Gear noise evaluation through multibody TE-based simulations · PDF fileGear noise evaluation...

Gear noise evaluation through multibody TE-based simulations

A.Palermo1, D. Mundo1, A.S. Lentini2, R. Hadjit2, P. Mas2, W. Desmet3 1 University of Calabria, Department Mechanical Engineering Ponte Pietro Bucci, 87036, Rende, Italy email: [email protected] 2 LMS International, Interleuvenlaan 68, B-3001, Leuven, Belgium 2 K.U.Leuven, Department Mechanical Engineering Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

Abstract The possibility of estimating noise radiating from a gearbox is essential to achieve valid design solutions in shorter timeframes and to limit the testing phase, especially in those industrial fields, such as automotive, helicopter and wind turbine industry, with a strong demand for gear noise reduction. This paper presents a methodology for the calculation of gear bearing forces, useful for the acoustic analysis of gearboxes and applicable to spur as well as helical parallel gear systems. The methodology is based on the implementation of a procedure for the computation of the dynamic transmission error (DTE) in a multibody environment. The DTE is obtained from the static transmission error (STE), i.e. the static relative displacement between meshing teeth, which is variable along the mesh cycle. The adopted multibody technique enables to overcome the principal drawbacks of FEM, achieving good computational efficiencies, and of analytical models, avoiding to lump the system in one or few degrees of freedom. These goals are reached by means of a user-defined force element, acting as teeth meshing force, which stems from the integration of the multibody software, LMS Virtual.Lab Motion, with an external program specialized in gear meshing analysis. The multibody software captures the system dynamics and includes the nonlinear effects such as gear backlash, bearing clearances and stiffness; the specialized software enables to consider tooth microgeometry, assembly errors, global and contact tooth stiffness and also shaft deflections. The new feature introduced by the proposed technique is the ability to take into account the instantaneous torque with good computational efficiency.

1 Introduction

Gears are extensively employed in mechanical systems since they allow the transfer of motion in a wide range of working conditions, with a variety of gear ratios, and at reasonable production costs. The gear meshing is a complex process because it involves moving and multiple contact points, variable load sharing on the meshing teeth, contact mechanics (which is nonlinear), and all of them from a dynamic standpoint. Furthermore tooth microgeometry, manufacturing imperfections and assembly errors have relevant effects on the behavior of gear systems and cannot be ignored [1-3]. This complexity has to be faced in the design phase, which must address both endurance and noise requirements. For these reasons, a considerable amount of research works on gear dynamics is available in literature, but still many aspects remain unresolved. Moreover, today’s markets are highly competitive and therefore reaching valid solutions in shorter timeframes represents a clear advantage. In this context, numerical models and simulations allow to achieve solutions to improve the dynamic behavior of gear systems, and to limit the testing phase saving time and money. This explains why efforts continue to be spent in the gear dynamics research field, with applications especially in helicopter [1], wind turbine [4] and automotive [5]

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industries. Gear dynamics are mentioned because, at the occurrence, the proposed methodology enables to evaluate the dynamic meshing loads, which in transients can be several times higher than the static ones. From this perspective, the proposed technique will also allow the simulation of load sharing in planetary gear trains, which is currently a major issue for this kind of transmissions [6-8]. Coming back to gear noise purposes, the proposed methodology takes into account the dynamic transmission error (DTE), which is defined for a gear pair as the dynamic relative displacement between meshing teeth.

The transmission error is widely regarded as one of the main causes of gear noise [10, 11]. According to Munro [12], the transmission error was defined first by Harris [13] in 1958, who started its analytical investigation. Nevertheless references [14, 15] prove that this concept was already applied before, but using an empirical approach. The two main factors affecting the TE are the mesh stiffness, which accounts for tooth flexibility and number of meshing tooth pairs, and the tooth microgeometry, in terms of intentional modifications [10, 14-16] and manufacturing errors. Variations in the TE, during the gear meshing, trigger vibrations and then airborne noise [17, 18]. The present analysis will be focused on the case of involute parallel spur and helical gears, which are the most common ones.

Several methodologies are available in literature to simulate and estimate meshing vibrations, using analytical lumped parameter, Finite Element (FE), and multibody approaches. The analytical model proposed by Umezawa [19-22], later corrected by Cai [23, 24] to consider the influence on tooth stiffness of the gear tooth number, describes the gear meshing with a single degree of freedom (SDOF) system aligned along the line of action. Assuming a time-varying function for the mesh stiffness, defined within one mesh period (or, adimensionally, within one mesh cycle), and the damping, the equation of motion can be solved. These models allow to consider the effects of tooth microgeometry, assembly and manufacturing errors, lumping them on the line of action with a displacement-driven excitation for the SDOF system. With these assumptions it is impossible to consider the three-dimensionality of the contact problem, and the quality of the results that can be obtained depends on how realistic are the mesh stiffness function for the given gear pair and the displacement excitation. Different analytical models improved the accuracy of the results using a FE model to take into account shaft deflections and three-dimensional geometry [25-28], but the lumped parameters description is still suitable to analyze simple cases. Full FE models [29] allow more accurate representations of gear systems and avoid a-priori assumptions on the TE, but since tooth contact happens in a very small area, and it spans the teeth from root to tip, highly refined mesh or contact detection followed by remeshing is needed along the whole tooth face. This causes high computational costs to run a simulation. Moreover, it is also an issue to correctly describe the tooth three-dimensional microgeometry. In the FE field, an interesting technique is proposed by Parker et al. [9], who use a semi-analytical finite element formulation specifically devised for contact problems [30]. The tooth is divided in a contact zone (extending beneath the tooth surface) and a FE zone, separated by a matching interface. The contact zone is analytically solved by means of the Boussinesq’s solution. This solution is evaluated at the matching FE nodes and the obtained nodal parameters are used to solve the remaining FE part. However, FEM techniques are not able to consider with good computational efficiencies nonlinear entities such as gear lashes, assembly clearances, bearings, clutches, and other nonlinear effects which arise from large angle rotations. Such nonlinear effects require time-domain integration, which is typical of the multibody environment. Moreover, to assess the noise and vibration performances of a geared system it is usually desirable to test a wide range of working conditions (e.g., torques, regime run-up, …). With such demands, the use of large scale finite element models in time domain becomes computationally expensive and maybe impractical.

The technique proposed in this paper is an improvement of the Static Transmission Error method described by Morgan et al. in [31]. The basic idea of the method is to let a specialized, thus highly efficient, software for gear contact analysis (abbreviated GCAS from now on) execute the calculation of the static mesh stiffness, which can then be used in the dynamic multibody simulation. The gear meshing, in the multibody simulation, is in fact governed by the dynamic equilibrium of the contact forces applied to the gears, rather than the ideal kinematic contact ratio. Considering a single spur or helical gear pair described by the standard gear parameters, the static mesh stiffness can be calculated using GCAS which enables to take into account three-dimensional teeth microgeometric modifications and manufacturing errors, teeth global and contact stiffness, shaft deflections and assembly misalignments. This mesh stiffness is obtained, for one static working condition, as a function of the position along the mesh cycle.

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Once the static mesh stiffness is imported in the multibody software, the contact forces are calculated and applied to the gears by a user-defined force element which reads the instantaneous value of the mesh stiffness based on the actual position along the mesh cycle. In this way, the meshing complexity is captured avoiding the high computational cost related to the full-scale model, since the multibody gears and shafts models are rigid (while the bearings are compliant). The improvement brought by the current work is the capability to import and use the static mesh stiffness as a function of the instantaneous values of torque. In this paper, first the multibody model adopted for the gear system is described, then the Static Transmission Error (STE) and the Dynamic Transmission Error (DTE) are defined. Subsequently, the static mesh stiffness sensitivity to the main assembly errors is evaluated, in order to identify the most influent ones. A description of the new technique to consider the variable torque follows. Finally, the obtained results are discussed and compared to the ones obtained with the previous technique and the static GCAS values.

2 Multibody simulation of gear noise

2.1 System modeling

Since the contact analysis is captured in the instantaneous static mesh stiffness by GCAS, including three-dimensional teeth microgeometric modifications and manufacturing errors, teeth global and contact stiffness, shaft deflections and assembly errors, the gear system can be modeled as rigid in the multibody environment, achieving a good computational efficiency. The term “static” mesh stiffness indicates that GCAS calculation is based on the assumption that the gears reach the equilibrium under static torque. Nevertheless the static mesh stiffness is variable along the mesh cycle, for example due to a different number of contacting tooth pairs or due to a profile modification. With more detail, GCAS returns as an output the STE, which is defined along the line of action as the difference between the real and the ideal displacement of the driven gear (Figure 1).

Figure 1 : Transmission Error definition.

The static mesh stiffness can be easily obtained dividing the nominal contact force value by the static TE.

Since the mesh stiffness can be calculated as described, in every gear train, single gear pairs can be identified and their meshing can be analyzed independently in GCAS. Once the STE is calculated by GCAS for every meshing gear pair, a user-defined force element can then be defined for each pair in the multibody model. Since this procedure is repeated for every gear pair, it will be described and discussed hereafter for one example.

The analyzed gear pair has the specifications reported in Table 1 and is shown in Figure 2. Addendum and dedendum are standard, and respectively equal to the module and 1.25 times the module. No microgeometry is considered at this stage, since it does not add to the purpose of showing the procedure.

MULTI-BODY DYNAMICS AND CONTROL 3035

Parameter name Value

Driving gear tooth number 20 Driven gear tooth number 20

Helix angle 0° Pressure angle 20°

Module 4 mm Facewidth 25 mm Center line 80 mm

Contact ratio 1.557

Table 1: Gears specifications. Figure 2: Three-dimensional view of the

gearing in the multibody software.

The gears are rigidly connected to the shafts, 200 mm long, which are supported by compliant bushings. The axial and radial stiffness are chosen to be the same for all the bushings and equal to 108 N/m, however the multibody software enables to assign clearances and stiffness-displacement laws. A reasonable damping coefficient value was adopted for all the bushings to stabilize the simulation. A time-dependent angular position law is assigned to the driving gear. In particular, a regime run-up is performed from 0 to 955 rpm in 5 s, with a linear increase in angular velocity, at an angular acceleration of circa 20 rad/s2. A constant resisting torque of 100 Nm is applied to the driven gear, causing a nominal contact force of 2660 N.

The user-defined force element governs the gear meshing, applying the instantaneous contact force to the gears based on the instantaneous static mesh stiffness (imported from GCAS), a given value of damping, and the instantaneous TE calculated by the multibody software, namely the DTE. The contact force is applied at the operating pitch point, located at the intersection between the instantaneous gear center line and the common tangent to the base circles, at half the teeth facewidth. The whole system can be represented schematically as shown in Figure 3.

Figure 3: Schematic system representation.

Therefore the driving gear follows the imposed angular position law, while experiencing a variable load due to the DTE. The driven gear, instead, is subject to a constant applied torque, while experiencing an oscillating angular position due to the DTE. The dynamic excitation in terms of transmission error is shown in Figure 4.


The results of the multibody simulation are calculated solving the system of equations of motion, which can be condensed like in Equation 1.

(1)

Where a dot in accent position indicates the time derivative, x is the vector of the Lagrangian coordinates, M, C and K are respectively the mass, damping and stiffness matrices and F is the vector of the applied loads. Referring to this formulation and recalling the definition of TE, the dynamic formulation of gear meshing can be considered as the scalar Equation 2 which belongs to the vector Equation 1:

(2)

In the Equation 2 the inertial contribution is taken into account implicitly into the DTE, when resolving the system of equations of motion.

Two aspects about this equation are worthwhile to be mentioned in order to explain how the dynamic analysis is performed. The first is that stiffness k is the static mesh stiffness which is variable along the mesh cycle and is imported from GCAS. The second is that, since the Equation 2 is part of the Equation 1, the DTE and the contact force are both influenced by all the multibody model parts in terms of inertia, damping and stiffness.

The bearing forces, which can be user later for an acoustic analysis of the gear train, are part of the solution found for the Equation 1, hence they are available in the results of the multibody simulation.

2.2 Static and Dynamic Transmission Error

Based on the general definition of the TE (Figure 1), the STE, in output from GCAS, was considered to calculate the static mesh stiffness and it was used in the multibody simulation to obtain the DTE, which accounts for the system’s dynamics. The differences between STE and DTE are shown in Figure 4.

Figure 4: STE and DTE comparison for the analyzed gearing.

The STE (black curve in Figure 4) is higher when only one tooth pair is in contact, and lower when two pairs mesh simultaneously, being the contact ratio between 1 and 2 (Table 1). This is the reason why the STE has a square wave trend. The simulated response is typical of a second order system, and dynamic oscillations are around the STE values as expected. When the angular velocity of the gears is increased, the excitation frequency increases, and the delay in system response becomes more evident. This simple case of a single gear pair is shown as an example, but the response becomes more complex when a gear train is analyzed. The adopted multibody approach is the key to capture the complex dynamic behavior of


the system, considering the mutual interactions between several gear pairs and their interactions with the power source and the power user.

Vibrations generated during meshing are one of the main causes of gear noise, and the TE is their main excitation [10]-[11]. In particular, the peak to peak value of TE can be used as an indicator of the TE variability. A constant TE, in fact, would not cause vibrations nor noise.

3 TE sensitivity to the main positioning errors

Errors affect the relative positioning of the meshing teeth. These errors affect the peak to peak TE in different extent. They originate from assembly and tooth generation errors, and shaft deflections. The most common errors in parallel gears are mainly located at the bearings with deviations from the ideal positions, clearances and deflections, which affect the shafts orientation and therefore the teeth relative positioning. Manufacturing errors can be variable from one tooth to the other, in this paper this variability will not be addressed. Under this assumption, manufacturing errors will be considered within the description of meshing teeth relative positioning.

Looking at the meshing tooth faces, five possible relative displacements (the allowed rotation of the gears is of course excluded) can be identified in space to describe the relative teeth positioning. In order to understand their effects on the TE it is useful to appropriately choose the reference system to define these linear and rotational displacements (Figure 5). Since the TE is defined along the Line Of Action (LOA), the first axis is chosen in this direction. The second axis is chosen along the orthogonal direction to the line of action in a radial plane (Offline LOA). The third axis is along the rotation axis direction. In this reference system it is possible to identify the Plane Of Action (POA), which contains the LOA and is parallel to the rotation axis, and the Offline Plane of Action (OPOA), which contains the OLOA and is parallel to the rotation axis.

Figure 5: Reference system and nomenclature for defining positioning errors.

An axial displacement results in a reduction of the effective facewidth. Good positioning accuracies are also normally achieved in the axial direction. For that reason axial displacement will not be considered in the present analysis. A uniform displacement along the LOA coincides with a constant additional TE, which does not alter the peak to peak TE value. For that reason LOA displacement will not be considered in the present analysis. A rotation in the POA alters the load distribution along the facewidth of the teeth, thus affecting the mesh stiffness and so the peak to peak TE. A uniform displacement along the OLOA can be considered as a center line variation which affects the contact ratio and so the peak to peak TE. A rotation in the OPOA is found to be ineffective on the contact area [32] and the tooth load distribution [2], so it is deemed to be ineffective also on the TE.


On the basis of the above considerations, the effects of rotations in the POA, commonly called (angular) misalignment, and variations in center line will be assessed hereafter by means of GCAS analyses. This is useful to decide if both the assembly errors have to be included in the dynamic analysis or one of the two is predominant.

3.1 Misalignment and Center Line variations

Two gear pairs without microgeometric modifications, belonging to an automotive and a wind turbine gearboxes, have been considered (Table 2).

Parameter name Automotive Wind turbine

Driving gear tooth number 39 40 Driven gear tooth number 40 80

Helix angle 26° 0° Pressure angle 20° 20°

Module 3 mm 4 mm Facewidth 25 mm 80 mm Center line 134.5 mm 240 mm

Nominal contact force 6000 N 13300 N

Table 2: Automotive and Wind turbine gear pairs specifications.

Realistic values of the center line variation are estimated in a range of ±0.2% of the whole center line, as confirmed by the values used in [2].

Since misalignment is a rotation in the POA, it can be defined in terms of a rotation angle, the slope associated to this angle, or the displacement caused by the rotation at a tooth face (Figure 6).

Figure 6: Schematic representation of misalignment and definitions

in terms of angle (α), displacement (δ) and slope (m).

Misalignment is affected by parameters such as shaft deflections, bearing positioning and clearances which cannot be assumed a priori. However the 1328 ISO Standard for Gear Quality provides the allowed ranges for slope errors along the lead direction. The slope error value has been used to estimate the range of misalignments for the considered gear pairs. Assuming an average gear quality (ISO 7), the allowed slope deviation is found to be for both gear pairs equal to 20 µm in terms of displacement. This value has been doubled because the slope deviation can be the maximum and add up for both the meshing teeth. The contribution due to shafts and bearings is then assumed to have the same magnitude of the slope deviation, so that the final misalignment is estimated to be in a range from 0 to 60 µm in terms of displacement.

The peak to peak STE percentage variation has been evaluated for center line variations and misalignments within the defined ranges (Figure 7).


Figure 7: Effects on peak to peak STE value due to misalignment and center line variation.

The peak to peak STE value increases for both the gear pairs if misalignment or center line are increased. The maximum increase of the peak to peak STE caused by the misalignment is about 100% for the wind turbine gear pair and 600% for the automotive gearbox gear pair. Considering the center line variation, the maximum increase is below 15% for both the gear pairs. The gear pair from the automotive gearbox, which is helical, shows a higher sensitivity to both misalignment and center line variation. Considering the maximum values, the misalignment can be considered as dominant with respect to the center line variation. Although, at low misalignment values, the center line variation effects prevail on misalignment, especially for the wind turbine gear pair. It has also to be pointed out that the sensitivity to misalignment is increased by the absence of lead modifications, which are usually adopted when misalignments are significant as in this case. Based on these considerations it is worthwhile to extend the technique to take into account both the dynamic misalignment and the center line variations.

4 Variable torque modeling

Up to now, the static mesh stiffness has been calculated by the user-defined force element dividing the nominal contact force by the STE obtained from GCAS as a function of the position along the mesh cycle. Therefore the nominal contact force, which has essentially the same meaning of the applied torque since the base radius is a constant, is assumed to be known and constant. The implication is that the effects of a variable applied torque are not included in the dynamic simulation. As a consequence, the unequal load sharing in multi-mesh gear trains (e.g. planetary stages [4, 7, 8]), which is a key point in their design, cannot be considered. The procedure described in the next paragraph has been implemented to overcome this current issue.

4.1 Mathematical modeling

The implemented procedure is based on the assumption that for a given position along the mesh cycle (abbreviated with POS), which is a function of the instantaneous angular position of the driving gear, there is a one-to-one correspondence between the instantaneous DTE and the instantaneous contact force (higher DTE for a higher contact force, here the damping contribution is not accounted for). Namely, given a POS and the actual DTE, the actual contact force can be calculated as a function of two variables (Equation 3). If this assumption is verified, the procedure can be used in the multibody software to calculate the instantaneous applied contact force.

, (3)


This two-variables function is described by a database of STE values calculated through GCAS considering different working conditions. In particular, the database is constituted by a set of STE curves calculated for a range of contact forces (resulting from a range of applied torques). In this way, given a POS, the STE is available as a function of the applied force (Figure 8).

The instantaneous DTE is used in place of the STE to enter the database, together with the instantaneous POS, during the multibody simulation, so that the instantaneous contact force can be calculated. This operation will be mentioned as “extraction”. The properties of the extracted contact force are discussed in the next paragraph. Keeping the focus on the mathematical aspects of the procedure, the one-to-one correspondence between applied contact force and STE is verified with the graphs in Figure 9. The one-to-one correspondence was also verified applying a misalignment and microgeometric modifications (lead and profile).

Figure 8: STE calculated through GCAS as a function of POS and

the applied contact force.

Figure 9: Applied force as a function of STE. (sections of the left figure surface

in the Fcontact - STE plane)

It has to be noted that GCAS calculates the STE in a discrete number of POS and applied torque values, therefore the database is discrete and requires interpolation (Figure 10).

Figure 10: Interpolation steps for contact force extraction.

Smooth interpolation (C1 continuity) is needed to avoid inducing stability problems for the multibody solver, therefore a simple linear interpolation cannot be used. Cubic spline interpolation ensures the C1


continuity but produces overshoots when rapid variations in STE values happen (e.g. variations in square wave trend of Figure 4). To avoid this problem, quadratic or Akima interpolating schemes were adopted.

4.2 Multibody and dynamic considerations

An important consideration is needed about the properties of the extracted contact force. The dynamic TE is used to enter a static TE database (Figure 11).

Figure 11: Contact force extraction procedure.

This implies that the extracted contact force is equal to the static part of the dynamic contact force. From a mathematical point of view, the DTE time derivative which appears in the Equation 2 goes to zero. This force is applied by the user-defined force element to the gears at the pitch point, and then the solver includes the contribution of damping and inertia.

5 Numerical results using the proposed methodology

In this paragraph the results obtained from the simulations are shown and discussed. The simulations are run for the gear pair described in paragraph 2.1.

The radial bearing forces for one bushing are shown in Figure 12 in a time window spanning from 4.5 s to 4.75 s during the run-up simulation.

Figure 12: Radial bearing forces in a time window from 4.5s to 4.75s during the run-up simulation.

The regimes corresponding to these values of time are respectively 860 rpm and 907 rpm, namely 14.33 Hz and 15.12 Hz in terms of angular frequencies. Multiplying the angular frequencies by the number of gear teeth (Table 1), the meshing frequencies are 287 Hz and 303 Hz. The bearing forces spectrum calculated for the considered time window shows the fundamental frequency, correctly located between the two extreme meshing frequencies, and its harmonics (Figure 13).


Figure 13: Radial bearing forces spectrum in a time window from 4.5s to 4.75s

during the run-up simulation.

The simulation time to execute the run-up described in paragraph 2.1 was of 307 seconds for the constant applied torque technique and 419 seconds when taking into account the variable applied torque (CPU: Intel Core Duo 2.0GHz, RAM: 2 GB @667MHz). This good computational efficiency is an important feature of the proposed methodology.

The accuracy of the results has been evaluated numerically, comparing the dynamic results obtained by the two simulation techniques and the ones statically predicted by GCAS.

5.1 Variable torque

To verify the variable torque technique, a constant torque is applied to the driven gear shaft and the obtained results are compared with the ones obtained using the constant torque technique. The results, shown in the next graphs, match each other while the variable torque technique relaxes the a-priori assumption of a constant torque. This proves that the extracted static contact force is actually the static component of the dynamic contact force and is a consistent method to excite the system.

The simulated DTE obtained using the variable torque technique (contact force extraction) perfectly traces the one from the constant torque technique (Figure 14).

Figure 14: DTE with constant and variable torque techniques for different regimes.

Also the dynamic contact forces overlap for the two techniques (Figure 15). The dynamic contact force oscillates around the value of the nominal applied contact force, which is equal to 2660 N. The extracted static contact force shows impulses at tooth pair handovers due to system response delay (peaks in Figure 15). In particular, the impulse forces are higher than the nominal contact force when the second tooth pair comes into contact, since the mesh stiffness increases suddenly while the DTE is still high (see Figure 4) because of system response delay. The opposite happens when the second tooth pair leaves contact, since the mesh stiffness suddenly decreases while the DTE is still low.


Figure 15: Dynamic contact force with constant (smooth line) and variable (marked line) torque techniques; static contact force with variable torque technique (dashed line).

6 Conclusions

In this paper a technique which enables to calculate the bearing forces aimed at the acoustic analysis of a gearbox has been proposed. This technique integrates a multibody software (LMS Virtual.Lab Motion) and a specialized software for gear meshing analysis (GCAS) to simulate gear dynamics (DTE and contact force) taking into account detailed tooth contact. In particular, the multibody analysis allows the time-domain integration of the solution, which captures the nonlinear effects of bearing stiffness and clearances, gear backlash, large rotations and other nonlinear phenomena, while the specialized software allows to include the three-dimensional tooth microgeometry, global and local tooth stiffness and relative positioning of the meshing teeth. The proposed technique enables to perform run-up analysis with good computational efficiency and the obtained results agree, in terms of TE and contact force, with the static ones predicted by GCAS. Since the effects of a variable torque (or contact force) are part of the solution, the proposed technique can also be used to predict the unequal load sharing in multi-mesh gear trains.

The technique is currently being improved to take into account the effects of center line variations, the dynamic misalignment and the variable position of the contact force point of application.

Future work includes the validation of the proposed method against experimental data.

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