Belt C-C distance to Tension

[RodrigoBlakenship]

TL;DR

Is there a known equation relating tension to C-C distance?

Hi all,
I've looked through design manuals (SDP/SI, Gates, Naismith) and textbooks like Shigley's, but haven't found an equation linking initial belt tension to pulley center-to-center distance for a timing belt. I understand initial tension is typically based on the applied load, but I'm curious:
A) Is there a known equation relating tension to C-C distance?
B) If not, how might one go about deriving it?

Thanks in advance!

 

[Baluncore]

A) Is there a known equation relating tension to C-C distance?
B) If not, how might one go about deriving it?

In theory, given the C-C distance, the zero length of the belt path can be computed. Knowing the initial belt length, Hook's law, and also the spring constant, allows tension to be computed, in proportion to extension.

In practice, the tension will be different in different parts of the belt, and there will be dynamic harmonics present. The belt will be a V-belt, and the sectional width of the V will be reduced by tension, so that it will drop and wedge deeper into the V of the pulley, increasing grip, while changing the computed zero tension length. That will rapidly become too complex to compute.

In real applications, a mechanical tensioner is employed with belts to maintain a specified tension in the belt, independent of variations in belt path length, wear, or aging of the belt.

 

[Lnewqban]

TL;DR: Is there a known equation relating tension to C-C distance?
A) Is there a known equation relating tension to C-C distance?
B) If not, how might one go about deriving it?

Thanks in advance!

Welcome!
What makes you believe that a relation exists between those two parameters?
The C-C distance is determined by the internal architecture of the engine (locations of crank and cam shafts).

 

[RodrigoBlakenship]

In theory, given the C-C distance, the zero length of the belt path can be computed. Knowing the initial belt length, Hook's law, and also the spring constant, allows tension to be computed, in proportion to extension.

In practice, the tension will be different in different parts of the belt, and there will be dynamic harmonics present. The belt will be a V-belt, and the sectional width of the V will be reduced by tension, so that it will drop and wedge deeper into the V of the pulley, increasing grip, while changing the computed zero tension length. That will rapidly become too complex to compute.

In real applications, a mechanical tensioner is employed with belts to maintain a specified tension in the belt, independent of variations in belt path length, wear, or aging of the belt.

Dear Baluncore,

I am trying your approach involving Hooke's law and I was wondering if I could please have some feedback on my method? The belt is composed of several fibreglass cords running along the length of the belt covered by neoprene. I am calculating the Young's modulus of the belt using EC = EF VF + EM VM , where VF and VM are the volume fractions of fibers and matrix (neoprene) respectively and EC is the Young's modulus of the belt.
I took this method from a free textbook on composites (https://www.princeton.edu/~maelabs/hpt/materials/composites.htm). Images of the internal structure of the belt can be found here: https://www.gates.com/in/en/power-t...r-synchronous-belts.p.9356-000000-000000.html

Thank you for explaining about the change in belt tension due to the wedging of the V of the pulley and the effect of harmonics. I will adapt that knowledge to timing belts and write that as a disclaimer/limitation of my theoretical model.

 

[RodrigoBlakenship]

Welcome!
What makes you believe that a relation exists between those two parameters?
The C-C distance is determined by the internal architecture of the engine (locations of crank and cam shafts).

Dear Lnewqban,

This is just a general model that applies to two generic timing pulleys (GT3 profile). I just wanted a theoretical model linking belt tension to C-C distance. In this use case there is no tensioner and pre-tension is only provided by the C-C distance. Although the belt will lose this initial tension, this is not important for this use case.

 

[Baluncore]

The GT3 is a stepped belt so it should run flat on the pulley, indexed by the rounded teeth. As you increase belt tension, the rubber between the pulley surface and the fibre layer may be crushed slightly. That will be most significant on small diameter pulleys, where the area of face contact between the belt and pulley is small, so the pressure is higher.

It is likely that the pulley mounting structure will deflect more than the changes in the length of the fibre reinforced belt.