Numerical and experimental analysis on helicopter’s main rotor

transmission for predicting structure-borne noise (SBN) using

CB-TPA methods Wafaa El Khatiri 1 Mechanical Engineering, Université de Sherbrooke, Centre de recherche acoustique-signal-humain de l'Université de Sherbrooke (CRASH-UdeS) 2500 Boulevard de l'Université, Sherbrooke, QC, J1K 2R1, Canada

Raef Cherif 2 Mechanical Engineering, Université du Québec à Rimouski 300 All. des Ursulines, Rimouski, QC G5L 3A1 Canada

Khalid El Bikri 3 Mechanical Engineering, Ecole Nationale Supérieure des Arts et Métiers ENSAM, Centre de recherche M2SM, Université de Mohammed V Rabat, Maroc Noureddine Atalla 4 Mechanical Engineering, Université de Sherbrooke, Centre de recherche acoustique-signal-humain de l'Université de Sherbrooke (CRASH-UdeS) 2500 Boulevard de l'Université, Sherbrooke, QC, J1K 2R1, Canada

ABSTRACT A large number of a vehicle’s mechanical systems are responsible for tonal vibrations, which propagate through the connected structures to radiate structure-borne noise into the cabin. In the literature, transfer path analysis (TPA) methods make it possible to solve vibro-acoustic problems using sub-structuring applications. This paper presents a case study of a heavy-active component connected to a plate backed cavity, using Component-Based transfer path analysis methods. The studied academic system is representative of a helicopter’s main transmission. Both numerical and the experimental characterization are used to discuss the effect of several parameters, such as coupling (in-situ) vs decoupling (sub-structuring), completeness of the used transfer function matrix, the accuracy of the inversion method, as well as the rigidity of the test bench used to identify the equivalent forces. It is shown both numerically and experimentally that by using part of the frequency response functions matrix, one can reconstruct the response of both vibration and acoustic target’s locations, even by decoupling the system and characterizing the equivalent forces on a test bench.

Keywords : Transfer path analysis, Structure-borne noise, Component-based TPA

1 Wafaa.el.khatiri@usherbrooke.ca 2 Raef_cherif@uqar.ca 3 k.elbikri@um5r.ac.ma 4 Noureddine.atalla@usherbrooke.ca

worm 2022

1. INTRODUCTION

The helicopter is essential above all for its versatile qualities and its use in various fields, and the notion of comfort has become a key parameter in any design. Helicopter still concerned about cabin noise, the experimental measurements carried out on it have shown that the structure-borne noise is dominant, where the mechanical systems constituting the helicopter, produce mechanical vibrations which are propagated inside the cabin via the fuselage. In fact, it crosses the whole structure by the struts which support the main rotor transmission [1].

The main rotor transmission is considered as the main source of noise because it contains various elements which produce tonal components at meshing frequencies, this mechanism generates noise which affects the comfort of passengers in the cabin and decreases the quality of communication between them.

The main rotor transmission considered in the context of this article is that of the helicopter, it is fixed to the vehicle using two pylons. Each one is fixed to the roof of the helicopter at 8 points, constituting the transfer paths responsible for the transmission of vibrations and noise into the cabin. In total, the main transmission contains two pylons therefore we have 16 transfer paths. The resulting vibrations of the main rotor transmission box generate forces applied to the entire roof and the connected structure. In this article we are interested in the characterization of these forces, using the component-based transfer path analysis CB-TPA methods. The frequency band of interest is that which corresponds to the frequencies of the tonal components of meshing (500 Hz to 3 kHz) [2].

In this article, two studies were carried out to identify the vibratory behavior of the system. First, a numerical study based on a representative model of the structure of the Bell 407 helicopter, where the effect of the incompleteness of the mobility matrix is studied. Second, an experimental study was performed on the system of interest by performing acceleration and (input, transfer) mobility measurements, on coupled and decoupled systems, to test the CB-TPA methods. The results of the two studies are documented for several configurations of the mobility matrix, which are subject to inversion to characterize the equivalent forces. These studies made it possible to see the repercussions of the numerical results on the experimental study, to verify the robustness of the used methods, as well as the precision of the obtained results.

2. BACKGROUND THEORY

2.1. Framework for the Component Transfer Path Analysis Method Transfer path analysis TPA methods are commonly used to dynamically analyze complex structures and the transmission of mechanical vibrations [3], their utility lies in understanding how the response to noise and / or vibration at the receiver appeared [4]. The idea of working with the source-transfer- path-receptor model dates to the 1980s [5], and TPA methods provide a way to detect dominant transfer paths, which negatively impact NVH behavior in the compartment of a vehicle, thus it allows engineers to focus on problem locations to provide optimal solutions.

One of the TPA family methods, the CB-TPA methods who seeks to express the excitation force as a function of the components which are characteristic only of the active subsystem. When modifying the design of the passive subsystem, it is not necessary to perform the operational tests again, hence the usefulness of the component based TPA. The dynamic at the receiving side can be obtained by applying the identified equivalent forces, to the FRFs of an assembled system with the active part shut down [3][6][7]. It is an approach which makes it possible to estimate the contributions of the source components in advance, from the measurements carried out on dedicated test benches [6][7][8], then the results are combined with the functions transfer, to predict the vibratory and / or acoustic behavior perceived by the user of the system of interest.

This article discusses two approaches of CB-TPA methods: coupled and decoupled methods applied on coupled and decoupled systems respectively. The methods of interest are the In-Situ method and the In-Situ Hybrid method, where we use a test bench to characterize the equivalent forces.

2.2. CB-TPA In-Situ Coupled Method The In-Situ method applied in original assembly, helps to characterize the active subsystems independently of their assemblies. After having defined the three elements, source-interface-receiver, it is possible to identify the blocked forces, which represent the forces necessary to eliminate the equivalent forces at the level of the interface.

The choice of the source-interface couple is based on the operation of the study structure, if an element participates in the transmission of noise and vibrations, it will be included in the active subsystem [9].

The equivalent forces can be obtained by the relation: 𝑓 2

eq = ሺ Y 32

AB ሻ −1 u 3 (1) Where AB designates the coupled system, points 2 and 3 refer to the interface points and the indicator points respectively. 𝑓 2

𝑒𝑞 , Y 32

AB and u 3 are the equivalent force at the interface point, transfer mobility between indicator-interface points on coupled system AB, and the response at the indicator points respectively.

And thus, the response of the target points is given by the following expression: u 4 = Y 42

eq (2) Where points 4 refer to the target points and the indicator points. u 4 and Y 42

AB 𝑓 2

AB are the response at the target points, and transfer mobility between target-interface points on coupled system AB respectively.

2.3. CB-TPA In-Situ Hybrid Coupled Method To avoid certain limitations in the implementation of CB-TPA methods, it is desirable to perform operational tests on a test bench. To apply this method, the active structure (A) which is the source, is coupled with the test bench R.

The equivalent forces can be measured indirectly by measuring the displacement u 3 , this approach has also been suggested in [7].

Measurements of FRFs should be performed, and the equivalent forces can be obtained by the following formula:

𝑔 2

AR = ሺ Y 32

AR ሻ −1 u 3 (3) Where AR designates the coupled system on the test bench. 𝑔 2

𝐴𝑅 and Y 32

AR are the equivalent force characterized on the test bench, and transfer mobility between indicator-interface points on coupled system AR respectively.

Once we find the equivalent forces, the vibroacoustic response of the target points is expressed as following:

u 4 = Y 42

AB 𝑔 2

AR (4) Where, u 4 is the response of the target points using force characterized on the test bench.

2.4. CB-TPA In-Situ and In-Situ Hybrid Decoupled Method TPA methods are mainly used as troubleshooting tools to identify the origins of noise, vibration, and harshness (NVH) in vehicles. However, these methods are not well adapted for design phases since the vibration paths provided by TPA methods are only valid for the measured “vibrating system/receiving structure” assembly.

To predict the dynamics of the assembly in a modified situation regarding the vibrating system, the components of the assembly should be characterized independently and combined afterwards to build the overall dynamic model.

A complex assembly can be difficult to study and to recover some of its dynamic properties, hence the need for sub-structuring. Indeed, a complex system can be decoupled into independent subsystems, where the work is easy, and the characteristics can be determined in a simple and efficient way. This is done by considering the dynamics at the level of the interfaces of the subsystems, and their properties make it possible to obtain the dynamics of the assembled system.

A coupled system AB can be decoupled into active subsystem A and passive subsystem B. the measurement of the sub-mobilities of the subsystems makes it possible to determine the global transfer mobility.

𝐴𝐵 , can be the admittances of the subsystem A and subsystem B [3],

The admittance of the assembled system 𝑌 42

Y 42

AB = 𝑌 42

𝐵 ሺ𝑌 22

𝐴 + 𝑌 22

𝐵 ሻ −1 𝑌 22

𝐴 . (5) Where A and B designate the active subsystem and the passive subsystem respectively. Y 42

B , Y 22

A and Y 22

B are the transfer mobility between target-interface point on subsystem B, input mobility between interface-interface points on subsystem A, and input mobility between interface-interface points on subsystem B respectively.

To find the responses of the target points using the sub-mobilities, equation 2 takes the following form:

−1 𝑌 22

eq (6) In the hybrid case, since we decouple the active structure for mounting it on the test bench to recover the equivalent forces, equation 4 takes the following form:

𝐵 ሺ𝑌 22

𝐴 + 𝑌 22

𝐵 ሻ

𝐴 𝑓 2

u 4 = 𝑌 42

u 4 = 𝑌 42

𝐵 ሺ𝑌 22

𝐴 + 𝑌 22

𝐵 ሻ −1 𝑌 22

𝐴 𝑔 2

AR (7)

2.5. Matrix Completeness To describe the dynamic behavior of a structure, mobility matrices are used, which represent the relationship between the forces applied and the vibroacoustic responses resulting from the excitation. A mobility matrix contains the frequency response functions FRF between the degrees of freedom studied in the system.

6Dofs characterize the behavior, namely 3 translational TDOFs and 3 rotational RDOFs. Numerically, the RDOFs are computed from the FE model [10], as described in section 3. However, from experimentally, sometimes, they are difficult to measure, due to the limitations of access to the junction’s points and the complexity of the studied systems. Hence, they are usually neglected.

Given the lack of devices that directly measure RDOFs, or the difficulty of accessing interface points, the global mobility matrix is subject to several simplifications. And even by neglecting certain terms of the matrix, several authors manage to reproduce the responses measured experimentally and numerically in a way closer to the real response [11] [12] [13].

In this article, we demonstrate that the neglect of certain terms does not influence the quality of the results. The numerical and experimental studies are carried out using only excitation forces.

AEE EGE Beiaee bye ee

(a) (b) Figure 1: Completeness considered matrix of (a) 1 Dof, 3 Dofs and (b) 6 Dofs

Figure 1 presents the various completeness levels studied in this paper. We choose matrix configurations which differ from each other, and we analyze the effect of the completeness of each one on the precision of the results.

3. NUMERICAL CASE STUDY

In this section, numerical studies concerning the application of the CB-TPA In-Situ and In-Situ Hybrid methods on coupled and decoupled systems are presented. The numerical model was developed with Simcenter 3D-Siemens software, and the simulations have been performed using Modal Frequency Response solution, with a frequency step of 1 Hz over a frequency range from 0 to 3000 Hz.

(a) (b) (c) (d)

Figure 2: Numerical study of (a) Coupled system, (b) Coupled system on test bench, (c) Active

subsystem, (d) Passive subsystem

The coupled system as shown in Figure 2(a), contains two pylons with an added mass, all considered as the active subsystem, and a plate embedded in a concrete cavity considered as the passive subsystem, and the interface is between the feet of the pylons and the aluminum plate.

To model the dynamic behavior of the helicopter’s roof, we use an aluminum plate clamped on its 4 edges to an air-filled cavity, and the dimensions are (1150 x 686 x 12.7 mm³). The used mechanical properties of the aluminum: young’s modulus of 72 GPa, density of 2780 kg/mᶟ, Poisson's ratio of 0.33, and damping of 2%. The air cavity represents the interior of the helicopter cabin, its dimensions are (1150 x 870 x 960 mmᶟ).

The system is excited by 3 forces in the 3 directions, which the X axis dominating, this choice is made to conform with the experimental study presented in the following section. The use of a test bench R is essential to apply the hybrid method, and numerically as shown in Figure 2(b), it is modeled by a steel plate with fixed boundary conditions imposed on its 4 edges, and whose dimensions are equal to those of the receiving structure, with a difference in thickness which is equal to 50mm. The mechanical properties of the steel: young’s modulus of 206.94 GPa, density of 7829 kg/mᶟ, Poisson's ratio of 0.288, and damping of 2%. To characterize the equivalent forces, whether on the receiving structure B, or on the test bench R, 32 indicator points were chosen around the 16 connection points.

For CB-TPA decoupled methods, each subsystem must be studied separately, to be able to measure each sub-mobility. Numerically, this amounts to performing the necessary measurements, on each

element alone. First, the two pylons as shown in Figure 2(c), to find the input mobility of the active subsystem, then the plate with the cavity as shown in Figure 2(d), to find the input and transfer mobility of the passive subsystem.

4. EXPERIMENTAL CASE STUDY

This section presents the experimental study carried out on the coupled and the decoupled system. The laboratory setup is a simplified representation of the main transmission roof cabin system. Measurements were carried out at the University of Sherbrooke Acoustics Group (GAUS laboratory), with a frequency step of 1 Hz over a frequency range from 0 to 3000 Hz.

(a) (b) (c) (d)

Figure 3: Experimental study of (a) Coupled system, (b) Coupled system on Test bench, (c) Active

subsystem, (d) Passive subsystem

The system as shown in Figure 3(a), is excited by a shaker, its excitation is dominated by the X axis. The test bench used as shown in Figure 3(b), is a heavy steel plate, bonded to a concrete block. Its rigidity has been studied to ensure a good quality of the tests and results. On the test bench, we have defined 48 indicator points.

The active subsystem as shown in Figure 3(c), consists mainly of the two pylons with the main axis of the transmission, the isolators, connection cables. While the passive subsystem as shown in Figure 3(d), consists of an aluminum plate, clamped on its 4 edges, on a concrete cavity full of air.

The plate is made of aluminum whose dimensions are (1150 x 686 x 12.7 mm³), it is the same dimensions adopted for the numerical study. And the cavity’s dimensions are (1150 x 870 x 960 mmᶟ).

The application of the decoupled methods requires the separation of the system to be able to recover each data alone. To recover the input mobility of the active subsystem, it is suspended, to be able to access the connection points. Similarly, to find the input and transfer mobility of the passive structure, it must be disconnected from any other subsystem.

5. RESULTS AND DISCUSSION

In this section, the results of simulations and measurements performed on the coupled and decoupled system will be presented.

5.1. Numerical Results and Discussion In this part, we present the results of the numerical study, using several configurations of the FRFs matrices as shown in Figure 1, to see the effect of their completeness on the precision of the results.

Figure 4: Acoustic target N°1 using CB-TPA

Figure 6: Structural target N°3 using CB-TPA

In-situ Coupled Method

In-situ Coupled Method

Figure 5: Acoustic target N°1 using CB-TPA

Figure 7: Structural target N°3 using CB-TPA

In-situ Decoupled Method

In-situ Decoupled Method

The figures above represent the numerical results of the CB-TPA In-situ method applied to the coupled vs. the decoupled system.

Figure 8: Acoustic target N°1 using CB-TPA

Figure 10: Structural target N°3 using CB-TPA

In-situ Hybrid Coupled Method

In-situ Hybrid Coupled Method

Figure 9: Acoustic target N°1 using CB-TPA

Figure 11: Structural target N°3 using CB-TPA

In-situ Hybrid Decoupled Method

In-situ Hybrid Decoupled Method

The figures above represent the numerical results of the CB-TPA In-situ Hybrid method applied to the coupled vs. the decoupled system. Five matrix configurations were used from 1 Dof to 6 Dofs as shown in Figure 1.

Each curve represents the result obtained using each matrix configuration; the green curve refers to the use of a 1 Dof configuration as described in Figure 1-(a) in section 2.5, the blue and red curves are subject to the use of 3 Dofs configurations as shown also in Figure 1-(a) in section 2., while the cyan and magenta curves represent the result obtained using a matrix configuration based on 6 Dofs as described in Figure 1-(b) in section 2.5.

The results prove that it is possible to predict the vibroacoustic response except when using the complete 6 Dofs matrix. Indeed, the divergence between the result and the reference is due to the pseudo-inversion FRFs matrix using a singular value decomposition (SVD). This matrix is ill- conditioned since the terms of the diagonal differ from the off-diagonal terms, especially from the terms representing translational-rotational coupling. Considering the configurations from 1 to 3 Dofs, leads to good results.

5.2. Experimental Results and Discussion In what follows, the results of the experimental study will be presented using the 3 matrix configurations as shown in Figure 1-(a) to evaluate the precision of the results.

Figure 12: Acoustic target N°1 using CB-TPA

Figure 14: Structural target N°4 using CB-TPA

In-situ Coupled Method

In-situ Coupled Method

Figure 13: Acoustic target N°1 using CB-TPA

Figure 15: Structural target N°4 using CB-TPA

In-situ Decoupled Method

In-situ Decoupled Method

The results are presented in figures 12 to 15. It is observed that with the three matrix configurations chosen, good predictions of the vibroacoustic responses are obtained.

Figure 16: Acoustic target N°1 using CB-TPA

Figure 18: Structural target N°4 using CB-TPA

In-situ Hybrid Coupled Method

In-situ Hybrid Coupled Method

Figure 17: Acoustic target N°1 using CB-TPA

Figure 19: Structural target N°4 using CB-TPA

In-situ Hybrid Decoupled Method

In-situ Hybrid Decoupled Method

Figures 16 to 19 show the same results suing the CB-TPA In-Situ method. The green curve refers to the use of a 1 Dof configuration, while the blue and red curves are subject to the use of 3 Dofs configurations as shown in Figure 1 in section 2.5. Again, good results are obtained even for the 1 Dof configuration. These test results conform that for this specific application, rotational Dofs can be ignored, thus leading to quicker and simple test setups.

6. CONCLUSIONS

This paper investigated the accuracy of CB-TPA methods to predict the vibroacoustic behavior of a simplified representative transmission-roof-cabin system of a helicopter. Two methods were used and compared, the first is CB-TPA In-Situ method applied on the original assembly. The second is the CB-TPA In-Situ Hybrid method where we use a test bench to characterize the equivalent forces. the two methods were applied on coupled and decoupled systems using several matrix configurations from 1 to 6 Dofs. Various numerical and experimental investigation based on matrix completeness were used to assess the accuracy of the selected CB-TPA methods.

The vibroacoustic responses are generally well predicted, most of the peaks being in good agreement with the reference measured directly. The results show that we can predict the response of the vibracoustic targets even if the system is decoupled, and only translational Dofs are used. We also show that even by characterizing the equivalent forces using a test bench, good results are obtained. Future work will apply the same methodology on a real helicopter.

7. ACKNOWLEDGEMENTS

We would like to thank Bell Helicopter Textron Canada Limited for providing us with the main gearbox, which represented the active study structure of our study. The authors would like to thank NSERC (Natural Sciences and Engineering Research Council of Canada) and CRIAQ (Aerospace Research and Innovation Consortium in Quebec) for their financial support.

8. REFERENCES

1. M. Bebesel, R. Maier, et F. Hoffmann. Reduction of interior noise in helicopters by using active

gearbox struts – results of flight tests . 27th European rotorcraft forum . September 11 - 14 (2001). 2. W. Gembler et H. Schweitzer, « Smart Struts - The Solution for Helicopter Interior Noise

Problems ». Twenty-fifth European rotorcraft forum. September 14-16, (1999) 3. M.V. van der Seijs, D. de Klerk and D.J. Rixen,” General framework for transfer path analysis:

history, theory, and classification of techniques”, Mechanical Systems and Signal Processing 68- 69, pp. 217-244, (2016). 4. C. V. Karsen, G. Gwaltney, et J. Blough, « Applying Transfer Path Analysis to Large Home

Appliances », p. 6. January 2000 5. J. Verheij, Multipath sound transfer from resiliently mounted shipboard machinery, PhD

Dissertation. Technische Physische Dienst TNO-DH, Delft, 1986. 6. D. de Klerk et D. J. Rixen, « Component transfer path analysis method with compensation for

test bench dynamics », Mechanical Systems and Signal Processing , vol. 24, n o 6, p. 1693‑1710, août 2010. 7. M. van der Seijs, E. Pasma, D. de Klerk, D. Rixen « A robust Transfer Path Analysis method for

steering gear vibrations on a test bench », p. 14. September 2014 8. M. V. van der Seijs, E. A. Pasma, D. de Klerk, et D. J. Rixen, « A Comparison of Two

Component TPA Approaches for Steering Gear Noise Prediction », dans Dynamics of Coupled Structures, Volume 4, M. Allen, R. L. Mayes, et D. J. Rixen, Éd. Cham: Springer International Publishing, 2015, p. 71‑79. 9. J. W. R. Meggitt, A. S. Elliott, A. T. Moorhouse, A. Jalibert, et G. Franks, « Component

replacement TPA: A transmissibility-based structural modification method for in-situ transfer path analysis », Journal of Sound and Vibration , vol. 499, p. 115991, May 2021, doi: 10.1016/j.jsv.2021.115991. 10. S. Helderweirt, L. Bregant, et D. Casagrande, « Application of accelerometer-based rotational

degree of freedom measurements for engine subframe modelling », p. 7. January 2001 11. Ortega Almirón, J., et al. "Predicting vibration levels on an experimental test case by using

invariant loads (eg blocked forces) as source characterization." Proceedings of ISMA 2018- International Conference on Noise and Vibration Engineering and USD 2018-International Conference on Uncertainty in Structural Dynamics. KU Leuven, 2018. 12. Van der Seijs, Maarten V., et al. "Validation of current state frequency based substructuring

technology for the characterisation of steering gear–vehicle interaction." Topics in Experimental Dynamic Substructuring, Volume 2. Springer, New York, NY, 2014. 253-266. 13. Rixen, Daniel J. "How measurement inaccuracies induce spurious peaks in frequency based

substructuring." Proceedings of the Twenty Sixth International Modal Analysis Conference,Orlando, FL. Society for Experimental Mechanics, Bethel, CT. 2008