Static Analysis of Selected Design Solutions for Weight-Reduced Gears

The paper presents an analysis of selected design solutions for spur gears with reduced weight, in which the re sults obtained are compared to a solid gear without modifications. The reduction of the weight of the gears is of particular importance, among others in the automotive and aviation industries, where it reduces energy consump -tion, and thus CO 2 emissions. It is also important to remember to maintain the required strength parameters when reducing the mass of the gear. This article focuses on the analysis of deformation and stress due to a given load on the considered weight-reduced gears. The values of the obtained static analysis results of the reduced-weight gears were also compared to the base gear.


INTRODUCTION
Gears are the most important element in the field of mechanical engineering especially for industry, automotive, and aviation, mainly in power transmission where variable load and speed are required to operate machines [1][2][3][4][5]. Straight gears, or cylindrical gears, are the most common type of gear because of the simplicity of design and manufacture, and because they are cost-effective and require less maintenance [2,[6][7][8]. They are usually used to transfer power to parallel shafts [9][10][11]. Many analyses are presently being conducted to determine when damage or defects occur due to gear bending. From these, it can be concluded that bending failures usually occur when the yield strength of the material is exceeded [3,6,9,12]. With the increase in interest, as well as the frequency of use of gears, designing a gearbox for the required conditions has become a demanding task [13][14]. Today, automotive components, particularly rotating parts in the powertrain-including gears-are designed to be as light as possible, to reduce fuel consumption and thus CO 2 emissions, among other things [15][16]. First of all, reducing the weight of the gear transmission is important in household appliances, where the gears are obtained by injection methods from polymer materials, in which reducing their weight is important for saving material while maintaining durability and wear resistance while ensuring proper cooperation, where the disc affects the meshing conditions. Of particular importance in the automotive and aviation industries is the reduction of the weight of the gear transmission, which improves the ratio of the weight of the transmission to the transmitted power and reduce the negative impact of the dynamic phenomena occurring during its operation [1,4]. Increasing the efficiency of the gearbox by obtaining lower friction loss, lower emissions, and lower lubrication consumption is an area of intense research [17]. By using lightweight gear materials (e.g. polymers), optimizing lubrication consumption, and modifying the gear (e.g. optimizing tooth thickness [18]), higher efficiency can be achieved in gearboxes [19][20][21]. It is also possible to observe an improvement in the properties of modern materials in the area, e.g. plastic deformation and the application of additional coatings [22]. Taking into account the above considerations and the conducted literature analysis, the paper undertakes an investigation of the effect of changing the design of the gear disc between the hub and the toothed rim, to reduce the weight of the design in relation to the solid gear. On the basis of the conducted review, it was found that the most frequently used variants of gears with a reduced mass are gears with recesses, which allow obtaining a gear with a disk placed symmetrically in relation to the toothed rim. Relief holes are also used. Typical solutions of gears with a reduced mass are shown in Figure 1. The presented analysis aims to present the typical solutions of gears with reduced mass, used in machine building (such as MB2, MB3, MB6 and MB7) and non-typical (such as MB4 and MB5), that are possible to produce using incremental technologies with decremental methods. The article focuses on the analysis of the stresses occurring in the disc. This investigation provides the basis for another static analysis of gears manufactured from polymeric materials using additive manufacturing (AM) technology. The rationale for the analysis is the growing interest in polymers and AM in the industry. Additive printing allows you to generate complex shapes that are impossible to achieve with conventional methods. Polymer materials are used for gears (including low-, medium-, and even high-power transmission in automotive engineering in particular) because of their lighter weight, better noise dampening capability, and low manufacturing costs compared to metal gears [23][24]. Increasingly, additional fillers are added to basic polymer materials, due to fact that composite materials generally show better properties (e.g. improved hardness and tensile strength) than homogeneous and pure materials [25][26]. When constructing weight-reduced gears, symmetrical gears were selected for analysis as it was concluded, based on a review of publications, that such an arrangement of the disc would result in reduced deformation due to its design (its location relative to the rim and hub of the gear) [1,[27][28].

DEVELOPMENT OF RESEARCH MODELS OF SELECTED DESIGN SOLUTIONS
A spur gear with a modulus of 2.5 mm, a number of teeth equal to 35, an outline angle α equal to 20°, and a pitch diameter of 87.5 mm was selected for the analysis. The width of the gear rim was 27.5 mm. Figure 2 is an illustrative diagram showing the structure of the gear. Only the disc was modified-the hub and the toothed rim are elements that are constant in each of the gears analysed.
Preliminary topology optimization of the solid gear was carried out in the Autodesk environment, assuming a weight reduction of 30%. After the simplified analysis was performed, an example solution of the gear with reduced mass was obtained, which is presented in Figure 3B. During the analysis, it was noticed that it was   Figure 3A, therefore the obtained solution cannot be directly implemented to create gear with a reduced mass, because in the case of a disc, it is important to keep the geometry that is symmetrical relative to the axis. As can be seen from the presented solution, the program applied a solution in which the disc mass was reduced ( Based on the performed preliminary topological optimization and literature review, four design options for weight-reduced gears were selected and produced ( Table 1). The models were designed and built in the Autodesk Inventor Professional 2021 environment [1,[29][30].
The gear including the rim was created in Inventor 2021 using the generator [29][30]. In the following steps, the hub and disc were created using solid modelling related commands and modified to reduce their weight. It was decided to create a cylindrical opening instead of a spline in the hub as, during the preliminary analyses of selected design solutions for gears with reduced weight, singularities occurred in the place of creating the spline, the so-called numerical notches which did not disappear despite the mesh compaction, and which disturbed the results of the conducted analyses. Table 1 shows selected weight-reduced gears with the designations used.
Type MB1 (Table 1) is a solid gear, without transformations, which forms the basis for design modifications of the weight-reduced gear disc. Therefore, this gear provides a reference for comparing the effects of weight reduction, which is presented later in the article.
Type MB2 (Table 1) is a symmetrical recess gear with a 10 mm thick disc. The sharp edges between the disc and the rim and hub have been rounded off to a radius of 2 mm.
The MB3 gear (Table 1) represents a symmetrical gear with a 10 mm thick disc with six ribs arranged concentrically in relation to the gear's axis of rotation. The thickness of the rib is 3 mm. The sharp edges between the disc, rim, and hub structural components have been eliminated with a 2 mm rounding.
The MB4 gear (Table 1) is a hollow shell gear with symmetrical discs located outside the gear rim (distance between discs: 17.5 mm; thickness of each disc: 5 mm) with six ribs, each 3 mm thick, with inspection openings in the form of ø10 mm diameter cylindrical holes placed on a ø50 mm diameter circle. The sharp edges inside the shell structure of this gear are also eliminated by rounding 2 mm.
The MB5 gear (Table 1) presents a symmetrical gear wheel with a 10 mm thick disc, the structure of which resembles a "honeycomb" arranged concentrically with respect to the gear's axis of rotation. The thickness of the walls present in the disc was set to 1 mm, in which the sharp edges were rounded with a radius equal to 0.5 mm. The MB6 gear (Table 1) presents a symmetrical gear with three bean-shaped holes. The radius of the "bean" is 5 mm and the angle between the center of the rounds-off the bean is 70°. These holes were arranged on a ø50 mm diameter circle.
The MB7 gear (Table 1) is created on the basis of the MB3 gear, but additionally has ø10 mm diameter cylindrical holes placed on a ø50 mm diameter circle.

PERFORMANCE OF STATIC ANALYSIS USING MES
Static analysis in the Autodesk Inventor Nastran 2021 environment was performed on the test models shown in Table 1 by loading the tooth with the resultant force due to the meshing as a result of loading the gear with a moment equal to 50 Nm. In the analysis performed, the gear was locked while the calculated resultant force was applied to one tooth. Figure 4-A shows the method of applying the gear force to the side of a tooth. The force was applied to an ellipse-shaped fragment of the tooth side. Figure 4-B shows the gear locking by applying a pin mate at the midpoint of the gear axis of rotation. The same initial conditions were assumed for each gear option. For each gear analysed, a steel material was determined and selected from the list of materials found in the Nastran 2021 environment. The grid size of the gear was set to 1 mm (Fig. 5-A), while the grid was compacted to 0.5 mm at and near the point where the force was applied (Fig. 5-B). Figure 6 shows the applied convergence settings -the maximum number of refinements was set to 10, and the stop criteria to 2%.
The static analysis was designed to determine the effect of the weight-reducing gear design modifications analysed on strength properties. After the analysis, several parameters were taken into consideration such as: displacement (max) [ Table 2 shows the obtained results for the considered solutions along with a percentage comparison of the obtained parameter values with respect to the solid gear (MB1).
Based on the results presented in Table 2, graphs were prepared and are shown in Figures 7-12.
Comparing the displacement (max) values obtained, it was found that the MB3 (ribbed disc) and MB5 (honeycomb) gears suffered the least deformation compared to the MB1 solid gear (Fig. 7). Considering the displacement along the Z-axis (max), the most advantageous option of the weight-reduced gear are the MB3 ribbed disc gear (Fig. 8) and MB7 ribbed disc gear with holes (Fig. 8), since among the analysed gears it achieved the lowest value of the analysed parameter. The gear with the MB2 disc obtained the highest values of von Mises stress (max) (Fig. 9) and underwent the greatest volume reduction (Fig. 10) relative to the gear designs analysed. The values of the principal moment of inertia I3 of the MB2 (disc), MB3 (ribbed disc), MB4 (shell), and MB7 (ribbed disc gear with holes) gears are comparable (within 900÷950 kg/mm 2 ) and relative to the MB1 base gear, they experienced the greatest decrease (Fig. 11).
Analysing the graph shown in Figure 12, the values of displacement (max) [mm] for the MB2   However, for the MB3 ribbed disc gear and the MB5 honeycomb gear, the parameter under consideration takes on a value higher by 19÷25% in relation to the MB1 solid gear. The MB7 ribbed disc gear with holes takes on a value about 30% higher than the MB1 solid gear. The value of displacement along the Z-axis (max) [mm] for the gear MB2 with the disc and for the gear MB6 with bean-shaped holes is the smallest (relative to the MB1 base gear it is about 1% smaller) (Fig. 7, Fig. 12). However, for the MB3 ribbed disc gear, MB4 shell gear, and MB7 ribbed disc gear with holes, it is greatest-it is within ±7% relative to the MB1 solid gear without modification (Fig. 12).
The von Mises stress (max) value [MPa] relative to the MB1 base gear for the MB2 disc gear had the least reduction ( Fig. 9) of about 6% (Fig. 12). For the MB3 ribbed disc gear, MB4 shell gear, and the MB7 ribbed disc gear with holes, the maximum von Mises stress values are comparable (Fig. 9), and relative to the MB1 base gear, they decreased by about 16% (Fig. 12).
The MB2 disc, MB3 ribbed disc, MB4 shell, and MB7 ribbed disc with holes gears experienced the greatest volume reduction and thus   weight reduction (Fig. 10) of over 32% relative to the unmodified MB1 gear (Fig. 12). The 10 mm thick disc MB2, 10 mm thick ribbed disc MB3, shell MB4, and ribbed disc with holes MB7 gears achieved similar values of Principal moment of inertia I3 [kg/mm 2 ] (Fig. 11) and declined from the base gear MB1 by more than 20% (Fig. 12). The MB5 honeycomb gear and the MB6 bean-shaped holes gear had the  smallest decrease in principal moment of inertia relative to the MB1 base gear by 11% (Fig. 12). It can be seen from Figure 12 that the obtained values of the analysed parameters of the MB2 disc, MB3 ribbed disc, MB4 shell, and MB7 ribbed disc with holes gears are close to each other and reach similar values with respect to the MB1 base gear. Figures 13 and 14 show the obtained displacement and von Mises stress distributions for the analysed structural solutions.

CONCLUSIONS
Reducing the weight of the gear transmission is of particular importance in the aviation and automotive industries due to, among others, the reduction of the negative impact of dynamic phenomena occurring during the operation of the gear transmission. First of all, gears with reduced mass, made by injection methods from polymer materials, are used in household appliances, in which, thanks to their use, one can notice, among others, reducing material consumption, reducing the mass of the entire system, reducing dynamic surpluses, and damping vibrations. Such gears also allow for lubrication-free operation and are often used with metal gears whereby the sound intensity that occurs with the interaction of the metal gears is reduced. Knowledge of the strength properties of weight-reduced gears enables their proper design to guarantee the transfer of a given load under specified initial conditions. Also, static analysis in the MES environment will allow the determination of deformations and strains that can have a significant impact on the strength of the analysed gear, and which can be eliminated to some extent, or their impact can be minimized already at the design stage before their final production.
The research presented in this paper was aimed at the analysis of selected design solutions of weight-reduced gears with a changed disc geometry, keeping in mind a relatively constant stress level while reducing the weight as much as possible. It is worth mentioning that while designing the construction of the disc as well as the gear wheel itself, you should take into account the manufacturing process that will be used to produce the considered gears, the material from which the elements are to be made and the working conditions they are to meet. Based on the analysis of selected design solutions of gears with reduced weight, it was found that there is no linear relation between the reduction in weight and the reduction in the considered strength parameters. Considering the analysis performed, it can be noticed that the most favourable solution is the gear with a 10 mm thick MB3 ribbed disc due to the similar values of displacements obtained in relation to the MB1 solid gear (increase in displacement (max) by over 24% and decrease in displacement along the Z axis (max) by over 6% in relation to the MB1 base gear (Fig. 12)), as well as a significant reduction of the principal moment of inertia (by about 32% in relation to the MB1 solid gear (Fig. 12)) and decrease in weight, and thus volume (decrease by over 32% in relation to the MB1 base gear (Fig. 12)). The greatest reductions in weight, and thus volume, relative to the MB1 solid gear except for the MB3 gear (with a ribbed disc) were the MB2 gear with a 10 mm thick disc at the center (down more than 37%), the MB4 shell gear (down more than 34%) and the MB7 ribbed disc gear with holes (down more than 33%) (Fig. 12). The MB3 and MB7 gears are gears of similar geometry, but the MB7 gear also has cylindrical holes. Comparing the results of the analysed parameters for both gears, it can be concluded that the values of the displacement along the Z-axis (max), von Mises stress (max), volume and the principal moment of inertia are similar -only the MB7 gear achieves higher values of displacement (max) than MB3 gear (Fig. 12). In addition, further work should be undertaken to investigate the MB5 honeycomb gear, as this gear achieved the lowest difference in displacement (max) values (a decrease of about 20% relative to the MB1 solid gear (Fig. 12)) and one of the lower differences in displacement along the Z-axis (max) (a decrease of about 2% relative to the MB1 base gear (Fig. 12)) and von Mises stress (max) (a decrease of more than 10% relative to the MB1 solid gear (Fig. 12)) relative to the MB1 base gear among the gears analysed.