Dipartimento di Ingegneria Civile e Industriale
Corso di Laurea in Ingegneria Meccanica
Experimental and Numerical investigation of Helical
Gear Transmission Error under Misalignment
Prof. Leonardo Bertini, Università di Pisa
Dr. Daehyun Park, Siemens Industry Software
Dr. Ali Rezayat, Siemens Industry Software
The research results presented in this thesis have been obtained by the joint effort between the
industrial and academic partners in the framework of the Eco-Powertrain project. In particular,
the author gratefully acknowledges Punch Powertrain for providing test materials and
supporting the research. The author expresses his gratitude to Siemens Industry Software for
providing the advanced numerical analysis tools and the high precision gear test rig facility.
Table of Contents
Acknowledgement ... 1
Abstract ... 5
1 Introduction ... 7
2 State of the Art ... 10
2.1 Gear dynamics ... 10
2.2 Vibration assessment ... 12
2.3 Gear contact models ... 13
3 Test ... 15
3.1 Precision gear test rig ... 15
3.2 Transmission error (TE) ... 18
3.3 Test scenarios ... 19
3.3.1 Torque ... 19
3.3.2 Angular velocity ... 19
3.4 Measurement campaign ... 20
3.4.1 Tooth profile and lead ... 20
3.4.2 Gear misalignment ... 20
3.4.3 Transmission error ... 22
3.5 Post processing ... 22
3.6 Summary ... 25
4 Simulation ... 27
4.1 Gear contact simulation ... 27
4.1.1 Assumption on the multibody model ... 27
4.2 Simulation process ... 28
4.2.1 Misalignments ... 29
4.2.2 Center distance shifts ... 29
4.2.3 Shaft deflection ... 30
4.2.4 Influence of total displacement ... 32
4.2.5 Graphical representation of gear angular misalignment ... 33
4.2.6 Gear angular misalignment identification ... 35
4.3 Tooth modification ... 43
4.3.1 Flank line modification on the misaligned gear pair ... 43
4.4 Summary ... 48
5 Validation ... 49
5.1 Case study: Quasi-static ... 49
5.1.1 Result comparison in angle domain ... 50
5.1.2 Result comparison in spectrum analysis ... 52
5.2 Case study: Dynamics ... 53
5.3 Summary ... 55
A mio padre Sandro, mia madre Gloria e mio fratello Lorenzo. A Camilla.
Gears are key components of any mechanical systems containing transmissions and drivelines. Their range of application goes beyond automotive, marine and aerospace industries. The dynamic behavior of gears in operating conditions usually involves complex phenomena; therefore, it is not straightforward to create an accurate model for an analytical study. The reliability of gear transmission systems can be ensured through the accuracy and robustness of the modeling techniques in a design stage. Performing the optimization task for low vibration and high efficiency is therefore essential, and it can be achieved by the numerical gear contact computation tool in multibody environment. The main purpose of the study for this thesis is to validate the gear contact model against the measurement data from the experiment. Both numerical gear contact models and a precision gear test rig have been developed by the Driveline research team in Siemens Industry Software. To employ the simulation tool for the real engineering problem solving, the evaluation of the dynamic behavior of transmission systems in the various operating condition must be a prerequisite for an accurate prediction of the gear noise, vibration and durability. In this study, the Transmission Error (TE) of a helical gear pair will be mainly examined both quantitatively and qualitatively because it is widely recognized as the main internal source of vibration in power transmissions. Transmission error is defined as the deviation in gear motion against the one in the perfect gear conjugation.
This thesis comprises three main chapters to deal with the process and post-processing for the purpose of the research, dealing with gear test campaign, numerical simulation and its validation. Firstly, in chapter 3, test and measurement procedures have been described to generate repeatable data acquisition from the gear test rig. High precision sensors such as encoders and toque sensors have been deployed to ensure the quality of the data acquired especially for the repeatable TE measurement with tight confidence bounds. The raw data has been post-processed using both frequency and order analyses.
Secondly, in chapter 4, numerical gear contact model in conjunction with a gear system multibody modeling tool (Transmission Builder), recently developed in Siemens in the multibody environment (LMS Virtual.Lab Motion), has been used for better understanding the gear contact behavior under misaligned conditions, as observed in the test rig setup.
Lastly, the comparison between numerical and test results has been conducted to evaluate the level of correlation and understand the limits of the numerical gear contact analysis with the predefined
boundary conditions. Validation results have been shown in this chapter, through various post-processed data such as TE plot and spectrum analysis. The possibility of an accuracy improvement in terms of the level of modeling complexity was then discussed.
In the end, the conclusions of the research activities conducted for this thesis have been summarized, and remarks for the future task were proposed in chapter 6.
Gear systems are easily found not only in everyday life but also in various industries. From the smallest and precise gears in wrist watches, as seen in Figure 1, to the ones used in large size and heavy duty applications such as wind turbine or minding industries. The usage of gear systems are countless and versatile. Furthermore, gear are used in the fastest machines on the ground such as transmission in Formula-1 cars, in the water such as speed reducers to drive propellers, in the skies on planes where they require operation at thousands revolutions per minute and in the slowest applications such as lifting system for bridges where they take a minute to complete a single revolution with a heavy loading condition.
Figure 1: Watch movement
The world’s earliest acknowledged application already dates in the first hundred years Before Christ. In living creature, a single known case has been reported in the planthopper, where the extremities of the legs of the insect present cogs that allow the synchronization of the two legs during the jump , making the insect jump straight (Figure 2).
Figure 2 : The cog wheels connecting the hind legs of the planthopper as a gear system. The cogs allow precise synchronization of the insect’s hind legs when it jumps. Photograph: Burrows/Sutton/PA. Gear’s role in mechanical machinery could seem one of the simplest among all the other tasks accomplished by other devices. Gears have just to transmit the power. However, they can do it in many different ways as the rotational speed can be varied with different diameter combination in a pair with a corresponding torque change. One interesting aspect of gear system is the fact that they can even transmit power between rotating shafts that are not aligned as intended in design. Also, given the fact that there is high number of combinations in gear shapes, arrangements and sizes in the application, gears can be said one of the most versatile mechanism, adapting various applications in many different areas. From this perspective, on a survey  published in the Journal of Mechanical Design, gears are perceived as the icon of Mechanical Engineering.
Although gear mechanism has been used for a long time in various applications, the complexity related to the physics in gear meshing still yields research topics due to lack of understanding of the gear contact behavior. As the design criteria and requirement of gear noise and vibration, especially for the electric vehicle demanding quite operational condition, the investigation of the dynamic behavior of gear and interaction with the surrounding structures in gearboxes is becoming more important. Since dynamics loads generated from gear meshing constitute the major excitation sources resulting from the vibration caused by the gear meshing, in this context, to approach to in-depth understanding, there are series of factors to be evaluated; for example, manufacturing and assembly and geometrical tolerances, working conditions, mechanical properties, wear phenomenon, etc. Among these influencing components, the most important geometrical parameter is a tooth surface modification, often called “microgeometry”, which enabled to make the favorable gear contact condition by compensating teeth deflections, mounting or manufacturing errors coming from the real application.
To investigate the role of the major factor impacting the gear contact ultimately to the gear system noise and vibration performance under a wide range of variability, a gear test campaign plays a key role in the characterization of static and dynamic behavior of structures. However, it is a challenging task to acquire meaningful data in a measurement due to the poor accessibility of the transmission in machinery, lack of available space for reliable sensor mounting point needed for accurate measurements. Moreover, it is also mandatory to ensure well controlled boundary conditions such as high bearing and coupling stiffness, secured bolted connections to avoid coupling misalignment and use the rigid shaft to avoid excessive deformations. In other words, the behavior of the same gear pair can be affected by the relative position changes under load due to the surrounding components and structures.
Furthermore, the quantity related to the transmission performance are very small as an order of micro meters (or micro radians) resulting in the difficult situation for an accurate measurement requiring the need for very precise devices to capture the reliable results. A gear test rig should be able to reproduce a working scenario for the transmission under investigation. In order to achieve good energy efficiency, torque recirculation type of test rigs (i.e. the small driving motor only overcoming the friction loss is required), over the end to end type, are widely used in transmission research laboratories. Torque recirculation test rigs provide steady torque to the rotating shafts, even if there is a limitation to emulate the dynamically varying loading cases.
2 State of the Art
2.1 Gear dynamics
The conjugate gear mesh only takes place under the ideal condition, assuming the gears to be perfectly aligned, to have theoretical perfect involute profile and to be infinitely rigid. When two profiles are conjugated, there is no relative velocity component along the common normal. However, in real applications, when load is applied, tooth flanks deflect and contact areas deform during the power transmitting period. In addition to those effect, various manufacturing and assembly tolerances make the profiles to be no more conjugated and the kinematic imperfect.
This kinematic imperfection generates a dynamic excitation of the transmission with the vibration of the teeth themselves. Therefore, all the mechanical geared transmissions generate a certain level of dynamic amplification interacting with structures, due to their internal kinematics and dynamic characteristics in gear contact. Gear vibration mainly depends on fluctuation in meshing stiffness and sometimes in loss of contact between the contacting surfaces. The former case introduces a tonal component noise in the meshing frequency known as “gear whine”. If tonal frequencies meet the natural frequencies of a structure, there could be unfavorable noise and vibration issues or even gear failures in extreme cases. Loss of contact during the meshing causes a series of impacts between the teeth that excite the supporting structure with a broadband frequency. This type of phenomenon is called “gear rattle”. This particular effect affects gear tooth durability. The reason can be found in the repetitive impact load that occur in the meshing, causing a dramatic reduction of the working life of gears, which results in catastrophic failure as the tooth breakage shown in Figure 3.
No matter what the gear types are, a gear pair engages before the leading pair has left contact, when the two subsequent tooth pairs are observed. This is a direct consequence of achieving a contact ratio always greater than unity to avoid detrimental impacts. When it comes to different types of gears, such as spur and helical, the handover between tooth pairs is more gradual for helical gears than the one for spur gears as generating a higher mesh stiffness fluctuation with discontinuity the spur gear case. An engagement or disengagement for a tooth pair takes place over the full active face width in the spur gears. The helical gears barely suffer from gear rattle as compared with spur gears thanks to this gradual gear engagement. The observation shows that the contact starts from a corner and extends progressively on the tooth surface, terminating in the opposite corner as illustrated in Figure 4. This characteristic of the helical gears makes them less susceptible of rattle problems . In this study, since helical gears have been examined for the gear dynamic behavior investigation, the rattle noise phenomenon is not of interest.
2.2 Vibration assessment
Transmission error (TE) is a calculated quantity that can be considered as the major source of vibration and used as the vibration assessment criteria. A first definition of TE appeared in a paper based on the Ph.D. dissertation of Munro , where it was defined as "the angular displacement of the mating gear from the position it would occupy if the teeth were rigid and unmodified". The transmission error of a gear pair is usually in the order of a few microns in displacement or angle and it can be caused by an imposed displacement (e.g. distortions, manufacturing or assembly errors) and by the applied load, which also causes deflections . The possible causes of TE are listed in Figure 5.
Figure 5: Main concurrent factors affecting TE 
The largest contribution for TE comes from the elasticity of the gears, and it can be split between the gear blank and local contact region of meshing teeth. The algebraic expression of TE in this study can be described as the difference between the displacements caused along the Line of Action (LOA) by the relative rotation between the driving and the driven gear  as seen in Figure 6;
𝑇𝐸 = 𝑟𝑏1𝜃1− 𝑟𝑏2𝜃2
𝜃𝑖 = real angular position from each gear (𝑖=1, 2)
Figure 6: Parameters needed to define TE as a displacement along the line of action. 
Moreover, it is necessary to distinguish between two types of TE; static (STE) and dynamic (DTE). The STE includes geometric deviations from the ideal case, deflections and contact deformations under the quasi-static assumption (at low rotating speed not influenced by the resonance with structures). The DTE contains the inertial and damping effects related to a variable contact force alongside of the quasi-static phenomena. Throughout in the context, when TE is mentioned, DTE has been only considered when speed sweep testing and simulation were performed, otherwise, STE has been assumed.
2.3 Gear contact models
When the gear contact multibody simulation is performed, various gear contact models are available in the literatures, depending on the level of fidelity to be achieved. The simple 1D model comprising a linear (or rotational) spring-damper system considering only the torsional vibration along the line of action can be the most computationally efficient way of investigating the gear contact behavior. This assumption of the method may include that the compliance of all supporting structures such as shafts, bearings and couplings are neglected and any kind of misalignments are not included, except the existence of gear mesh stiffness in a constant or time varying fashion.
As seen in Figure 7, this model can be achieved in the lumped parameter system definition to have a lower computational effort but less accuracy while it still reproduces a fair level of gear whine and rattle. However, since translational displacement and out of plane rotation are not considered, the misaligned effect on the TE or dynamic behavior cannot be captured. For example, a model from Houser and Ozguven  detects the loss of contact occurred showing that it happens at a meshing frequency equal to the teeth pair natural frequency, resulting in the nonlinear gear rattle. Kahraman and Singh  extended the model to have the clearance between teeth, setting the mesh stiffness to zero where TE amplitude was below backlash values.
Similarly, Cai  investigated the lumped parameter model with the gear mesh stiffness calculation with combination of tooth stiffness at a pitch point and the variable contents in time. A schematic representation of this model is shown in Figure 7.
Figure 7: One dimensional gear pair rotational model 
A linear spring-damper system is used to model the meshing stiffness. The meshing stiffness at the pitch point was computed by Cai using the formulation from the ISO standard 6336  and an exponential function was used to represent its variation during the rotation.
More advanced tooth stiffness calculation approaches have been proposed by Andersson and Vedmar  where the meshing stiffness is computed from the contribution of both analytical contribution in contact deformation and tooth bulk compliance computation based on Finite Element (FE) approach. At Siemens Industry Software, one of the most advanced gear contact computational formula in the market is available, called “FE Preprocessor”  in conjunction with Transmission Builder , an interface to create 3D multibody gear models including gear contact forces and joints. This combined package with an advanced model consider the lightweight gear body (thin rims or multiple holes existing cases) to take into account the overall flexibility of the gear body and tooth blank in the gear pair. In this model, tooth bulk compliance is computed with conventional FE method with model reduction technique, and tooth contact deformation is calculated by the analytical solution introduced by Weber and Banaschek . This hybrid contact model scheme enables the contact model to be computed in an efficient way maintain the
In this study, for the validation purposes of the numerical gear contact model against the tested gear pair in the test rig, the analytical model combined with an ISO method for pitch point stiffness calculation and CAI method  for varying stiffness term has been employed in the multibody environment (LMS Virtual.Lab Motion). This choice mainly aims at less expensive computation but with fairly accurate results, because it takes into account 3D contact detection capability with a novel tooth slicing technique to account for the microgeometry and misalignments.
The evaluation of mechanical transmission behavior is a prerequisite to ensure good results in noise and vibration (N&V) and durability applications. Issues in both areas are often related to gear transmission error (TE), which, in a definition given by Smith  is “The difference between the angular position that the output shaft of a drive would occupy if the drive were perfect and the actual position of the output”. This deviation causes a vibration that excites the structure.
3.1 Precision gear test rig
The general configuration of gear test rigs can be divided in two categories; power absorption and power recirculation layout. The first category includes the test rigs where in the power absorption side, an electric generator is used as load for the system. The second category includes the test rigs where one source (driving motor) is used only to overcome the energy loss in the system, and no external loading is required. The arrangement of Siemens’ test rig discussed in this study belongs to this power recirculation layout . In fact the load on the gears teeth is imposed by setting a torque preload. Between the different arrangements enabling to accommodate this functionality, the simplest concept consists in a manually-imposed twist of the shafts converted in a torque preload. In the CAD model of Siemens' test rig seen in Figure 8, it is possible to distinguish two main sides: the test side on the left, where the measured gears are placed, and the reaction side on the right, typical of the power recirculation arrangement, needed to react the torque imposed in the system. The gear pair in the reaction side introduces additional excitations in the structure that have to be separated from the dynamic excitation of the gears in the test side by using flywheel and flexible coupling.
Figure 8: Siemens' test rig CAD representation: 1. Test gears, 2. Reaction Gears, 3. Bearings support plates, 4. Flexible couplings, 5. Flywheel, 6. Clutch flange for preload 
The test rig is mounted on a concrete base suspended on four air springs in the corners that allow to isolate the overall system and separate the two sides to avoid propagation of stray vibrations between them, where such vibrations are damped through the concrete base. With the actual test rig configuration, it is possible to set the different testing condition by using the various parameters as shown in Table 1. The parameters include the speed, torque, relative misalignment between the test gears as the angular and parallel misalignments.
Table 1 Test rig specifications
Parameter Range Uncertainly
Speed 0 to 4500 rpm (0 to 75 Hz) Measured
Torque 0 to 500 Nm 0.05 %
Angular misalignments 0 to 2 mrad 0.1 mrad Parallel misalignments 0 to 0.3 mm 0.020 mm
A relative displacement of the gears can be imposed by employing the rotatable and eccentric bearing housings in the test side. By rotating the bearing caps, a two dimensional displacement is imposed to the both end position of each shaft resulting in a change of the shafts orientation as illustrated in Figure 9. By imposing the same angle in the same direction on the bearing caps on the same shaft, it is possible to change the parallel misalignment between the gears. A rotation in the opposite direction results in a relative angular misalignment.
Figure 9: Angular misalignments on the left and Parallel misalignment on the right 
As illustrated in Figure 10, the bearings layout is designed to support all the loads generated operation of the test rig. Their choice enables the stiffness of the system to increase and to allow the test gears to be placed in the desired relative position without adding reaction forces. Four high-precision spherical roller bearings are placed to support the two test side shafts in all the working conditions . These bearings, indeed, can support radial and axial loads allowing an angular misalignment
between the axis of the inner and the outer races. The high-precision level of the bearings limits the additional eccentricity contribution in the shaft rotation. On the reaction side, different loads act as the radial load of the flywheels and the loads coming from the reaction gears. The radial loads of the flywheels are supported by Y-bearings units and by wide-face single-row cylindrical roller bearings. The Y-bearings do not require a precise alignment between the shaft and housing allowing an easier installation. The cylindrical roller bearings contribute also in supporting the radial loads originated by the test gears. Double-row tapered roller bearings help in the previously mentioned function and also supporting the axial thrust coming from the helical reaction gears.
Figure 10: Bearings arrangement for one shaft branch of the test rig: 1. High precision spherical roller bearing, 2. Y-bearing unit, 3. Wide-face single-row cylindrical roller bearing, 4. Double-row tapered roller
For the test gears, the main requirements can be found in providing a repeatable meshing excitation which allows, at the same time, the desired gear pair resonance. The requirement is satisfied by imposing a gear ratio equal to 1 and in this way one tooth will always mesh with same one on the corresponding gear. This solution permits to exclude deviations introduced by manufacturing variability. The gears that have been tested and that are going to be analyzed in this work are helical gear pair, and their specification are summarized in Table 2:
Table 2: Gears specifications
Parameter Test gears
Normal module 2.50 mm Normal pressure angle 20 deg Helix angle 20 deg Number of teeth 56 Facewidth 23 mm Tip diameter 154.97 mm Root diameter 143.78 mm Total contact ratio 2.47
3.2 Transmission error (TE)
One of the main purposes of measurement campaign is to acquire the static and dynamic transmission errors, which are directly linked to the noise and vibration behavior of the gear system. A typical transmission error measurement in quasi-static conditions is reported in Figure 11. It is easy to decouple the two main contributions of the raw TE signal by the so-called Run out and tooth passing components.
Figure 11: Typical transmission error curve
The run out has a typically low frequency and high amplitude, with a periodicity of one per revolution. It is caused by the eccentricity error in the gears or shafts with respect to their own supporting elements. On the other hand, the tooth passing contribution has smaller amplitude and a typical higher frequency due to the fact that every peak is caused by a single tooth deflecting during every meshing cycle.
Run out component in shaft frequency Tooth passing component in mesh frequency Raw signal of transmission error
3.3 Test scenarios
To acquire all the different measurement wanted, several configurations have been set in the test rig for the data acquisition campaign . For each test scenario, the power inverter was programmed to drive the electric motor at the desired angular velocity (rpm), and the torque loop system was set at the chosen value by fine-tuning the angular displacement of the clutch system. The main objective of this test campaign was the evaluation of TE, and for this reason it was decided to measure the following quantities: angular position of the two gears, angular velocity and torque. Any strain and acceleration acquisition was not performed in this TE study.
By varying the torque, it is possible to investigate the effects of the applied load to the static TE, and also especially possible to appreciate a certain amount of the change of TE in shape and amplitude variation. Taking in account only the torque contribution to TE, it is possible to consider only the elastic contribution, avoiding analyzing other effects that can be neglected in the setup used for the measurement. Since these helical gears were designed with solid blanks, it was also possible to neglect the contribution to the elastic part of the TE coming from the gear body deformation, therefore assuming that the entire meshing stiffness contribution to the TE comes from the teeth compliance. In the measurement campaign performed for these test gears, it has been decided to impose six different torque preloads, starting from a value of about 0 Nm for analyzing the pure kinematic contribution to the TE avoiding load dependent effects up to 250 Nm in steps of 50 Nm. Every time the torque was changed before running the test, the torque sensor was checked in quasi-static condition set by manually rotating the shafts at low angular velocity to assure that the correct value of torque has been applied to the gear pair.
It is important to distinguish between static TE (STE) and dynamic TE (DTE); the first one includes geometric deviations from the ideal case, deflections and contact deformations under quasi-static condition. On the other hand, DTE includes the inertial damping effects related to a variable contact force on top of the quasi-static phenomena, and it is also affected by resonances of the system as it is more relevant for dynamics. To acquire data in order to analyze static TE, it was necessary to run test at low speed by rotating the shaft at around 10 rpm. For the dynamic TE evaluation, it is necessary to perform the data acquisition for a wide range of angular velocities up to 1500 rpm. For this reason the power inverter was programmed to gradually accelerate from 0 to 1500 rpm in 2 minutes to have enough samples at all the different velocities.
3.4 Measurement campaign
For gear system test in this study, the most important measurement includes the gear tooth profile and lead accuracy check with a Coordinate Measurement Machine (CMM), shaft & gear misalignment and Transmission error at various loading and speed condition.
3.4.1Tooth profile and lead
Manufacturer of the helical gear pair used in the test provided the following tooth profile and lead accuracy measurement result. It is seen that this gear pair is finished with a grinding process and the measurement results shows that the accuracy of the tooth surface is ensured. (Max. 5 micron deviation)
Figure 12: Tooth profile (upper chart) and lead (bottom chart) topography measurement results
The following measured graphs represent the gear axial deviations during the a single revolution of each gear (left and right) measured at the gear side blank, right below the tooth root diameter with a precision dial gauge. As seen in the figures, a fair amount of deviation (up to about 50 µm) is observed.
The main cause is the presence of a SIT-LOCK clamping device, which introduces a manufacturing / assembly error between shaft and gear. This amount of axial error can be interpreted to the angular misalignment error between shaft and gear up to 0.2 degrees. This misalignment will generate the modulated TE signal throughout the shaft revolution. The detailed investigation about this misaligned condition is presented in Chapter 4.
Figure 13: Measurement of axial deviation of gear blank side with a dial gauge
After observing the gear misalignment for the test gear pair, an additional check has been made by applying a special purpose paint to the contacting tooth surface to check the contact position under different level of loading condition  as seen in Figure 14. The left side image shows the contact patch at T=50 Nm and right side image shows the contact patch at T=150 Nm. Since the helical gear used in the test do not have any modification in flank line direction, it is seen that the contact position under light loaded condition is very sensitive, and as load increases as seen in the right side of image, the contact patch gets more distributed over the tooth flank as a consequence of teeth deflection.
Figure 14: Contact pattern check: (a) T=50 Nm, (b) T=150 Nm
-10 0 10 20 30 40 50 60 0 2 4 6 8 10 12 m icr o n s Point number
Axial deviation at base circle - G2
Gir-R2toL Poly. (Gir-R2toL) -25 -20 -15 -10 -5 0 5 10 15 20 25 0 2 4 6 8 10 12 M ICRONS POINT NUMBER
Axial deviation at base circle - G1
Gir-R1toL Poly. (Gir-R1toL)
An accurate measurement of the static TE is a challenging task because the environment conditions and the assembly accuracy can greatly affect the measured signal, lowering the high precision level of the system and data acquit ion. The goal of the experimental campaign is to obtain high quality data to be used for the validation of numerical gear contact models; therefore, several practical precautions for the smallest assembly details have to be taken before measurement.
The plot shown in Figure 15 shows examples of transmission error measurement with different loading condition from 50 to 250 Nm under gear misaligned condition, where gear pair tooth are unmodified. The low frequency run out components are filtered out so that only tooth passing frequency components are displayed to observe the TE amplitude variation in time (or shaft angular position). Typically TE amplitude keeps increasing when torque becomes higher for the unmodified gear pair in well aligned condition; however, it is not true when angular misalignments for both gears are imposed. To get more comprehensive understanding for this behavior, a detail contact pattern (or load distribution) analysis is essential. More detailed result analysis will be explained in chapter 4.
Figure 15: Transmission error plot in various torques in quasi static operation
3.5 Post processing
The measurement of the TE signal is performed in time domain, at a given (constant) sampling rate. The acquired data has been post-processed using a MatLab script. The script allows separating the contribution of different orders, setting boundaries for lower and upper orders. By this separation it is also possible to compare the curves in different operating conditions.
For rotating elements such as gears, the analysis of the numerical and experimental results is usually done in angle domain. Adopting the angle domain analysis makes it possible to have a uniform spacing in the tooth passing contribution due to the fact that the angular pitch is constant for all teeth in theory. In contrast with the angle domain, time domain signals might be unevenly spaced in case of having varying shaft speed configurations.
When time domain is processed, the Fourier transform provides the frequency spectrum. On the other hand, if the raw data is in angle domain, the result of the Fourier transform is the order spectrum. Order spectrum allows seeing a specific harmonic component which is related to a certain phenomenon for the specific order corresponding to the mechanical component. For example, the first order (harmonic) usually relates to unbalance of the machine meaning that it occurs one time per revolution; the second harmonic often relates to eccentricity and gear angular misalignments. Both the frequency and order spectrum provide meaningful information about the characteristics of the rotating system.
The acquired time series usually include several revolutions of the shafts. Having multiple revolutions allows increasing the quality of the results by averaging over the different revolutions. The encoder emits a pulse at the starting angle of every revolution. These pulses constitute the tacho signal. The knowledge of the tacho signal is necessary for splitting the raw data into multiple revolutions. The followed methodology for processing the raw TE data is illustrated in Figure 16. The processing scheme is summarized as follows:
1. The raw TE data is divided into different revolutions using the tacho signals of the encoders. 2. The time domain data is converted to angle domain by resampling.
3. The angle domain data segments (corresponding to each revolution) are converted to order domain, using the Fourier transform.
4. The spectrum data of the different revolutions are averaged in order domain.
5. The averaged signal is (band-) filtered, resulting to two main parts: The run out and tooth passing components.
6. The separated components are converted to angle (or time) domain by an inverse Fourier transform.
Figure 16: Post processing scheme
One of the advantages of this filtering method is the possibility of observing a full revolution data, representing all of the gear teeth. As a consequence, the effect of misalignment for example can be visible in the final signal result. Additional benefit is the ability to average the revolution data. The averaging allows increasing the signal to noise ratio of the results.
Another useful feature to analyze the behavior of a meshing gear pair is to stack up frequency spectrum analysis plot at subsequent speeds , obtaining a three-dimensional diagram which is called waterfall plot as seen in Figure 17. In order to be able to have the data for this types of plots, it is necessary to run test with a slow speed ramp up with the aim of covering all the spectrum at different angular velocity at the given constant torque. More detailed description of the waterfall (or Campbell diagram) can be referred at the Chapter.
Revolution 1 Revolution 2 Revolution 3
Revolution 1 Averaged Revolution Revolution Time Frequency/ Order Time/ Angle Tacho pulses Average
Fourier Fourier Fourier
Revolution 2 Revolution 3
Figure 17: 3D waterfall plot of speed sweep experiments with constant torque of 100 Nm
Test campaign has been conducted on the precision test rig that is designed for the purpose of measuring the vibration behavior. The final goal in this study of the measurement point of view is to obtain the transmission error, which impact the gear system dynamic behavior by ultimately introducing the gear tooth excitation forces. The following items studied in this chapter are summarized as follows;
High precision test rig with high stiffness structures (shaft, bearing and housing) was introduced and the capability of the test rig is described. Moreover, the accuracy of the gear profile and lead topology was shown with the CMM measurement results.
Gear angular misalignment was detected caused by the SIT-LOCK device assembly error. This error was measured with dial gauge and checked with tooth contact patch examination under light and medium torque. Due to the nature of unmodified gear, severe edge contact is detected under light load (50 Nm), but the contacting area gets more uniform when higher load (150 Nm) is applied.
Transmission error under misalignment condition was measured with various torque ranges (T=50 to 250 Nm) at low rotation speed (10 rpm) to ensure the gear contact takes place at quasi-static condition.
Post processing technique and work flow to generate the TE and spectrum plot in angle or order domain to ease the analysis. Also, by removing the low frequency component from the raw data, the investigation of TE amplitude variation became more straightforward.
To validate the gear contact multibody simulation tool against the test results, all assumptions of the model established should be considered within acceptable ranges to capture any important static or dynamic behavior of the gear contact. On the other hands, it is recommended to make the model simpler to accelerate the simulation speed for the parameter study.
4.1 Gear contact simulation
In the gear contact simulation, the most important aspect to consider is which gear contact model has to be used. Among the various tooth stiffness computation models having different levels of fidelity, it is recommended to select the gear contact model that computes the tooth stiffness accurate enough to capture TE variation in time on efficient way for parameter study in different contact conditions.
4.1.1Assumption on the multibody model
The gear contact model used in this simulation is an analytical model where gear meshing stiffness computation is based on the literature  with slicing technique suitable for helical gear contact analysis. Also, the inclusion of slicing scheme allows the contact model to handle the case of tooth flank modification and misalignment on efficient way. This gear contact model also computes the time varying gear mesh stiffness to investigate the TE variation and its amplitude influence to the NVH study. Since the test rig has been designed to have highly rigid structures including shafts, bearings and bearing housings, all the structures and mechanical components are assumed to be rigid except the contact condition in a pair. Test rig has two gear pairs (test and reaction gear pairs) but this design is to apply a certain amount of constant torque to the test gear by using the torque loop concept. Since test gear pair and reaction gear pair are practically isolated by using a large inertia flywheel and flexible coupling, it was decided to model only test gear pair for the simple multibody model. A render of the CAD model is shown in Figure 18 below.
Figure 18: 3D CAD render of the multibody model
In order to apply the constant torque to the model, torque driver element is applied to the gear-1 (driving gear), and to apply the resistance, a certain amount of damping has been applied to ensure the system to be rotated in a constant speed. This simulation has been performed both for quasi static and dynamic conditions in different speed ranges. However, since structures such as shafts and housings are assumed to be rigid, only the limited ranges of the natural frequencies are supposed to be captured as compared to the measurement results in the test rig.
4.2 Simulation process
Model development process starts with Transmission Builder. This software helps creating a complex transmission system multibody model in a fast but accurate way. First of all, the geometry parameters of the model are chosen followed by a gear meshing parameters settings. Once every desired parameter is set, Transmission Builder creates the model inside Virtual.Lab Motion. Here, boundary condition of the system has to be selected, by choosing how to apply torque and velocity to the system and by setting the misalignment condition desired to the gear pair. Once simulation parameters are chosen, in Transmission Builder it is possible to select the gear contact model and to set meshing parameters. It is also allowed to impose micro geometry modification for the gears. Transmission Builder solves the model and commanded to import the results to Virtual.Lab. from the gear contact solver. It is now possible to post process the raw data in MatLab, setting the desired filters for the orders and plotting the results. A work flow of this simulation process is shown in Figure 19.
Figure 19: Work flow diagram of simulation process.
To accurately reproduce the gear pair system in a simulation, it is important to understand what types of misalignment can be introduced and how much the amount and the influence to the behavior would be. The potential cause of misalignments can be center distance changes and angular misalignment.
4.2.2Center distance shifts
Center distance shifts may occur in real applications due to several reasons  as follows:
• Shaft, bearing, and housing deflections will usually cause an increase in the center distance. • Mounted gear run out includes the effect of manufactured eccentricity as well as mounting
eccentricity. These eccentricities have the effect of increasing and then decreasing the effective shaft center distance at a period of one shaft rotation per each gear.
• Bore parallelism error may be obtained from the housing engineering drawing and might cause either a contraction or extension of the housing center distance.
Since the test rig used in this study was designed with tight tolerances and high precision, the effect of the mechanical system precision induced errors will not be considered, but the combined effect of shaft, bearing deflection and SIT-LOCK related assembly errors are the primary focus.
To evaluate shaft deflection, it was chosen a model of a simply supported beam with hinges under point load as seen in Figure 20, because the spherical roller bearings used in the test rig have rotational degree of freedom to allow non-parallel shaft configuration.
Figure 20: A simply supported beam model with eccentric point load
To apply the load to this beam model, tangential force 𝐹𝑡 on the base circle (see Figure 21) is calculated
for the condition of highest torque value (250 Nm) to accommodate the worst scenario.
Figure 21: A schematic of two meshing gears along the line of action
= 3.6 𝑘𝑁 where,
31 𝑇 = 250 𝑁𝑚 𝑟𝑏= 𝐷𝑏 2 = 138.94 2 = 69.47 𝑚𝑚 𝐷𝑏= 138.93 𝑚𝑚
The deflection of the beam introduced is evaluated as
𝛿𝑠ℎ𝑎𝑓𝑡 = 𝐹𝑟 𝑏 (𝑙2− 𝑏2) 3 2 ⁄ 9√3 𝑙 𝐸 𝐼 = 54 𝜇𝑚 where, 𝐸 = 206 𝐺𝑃𝑎 𝐹𝑟 = 1.4 𝑘𝑁 𝐼 =𝜋 ∙ 𝐷 4 64 𝑎 = 145 𝑚𝑚 𝑏 = 123 𝑚𝑚 𝑙 = 𝑎 + 𝑏 = 268 𝑚𝑚 𝐷 = 40 𝑚𝑚 220.127.116.11.1 Bearing deflection
In the test rig, there are four spherical roller bearings (SKF 22308 EK) as seen in Figure 22 to support the two shafts containing test gear pair.
Figure 22: SKF 22308 EK precision bearing
The evaluation of bearing radial deflection was performed using the radial stiffness value given by the manufacturer, and the deflection of each bearing is expressed as
32 𝛿𝑏𝑒𝑎𝑟𝑖𝑛𝑔= 𝑃 𝐾𝑟𝑎𝑑𝑖𝑎𝑙 = 2.9 𝜇𝑚 where, 𝐾𝑟𝑎𝑑𝑖𝑎𝑙= 6.12 × 108 𝑁 𝑚 𝑃 =𝐹𝑡 2 = 1.8 𝑘𝑁
18.104.22.168.2 Concentricity error on SIT-LOCK
In the gear precision test rig, the only mechanical component where the manufacturing error cannot be negligible is the SIT-LOCK (see Figure 23) that clamps a gear to the shaft by friction. The potential assembly error is described in the manufacturer catalog  as “For self-centering locking assemblies, the clamping element has a centering effect and the concentricity error can be considered 20 - 40 µm”.
Figure 23: SIT-LOCK clamping element and its working principle.
4.2.4Influence of total displacement
Shaft and bearing deflection contribution to center distance shift is constant due to the fact that static deflection caused by the tangential force remaining approximately constant during the gear pair meshing. On the contrary, an eccentricity error induced by SIT-LOCK results in a sinusoidal center distance variation during a full gear revolution.
Considering the worst case scenario where the two clamping element errors from each gear in a pair are in phase and aligned with the static deflection of the shafts and bearings, the total displacement is expressed as
To evaluate the impact on TE of this center distance shift in the worst case scenario, gear contact simulation was conducted at the nominal condition (no misalignment) under the load T=250 Nm with 0.2 mm larger center distance. As the result is seen in Figure 24, the alteration of TE is low. Even when compared to the TE introduced by angular misalignments (see next chapter 4.2.5), the magnitude changes are negligible. For this reason, center distance variation is considered to be negligible and not taken into account in the simulations.
Figure 24: TE comparison at nominal condition (orange line) against the case where the center distance shift is considered under load (blue line) [T=250 Nm, Ω =10rpm]
4.2.5Graphical representation of gear angular misalignment
Before performing multiple simulations for different types of angular misalignment cases due to the SIT-LOCK, categorizing of the gear misalignment would be a useful practice to sort out the minimum number of cases. For this, firstly establishing a convention to describe the angular misalignment should be necessary for the systematic approach. By imposing a reference frame to the center of the gear body as shown in Figure 25, it is possible to define the angular misalignments which occur along the two axes (X and Y) in a schematic way.
It is possible to describe each possible misaligned case in a way that the axis related to the current misalignment for gear-1 (driving) is written with the separator “/” followed by the axis related to the misalignment for gear-2 (driven). The positive sign at the name of axis indicates the angular misalignment occurs in the counter clockwise direction (right handed rule) and the angular amount (degree) can be added with parenthesis. For example, if the gear-1 is perfectly aligned along the X axis but misaligned along the Y axis by 0.1 degree, it is written as “+Y(0.1˚)” for gear-1 misalignment. In addition, If gear-2 is misaligned along the X axis by 0.2 degree and along Y axis by -0.1 degrees, then it
is written as “+X(0.2˚)–Y(0.1˚)”. Note that for the combined configuration for both gears, it is written as “+Y(0.1˚) / +X(0.2˚)–Y(0.1˚)”.
Figure 25: Gear reference frame
In addition, to describe the combined misalignment assembly condition imposed to both gears, a simple graphical representation would be a useful tool to identify the current configuration at a glance. Figure 26: shows examples of the graphical representation for the different misalignment condition detected in the gear pair (g1 and g2). It is noted that the amount of the angular misalignment in the CAD model has been exaggerated for description purposes with much greater amount than the one found in the test rig.
Figure 26: Gear angular misalignment graphical representation for
+X(5˚) / +X(5˚) (left) and +Y(5˚) / -Y(5˚) (right) (exaggerated values with respect to reality for graphic purposes)
+Yg1 g2 g1 g2 Z Y Z Y Z Y Z Y
4.2.6Gear angular misalignment identification
The angular misalignment caused by the SIT-LOCK to each gear may have a different angular amount condition with different direction and magnitude of an angle as described in the previous chapter. Depending on the view from a certain axis or at a certain moment of the shaft rotational angle, the assembled gear pair looks different even if the assembled condition remains same during the gear rotation. To identify the unique sets of misalignment condition before gear contact simulation, examining the misalignment shape at every 90 degree rotational angle can provide us with some intuitive way of analysis.
In order to simplify the treatment, a fixed amount of misalignment along each gear reference axis of rotation is taken into account without specifying the misalignment magnitude. For example, by considering a misalignment applied only along X axis for both gears, an expression +X/+X can be used to describe the staring position of gear rotation as seen the zero degree position in Figure 27.
Figure 27: Changes of misalignment condition during one complete revolution (initial setup = +X/+X) During a complete revolution of each gear in opposite direction around the Z axis, the schematic gear assembly figure continuously changes. While maintaining the initial misaligned assembly condition during the rotation, this assembled condition is considered as one unique set. To find out the all possible unique assembly condition used for the meaningful gear contact simulation, the initial assembly condition can be changed in combination from the individual misaligned condition for each gear. For example, instead of the first example of assembly condition (+X/+X) shown in Figure 27, the initial assembly condition can be different combination of misalignment of each gear with (+X/-Y) for instance.
All candidate combinations from the each misaligned gear, one could find total 81 cases, by considering the nominal case (no misalignment) and 8 angular misalignments (+X, -X, +Y, -Y, +X+Y, +X-Y, -X++X-Y, -X-Y) for each gear. As highlighted in Table 3 the totally unique cases are 30, but in the case of gear with the same number of teeth as the gear pair in the test rig, due to the symmetry, the unique number of combination becomes reduced to 15 cases. These cases were easily analyzed by performing the simulations in Siemens LMS Virtual.Lab Motion.
Table 3: Number of unique misalignment cases for a gear pair.
22.214.171.124 Gear contact simulation for various misalignment gear pairs
Gear contact simulations were conducted for 15 configurations of misaligned gear pairs as found in chapter 4.2.6, and transmission error (TE) results has been generated to explain the gear vibration behavior. The input and the boundary conditions are as follows;
Angular misalignment: 0.1˚ along the X and Y gear reference axis Torque: 50 – 250 Nm with a 50 Nm step
Angular velocity: 10 rpm for all simulation to ensure a quasi-static condition Damping coefficient for gear contact: 10,000 kg/s
Number of slices along the flank line: 20 slices Tooth modification: none
The simulated TE (in displacement [mm]) results are shown in Figure 28 to Figure 31 in the time domain ([sec]). Compared with the nominal assembly condition (no misalignment), all misaligned combination of gear pairs exhibit a modulated TE signal with order 2 during one revolution with mean value changes. This phenomenon is caused by the gear mesh stiffness changes not only at every mesh cycle but also throughout the every gear (or shaft) revolution. Depending on the misaligned configuration for each gear pair, the peak-to-peak TE values vary as well affecting the gear excitation forces transferred to the housing though the bearings to generate the airborne noise as a result.
Figure 28: TE for misaligned gears [T=250 Nm, Ω=10rpm]
Figure 30: TE for misaligned gears [T=250 Nm, Ω=10rpm]
Figure 31: TE for misaligned gears [T=250 Nm, Ω=10rpm]
126.96.36.199 Tooth contact pattern
After TE results are investigated for the 15 different misalignment combinations, one set of misalignment cases having very similar shapes of TE was detect as seen in Figure 32. Having the similar TE peak-to-peak magnitude over time, the comparison plot exhibits the different phase of the low frequency wave pattern.
Figure 32: TE of [+X+Y/+X+Y] and [+X+Y/-X+Y] [T=250 Nm, Ω=10rpm]
In order to clarify this unexpected case in detail, the contact pattern comparison has been conducted to conclude the feasibility of this occurrence. During one mesh cycle, the instantaneous load distribution was examined at each time step over the base pitch angle. Since the total contact ratio of the helical gear used in this simulation is 2.47, two teeth or three teeth are in contact at the same time during the mesh cycle, and the total length of contact lines on the tooth surface continuously changed. Due to this fact, the load taken by the contact lines were continuously changed. In the misaligned condition as being studied in this chapter, the contact pattern will not be positioned over the tooth flank as opposed to the gear contact with non-misaligned condition.
A simple way to examine the contact pattern on the helical gear tooth surfaces is using a planar projection surface instead of the tooth flank surface for load distribution plotting purpose. As seen in Figure 33, the projection plane can be established by rotating the four vertex of the tooth flank from 3D Cartesian coordinate system (𝑥, 𝑦, 𝑧) to planar coordinate system (𝑟 = √(𝑥2+ 𝑦2), 𝑧).
Figure 33: Definition of a projection plane
Since two misaligned configurations being compared in Figure 32 has a phase shift of 45 degree, it is reasonable to plot the contact pattern at the corresponding angular position (e.g. starting at 0 degree for the first case, and 45 degree for the second case) for the comparison. In this manner, the four contact patterns were plotted at every 90 degree rotation for each case as shown in Figure 34 and Figure 35 respectively. From this comparison plots, the two different misaligned assembly cases exhibited almost identical contact patterns, which confirms that TE curves examined in Figure 32 are sensible results. x y z z r
Figure 34: Contact analysis in a mesh cycle during a complete revolution for +X+Y/+X+Y case [T=250 Nm, Ω=10rpm]
Figure 35: Contact analysis in a mesh cycle during a complete revolution for +X+Y/-X+Y case [T=250 Nm, Ω=10rpm]
4.3 Tooth modification
The term “tooth microgeometry” refers to deviations of the tooth surface from a perfect and correctly aligned involute. Microgeometry modifications along profile and flank line of the tooth surface can achieve a significant reduction of gear vibration . Microgeometry modification also improves the durability of the gear from uniform contact stress distribution.
Figure 36 shows both profile and flank line modification on the gear tooth flank in a parabolic fashion.
Figure 36: Tooth modification of both profile modification (t) and flank line modifications (c) 
4.3.1Flank line modification on the misaligned gear pair
In this chapter, an attempt to apply the tooth modification to mitigate the uneven load distribution and TE modulation fluctuation in effective way. Among these two types of tooth modification (profile and flank line tooth modifications), flank line modification is well known to be effective to reduce the sensitivity to misalignment. Lead slope flank line modification can be efficient when the misalignment condition is predefined  but to apply for the various combination of the misalignment cases in this study, parabolic crowning flank line modification was applied to shift the contact pattern from edge towards center position of the tooth flank.
4.3.2Case study for flank line modification effect on TE
To validate how flank line crowning modification improves the performance of the gear pair in terms of reducing peak-to-peak amplitude of TE and to mitigate the modulation due to the angular gear misalignment, three sets of test cases that show different level of TE magnitude variation and modulation phenomena were selected. The example set of TE is obtained as seen in the following summary;
Case B: 0/+X (gear-1 is perfectly aligned and gear-2 contains angular misalignment along the positive X direction by 0.1˚).
Case C: 0/+X+Y (gear-1 is perfectly aligned and gear-2 contains angular misalignment along both positive X and Y directions by 0.1˚ each).
188.8.131.52 TE changes with different amount of flank line modification
In the three configurations, two different values for the flank line crowning were applied to see the TE changes with load distribution. Figure 37 shows the TE without tooth flank line modification as baseline TE curves under different misaligned condition with T = 150 Nm and 10 rpm rotational speed.
Figure 37: TE unmodified gear pair [T=150Nm, Ω =10rpm]
Figure 38 shows TE changes when 10 and 20 μm flank line crowning modifications applied to case B respectively.
Figure 38: TE of case B with increasing flank line crowing on gear-2 [T=150Nm, Ω =10rpm]
Figure 39 shows TE changes when 10 and 20 μm flank line crowning modifications applied to the most severe misaligned condition (case C).
Figure 39: TE of case C with increasing flank line crowing on gear-2 [T=150Nm, Ω =10rpm]
As seen in the TE curves from the simulation of misaligned cases having different level of severity, significant amount of TE changes were examined whereas the low frequency component (2nd order of
shaft frequency) remains similar. It is noted that this low frequency component does not have influence in the NVH perspective Errore. L'origine riferimento non è stata trovata..
184.108.40.206 Modulation reduction in TE
To examine the modulation reduction when two different amount (10 and 20 μm) of flank line modifications are applied to the most severe case (case C), low frequency components below 30th order
are filtered out, and also higher frequency components above 500th order were removed. The tooth passing component is then converted from time domain to the angular domain as seen in Figure 40 .
Figure 40: Tooth passing component of TE (30-500th order) of case C in angle domain
The comparison result plot shows that when larger amount of flank line crowning is applied, the maximum peak-to-peak TE reduced where the minimum peak-to-peak TE increased. In this condition changes improved the highly modulated condition of the unmodified gear pair contact, resulting in more uniform TE fluctuation.
220.127.116.11 Contact pattern changes
To gain a comprehensive understanding on how the flank line crowning makes TE curve more uniform, multiple load distribution at the critical angular position has been compared for nominal and case C as seen in Figure 41 and Figure 42. The selected angular position is the one exhibiting the minimum and maximum TE during one complete gear revolution. It is clear to see that an inferior level of edge contact condition was detected at the angular position showing the maximum TE in case A while the load distribution look well distributed over the flank at the position showing the minimum TE. After flank line crowning modification on case C gear pair, the load distribution at the angular position showing the maximum TE gets better distributed with lower loading amplitude. It is noted that respect to the unmodified case, where the load distribution was uniform at the angular position with minimum peak-to-peak (green line in picture), the application of the flank line modification reduce the area of contact, increasing the load. However, the general peak-to-peak amplitude and modulation was considerably reduced with more uniform load distribution over the entire gear revolution.
Figure 41: Tooth passing component of TE of case C (unmodified) with contact analysis for a mesh cycle
Figure 42: Tooth passing component of TE of case C (20µm flank line crowing) with contact analysis for a mesh cycle
From the parameter study for the TE and load distribution of helical gear pairs under various misalignment conditions, the following aspects were observed.
Due to the shafts and bearings deflection caused by reaction force at tooth contact, center distance of gear pair increases; however, it results in small TE peak-to-peak value changes if compared against TE fluctuation under angular misaligned condition caused by SIT-LOCK assembly error.
The simulation results for unmodified and modified (flank line) tooth flank show the amplitude modulations with two mountain/valley shapes over the one gear revolution due to the nature of the gear misalignments with respect to the shafts.
The maximum amplitude of the peak-to-peak TE becomes smaller (about 30%) with a 20 µm tooth flank line crowing modification. In addition, the overall shape of the load distribution becomes more uniform due to the fact that the tooth modification brings the contact to the center region of tooth flank.
While the TE at some angular region is higher than the one of unmodified gear pair, the modulation phenomenon significantly reduced resulting from the favorable load distribution by tooth modification.
The main goal of the validation process is to evaluate the accuracy and robustness of the modeling techniques (simulation data) using test measurement data. The numerical analysis (gear contact) is performed with rigid structures (shafts and bearings) using gear contact element solver by linking with Siemens LMS Virtual.Lab Motion. The data obtained from the simulations are processed and compared to the experimental results processed in the same way. This chapter provides the in depth analysis of some validation examples, and the numerical and experimental results are evaluated.
5.1 Case study: Quasi-static
In this case study, the quasi static conditions are applied to the gear set. The shafts rotate at low speed, and a constant speed of 10 rpm with best practice has been applied to the system by manually rotating the flywheel. A medium level of torque value of 100 Nm has been applied, and the experimental data been post-processed and analyzed. The post-processing of the raw data has been done by following the methodology introduced in previous section. The results in angle domain provide an overall insight on the TE at different angular positions and it is very useful for investigating misalignment issues when rotating speed is not strictly constant during the test of low rotating speed. The spectrum  makes it possible to investigate the mesh frequency between test gear pair, and the coupling effects between the two gear pairs (test and reaction side gear pairs) by analyzing the mesh harmonics and sidebands form the modulation .
5.1.1Result comparison in angle domain
Figure 43 represents the tooth passing component (30th - 500th order) of the filtered TE over the rotation
angle. As illustrated in this figure, it is found that a modulation phenomenon is present. In the figure, the orange curve indicates the second order modulation of the TE signal because it highlights an event occurring two times per revolution, whereas the blue curve indicates the presence of a phenomenon occurring 8 times per revolution (8th order modulation).
Figure 43: Tooth passing component of measured data [T=100Nm, Ω =10rpm]
The reason behind this 8th order modulation has to be investigated. To analyze how this effect occurs, by
considering tooth passing contribution as the carrier of the information of TE and associating it a wave form (sine wave) called 𝑐(𝑡) of frequency 𝑓𝑐 and amplitude 𝐴 given by:
𝑐(𝑡) = 𝐴 ∙ sin(2𝜋 𝑓𝑐 𝑡)
Let 𝑚(𝑡) represent the modulation waveform, coming from the coupling effect with the rest of the structure of test rig. For simplicity, the modulation effect will be considered as a sinusoidal wave by the principle of Fourier decomposition, and 𝑚(𝑡) can be expressed as the sum of a number of sinusoidal waves of various frequencies, amplitudes, and phases described as.
𝑚(𝑡) = 𝑀 ∙ cos(2𝜋 𝑓𝑚 𝑡 + 𝜑)
where, 𝑀 is the amplitude of the modulation and it has to be 0 < 𝑀 < 1, so the quantity (1 + 𝑚(𝑡)) is always positive. Amplitude modulation occurs when the signal carrier wave 𝑐(𝑡) is multiplied by the positive quantity(1 + 𝑚(𝑡)):
𝑦(𝑡) = [1 + 𝑚(𝑡)] ∙ 𝑐(𝑡) = [1 + 𝑀 ∙ cos(2𝜋 𝑓𝑚 𝑡 + 𝜑)] ∙ 𝐴 ∙ sin(2𝜋 𝑓𝑐 𝑡)
Using Prosthaphaeresis identities, the resulting wave𝑦(𝑡) can be described as:
𝑦(𝑡) = 𝐴 ∙ 𝑠𝑖𝑛(2𝜋 𝑓𝑐 𝑡) +
2 [sin(2𝜋(𝑓𝑐+ 𝑓𝑚)𝑡 + 𝜑) + sin(2𝜋(𝑓𝑐− 𝑓𝑚)𝑡 − 𝜑)]
Therefore, the modulated resultant signal has three components: the carrier wave 𝑐(𝑡) which is unchanged, and two other waves, known as sidebands.
It should be noted that the additional modulation attribute having the 8th order of shaft frequency was
measured. The source of the modulation wave can be found in the coupling effect between test gear pair and reaction gear pair of the test rig. Reaction gear pair comprises helical gears with the same number of teeth (𝑧𝑟 = 64) while test gear pair also comprises helical gears with the same number of teeth (𝑧𝑡 =
56). Since all shafts rotate with the same speed, these two gear pairs have a common factor of 8 meaning that a certain form of periodic signal (8 times per revolution) can be generated influencing the test gear mesh to some extent even if two sets of gear pairs are isolated with flexible coupling and flywheel to minimize this effect.
Once the identified types of misalignments with rotational axis and the amount are introduced in the gear contact multibody analysis, the resulting TE signal showed the TE fluctuation in tooth passing frequency and modulated with 2nd order of shaft frequency. Figure 44 shows the resulting tooth passing components
of TE for experiments (in orange) and simulation (in blue) under the quasi static condition with 100 Nm torque.