Modified Beam Bending Equations

[cisggy]

Hi everyone,

I am attempting to model the stiffness of a gearbox using none FEA methods, and the way I have approached it so far is to break down the gear into a load of separate elements to analyse each element's contribution to the stiffness, e.g. torsional stiffness of the bulk of the gear, contact stiffness (hertzian contact)

Anyway, as part of this I am thinking of modelling the gear teeth as simple beams, with a load applied at some point along the beam (at PCD). However, the formula that I was going to use:

Y_max = (Wa^3/6EI)((3L/a) - 1))

is not valid for such 'stubby' beams as would be found on gear teeth - my data book tells me the equation is only valid for lengths twice that of the depth

Do any of you know of a formula that may be more appropriate? or even a modification of the above formula?

Thanks for your help

 

[btrueblood]

Model the stubby beam using simple shear stress, and add that deflection to the calculated bending deflection. Also, get a copy of Roark's Formulas for Stress and Strain, the section on short beams includes a formulas derived by Heywood for the deflection of gear teeth due to point loads.

 

[rb1957]

i'd've started with assuming that the teeth are rigid, aren't there much larger influences on the stiffness of a gearbox (than the bending stiffness of the teeth) ?

 

[cisggy]

Thanks for the help,

rb1957, you are right - the bending of the teeth is insignificant compared to other factors - I think that the stiffness of the gearbox is mostly affected by the hertzian contact of the teeth. I just wanted to cover all bases in my model to start off with really. Once I have some experimental data I may simplify it all (or perhaps more likely just scrap it and try FEA!)

btrueblood: Used Roark's book - pretty helpful although sometimes confusing to navigate... ending up using a modified lewis equation for bending stress to find the deflection under load of a gear tooth

thanks