Mathematical Definition of an Involute, and Practical Gear Pump design

We're going to use the approach of "unwrapping a piece of string from a cylinder". Consider the following diagram. We will take vectors A and B, break them into components, and add them. Note that "theta" is an arbitrary parameter, in radians, not the pressure angle of the gear.

Also note that we start the operation at the "base circle", which is the pitch circle * the cosine of the pressure angle.

The math follows from this:

So the vector B is a point on the involute, which will trace the path of the involute curve for you as parameter "theta" is varied. "Theta pressure" is the pressure angle of the gear. You can then Apply the normal gear parameters for addendum and dedendum circles out of Machinery's handbook or Shigley, etc.

Practical Hints

CAD Modeling

If you are using the CAD model to determine the displacement of the gear pump, be careful to understand that some of the gap between gear teeth does not represent the volume pumped: some of this oil volume, at the root between teeth and the mating tooth from the other gear, goes around a full 360! Ask yourself what would happen to the volume pumped if you machined a large undercut at the root of the teeth: would this increase the pump displacement? No.

Contact Ratio

The contact ratio (the average number of gear teeth in contact at any time) should be always be above 1, but as close to 1 as possible.

AGMA Quality Grade and Gear Metrology

You will need to specify an AGMA quality grade on your drawing. Instead of just calling for "AGMA 8", you should specify what aspect of the gear metrology is referred to.

Lead

For gear pumps, the most critical parameter by far is the "lead" tolerance, which is the deviation in the axial direction of the gear, in other words, how close it is to being a perfect extrusion. You might need to consult with your gear supplier since they typically "crown" their gears to compensate for deflection during loading... but this would create a leakage path when used in a gear pump!

Profile

Second up is "profile" tolerance, which is what you imagine it to be, similar to the "profile" tolerance in GD&T. The reason for this requirement is that at very high pressures, such as the contact stress of a gear mesh, the viscosity of oil increases exponentially. If there is substantial sliding motion (due to poor involute gear profiles), this will cause a large friction loss, for gear pumps operating at high loads.

Be careful to test a new supplier's gear using a professional gear metrology service: most suppliers who claim to work to your drawing will not be able to meet your AGMA spec's. Testing new suppliers is mandatory!

Contact Stress

The gear mesh contact stress causes another problem, again mainly for high pressure gear pumps: the contact stress can yield the material. Use your CAD system to determine the instantaneous radii of curvatures of the gear faces, calculate the Hertzian contact stresses, and specify material hardness accordingly. This will often require the gear to be hardened before final machining and grinding, which adds to the expense.

A few remarks about Gear Pump Housings

• Gear shafts need to vent back to intake or reservoir, or simply locate the gear pump in the tank.
•  Don't forget to specify "DO NOT BREAK THIS EDGE: LEAVE EDGE SHARP" on the housing drawing, 2 places, at the edges of the gear pockets, or you will have a large leakage path between inlet and outlet, due to broken edges!
• Rotary shaft seals continuous service life depends on the PV ratio that the seal elastomer can withstand. These seals are not normally designed to take any real pressure, but select a "pressure seal" if you purge the hydraulic system using pressure!
• It is a very good idea to integrate the seal into the pump's pressure plate, so you can hold the seal manufacturer's concentricity spec., which can otherwise be a very long tolerance stackup.
• This can be difficult to do due to the diameter of the seal encroaching on the idler shaft. The Creavey company makes a nice little small-diameter Teflon seal energized with a stainless steel garter spring. This can also be found on McMaster-Carr.
• If the seal is integrated into the pump's pressure plate, then ensure that the drive gear shaft bore is:
• Stepped up to prevent the creation of an accidental journal bearing (which will create enough pressure to blow the seal!)
• Vented to tank
• Use a narrow washer underneath the screws which fasten the pressure plate to the gear housing, to minimize manufacturing-induced side loads on the gear shafts.
• Housing and pressure plate flatness is more important than surface finish. For high-pressure gear pumps, these surfaces can be lapped.