Forces acting on shafts, driving case

Another important output of calculation performed by Kisssoft software is represented by system of forces exchanged by gearwheels during operation. Actual Chapter 7.1 analyzes system of forces generated by gears during driving condition, that occurs when motors supply power.

Such system of forces is derives by application of , which is the maximum torque which is possible discharge to the ground, as widely explained at Chapter 4.1. In order to proceed in a organized way, It’s necessary to study each shaft by a free-body plot. By virtue of that, It’s necessary proceed studying forces inducted by gear meshing on shaft 1.

Plot 7.1-1: Gear meshing forces on Shaft 1, driving condition.

Analyzing Plot 7.1-1, It’s possible to notice two different views of the shaft that are useful to display all meshing forces acting on drive wheel of first stage of reduction. Like It was declared in advance at Chapter 4.2.3, meshing of bevel gears generates a force which needs to be fractionate on 3 components: tangential, radial and axial.

 is “tangential force of first gear in driving condition” which driven wheel 2 transmits to drive wheel 1. It’s defined by relation Eq. 5.4-2.

 is “radial force of first gear in driving condition” which driven wheel 2 transmits to drive wheel 1.

 is “axial force of first gear in driving condition” which driven wheel 2 transmits to drive wheel 1.

Basically, previous forces represent components of reaction force which driven wheel 2 opposes to drive wheel 1 during gear operation.

Eq. 7.1-1

Eq. 7.1-2

Eq. 7.1-3

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The same way to operate can be applied to shaft 2, which is depicted by Plot 7.1-2. By exam of the plot, first aspect to take into account is that shaft 2 integrates two gearwheels, for this reason system of forces is more complex than previous case. As declared in advance, every bevel gearwheel is subject to a force that needs to be fractionate on 3 vectors.

Plot 7.1-2: Gear meshing forces on Shaft 2, driving condition.

In the specific, vectors need to be opposite in sign respect to vectors displayed on Plot 7.1-1. Other gearwheel acting on shaft 2 is spur, for this reason meshing force needs be fractionate on 2 only vectors, tangential and radial. It’s necessary to notice that offset of second stage of reduction features a diagonal position referred to car axis system. Plot 7.1-2 display meshing force of second stage of reduction, fractioned along gear offset and perpendicular direction. Therefore, intermediate shaft is subject to 5 gear meshing vectors in total:

 is “tangential force of first gear in driving condition” which drive wheel 1 transmits to driven wheel 2.

 is “radial force of first gear in driving condition” which drive wheel 1 transmits to driven wheel 2.

 is “axial force of first gear in driving condition” which drive wheel 1 transmits to driven wheel 2.

 is “tangential force of second gear in driving condition” which driven wheel 4 transmits to drive wheel 3. It’s defined by relation Eq. 5.4-2.

 is “radial force of second gear in driving condition” which driven wheel 4 transmits to drive wheel 3.

Basically forces acting on bevel gearwheel represent forces which propel intermediate shaft, while forces acting on spur gearwheel represent reaction which driven wheel 4 opposes to drive wheel 3.

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Eq. 7.1-4

Eq. 7.1-5

Eq. 7.1-6

Eq. 7.1-7

Eq. 7.1-8

Observing Plot 7.1-2, It’s possible to notice that reaction forces on gearwheel 3 aren’t parallel to conventional axis. For this reason, It’s clear that gear meshing forces acting on gearwheel 3 need to be decomposed along, X and Z axis in order to be parallel to corresponding forces applied on gearwheel 2.

By virtue of that, first step is composition of , displayed by Eq. 7.1-7, and , displayed by Eq.

7.1-8, along line of action of second stage gear.

Plot 7.1-3: Composition of forces along line of action, driving condition.

Result of composition is displayed by Plot 7.1-3, which is the “total force of second gear in driving condition”. Its module can be calculated thanks to knowledge of following geometric parameter:

 is “normal pressure angle of second stage” defined at 132Chapter 6.7.

Calculation of module can be performed:

Eq. 7.1-9

Once and are composed in vector, It’s necessary doing the opposite operation, decomposing along X and Z axis. Result of such operation is represented by:

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 is “component along x of total force of second gear in driving condition”. It’s measured in [N] and It’s displayed by Plot 7.1-4.

 is “component along z of total force of second gear in driving condition”. It’s measured in [N] and It’s displayed by Plot 7.1-4.

Observation of Plot 7.1-4 indicates sense of previous vectors, sign is ruled by convention depicted in green. In order to calculate modulus of and , It’s necessary to present another geometric parameter:

 is “inclination of second stage” respect to XY plan.

Plot 7.1-4: Decomposition of gear meshing force along XY axis, driving condition.

Calculations can proceed as follows:

Eq. 7.1-10

Eq. 7.1-11

Eq. 7.1-12

By an accurate analysis of Plot 7.1-4, It’s possible notice other two elements useful for free body diagram set up, levers of forces acting on gearwheel 3:

 is “lever parallel to x”.

 is “lever parallel to z”.

Finally, gear meshing forces acting on shaft 3 can be analyzed. One gearwheel only is connected to shaft 3, the cylindrical spur gearwheel 4. By virtue of that, gear-train acts on the shaft with only two components, tangential and radial:

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 is “tangential force of second gear in driving condition” which drive wheel 3 transmits to driven wheel 4.

 is “radial force of second gear in driving condition” which drive wheel 3 transmits to driven wheel 4.

Such components derive by the fraction of the force which propels the output shaft.

Eq. 7.1-13

Eq. 7.1-14

Obviously, propelling forces features contrary signs respect to opposing forces.

Plot 7.1-5: Gear meshing forces on Shaft 3, driving condition.

As depicted in the previous case, component of forces parallel to X and Z axis are needed to perform calculations.

Plot 7.1-6: Gear meshing forces on Shaft 3, along X and Z axis, driving condition.

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By analysis of Plot 7.1-6, It’s necessary to observe that displayed vectors are equal, to vectors displayed by Plot 7.1-4 by point of view of module and direction, signs are opposite. By virtue of that:

 is “component along x of force transmitted by gearwheel 3 to 4 in driving condition”. It’s measured in [N].

 is “component along z of force transmitted by gearwheel 3 to 4 in driving condition”. It’s measured in [N].

Eq. 7.1-15

Eq. 7.1-16

Anyway, as declared in advance, shaft 3 integrates rear brake disk too. For this reason, It’s necessary to take into account effect of braking events on the entire system.