Calculation of drive-related structure-borne sound in electric driven trains using elastic multi-body simulation

Sascha Noack 1 Technische Universität Dresden – Institute of Solid Mechanics Marschnerstraße 30 01307 Dresden Germany

Prof. Dr.-Ing. Michael Beitelschmidt 2 See above

ABSTRACT This paper introduces a new calculation tool for drive related structure-borne sound in electric driven trains. Basics of elastic multi-body simulation are explained. The development of a model-kit based on a database concept, that is necessary because of the vast number of model elements needed to describe a train, is introduced. Important drive-related excitation mechanisms for the acoustic fre- quency range (gear meshing, rotating magnetic field in asynchronous machine) are identified using measurement data of an acceleration ride of a train. The necessary level of detail for implementation of model elements for excitation mechanisms and model elements describing the transfer path is pre- sented. A simulation run of the whole train model is carried out and simulation results are compared to measured vibration values.

1. INTRODUCTION

The structure-borne sound coming from the drive system defines, especially for transient train ma- noeuvres, due to its tonal character the experienced interior noise of modern railway traction vehicles. Therefore, a deeper look at the origin and the transmission of structure-borne sound is of great im- portance to increase passenger comfort in newly developed trains. Depending on the application, statistical energy analysis (SEA), the finite element method (FEM), ray tracing or a combination of these methods are used to calculate the acoustic properties of the vehicles [1]. Disadvantages of the methods currently used are, for example, that the influence of structure-borne sound is only included in a simplified way in the calculation of interior acoustics (SEA, ray tracing) or that only linear system properties can be taken into account for large systems in practice (FEM). To overcome these disad- vantages, the possibility of modelling the vibration behaviour of an electric multiple unit train (EMU) up to the acoustic frequency range using elastic multi-body simulation (EMBS) is investigated in a joint research project of the Chair of Dynamics and Mechanism Design at Technische Universität Dresden and the company Alstom.

1 sascha.noack@tu-dresden.de

2 michael.beitelschmidt@tu-dresden.de

worm 2022

2. APPLICATION OF ELASTIC MULTI-BODY SIMULATION

During the development of new rail vehicles, multi-body simulation (MBS) is used to calculate the dynamic vehicle behaviour. Using this method, the vehicle is broken down into a discrete number of rigid bodies whose degree of freedom in relation to each other and to the surroundings is defined by joints and between which forces can act. To expand the validity to the acoustic frequency range, the structural flexibility and dynamic behaviour of components taking part in the transmission of struc- ture-borne sound has to be considered by integrating flexible bodies.

In Simpack, the commercial software that is used in this project, floating-frame of reference (FFRF) formulation is used to describe the EMBS: The position of a point on an elastic component is calculated as a sum of a large non-linear rigid body motion and a small, linear elastic deformation:

𝒓ሺ𝑡ሻ= 𝒓 1 ሺ𝑡ሻ+ 𝑹ሺ𝑡ሻ𝒓 𝑃 ሺ𝑡ሻ+ 𝑹ሺ𝑡ሻ𝒖ሺ𝒓 𝑝 , 𝑡ሻ (1)

where 𝒓 1 ሺ𝑡ሻ is the translation vector of the elastic body reference frame, 𝑹ሺ𝑡ሻ the rotation matrix that is used to transform undeformed position 𝒓 𝑃 ሺ𝑡ሻ and the linear elastic deformation 𝒖ሺ𝒓 𝑝 , 𝑡ሻ from the reference frame of the moving elastic body to the global reference frame. Linear elastic deformation 𝒖ሺ𝒓 𝑝 , 𝑡ሻ is approximated by a global Ritz-approach that separates the solution into a set of location- dependent shape function 𝜙 𝑘 ሺ𝒓 𝑝 ሻ and a function of time 𝑞 𝑘 ሺ𝑡ሻ that is called elastic coordinate:

𝑛

𝒖ሺ𝒓 𝑝 , 𝑡ሻ≈෍𝜙 𝑘 ሺ𝒓 𝑝 ሻ𝑞 𝑘 ሺ𝑡ሻ

(2)

𝑘=1

Finite-element models of components like the gear housing or bogie frame usually have a very large degree of freedom (DOF). For computationally efficient time domain simulation of large sys- tems using the FFRF method, their DOF has to be reduced by means of model-order-reduction (MOR) techniques. Different MOR approaches are explained in [2]. The application of the Krylov Subspace Method (KSM) and the fixed Component Mode Synthesis (CMS) to a bogie frame and comparison of results is shown in [3]. KSM produces similar results at lower model size, but projection to non- physical subspace is used. Back transformation to physical subspace is not included in SIMPACK software package and applicability of boundary conditions using KSM in Simpack is unsure. Thus CMS, which is also included in most FE-software packages, is used for MOR. From the reduced order model and applied boundary conditions, Simpack generates shape functions 𝜙 𝑘 ሺ𝒓 𝑝 ሻ used to describe the linear elastic deformation 𝒖ሺ𝒓 𝑝 , 𝑡ሻ .

Simpack converts the modelling elements (rigid and elastic bodies, linear and nonlinear forces, joints) and the model structure into a set of nonlinear ordinary differential equations (ODE). Models containing constraints in closed kinematic loops and algebraic states require an additional set of al- gebraic equations. The classification of equations would then be differential-algebraic equations (DAE) which are usually computationally more expensive. For elastic multi-body systems, it has proven useful to include all bodies with six DOF joints to take into account their rigid body movement and use no constraints to avoid DAE. Therefore, forces between bodies can only be caused by force elements. The general form of an ODE that can be used to calculate the generalized coordinates 𝒒 in means of a vector of the pose of bodies and elastic coordinates is shown in equation (3).

worm 2022

𝑴ሺ𝒒ሻ𝒒ሷ+ 𝒉ሺ𝒒, 𝒒ሶሻ= 𝒇ሺ𝒒, 𝒒ሶ, 𝑡ሻ (3)

𝑴 represents system matrix for mass, 𝒉 the vector of gyroscopic terms and f the vector of acting forces. State-space representation is used to reduce the order of the equation system to one. Solving this kind of nonlinear ODEs cannot be carried out analytically, thus integration must be done numer- ically. Simpack solver SODASRT2 [4], which is a BDF integrator [5] and has proven to be robust and fast for solving the present stiff ODEs can be used to calculate 𝒒 over time to get information about the deformation and vibration behaviour of the analysed system. 3. MODEL-KIT BASED ON A DATABASE CONCEPT

The model build-up starts from a simulation model which is state of the art for calculation of vehicle dynamics up to a frequency range of 30 Hz. Step by step the level of detail of modelling elements is increased to raise the theoretical validity of calculations to 1000 Hz. To be able to • test the influence of degree of detail of single model elements on calculated results, • analyse assemblies and parts of the model • and easily save and swap differently parametrized model elements a modular model concept was introduced [6]. Due to the large number of model elements to build a train model it is advisable to use a database concept for systematisation and standardization. Also automation of model generation is then possible. The developed model-kit is based on a database structure also used in Simpack Wizard, which is a modelling and simulation environment for non- expert users. Hence, the developed model-kit can also be used in that environment.

worm 2022

The grey part of Figure 1 displays the structure of the database. Components are stored in one folder. For each individual component, different implementations can exist, which are stored in sub- folders. These can be implementations of components with different levels of detail, e.g. an elastic and a rigid bogie frame, or physically different configurations such as a tapered roller bearing and a ball bearing.

Figure 1: Calculation process and structure of database

eee eee

Components have access to a folder in which CAD geometries, flexible bodies and non-linear descriptions of all components are stored. Parameters are defined and assigned in components. Mod- els are composite groups of components and models. This makes it possible, for example, to install the engine-gearbox assembly several times in the motor bogie and the motor bogie several times in the middle wagon. Loadcases can also consist of components and models, have parameters and have access to a folder in which, for example, speed profiles of a test run of the train can be stored. Load- cases and models can be combined to a scenario to do the actual time domain simulation. Results can be evaluated afterwards through Simpack-Postprocessing or the software Matlab.

4. IDENTIFICATION OF RELEVANT STRUCTURE-BORNE NOISE SOURCES BY MEASUREMENTS

Measurements on a commuter train on track have been carried out to identify relevant noise sources and get vibration time series for model validation. Most excitation mechanisms are directly connected to the main motion of the machine, in this case the rotation of the shafts. These excitation mechanisms cause excitation frequencies that are multiples of the rotational velocity, the so called orders. Param- eters like • the number of teeth on gears, • switching frequencies and pulse width modulation (PWM) strategies of the converter feeding the electric motor • number of pole pairs of asynchronous machine of the installed powertrain can be used to calculate expected frequencies of known excitation mech- anisms during run up like • parameter excitation at tooth-contact due to fluctuating gear stiffness • and Maxwell forces acting on stator and rotor due to the rotating magnetic field in the air gap of the electric machine. Excitation coming from tooth-contact and magnetic field is load dependent. Usually the vibration excitation is larger at higher load [7]. Structure-borne noise is more relevant at lower to mid train speeds. Therefore, measured vibration values of a train ride with maximum possible acceleration to 80 km/h was used for identification of relevant excitation mechanisms. The yaw damper is an im- portant transmission path of structure-borne noise [8] because of its elastomer bushings with high stiffness at attachment points to car body and bogie frame. Figure 2 shows order spectra of the meas- ured acceleration signal at yaw damper attachment point at motor-bogie frame in vertical direction.

worm 2022

Figure 2: Order spectra of measured acceleration (vibration) signal in the acoustic frequency range at yaw damper attachment point at motor-bogie frame in vertical direction

Whole Signal Tooth Contact Electric Motor Rest of Signal high Ss i 100 200 300 400 500100 200 300 400 500100 200 300 400 500100 200 300 400 500 Rotation Speed Wheelset / 4/min

Using Vold-Kalman filter [9] orders corresponding to tooth contact (first and second order of tooth-meshing frequency of the two gear stages) and rotating magnetic field in the air gap of the electric machine (orders that could not be eliminated for applied synchronous PWM in power con- verter) are separated from the rest of the signal. Through calculation of the Root Mean Square (RMS) values of the shares of the vibration signals of the different excitation mechanisms, the plot in Figure 3 is generated. It can be seen that tooth contact and excitation coming from the electric motor lead to the highest measured vibration values in the acoustic frequency range during acceleration of the train to 80 km/h.

Figure 3: Shares of excitation mechanisms of measured acceleration (vibration) signal in the acoustic frequency range at yaw damper attachment point at motor-bogie frame in vertical direction

Also for other directions (lateral, longitudinal) and 11 other measurement points, tooth contact and excitation coming from the electric motor are the most important excitation mechanisms. The rest of the signal is coming from asynchronous PWM of converter, rail-wheel contact, bearings, auxiliary units, nonlinear effects in hydraulic dampers and side bands of tooth meshing orders. This research focusses on transient train manoeuvres. At constant train speeds with no or low load, rail-wheel con- tact will be more dominant. If bearing damage is present, the excitation coming from the rolling bearings will increase. These excitation mechanisms are neglected in the model. The results are only valid for the investigated train and should be investigated for further electric multiple units to derive general statements. 5. NECESSARY LEVEL OF DETAIL OF SELECTED MODEL-COMPONENTS

worm 2022

5.1. Model-Elements Relevant for Excitation The gear contact and the related stiffness is calculated using the Gearpair 225 algorithm of Simpack [4]. This algorithm performs an analytical contact calculation and uses gear stiffness definitions ac- cording to DIN 3990. The model calculates tooth contact stiffness at each time step to account for parameter excitation at tooth-contact due to fluctuating gear stiffness. Input parameters include flank shape corrections. For consideration of changes of the contact pattern under load due to deformation in contact zone and the surroundings (shafts, bearings, housing) [10], a width discretization of the teeth is carried out. The tooth stiffness calculation takes place at 100 slices across the tooth width and is summed up afterwards. Imbalances of all shafts are implemented according to DIN ISO 1940-1. Excitation frequencies of imbalances are not relevant at low to mid train speeds for vibration in the acoustic frequency range but lead to side bands of tooth meshing orders.

Power converters for trains have to deal with high voltages. In order to achieve high efficiency, the proportion of switching times (correlates to switching frequency) should be as low as possible.

RMS(Acc) / m/s? [Share Rest IEEE Share Tooth Contact [EGE Share Electric Motor RMS Whole Signal 10 20 30 40 50 Train Speed / km/h 60 BM-YD-z 70 80

Ratio of switching frequency to fundamental frequency 𝑓 1 = 𝑝∙ሺ1 −𝑠ሻ∙𝑛 with 𝑝 being number of pole pairs, 𝑠 the slip and 𝑛 the rotation speed should be at least larger than 10 for asynchronous PWM to avoid large amplitude subharmonics which lead to high torque ripple resulting in lower structural durability of the power train. This is why starting from a certain rotor speed synchronous PWM is used. Synchronous PWM leads to harmonics in the supply voltage that are nearly rotation speed de- pendent. Time domain description of voltage PWM signal can be used as input for ideally rotating transformer (IRTF) model [11] to calculate currents in stator windings and rotor bars including PWM caused harmonics. Through a spatial model of the magnetic field over the circumference of the air- gap, radial and tangential force field can be calculated [12]. Integrating the forces over sections of the circumference leads to discrete forces that can be applied to elastic bodies of motor housing and rotor shaft. This model has been implemented in Simulink and integrated into Simpack using MatSIM [4], which translates Simulink models to C++ code and makes them available for Simpack. Orders that were found to be dominant in Figure 2 are also dominant in calculated discrete forces. Integration to the model-kit is currently being worked on.

5.2 Model-Elements for Structure-Borne Noise Transmission All bodies of the powertrain are represented through flexible bodies using Timoshenko beam ap- proach (inter shaft, wheelset shaft) or built by FEM and integrated to EMBS via MOR using CMS (drive shaft, motor gear housing, hollow shaft). Coupling of interface nodes with interpolation ele- ments (RBE3) was used to minimize artificially added stiffness. All bodies have modal coordinates in a frequency range up to at least 2000 Hz to have model validity up to 1000 Hz. Special attention is given to the motor gear housing, which has been validated experimentally [13] and drive shaft, whose material parameters of sheet package and rotor bars have been validated experimentally.

Roller Bearings are represented using the Rolling Bearing 88 algorithm of Simpack, which calcu- lates load dependent stiffness matrices with consideration of clearance and a logarithmic roller profile function through sliced contact model according to DIN 26281. A distributed force law (forces around the circumference for each roller) with complex model creation similar to investigations in [14] was tested, but turned out to have a small influence on model results [15]. Therefore, point-to-point force law between shaft and housing at the centre of bearing is used. Elastomer bushings are modelled using linear force law for translations and rotations with multiplication factor for consideration of dynamic stiffening according to [16]. Rubber wedge coupling is modelled through spatially resolved linear force elements for each of the rubber wedges.

The generation of the experimentally validated elastic body of bogie frame has been explained in [3]. A new hydraulic damper model based on a multi mass oscillator with integrated flexible bodies for container and piston suitable for simulation of structure-borne sound has been developed and experimentally validated in [8]. Anti-roll bar is modelled using Timoshenko beam elements and linear force elements for bearings. The rest of the model elements (car bodies, primary spring, secondary spring, wheels, bumpstop, traction pivot, brakes, axle box housing) are used unchanged from state of the art vehicle dynamics model. Modelling guidelines can be taken from [17]. The car body is mod- elled using a reduced order flexible body with eigenfrequencies up to 30 Hz. Considering all eigenmodes up to a frequency range up to 2000 Hz would lead to computation effort which cannot be realized as of today. Integrating measured receptance in time domain simulation as shown in [18] would be an option, but in this research system boundary was drawn at connection points to car body. Blocked forces at connection points which can later be used to calculate structure-borne sound power

worm 2022

and vibration values at powertrain and bogie frame are accepted to be sufficient as simulation model results.

6. SIMULATION RUN AND COMPARISON OF SIMULATION TO MEASUREMENT RESULTS

Only one motor bogie of the whole train model is equipped with model elements suitable for calcu- lation of structure-borne sound results to save computation cost. Maximum torque according to the datasheet of the electric motor is applied directly at the drive shafts to accelerate the train to 80 km/h. The resulting speed over time curve is found to be almost the same as measured in experiment ex- plained earlier.

Due to pending integration of the electric motor model suitable for structure-borne noise calcula- tion into the model-kit, the gear meshing frequency is yet the only relevant excitation mechanism in the model. Figure 4 shows that calculated results filtered for the first meshing order of the faster of the two gear stages have similar vibration amplitudes as measured results. The peak of measured results (resonance) occurs at 400 rpm, for simulation at 360 rpm. Similar results can be observed for other measurement points and spatial directions. Rotation speeds of resonances do not always match as good as for shown results in Figure 4.

worm 2022

MP11

x z

y

MP10

MP2

Figure 4: Comparison of measured and calculated results of acceleration (vibration) signal filtered for the frequency of the first meshing order of the faster gear stage at yaw damper attachment point at motor-bogie frame in vertical direction 7. OUTLOOK

Adding the electric motor model suitable for structure-borne noise calculation to the model-kit is currently worked on. Measurements and simulations are being carried out at another existing electric driven train for validation purposes. Integration of the presented structure-borne noise calculations into the development process at Alstom for optimization of gear design, PWM strategies and transfer paths with the goal of lowering drive related structure-borne sound and prognosis of drive related structure-borne sound power is planned.

8. ACKNOWLEDGEMENTS

This research was funded and supported through regular meetings and professional exchange by the company Alstom. We are thanking Dr. Johannes Woller and all students that worked under our su- pervision for their contribution.

9. REFERENCES

Please feel free to translate German publications with deepL or other translation program. 1. M. Glesser, J. Sapena, T. Kohrs, A. Guiral, E. Iturritxa and B. Martin, "Review of state of the art

for industrial interior noise predictions," FINE1/Shift2Rail, Report D7.1, Paris, 2017. 2. P. Koutsovasilis, Model Order Reduction in Structural Mechanics - Coupling the Rigid and Elas-

tic Multi Body Dynamics, Dresden: Institut für Bahnfahrzeuge und Bahntechnik, 2009. 3. J. Woller, C. Lein, R. Zeidler and M. Beitelschmidt, "Numerical Study of the Structure-borne

Sound Transmission of a Bogie," Cagliari, 2016. 4. D. S. S. Corp., "Simpack Assistent 2022," 2022. 5. K. Brenan, S. Campbell and L. Petzold, "Numerical Solution of Initial-Value Problems in Differ-

ential-Algebraic Equations," Journal of Computational and Applied Mathematics 87, pp. 135- 146, 1997. 6. S. Noack, J. Voigt, J. Woller and M. Beitelschmidt, "Entwurf eines Modellbaukastens zur Erstel-

lung von Mehrkörpersimulationsmodellen von Schienenfahrzeugen," in 17. Internationale Schienenfahrzeugtagung Dresden, Dresden, 2020. 7. J. Woller, S. Noack and M. Beitelschmidt, "Load Dependent Structure Borne Sound Excitation

in Rail Vehicle Drive Trains: Simulation Approach and Experimental Insight," in Keynote on the 25th International Congress of Theoretical and Applied Mechanics, Milano, 2020. 8. S. Noack and M. Beitelschmidt, "Entwicklung eines Dämpfermodells für die elastische Mehrkör-

persimulation zur Berechnung von Körperschallgrößen," in 18. Internationale Schienenfahrzeug- tagung, Dresden, 2021. 9. H. Vold and J. Leuridan, "High Resolution Order Tracking at Extreme Slew Rates Using Kalman

Tracking Filters," Shock and Vibration Vol. 2, p. 507–515, 1995. 10. S. Noack, J. Woller and M. Beitelschmidt, "Anwendung der elastischen Mehrkörpersimulation

zur Berechnung von Körperschallgrößen in elektrischen Triebfahrzeugen," in 2. VDI-Fachtagung Schwingungen, Würzburg, 2019. 11. A. Veltman, D. Pulle and R. De Doncker, Fundamentals of Electrical Drives, Madison: Springer,

2016. 12. W. Bischof, Hennen, M. and R. Kennel, "Synthesized Magnetic Flux Density in Three-Phase

Cage Induction Machines with MATLAB & Simulink," in 18th International Conference on Elec- trical Machines and Systems (ICEMS), Pattaya City, 2015. 13. J. Woller, Durchgängiger Berechnungsansatz für die Körperschallprognose des Antriebsstrangs

eines Triebfahzeugs, Dresden: Dissertation, Institut für Festkörpermechanik, 2020. 14. M. Cardaun, R. Schelenz and G. Jacobs, "Calculation of structure-borne sound in a direct drive

wind turbine," Forschung im Ingenieurwesen 85, p. 165–171, 2021. 15. S. Noack, M. Andersch and M. Beitelschmidt, "Räumlich aufgelöste Wälzlagermodellierung für

die elastische Mehrkörpersimulation," in 3. VDI-Fachtagung Schwingungen, Würzburg, 2021. 16. F. Göbel, Gummifedern: Berechnung und Gestaltung, Berlin: Springer, 1969. 17. S. Bruni, J. Violas, M. Berg and S. Stichel, "Modelling of Suspension Components in a Rail

Vehicle Context," Vehicle System Dynamics Vol. 49, pp. 1021-1072, 2011. 18. J. Woller and M. Beitelschmidt, "Integration of measured receptance into a time domain simula-

tion of a Multi Body Model using SIMPACK," in European Multibody Simulation UGM, Braun- schweig, 2017.

worm 2022