Predicting Rate of Fire for Various Gearsets

In this artcle, the science of gears and gear trains is discussed in relation to AEG gearsets, and a formula is derived for the prediction of rate of fire which displays < 2% error when tested with real world results.

Introduction:

Everyone knows what gears are and that there are gears in an AEG, but just how these gears function to affect your rate of fire and drive higher power springs falls within the realm of conjecture and speculation. Solid answers are hard to come by when it comes to determining just which gearset to go with for a specific application or how exactly the different gearsets available on the market will affect your rate of fire. Fortunately, this state of confusion need not be permanent. Gears are mechanical components with well known properties and calculable characteristics which transcend their narrow use as an AEG transmission. In this article, we hope to clarify many of the myths and misconceptions regarding AEG gearsets and their function, and introduce a mathematical method for the comparison of gearsets and the prediction of rate of fire.

Gear Fundamentals:

*For a more in-depth discussion of gear types, check out the excellent gear intro at HowStuffWorks.com

Gear Ratios:

*the colored cells delineate each gear intersection: blue = bevel, orange = spur, green = sector.

Bevel to Motor Pinion Ratio: 30:10 = 3:1
Spur to Bevel Ratio: 39:10 = 3.9:1
Sector to Spur Ratio: = 32:20 = 1.6:1
Final Ratio = 3 x 3.9 x 1.6 = 18.72

By altering the standard ratio of teeth between the various gears in the mechbox, aftermarket parts manufacturers can choose to enhance the final torque output as in the case of “torque-up” gearsets or the final output speed as in the case of “hi-speed” gearsets.

Rate of Fire:

From the discussion on gear ratios, it is easy to see that the final ratio of an AEG gearbox is a determining factor of the maximum rate of fire for a gearbox. Gearsets which result in a larger final ratio to drive stronger main springs will correspondingly result in a lower maximum rate of fire since the torque multiplication is achieved through a sacrifice in output speed. Conversely, gearsets which result in a lower final ratio to attain an increased rate of fire will result in a lower final torque output, limiting the strength of the main spring which can be used in the combination. Given the ease with which gear ratios can be determined, it would stand to reason that the maximum rate of fire for any gearset combination can be easily calculated with some simple math. As long as the output RPM of a motor is known, it seems obvious that simply dividing this output RPM by the gear ratio would result in the the final RPM and rate of fire. Using the published 27552 RPM* of the standard Marui EG1000 motor and the various ratios available from the Systema, Prometheus and Tokyo Marui gearsets, the following graph of RPM vs Gear Ratio can be generated:



The graph shows the characteristic hyperbolic decay curve which is expected given a plot of y=27552/x, but the raw numbers are obviously all wrong. Standard Marui ratio AEGs certainly do not shoot at 1472 RPM, and “super torque-up” gearsets definitely do not yield rates of fire in the 1002 RPM range. What are we missing in the equation?

The Effect of Motor Loading:

DC electric motors, such as the EG1000 motor, do not provide the same output RPM under all conditions. Instead, output RPM is linearly proportional to the torque load it sees. As the torque load increases, such as with a stronger main spring or heavier piston, the motor output speed will decrease. Even though the maximum given speed of the EG1000 motor is given at 27552, this is the “no-load” speed, meaning the motor only achieves this rate of rotation when it freely rotates without a torque load. Since even the weakest piston and spring combination will add some load, the maximum 27552 rate of rotation is never attained. To determine the actual output RPM of a motor under a given load, the motor's torque/speed curve must be consulted. Fortunately, this curve is linear for DC motors, and can be easily determined from a motor's rated maximum RPM and torque output. The X intercept of the plot is the maximum torque rating while the Y intercept is the maximum rated RPM. Again using published* ratings for the EG1000 motor, we get the following torque/speed curve:



One caveat of note here is that motor torque/speed curves must be qualified with a specific rated input voltage, since an increase in supply voltage to a DC motor will result in a change in its characteristics. Unfortunately, the specifics of AEG motors are hard to obtain, including the rated voltage attached to the curve above.

*Motor ratings for the EG1000 motor were obtained from a technical post at FilAirsoft.com The original source of these published numbers were from a printed Systema ad in an issue of ARMS magazine circulated in Japan. Because the source of these numbers is a party with vested interest in their interpretation, it must not be presumed to be 100% accurate.

Putting it Together with some Math:

From the previous Gear Ratio discussion, we know we can easily plot a graph of expected rate of fire vs all available gear ratios, as long as we know the motor output RPM. From the Motor Loading discussion, we also know that we can easily determine the motor output RPM by consulting the motor torque/speed graph plotted from a motor's maximum speed and torque ratings, as long as we know the load the motor is seeing. Putting these two facts together mathematically, we can arrive at a formula for expected rate of fire given a motor torque load and gear ratio.

ROF = motor output RPM/gear ratio
EG1000 output RPM = (27552/1407)*(1407 – motor torque load)

Combining the formulas above we get:

Expected ROF with EG1000 = ((27552/1407)*(1407 – motor torque load))/gear ratio

Since the torque load the motor sees varies with the ratio of the gear train between the motor and the loading components of the mechbox, we must isolate the effect of the gearset from the motor torque load and determine the “mechbox torque load” so that we can predict the rate of fire given different gear ratios. This can easily be incorporated into the equation because:

mechbox torque load = motor torque load*gear ratio

With a little more algebra, we derive our final equation:

Expected ROF with EG1000 = ((27552/1407)*(1407 – (mechbox torque load/gear ratio)))/gear ratio

Mechbox Torque Load, the Last Piece of the Puzzle:

The equation for expected rate of fire as derived allows us to calculate an expected rate of fire given a specific mechbox torque load and gear ratio. This is powerful in that if we know what load our mechbox currently exerts on the motor, the equation allows us to predict what our rate of fire will be with any other gear ratio. But just how do we figure out what torque load our mechbox exerts? Predicting the exact torque load is difficult given the multitude of parts which effects the ease with which the mechbox cycles. Different pistons, piston heads, cylinders, springs, bearings, gears, shims, lubricant, etc could all easily effect the overall torque load which the motor sees, making prediction of the exact torque load purely from a parts listing nearly impossible. Does this fact render the previously derived formula useless? Not quite. The same formula can be reconfigured to solve for torque load given a known gear ratio and a known rate of fire:

Starting with:

ROF = ((27552/1407)*(1407-(mechbox torque load/gear ratio)))/gear ratio

We can solve for torque load and get:

mechbox torque load = (-(ROF*1407*gear ratio)/27552 + 1407)*gear ratio

This gives us the critically missing piece of the puzzle. With the mechbox torque load now calculable for a specific combination, we can predict expected ROF with the same combination when different gear ratios are used. Below is a sample plot showing predicted rates of fire for 3 different arbitrary loads and 4 available Prometheus gear ratios.



It's critical to note here that at high mechbox loads such as with a strong main spring (yellow line), the shape of the curve takes a down turn at lower gear ratios, exhibiting behavior opposite that of medium and low mechbox loads. The apex of the downturn signifies the point where using a lower ratio/higher speed gearset sacrifices too much torque multiplication and ceases to improve the rate of fire.

Practical Application

What exactly does all this math and theory really get us? Why predict theoretical rate of fire? The answer is best described anecdotally. Suppose you are in possession of an AEG which has been pre-upgraded with a powerful spring but retains the standard ratio gearset. The range and accuracy of the AEG serves you well, but the low rate of fire is unsatisfactory. You decide to improve your rate of fire by purchasing a new gearset, but which gearset do you go with? On the one hand, you can purchase a high speed gearset to reduce your gear ratio and increase the rate of fire, but on the other hand, if the main spring and combination of parts in your mechbox result in too much load for the motor, a high-speed gearset could decrease your rate of fire. Conversely, a “torque-up” gearset could improve your rate of fire if there is already too much load for your motor to handle, but could also decrease your rate of fire if that isn't the case.

Therein lies the dilemma many have encountered when faced with the decision of which gearset to purchase. With the formulas derived above however, a solution to the dilemma is simple:

Step 1: Determine Current Rate of Fire
By using the audio-analysis method, the rate of fire for any AEG can be easily and accurately measured. It's important to note that the measured rate of fire is dependent on the output voltage of the battery, hence the measured value can change significantly with batteries in different charged states.

Step 2: Calculate Theoretical Mechbox Torque Load
Using the formula:

mechbox torque load = (- (ROF*1407*current gear ratio)/27552 + 1407)*current gear ratio

Calculate the theoretical mechbox torque load given your measured rate of fire and current gear ratio.

Step 3: Calculate Predicted Rate of Fire
Once the mechbox torque load has been determined, calculate the predicted rate of fire with your new gear ratio by using:

ROF = ((27552/1407)*(1407-(mechbox torque load/new gear ratio)))/new gear ratio

*The equations above assume that the EG1000 motor is being used. Calculations with other motors will require the substitution of that motor's operating specifications into the equation. Operation on voltages other than 8.4v will also require scaling of the equations to properly model the change in output of the motor which will be discussed in the upcoming AEG Motor article.

Experimental Results

The results show a less than 2% error between the predicted results and the actual experimental results, supporting the validity of the equations. The error may be reduced further if the difference between measured battery voltage and rated battery voltage for the motor speed/torque curve are taken into consideration, but that discussion will be reserved for the upcoming AEG Motor Science article.

Conclusion