**1. Introduction**

Large size spiral bevel gears are frequently used in applications [1,2] that require smooth and silent high-power transmission. This is the case for equipment dedicated to thermal energy generation, ship propulsion systems, wind turbines or power transmission in the aeronautical sector, among many others. Nowadays, there is a continuous demand for energy, and consequently, there has been an increase in the amount of equipment dedicated to energy generation and its components, such as large sized spiral gear. Traditionally, these types of gears have been manufactured with specific gear cutting machines. There are different methods for traditional gear cutting, for example, some of the most commonly used are: (1) gear hobbing with perimeter cut (Gleason) [3]; (2) continuous generation by spiral hobbing with perimeter cut (Cyclo-Palloid from Klingelnberg and Oerlikon) [4]; and (3) continuous generation by spiral hobbing with conic type cut (Palloid from Klingelnberg) [5].

However, the eruption in the market of multitasking or multiprocess machines [6], and the continuous improvement experienced in the area of numerical controls and CAM software, has led to the appearance of a suitable medium for the manufacturing of these complex geometric elements in general purpose machines and with standard tools [7,8]. This type of technology is especially interesting for the manufacture of high module gears (4–12 mm), where it is not so common to find

specific gear cutting machines as in the case of lower value modules. The use of standard tools is also an advantage given the reduction in both cost and delivery times, which are parameters of vital importance in production. Special tools for gear manufacturing are also available [9], thus providing flexible alternatives for producing small or medium batches of large bevel gears using a five-axis machine and a disk tool cutting method. This methodology also allows the manufacture of gears of varied geometries, for example, straight gears, helical gears, double helical gear, bevel gears and hypoid gears. The manufacture of gears in multitasking machines [10–17] is seen as an increasingly widespread solution, especially given their high flexibility [18]. Four-axis [19] and 5-axis [20,21] CNC machining can be performed for spiral bevel gears manufacturing. Some of the advantages of this method include an increase in the versatility of the manufacturing process, both in terms of typology and size, allowing the realization of arbitrary modifications of the different gear teeth. Surface quality and the structure of the materials are also important for gear life, as studied in [21]. In order to guarantee the quality of the manufactured components and gear contact [22–26], the machining process of the gear surfaces requires special attention. Surface morphology [27] will determine machining strategies, making it possible to machine gear sculptured surfaces [9], classified as developable ruled surfaces [28], with flank milling strategies [29,30].

On the other hand, gears, and more specifically spiral bevel gears (bevel gears with helical teeth), are geometrically complex components. Once the feasibility of manufacturing these components in multi-axis general-purpose machines has been demonstrated, it is necessary to evaluate whether the number of the machines axes involved in the machining strategy influence the resulting gear surface quality. Since the transmission of movement and power between different axes is the main function of this type of gears, there is a greater contact surface between the pinion and gear compared to a pair of straight bevel gears. Due to the helix angle, spiral gears work in a gradual way, operating with greater smoothness and more silently, allowing work at higher speed ranges. However, this type of gear presents a greater sensitivity to contact errors than other types of gears.

Therefore, in order to ensure good gear contact, the surface roughness parameter is a parameter to be considered and studied in the manufacture of spiral bevel gears by multi-process machines. Optimal surface roughness values ensure good contact [31], which is translated into the correct transmission of both movement and power, increasing the useful life of the element. It is worth mentioning that both excessive surface roughness and polished surfaces are harmful for gear contact. On one hand, gear rough surfaces influence component life, and, on the other hand, gear polished surfaces hinder proper lubrication.

In this work, a predictive model of surface roughness for spiral bevel gears manufactured by multiprocess machines with ball end mills was developed and validated. The model estimates surface topography for each gear surface based on parameters such as tool inclination and orientation, the geometrical cutting parameters, and mill feed and speed values. The gear machining finishing process is optimized by the simulation of different machining conditions. Thus, it is not necessary to perform trial and error tests, which results in cost and time savings. This optimized process also adjusts cutting parameters depending on the required surface quality, without having to machine a greater number of passes than strictly necessary. This also reduces machining time and tool life.

#### **2. Spiral Bevel Gears Manufacturing Process in Multitasking Machines**

The gear manufacturing process consists of several stages. First, the geometry of the component, which directly influences the subsequent manufacturing stages, is defined.

#### *2.1. Design*

There are different options for the design of gear geometry [32–34]: Standard CAD/CAM software (CatiaV5, Siemens NX12), specific gear design module inside standard CAD/CAM software (GearTrax module for Solid Edge, SolidWorks and Inventor), specific CAD/CAM software (EUKLID), software developed by machine tool manufacturers (gear MILL from DMG-MORI, GearPro from Mag), and, software developed by tool manufacturers (InvoMilling and Up-Gear Technology from Sandvik).

In this particular case, the "3d spiral bevel gear software" was used for the design of the gear geometry. The main reason for choosing this software was the reduced cost of the license in comparison to other software. Specifically, this program allows the design of the spiral bevel geometry, with the option to choose between Gleason and Klingelnberg manufacturing methods.

A spiral bevel gear geometry was selected (Table 1) according to the Gleason method, since it is the most used method. The objective was to choose a complex and large dimension geometry to validate the capacity of general purpose multiprocessing machines for gear manufacturing.


**Table 1.** Spiral bevel gear geometry and parameters.

The selected material is a commonly used steel for manufacturing gears, F-1550 (18CrMo4) (C 0.186%, Si 0.259%, Mn 0.805%, P 0.011%, S 0.028%, Cr 1.071%, Mo 0.155%), and it reaches values up to 47 HRC (Rockwell Scale of Hardness, part C).

First, geometric parameters were introduced into the design software (module, gear ratio, gear direction, teeth number, pressure angle, etc.). With this information, the software generated the geometry of one of the teeth, from which the three-dimensional gear was generated.
