# Driveline Stiffness as a function of Gear Ratio

1. Sep 10, 2013

### b7labelle

Hi Everyone,

Obviously first time poster, very encouraged with the discovery of this forum.
And yes, I have used the search function and Google a lot today. :-)

I work at a transmission company, and in discussions with other more experienced engineers this statement was mentioned: "driveline stiffness is a function of gear ratio squared."

Can anyone point me in the direction to better understand this statement? I'm sure I will have one of those "ah-ha" moments...

Thanks!

Mods:
I posted this in the Mechanical Engineering forum, but decided this may be more relevant in the AE forum. I do not believe users have a "delete thread" option..sorry in advanced.

2. Sep 18, 2013

### sgb27

Consider two shafts (1 and 2) are joined with gears of radius R1 and R2, giving a gear ratio R1/R2. If we hold one free end fixed and twist the other we should be able to calculate the stiffness in terms of the stiffness of the shafts and the gear ratio.

The torque in each shaft (T1 and T2) is not the same (the difference will be provided by the gearbox mounting), however the force, F, between the gear teeth is. The relevant equations relating the torques, stiffnesses and forces are:

T1 = F . R1
T2 = F . R2

A1 = F.R1/K1
A2 = F.R2/K2

(K1 and K2 are the stiffnesses of the shafts)

Next to find the "twist" in the first shaft, A1. If the free end is turned by A1f radians and the end with the gear by A1g radians, then:

A1 = A1f - A1g

Similarly for A2, but we fix the free end (so A2f=0):

A2 = 0 - A2g

However A1g and A2g are related geometrically by the gear ratio:

A1g . R1 = - A2g . R2

And we want to know how far the free end (A1f) rotates in terms of the applied torque (T1):

A1f = A1 + A1g
= A1 - A2g . R2/R1
= A1 + A2 . R2/R1
= F.R1/K1 + F.R2/K2 . R2/R1
= F(R1/K1 + R2/K2 . R2/R1)
= T1/R1 ( R1/K1 + R2/K2 . R2/R1)
= T1 ( 1/K1 + 1/K2 . R2^2 / R1^2)

Thus the overall stiffness, T1/A1f, in this case is:

T1/A1f = 1/( 1/K1 + 1/K2 . (R2/R1)^2 )

So the stiffness of the shaft (and anything else) "on the other side" of the gearbox affects the overall stiffness by the gear ratio squared.