Contact Ratio for Helical Gear

1- is it a home work?

2- if you deal in detail with gears the best is to get a book about the problems since it will not be easy to obtain detailed informations here. dealing with gears requires as well sketches and those are better found in books.

3- the higher the angle the longer the contact but also the higher the quality constrains for the teeth direction since the longer the tooth the higher can be the dynamic load due to pitch and direction errors.

4- it is not a min -max value but what the transmission you design requires.

Some tips:

-In general don't spec a helix angle higher than 30 degrees.

-For smooth running a contact ratio close to whole numbers (slightly above) is best. I.e. 2.2 or 3.2.

-The higher your helix angle the higher the contact ratio but also the higher the axial load on the bearings that support this gear (and it's mating gear).

-The wider the gear is the more it is sensitive to misalignment, also think about the span of the bearings supporting the gear as this also affects the gears sensitivity to misalignment. As these affect the load distribution on the gear tooth flank this also affects the smooth running of the gears.

-Smooth running is not just defined by the contact ratio, as Nick Name said pitch errors can drastically affect your smooth running. So you might find yourself designing your gear to a higher quality (and higher cost)

-Don't forget the (viscosity of the) lubrication also affects the smooth running of a gear.

-I guess this also highlights Nick Name's comment to get some literature on the subject. For example: Handbook of Practical Gear Design (Mechanical Engineering (CRC Press Hardcover)). by Darle W. Dudley

I would like to add that also the position on the gear along the shaft between the bearings and the shaft stiffness are to be considered since the higher the longer the tooth the higher its sensitivity to angular misalignments (as mentioned in other comment). The best is to have both gears so that the shaft angular deflection will minimize the relative angle of flanks. The deformation of a shaft is directly proportional to span^3 and inverse proportional to d^4.