Compound Gear Train with Rack-Gear Forces?

[mbuis]

It has been a long time since I have dealt with gear trains. Probably since my college days and I am now 43 years old. Anyway, I am working on something and I would like someone with more experience with gears to offer their input. Attached is a PDF showing my compound gear train interacting with a rack. I included the pitch diameters of the spur gears, although these are just randomly picked gears sizes with a diametral pitch of 20, for this example.


Allow me to give a verbal description of the gear train. I have a drive gear (Gear 1, Transparent) which meshes with another gear of equal size (Gear 2, Transparent) and a smaller gear (Gear 5, Grey). Gear 2 meshes with a similar in size as gear Gear 5 (Gear 6, Grey). Gear 5 and Gear 6 form compound gears respectively with Gear 3 (Blue) and Gear 4 (Blue). Gear 3 and 4 mesh with the rack (Red).


My question is what is the amount of the linear force imparted to the rack?

I started my evaluation by assuming an input torque of 100 in-lbs and using the equation W_t=2*T/d. With "W_t" being transmitted force, "T" is torque and "d" is the pitch diameter. I only evaluated half of the gear train (Gear 1, Gear 3, and Gear 5) because I assumed if I had done the free body diagram I would have ended up with half of the force on Gear 3 and half of the force on Gear 4 then adding the two together to get the total force imparted to the rack. This might be an incorrect assumption, but I do not believe it is incorrect.

For Gear 1 this gives a W_t equal to 114.3 lbs. Then using half of the pitch diameter for Gear 5, I calculated a torque of 40 in-lbs imparted to the compound gear of Gear 3 and Gear 5. Again using the formula for transmitted load, Gear 3 imparts a linear force of 64 lbs to the rack.

Is my reasoning for the problem valid? Is the solution valid? If I am wrong, please advise.

 

[tbuelna]

In reality the load distribution between the blue gears/racks will not be balanced. In fact, depending on how the gears/rack are constrained, how they are assembled, and how precise they are manufactured, it is quite possible that all the load will be transmitted thru just one gear path.

 

[mbuis]

tbuelna,
Thank you for the response. I understand your statements about the precision and quality of the items manufactured. I can certainly see the rationale for your statement. Do you have experience with rack and pinions? If you do, then experience wins out over something read in a book. However, if you are basing this on theory I would disagree...respectfully. As I mentioned in my post, I have not had any real experience with gear trains since my college days, and even then it was only on paper. I completely agree, how the gears are constrained and how precise the manufacturing will be has an impact. In fact, my concerns over these issues prompted this design. I was concerned if the rack was driven on one side the item it was attached to would bind in its travel because all of the load would be on one side; therefore, inconsistencies in the manufacturing process may compound the issue. I thought having the setup like this would essentially "balance" the rack, within the manufacturing tolerances.

Again, I am not emphatically stating my theory is correct and I believe practical experience trumps "book learning"; however, what I am doing is opening a dialogue in which to explore the merits of this design.

 

[gruntguru]

Counting teeth gives the same result. 114.3 x 15/25 = 64.08 lbs.

The load may well be shared equally depending on how the gearbox is constrained eg if the entire geartrain is free to rotate about an axis parallel to the shafts and the force is applied along the centreline between the two racks, the gearbox will align to produce an equal force in each rack.

je suis charlie

 

[tbuelna]

mbuis,

As noted above, it may be possible to balance the mesh contact forces on both sections of rack teeth using some arrangement of positional constraints and/or compliance in the geartrain. Another important thing to consider is the effect of contact frictions in the system.

 

[mbuis]

I am very appreciative to the both of you for commenting. Thank you gruntguru for verifying my calculations. Also, for the suggestion (it has given me an idea) and for tbuelna for seconding the suggestion.

 

[gruntguru]

If you are thinking about balancing the loads by allowing the gearbox to rotate it would help if you could move gears 3 and 5 to the upper side of gear 1. This gets gear 3 closer to the level of gear 4.

je suis charlie