Shaft 3

Plot 7.6-1: Details of assembled Shaft 3.

Shaft 3 represents an assembly of components with the function of linking second stage of reduction with half-shaft. By virtue of that, shaft 3 needs to integrate gearwheel 4 and “tripod joint housing”.

Tripod is a kind of constant velocity joint which is mainly based on the sliding of three spherical rollers into their housing, how depicted by Plot 7.6-2. It is usually employed where It’s necessary to compensate small angles caused relative movement between chassis and un-sprung weights. Despite its relatively low weight, It’s not suitable for steering applications.

How It’s possible to observe by Plot 7.6-1, “gearwheel 4” is realized by a thin steel toothed ring provided of eight threaded holes. Steel component is fastened to an Al 7075 aluminium core, “shaft 3 hub”, which bears the gear ensuring an important save in weight and momentum of inertia. Housing of

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the tripod joint, is integrated into a slot centred inside the hub, realizing a solution very similar to those displayed by Picture 1.5-1, Picture 1.5-3 and Picture 1.5-6. That leads to a reduction of axial dimensions with an important simplification of layout, which reduces overall number of components and deletes issues related to assembly. Main drawback of this solution are substantially two, one is the employ of large bearings which aren’t the best solution in order to reduce weight and inertia. Other is the employ of large dimension “rotary seals” which aren’t the best solution by point of view of efficiency, due to high circumference of friction.

Other task which shaft 3 needs to fulfil is integration of rear “brake disc”. While brake caliper is fastened rigidly to gear-box structure, floating element needs to be the disc. In order to achieve floating effect, disc is fixed to a lightweight aluminium flange known with name of “brake bell”. Linkage between disc and bell is realized by six special bushings which guarantee a floating constraint due to assembly tolerances which realizes backlash. Another important benefit of floating linkage between brake disk and bell is that axial loads due to translation of brake pads, aren’t transmitted to associated gear-box components. By virtue of that, axial loads on shaft 3 can be neglected.

Plot 7.6-2: Details of shaft 3 inside gear-box case.

How It’s highlighted by yellow circle of Plot 7.6-2, precise coupling between brake bell and shaft 3 hub is ensured by a centring diameter machined with tight tolerances on both components. While fastening is ensured by three “I.S.O. 4762 Screw M6x45”, as depicted by Plot 7.6-1. Nuts located on the on the internal side of the transmission ensure a quick and easy replacement of the disc/bell assembly.

Analogously to previous cases, a couple of angular contact bearings have been chosen to sustain shaft 3, even if axial loads are neglected. That’s why a couple a large dimension radial bearings features a large axial backlash which may compromise operation and feeling on rear brakes.

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Due to large dimensions of bearings, the only one solution to reduce weight and inertia is employ of a couple of high precision hybrid bearings, Ref.[15]. Such bearings combine high precision steel raceways with ceramic spheres, which are considerably lighter than steel ones. About bearings chosen for shaft 3, each bearing allows a weight save of about 15% compared to steel conventional solution. Anyway, most important benefit regards inertia of spinning elements which is reduced of about 30%. On the other hand, ceramic compared with steel, features lower friction when is contact with steel. That result is very important to reduce loss of efficiency and overcome issues deriving by poor lubrication. Poor lubrication is a scenario which may more likely happen, due to position of bearings in relation with other components of the transmission.

How It’s possible to notice by Plot 7.6-2, couple of shaft 3 bearings is assembled in “X” configuration, even if “O” configuration would be applicable too. By virtue of that, external raceway of “E bearing” is in contact with “internal gear-box case” while internal is in contact with “shaft 3 hub”. Internal raceway of “F bearing” is in contact with shaft 3 hub, while external is in contact with “shaft 3 preload cap”. Both bearings are coupled with hub by interference on the internal raceway. How It was explained by Chapter 7.5, “X” configuration allows an easier assembly compared to “O” configuration.

Moreover preload of bearings couple can be easily tuned by dimension of “preload control thickness”, displayed by Plot 7.6-2.

How it was performed in previous chapters It’s necessary to study constraint reaction forces acting on shaft 3 hub. By virtue of that, free-body diagrams of shaft 3 have been set up on the base of three different conditions: driving, energy recovery braking. Forces related to each condition are widely described at Chapter 7.1 Chapter 7.2 and Chapter 7.3, while geometric parameters of shaft 3 needs to be presented as follows:

 is “distance of points K-F”.

 is “distance of points K-E”.

 is “distance of points E-L”.

 is “distance of points L-F”.

How seen in previous Chapter 7.4 and Chapter 7.5, by diagrams Plot 7.6-3 and Plot 7.6-4, It’s easy calculate reaction forces acting on constraints during driving condition:

Eq. 7.6-1

Eq. 7.6-2

Eq. 7.6-3

Eq. 7.6-4

Positive sign of vectors according to signs displayed by Plot 7.6-3 and Plot 7.6-4.

By observation of magnitude of reaction forces, It’s clear that driving condition stresses “F” bearing in particular . Solicitations on “E” bearing represent less than 10% in magnitude respect to “F”.

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Plot 7.6-3: Free body diagram of shaft 3 in driving condition, vectors lay on XY plan.

Plot 7.6-4: Free body diagram of shaft 3 in driving condition, vectors lay on YZ plan.

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Plot 7.6-5: Free body diagram of shaft 3 in energy recovery condition, vectors lay on XY plan.

Plot 7.6-6 Free body diagram of shaft 3 in energy recovery condition, vectors lay on YZ plan.

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Plot 7.6-7: Free body diagram of shaft 3 in braking condition, vectors lay on XY plan.

By free-body diagrams of Plot 7.6-5 and Plot 7.6-6, It’s possible to calculate reaction forces acting on constraints during energy recovery condition:

Eq. 7.6-5

Eq. 7.6-6

Eq. 7.6-7

Eq. 7.6-8

Positive sign of vectors according to signs displayed by Plot 7.6-5 and Plot 7.6-6.

By brief analysis of magnitude of reaction forces acting during energy recovery condition, It’s clear that most stressed bearing is “F”, in this case too. In particular, highest magnitude of force shifts from x to z direction, that’s due change in sign of tangential component of gear meshing force. Anyway, magnitude of loads acting on bearings is sensibly lower during energy recovery condition.

By free-body diagram of Plot 7.6-7, It’s possible to calculate reaction forces acting on constraints during braking condition:

Eq. 7.6-9

Eq. 7.6-10

Positive sign of vectors according to signs displayed by Plot 7.6-7.

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Differently from previous cases, by brief analysis of reaction forces acting during braking condition, It’s possible to notice that forces act on xy plan only. By analysis of magnitude, It’s clear that most stressed bearing is “E”.

By virtue of elements shown in previous dissertation, It’s possible to state that worst condition for “E”

bearing is braking, while worst condition for “F” bearing is driving. This fact is fundamental to perform a more accurate verification on safety and duration of bearings.

Due to complex geometry of shaft 3 hub, It’s not convenient set up an empirical structural model like those analyzed by Chapter 7.4 and Chapter 7.5. Cases of driving, energy recovery and braking conditions are going to be analyzed through a more complex F.E.M. model. Most important result of structural simulations is the crown of axial holes which is possible to observe by Plot 7.6-1. Like It was declared in advance, axial holes are a simple and cheap solution to reduce total weight and inertia of a mechanical component. All portions of shaft 3 hub, work in good safety condition, even if tripod housing zone exploits a spike of stress due to roller contact, like It’s displayed by Picture 7.6-1. It’s known that aluminium from which shaft 3 hub is machined doesn’t bear so huge stresses like steel, even if an hard oxidation coating is performed on the component.

Picture 7.6-1: F.E.M. model of shaft 3 hub in driving condition.

By virtue of that, solution was a quenched “steel insert” housed into shaft 3 hub, displayed in yellow by Plot 7.6-1.

At this level of detail, dissertation about shaft can be considered concluded. It’s necessary study bearings and, if verifications are satisfying, design of shafts can be considered frozen.

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