Improving the wear resistance of 3D printed spur gears through a free-form tooth flank optimization process

. Involute gears have traditionally been the preferred choice for gear transmission systems due to their simplicity and interchangeability. However, there are applications where they do not provide the most durable and efficient solution. While the cost of implementing optimized non-involute gears in most applications often outweighs their comparative advantages, the advent of additive manufacturing has opened up possibilities for designers to explore alternative gear tooth profiles. This is particularly relevant in the realm of plastic gears, where optimized non-involute gears produced through 3D printing can address their primary drawbacks, such as surface durability and wear resistance. In this study, a comprehensive free-form optimization process was conducted to determine the optimal tooth profile that minimizes wear on 3D printed spur gears during operation. The tooth flank geometry was represented using a 4th order B-spline curve, and a genetic algorithm was employed to determine the optimum positions of the control points aiming to minimize wear depth across the tooth flanks. The spur gears were manufactured using Fused Deposition Modeling (FDM) with PLA material. The parameters of the additive manufacturing process were experimentally fine-tuned to achieve the best possible accuracy. To evaluate the performance of the optimized free-form gears, two case studies were implemented, demonstrating that the optimized gears achieved a remarkable reduction of average wear depth by more than 50% and a reduction of maximum wear depth by more than 69% compared to standard involute gears. To further validate the effectiveness of the optimization method, experiments were carried out using an FZG test rig. The profiles of the tooth flanks were measured on a Coordinate Measuring Machine (CMM) before and after the experiments to compare the wear depth against the standard involute gears. The results revealed a significant improvement in the wear resistance of the tooth flanks, with a reduction of wear depth of 44.1%.


Introduction
Gears are widely regarded as the most commonly utilized mechanical transmission system across various industries, including automotive, aerospace, agriculture, mining, and energy [1].Despite their diverse applications with distinct requirements, the vast majority of gear transmissions feature the same tooth flank profile, which follows the involute curve.Involute gears offer several advantages, such as insensitivity to variations in center distance, simplicity and robustness in manufacturing [2,3,4,5].However, depending on the specific application, they also possess certain drawbacks such as limited load capacity due to their relatively poor pitting resistance, suboptimal lubrication characteristics, and proneness to interference [2,6,7].To address these drawbacks, researchers and gear designers have proposed alternative gear profiles that can enhance the performance of gear transmissions by increasing efficiency and load capacity or reducing noise and vibrations.Among the well-known alternatives are cycloidal or circular-arc gears (i.e., Novikov gears) that offer improved efficiency and load capacity, respectively.However, their performance in practical applications is often worse than that of involute gears due to their high sensitivity to manufacturing and assembly errors [2].In addition to cycloidal and circular arc gears, researchers have introduced a wide variety of tooth profiles.Komori et al. [6] proposed the use of a rack-cutter profile composed of many continuously divided minute involute curves to create the "LogiX" tooth profile, which offers twice the durability of involute gears.Tsai & Tsai [8] developed a method for designing high-contact-ratio gears based on a quadratic function of the pressure angle to improve load capacity and NVH characteristics.Litvin et al. [9] implemented a parabolic rack-cutter profile to produce a modified Novikov-Wildhaber double-crowned helical gear drive, resulting in reduced noise and vibrations and enhanced load capacity.Zhang-Hua et al. [10] proposed a novel tooth profile based on a hybrid line of action, which incorporates straight line and circular-arc segments to control curvature, sliding velocities, and contact ratio of the meshing gears.Luo et al. [4] introduced a cosine gear profile that exhibits lower sliding coefficients and reduced contact and bending stresses compared to standard involute gears.Wang et al. [11] developed a tooth flank profile based on a parabolic line of action, aiming to reduce contact and bending stresses and improve efficiency.All of the aforementioned approaches aim to enhance the performance of gear pairs by initially introducing a predetermined tooth flank profile and subsequently evaluating its performance.However, it would be advantageous for gear designers to have direct control over the optimization of specific performance objectives.In this regard, Liu et al. [12,13] developed tooth profiles based on a predefined distribution of relative curvature, resulting in improved bearing capacity and lubrication characteristics.Additionally, Wang et al. [14] proposed a gear design method that focuses on achieving higher efficiency by utilizing given sliding coefficients.Taking a further step, Yeh et al. [2] proposed an optimization methodology that allows for control over the shape of the tooth flank to improve multiple performance objectives, including load capacity and efficiency.This approach utilizes a deviation function technique based on Non-Uniform Rational B-Splines (NURBS) to control the shape of the tooth profile.Yu & Ting [5] introduced a free-form conjugation modelling technique that enables the modification of the rack-cutter profile or the contact path to achieve desired performance characteristics.Bspline curves are employed to model these profiles, and certain modelling challenges are discussed, such as undercutting or cusps in the conjugate profiles.Furthermore, Kalligeros et al. [15] presented a comprehensive free-form optimization method aimed at enhancing the pitting resistance of spur gear tooth flanks.The tooth profiles are represented as B-spline curves, and the positions of their control points are optimized to achieve a minimum and uniform distribution of relative curvature along the contact path.Despite these advancements in alternative gear designs, the industry has shown reluctance to incorporate them into their applications.This reluctance primarily is derived from the substantial effort and cost associated with changing and fine-tuning the manufacturing processes, which are predominantly tailored for involute gears.These obstacles often outweigh the potential benefits gained from utilizing non-involute gear designs [5,16].However, the emergence of additive manufacturing and 3D printing as viable alternative manufacturing processes can address these challenges.Unlike traditional manufacturing methods, which often require complex machining processes, 3D printing enables the production of gears with intricate, free-form geometries that were previously difficult to achieve [17].This capability allows engineers and designers to optimize gear designs for specific applications, resulting in improved performance, reduced weight, and enhanced functionality [18].Versatility and design freedom provided by 3D printing technology are predominant advantages of plastic gears, along with lightweight construction, self-lubrication and reduced manufacturing costs [19].However, alongside the advantages of plastic 3D printed gears, wear emerges as a major concern [20].Wear is the gradual loss of material that occurs when two surfaces slide or roll against each other and it can have a profound impact on the performance and longevity of plastic gears.In the context of 3D printed plastic gears, the unique material properties, manufacturing processes, and gear geometry all play significant roles in the resulting wear.In an attempt to minimize wear, Zhang Ye et al. optimized 3D printing parameters (printing speed and temperature, infill, and bed temperature) using ANNs, achieving a remarkable 300% reduction in wear [21].Tunalioglu et al. introduced models to predict the wear of FDM plastic gears [22].However, determining the wear coefficient for plastic gears in the case of 3D printing remains an open issue, as standard tests such as pin-on-disc tests are not adequate [23].This study focuses on the geometrical optimization of gear tooth flanks to maximize their wear resistance.A comprehensive optimization algorithm is developed, treating the tooth profiles as B-spline curves and determining the optimal placement of their control points through a genetic algorithm to produce flanks with desirable wear characteristics.The optimized free-form curves achieved a remarkable reduction of wear by over 50%.The results were validated through experiments on both standard involute and free-form gears using an FZG testing machine and a CMM measuring machine.The experimental results revealed that the optimized gears experienced 44.1% less wear than their involute counterparts.

Free-form optimization algorithm
Building upon the research presented in [15], the objective of the optimization process is to establish a systematic algorithmic procedure capable of generating optimized tooth flanks that adhere to the fundamental principles of gearing (Fig. 1).These optimized profiles should ensure the absence of undercutting, interference, cusps, or any other geometric or operational issues that may arise.
The optimization process is structured as follows: 1.The tooth flank profile is represented by a B-spline curve with a specified number of control points and polynomial curve order.2. The positions of the control points are adjusted using a genetic algorithm.3. The resulting profile is evaluated to determine if it satisfies the law of gearing at each point.4. If the profile meets the requirements, the conjugate profiles (i.e., the conjugate gear profile, rack-cutter profile and contact path) are calculated implementing the established analytical theory of gearing and the involutization method.5. Based on these profiles, the objective function is calculated.6.In case the produced profiles violate any constraints related to their geometric or operational characteristics, a penalty is applied to the objective function.7. The entire procedure is iteratively repeated until convergence is achieved.

Modelling of tooth profile
In non-involute gear design, it is often preferred to optimize the rack cutter or the contact path profile to indirectly enhance the performance of the conjugate tooth flanks [5].While this approach can be meaningful in certain cases (i.e., to ensure an easy-to-manufacture rack cutter profile), there is a desire among designers to directly control the shape of the tooth flank, which in turn directly affects the desired performance improvements.However, a common challenge arises when applying Buckingham's analytical theory of gearing [24], as it requires solving an implicit equation to calculate the conjugate profiles [15].Incorporating a numerical convergence algorithm to solve this equation would significantly increase computational cost and potentially undermine the reliability and robustness of the optimization process.This is one of the reasons why gear designers tend to avoid utilizing the tooth flank as the primary geometry for optimization.This limitation can be overcome by employing the involutization method introduced by Spitas et al. [25,26].In this method, every point of the tooth profile is associated with a local involute curve.By applying the fundamental principles of gearing and utilizing the inherent properties of involute curves, it becomes possible to explicitly compute the contact path of the gear pair.
Once the contact path is determined, all the corresponding conjugate profiles can be derived utilizing the analytical theory of gearing [24].More specifically, when considering a random point G on the tooth flank characterized by  = () that has an inclination of   ⁄ |  , it is possible to associate a local involute curve with G.This local involute curve is defined by a local base circle and a local pressure angle that intersects with G and shares locally the same inclination.The calculations for the local base circle and pressure angle are as follows [25]: where   and   the local base circle and pressure angle respectively,   , (  ) the coordinates of G and  0 the pitch circle of the pinion.Based on the local involute curve, the coordinates of the contact point that corresponds to G can be determined as: where   ,   the coordinates of the contact point and   the radius of G.By applying this process to each point along the tooth flank, the complete contact path can be generated.Subsequently, the conjugate gear profile can be explicitly calculated using the analytical theory of gearing, as outlined below: where  2 ,  2 the coordinates of the conjugate gear profile,   and  2 the pitch circles of the pinion and the conjugate respectively,  and  2 the rotation angles of conjugates points of the pinion and the conjugate gear respectively in order to mesh.
To establish a truly free-form tooth flank optimization process, it is necessary to find an efficient and convenient method for modifying the tooth profile.One highly effective approach is to represent the tooth flank as a parametric curve, such as a B-spline or Bezier curve, which can be easily adjusted by manipulating the positions of a small number of control points.These control points serve as the variables in the optimization problem, with the objective of the optimization being to determine their optimal placement.For the optimization problem at hand, B-spline curves are employed due to their superior local control compared to other parametric curves (i.e., Bezier curves).Fourth-order polynomials are utilized whenever possible, as higherorder B-spline curves yield smoother curves -a desirable characteristic for tooth flanks [2].The determination of the appropriate number of control points is specific to each case and their position should be subject to certain constraints, which will be examined in the subsequent section.As illustrated in Fig. 2 only the portion of the tooth flank that engages with the conjugate gear profile is modeled since it is the section of interest for this optimization problem.The lower part of the tooth is generated once the optimization process is finalized.In Fig. 2, it is also demonstrated how changing the position of the control points can affect the tooth profile.Even an adjustment to just one control point can result in a significant alteration in the shape of the tooth profile.

Fig. 2.
Influence of the position of the control points of the Bspline curve to the tooth profile.

Optimization constraints
The gear design community has faced a challenge in establishing a well-defined set of constraints to effectively assess the suitability of the generated free-form tooth flanks [5].Firstly, the produced tooth flanks should be free from issues like undercutting or interference during operation.Additionally, the generated tooth profiles should not result in conjugate gear geometries that exhibit cusps or other discontinuities.At the same time, it is essential to ensure that the contact ratio remains within an acceptable range and that the profiles adhere to the law of gearing at each point.
To prevent interference during operation, it is crucial to ensure that the B-spline curve representing the tooth flank includes the lowest point of tooth contact of the pinion (i.e., the generating curve).This enables the creation of a corresponding conjugate point on the gear profile, thus avoiding interference.Since the radius of the lowest point of tooth contact varies for each case, the control point at the lower end of the tooth profile should be positioned to accommodate all possible scenarios.If n is the number of control points and the n th control point is located at the lower point of the tooth flank, the first constraint for its radius can be expressed using the following relation. , =  12 −  ,2 (5) where  , is the radius of the n th control point,  12 is the center distance and  ,2 is the addendum circle of the conjugate gear.Given the addendum circle of the gear, no contact can occur between the meshing gears below this radius ( , ).To prevent undercutting, two measures are implemented.Firstly, the addendum and dedendum coefficients of both gears can be adjusted to address potential undercutting issues.If these adjustments are insufficient, an upper limit should be imposed on the x-coordinate of the n th control point.The precise value for this limit can be determined after a few optimization iterations.Furthermore, setting an upper bound on the x-coordinate of the n th control point helps prevent the generation of a "pointy" tooth (i.e., a tooth with an excessively narrow top land width) in the conjugate gear.To address the same issue in the pinion, it is necessary to establish a lower bound for the x-coordinate of the first control point located at the tip of the tooth flank.According to [27], the narrowest width of a tooth top land should be 20% of the gear module.Consequently, if the y-axis of the absolute coordinate system aligns with the tooth's axis of symmetry, the lower bound should be set at 0.1m, where m is the module of the gear pair.For the remaining control points, while specific bounds may not be necessary, leaving them entirely unconstrained can result in convergence issues.This is because in such a case the majority of generated profiles tend to not adhere to the optimization constraints.After a large number of optimization iterations, a robust approach to setting bounds for the control points is as follows: an involute curve is initially generated based on the gear parameters, which is then interpolated using a B-spline curve with the desired number of control points.The bounds for the control points are then defined as a range around these values.This procedure ensures that a sufficient number of the generated curves comply with the constraints, allowing the optimization algorithm to converge.However, it is essential to ensure that the bounds are large enough to avoid guiding the optimization towards involute-like curves.Alternatively, different arbitrary bounds can be implemented, which it was proved that they lead to similar results after sufficient iterations of the optimization algorithm.Another crucial constraint that needs to be considered is the contact ratio ε of the free-form gears.For a spur gear pair, the contact ratio should exceed 1, with ISO and AGMA standards recommending values above 1.2 or 1.3 to ensure smooth operation [28,29].However, achieving a sufficient contact ratio for optimized free-form gears has proved to be a challenge, as it is typically a contradicting parameter with efficiency.To ensure an appropriate contact ratio for the optimized gears, a penalty is incorporated into the objective function.This penalty becomes higher for lower contact ratio values.After experimenting with various expressions, the following one was selected that has the distribution illustrated in Fig. 3 and is added as a factor in the objective function.
In addition to the aforementioned well-known constraints that apply to standard involute gears, when dealing with free-form gears it is necessary to establish additional constraints to ensure that the generated profiles adhere to the law of gearing and are manufacturable, without any cusps or other discontinuities.After generating a large number of free-form profiles, two major problems have emerged during the calculation of the conjugate geometries.Firstly, some points encounter calculation issues as they lead to negative radicands.Secondly, conjugate profiles with cusps and C1 discontinuities are frequently generated, as depicted in Fig. 4. When the calculation is impossible at certain points, it indicates that these points cannot satisfy the law of gearing.In other words, regardless of the point's rotation, its normal vector will never intersect the pitch point of the pinion.Free-form profiles that contain such points cannot function as tooth flanks and should be heavily penalized by the optimization process.
To address the problem of discontinuities in the calculation of conjugate profiles, certain measures are implemented.As depicted in Fig. 4 when such issues arise, the resulting conjugate profile appears to consist of multiple distinct segments.For each segment, an additional penalty is applied as a factor to the objective function, since a high number of these segments indicates that the free-form profile deviates significantly from being a valid tooth flank.

Wear model
As previously stated, the primary objective of the optimization process is to propose alternative tooth flank profiles that exhibit improved wear resistance during operation.To establish a comprehensive objective function, the wear model presented in [30,31] is incorporated.
The wear model is based on the well-known Archard's wear equation that is typically expressed as: where V is the volume of the worn profiles being removed, s is the sliding distance between the contact surfaces, K is the dimensionless wear coefficient, W is the normal load and H is the surface hardness of the profiles.For the case of meshing gears, the expression takes the following form: where p is the Hertzian pressure at the contact point, k is the dimensional wear coefficient and h is the wear depth that needs to be minimized.For a given contact point P, the wear depth ℎ  can be calculated as: The wear depth of P after n (-already used for number of control points) meshing cycles can be determined by the following expression ℎ , = ℎ ,−1 +  ,−1   (10) where ℎ , the wear depth after n meshing circles, ℎ ,−1 and  ,−1 the wear depth and the contact pressure respectively after n-1 meshing cycles and   the relative sliding distance of the contact point P The sliding distance of the points P1 and P2 of the pinion and the gear respectively that corresponds to the contact point P can be calculated as where   is the hertzian half-width and  1 ,  2 are the sliding velocities of P1 and P2 respectively.After combining the above equations, the wear depth of the contact point P after N meshing cycles can be determined as: ℎ  =   (13) The hertzian half-width can be calculated by the following relations: where   the circumferential load on the tooth flanks,   the output torque, d0,2 the gear's pitch circle, b the gear wheels' width, R* the relative radius of curvature at each contact point, R1, R2 the radius of curvature of the pinion and the gear respectively at each contact point,  * the equivalent modulus of elasticity,  1 ,  2 the modulus of elasticity of the pinion and the gear respectively and v1, v2 the Poisson ratio of the pinion and the gear respectively.
To accurately model the load sharing process between meshing teeth without introducing a time-consuming numerical process into the optimization algorithm, the following approach was implemented.After calculating the highest and lowest points of single tooth contact for each pair of conjugate profiles, the entire load is applied to each tooth flank during the single-tooth contact period.However, during the double-tooth contact, the load is reduced as it is distributed among multiple pairs of teeth.
To ensure a smooth distribution of wear depth across the tooth flank, the load-sharing model presented in Fig. 5 was incorporated.

Optimization algorithm
The majority of published studies, particularly in the last two decades, focused on gear optimization have employed evolutionary algorithms due to their capability to effectively avoid local minima and their relatively lower computational cost compared to earlier approaches [1].Among the various evolutionary algorithms, the genetic algorithm [32] is the most widely used.
For this optimization problem, the genetic algorithm (GA) incorporated in MATLAB was utilized.To obtain the best possible results, the algorithm's parameters were systematically investigated.The tournament selection method was chosen to determine the 'parents' for the subsequent generations in the optimization process.The mutation function employed was 'mutationadaptfeasible' since the optimization variables were subject to constraints.Additionally, due to the presence of constraints in the optimization problem, the 'crossoverintermediate' was selected as the crossover function.The specific values for the crossover fraction, elitism, maximum generations, and population data were independently determined for each individual case.Another significant parameter for the optimization algorithm is the number of variables, which is directly related to the number of control points.More specifically, each coordinate of the control points serves as an optimization variable, except for two cases; one of the coordinates of the first and last control points is determined directly to ensure that the lower and upper radii of the tooth flank are fixed.
As for the number of the contact points, apart from the fact that a high number of control points and thus variables could lead to convergence issues for the algorithm, it was found that as the number of control points increases, the number of the produced curves that actually adhere to the constraints reduces significantly.For the cases of 5 and 7 control points, 10,000 free-form curves were produces for each case and tested to determine if they can firstly comply to the law of gearing and if yes then if they have cusps/discontinuities or not.
As it can be seen by the results the percentage of the freeform curves that are valid tooth flank profiles falls from 30.9% for the case of 5 control points to only 2.5% for 7 control points.
Regarding the number of contact points, it was observed that a high number of control points and variables could not only potentially lead to convergence issues for the optimization algorithm, but it also results in a significant reduction in the proportion of generated curves that actually comply with the constraints.To investigate this, 10,000 free-form curves were generated using 5 control points and another 10,000 using 7 control points and were tested for compliance with the law of gearing and the presence of cusps or discontinuities.The results (Table 1) indicate that the percentage of valid tooth flank profiles decreases from 30.9% for 5 control points to only 2.5% for 7 control points.Consequently, the exact number of control points should be also selected for each case independently.
Table 1.Distribution of profile curves regarding their adherence to the law of gearing and the geometrical constraints for the case of 5 and 7 control points (10,000 profiles each).

Results
The optimization process was conducted on two distinct case studies, as outlined in Table 2.The first case study was chosen based on the specifications of the FZG test rig implemented for the experimental evaluation of the optimized gears.A second case study that differs significantly in terms of the transmission ratio was examined to allow for a comprehensive assessment of the optimization procedure's effectiveness under varying conditions.

Case study #1
The first case study involves a spur gear pair with a transmission ratio of 1.033, which is utilized in the FZG testing machine described in Section 4. To achieve the best possible outcome, the optimization settings were parametrically explored.Various numbers of control points were employed, and different values for optimization settings such as crossover fraction and population size were tested.The geometric and operational parameters utilized in this case study are summarized in Table 3, while the optimization settings employed are listed in Table 4.A range of control points from 4 to 7 was implemented in the optimization process, and the results are presented in Table 5.The corresponding generated profiles can be observed in Fig. 6.Two objective functions were considered; the first one focused solely on the integral of the wear depth along the contact path of the meshing tooth flanks (Eq.18), while the second one incorporated the maximum absolute values of the wear depth in addition to the integral (Eq.19).Unless specified otherwise, the second objective function was employed for the optimization processes as it yielded slightly improved and notably smoother wear depth distributions.
As mentioned in Section 2.2, the addendum and dedendum coefficients of the optimized gears were adjusted.As depicted in Fig. 7, the optimized profiles exhibit a steeper slope along their length, which raises undercutting issues.To address this, both the addendum and dedendum coefficients are reduced, as indicated in Table 3.
The results presented in Table 5 and Fig. 8 demonstrate that the generated non-involute profiles exhibit significantly improved wear resistance compared to their standard involute counterparts.The average wear depth was reduced by up to 53.6%, while the maximum wear depth was reduced by up to 69.1%.The reduction in the addendum and dedendum coefficients played a role in both the average wear depth reduction, as it decreased the length of the contact path, and the maximum wear depth reduction, as the wear mechanism is more active at the extremities of the tooth profiles.However, this reduction in coefficients also led to a slight decrease in the already -anticipatedly-reduced contact ratio, which is slightly below 1.2 for all cases.
Even though all the generated non-involute curves demonstrate improved wear performance, those created from a higher number of control points exhibit a more wavy profile.While wavy profiles have the potential to yield even better results, their practical implementation is often challenging due to their sensitivity to assembling errors, such as center distance variations.Therefore, the smoother non-involute profiles generated with 4 control points (Fig. 6) are considered the most suitable for this application, as they also offer the best results in terms of average wear depth and exhibit the higher contact ratio.

Case study #2
For the second case study, a gear pair with a transmission ratio of 2.579 was employed.The material of the gears remained the same as in the previous case study, while the operational characteristics, specifically the input speed and the output torque, were altered as outlined in Table 6.Slight variations in the optimization settings were implemented to obtain the final results, as presented in Table 7.In this case study, even greater performance improvements were achieved.The average wear resistance was reduced by up to 56.6%, and the maximum wear depth was reduced by as much as 79.7%.The noninvolute curves generated from different numbers of control points, as shown in Fig. 9, exhibit remarkable similarity, demonstrating the robustness of the optimization process.Among these curves, the one generated from 5 control points was selected due to its slightly higher contact ratio value.

Experimental validation
All spur gears were manufactured through Fused Deposition Modeling (FDM) using a Creality Ender CR10-S5 printer, which was chosen for its large bed, allowing both gears to be printed simultaneously.STL files were generated from the original CAD models, and the final G-code was created using CURA Slicer Software, with the printing properties specified in Table 9.These parameters were selected after a large number of trials to achieve the best possible accuracy for the tooth flank profile.The produced gears are illustrated in Fig. 12.In order to evaluate the main form errors of the gear teeth in relation to the nominal CAD model, a Coordinate Measuring Machine (CMM) is utilized (Fig. 13).The CMM model and its specifications are shown in Table 10.

Involute
The process begins with importing the CAD model of the gear into the graphic environment of PC-DMIS.Additionally, a point cloud is imported into the PC-DMIS software to guide the CMM.A total of 50 points are used on each side of the tooth, positioned at a specific vertical distance from the upper surface of the gear.Once the experiments at the test rig are completed, the gears are measured again to calculate the final wear.
Using the same PC-DMIS guide program, precise measurements of the final wear on the gears are ensured.
To enhance repeatability in measurements, jigs are designed and 3D printed for each gear (Fig. 14).This approach eliminates the need for realignment after completing the measurements, as it matches the setup done before.The measurement data is exported as text files and then imported into MATLAB software to calculate the wear depth.For the experimental evaluation of Case Study #1, the FZG test rig illustrated in Fig. 15 is used.Both standard involute and optimized free-form spur gears are tested for comparison.As stated before, the optimized profile generated by 4 control points was selected as the freeform tooth flank.Following the procedure described above, the gears are manufactured and then measured by the CMM to obtain the initial profile of the tooth flanks.Subsequently, the spur gears are subjected to a load of 10 Nm for 180 minutes using the FZG test rig.Afterward, they are measured again in the CMM machine and the wear depth at each point is determined.Some representative results are presented in Fig. 16.Firstly, it is evident that the distribution of the wear depth along the tooth flank does not precisely follow the theoretical distribution presented in Figs. 8 and 11.This discrepancy is mainly due to the geometrical deviations of the tooth flanks caused by 3D printing accuracy.Although the achieved accuracy was not poor for 3D printed parts, the resulting profiles exhibited a wavy pattern and the wear depth distribution followed this profile as illustrated in Fig. 17 (right).Geometrical deviations were more pronounced in some teeth than others, as depicted in Fig. 17 (left), resulting in local increases in wear depth, such as in the tip regions of teeth #2 and #20 of the free-form conjugate and the involute pinion respectively (last two diagrams in Fig. 18).However, the majority of the tooth flanks, despite containing geometrical deviations, yielded representative wear depth distributions that allowed for a valid comparison between the free-form and the involute profiles.

Conclusions
In this study, a comprehensive free-form optimization process was conducted to determine the optimal tooth profile that minimizes wear on 3D printed spur gears during operation.The tooth flank geometry was represented using a B-spline curve, and a genetic algorithm was employed to determine the best positions for the control points, aiming to minimize wear depth across the tooth flanks.The major outcomes of the study are summarized below: • Development of a robust and comprehensive freeform tooth flank optimization algorithm capable of handling the complex geometric constraints of tooth profiles and generating conjugate flanks with improved performance.

•
Implementation of two case studies to minimize the wear depth of spur gears manufactured from PLA through 3D printing.In these cases, the optimized free-form gears achieved a significant decrease in average wear depth, with reductions of 53.6% and 56.6% compared to their standard involute counterparts.Additionally, the maximum wear depth was reduced by 69.1% and 79.7% respectively.

•
Conducted experiments using an FZG test rig and performed measurements on a CMM both before and after testing.The results showed that the optimized free-form gears experienced 44.1% less wear than their standard involute counterparts.

Fig. 8 .
Fig. 8.Comparison of wear depth along the tooth flanks of the free-form gears generated with 4, 5, 6 and 7 control points and their standard involute counterpart for case study #2.

Fig. 11 .
Fig. 11.Comparison of wear depth along the tooth flanks of the free-form gears generated with 4, 5, 6 and 7 control points and their standard involute counterpart for case study #2.

Table 2 .
Geometrical parameters of case studies

Table 3 .
Geometrical, material and operational parameters of case study # 1

Table 4 .
Optimization parameters of case study #1.

Table 5 .
Optimization results for case study #1.

Table 6 .
Geometrical, material and operational parameters of case study #2.

Table 7 .
Optimization parameters of case study #2.

Table 8 .
Optimization results of case study #2.

Table 11 .
Experimental results regarding the average wear depth of standard involute and optimizes free-form gears.