The word “optimization” is becoming fashionable, also with regard to gear design. It is applied to both macro-geometry and micro-geometry. The approach can be of various types: analytical pre-optimization with different objectives, bulk generation of variants, multi-objective and multi-disciplinary commercial optimizers, generative optimization, and even artificial intelligence. Sometimes, the best solution is presented directly; other times, the choice is left to the user according to multiple criteria. However, these are all scenarios that assume the manufacturer will accept any geometry indicated by the designer. This is certainly not the case with the industrial gearboxes on catalog for which standard cutting tools are used to reduce cost and keep available the interchange of suppliers, nor with special gearboxes, “goods to order,” in which the producers try to use cutting tools already in the tool room. Even in the automotive industry, manufacturers try to use existing cutting tools as much as possible, at least during prototyping and for small batches.

After presenting some design optimization techniques adopted in different companies, the focus of the article shifts to some business scenarios where manufacturing has been equipped with a software for a semi-automatic selection of hobbing- and pinion-type tools starting from the macro-geometry of the gear. In particular, it will look at the case where a paper database of more than 10,000 hobs, with different dimensioning modes, has requested to be harmonized into a single computer database. The software allows the search for a hob even with “modified rolling,” a method very widespread in the automotive industry, practically “unknown” for industrial gearboxes.

Finally, for companies that have both design and manufacturing departments, a design optimization with a list of cutting tools as a main boundary will be presented.

### 1 Introduction

The key issues of this article are design and manufacturing. So, our starting point will be the opening words of two classic university books focusing on these issues:

• “The main task of engineers is to apply their scientific and engineering knowledge to the solution of technical problems, and then to optimize those solutions within the requirements and constraints set by material, technological, economic, legal, environmental and human-related considerations.” [1]

• “Machine tools are used for the purpose of manufacturing parts, which meet design requirements concerning shape, size tolerance, and surface characteristics from both a technical and economic viewpoint.” [2]

It is clear how the three requisites — material, technological, and economic — listed in the work focusing on design are linked to manufacturing, and, reciprocally, manufacturing refers to design requirements.

The need for increasing integration of these two phases is also pursued by CAD/CAM system developers to the extent that books such as “Integrated Design-to-Manufacturing Solutions: Lower Costs and Improve Quality” [3] are distributed online by these types of companies.

Moreover, the term “optimization” is becoming increasingly popular, especially in papers presented at various conferences, such as AGMA’s Fall Technical Meeting.

So, firstly, we have to focus individually on the four terms found in the title of this article (integration, optimization, design, and manufacturing), clearly restricting ourselves to the field of gears. We will look at them in “chronological” order:

• Firstly design, because man is above all homo sapiens, a thinker, able to plan and project.

• Followed by manufacturing, in other words, the ability to construct, which is a hallmark of homo faber, a Latin expression that became popular once more during the Renaissance.

• Lastly, optimization and integration, which are new words.

We will limit ourselves to cylindrical gears, which are the most common. We will put to one side wormgears, which I have already covered in other publications [4] [5], and bevel gears, which are highly branded [6].

### 2 Design

Generally speaking, gear design is lifetime-based: The aim is to transmit a specific load for a set period of time. The ways in which tooth failure occurs are taken into account in order to satisfy this requisite.

Recent updating of document numbers ISO 6336 allows for an easy overview of the main ones (bending, pitting, micropitting, scuffing, TFF) and stresses the importance of focus on the type of failure in order to achieve correct sizing.

The terms used in the title of the standards (Table 1 and Table 2) play on the nuances that can be given to design goals: calculation or rating, strength, load capacity, durability, resistance.

From a historical viewpoint, the geometrical principles of toothing were established first of all, especially involute toothing, and then rating criteria, above all for bending and surface fatigue [7, 8]. The formulas contained in the various standards and bibliographical references were then implemented in manual calculation sheets (Figure 1) and subsequently in electronic spreadsheets and software (Figure 2) to simplify the work of designers.

One of the first areas of focus in all publications dating from the last century was the definition of the proportions to be given to gears following the rules cited at the beginning of this article, in other words “within the requirements and constraints set by material, technological, economic considerations.” Here are just a few examples:

• Dudley, in his book unmistakably titled “Practical Gear Design,” later re-titled “Handbook of Practical Gear Design” [9].

• Niemann [10] with his formulas to split the transmission ratios of a parallel-axis reducer so as to minimize the costs of gear materials and housing (a concept further developed by Schlecht [11] at a later date).

• Severin [12] with his translation of the Russian work titled “Increasing the load on gearing and decreasing its weight.”

From a standardization viewpoint, the ISO documents listed in Table 1 provide methods to verify gears whose geometry is known. In some of the AGMA documents listed in Table 2, design suggestions depending on the application are also provided. There are no universal criteria. While in the automotive field small b/d ratios are common, in rolling plants, b/d ratios are often greater than 1.

### 3 Manufacturing

We will limit ourselves to looking at metal cylindrical gears, cut mostly using hobs, pinion-type cutters or power skiving, with possible grinding for finishing, to correct distortion error caused by any thermal or surface treatments or to define micro-geometric modifications [13].

For obvious reasons of space, we will put to one side gears boasting a “free” geometry: plastic, sintered, obtained by additive manufacturing, 5-axis milling, or form cutting.

Therefore, the main job of the person who receives the gear drawing, such as the one in Figure 3, is to define the dimensions of the most suitable tool, in this case a hob, trying not to buy a new one, but to choose from those already available (Figure 4).

Let us now try to describe some atypical situations that can occur in the hob’s dimensioning that may result in a difficult interpretation of the geometry for the reader of the drawing in Figure 5, which is often not even to scale. There is no standard that regulates a single method of dimensioning for these tools.

When there is protuberance:

• Only two out of its three dimensions are independent.

• If the wording “full-radius” is included for the hob’s tip radius, an iterative calculation is needed to calculate the value of the root radius.

• The reference line, in relation to which the other dimensions such as addendum and dedendum are provided, may not be the line that divides the space thickness the same as the tooth thickness as, instead, assumed by some calculation software.

• Semi-topping can have a double inclination or radius not dimensioned in the change of the pressure angle.

As for design, the focus in this case is also on aim and criteria. The aim is the one cited in the introduction “to manufacture parts that meet design requirements concerning shape, size tolerance, and surface characteristics from both a technical and economic viewpoint.”

The choice of hob, which allows for the required shape to be obtained, can be made by entering the data of the required geometry and the data of the hob (uniquely established, as we said) into specific calculation software (Figure 6) and superposing the calculated geometry with the one produced via enveloping (Figure 7). For example, in the case of the use of a pre-grinding tool with no protuberance, it is easy to note the grinding notch. The grinding notch is accepted for small-size industrial gearboxes, clearly not in the case of automotive or aerospace gears.

Once the technical aim has been achieved, there is not a single criterion for the most economic choice. For example, it could be attempted to obtain the maximum efficiency from the hob K [14]

where

K is the efficiency of the hob; in m/tooth.

p is the number of gears (pieces).

z is the number of gear teeth.

l is the face width, mm.

t_{os} is the axial pitch of the hob, mm.

i_{os} is the number of hob gashes or flutes.

β is the helix angle of the gear.

b_{1} is the working length of the hob (Figure 8).

The efficiency K should be between 4 and 5 m/tooth in order to be assessed as good. Before calculating K, the level of wear of the hob to be reached prior to replacement needs to be set and the cost of the tool and grinding taken into account.

Even if more advanced methods have been proposed [15], Hoffmeister’s formula can still be used to calculate the chip’s maximum thickness given a set progress for each part revolution.

where

h_{1,max} is the maximum chip thickness.

m_{n} is the standard module.

β_{0} is the angle of the hob’s helix.

x_{p} is the addendum modification factor.

f_{a} is the axial feed.

d_{a0} is the hob head’s diameter.

i_{0} is the number of gaps.

z_{0} is the number of threads.

h is the cutting depth.

As regards to cutting parameters, it must be remembered that it is possible to estimate profile ε_{1} and helix ε_{2 }deviations caused by the progress value (Figure 9).

where

ε_{1} is the profile deviation.

ε_{2} is the helix deviation.

z_{0} is the number of hob teeth.

i_{0} is the number of hob starts.

z is the number of gear teeth.

R_{p} is the pitch radius of the hob; mm.

f_{a} is the progress per part revolution; mm/rev.

β_{0} is the angle of the hob’s helix.

α is the pressure angle.

Therefore, with the same reference profile of the hob, the choice of hob is determined by:

• Other geometric characteristics of the hob, such as the number of cutters, external diameter, number of principles, helix angle length.

• Cutting parameters, such as cutting speed and progress/tooth, the recommended values for which can be found in the bibliography [14].

• Number of parts to be cut.

All these values can be used in the Equation 2 and 1, in order to check that:

• The chip thickness is not excessive.

• Efficiency falls into the 4-5 m/tooth interval.

But this is not the only criterion for assessing the advantageousness of specific working conditions. For example, the choice of favoring an increase in cutting speed and hence a reduction in cutting time is commonplace, resulting in the waiver of a good level of hob efficiency.

We have tried to present simple formulas with a deep educational value [14]. Some other examples are in [16] and [17]. A more precise approach can be found in [18]. For pinion-type cutter, see [19].

### 4 Optimization

We will focus on the optimization of design and the optimization of manufacturing

as separate, independent activities: The former to be adopted in the technical department and the latter in the workshop, even in the case of two different companies,

i.e., an engineering company and a subcontractor.

#### 4.1 Optimization of design

As stated in the introduction, design consists of a choice of variants, while generating and selecting them forms part of the optimization process. Without going into detail, the notion of optimization is based on three concepts: objective/s, constraints, variables. Once these three concepts have been established, a multitude of variants are generally obtained and the optimal solution chosen among these, based on well- defined criteria.

Let us have a look at some cases of optimization applied solely to gear design:

#### 4.1.1 Analytical optimization

Some years ago, Schöler [20] presented an evolution, hence an optimization, of the traditional proportioning and pre-dimensioning formulas. The paper refers to beveloid gears, but it offers a clear idea of what has also been done with regard to cylindrical gears.

#### 4.1.2 Fast generation of variants

Kissling [21] has shown how quick the generation of macro-geometry variants can be, using software already widely adopted in technical departments (Figure 10). The numerous variants generated (Figure 11) are then selected by the designer with the help of filters and graphs (Figure 12). The choice is up to the designer. The same approach is used to generate micro-geometry variants, as presented in a recent FTM [22].

#### 4.1.3 Multi-objective commercial optimizers

Bonfiglioli [23] and Noesis [24] presented the use of a multi-objective optimizer interfaced with gear calculation software. ModeFrontier and Optimus took care of the experiment design (DOE) while KISSsoft calculated each individual variant. The variant generation criterion performs better, and reporting is more functional in the face of longer processing times.

#### 4.1.4 Optimizers for supercomputers

UniMoRe has recently made available to some companies [25] a genetic algorithm optimizer developed by the university [26], which works exclusively on supercomputers.

#### 4.1.5 Artificial intelligence

Schlecht [27] even decided to make use of artificial intelligence in order to find the optimal flank modification for a pair of cylindrical gears. Compared to all the methods described earlier, a training phase for the AI engine is needed in this case, but advanced contact analysis software can be done away with.

#### 4.2 Optimization of manufacturing

The same concepts seen for the optimization of design can be applied to manufacturing.

In this case, too, optimization involves the choice of the best variant, in other words, the choice of the hob that “copies” the geometry of the gear under design at the lowest cost, also taking into account the stock allowance.

As mentioned previously, the aim could be the hob’s performance, which falls within the values listed earlier. The constraints concern generation of the desired profile and the maintenance of cutting parameters within the recommended ranges. The only variable is the hob. Software able to perform the calculations listed under point 3 is obviously required. An example of hob selection from database in a gear software calculation is shown in Figure 13.

#### 4.2.1 Tool database

Before explaining the hob’s optimized selection process, let us take a deeper look at the tool database. It is necessary to have a computer database containing all the hobs’ characteristics. The platform used can range from a straightforward Excel spreadsheet to PLM.

There are workshops that cut gears for medium-size reducers (with a module from 0.5 to 7 mm), that have 400 hobs entered into an Excel spreadsheet (Figure 14), and there are workshops working especially for the automotive industry that have 650 hobs in an Oracle database that also lists the resharpening (Figure 15).

There are also workshops working for the automotive and agricultural industries that handle more than 10,000 hobs for which only printed information sheets are available. Therefore, the first step is to enter data into a computer database. An Excel spreadsheet has been prepared with some formulas in order to harmonize the various ways of sizing the hobs mentioned previously. The enormous amount of work involved in compiling the database can only be justified by the savings, in economic terms, obtained by the process listed in the paragraph below.

In any case, the hobs database must contain this information:

• Reference profile (module, pressure angle, addendum, dedendum, tip and root radius, protuberance, semitopping).

• Geometric characteristics (hob diameter, cutting edge length, helix angle and hand, number of gashes, number of starts, material, coating).

• Working conditions (recommended stock).

Sometimes, the database also includes data [28] or drawings of dressers (Figure 16). So, this method is also used to determine the choice of roll in order to dress the grinder wheels and obtain the tip-relief listed in the drawing.

#### 4.2.2 Workflow

The process to be followed in order to generate a list of hobs used to cut the required toothing is shown in the flowchart in Figure 17, taking into account also the stock allowance.

Among the proposed variants, the optimal solution is the one that best meets the set criteria. Similarly to what we saw in Point 4.1.2, the Pareto front must also be adopted in this case.

The process can be more advanced and taken into account “modified rolling” or “short pitch tool” [29]. In this case, the hobs will not be strictly filtered on the basis of module and pressure angle initially, but also by the base pitch, optionally inside a tolerance range.

The short pitch tool usually is selected to reduce undercut when there is the protuberance, to achieve smaller root form circle after grinding and increase the lifetime of the hob. The tooth form changes only in the root, and this change should be considered in the strength calculation. In this case, both the tool and gear have the same base pitch.

In another case, the tool can have a base pitch different from the gear. Checking of the geometry obtained via enveloping will not be solely of the tip and root diameters, but also of the profile deviation, which must remain within values that can be removed by grinding. This operation can be performed only if the first selection failed to result in a solution or if the workshop normally adopts modified rolling for cutting or if a prototype or small batch is being manufactured.

In (Figure 18), there is the same gear with m = 2.5 mm and α = 20°, hobbed and ground completely (flank and root). The hob in (A) has the same base pitch of the gear (m = 2.4701 mm, α = 18°); the hob in (B) has a different base pitch (m = 2.5 mm, α = 18°); the grinding allowance is not constant, but it’s acceptable for a prototype.

### 5 Integration

The meaning of the term “integration” in this article goes beyond the one adopted by Norton in the title of his book “Machine Design: An Integrated Approach” [30] where, instead, he refers to the educational approach. The approach tackles numerous machine parts within the same whole that are often mutually dependent.

As mentioned in the introduction, the integration we are focusing on is that of design for manufacturing; this has become a must, or at least a leitmotif for many companies.

Indeed, design decisions have a significant impact on manufacturing costs and product quality; 70 to 80 percent of the end manufacturing costs and 80 percent of the work that affects product quality are established by the end of the design phase (Figure 19). Moreover, the further along you are in the development phase, the more expensive it becomes to make modifications (Figure 20). For example, once the hob has been ordered, any geometric modifications to the design have an extremely costly impact.

### 6 Integrated optimization

We do not need to go deeper into the importance of integration. We have reached the apex of this ascent of the four terms listed in the article’s title. It is just a small step to achieve integrated optimization of design and manufacturing. The following are necessary:

Adoption of a single gear calculation software in the technical department and in the workshop. Usually, it is first chosen by the technical office and then adopted by workshop.

Sharing of the same hobs database by the design and manufacturing divisions. If, as listed earlier, the first step is taken by the technical office, then it is the workshop that must share its information.

In the design software, the DOE of the optimizer searches for solutions limited to those obtained by the hobs available in the database (Figure 21) [21]. For each found variants (Figure 11), the efficiency of the hob could be added as a result to help the designer in the selection of the best solution — “best” for the designer and “best” for the workshop.

The advantages are for the whole company:

• Saving money in the purchase of new hobs and time in supplying, because the designer tries to limit himself to proposing geometries generated using just the hobs available in the workshop, rather than coming up with geometric variables at a mathematical level only (e.g. pressure angle, module, addendum, dedendum).

• The designer has a greater awareness of what will be produced, even at the level of efficiency of the hob, pre-grinding quality and grinding twist, especially if the software used conveys the skills of gear designers and machine tool manufacturers [21].

• The workshop already has the files with toothing and hob data; it does not have to interpret the drawing or enter data related to the hob, if chosen from the database. Therefore, it can focus exclusively on the technological aspects.

### 7 Conclusions

The sharing of information and the desire to network, which is the same as the goal of AGMA, and especially of the FTM, is the spirit that lies behind the drafting of this article. No new formulas or technologies are presented in this article. The state-of-the-art, good practices, and some real cases encountered in various situations and inside companies are presented in order for us to draw from them. “Uncomplicated” instruments that are already on hand have been described:

• To some designers in order to see whether there is already a tool to manufacture the gear wheel they have in mind.

• To the relative workshops in order to avoid having to spend time re-interpreting designs and to speed up the search for the ideal tool.

If a drawing is a way to encode design information and reading of the drawing represents decoding, an example of CODEC (an IT term used in relation to audio and video meaning COde-DECode) involving design and manufacturing is shown.

### 8 Acknowledgements

The author wishes to thank KISSsoft (a Gleason company) for the software. Thanks also to the companies Varvel — Mechnology, CIMA (Coesia group), Graziano (now parts of Dana group), and CEI, that provided some pictures for this article and, over the years, have dedicated resources for the integrated optimization of design and manufacturing. They have shared among themselves the same software and same database of hobs and pinion-type cutters in design and manufacturing, not without some initial problems.

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^{Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented November 2021 at the AGMA Fall Technical Meeting. 21FTM13}