Optimized double helical gearing designs can help to save weight, assembly space and costs of turbo gear units, increase efficiency, and ensure low-noise operation. Double helical gears are becoming increasingly relevant, especially in the fields of turbomachinery, aerospace, and electro mobility. The reasons for this are the elimination of axial forces, which means bearings can be dimensioned smaller. Small gap widths between the gear halves are desirable in order to optimize the power density. Interaction effects between the two halves of the gear are thus becoming increasingly important. The apex point is an important parameter for tolerancing in these applications. It describes the intersection of the extension of the tooth flanks of the left and right half to an axial reference plane. In the current state of the art design processes, the interactions between the individual gear meshes of the left and right half are neglected. Thus, no specific design of the apex point position is possible.
This article, therefore, presents a method for considering the quasi-static stiffness behavior of coupled gear meshes, e.g. double helical gears for gear design. An FE-based approach is used to derive design and tolerance recommendations for coupled gear meshes. The developed method is validated by means of experimental studies. The focus is on the investigations of the apex point and the influence of manufacturing deviations on the excitation and contact behavior. The validated method allows to derive design and tolerance recommendations for double helical gears in order to optimize the excitation behavior.
1 Introduction and Motivation
In the field of driveline technology, it has been possible to massively increase the power density of gearboxes in recent decades. This has enabled the gear widths to be significantly reduced. With the resulting enhancement in the compactness of gear units, important comfort features such as noise behavior are now increasingly coming to the focus. Due to the proximity of individual meshes to one another, mutual interaction is intensified by excitation mechanisms such as axial vibrations in double helical gears. The axial vibrations result from stiffness-related axial force fluctuations between neighboring gear halves. In today’s gear design using tooth contact analysis, which represents the state of industrial practice, these excitation mechanisms are not taken into account. In order to achieve the full potential of parallel gear meshes in double helical gears, methods are needed that allow a stiffness-true representation of the entire gear chain in the FE-based tooth contact analysis. Thus, an optimized design of the gear geometry with consideration of the interactions shall be achieved with regard to the operational behavior.
Double helical gears are frequently used in turbo transmissions. Due to the high speeds, hydrodynamic bearings are usually used. In order to reduce the axial load on the hydrodynamic bearings and thus the power loss of the bearing, the axial force compensating property of double helical gears is exploited. In addition, due to the large dimension and low production volume of turbo gearboxes, single-part production of the gears is carried out. Precise knowledge of the excitation behavior should be obtained in order to avoid additional costs. Tolerancing the position of the apex point is one way of influencing the excitation behavior. Currently, one approach is to use a simplified calculation model to calculate double helical gears by reducing the double helical gear to one helical gear and loading it with half the torque. These simplifications neglect essential properties of double helical gears (elasticity, parallel gear mesh, manufacturing influences). The excitation behavior and a calculation of load-bearing capacity-determining variables such as pressure and load distributions can therefore not be represented accurately.
2 State of the Art
The design and analysis of neighboring meshes is often carried out separately for each individual mesh. In the design, which is usually based on quasi-static approaches, the macro geometry is determined first and then the micro geometry. The choice of the macro geometry is particularly relevant for acoustically sensitive applications, since it is decisive for the excitation behavior . The minimization of gear-set volume, gear mass, total center distance, manufacturing costs, or mass moments of inertia are described in the literature as target variables . TENBERGE applied mathematical and knowledge-based methods for macro geometry design of cylindrical gears . RÖMHILD, BANSEMIR and PARLOW also focus on the design of the macro geometry [4–6].
Once the macro geometry has been defined, the micro geometry is usually determined. For the design of the micro geometry of double helical gears, approximate equations exist so far to compensate for the misalignment . A design of the apex point with consideration of load- and manufacturing-related deviations can thus not be carried out. Instead, it is shown that, due to the interactions between the gear halves, the analytical approach causes inaccuracies and a FE-based approach should be used instead [8, 9]. Detailed analysis of this type of gears based on the FEM can currently only be performed with fully detailed FE models, see . However, the modeling effort and computation time are very high. In the case of double helical gears, the axial force components on one gear body ideally balance each other out. Nevertheless, axial forces can occur due to meshing errors caused, for example, by manufacturing deviations . As a result, axial force vibrations come to the foreground, which are currently not taken into account in the design. The axial vibrations were investigated computationally and experimentally by WANG ET AL. Under quasi-static conditions, there is good correlation in the comparison between measured and calculated axial displacements, but there is no integration into the design .
Various program systems have been described in the literature for modeling the quasi-static operational behavior of gears. In the FE-based tooth contact analysis STIRAK prerequisites have already been created for the calculation of multiple gear meshes in planetary gear stages [13, 14]. Another way to model multiple gear meshes in tooth contact analysis can be done using the Software RIKOR . RIKOR is a tooth contact analysis based on an analytical, plate-theoretical approach according to WEBER and BANASCHEK for modeling the tooth stiffness . With this approach, double helical gears can already be analyzed based on the load distribution. The input is done separately for each gear mesh. The solution of the contact problem and the calculation of the load distribution is performed within the load and deformation analysis in the overall gear system, but each gear mesh is calculated separately and not coupled.
In addition, KANG is using experimental investigations to examine the influence of the phase offset between two gear halves of a double helical gear on the quasi-static and dynamic operational behavior. The measurement data obtained are used to validate the simulation method developed. The simulation method is based on an analytical approach. There is no consideration of an FE-based approach. 
Conclusion: The excitation behavior of double helical gears is characterized by increasing relevance and by interactions between the gear meshes and can so far only be calculated efficiently on the basis of analytical approaches or simplified models. Predicting the excitation behavior, as accurately as possible at an early stage of the design process reduces costs and effort in development. The design of gears is divided into the design of the macro and micro geometry. With a focus on the excitation behavior, an FE-based approach is useful for accurate calculation. In the design of double helical gears, there is a lack of investigations into the influence on the excitation behavior of gear-specific variables such as the apex point.
3 Introduction of the Developed Simulation Model
In the first step, a simulation method was developed that allows interactions between adjacent tooth meshes to be taken into account and thus the influence on the operational behavior to be determined. In this context, a parametric input, such as the pitch offset of the two tooth meshes, was provided, and an adaptation of the calculation approach was carried out. Consequently, the operational behavior of double helical gears can be calculated with the FE-based calculation method.
3.1 Parametric Input and Output
Up to now, in the text-based input file, each gear mesh is defined as a pairing of driving and driven gear. To consider double helical gears, the input file has been extended so that the gear halves are linked and consequently two gear halves act as the input and two gear halves as the output. The parameterization of the gear halves is done separately in the gear definition area. This means double helical gears can be defined with slightly different macro geometric parameters in addition to the different sign of the helix angle. Further, additionally required inputs concern the pitch offset between the gear halves on a shaft, as well as the definition of the apex point with and without consideration of the lead line angle deviation of the two gear halves. In addition, further constraints are implemented to consider the load-induced torsion along the tooth width, which can also be specified in the input parameters. For the double helical gear, two separate models are built and meshed for each gear half. The models are then combined into an overall model and the calculation results for the entire double helical gear are determined in one plane of action. A finite element-based approach is used for the calculation of the tooth contact. 
3.2 Implementation of the Plane of Action Model
When calculating and analyzing double helical gears with FE-based tooth contact analysis, the gear halves could previously only be analyzed separately, see Figure 1 on the left. First of all, an assumption must be made how the total torque is distributed between the two halves of the gear set. Often, an equal distribution is assumed in the first step, so both meshes participate equally in the torque transmission. This does not represent reality. Subsequently, both halves are considered like an individual gear mesh. Consequently, the FE models are set up as segment gears. Here, the constraint is applied to the interfaces, as shown in Figure 1, left. Thus, no interactions between the gear halves can be considered.
The modification of the Plane of Action Model is intended to compensate for the aforementioned deficits. The segment gears are completed with toothless segments to form solid gears. The individual models are then combined to form an overall model for each double helical gear. All load cases and constraint conditions are transferred to the overall model. Both the input and output shafts are constraint at the separation plane at the center of the gap. The constraint is applied on the input side at the point of force application and on the output side at the point of force dissipation. The one-sided constraint as shown in Figure 1 on the right would cause the shaft to bend strongly, which does not correspond to the displacement that actually occurs. Therefore, a bearing element is inserted that locks the displacements but has no stiffness in the direction of rotation. This allows torsion along the gear, which is suppressed by the very stiff restraint on the cross sections of the segment gear in the previous calculation method.
The program sequence is basically divided into four program modules: The inputs specified by the user are read in as a first step. Here, for example, information about the gear geometry or characteristic values for controlling the program sequence are read in and processed. The second program module then generates the FE structure of the gears to be calculated. The meshing is done automatically on the basis of the initially set parameters. The third program module controls the FE solver. The calculation kernel determines the displacement influence coefficients for characterizing the gear tooth stiffness by solving systems of equations. The last program module evaluates the calculation results on the basis of the initially set parameters and optionally outputs the relevant characteristic values with regard to the gear excitation.
Before the mathematical spring model in program module No. 3 can be solved, the influence coefficients are needed, see Figure 2 on the left . For this purpose, the influence coefficients are determined separately for the combined FE models of the input and output shafts. The adapted calculation method is shown in Figure 2, top. When a force is applied to half 1 (H1), only deformations in H1 are evaluated. The same applies to half 2 (H2). By calculating the shafts in the overall model, the areas H2-H1 and H1-H2 are now also taken into account, reflecting the cross influences between the gear halves. A force on H1 now also causes a deformation in H2 and vice versa. In addition to the influence coefficients determined in this way, the load and the contact distances are still required to build up the entire spring model in order to solve the system of equations as a function of the rolling position and to determine the result variables.
4 Design and Manufacturing of the Test Gears
This chapter first discusses the design of different variants for double helical gears. The manufactured gear variants are always based on a reference variant, which usually has lead crowning of Cβ = 4 µm and tip relief of Cα = 10 µm, but no other modifications. The reference variant is of importance for subsequent experimental investigations, as it often serves as an unoptimized comparison variant.
4.1 Gear Design and Parameter Studies
A selection of the gear variations to be investigated for the analysis of the excitation behavior of double helical gears are listed in Figure 3. In addition to the reference variant, the first category of investigation is the variation of the apex point. Here, two variants with modified micro geometry are provided. Due to the changed micro geometry, each half transmits a higher or lower share of the load, respectively. Another variant is manufactured with a modified pitch offset between the two halves to shift the stiffness variations to achieve a lower excitation response. The change in the gap width between the two gear halves is intended to make the influence of the cross influences assessable. The goal of the last variant is a balanced load distribution in both halves, as calculated with the method of combined gear halves. The newly developed method is used to design the modifications. For this purpose, parameter studies will be carried out in a next step.
The calculation method is capable of modeling the theoretical operational behavior of double helical gears and improving the operational behavior with newly designed micro geometry modifications. In the actual production of gears, however, process-related manufacturing deviations occur, so deviations from the nominal geometry appear. In fact, the manufactured gear geometry scatter around the nominal geometry and thus influence the application behavior, which can lead to a deterioration of the target values. The previous focus on the nominal geometry alone neglects these effects. The influence of manufacturing scatter on the total transmission error at a particular design point is shown in Figure 4.
The parameters kept constant are shown in the left part of Figure 4 (Cα,1st half = 0 µm, Cα,2nd half = 0 µm). On the global axes, the values of the lead crowning are varied (abscissa — 1st half, ordinate — 2nd half). In contrast, on the local axes of each tile the values of the lead angle deviations are shown (abscissa — 1st half, ordinate — 2nd half). The unmodified reference variant (REFERENCE) is marked by the blue dot. As an example, the total transmission error is evaluated as the target value. The rotational transmission error is specified here as a path deviation. First, the differences between the real rolling positions and the ideal rolling positions calculated by the given transmission ratio are calculated as angular deviation. After subsequent multiplication by the base circle radius of the reference gear, the resulting rotational transmission error is obtained. To avoid dynamic effects, this is considered for the quasi-static case. Areas with low transmission error are shown in white. Variants with the largest transmission errors are marked in purple. Depending on the micro geometry, variants can be graphically identified in which areas around a possible nominal design point also exhibit low excitation behavior and consequently appear suitable as a possible nominal design.
In addition to micro geometric variation variables, the method can also be used to investigate the influence on the excitation behavior when the two gear halves are rotated relative to each other. The study shown in Figure 5 illustrates this influence for the load-free condition. In the simulation series, the second half of the gear was varied in a range of dp = 0.0 – 1.0 pitches in a step size of Δdp = 0.1 pitches.
In the upper part of Figure 5, the load-free rotational transmission error for the different pitch offset simulations is plotted over the rolling position. For pitch offsets close to dp = 0.5 pitches, the prominent maximum near the 18th rolling position decreases significantly and a minimum rotational transmission error is achieved. The fluctuation range shown in the lower part of the figure in the form of the load-free peak-to- peak transmission error confirms this observation.
4.2 Manufacturing of the Test Gears
Experimental investigations are carried out to validate the method, whereby the gears have to be manufactured. In order to achieve the smallest possible gap width between the gear halves, the complete machining was specified in the 5-axis milling process. Quenched and tempered steel is used as the material for the gears. In contrast to conventional process chains, the manufacturing effort is reduced here because the hardening process steps and the hard fine machining of the gears are omitted. Production deviations resulting from the clamping process can also be avoided because the gears are produced in a single clamping operation. On the basis of this process chain, a reference gear was designed that, due to the module mn = 5 mm, can be produced with commercially available 5-axis milling cutters. It was ensured the necessary cutting speeds can be achieved on commercially available machines. The gap width for the reference variant is bGap = 17 mm.
With the 5-axis milling process, all designed gear variants of the double helical gear were produced. Good qualities within the quality class A = 5 of the gears were achieved, particularly for the reference gears . In the following, the reference variant and the variant with relative rotation between the two gear halves are discussed in more detail. The position of the flank intersection point of the double helical gears (Apex Point) was produced with a deviation of
ΔxApex,Ref,Pinion = -16.7 µm for the reference pinion variant and with a deviation of ΔxApex,Ref,Gear = +88.0 µm for the reference wheel variant within the specified tolerance of ΔxApex,Target = ±100µm. The deviation of the apex point ΔxApex is defined here as the distance between the imaginary intersection of the two flank lines of the two gear halves and the plane formed by the midline of the gear. For the pitch offset pinion variant, the two gear halves, both of the pinion and the gear had to be manufactured with an angular offset of Δϕ = -10°. The measured deviation of the pitch offset is ΔϕDeviation,Pinion = -0.0247° and the deviation of the apex point is ΔxApex,PitchOffset,Pinion = +20.8 µm. The measured deviations of the gear are ΔϕDeviation,Gear = -0.0449° and ΔxApex,PitchOffset,Gear = -49.0 µm.
5 Experimental Investigation of the Excitation Behavior
This section presents the experimental investigations on the manufactured double helical gears. A test rig specially adapted to the application was used for this purpose. This was specially designed for measuring high-precision angular position in order to be able to derive conclusions about the excitation behavior of gears on the basis of the transmission error. The test rig is characterized by a rigid design, so load-induced displacements of the gears are reduced. The test rig concept for mounting the pinions of the double helical gear stage is shown in Figure 6. On the one hand, the test rig is shown as a model and, on the other, as a photo of the actual setup.
High-precision optical encoders ERA 180 manufactured by Heidenhain GmbH are in the main bearing blocks on the input and output sides. The rotating encoder disks contain 18,000 increments over their circumference allowing very accurate measurement of the angular rotational position. On the input side, a bearing unit supports the pinion shaft. The half-shell design allows the bearing concept to be adapted and the gears to be changed quickly. The metal bellows coupling allows axial displacements of the drive shaft if an appropriate bearing concept is provided. At the same time, the metal bellows coupling enables stiff rotary transmission behavior. On the output side, the end shield supports the free shaft end of the output shaft. Externally preheated oil is continuously supplied to the gear meshes and bearings, ensuring repeatable measurement results under stationary conditions.
A special feature is the bearing concept of the pinion shaft. In the application, the pinion of the double helical gears is often mounted on a floating bearing. This allows axial compensation movement of the shaft, and load compensation between the halves, takes place independently during operation. A distinction is therefore made between two types of bearing arrangement for the pinion shaft: the fixed/floating bearing arrangement and the floating bearing arrangement. In the case of the fixed-floating bearing arrangement, one bearing is fixed so no axial compensation movement is possible, whereas this is permitted in the case of the floating bearing arrangement. For this reason, the pinion shaft is mounted in a separate bearing unit in which the different bearing concepts can be investigated. This is characterized by the highest possible torsional stiffness while simultaneously releasing the axial degree of freedom.
The reference gear was investigated in a series of tests. Furthermore, the excitation behavior of one variant was investigated in which the two gear halves were rotated relative to each other by half a tooth pitch both at the pinion and at the opposing gear, see Figure 7. The amplitude of the total transmission error is plotted over the drive torque for the first two gear mesh orders. This variant proved to be particularly low in excitation in the load-free calculation and was therefore used for the experimental investigations.
First, it can be seen that its significant amplitude reduction occurs between the first and second tooth meshing frequencies. Furthermore, this can be understood in Figure 8, where a corresponding amplitude spectrum is shown with respect to the order. Also, a significant reduction in the transmission error amplitude of the first gear mesh order can be seen, particularly at medium to high loads, compared with the reference variant. However, since there are now theoretically 36 meshes per pinion revolution instead of 18, it is reasonable to assume that increased excitation of the second gear mesh order or sidebands are to be expected.
In this case, the number of the doubled meshing frequency results from the number of teeth. On the drive side, the pinion has z = 18 teeth. However, since the tooth meshes are offset by half a pitch over the circumference, there is also a time offset between the mesh starts with the mating gear. This results in a doubled excitation frequency
f = 2 x 18 = 36 meshes per revolution. Nevertheless, a closer look at the second gear mesh order reveals that the excitation at medium to high loads is only slightly higher or even lower than the excitation of the reference variant. The order spectra for two different drive torques further confirm this statement, see Figure 8.
The first and second gear mesh order were marked correspondingly. It can be clearly seen in the spectrum that no sidebands occur either. In summary, it can be said the gears with a pitch offset of half a pitch in the present application have a significantly reduced excitation of the first gear mesh order compared to the reference variant and, in addition, do not negatively influence any other known excitation mechanisms in the experiment. The effect can be explained by a more balanced meshing stiffness in the pitch-offset variant. The fact the teeth on the neighboring gear halves are in mesh offset by half a pitch reduces the maximum fluctuation in mesh stiffness. This confirms the method of design.
6 Validation of the Methodology
The results of the test rig investigations on the double helical gears are used in the following to validate the developed method. The validation is carried out on the one hand by comparing the measured and simulated transmission errors and on the other hand by the recorded and simulated contact patterns.
Figure 9 shows both the measured and the simulated contact patterns using the simulation method. The contact patterns were generated in the simulation and in the test rig at the same torque. The results for the reference gear set are shown in the center of the figure. In the test, a contact pattern was determined that is slightly off-center on both gear halves. The shift of the bearing patterns to one side indicates a displacement in the system. Also, slight spreading of the contact pattern can be seen in the tooth tip areas. The same is also evident for the simulated contact patterns of the reference gear set. Measured topography data of the gears were used for the calculations. In addition, the experimental test setup showed load- related displacements and deflections of the shaft-bearing system. These displacements have been taken into account in the simulation method. The displacements from the system were converted into the tooth meshing plane as deviations. In summary, it can be said the method reproduces the meshing conditions of the double helical gearing well on the basis of the contact patterns.
Figure 10 shows both the measured and the simulated transmission error curve vs. torque. The simulation was carried out with measured tooth flank topographies in order to be able to represent possible manufacturing influences. It can be seen that the simulation slightly underestimates the measurement for the first gear mesh order almost over the whole torque range. The curve of the measured data shows a large gradient that is caused by the load-related displacement at the test bench. This displacement behavior has been taken into account in the plane of action when calculating the rotational transmission errors. Quantitatively, the simulated rotational transmission error amplitudes of the first tooth meshing frequency reproduce well the measured data from the experimental investigations.
The second tooth meshing frequency could also be well represented by the simulation method. The simulated characteristic curve of the rotational error over the torque is similar to that from the measurement. Furthermore, the quantitative values of the amplitudes have also been well mapped by the simulation. However, the value at MIn = 400 Nm is overestimated by the simulation compared to the measured data.
In addition to the reference variant, the pitch-offset variant was also simulated. The simulated and measured rotational transmission errors for the pitch-offset variant are shown in Figure 11. An examination of the first tooth meshing order shows that the rotational transmission error curve is well reproduced up to a drive torque of MIn = 250 Nm. Only, above this value, the simulation overestimates the data from the experimental measurements.
Similar behavior — as can be seen for the first tooth meshing order — can also be observed for the second tooth meshing order. There is a constant underestimation of the measurement by the simulation. Up to a drive torque of MIn = 250 Nm, however, there is a qualitative similarity of the curve shapes between measurement and simulation. Both the investigations on the reference variant and the pitch-offset variant have shown the simulation method is able to represent the quasi-static excitation behavior by calculating the peak-to-peak total rotational transmission error. The load-induced misalignments were taken into account.
7 Summary and Outlook
In recent decades, it has been possible to massively increase the power density of gearboxes. This has made it possible to significantly reduce the axial distance between individual gear stages and the gear widths. With the increase in the compactness of gear stages thus achieved, important comfort features such as noise behavior are now increasingly coming into focus. Due to the increased proximity of individual meshes to each other, mutual interaction through excitation mechanisms is intensifying. In today’s quasi-static gear design by means of tooth contact analysis, which represents the state of industrial practice, these excitation mechanisms have not yet been taken into account.
The aim of the research was therefore to develop a method for taking into account the quasi-static, interactive stiffness behavior of double helical gears in gear design. Therefore, first a simulation model was set up and then the calculation method was presented. With the use of parameter studies, it was possible to determine influences that have different effects on the excitation behavior of double helical gears. A selection of the variations determined in this way was then manufactured using the 5-axis milling process and checked for dimensional accuracy. The experimental investigations of the manufactured gears were carried out using a test rig adapted to the investigation requirements. Finally, the developed method was validated on the basis of the test rig data obtained.
First, contact pattern comparisons at low loads were performed to analyze the system behavior. Closer analysis of the contact patterns initially indicated a displacement in the test rig system. These could be considered and confirmed using the simulation method. When comparing the quasi-static transmission error data for the double helical gears, a good match was noticed when considering the curves of the first gear mesh order of the reference and the pitch-offset variant. Similarly, the second tooth meshing frequency could be well represented by the simulation. In general, however, the method frequently underestimates the measured data.
The developed method can be used to support during the design process. In the future, this will make it possible to take into account excitation effects caused by the coupling of several interferences at an early stage. Up to now, gears could only be considered as individual uncoupled gear sets in tooth contact analysis. Thus, it was previously not possible for cross influences from the neighboring tooth meshes to be taken into account during simulations.
Looking forward, an investigation of the dynamic excitation behavior of double helical gears is useful. This could be accompanied by the creation of a multibody simulation model (MBS). With the aid of modal analyses of the test rig components, this can be used to accurately predict dynamic excitation effects. Measurement data of the rotational acceleration could be used to validate the MBS model. In addition, psychoacoustic analysis under dynamic operating conditions is also useful. In particular, variants with pitch offset should be investigated in this regard. In this context, it would be useful to consider axial forces, as these can also lead to additional excitation effects.
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Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2022 at the AGMA Fall Technical Meeting. 22FTM04