There are many variables and considerations that must be taken into account when you try to determine the optimum ratio spread, which are defined by the individual ratios required to produce the required overall ratio. This article does not intend to address the requirements of a multi-speed, multi-shaft gearbox. In that type of design, each shaft will have a required ratio, and thus each ratio-step will be defined by the gearbox requirements.
Of course, there are also the requirements of packaging to consider, along with all the practical aspects of gearbox design and development, such tooling and gearbox size and shape, etc. Although those parameters are important and must be adhered to, this article will focus purely on the aspect of ratio spread as a function of the overall required ratio as an optimization exercise.
As we know, the overall ratio is the individual step-ratios multiplied together. The task then is to determine how to determine the individual ratios. There are a number of factors to consider (beyond those listed above). Generally, it is a good idea to attempt to generate the individual step ratio based on a combination of tooth numbers based on prime numbers. Ideally, you would want to achieve the ratio requirements using a prime number of teeth in each of the driver and driven members, or a full hunting tooth design. Sometimes, of course, you cannot use any prime number tooth count due to ratio requirements, thus you may be forced into a semi-hunting tooth design wherein only one of the two members has a prime number of teeth and the other does not.
Let’s add some additional constraints. If we know the torque requirements and some idea of input or output rotational speed, then we can start to define the bearings we are likely to use. We may not be able to select an exact bearing part number, but we can select a size and series of bearing. With a bearing identified, we know the number of rolling elements, therefore we can calculate the bearing pass frequency (e.g. a reasonable estimate of shaft rotational speed times the number of rolling elements converted to revolutions per second is the bearing pass frequency). We want to make sure that our gear mesh frequency, at least for that stage, is a very different frequency and / or not a multiple of the bearing pass frequency.
Now we have some parameters to start to design our gear ratios. Briefly, I would like to present a formulation for the design of a multistage gear train that selects stage ratios in a manner that maximizes transmission efficiency, maximizes rotational acceleration, and / or minimizes mass for a given desired total transmission ratio. The problem is basically a constrained multivariate optimization problem for any number of stages and can be used with a wide variety of stage scaling criteria. The scaling criteria will be shown to provide constant tooth stress for each stage. We also need to remember that for a single parallel axis gearset the maximum ratio is probably no more than about 5.0 : 1.0 for reasonable tooth form. Thus, as an example, if we wanted an overall ratio of 35.0 : 1:0, we would need at least three stages. We might try to package the 35:1 into just two stages, and if the gears are large enough in diameter, we may find that we could actually make this work. However, for our discussion, we will state that we need three stages. The simplest way to develop the individual stage ratio would be to take the overall ratio and simply take the cube-root of it. For our example, that would be the cube-root of 35, or 3.271 : 1.0. A quick calculation provides several tooth combinations that give us a ratio near the target (e.g. 72 X 22, 3.2727 : 1.00, as an example, with an overall ratio of 35.052 : 1.00). However, this will not provide an optimized gear design as a function of weight or stress equalization in each stage of the design.
To address this, and as a rule-of-thumb, I suggest the following ratio distribution. Start with the cube-root of the overall ratio, or the Ratio Root Coefficient. Staying with our example, that yields the cube-root of 35, or 3.271.
Cube Root of Required Ratio or the Ratio Root Coefficient,
CR3 = 350.3333 = 3.271
Starting with the third stage, the Third Stage Ratio Coefficient is given by the inverse of the natural log of overall ratio:
Inverse of the Natural Log of the Required Overall Ratio,
CRStage3_Coeff = Inv Ln CR3 = 1 / Ln (35) = 0.2813
The Second Stage Ratio Coefficient is given by simply the required overall ratio divided by the number of stages:
Overall Ratio Divided by the Number of Stages,
CRStage2_Coeff = 1 / (3) = 0.3333
Finally, the First Stage Ratio Coefficient is given by the difference of the other two ratio coefficients and unity:
One Minus the Sum of Stage 1 Ratio Coefficient and Stage 2 Ratio Coefficient
CRStage1_Coeff = 1 – (CRStage3_Coeff + CRStage2_Coeff) = 0.3854
Now, let’s calculate the individual stage ratios for the low limit of our design. Remember, that we are given approximately a five percent ratio spread to design to, thus we have an implied lower limit of ratios and an upper limit. The basic form of the equation is the Ratio Root Coefficient multiplied by the individual Stage Ratio Coefficient multiplied by the Number of Stages.
So, for the third stage lower ratio limit:
Third Stage Lower Ratio Limit,
CR3rd_LRL = (3.271) • (0.2813) • (3) = 2.760 : 1.00
Second stage lower ratio limit:
Second Stage Lower Ratio Limit,
CR2nd_LRL = (3.271) • (0.3333) •(3) = 3.271 : 1.00
Finally, the first stage lower ratio limit;
First Stage Lower Ratio Limit,
CR1st_LRL = (3.271) • (0.3854) • (3) = 3.782 : 1.00
Which gives us a low limit of the overall ratio
(CR_OLRL) of; 34.146 : 1.00.
To calculate the upper stage ratio limits, we simply multiply the individual lower ratio limits by the Required Overall Ratio divided by the Overall Ratio Lower Limit.
Third stage upper ratio limit:
Third Stage Upper Ratio Limit,
CR3rd_LUL = (2.760) • (35.000 / 34.146) = 2.892 : 1.00
Second stage upper ratio limit;
Second Stage Upper Ratio Limit,
CR2nd_LRL = (3.271) • (35.000 / 34.146) = 3.353 : 1.00
Finally, the first stage upper ratio limit:
First Stage Upper Ratio Limit,
CR1st_LUL = (3.782) • (35.000 / 34.146) = 3.877 : 1.00
Which gives us a low limit of the overall ratio
(CR_OLRL) of: 36.773 : 1.00.
We now have a bracketed target for the overall ratio of each stage, as well as a bracket of acceptable ratios for the overall gearbox. What follows now is the process of selecting the appropriate tooth numbers, performing all calculations and checks of performance predictions against design requirements, etc. As stated, we can further optimize our design by comparing predicted life of each stage to each of the other stages and working to balance then (no sense one stage wearing out significantly before any other), as well as developing packaging and spacing considerations. I’ll leave that to you.
