kisssys 03/2015 selected topic - calculation … / 20 1. introduction 1.1 structure of the model...

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KISSsys 03/2015 – Selected Topic

Bevel gear differential coaxial

01/04/2015

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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 .................................................................................................................................. 8 2.5 Adding the KISSsoft calculations .................................................................................................... 9

3. Modeling, shafts and gears .................................................................................................................... 10 3.1 Add input and output .................................................................................................................... 10 3.2 Adding gear data .......................................................................................................................... 11 3.3 Modeling the shafts and bearings ................................................................................................. 12 3.4 Positioning .................................................................................................................................... 15

4. Kinematic calculation ............................................................................................................................. 17 4.1 Input load ...................................................................................................................................... 17 4.2 Speed condition ............................................................................................................................ 17 4.3 Different speeds on the two shafts ............................................................................................... 19

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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 1.2-1 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 2.1-1 Starting the model

2.2 Shafts

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

Figure 2.2-1 Adding the coaxial shafts to group Out

Now add the input shaft sin to the group All

Figure 2.2-2 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.

Figure 2.2-3 Adding the planet bevel shaft sp underneath sdiff

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2.3 Other elements

We now add all the bevel gears as shown below.

Figure 2.3-1 Adding the bevel gears to the shafts

Now, we add the input and output couplings.

Figure 2.3-2 Adding input and output coupling

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

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Figure 2.3-3 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.

Figure 2.3-4 Adding connecting bearing. Select inner and outer race reference shaft.

Figure 2.3-5 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

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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.

Figure 2.3-6 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 2.3-7 Adding a support to the shaft sp

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

Figure 2.3-8 Adding the planetary coupling to the shaft sdiff

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

See file KISSsys-ANL-14-910-Bevel-Diff-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 2.4-1 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).

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Figure 2.4-2 Gear meshes definition for zpzl and zpzr

Figure 2.4-3 Resulting model

We have now defined all kinematic constraints.


2.5 Adding the KISSsoft calculations

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 2.5-1 Adding shaft calculations

Now, add the bevel gear calculations to all bevel meshes:

Figure 2.5-2 Adding the bevel gear calculations

With this, we have added all necessary calculations.


3. Modeling, shafts and gears

3.1 Add input and output

Add the input and output elements and connect them to the corresponding couplings:

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Figure 3.1-1 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 3.2-1 Adding gear data for mesh z1z2

Now, add gear data for zpzl and zpzr.

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Figure 3.2-2 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.

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Figure 3.3-1 Model of input shaft

Figure 3.3-2 Bevel planet shaft. Note that the support is constrained in all degrees of freedom

excep for the rotation around y axis (so that the shaft can rotate).

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Figure 3.3-3 Output shaft

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Figure 3.3-4 Modelling of connections between output shafts sl and sr and gear shafts szl and

szr.


3.4 Positioning

Let us position the input shaft sin with respect to the output shaft:

Figure 3.4-1 Positioning of input shaft sin with respect to output shafts using z2.

And let us position shaft sp:

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Figure 3.4-2 Positioning of sp with respect to output shaft using gear zl

Now add the 3D graphics to the model and press “Refresh”

Figure 3.4-3 3D graphics

Use the settings to change the graphics:

Figure 3.4-4 Cut view

We do some fine-tuning

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- Add inner diameter on the bevel gear z2

- Increase shaft size of sp

Figure 3.4-5 Model after some tuning


4. Kinematic calculation

4.1 Input load

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

Figure 4.1-1 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.

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 4.2-1 We define that the output (left) speed is constrained

While the right side is not constrained

Figure 4.2-2 Right side is not constrained

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

Figure 4.2-3 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:

Figure 4.2-4 Resulting speeds on the two outputs.

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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 4.3-1 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 4.3-2 Multiplying the speed of the left side shaft by factor k

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

Figure 4.3-3 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)

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Figure 4.3-4 Speed of shaft sdiff


kisssys 03/2015 selected topic - calculation … / 20 1. introduction 1.1 structure of the model...

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