plants or organs of plants. This is now an interdisciplinary area of biology, mathematics and computer graphics, which extends from the theoretical discovery of newer, more real models based on nature to the three- and four-dimensional visualization of the results. The results are used in the film industry and for pattern description in mathematics.

fig:B.1

development of a filament of Anabaena catenula

\[\omega \enspace : \enspace a_r\\ p_1 \enspace : \enspace a_r \enspace -> \enspace a_l b_r\\ p_2 \enspace : \enspace a_l \enspace -> \enspace b_l a_r\\ p_3 \enspace : \enspace b_r \enspace -> \enspace a_r\\ p_4 \enspace : \enspace b_l \enspace -> \enspace a_l\]

\(\omega\) is called an axiom and \(p_1 \enspace ... \enspace p_n\) are called production rules. If one now starts the growth with a single cell \(a_r\), the following sequence results:

\[a_r\\ a_l b_r\\ b_l a_r a_r\\ a_l a_l b_r a_l b_r\\ b_l a_r b_l a_r b_r a_r b_l a_r b_r\\ ...\]

This sequence of letters is certainly sufficient to describe the growth of one-dimensional plants. However, for more highly developed plants with branches, fruits and spreading in two or three dimensions, a different graphical representation must be found. The most common representation of two-dimensional figures is the so-called turtle graphic. Imagine a turtle whose tail has been painted with ink. As it moves forward, it leaves a mark on the sheet of paper. The character string of the L-system is used as instructions for the turtle. As an example for interpreting the character string of an L-system, you can use the following three statements:

\[F \enspace | \enspace Take a step forward while drawing a line\

+ \enspace turn to the left by a defined angle\\
- \enspace turn a defined angle to the right\\]

Graphically this is shown in picture B.2 on the left.

fig:B.2

Graphical interpretation of the characters F, + and - (left) and example of the interpretation of a string with a turtle (right)

\[\omega \enspace : \enspace F - F - F - F\\ p \enspace : \enspace F \enspace -> \enspace F - F + F + FF - F - F + F\]

fig:B.3

generating a cooking curve with the help of an L-system, partial picture a is the interpretation of the axiom, b after the first application of the production rule, c after the second, etc.

fig:B.4

tree structure with branches, created by Turtle graphics (left) and associated string with brackets from an L-system, which served as a basis (right)

All illustrations, examples and basics in this section are taken from (PRL, 35).

C Tables and worksheets according to Mueller

The following tables are taken from the textbook Mueller, ‘die Umlaufgetriebe’. They contain all important formulas for the calculation of speeds, static transmission ratios and efficiency of recirculating gears with double kinematic and single static degree of freedom, with two or three rotating shafts.

thebibliography00

LITERATURE DIRECTORYLITERATURE DIRECTORY

[ARN] Arnscheidt, Heinz-Uwe: Ergononmische Gestaltung von CAD-Programmen am Beispiel der Analyse von Umlaufraedergetrieben, Dissertation Universtitaet der Bundeswehr Hamburg, 1991

[BRA] Brandenberger, H.: Wirkungsgrad und Aufbau einfacher und zusammengesetzter Umlaufgetriebe, Masch.- Bau/ Betrieb Bd. 8(1929), S.249/253 und 290/294

[BRO] Bronstein, Semendjajew: Taschenbuch der Mathematik,B.B. Teubner Verlagsgesellschafft, Stuttgart - Leipzig und Verlag Nauka, Moskau, 1991, 25. Auflage, S.735ff

[DOM] Domian, Hans-Joerg: Systematische Synthese von Getriebestrukturen der Vorgelegebauart, Dissertation Technische Universtitaet Muenchen, 30.11.2000

[DRE] Dreher, Klaus: Rechnergestuetzte Optimierung von Planetenkoppelgetrieben, Dissertation Darmstadt Juni 1983

[GKK] Ingrid Gerdes, Frank Klawonn, Rudolf Kruse: Evolutionaere Algorithmen. genetische Algorithmen - Strategien und Optimierungsverfahren - Beispielanwendungen Wiesbaden. Vieweg 2004.

[GRUE] Grueschow, Gerhard: Optimierung von Planetengetrieben. Dissertation am Lehrstuhl fuer Maschinenelemente und Foerdertechnik der TU Braunschweig 1986.

[HAR] Harary, Frank: Graphentheorie. R. Oldenbourg Verlag M"unchen, Wien 1974.

[JON] de Jong, Kenneth A.: Evolutionary Computation. A Unified Approach. A Unified Approach. MIT Press 2006.

[KAH] Kahraman, A.: A kinematics and power flow analysis methodology for automatic transmission planetary gear trains. Journal of Mechanical Design, Vol. 126, November 2004

[KLE] Klein, Hugo: Die Planetenrad Umlaufraedergetriebe. Carl Hanser Verlag Muenechen 1962

[KUR] Kuratowski, K.: Sur le probleme des courbes gauches en topologie. Fund. Math. 15, 1930, S.271

[KZB] Kutzbach, K.: Mehrgliedrige Radgetriebe und ihre Gesetze. Maschinenbau 6 (1927) H. 22, S. 1080/1083

[MUE] Herbert W. Mueller: Die Umlaufgetriebe, Springer Verlag, Berlin, Heidelberg, New York, 2001

[NIS] Nissen, Volker: Einfuehrung in evolutionaere Algorithmen. Optimierung nach dem Vorbild der Evolution Braunschweig. Vieweg 1997.

[POH] Pohlheim, Hartmut: Evolutionaere Algorithmen. Verfahren, Operatoren und Hinweise fuer die Praxis Berlin. Springer 1999.

[NOL] Noltemeier, Hartmut: Graphentheorie mit Algorithmen und Anwendungen. Walter de Gruyter Verlag, Berlin, New York 1976

[POPP] Poppinga, Reemt: Stirnrad Planetengetriebe. Habilitation Berlin-Charlottenburg 1942, Frankh’sche Verlagshandlung, Abt. Technik, Stuttgart 1949

[PRL] Przemyslaw Prusinkiewicz, Aristid Lindenmayer: The algorithmic beauty of plants. Springer Verlag, New York 1990, reprinted 1996.

[REC] Rechenberg, Ingo: Cybernetic solution path of an experimental problem. Royal Aircraft Establishment, Library Translation 1122, Farnborough, 1965.

[REU] Reuleaux, Franz.: theoretische Kinematik. Vieweg Sohn, Braunschweig 1875

[SCHTZ1] Schnetz, Klaus: Optimierung zusammengesetzter Planetengetriebe. Fortschrittsberichte der VDI Zeitschriften, Reihe 1, Nr. 30, VDI-Verlag GmbH Duesseldorf 1971

[SCHTZ2] Schnetz, Klaus: Reduzierte-Planeten Koppelgetriebe, eine Wirkungsgrad und Durchmesseroptimierung. Dissertation, Darmstadt Juni 1976

[SToe] Stoelzle, Karl: Leistungsuebertragung in Planetengetrieben bei statischem und dynamischen Betrieb; Berechnung, Optimierung, Versuchsergebnisse, Dissertation Muenchen Juni 1969

[STR] Strauch, Helmar.: Theorie und Praxis der Planetengetriebe. Vieweg 1970, S. 15

[VDI1460] Prof. Dr. Ing. J. Loomann, VDI-Berichte: Berechnung reduzierter Planetengetriebe, Verein Deutscher Ingenieure, VDI 1460, 1999

[VDI2157] Gemeinschaftsarbeit der VDI-Gesellschaft Konstruktion und Entwicklung (VDI-GKE): Planetengetriebe. Begriffe, Symbole, Berechnungsgrundlagen. Verein Deutscher Ingenieure, VDI 2157, 1978

[VDI672] Prof. Dr. Ing. J. Loomann, Tagung Bad Soden, 16./17. Maerz: Planetengetriebe. Eine Leistungsfaehge Komponente der Antriebstechnik. Verein Deutscher Ingenieure, VDI 672, 1988

[WEI] Weicker, Karsten: Evolutionaere Algorithmen. Stuttgart, Teubner 2002

[WOL] Wolf, Albrecht: Die Grundgesetze der Umlaufgetriebe. Vieweg 1954

[XIY] Li, Xingyuang: Systematische Synthese mehrgaengiger Planetengetriebe. Dissertation der Rheinisch Westfaelischen Technischen Hochschule Aachen, November 1993

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