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Dieser Artikel geht auf die wachsende Bedeutung der Optimierung des NVH-Verhaltens von Stirnrädern ein, insbesondere im Zusammenhang mit elektrifizierten Automobilgetrieben. Es untersucht den Einsatz gezielter Mikrogeometrie-Streuung als Mittel zur Reduzierung von Tongeräuschen und zur Verbesserung der akustischen Leistung. Der Text stellt eine neuartige Methode zur Gestaltung von Schleifschneckengeometrien vor, die die Herstellung von Zahnrädern mit unterschiedlichen Mikrogeometrien ermöglicht und sich den Herausforderungen wiederkehrender wellenförmiger Topographieabweichungen stellt. Durch einen detaillierten Prozess von Rückwärts- und Vorwärtsberechnungen zielt die Methode darauf ab, geometrische Überschneidungen zu minimieren und die Qualität der Getriebefertigung insgesamt zu verbessern. Der Artikel enthält auch eine Verifizierung der Methode mittels quasi-statischer Zahnkontaktanalyse, die das Potenzial für ein verbessertes NVH-Verhalten im Vergleich zu Standardkonstruktionen aufzeigt. Darüber hinaus wird die praktische Umsetzung der Methode mit Verbandgeräten diskutiert und zukünftige Forschungsrichtungen zur weiteren Optimierung und experimentellen Validierung skizziert.
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Abstract
This report presents a new method for calculating and designing grinding worm geometries for continuous generating grinding in order to realize targeted micro geometry scattering on gear flanks. The method uses the idea to transfer the ideal micro geometry onto the worm topography utilizing a reversed rolling calculation. First, the reversed calculation of the grinding worm flanks is carried out by gradually breaking down a specified gear geometry with varying micro geometries into individual profile cross-sections. Afterwards, these profile cross-sections are transformed into a grinding worm geometry whose starts can manufacture different modifications. The grinding worm topography is calculated using an exemplary application in which targeted micro geometry scattering on a cylindrical gear is reverse generated onto the generating profile and then onto the grinding worm body. The example serves to verify the method. The calculated grinding worm geometry is used for forward generation of the flank geometries and kinematic differences between the nominal and actual geometry of the gear flanks are shown as a comparison.
A key advantage of this new approach is the avoidance of periodically repeating ripples on the tooth flanks, which occurred when using previous approaches to manufacture targeted micro geometry scattering in the generating grinding process due to differently dressed grinding worm starts. The new approach uses a dressing gear for shaping the grinding worm geometry. It achieves an improved NVH behavior of the gear pair concerning the tonality through varying flank modifications by utilizing the topographically shaped grinding worm starts. This was verified as part of a quasi-static tooth contact analysis in which the derived grinding worms were used to generate replica of dressing gears and investigate the acoustic excitation behavior. The simulation results show significantly reduced levels of the transmission error in the gear mesh orders and at the same time lowered tonal noise components, which indicates a subjectively more favorable noise behavior.
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1 Introduction and motivation
The growing relevance of noise emissions in the automotive sector, increased customer requirements and reduced noise masking as a result of the electrification of powertrains are causing the acoustics of transmissions to become increasingly important [1, 2]. In particular, the tonal acoustic behavior of high-speed transmissions in electromobility are perceived negatively by customers [3]. The noise behavior is essentially determined by the torsional vibration excitation in the gear mesh, which significantly affects the product quality [4].
A key objective in the design of gear drives is to reduce the excitation in the gear mesh. Gear noise can be improved by flank modifications such as tip relief and helix angle modifications as well as topological modifications [5]. Nevertheless, the NVH behavior can be perceived as disturbing to the human hearing even at a low excitation level. [6, 7]. Customers evaluate the acoustic quality not only on the basis of the noise level, but perceive noises as particularly negative if they are considered incompatible with the product itself [8]. The subjective auditory experience of technical systems is described by psychoacoustics [9]. In particular, tonal components (noise at one single frequency) in the hearing spectrum contribute to a noise characteristic that is perceived as disturbing [1]. They result in particular from repeating events such as the periodical single gear mesh. To reduce these noise components, targeted micro geometry scattering can be used when designing gear pairs [10]. Micro geometry scattering is a combination of conventional profile and lead modifications, which are selectively distributed on the different flanks of the gear around the circumference.
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With the increasing demands on the excitation and noise behavior, the demands on the precision of the tooth flank geometries to improve the operational behavior are also increasing. Conventional modifications such as crowning or tip relief are carried out during the hard fine machining of gears [7, 11, 12]. The continuous generating grinding process offers the possibility of producing tooth flank geometries with high process stability [5]. Due to its high productivity, this process is particularly suitable for high volume gear production in the automotive industry [13]. In the series production of small and medium-sized gears, the continuous generating grinding process is generally the most common method [14, 15]. On the production side, targeted micro geometry scattering on gears has so far only been realized in the discontinuous profile grinding process [10]. In contrast, continuous generating grinding offers a more productive and suitable hard fine machining process, which, however, has geometric and kinematic limitations due to the process itself. The production of targeted micro geometry scattering is not yet possible in this process due to a limited machine configuration and lack of methods for practical implementation. For this reason, the manufacturability of such gears must first be taken into account when designing them using the corresponding rolling process. Preliminary studies show that a scattering of geometric properties on the tooth flanks from tooth to tooth can be achieved by using multi-start grinding worms with differently dressed grinding worm starts. In order to produce different defined micro geometries, complex profiling of the grinding worm starts along the start width is required [16]. As the necessary line dressing process is very time-consuming, it would significantly reduce the productivity advantage of generating grinding compared to profile grinding. One of the manufacturing objectives must therefore be to know the exact grinding worm topography to be dressed and how this can be applied efficiently to achieve an adequate production result for micro geometry scattering. The rolling coupling and the point contact in the meshing between grinding tool and dressing tool, as well as gear and grinding worm, are the challenges that must therefore be addressed in more detail.
2 Dressing process with gear-like dressing tools
The effectiveness of targeted micro geometry scattering to optimize the acoustic performance has been proven both mathematically and experimentally [10]. In previous work, a method was also developed which generates the geometry of multi-start grinding worms individually for each start and generates a gear geometry using the entire start width [16]. A precise gear rolling simulation and the pre-calculation of the meshing conditions between tool and workpiece are essential here. The aim was to generate micro geometry scattering through increased flexibility in the design of the grinding worm geometry. Despite the successful implementation and verification of the method, geometrically recurring systematics and wave-shaped topography deviations arise as a result of the overlaps of the grinding worm profiles of different geometric designs meshing with the tooth space [16]. Recurring patterns and wave-like topographies on the tooth flank are considered to have a negative influence on the acoustic performance due to increasing tonality of the noise impression as well as an increase in modulation frequencies and should therefore be avoided.
No countermeasures were derived to reduce the geometric overlapping of the gear topography in relation to the design of the grinding worm. For this purpose, it is particularly important to consider the contact conditions during the dressing process and to know the required grinding worm topography. In the following, therefore, dressing methods that have a geometric similarity to the target workpiece as a dressing tool are discussed.
The grinding worm is dressed inside the machine using special profiling equipment such as form dresser rollers, dressing gears or profile dresser rollers. These conventional dressing systems work according to the start rotation principle, whereby each start of the multi-start grinding worm must be dressed individually. This leads to an increase in dressing time as the number of starts increases and can partially cancel out the time advantage achieved by using multi-start grinding worms. [17]
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As an innovative alternative, dressing with a diamond-coated dressing gear offers considerable advantages. In this generating process, the principle of “worm manufactures gear” is reversed by integrating a diamond-coated master wheel into the dressing process. During the rolling of the dressing gear and grinding worm, the required reference profile is generated in the normal cut of the grinding worm. As the dressing gear and the grinding worm are positioned at the required center distance from each other under crossed axes according to their lead or helix angles, the profile generation takes place in the shared plane of action. A major advantage of this method is that the dressing time is independent of the number of grinding worm starts. As a result, the productivity increases that can be achieved by using high-speed grinding worms are fully utilized. In addition, dressing with a dressing gear enables higher grinding worm speeds during the dressing process, as the speed is not limited by the maximum feed rate of the dressing tool. This leads to more consistent conditions between the dressing and grinding process and minimizes differences in worm expansion and balancing conditions. [17]
Geometrically, the reference profile of the dressing gear should match the gear to be machined. Profile modifications such as crowning or tip and root relief must be integrated into the dressing gear, similar to conventional dressing tools. The implementation of this process does not require any additional drives on the machine, meaning that only the existing NC axes can be used. The dressing gear can be permanently integrated into the clamping device of the workpiece, which makes handling easier and ensures high repeat accuracy. This simple and efficient design also contributes to the cost-effectiveness of the process. However, it may be necessary to make adjustments to the machine due to the deviation of the process sequence from conventional dressing methods. This includes, for example, precise positioning and synchronization with the machine axes, as well as an adjustment of the machine kinematics. [17]
Overall, dressing with a diamond-coated dressing gear represents significant potential in continuous generating grinding. It enables more efficient and precise conditioning of the grinding worms. This makes a significant contribution to increasing productivity and quality in gear production. Disadvantages are the high purchase costs of the dressing gear due to the high precision requirements of each tooth flank as well as a lack of flexibility compared to a form roller dresser due to specific axis coupling.
Liebherr Verzahntechnik has extended the basic principle of the dressing gear. In the adapted principle, the diamond-coated dressing gear is modified in order to save manufacturing costs. The process addresses this challenge by using a specially designed dressing gear with defined edge areas. With this approach, the teeth of the dressing gear have transitions from the face to the flank, which actively machine the grinding profile. The main advantage of this process lies in the possibility of creating modifications to the grinding worm surface by specifically positioning or varying the movements of the dressing gear during dressing. During the dressing process, the grinding worm is moved along its axis of rotation so that the defined edge areas of the dressing gear are guided along the entire width of the grinding worm. The exclusive use of the edge areas for material removal results in a theoretical point contact between the dressing gear and the grinding worm, which enables high precision in shaping the grinding worm threads. Micro geometries that vary from tooth to tooth have not yet been considered for this process. [18]
So far, it has not been investigated how micro geometries that vary from tooth to tooth can be realized with the use of dressing gears. The shown approach by Liebherr does not take into account individual modifications of the micro geometry for individual teeth over the gear circumference and the resulting worm topography. The development of a method for taking into account micro geometry scattering on a dressing gear when calculating the grinding worm topography for the production of locally varying modifications therefore remains an open research task.
3 Objective and approach
The state of the art shows that targeted micro geometry scattering is an effective measure for optimizing acoustic behavior of cylindrical gears [10]. Methods exist for generating varying tooth flank topographies in the generating grinding process using multi-start grinding worms and differently dressed start geometries. However, this results in recurring, wave-shaped topography deviations and, as a result, tonalities that have a negative impact on noise behavior [16]. In addition, conventional dressing processes with dressing rollers only allow the grinding worm geometry to be adapted cost-effectively to a limited extent in order to produce targeted micro geometry scattering.
Innovative concepts such as dressing with diamond-coated dressing gears shorten the process time and increase precision, but do not yet take into account micro geometries that vary from tooth to tooth. The potential of using a dressing gear to produce targeted micro geometry scattering on the gear to be manufactured by selectively modifying the grinding worm topography has not yet been investigated and is therefore the approach presented in this paper. The process outline for a possible, practical approach shows Fig. 1. A possible realistic process can be imagined by shaping a grinding worm with the use of a dressing gear, which consists of tooth flanks with targeted micro geometry scattering.
Fig. 1
Approach for the calculation method: dressing with a dressing gear
The aim of this report is therefore to develop a method that can be used to calculate the grinding worm geometry for targeted micro geometry scattering. This is to be modeled using a reverse grinding process. The grinding worm is therefore machined by the dressing gear during the normal generating grinding process. The starts are individually shaped by simultaneous continuous shifting. During subsequent machining, the micro geometry modifications are copied from the dressing gear to the blank. The challenges here are the calculation of the grinding worm topography via the reverse process and the resulting individual topographies of each individual tooth flank on the final gear. The procedure developed for this report is divided into three subsections.
In the first step, the methodological development for the calculation of grinding worm topographies as a result of targeted micro geometry scattering on a cylindrical gear in the continuous generating grinding process is carried out. For this purpose, the gear geometry is to be transformed in reverse to the generating profile tooth space by tooth space. Geometric contact conditions are then used to assign the generating points on the grinding worm base body in order to model the topography. In the next step, the method is applied to an example gear with micro geometry scattering. This is converted into a grinding worm geometry and rolled in the forward direction again. Here, kinematic differences between backward and forward gear grinding in relation to the geometry comparison (nominal and actual geometry) are discussed. The final evaluation of the tooth flank geometries and a comparative tooth contact analysis against the ideal reference are intended to classify the process as a possibility to produce targeted micro geometry scattering with an optimized NVH behavior in the generating grinding process.
4 Method development for the calculation of the grinding worm geometry
The approach developed in this report runs in the opposite direction to gear production in the real process of continuous generating grinding. The objective is to answer the question of how a grinding worm topography must be designed so that targeted micro geometry scattering can be manufactured in the continuous generating grinding process. In the past, wave-shaped topography deviations occurred on the tooth flanks due to the overlapping of grinding worm profiles with different geometric designs. This challenge is to be addressed by designing the grinding worm specifically and selecting a suitable shift strategy. For the practical comparison, the production of a grinding worm with a dressing gear is imaginable.
The core element of the developed method is the rolling calculation according to Litvin [19]. It is embedded in upstream and downstream calculation steps. The calculation process is shown schematically in Fig. 2. The first step in the calculation sequence is to specify the dressing gear. The macro geometry of the dressing gear automatically defines the geometry of the gear to be produced—i.e. the replica. At least the number of teeth z, the normal pressure angle αn, the normal modulus mn, the tooth width b and the helix angle β on the macro geometry side of the gear must be known to the program sequence. The micro geometry of the dressing gear can be specified individually for each tooth space. In the next step, the geometry of the dressing gear is generated from this data. At the end of this step, the data is available as rolling paths in the form of point courses for each individual cut.
A cut represents the contact line between the grinding tool and the tooth surface in three-dimensional space, as shown schematically in Fig. 3, left and center. They generally run diagonally across the tooth flank and have a turnaround in the tooth root area. The individual cuts or contact lines wind in a helix along the cylindrical gear body, see Fig. 3, right. When the gear rotates, they are spaced at the distance of the axial feed fa within a tooth space. Each gear rolling path also contains individual information about the micro geometry via the 3D coordinates of each individual point on the point course in accordance with the specifications made at the beginning. The points and their normal vectors are clustered for each profile section and passed on collectively to the backward calculation, see Fig. 2, right.
Fig. 3
Backward calculation of the tool profile using rolling calculation
The grinding worm profiles are generated during the backward calculation step. To create these profiles, the geometry parameters of the tool must be known and specified in addition to the geometry parameters of the dressing gear during the specification step. This includes the number of starts z0, the workpiece modulus mn0, the outer diameter da0 and the pitch direction sign(γ). For the positioning of the grinding worm in relation to the gear, the tool head height haP0 must also be specified to determine the position of the reference profile line and the generation profile shift factor xe*. To generate each tool profile point, the rolling condition according to Litvin is solved for each point of the point courses [19]. This creates a tool profile for each cluster of rolling paths. According to a real process, additional rolling kinematics can be taken into account for the reverse process. This allows, for example, lead crowning Cβ and flank line angle modification CHβ to be corrected. During the backward rolling process, the corresponding movement is then carried out by changing the center distance and a resulting profile is calculated. However, the movement is taken into account for all profiles according to the physical condition of a coherent grinding worm body. In addition, a theoretically ideal, continuous shifting is used during the backward calculation. In practice, this means that all rolled-out profiles are positioned on the grinding worm without overlapping. A defined shift amount is taken into account in the topography calculation of the grinding worm in a later step.
The left part of Fig. 3 shows the transformation from the flank points of a gear flank into the generating profile. The points from the gear generation process are arranged along the generated rolling path for each tooth. These are helically lined up on the base body with the axial distance of the feed. The backward calculation from the workpiece points to the respective generating profiles is implemented according to the adapted rolling kinematics from the continuous generating grinding process corresponding to the reversed process. For this purpose, each point on the rolling path is transformed from the workpiece coordinate system into the tool coordinate system. This is done using the zero-condition according to Litvin [19]. The contact point with the tool (i.e. the grinding worm profile) is calculated using the scalar product of the velocity and normal vectors of the generating point (from the rolling path).
The generated straight-edged tool profile is inclined by the helix angle γ of the grinding worm. For verification purposes, a geometric comparison between the ideal tool profile for the forward calculation and the calculated tool profile was carried out. This showed no geometric deviations, meaning that the closed loop from the forward and backward calculation can be considered as verified with regard to the kinematic transformations.
The generated tool profiles are collected after the loop shown in Fig. 2. A plausibility check ensures that all points that lie on the gear flank can be assigned to a point on the generating profile in accordance with the Litvin condition. A grinding worm is then generated as an enveloping body from all the calculated individual profiles, taking into account the program specifications. It is used only for visualization purposes, as only the individual profiles are required as point clouds for subsequent program steps. The generated tool profiles of the idealized grinding worm now contain the information of each cut between tool and workpiece. The topography information of each tooth space is also included. It should be noted that the influence of the kinematics from the backward calculation is also included and can change the geometric shape of the profile. This effect is relevant for gears that have tooth flanks with different lead crownings or flank line angle deviations (micro geometry scattering) over the gear circumference. The backward calculation can only take into account one of the modification values for the calculation of the rolling kinematics. It is therefore not possible to reconstruct the exact tool profile that originally generated the tooth flank geometry on all flanks using the backward calculation.
Once all the individual tool profiles of the grinding worm have been generated, a forward calculation is carried out to generate a new workpiece as a desired replica of the dressing gear in the last step of the method sequence shown in Fig. 2. The objective is to generate all tooth spaces and then identify geometric deviations from the ideal target geometry of the dressing gear. For this purpose, the tool profiles are read in one after the other and transformed to the tooth flank geometry using forward rolling according to Litvin, so that the generated point clouds can be used for a flank comparison. Possible deviations from the target geometry can result from the aforementioned differences from the kinematics on the one hand and can be influenced by geometric deviations on the grinding worm and positioning differences on the other. Geometric overlaps due to interference on the grinding worm are not taken into account in the current state of the calculation method. However, for an initial estimate due to possible overlapping contact zones between the grinding worm and gear, the contact surfaces (contact ellipses) are considered on a geometric basis. This is to ensure that the contact zones do not interfere with each other and is presented in more detail in the following chapter. The entire method is then applied to an example gear pair.
5 Use and verification of the method
The method presented for calculating generating profiles for the continuous generating grinding process is used in the following to first model a grinding worm from gear geometries of cylindrical gears with targeted micro geometry scattering in order to then use it again to generate a gear. In the first step, the macro and micro geometries and their scattering of the example gears are presented. In the following sections, the gear data is used as a comparison geometry to verify the geometry generation of the grinding worm and the newly generated gears with the developed method.
5.1 Introduction of the gear geometry with micro geometry scattering
The gears used for the analysis were selected based on a series of tests by Kasten et al. [10]. This involves a gear pair design (z1=20 and z2=33) that has a targeted micro geometry scattering on both the pinion and the gear. In their studies, Kasten et al. show the positive effects on the acoustic behavior of this particular gear set. The geometry should therefore serve as a reference with the ideal micro geometry scattering for the comparison of subsequent calculations. The geometry data and the tool data used are shown in Fig. 4.
Fig. 4
Example gears with targeted micro geometry scattering
Gears in the range with normal modulus mn≈4mm are used in commercial vehicle transmissions, for example, and are therefore produced in large-scale manufacturing processes. Equal tool parameters are used for the computational manufacturing of both gears. A special feature of the gears is the distribution of different micro geometry modifications over the circumference of the gears. The tables on the right-hand side of Fig. 4 show the values of the modifications. Profile and lead crowning Cα and Cβ were used, as well as profile and lead angle modifications CHα and CHβ on both gears.
The corresponding start and end diameters dC differ depending on the pinion and gear. It is noticeable that a total of four different modifications are distributed on the pinion (z=20), while there are only a total of three modifications on the gear (z=33). Modification 1 is identical on both gears with Cα=4µm and Cβ=6µm. These values vary for the other topographies. In addition, the profile and flank line angle modifications are added with different values. The topographies formed in this way are distributed in the tooth spaces around the circumference of the two gears. The distribution is based on the design according to Kasten et al. who produced these gears using the profile grinding process. It is crucial for the acoustic performance that the distribution sequence is precisely maintained.
5.2 Calculation of the grinding worm topography under consideration of micro geometry scattering
For the following calculation of the grinding worm, the previously presented pinion is used as an example. In theory, the grinding worm for the gear is created in a similar way with regard to the subsequent calculation of the quasi-static excitation behavior of the gear set in the tooth contact analysis. For this purpose, the gear and pinion are each given a micro geometry scattering. The gear data and the tool data used are shown in Fig. 5 on the left. These were transferred to the developed calculation method as specifications. A single start grinding worm is created for this application. This means that coprime grinding takes place between the grinding worm and the gears presented.
Fig. 5
Transformation of the grinding worm points for base body creation
From this, the meshing conditions and the gear body were pre-calculated macro-geometrically in accordance with the methodical procedure in Fig. 2. The additional point data required to generate the generating profiles were calculated using the upstream GearGenerator method. Here, the micro geometry was taken into account in accordance with the scattering shown above. The result are the flank geometries of a total of z=20 tooth spaces, which have an individual shape according to their rolled contact paths. The points are read in taking into account the inputs and the calculated kinematics are rolled backwards, so that transformed generating tool profiles are created. The points generated in this way already contain the information about the initial micro geometry.
To obtain the grinding worm body, the generating profiles are processed one after the other. To do this, they are first transformed into the axial section of the grinding worm and positioned in the meshing conditions to the gear. The calculations are carried out in the normal section. As shown in the middle of Fig. 5, the tool profile is then moved translationally according to the helical-rolling coupling in the reference system of the gear. The tool points intersect the lines of contact one after the other. The contact lines are tangent to the base circle of the gear at the normal contact angle αn=18.0°. Analytical correlations are used to calculate the displacement distances for the translational movements for which each point on the tool profile intersects the line of contact. From this, the necessary transformation of the profile point for repositioning can be calculated via the helical movement of the grinding worm. The displacement distances dx are calculated according to Eq. 1 for a point (xi,yi) and a tangent:
$$\mathrm{dx}=\frac{c-y_{i}}{m}-x_{i}$$
(1)
Where:
dx =
Displacement distance/mm
c =
y‑axis section/mm
yi =
y‑coordinate of the profile point/mm
xi =
x‑coordinate of the profile point/mm
m =
Tangent gradient/–
The points on the grinding worm are rotated over the elapsed path dxi. In this way, the tool points form a helical course of points along the grinding worm path. A rotational offset is calculated from the shift amount for each new profile in order to prevent point overlap. The offset is calculated from the total width of the grinding worm, which was assumed here to be b0=160mm. This corresponds to a continuous shift movement across the entire grinding worm. A possible influence of the profiles on each other will be examined in a later step. For the left and right flank of the gear or the grinding worm start, this results in the shape of the grinding worm body shown in Fig. 5.
In order to evaluate the effect of the targeted micro geometry scattering on the pinion on the grinding wheel, the topography of a gear section must be examined more closely. Figure 6 shows the grinding worm topography of the left flank for an angle section of ϕ=360°. The topography results from a distance calculation in the normal direction with an unmodified grinding worm. The deviations determined in this way are shown both as a 3D representation (left) and in the projected plane (center). A closer look at the 2D projection reveals a dark horizontal area. This area results from the profile modification on the gear, which has an almost identical amount for each tooth. This means that there are hardly any deviations between the point courses on the tool flank. In contrast, lighter areas can be seen at the top and bottom of the section. These are traversed by diagonal paths. They can be assigned to those modifications on the gear that show large deviations from tooth to tooth.
A magnified section of the point courses is sketched in the bottom right-hand section of Fig. 6. It shows the resulting paths of different deviation values that run parallel to each other. This observation applies to the straight-edged area of the generating profile. For the case under consideration, the point courses have a distance of ∆rn=0.14mm from each other. The value results from the number of generating profiles nPr rolled off and the specified grinding worm width b0=160mm, or the resulting shift path. The shift strategy provides for continuous shifting so that the point courses run parallel to each other.
As there is no pure point contact between the dressing gear and the grinding worm, but rather a contact ellipse between the curved surfaces depending on the magnitude of the modification, the half-axis lengths of a contact ellipse were calculated here. The purpose is to determine and derive whether the point courses influence each other and whether this could result in a change in the target geometry during forward rolling. If the half-axes touch a neighboring line, the geometry of adjacent lines is influenced by overlapping during dressing. For the worst-case consideration, an infeed of ∆a=70µm is assumed, which corresponds to two adjacent point trains with maximum deviation from each other, see topography scale in Fig. 6. The relevant diameter of the contact ellipse is determined to be de≈0.25mm by calculating the radii of curvature and is therefore not critical for the case under consideration. In order to further increase the distance between the point courses, approaches such as increasing the shift path or dividing the profiles into several starts can be considered. If the design allows it, modifications can also be cleverly arranged from tooth to tooth so that there are small gradients in the modification values between the point courses. In addition, it could be advantageous to calculate an average value for the lead crowning and the flank line angle deviation and to take these into account kinematically. Minimizing the difference to all scattered flank modifications could also reduce the modification gradients.
5.3 Generation of the gears with newly designed worm geometry
Based on the profile geometries presented above, a forward calculation is carried out for the grinding worm and the test gears in the continuous generating grinding process for both the pinion and the gear. The objective of the analyses is to verify the closed loop from the specified reference geometry and “replicated” geometry with the intermediate calculation method presented above.
The profile points generated for both the pinion and the gear were fed into the forward rolling process, so that flank geometries in the form of point clouds are eventually calculated again. Similarly, geometry files for the target modifications of the individual tooth spaces are calculated according to Kasten et al. The geometric deviations are compared with the ideal, unmodified involute shape using a mathematical flank comparison. The evaluation of the data for the pinion is shown in Fig. 7. For illustration purposes, the modification specifications are listed on the left-hand side of the figure. The modifications are considered separately in the profile and flank line evaluation. The variants are given different line types.
Fig. 7
Virtual measurement diagrams on the pinion (z = 20)
The upper section of Fig. 7 shows the profile line measurements for the center profile line. The lower section shows the flank line measurements for the center flank line on the pinion. Three measurements are shown per diagram. The reference geometry with target modification is shown by the solid line. Furthermore, two geometries calculated using the developed method are shown: without kinematics (dashed line) and with kinematics (dash-dotted line).
The use of kinematics here means that an average lead crowning of Cβ,off=2µm was applied during the backward calculation to generate the tool profile—i.e. an additional kinematic movement. This ensures that the offset to the target Cβ values is reduced and therefore fewer geometric changes to the profile shape are required. During the backward calculation, a homogeneous, shared kinematic movement is carried out for the entire grinding worm, as would also be the case in the real process for a dressing process with a dressing gear. Different crownings due to different kinematic sequences are therefore not possible. According to the legend, without kinematics means that no additional movement in the form of center distance variations was performed during the backward calculation to maintain a crowning. The crowning is therefore applied solely via the profile geometry, which leads to positioning deviations. This results in a greater offset than with the use of additional kinematics.
The same procedure was used to generate the geometry for the mating gear (z=33). The evaluated profile and flank line measurements are shown in Fig. 8. In contrast to the pinion, only three different topographies are applied to the gear. Similar to the pinion, the positive influence of the kinematic movement during the backward calculation can also be observed here. In the profile direction, the effect is again significantly smaller. It is also noticeable here that the use of the additional kinematics causes a reduction in the offset of the lead crowning to the target geometry. This means that a positive effect can be observed overall.
Overall, it can be stated that the resulting geometries are a good approximation of the required target geometries after utilizing the developed calculation method. Additional influencing with the help of kinematic movements can even have a positive effect on the resulting geometry for further approximation to the target geometry. The effect becomes particularly clear when looking at the flank line measurements. As each lead crowning can theoretically be set individually for each tooth space in the profile grinding process, the movement for each tooth space is coupled in the continuous generating grinding process due to the contiguous grinding worm. One possible cause for the determined offset amounts can therefore be kinematically induced positioning differences. Accordingly, the selection of an average lead crowning for the present case of both gearings ensures an approximation to the target geometry and thus has a robustness-enhancing effect against these deviations. An approximation to an average lead crowning is therefore appropriate for this application, as the flank line measurements show. The deviations from the target geometry are greatest for the variant without kinematics in terms of shape and amount, while the variant with kinematics shows a recognizable convergence to the target geometry. Overall, there is good agreement with the target modifications for the pinion, particularly for the profile measurement records. Strongly differing lead modifications per micro geometry seem to lead to higher deviations from the target geometry due to the kinematic coupling.
5.4 Generation of the gears with calculated worm geometry
For the test gear sets with targeted micro geometry scattering according to Kasten et al., it was possible to achieve positive acoustic performance characteristics compared to gear sets with standard modifications. For this reason, the calculated quasi-static transmission error spectra are compared. Figure 9 shows the variants of the standard design, the profile-ground design according to Kasten et al. and the geometries calculated using the method developed in this report applying the virtual continuous grinding process. The excitation behavior is evaluated for a torque of M2=25Nm on the gear using total transmission error spectra.
Fig. 9
Comparison of loaded total transmission error spectra
The basic design is characterized by the fact that it has no micro geometry scattering but has an identical standard modification of a crowning Cβ and a crowning Cα on each tooth. The calculated transmission error spectrum has clearly prominent amplitudes without random noise in the form of side orders due to the systematics that are repeated from tooth to tooth. The exciting order corresponds to the gear mesh order and their higher harmonics. In contrast, the order spectrum of the optimized, profile-ground variant according to Kasten et al. shows a clear broadband background noise with a simultaneous reduction in the amplitudes of the gear mesh orders compared to the basic design, which has a positive influence on the acoustic performance [10].
The variants generated by the developed method are shown in the right-hand diagram in Fig. 9 with and without the additional kinematics applied. It can be seen that the spectrum without kinematics has slightly higher levels at the gear mesh orders than the variant with the target modification. Nevertheless, there is a significant level of noise in the overall spectrum. In addition, the improved geometry with kinematics again shows significantly reduced levels at the gear mesh orders with a consistently strong noise component and thus comes very close to the transmission error behavior of the target modification.
In addition to the evaluation of the total transmission error spectra for the torque shown, the amplitudes of the gear mesh orders were evaluated for six further load points and shown in Fig. 10. The level of the first gear mesh order and its first higher harmonics is often used as an indicator of the potential of the overall excitation of the gear pair. For this reason, a comparison of the first and second gear mesh order (1st fz and 2nd fz) is carried out among the different variants.
Fig. 10
Loaded total transmission error as a function of the torque
The basic design variant is the least optimized variant and accordingly has the highest levels for the first and second gear mesh order for large parts of the torque range. For the 2nd fz in particular, it dominates the entire spectrum. As expected, the variant with target modification shows consistently lower levels due to the ideally scattered micro geometries. Although the variant without kinematics calculated using the developed method shows a similar characteristic curve to the variant with target modification, it excites the 1st fz and 2nd fz significantly more for all torques with the individual levels. In the upper torque range and in the low-load range, the excitation levels are even higher than those of the basic design. This may be due to the sometimes significant geometric deviations from the target modification variant. The design without kinematics is therefore only suitable for noise reduction in practical applications to a limited extent. In contrast, the calculated variant with kinematics not only shows the same characteristic curve, but also a consistently reduced excitation behavior of the levels for all torques of the 1st fz and 2nd fz. In addition to the well-defined micro geometry scattering, a possible cause here could be an additional variation of the eccentricity of the lead crowning points. This is also present in the variant without kinematics, but with much smaller amounts, so that these have an excitation-optimizing effect. The effect comes close to an additionally scattered pitch deviation.
Overall, it is evident that the gear geometries calculated using the developed method for the investigated gear set exhibit improved acoustic excitation behavior compared to the basic design and can theoretically be produced using the generating grinding process. The variant with additional kinematics applied during backward rolling even shows a qualitatively and quantitatively similar spectral image to the profile-ground design according to Kasten et al. [10]. The method developed and its application can therefore be regarded as verified.
6 Conclusion and outlook
This report presents a method that can be used to design grinding worm geometries for continuous generating grinding in order to generate targeted micro geometry scattering on gears. Gear geometries are transformed backwards to the generating profiles so that specific tool profiles are created for each individual tooth space. These profiles can then be converted into a grinding worm geometry whose starts have different topographical properties depending on the micro geometry scattering applied.
The essential core of the approach is the avoidance of periodically recurring waviness and systematics on the tooth flanks, which negatively influence the noise behavior. This effect occurs during relatively prime grinding with differently dressed grinding worm starts. The newly developed method achieves this objective, but deviations from the target geometry have been identified. However, these could be reduced by using compensating additional kinematics. A broadband noise spectrum is generated through the targeted variation of the micro geometry on the teeth, which improves the subjectively perceived gear noise. The final verification is carried out using a quasi-static tooth contact analysis. This shows that the extensively modified micro geometry was successfully “replicated” using the calculated grinding worm. With additional kinematic adjustments, such as the specific variation of the center distance by maintaining an average lead crowning, the deviations from the target geometry can be further reduced. As a result, the transmission error spectrum shows a higher base noise and significantly reduced levels at the gear mesh orders, which indicates an improved NVH behavior compared to a standard design without micro geometry scattering due to less tonality. This optimized design can be theoretically manufactured in a generating grinding process. However, the state of the art shows that continuous generating grinding using dressing gears has already been introduced in practice. For the practical implementation of the objective presented in this paper, only a dressing gear with existing target microgeometry scattering is required. One possible approach here could be the so-called “negative method” in which the reverse profile of the dressing tool is used to set the diamonds. Process characteristics and sequences from the conventional generating grinding process could remain unchanged.
Looking forward, further research could investigate how overlapping topographies on the grinding worm and additional variations of the kinematics affect the resulting gear geometry. In this context, it would be useful to implement an automated optimization routine in order to determine the distribution of individual modification parameters even more precisely and thus keep the topography transitions as low as possible. Experimental grinding studies with a dressing gear would validate the process in its entirety.
Acknowledgements
The authors gratefully acknowledge financial support by the WZL Gear Research Circle for the achievement of the project results.
A. Mann, C. Westphal and C. Brecher declare that they have no competing interests.
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