Noise Reduction in Spur Gear Systems

This article lists some tips for reducing gear case noise. With this aim, a static analysis was carried out in order to describe how stresses resulting from meshing gears affect the acoustic emissions. Different parameters were taken into account, such as the friction, material, and lubrication, in order to validate ideas from the literature and to make several comparisons. Furthermore, a coupled Eulerian–Lagrangian (CEL) analysis was performed, which was an innovative way of evaluating the sound pressure level of the aforementioned gears. Different parameters were considered again, such as the friction, lubrication, material, and rotational speed, in order to make different research comparisons. The analytical results agreed with those in the literature, both for the static analysis and CEL analysis—for example, it was shown that changing the material from steel to ductile iron improved the gear noise, while increasing the rotational speed or the friction increased the acoustic emissions. Regarding the CEL analysis, air was considered a perfect gas, but its viscosity or another state equation could have also been taken into account. Therefore, the above allowed us to state that research into these scientific fields will bring about reliable results.


Introduction
Under working conditions, spur gears are simultaneously subjected to mechanical and thermal loads, both of which are strictly related to noise emissions. Specifically, the transmitted torque affects the maximum stress on the teeth, rotational speed affects the cyclic loading, and friction leads to increased temperature in the gears and surrounding air. In [1], the interactions between gears and the air is considered, whereby the modeling air is treated as a one-dimensional ideal gas with constant entropy (ds = 0) and an adiabatic flow. The unification of the laws of thermodynamics and Newtonian mechanics has been pursued by many scientists in the last century. The principles are based on using entropy as a bridge between mechanics and thermodynamics [2]. Mechanothermodynamics (MTD) combines two branches of physics-mechanics and thermodynamics-to study the evolution of complex systems. For example, the theory of elasticity assumes that there is no entropy generation at the material level [3]. As a result, everything is reversible, which violates the second law of thermodynamics. The above approaches and models for the energy and stress-strain states of complex systems under thermodynamic and mechanical loads are considered in [4][5][6]. Damage and entropy

Materials and Methods
To perform the noise evaluation for the spur gears, FEM modeling and numerical simulation were performed. The first step in the FEM analysis was designing a spur gear pair. This was achieved using SolidWorks. The design parameters are shown in the following table (Table 1).

FEM Static Analysis of a Spur Gear Pair
To perform the FEM static analysis of the spur gear pair, the Abaqus/CAE 6.14-5 program code was used. The considered model was discretized using C3D8 brick elements, giving an 8-node linear brick (Abaqus Manual). The following picture (Figure 1) shows the modeled spur gears. report, through the mean analysis performed using the FEM program code, it was possible to describe how stresses resulting from meshing gears affected the acoustic emissions [34][35][36][37].

Materials and Methods
To perform the noise evaluation for the spur gears, FEM modeling and numerical simulation were performed. The first step in the FEM analysis was designing a spur gear pair. This was achieved using SolidWorks. The design parameters are shown in the following table (Table 1).

FEM Static Analysis of a Spur Gear Pair
To perform the FEM static analysis of the spur gear pair, the Abaqus/CAE 6.14-5 program code was used. The considered model was discretized using C3D8 brick elements, giving an 8-node linear brick (Abaqus Manual). The following picture (Figure 1) shows the modeled spur gears. Regarding the FEM static analysis of the spur gear pair, five conditions were considered in order to evaluate the stress resulting from the meshing of the gears, since their acoustic emissions are influenced by the stresses resulting from the contact point [38,39]. Different parameters were taken into account for each condition considered, as shown in Table 2. Regarding the FEM static analysis of the spur gear pair, five conditions were considered in order to evaluate the stress resulting from the meshing of the gears, since their acoustic emissions are influenced by the stresses resulting from the contact point [38,39]. Different parameters were taken into account for each condition considered, as shown in Table 2.

Condition Friction Material Parameters Lubricated Friction
An exponential decay friction model (µ) was adopted ( Figure 2). It was necessary to insert the static friction coefficient µ s , the kinetic friction coefficient µ k , and the decay coefficient d c as well. This model was based on the following equation, where Υ is the slip rate: An exponential decay friction model (µ) was adopted ( Figure 2). It was necessary to insert the static friction coefficient μs, the kinetic friction coefficient μk, and the decay coefficient dc as well. This model was based on the following equation, where ϒ is the slip rate: Figure 2. Exponential decay friction model.

Results
Condition A: The first analysis involved a steel spur gear pair and was characterized by frictionless contact. The results are shown in Figures 3 and 4.

Results
Condition A: The first analysis involved a steel spur gear pair and was characterized by frictionless contact. The results are shown in Figures 3 and 4.  An exponential decay friction model (µ) was adopted ( Figure 2). It was necessary to insert the static friction coefficient μs, the kinetic friction coefficient μk, and the decay coefficient dc as well. This model was based on the following equation, where ϒ is the slip rate: Figure 2. Exponential decay friction model.

Results
Condition A: The first analysis involved a steel spur gear pair and was characterized by frictionless contact. The results are shown in Figures 3 and 4.                                As we can see, the maximum von Mises stress equaled 412 MPa, while the contact pressure on the tooth amounted to 332 MPa.

First Analysis (Condition A) versus Second Analysis (Condition B)
The second analysis (condition B) showed higher von Mises stress and contact pressure values than the first one, so the presence of friction made the stress increase together with the sound excitation, as previously mentioned for this parameter.
Third Analysis (Condition C) versus Fourth Analysis (Condition D) As we can see in the third and fourth analyses, the ductile iron spur gears had lower von Mises stress and lower contact pressure values, in agreement with comments regarding gear materials and ductile iron properties. Therefore, the sound excitation was lower as well.

Fifth Analysis (Condition E) versus First Analysis (Condition A)
The fifth analysis showed that lubricated friction made the stress decrease together with the contact pressure, leading to agreement that the oil coating reduced the sound excitation of the gears.

First Two Analyses (Conditions A and B) versus Third and Fourth (Conditions C and D) Analyses
The first two analyses (conditions A and B), which involved steel spur gear pairs, showed higher stress values than the third and fourth analyses, which involved ductile iron spur gears. These results again validated the previously mentioned theories about noise in gears.
In Figure 13, the diagram shows the von Mises stress trends between the different analyses. As we can see, the maximum von Mises stress equaled 412 MPa, while the contact pressure on the tooth amounted to 332 MPa.

First Analysis (Condition A) versus Second Analysis (Condition B)
The second analysis (condition B) showed higher von Mises stress and contact pressure values than the first one, so the presence of friction made the stress increase together with the sound excitation, as previously mentioned for this parameter.
Third Analysis (Condition C) versus Fourth Analysis (Condition D) As we can see in the third and fourth analyses, the ductile iron spur gears had lower von Mises stress and lower contact pressure values, in agreement with comments regarding gear materials and ductile iron properties. Therefore, the sound excitation was lower as well.

Fifth Analysis (Condition E) versus First Analysis (Condition A)
The fifth analysis showed that lubricated friction made the stress decrease together with the contact pressure, leading to agreement that the oil coating reduced the sound excitation of the gears.

First Two Analyses (Conditions A and B) versus Third and Fourth (Conditions C and D) Analyses
The first two analyses (conditions A and B), which involved steel spur gear pairs, showed higher stress values than the third and fourth analyses, which involved ductile iron spur gears. These results again validated the previously mentioned theories about noise in gears.
In Figure 13, the diagram shows the von Mises stress trends between the different analyses. In Figure 14, the diagram shows the contact pressure trends between the different analyses. In Figure 14, the diagram shows the contact pressure trends between the different analyses. Finally, we can see that the contact pressure distribution was correct, which allowed us to say that the analyses were carried out in a proper and accurate manner, providing reliable results. Finally, we can see that the contact pressure distribution was correct, which allowed us to say that the analyses were carried out in a proper and accurate manner, providing reliable results.

Difference Static and Dynamic Analyses
The static analyses were mainly carried out with the aim of setting all of the basic parameters necessary for the simulations (the type of material, friction, constraint conditions, and contact algorithms used). Once all of the basic parameters had been appropriately set (by checking for the presence of any penetration and assessing that the distribution and stress values were acceptable), the model was further complicated by inserting an airbox and carrying out coupled Eulerian-Lagrangian (CEL) analyses [40][41][42]. The parameters that affected this second type of analysis were related to the evaluation of the acoustic emissions, for which we focused on the friction, lubrication, and rotation speed. The second type of material was not included among the variables, as the required number of simulations involving analyses requiring long computation times would have greatly increased [43,44].

Coupled Eulerian-Lagrangian (CEL) Analysis of a Spur Gear Pair
CEL analyses are very demanding in terms of the computation time; therefore, the box sizes were as small as possible in order to contain the gear wheels and limit the number of elements.
Constraint conditions were applied to avoid any disturbing effects; for example, waves could be reflected on the edges of the box walls. Indeed, the air domain was assumed to be practically infinite. The keyword used in the Abaqus input file that allowed this condition to be achieved was: Within the air domain, a node was chosen near the walls in order to measure the sound pressure level. Its exact position was 80 mm below the midpoint of the wheelbase of the two wheels. Both the distance of 80 mm as that of a meter has no particular meaning because they are used only as reference points to study the sound variations [45][46][47]. These values were only set as references to allow a comparison between the various analyses under equivalent conditions. The pressure values obtained from these analyses were then extrapolated to a distance of 1 m (with the following formula) with respect to the reference point ( Figure 1). SPL = SPL re f − 20log 10 r r re f (2) where SPL is the sound pressure level at a distance of 1 m; SPL ref is the sound pressure level at a distance of about 80 mm; r is a distance of 1 m; r ref is a distance of about 80 mm. In this way, in order to evaluate the noise level deriving from the meshing spur gear pair, the pair was modeled with a Lagrangian formulation. The gears were in contact during their operation and were considered to be "immersed" in an air box (modeled in an Eulerian way) in order to allow the propagation of mechanical and acoustic waves inside the considered mean.
The airbox dimensions were as follows: X = 350 mm; Y = 200 mm; Z = 100 mm. The airbox was modeled with EC3D8R brick elements; that is, an 8-node linear Eulerian brick, reduced integration, and hourglass control (Abaqus Manual). The number of Eulerian elements used was 155,382. Therefore, in order to evaluate the sound pressure level resulting from the meshing spur gear pair, different coupled Eulerian-Lagrangian analyses were carried out, putting the gears in an Eulerian mesh, which represented air (modeled as a perfect gas), since noise is characterized by pressure waves that propagate through a medium. A noise, vibration, and harshness (NVH) analysis could not be properly performed, since it could only be used to evaluate the noise resulting from rolling bearings and supporting system vibrations [48][49][50][51]. The following table (Table 3) shows the parameters that characterized the different examined conditions. As explained in [1], air was modeled as an ideal gas and entropy variation was imposed as zero (ds = 0). Despite friction being taken in account, no energy degradation was considered, meaning the temperature remained constant and equal to room temperature.
Regarding the friction model, the same exponential decay friction model was used. It was necessary to insert the static friction coefficient µ s , the kinetic friction coefficient µ k , and the decay coefficient d c as well. This model was based on Equation (1), in which Υ is the slip rate ( Figure 15).

10
As explained in [1], air was modeled as an ideal gas and entropy variation was imposed as zero (ds=0). Despite friction being taken in account, no energy degradation was considered, meaning the temperature remained constant and equal to room temperature. Regarding the friction model, the same exponential decay friction model was used. It was necessary to insert the static friction coefficient μs, the kinetic friction coefficient μk, and the decay coefficient dc as well. This model was based on Equation (1), in which is the slip rate ( Figure 15)

Discussion
Condition 1: The first analysis involved a steel spur gear pair and was characterized by frictionless contact and a rotational speed of 500 RPM. Figure 16 shows the sound pressure level of the meshing gears at a distance of 1 m. Finally, the sound pressure level resulting from gear meshing was evaluated at a distance of 1 m thanks to Equation (3).

Discussion
Condition 1: The first analysis involved a steel spur gear pair and was characterized by frictionless contact and a rotational speed of 500 RPM. Figure 16 shows the sound pressure level of the meshing gears at a distance of 1 m. As we can see, after the initial transient peak, the SPL stabilized between 80 and 85 dB. Condition 2: The second analysis involved a steel spur gear pair and was characterized by the mentioned exponential decay friction model and a rotational speed of 500 RPM. Figure 17 shows the sound pressure level of the meshing gears at a distance of 1 m. As shown, after the initial transient peak, the SPL stabilized at over 100 dB Condition 3: The third analysis involved a steel spur gear pair and was characterized by the mentioned exponential decay friction model and a rotational speed of 1500 RPM. Figure 18 shows the sound pressure level of the meshing gears at a distance of 1 m. As we can see, after the initial transient peak, the SPL stabilized between 80 and 85 dB.

Condition 2:
The second analysis involved a steel spur gear pair and was characterized by the mentioned exponential decay friction model and a rotational speed of 500 RPM. Figure 17 shows the sound pressure level of the meshing gears at a distance of 1 m. Finally, the sound pressure level resulting from gear meshing was evaluated at a distance of 1 m thanks to Equation (3).

Discussion
Condition 1: The first analysis involved a steel spur gear pair and was characterized by frictionless contact and a rotational speed of 500 RPM. Figure 16 shows the sound pressure level of the meshing gears at a distance of 1 m. As we can see, after the initial transient peak, the SPL stabilized between 80 and 85 dB.

Condition 2:
The second analysis involved a steel spur gear pair and was characterized by the mentioned exponential decay friction model and a rotational speed of 500 RPM. Figure 17 shows the sound pressure level of the meshing gears at a distance of 1 m. As shown, after the initial transient peak, the SPL stabilized at over 100 dB Condition 3: The third analysis involved a steel spur gear pair and was characterized by the mentioned exponential decay friction model and a rotational speed of 1500 RPM. Figure 18 shows the sound pressure level of the meshing gears at a distance of 1 m. As shown, after the initial transient peak, the SPL stabilized at over 100 dB Condition 3: The third analysis involved a steel spur gear pair and was characterized by the mentioned exponential decay friction model and a rotational speed of 1500 RPM. Figure 18 shows the sound pressure level of the meshing gears at a distance of 1 m. As we can see, after the initial transient peak, the SPL stabilized at around 100 dB Condition 4: The fourth analysis involved a steel spur gear and was characterized by the mentioned exponential decay friction model and a rotational speed of 3000 RPM. Figure 19 shows the sound pressure level of the meshing gears at a distance of 1 m. As we can see, after the initial transient peak, the SPL stabilized at around 100 dB Condition 4: The fourth analysis involved a steel spur gear and was characterized by the mentioned exponential decay friction model and a rotational speed of 3000 RPM. Figure 19 shows the sound pressure level of the meshing gears at a distance of 1 m.
As we can see, after the initial transient peak, the SPL stabilized at around 100 dB Condition 4: The fourth analysis involved a steel spur gear and was characterized by the mentioned exponential decay friction model and a rotational speed of 3000 RPM. Figure 19 shows the sound pressure level of the meshing gears at a distance of 1 m. Figure 19. Condition 4: Sound pressure level spectrum.
As shown, after the initial transient peak, the SPL stabilized between 100 and 110 dB. Condition 5: The fifth analysis involved a steel spur gear pair and was characterized by the mentioned exponential decay friction model (lubricated friction) and a rotational speed of 500 RPM. Figure 20 shows the sound pressure level of the meshing gears at a distance of 1 m. As shown, after the initial transient peak, the SPL stabilized between 100 and 110 dB.

Condition 5:
The fifth analysis involved a steel spur gear pair and was characterized by the mentioned exponential decay friction model (lubricated friction) and a rotational speed of 500 RPM. Figure 20 shows the sound pressure level of the meshing gears at a distance of 1 m. As we can see, after the initial transient's peak, the SPL stabilized between 85 and 100 dB. In the first analysis, it was possible to see that the SPL stabilized between 80 and 85 dB, while in the second one it stabilized at over 100 dB because of the friction's presence, which meant the gears' acoustic emissions were higher. Therefore, we can agree with the comments about the influence of friction on gear noise. Figure 21 shows a comparison between the respective sound pressure levels of the first and second analyses. As we can see, after the initial transient's peak, the SPL stabilized between 85 and 100 dB.

First Analysis (Condition 1) versus Second Analysis (Condition 2)
In the first analysis, it was possible to see that the SPL stabilized between 80 and 85 dB, while in the second one it stabilized at over 100 dB because of the friction's presence, which meant the gears' acoustic emissions were higher. Therefore, we can agree with the comments about the influence of friction on gear noise. Figure 21 shows a comparison between the respective sound pressure levels of the first and second analyses. 13 As we can see, after the initial transient's peak, the SPL stabilized between 85 and 100 dB. In the first analysis, it was possible to see that the SPL stabilized between 80 and 85 dB, while in the second one it stabilized at over 100 dB because of the friction's presence, which meant the gears' acoustic emissions were higher. Therefore, we can agree with the comments about the influence of friction on gear noise. Figure 21 shows a comparison between the respective sound pressure levels of the first and second analyses.

Third Analysis (Condition 3) versus Fourth Analysis (Condition 4)
In the third analysis, after the initial transient peak, the SPL stabilized at around 100 dB, while in the fourth one it stabilized between 100 and 110 dB, because of the increased rotational speed (it

Third Analysis (Condition 3) versus Fourth Analysis (Condition 4)
In the third analysis, after the initial transient peak, the SPL stabilized at around 100 dB, while in the fourth one it stabilized between 100 and 110 dB, because of the increased rotational speed (it went from 1500 RPM to 3000 RPM), as well as the friction. Therefore, we can agree again with the idea that the increasing rotational speed affected the acoustic emission of the meshing gears. Figure 22 shows a comparison between the respective sound pressure levels from the third and fourth analyses. went from 1500 RPM to 3000 RPM), as well as the friction. Therefore, we can agree again with the idea that the increasing rotational speed affected the acoustic emission of the meshing gears. Figure  22 shows a comparison between the respective sound pressure levels from the third and fourth analyses.

Fourth Analysis (Condition 4) versus Fifth Analysis (Condition 5)
In the fourth analysis the SPL stabilized between 100 and 110 dB, while in the fifth analysis it stabilized between 80 and 100 dB. This was due to the presence of lubricated friction and a lower rotational speed. Figure 23 shows a comparison between the respective sound pressure levels of the fourth and fifth analyses.  In the fourth analysis the SPL stabilized between 100 and 110 dB, while in the fifth analysis it stabilized between 80 and 100 dB. This was due to the presence of lubricated friction and a lower rotational speed. Figure 23 shows a comparison between the respective sound pressure levels of the fourth and fifth analyses.

Fourth Analysis (Condition 4) versus Fifth Analysis (Condition 5)
In the fourth analysis the SPL stabilized between 100 and 110 dB, while in the fifth analysis it stabilized between 80 and 100 dB. This was due to the presence of lubricated friction and a lower rotational speed. Figure 23 shows a comparison between the respective sound pressure levels of the fourth and fifth analyses.

Second Analysis (Condition 2) versus Fifth Analysis (Condition 5)
In the second analysis the SPL stabilized at over 100 dB, while in the fifth analysis it stabilized between 85 and 100 dB. This was explained by the presence of lubricated friction in the fifth analysis, meaning we found a lower level of acoustic excitation. Figure 24 shows a comparison between the respective sound pressure levels of the second and fifth analyses. In the second analysis the SPL stabilized at over 100 dB, while in the fifth analysis it stabilized between 85 and 100 dB. This was explained by the presence of lubricated friction in the fifth analysis, meaning we found a lower level of acoustic excitation. Figure 24 shows a comparison between the respective sound pressure levels of the second and fifth analyses.

Final Remarks
In the model adopted for the CEL analysis, air was considered a perfect gas, but it could be changed by taking into account the viscosity of air or another state equation. Therefore, different results could have been obtained, leaving an opportunity to do more research in this field.

Final Remarks
In the model adopted for the CEL analysis, air was considered a perfect gas, but it could be changed by taking into account the viscosity of air or another state equation. Therefore, different results could have been obtained, leaving an opportunity to do more research in this field.

Conclusions
Noise reduction, at least in Europe, has become a major concern for communities [52,53]. This concern has led to great societal pressure on policy-makers, thus giving rise to stacked legislations and regulations at various levels. In Europe, two directives address noise issues, the first from a general standpoint and the second one specifically in regard to noise-related operating restrictions at the community level. Both of these EC directives refer to notions that are now commonly handled by specific industries, such as noise mapping or dose-response curves. Each of the aforementioned regulations has triggered an ongoing effort by engine manufacturers and by their associated research centers to define and follow a path toward more silent devices.
In this work, the acoustic emissions generated due to the spur gears meshing was studied and various values of the operating parameters were considered. The work was organized into two stages: (1) static analysis of the meshing gears was conducted to set and evaluate the goodness of the parameters of the model and to study the stress levels reached in the teeth; (2) coupled Eulerian-Lagrangian (CEL) analysis was performed to evaluate the noise generation. In this stage, the spur gears were "immersed" in an air box at room temperature. In this way, the pressure waves generated due to the interactions of the gears could propagate up to a reference point, where the sound pressure level was measured. Many work conditions were taken into account in the second stage in order to study how they can affect noise generation. The air was modeled as an ideal gas at room temperature and no heat flux was generated due to the friction among the gears. The entropy was set as constant (ds = 0) and an adiabatic process was considered. The results were in good agreement with the literature.
Future developments in this area could include analysis of how the heat generated during the meshing affects the noise generation. In this way, a damage detection model that considers the interactions between the noise generation and the entropy formation could be developed and tested on an experimental apparatus.
Author Contributions: All the authors equally contributed to carrying out the research, whose results are reported in this work. All authors have read and agreed to the published version of the manuscript.