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.

Simulating any PMDC motor during Hydraulic Pump testing

A practical problem will be the lead time between specifying and receiving PMDC motor samples, particularly if specifying a custom motor. Custom motors are best specified as slightly modified versions of existing motors which the supplier is tooled up to produce in volume. The motor manufacturer will probably be able to inexpensively produce a custom motor curve by changing the motor stack height and windings. Be aware that changing brush compounds and even the wiring harness can also have a large impact on motor performance.

The motor samples you have access to will likely have dynamometer curves, probably for that specific sample. Recall what was mentioned in the PMDC motor post: the rpm vs. torque curve of a motor can be varied arbitrarily and systematically simply by changing the no-load rpm (by changing supply voltage): the slope of this curve is unchanged, and the current vs. torque curve can be assumed to be unchanged.

Since you have a motor of known performance, you can vary its rpm from baseline and still know its efficiency. This is important since it allows you to divide out the effect of motor efficiency and thus have complete knowledge of the pump input and output parameters: Power In is torque * omega, Power Out is pressure * flow rate.

Better still, by varying the supply voltage, you can make any PMDC motor simulate any other motor, within reason. You will need to use your mathematical model of the motor to determine the rpm as a function of voltage and torque, and to vary the voltage for each data point (use the amps to infer the rpm). You can also put a strobe light on the motor and measure rpm directly a a check.

This simple trick saves you a huge amount of time, particularly if the motor supplier can generate predictive curves of their motor using their own software, thus allowing you and the supplier to work together on a design before a motor sample needs to be built.

Matching of a pump to a motor is easy if you have power to burn, but more challenging if you are on a power budget, which is common for PMDC applications.

You might be tempted to save having to add a relief to your system by stalling the motor at some relatively low pressure. It is OK to do this by deliberately designing a "leaky" pump, which deadheads against a pressure that is safe for your system, but it is definitely not acceptable to stall a motor in order to stall a pump. Depending on the design, damage to the motor  is likely.

Hydraulic Pump Journal Bearing considerations

The pressure across a gear pump can be thought of as acting on the projected cross-sectional area of the two gears * the gear depth. This substantial load needs to be supported by bearings which control shaft displacement such that the gears are constrained from translating by less than their radial running clearance. Also, the gear shafts create a potential leakage path. For these reasons, journal bearings are the bearing of choice for gear pumps.

Journal bearings can be the most critical aspect of the design. The bearings are designed to operate in hydrodynamic lubrication (HDL) where there is no metal-to-metal contact. If HDL is lost, then performance penalties and severe damage will likely result. Conversely if HDL is maintained, gear pumps can last a surprisingly long time with minimal wear.

Journal bearings are designed to operate in the region between the dotted lines:

Chart from Shigley & Mischke

Note that "P" is not pump pressure! It is applied load / projected total bearing area (and you have 4 such bearings). Applied load is pressure * gears cross-section.

This chart provides the minimum film thickness as a function of original radial clearance. As can be seen from the chart, for low bearing characteristic (Sommerfeld) numbers, the film thickness can be very small, considering that the radial clearance can easily be only a few tenths of a thousandths of an inch.

In theory, the friction variable is near-linear, converging on zero as the bearing is loaded right to contact:

Chart from Shigley & Mischke


Loss of Hydrodynamic Lubrication in Practice

In practice, this never happens.

This assumes perfect:

• Cylindricity
• Runout
• Perpendicularity, and
• Alignment

...between half-shafts and housings, as well as perfect surface finish and no bending moment applied to the shafts.

Instead, the above chart behaves like this (the Stribeck curve):

In practice, the transition point where HDL is lost will depend on practical tolerances and surface finish where metal-to-metal contact starts.

The Effect of Heating on Viscosity

There is one important exception to this: note that viscosity is a term in the Sommerfeld number. Even with the low friction coefficients in the order of .01, there is a power loss due to applied side force * friction coefficient * moment arm * rpm, which is converted to heat. Keep in mind that this heat goes into a film of oil the thickness of a sheet of paper. While there is a flushing of oil due to both pump leakage and also the journal bearing's natural (and considerable) pumping action, the heating of the oil can be considerable.

When this happens, the viscosity decreases. But note that viscosity decreases exponentially with increasing temperature: a 50 degree C increase in temperature can decrease viscosity by a factor of 10x or more. This process massively decreases the Sommerfeld number and can result in inexplicable journal bearing failures in design that appear sound on paper.

Be careful also of using an increase in viscosity to solve journal bearing problems: increased viscosity will increase viscous friction work and thus heating of the fluid, which decreases viscosity. Happily, the temperature increase and thus viscosity decrease due to HDL can be calculated easily and the design checked for this issue, before the drawings are released.

Temperature Control on Your Test Loop

Be very careful about this effect when testing: a typical hydraulic test loop consists of a small oil reservoir and a gear pump throttled back to produce ~ 1 kW of power into ~ 1 gallon of fluid, which is dissipated into the oil and raises its temperature.

This can easily result in a 50 degree C temperature rise in the fluid, which will render your test results useless!

Temperature must be recorded and controlled during testing.

Housing Material

One unexpected source to this effect is the choice of housing material: one of the reasons aluminum is so widely used is because its thermal conductivity is ~ 5x higher than that of steel or cast iron.

Shaft Aspect Ratio

In support of these practical considerations, it is recommended to keep the length / diameter aspect ratio of the half shafts (the shaft on either side of the gear) to between 0.8 and 1.2. Less than 0.8 results in poor sealing and limited carrying capacity of the bearing, and greater than 1.2 runs the risk of early metal-to-metal contact between shaft and housing.

This point bears repeating: particularly when adapting an existing design to operate at higher pressures, it is very temping to "save" the existing shaft diameter by increasing its length, thus preserving the existing production tooling and gauging for both shaft and housing bores. Don't do it. Exceeding an aspect ratio of 1.2 will lead to journal bearing crashes at high pressures.

The way to detect a journal bearing crash is to look at the pump curve generated in your testing: this should be a straight line. Particularly when driving the pump with a PMDC motor, which slows down under load, which further lowers the Sommerfeld number, will hasten the early onset of metal-to-metal contact. In the test data, this shows up as the straight-line pump curve taking a nose dive.

Shaft diameter vs. Gear diameter

Be careful about something else: increasing the journal diameter will increase the carrying capacity of the bearing, but the resulting friction coefficient will act on a larger moment arm, which increases the power consumed. This also eats into the gear face, which decreases the available sealing area there.

Be careful to allow for a grinding relief at the intersection of the shaft and gear face, since both the shaft and gear face are typically ground to achieve the required surface finish. These reliefs decrease the sealing area. The relief does not count toward the shaft aspect ratio.

Matching a hydraulic pump to a PMDC motor

Matching a gear pump to a permanent magnet DC (PMDC) motor presents a challenge because the motor speed decreases linearly with torque, but analysis is easy because both the pump curve and the motor curve are perfectly linear.

As discussed in an earlier post, manufacturer's pump curves can be adapted to predict torque requirements and volumetric efficiency at any combination of rpm, viscosity, and pressure. This information is required before proceeding because the volumetric efficiency of the pump will be strongly affected by rpm, which for a PMDC motor will decrease with load (pressure).

Motor manufacturers publish motor curves which are in the form of rpm and current draw as a function of flow rate. Be aware that the "motor constant" approach in electrical engineering textbooks is wrong: this assumes that current is zero at zero torque; in practice, current draw is not zero at zero load due to i2R, hysteresis, and windage losses. However, both rpm and current as a function of torque are linear functions. Consider the following motor curve from Allied Motion:

Units of torque are in oz*in.

Solving for Motor Curves

Take two data points from the motor curve such as above, for rpm "N" and current "I" at two values of torque "T", and solve as follows:

Where c1 is the no-load rpm and c2 is the slope of the rpm vs. torque curve; c3 is the no-load current and c4 is the slope of the current vs. torque curve. Note the correction to c1 for varying supply voltage, which is discussed below. Obviously, these equations will need to be corrected to keep the units consistent.

Once these constants have been determined, the motor curves can be plotted. In this case the curves are plotted at 12, 8, and 6 volts.

I have plotted a single current curve on the same graph, obviously this is scaled differently than the rpm curves.

Working from this, power out = torque * omega, power in = volts * amps, which allows motor efficiency to be determined.

Effect on Motor Curves of varying supply voltage

These curves are for a stated supply voltage, in this example 12 V. Note the high current that this type of motor can draw: the electrical resistance of the wiring harness and the tolerance on the supply voltage must be taken into account, as these can easily consume a few volts at high loads, which will have a disproportionate effect on the performance of the pump. Also be aware that the motor performance curves are subject to tolerances: consult with the supplier. The effect of changing the supply voltage is to change the no-load rpm, which varies linearly with voltage. The slope of the voltage vs. torque curve is unchanged, and the current vs. torque curve can be treated as identical, independent of voltage.

You heard correctly: at 8, 12, or 14 Volts, you can assume that the above current vs. torque curve does not change. The reason for this is simply the physics of torque creation: more current = more force on a moment arm, i.e. torque.

Max recommended operating point of PMDC motors

Before proceeding further, you must take into account that PMDC motors are not designed to be loaded steady-state above their peak efficiency. Therefore, use the rpm and current curves from the motor curve to solve for the unknown constants of the linear functions, and then plot the the efficiency and power output curves. Design your system to not load the motor above its peak efficiency point.

Producing a Pump Curve for the System

Your goal is to produce the flow rate vs. pressure curve for the pump-motor system. Start by choosing a candidate motor and pump. I'm assuming you have used the previous post to determine the pump displacement, friction variable, and leakage resistance at the viscosity of your desired fluid. Since the resulting flow rate vs. pressure curve is linear, you only need perform this calculation at two points.

The approach we will be taking is to understand that motor power output is used to produce 3 things:

1. Useful work
2. Internal leakage
3. Viscous friction

The power consumed by internal leakage is handled automatically, since the leakage fluid is pumped to the output, where it leaks back to input. So the power consumed is simply the pressure * the total flow rate, where the total flow rate is simply the pump displacement  * the rpm. However, we must take internal leakage into account when calculating the actual volume produced by the pump.

To simplify the math, use zero pressure as one of the data points since 100% of the motor torque is to combat viscous friction, with zero leakage losses.

Converting Manufacturers' Hydraulic Pump Data to a Pump Curve and Matching to an Application

The problem: when you select a hydraulic gear pump, the manufacturer's curves are for a certain viscosity and rpm which in general do not match your design.

Worse, if you are driving your gear pump with a permanent magnet DC (PMDC) motor, then the rpm will decrease with load and your actual performance curve will look nothing like the manufacturer's curves. The purpose of this article is to show you how to make those adjustments. I have used this technique in practice and found that it agrees surprisingly well with pump test loop experiments.

Note that the above curve spans a range of rpm's and pressures. This particular curve is from the Marzocchi pump company and is for a viscoity of 30 cSt. In practice, you will need to rework this information to construct a single pump curve where flow rate is graphed against pressure. Note that the above data does not go all the way down to zero pressure, however a flow vs. pressure curve for a properly designed and purposed gear pump is a perfectly straight line. This article will walk you through the steps required to generate your actual pump curve.

Breaking down the data into viscous friction vs. leakage

The observed flow rate vs. pressure depends on two things: viscous friction and leakage. Happily, gear pumps tend to operate at low-Reynolds numbers which makes analysis easy by using laminar flow equations. Our goal here will be to understand the friction and leakage equations only well enough to identify and quantify the contributions of factors we have control over. The most critical aspect of gear pump design, the determination of running clearances, is beyond the scope of this article but will be considered on this Blog in a future article.

Viscous Friction

Let's start with viscous friction. At zero pressure, the pump is doing no useful work, and all of the torque is used to combat viscous friction. Consider the definition of viscous shear. For a plate of area A moving with respect to a fixed surface separated by a gap t:

Clearly the friction torque is the sum of the contributions of viscous shear force elements acting on moment arms across the wetted rotating surfaces of the pump (i.e. you integrate). Right away, the zero-load torque can be adjusted from the manufacturer's value simply by a linear adjustment of the viscosity. Since the friction torque also depends linearly on speed, i.e. rpm, then the friction torque can likewise be easily adjusted for varying rpm also with a simple linear ratio.

Notice that we have not had to consider the actual internal geometry of the pump, which does not change as the viscosity and rpm vary: by giving us the no-load motor torque at a stated rpm and viscosity, the manufacturer has in effect done this for us.

A test against zero pressure tells us:

• The pump displacement
• The friction torque

Internal leakage

Next, the internal leakage needs to be considered. Moving parts require running clearances and these become fluid leakage paths from high to low pressure. As we did for fluid friction, we're going to idealize these leakage paths as simple rectangular gaps, which can in principle be integrated to obtain one overall "fluid resistance", analogous to electrical resistance, for the entire pump.

Again, by giving us the pump curves, the manufacturer has provided enough info to deconstruct this data and solve for the leakage at various pressures. So we will use the published flow data to determine the volumetric efficiency of the pump and thus the "resistance" term on the right of the above equation.

Volumetric efficiency

In other words, by giving us the actual flow rate at two rpm's, the manufacturer has given us the data we need to calculate volumetric efficiency. Since we know that the leakage flow rate at zero pressure must be zero, and thus volumetric efficiency is 100%, we can easily calculate both the pumps' leakage flow resistance (above) and also the pump displacement.

At non-zero pressure, you cannot directly measure viscous friction (counting journal bearing friction). However, if you know the rpm and the actual flow rate, you can calculate the volumetric efficiency and thus infer the viscous friction and seal friction losses.

Practical simplifying assumptions

We are going to make an assumption here that experiment shows holds up well in practice: assume that the leakage resistance depends only on the terms in the above equation, in particular those terms we have control over: the viscosity and the rpm. In theory, pump leakage resistance due to viscous shear depends on the speed of the fluid with respect to the surfaces of the leakage gap, and those surfaces are not stationary (the gear surfaces rotate), meaning that the actual leakage flow speed will depend on the magnitude of the vector sum of the leakage and moving surface speeds of the parts, and that the leakage flow down the journal bearing shafts is similarly complicated by both the surface speed and a changing eccentric gap. However, it can be shown experimentally that these effect can be ignored. 

Once the fluid resistance term above has been determined for your pump, this number can easily be adjusted for a different viscosity. Again, we assume that fluid leakage will vary from this term only due to changes in viscosity.

But be careful: volumetric efficiency depends on both leakage and pump output: the displacement and the rpm. At any pressure, the faster the rotation speed, the greater the volumetric efficiency. Changing the rpm's will dramatically effect the pump flow rate at pressures near pump stall.