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KISSsys 03/2018 – Selected Topic

Bevel gear differential coaxial

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Contents

1 Introduction ............................................................................................................................. 3

1.1 Structure of the model ............................................................................................................................. 3 1.2 Modeling hints ......................................................................................................................................... 3

2 Modeling, tree structure .......................................................................................................... 4

2.1 Groups .................................................................................................................................................... 4 2.2 Shafts ...................................................................................................................................................... 5 2.3 Other elements ........................................................................................................................................ 6 2.4 Gear meshes ......................................................................................................................................... 10 2.5 Adding the KISSsoft calculations .......................................................................................................... 11

3 Modeling, shafts and gears ................................................................................................... 12

3.1 Add input and output ............................................................................................................................. 12 3.2 Adding gear data ................................................................................................................................... 13 3.3 Modeling the shafts and bearings ......................................................................................................... 14 3.4 Positioning ............................................................................................................................................. 16

4 Kinematic calculation ............................................................................................................ 20

4.1 Input load .............................................................................................................................................. 20 4.2 Speed condition .................................................................................................................................... 21 4.3 Different speeds on the two shafts ........................................................................................................ 23

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1 Introduction

1.1 Structure of the model

Bevel gear differentials are commonly used in axles of vehicles. Below the schematic of such an arrangement is

shown.

Power is introduced into the model through the coupling cin, driving a bevel pinion shaft sin which is supported by

two bearings bin1 and bin2. The bevel pinion z1 is in mesh with the bevel gear z2 which is situated on the bevel

differential housing sdiff. The differential housing is supported on the two output shafts sl and sr by means of two

bearings bcl and bcr. The two output shafts sl and sr are again supported by two bearings each (bl1 and bl2 for sl

and br1 and br2 for sr). These bearings are mounted to the ground on their outer ring / the outer ring is not

rotating.

On the differential housing sdiff, we will add a planetary coupling cpc which is rotating the bevel planet shaft sp in

space. On the bevel planet shaft, the bevel planet zp is located. The planet zp is in mesh with the “sun” zl and the

“ring gear” zr. The planet gears zl and zr are having an inner diameter with a spline and are mounted on the

output shafts sl and sr by means of a spline connection. To model this spline connection, we will use two general

type connections per side, uzl1 and uzl2 for left side and uzr1 and uzr2 for right side.

Power output is on the left and right shaft output coupling cl and cr. We will define a condition that the speed of

one shaft is a function of the speed of the other shaft. If the vehicle is driving forward in a straight line, then, the

condition would be that the speed of sl is equal to the speed of sr. If the vehicle is moving through a curve, then,

the condition is that the speed of sl is equal a factor times the speed of sr.

For the torque, we need not give a condition, the output torque on the left side through cl and on the right side

through cr is given from the input torque and the nature of the differential where left and right side torque is equal.

1.2 Modeling hints

We will use two groups, a group All including the whole model and a group Out including all the coaxial shafts.

The shaft sin is not a coaxial shaft. Also, the shaft sp is not a coaxial shafts.

All other shafts (sl, sdiff, szl, szr, sr) are coaxial shafts.

The bevel gears zl and zr will be modeled from two parts: the hollow shafts szl and szr representing the gear

body and the gear elements zl and zr representing the gear teeth.

Note that when modeling a bevel differential, the kinematic calculation will only work AFTER all shafts

are positioned in space and AFTER shaft geometry is modeled!

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Figure 1. Structure of the model

2 Modeling, tree structure

2.1 Groups

Start KISSsys, go to administrator mode. Open a project by selecting a folder where you want to save your

model. Then, add the group All to the root of the model and inside the group All, add the group Out

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Figure 2. Starting the model

2.2 Shafts

Add the coaxial shafts sl, sdiff, szl, szr, sr to the group Out.

Figure 3. Adding the coaxial shafts to group Out

Now add the input shaft sin to the group All

Figure 4. Adding a shaft sin to the group All

Finally, we add the planet bevel shaft sp. The shaft sp should rotate in space with the speed of the differential

housing sdiff. Therefore, we locate the shaft sp underneath the shaft sdiff. Remember that this is a single shaft

which should be added to the differential shaft “sdiff”. That it is shown as a coaxial shaft is just a graphical issue.

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Figure 5. Adding the planet bevel shaft sp underneath sdiff

2.3 Other elements

We now add all the bevel gears as shown below.

Figure 6. Adding the bevel gears to the shafts

Now, we add the input and output couplings.

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Figure 7. Adding input and output coupling

Now, we add the bearings (those where the outer ring is not rotating):

Figure 8. Adding the bearings to the output shafts and the input shaft

Now we add the connecting bearings bcl and bcr that connect the differential housing sdiff to the two output

shafts sl and sr. These bearings are not placed underneath a shaft in the tree structure but directly in the group

Out as they are connecting two shafts in the same group.

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Figure 9. Adding connecting bearing. Select inner and outer race reference shaft.

Figure 10. Resulting tree structure in the model

Now we connect the shafts szl and szr (they represent the gear body) to the shaft sl and sr by means of the

general supports uzl1, uzl2 and uzr1, uzr2. Add the connections directly into the group Out as they connect two

shafts inside this group. In the dialog, define the inner and outer shaft that belong to this connection. Also, define

that the rotation around the y axis is fixed (the two shafts are rotating with the same speed) for one of the

connections. For the second (per side), define that it is free, otherwise, the system is overconstrained. Do this for

all four connections correspondingly.

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Figure 11. Adding general connections to connect the bevel gears to the output shafts

Now, we add the support usp to the shaft sp (this is needed to run the shaft calculation)

Figure 12. Adding a support to the shaft sp

And finally, we add a planetary coupling to the shaft sdiff:

Figure 13. Adding the planetary coupling to the shaft sdiff

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With this, we have now added all the elements to the model.

See file “202-Bevel-gear-differential-coaxial-Step-1.ks”

2.4 Gear meshes

We now define the gear mesh between the bevel pinion and the bevel gear. Add the gear connection in the

group All.

Figure 14. Adding gear mesh z1z2

Now, add the bevel planetary gear meshes zpzl and zpzr. Add these in the group Out. Note that the configuration

gear/planet means that Gear 1 is the output gear (e.g. zl) and Gear 2 is the planet (zp).

Figure 15. Gear meshes definition for zpzl and zpzr

Figure 16. Resulting model

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We have now defined all kinematic constraints.

See file “202-Bevel-gear-differential-coaxial-Step-2.ks”

2.5 Adding the KISSsoft calculations

Add the bevel gear calculations to all bevel meshes:

Figure 17. Adding the bevel gear calculations

Now, add a coaxial shaft calculation to the group Out and a normal shaft calculation to the shafts sin and sp

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Figure 18. Adding shaft calculations

With this, we have added all necessary calculations.

See file “202-Bevel-gear-differential-coaxial-Step-3.ks”

3 Modeling, shafts and gears

3.1 Add input and output

Add the input and output elements to the root directory and reference them to the corresponding couplings:

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Figure 19. Input and ouptut elements

3.2 Adding gear data

Add gear data for the gear mesh z1z2 as follows (double click on z1z2_calc, enter the data and close the window

again)

Figure 20. Adding gear data for mesh z1z2

Now, add gear data for zpzl and zpzr.

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Figure 21. Adding gear data for the two meshes zpzl and zpzr

3.3 Modeling the shafts and bearings

Now, model the shafts and the bearings, e.g. as shown below.

Figure 22. Model of input shaft

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Figure 23. Bevel planet shaft. Note that the support is constrained in all degrees of freedom except for the rotation around y

axis (so that the shaft can rotate).

Note that depending on in which order the shafts are added to the model tree, it can be necessary to change the position, inner and

outer geometry dimensions of each shaft in the calculation “Out_calc”. It is recommended to do this step shaft by shaft in such kind of

coaxial shaft calculations. Below is an example of a possible configuration:

Figure 24. Output shaft

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Figure 25. Modelling of connections between output shafts sl and sr and gear shafts szl and szr.

See file “202-Bevel-gear-differential-coaxial-Step-4.ks”

3.4 Positioning

The Groups “All” and “Out” should be our reference groups and they should be positioned with the “Own Input”

option at the origin of the system:

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Figure 26. Positioning of the groups “All” and “Out” which are the reference groups regarding positioning in the system

The input shaft can then be positioned with respect to the output group:

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Figure 27. Positioning of input shaft sin with respect to output shafts using z2.

And let us position shaft sp:

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Figure 28. Positioning of sp with respect to output shaft using gear zl

Now refresh the 3D graphics and open it with “show” if it is not opened already:

Figure 29. 3D graphics

Use the 3D settings to change the graphics:

Figure 30. Settings for 3D viewer

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Figure 31. Cut view

We do some fine-tuning

- Add inner diameter on the bevel gear z2

- Increase shaft size of sp

Figure 32. Model after some tuning

See file “202-Bevel-gear-differential-coaxial-Step-5.ks”

4 Kinematic calculation

4.1 Input load

For the input, we have defined the following load condition:

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Figure 33. Load condition on input

4.2 Speed condition

We now need to add a condition that the one of the output shafts is rotating with a speed that is equal to the

speed of the differential housing sdiff, times a factor. Note that it would be also possible to make the speed on

one output side dependent on the other output side. In this instruction, it is shown how to make one output speed

dependent on the bevel differential shaft speed.

Since in this case the entered speed depends on another calculated speed in the system, the iterative kinematic

calculation must be activated from the settings:

Figure 34. Settings for kinematic calculation

First, we assume the factor is 1, so, the differential housing speed is equal to the left output shaft speed which

again is equal to the right output shaft speed.

Let us define that the speed of shaft sl is equal to the speed of the differential housing.

First, we define for the outl (right mouse click on outl and select dialog)

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Figure 35. We define that the output (left) speed is constrained

While the right side is not constrained

Figure 36. Right side is not constrained

Now, we go to the properties of outl and ad an equation as follows:

Figure 37. Defining the left side speed as a function of the differential shaft speed

We can now run the kinematic calculation – which will take some time as some iteration takes place – to find:

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Figure 38. Resulting speeds on the two outputs.

See file “202-Bevel-gear-differential-coaxial-Step-6.ks”

4.3 Different speeds on the two shafts

We may also introduce a factor, let us call it “k”, to make the speeds different on each output shaft.

First, add a new variable “k” e.g. in the group Out. Let us use a value of k=0.5 to define that the output shaft sl

has a speed of 0.5 times the speed of the differential housing sdiff:

Figure 39. Adding a variable k of type real and value 0.5 to the group Out

Now we change the formula for the calculation of the left output speed:

Figure 40. Multiplying the speed of the left side shaft by factor k

If we now re-calculated the kinematics, we find the following speeds:

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Figure 41. Speeds on the two outputs.

Note that the bevel differential shaft sdiff speed is 266.6666RpM (use right mouse click on sdiff and select

properties to find the below informaiton)

Figure 42. Speed of shaft sdiff

See file “202-Bevel-gear-differential-coaxial-Step-7.ks”