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Content uploaded by Philipp Wagner

Author content

All content in this area was uploaded by Philipp Wagner on Dec 15, 2020

Content may be subject to copyright.

Application of dynamic substructuring in NVH design of

electric drivetrains

P. Wagner 1,2, A. P. H¨

ulsmann 2, M. V. van der Seijs 3

1AMITRONICS GmbH,

Am Technologiepark 10, D-82229 Seefeld bei M ¨

unchen, Germany

2BMW Group,

Knorrstrasse 147, D-80788, M¨

unchen, Germany

e-mail: arthur.huelsmann@bmw.de

3VIBES.technology,

Molengraaffsingel 14, 2629 JD Delft, The Netherlands

Abstract

This paper shows the advantages of frequency-based substructuring (FBS). It shows that FBS can bridge

the gap in full-vehicle responsibility between the conception phase and full-vehicle hardware testing in the

V-Model describing the full-vehicle development process. Substructures can be exchanged during rebalanc-

ing of subsystem conception and targets. In the presented case, FBS and component-based Transfer Path

Analysis (TPA) methods are used to analyze the drivetrain acoustics of an electrical vehicle. The model de-

scription for measurement-based and hybrid modeling, based on experimental data as well as on simulation

data from FE-models, will be shown. The excitation by equivalent forces, the mount models, the coupling

of the subsystems and the full-vehicle synthesis showing acceleration and sound pressure level results are

validated one by one. As a last step, a parameter variation as a powerful application example is mentioned.

1 Introduction

The automotive industry has to deal with great challenges for the future. The current global situation forces

the OEMs to accelerate the development process by reducing costs and limiting the number of hardware

prototypes. From an acoustic point of view it is important to predict the interior sound in an early design

phase and track design changes and their consequences. In addition, new acoustical phenomena of electriﬁed

and electrical drivetrains need to be designed for. Therefore powerful technical design tools are needed to

face these challenges.

1.1 State of art

A full-vehicle design department responsible for electrical drivetrain acoustics has the task to balance the

sound pressure level targets at the passengers ear and the dynamic behavior of the subsystems among them-

selves. In the early design stage, this responsibility is accounted for by determining subsystem targets. In

the middle stage of the design timeline, the departments responsible for the development of the subsystems

verify subsystem target fulﬁllment using subsystem simulation and subsystem test benches. In this stage,

the full-vehicle department cannot account for this full-vehicle design responsibility, which would be in re-

balancing of the subsystem targets. Only when the ﬁrst prototype car is ready to drive (i.e. in the last phase

of the design timeline) the full-vehicle point of view is considered again.

In these late stages of design, classical TPA methods are often used to identify the dominant vibrations

paths and inform design teams where to put the last efforts [1]. Apart from their late availability, results from

3365

Figure 1: The V-Model, describing the vehicle development process. Dynamic Substructuring can be used

to close the gap in full-vehicle analysis and responsibility between the early-stage conception phase and the

ﬁrst actual full-vehicle prototype in hardware.

classical TPA are also limited in use: they cannot simulate for design changes and cannot properly implement

results obtained on subsystem level, e.g. from component test benches. This underlines the need for a new

paradigm in automotive NVH design, that is better tailored towards a future of modular “platform-based”

end-products and shorter design timelines with fewer prototypes in hardware.

1.2 Advantages of Dynamic Substructuring

Dynamic Substructuring [2] enables us to collect the subsystem models and assemble them to a whole vehicle

model for structure-borne electrical drivetrain acoustics. The subsystem dynamics can either originate from

ﬁnite element models or from component or subsystem measurements, depending on the actual development

stage. The excitations can equally be obtained from full computer simulation or component test bench

trials. The combination of component-speciﬁc excitation proﬁles (e.g. equivalent forces, often referred to

as “blocked forces”) with substructured vehicle dynamics is often regarded as component-based TPA, and

relies on proper use of FBS subsystem models and operational spectra under the right conditions [1, 3, 4].

Some substructure models, like the drivetrain housing, axle carriers and the car body, are usually available

in an early design phase from ﬁnite element modeling. Models for rubber mounts and acoustical transfer

functions of the car body can either be derived from measurements on predecessors or from synthetic values.

Hence it is possible to analyze and design the interactions of all coupled components and the resulting

NVH-synthesis over all transfer paths, but also to do parameter variations with a reduced amount of efforts.

In a later design phase, hardware of the substructures will become available and can be measured. Test-

based models derived from these measurements can be interchanged with the virtual substructure models to

compare and to reﬁne model quality [5].

The full-vehicle model, simulated through dynamic substructuring, can be used to close the gap in full-

vehicle analysis and responsibility between the early-stage conception phase and the ﬁrst actual prototype in

hardware, see ﬁgure 1. Target levels which have not been fully met in one subsystem can be compensated

by target over-fulﬁllment on another subsystem, avoiding expensive and potentially unnecessary measures

on subsystems. Working on the full-vehicle analysis in cooperation with all involved departments will also

intensify the cooperation and the understanding between these colleagues.

1.3 Paper outline

This paper shows an application of aforementioned FBS and TPA methods on the simulation of the acoustics

of an electric drivetrain. First we will introduce the chain of components and FBS modeling approach in

section 2. Section 3 will discuss the modeling of the subsystems, as well as the modeling of the excitation

3366 PROCEEDINGS OF ISMA2020 AND USD2020

RAC

EDU

TMB

EDU mounts rear (2x)

RAC mounts (4x) EDU mounts front (2x)

𝐟1

𝐮2

𝐮3

𝐮4

𝐮5

𝐮6

(a) Schematic of the subystem topology.

𝐮2

𝐟1

A

𝐮6

E

𝐮3𝐮4𝐮5

C DB

TMBRACEDU

(b) Three-stage FBS model as a chain of subsystems.

Figure 2: Schematic overviews of the electric drivetrain assembly.

spectra. Remarks towards inclusion of secondary noises are also made. Results of assembly and validation

are discussed in section 4.

2 Modeling approach

2.1 Subsystem deﬁnitions

The electric power train assembly under consideration is shown schematically in ﬁgure 2(a). We distinguish

the following subsystems:

•EDU: the electric drive unit, connected to the rear axle carrier (RAC) at 4 coupling points.

•RAC: the rear axle carrier, connected to the bodywork (TMB) with 4 coupling points. Altogether this

carrier is modeled by 8 coupling points.

•TMB: the bodywork or “trimmed-body”, including the noise transfer functions (NTFs) to the interior

microphones.

A total of 8 coupling points can be identiﬁed: each of them is comprised by an elastic mount. Part of

building the transmission model is thus the identiﬁcation of the dynamic mount stiffness properties in the

relevant directions; this will be discussed in section 3.1.2.

Figure 2(a) shows the original full-vehicle assembly, for which operational run-up measurements have been

conducted. In this conﬁguration FRFs have also been measured, using a variety of shakers and impact

hammers; this is detailed in section 3.1.

Figure 2(b) shows the ultimate goal of applying dynamic substructuring: to create a modular transmission

model, allowing to interchange, optimize and re-use individual subsystems. This model will be referred to as

the three-stage FBS model. As we will use Lagrange-multiplier frequency-based substructuring (LM-FBS)

[2], all subsystems are to be described by linear(ized) FRF matrices and the excitation’s by frequency spectra.

2.2 Subsystem coupling

This section discusses the assembly of substructures by LM-FBS. The goal is to build the transfer functions

of the three-stage model using the substructure FRFs of the components EDU (YA), RAC (YC) and TMB

(YE). This assembled FRF can then be used to apply forces, such as internal excitation forces f1or equivalent

(blocked) forces feq

2:

u6=YDS

61 f1;u6=YDS

62 feq

2(1)

VEHICLE NOISE AND VIBRATION (N VH) 3367

In the following derivation, all subsystem denotations and DOF subscripts use the deﬁnitions as introduced

in ﬁgure 2(b).

2.2.1 Rigid LM-FBS coupling

The main equation of LM-FBS is well known and can straight-forwardly be set up for this assembly using a

block-diagonal organization of the three subsystem FRF matrices:

Ydual =Y(I−BTBYBT−1BY)(2)

where we have for the uncoupled subsystem FRFs, response DOFs and force DOFs:

Y,

"YA

11 YA

12

YA

21 YA

22#0 0

0"YC

33 YC

34

YC

43 YC

44#0

0 0 "YE

55 YE

56

YE

65 YE

66#

;u,

uA

1

uA

2

uC

3

uC

4

uE

5

uE

6

;f,

f1

f2

0

0

0

0

(3)

and for the Boolean compatibility matrix:

B,0 I

0 0−I 0

0 I0 0

−I 0 (4)

Note that the TMB subsystem also contains acoustic noise transfer functions (NTFs) to the interior positions;

these are part of YE

65. It is also important to note that the shown subsystem matrices are not all obtained

separately, but really by measurement or simulation of the subsystems, resulting in the groups of FRFs

within the brackets.

2.2.2 Compliant coupling using mount models

Standard rigid LM-FBS coupling requires strict coordinate compatibility and force equilibrium for each

matching DOF i. This would read for the compatibility and equilibrium between A and C:

Rigid coupling: (Bu =0=⇒uA

i−uC

i= 0

LTg=0=⇒gA

i+gC

i= 0 (5)

In the presented three-stage model, the subsystems are all connected through ﬂexible rubber bushings. There-

fore, we adopt the compliant coupling technique, given by:

Compliant coupling: (Bu =Υλ=⇒uA

i−uC

i= Υiiλi

LTg=0=⇒gA

i+gC

i= 0 (6)

We here deﬁne Υas the diagonal ﬂexibility matrix containing the inverse of the dynamic stiffness functions

(zii) for the rubber bushings at the relevant coupling DOFs i. As the matrix is diagonal, cross-coupling in

the bushings is neglected. The dynamic stiffness functions are estimated using an inverse substructuring

approach [6, 7] of which the practice is discussed in section 3.1.2.

It can be shown that incorporation of the weak compatibility condition of (6) in (2) yields the following

compliant coupling LM-FBS scheme:

Ydual =Y(I−BTBYBT+Υ−1BY)(7)

3368 PROCEEDINGS OF ISMA2020 AND USD2020

This equation, together with the deﬁnitions of (3) and (4) is used to create all substructured assemblies. For

our application, we used the VIBES Toolbox for MATLAB, which implements this method for compliant

LM-FBS coupling out-of-the-box.

2.2.3 Virtual points

Key to successful substructuring is having a well-deﬁned set of interface degrees of freedom (DOF). As the

goal is to be able to replace numerical components for test-based components, one needs a uniquely deﬁned

set of DOFs for both representations. In the following analysis we use virtual points (VP) [5, 8] for the

above-mentioned coupling points. Every VP is described by 6 DOF, i.e. 3 translations and 3 rotations. For

the acquisition and modeling of the experimental models, DIRAC is used (see section 3.1), which naturally

produces virtual point FRFs by transformation of the measured FRFs. For the models synthesized from

FE, RBE2 and RBE3 elements have been used in the reduced numerical models, which are effectively the

numerical equivalent of virtual points.

2.3 Equivalent force description

As a next step the equivalent force concept will be introduced. In many cases, measuring the internal excita-

tion force f1directly is not feasible. Instead it is possible to deﬁne a set of equivalent forces feq that have the

same characteristics as the source for what concerns the responses downstream of their points of application;

typically the interfaces. In other words, the equivalent forces at the interface of the full assembled system

leads to the same dynamic behaviour at the passive side like the excitation force at the active side does.

Hence the feq are more general and a property of the source, contrary to for instance interface forces g2. For

more information on this concept see [1].

In this paper an equivalent force approach based on the free velocities (or in practice: accelerations) is used.

The advantages and disadvantages of free velocities for this application will be discussed in more detail in

section 3.2.2. The equivalent forces are deﬁned as:

feq

2= (YA

22)−1ufree

2(8)

In this equation, (YA

22)−1can be understood as the dynamic stiffness of the freely-suspended EDU at the

active-side interfaces. The term ufree

2is the operational acceleration measured on the source acting in free-

free condition.

3 Implementation

3.1 Test-based modeling using DIRAC

Obtaining high-quality subsystem models from test is known to come with many challenges [9]. In this

project we have used DIRAC from VIBES.technology, which is designed to optimize the process of test-

based modeling from start to end and mitigate any of the traditional challenges. The process consists of the

following steps:

1. Off-line preparation: design of experiment. CAD geometry of the vehicle parts was loaded in the

3D Prepare environment of DIRAC. As virtual points are ultimately the desired outputs of the model,

these have been deﬁned ﬁrst. Next, sensors have been placed around the virtual points, as well as

excitation points for impact hammer testing and shaker testing. Suitable locations have been assessed

by cross-checking the 3D CAD environment with the actual car in the test facility.

2. On-line preparation: sensor placement and DAQ setup. In the test facility, sensors were connected to a

M¨

uller-BBM MK2 data acquisition system (DAQ). Two impact hammers were connected for alternate

testing with a light nylon-tip hammer and a heavier rubber-tip hammer.

VEHICLE NOISE AND VIBRATION (N VH) 3369

Figure 3: Test-based modeling of the trimmed-body using DIRAC with live 3D guidance.

3. On-line measurement: guided FRF impact testing. The Measure module of DIRAC offers a measure-

ment experience with live calculation of virtual points, such that the quality of the desired end-results

– virtual-point transformed FRFs at a multi-kHz range – can already be assessed during measurement.

Figure 3 shows the workﬂow for the measurement of the TMB subsystem. A measurement like this

can be executed in half a day.

4. Virtual point transformation & analysis. The Analyze module offers an in-depth view into the mea-

sured FRF data, by means of coherence overviews and ODS animations, as well as VP quality in-

dicators like sensor and impact consistencies and reciprocity. This is shown for the rear axle carrier

measurement in ﬁgure 4. The matrix on the left shows the overall reciprocity of the 8 virtual points

after transformation.

3.1.1 Subsystem modeling

The subsystems as listed in section 2.1 have been measured in several “test assemblies”. These test assem-

blies have a certain meaning for FBS and component-TPA purposes and are identiﬁed as follows:

1. Full vehicle: the original full-vehicle conﬁguration;

2. TMB + RAC: the vehicle on air springs, without the drive unit but with rear axle carrier;

3. TMB: the trimmed-body vehicle on air springs, without the drive unit and rear axle carrier;

4. RAC: the rear axle carrier freely suspended;

5. EDU: the electric drive unit freely suspended;

6. RAC + EDU: the sub-assembly of rear axle carrier and electric drive unit.

After measurement, an ﬁnal round of measurement optimization is done in DIRAC using the coherence

overviews and ODS animations, for instance to check for incorrect sensor orientations.

By applying the virtual point transformation for both sensor channels and excitations on the measured FRF

matrix, a new VP FRF matrix is generated that has a square size of N×6, where Nis the number of coupling

3370 PROCEEDINGS OF ISMA2020 AND USD2020

Figure 4: The virtual point FRF matrix for the 8 coupling points of the rear axle carrier.

points. Additional response points, such as validation sensors, microphones and seat-rail accelerations can be

added as additional untransformed response rows to this matrix [5]. The same applies for validation impacts

that can later be used to validate the results of FBS.

An additional beneﬁt of the modular substructuring approach is that subsystem FRFs/NTFs are typically

obtained at higher signal-to-noise ratios (SNR) then full-vehicle FRFs/NTFs, and thus better overall quality.

In the full-vehicle conﬁguration, an excitation on the electric drive renders hardly any signal at the driver’s

ear microphones. This is of course due to the two stages of isolation along this path. The trimmed-body

measurement on the other hand has much more direct transfer of vibrations from the RAC coupling points

to the driver’s ears. As a result, the coherence of the impact measurements is high even up to 4 kHz, as can

be observed in ﬁgure 3.

3.1.2 Mount modeling

The rubber mounts are characterized using the inverse substructuring technique [7]. Figure 5(a) shows the

measurement setup for one of the RAC mounts in DIRAC. Aluminum brackets have been machined that ﬁt

tightly around the mount’s outer cylinder. On the other ends, aluminum crosses have been connected that

allow for easy measurement on the ends that are normally connected to the vehicle bodywork (as was shown

in ﬁgure 3). A total of 14 tri-axial accelerometers can be identiﬁed: 6 for the center bracket and 2 times 4

for the crosses. This is quite some more than strictly needed for the process, but done to further study the

rigidness of the adapter brackets.

The general process of dynamic stiffness characterization is depicted in ﬁgure 5(b). After impact measure-

ment, the 14 ×3 = 42 response channels and 37 impact positions are transformed to 2×6 = 12 virtual

point DOFs, resulting in VP FRF matrix Yqm(ω). This matrix comprises 6 rigid body modes plus 6 ﬂexible

modes, as a result of the mount’s stiffness and the mass of the brackets1.

The obtained 12 ×12 VP FRF matrix is inverted to yield Zmnt(ω), which is now the dynamic stiffness

matrix of the assembly of bracket-mount-crosses. The inverse substructuring concept takes the off-diagonal

1The next mode that kicks in can be seen as an indicator for the upper frequency limit of the method: either caused by an internal

resonance of the mount or due to a ﬂexible mode in one of the adapters.

VEHICLE NOISE AND VIBRATION (N VH) 3371

(a) Model of mount characterization.

Modeling

DIRAC

3D definition of virtual points (𝐪),

sensors (𝐮) & impacts (𝐟)

Impact measurement

𝐘uf 𝜔

ODS / consistency / reciprocity /

passivity checks

VP transformation

𝐘uf 𝜔 → 𝐘qm 𝜔

VIBES Toolbox

Inverse substructuring

𝐙𝐦𝐧𝐭 𝜔 = 𝐘qm 𝜔−𝟏

Averaging &

LF extrapolation

Dynamic Bushing Stiffness

𝐙mnt 𝜔

Measurement

(b) Workﬂow of mount characterization.

Figure 5: Mount modelling.

(a) Comparison of the front RAC mount in X-

direction (radial).

(b) Comparison of the rear EDU mount in Z-

direction (radial).

Figure 6: Comparison of mounts characterization methods for different types and directions.

6×6blocks of this dynamic stiffness matrix, which approximates the dynamic stiffness terms of a just the

mount2. By choosing the virtual points in the mass/stiffness center of the mount, we minimize the effect of

cross-terms. Therefore the main diagonal of the two off-diagonal submatrices can be assumed to be a good

approximation of the dynamic stiffness of the mount before its ﬁrst internal resonances3. As a last step, the

6 pairs of reciprocal stiffness curves are averaged and extrapolated down to 0 Hz. The process was repeated

for the other mount types, resulting in a total of 8×6 = 48 dynamic stiffness curves, with amplitude and

phase information for all translational and rotational DOFs. These are in the right format to plug into the

compliant LM-FBS method governed by (7).

This application of inverse substructuring is validated against classical mount test bench measurements with

unidirectional pre-load in X- and Z-direction. As can be seen in picture 6, the different curves align well

for the different mount types, directions and calculation methods. Therefore it is possible to say that the

FBS stiffness model is valid for the linear range of the mounts, i.e. in a range that is most relevant for NVH

engineering.

2By deﬁnition of the terms Z12 and Z21: any connected structure has no contribution to these terms [6].

3Beyond these resonances, the method’s assumption of negligible mass is not satisﬁed anymore. A full FBS decoupling approach

is warranted here [7], which is beyond the scope of this paper.

3372 PROCEEDINGS OF ISMA2020 AND USD2020

3.2 Hybrid subsystem modeling

So far, the models discussed in section 3.1 are all originating from measurements. In this section, we will

combine test-based models with model descriptions originating from ﬁnite element models. The goal is to

be able to obtain each input from either measurements or ﬁnite element modeling and to interchange these

different model descriptions, depending on e.g. the progress of the development process of the vehicle.

3.2.1 Setup of the structure model

At the time of writing this paper, we were able to describe the dynamics of the drivetrain housing, the RAC

and the connecting points to the car body using ﬁnite element models. The noise transfer function of the car

body and the description of the dynamics of the rubber mounts are taken from measurement data as covered

in section 3.1.

The elasticity of the mounts could be modeled either as an elastic coupling between subsystems, or as

subsystems by their own [7]. We used the ﬁrst option as introduced in section 2.2.2 and equation (7). In

Y, the accelerance FRFs of each of the substructures are combined, which are all in free-free boundary

conditions. Because of the compliant coupling, we need a dual assembly. That means, after coupling all

degrees of freedom on both sides of the interface are present in the dynamic equations, to be able to describe

the dissimilar dynamics for each side of the interface.

3.2.2 Applying excitation

In the previous section, we built up the assembled system. In this section we will apply loads, i.e. dynamic

forces and moments on this system. There are two possibilities to do so: By applying the physical excitation

on the degrees of freedom that correspond with the locations where the loads act in reality, or by applying

equivalent loads on other degrees of freedom, using component-based TPA [1]. In our case of hybrid system

characterization we basically can do both. The drivetrain department can simulate the loads in the air gap of

the electric motor caused by the electromagnetics, as well as the loads on the gear shaft bearings caused by

the gear wheel contact. These loads can be applied directly on the corresponding degrees of freedom of the

ﬁnite element model of the drivetrain.

This solution has a few important disadvantages. First, the size of the model becomes very large, because

of the amount of DOFs surrounding the air gap and the number of shaft bearings. Secondly, the loads and

accelerances can be determined in ﬁnite element models, but not (with manageable effort) in a measure-

ment setup. Therefore the compatibility and exchangeability between simulation and measurement is not

warranted in this case.

Therefore, we decided to use the second possibility: we use component-based TPA and determine the excita-

tion using the free accelerations and the accelerances on the interface between the drivetrain and the rear axle

carrier, or more speciﬁcally on the drivetrain housing directly next to the rubber bushings; see section 2.2. As

described in [1], there are a few different concepts to determine equivalent forces, for instance the methods

based on blocked forces and the method based on free velocities, which corresponds to free accelerations.

The difference between these two methods is in the boundary conditions for the subsystem setup during the

determination of these forces or accelerations. To determine blocked forces, it is essential to ﬁx the interface

degrees of freedom, which works well for low frequencies but is the harder the higher the frequency. To

determine free accelerations, it is essential to leave all degrees of freedom of the system and in particular the

interface degrees of freedom free, which works well for high frequencies, but at low frequencies is difﬁcult

for an operating electrical drivetrain, because the static torque and its reaction forces in the rubber bushings

need to be supported.

We decided to use the free accelerations method rather than blocked forces, because this method is better

suited for the higher frequency range of the excitation of the electrical drivetrain.

Besides the disadvantages stated earlier, using free accelerations brings up an advantage related to the orga-

nization of our development processes: our department for whole vehicle acoustics derives targets for the

VEHICLE NOISE AND VIBRATION (N VH) 3373

different departments, each responsible for a different subsystem. The interface between the drivetrain hous-

ing and the rubber bushings ﬁts very well to this organizational implementation and the (free) accelerations

corresponds very well to a suitable target value for the excitation and the dynamic behavior of the drivetrain.

This target value can be very well simulated with ﬁnite element models of the drivetrain as well as measured

on a drivetrain test bench or in a whole vehicle measurement setup.

The difference between free accelerations and operational accelerations at this interface (at a test bench or

in a full vehicle) is depending on the dynamic stiffness of the bushings and the rear axle carrier compared to

the drivetrain housing itself and we assumed it to be small for the frequency range of the electrical drivetrain

acoustics, starting at about 200 Hz upwards. The rigid body modes of the drivetrain on the bushings and also

most of the rigid body modes of the rear axle carrier between the drivetrain and the car body are (far) lower.

Nevertheless we have to be aware of the possibility of dynamics of the rear axle carrier and especially of the

rubber mounts in the frequency range above 200 Hz.

Both as input for the calculation of the equivalent forces as well as for validation, we used the same opera-

tional accelerations from a full-vehicle measurement at a roller test bench.

3.2.3 Including airborne noise

Up to now, the hybrid model exist of a dynamic description of the structure -borne paths and the structure-

borne excitation for the acoustics of the electrical drivetrain. In addition, the airborne source and paths could

be included. To do so, the radiation at the surface of the drivetrain housing can be calculated in the ﬁnite

element model and the transfer paths can be measured on a full vehicle, which is not done in this study.

3.2.4 Including wind and rolling noise

Besides the drivetrain acoustics, other sources can be accounted for as well, such as wind noise and rolling

noise. This can be done for two purposes: either to analyze masking effects or to make an auralization for

all the different sources together.

One way to account for wind and rolling noise is to measure accelerations and sound pressure levels during

a roll-out test, with the electric motor in neutral, that is inactive and not in recuperation. These measured

values can than be added to the synthesis result for the electrical drivetrain acoustics.

However, it is important to realize which inputs have been used for determining the excitation for the synthe-

sis in the ﬁrst place. If these free acceleration data is determined by simulation of the drivetrain, no rolling or

wind noise sources are present. Also if these are measured on a subsystem test bench without axle and tires,

there is no problem with including these sources twice. But if measurements on a full vehicle or on a test

bench where the drivetrain is combined with the whole axle including the tires on a drum, adding additional

rolling noise measurements is problematic.

For the curves shown in this paper, measured accelerations from a roller test bench with a full vehicle are

used and therefore rolling noise is not added additionally. Separately determined wind noise, for instance

from a wind tunnel, is also not added here.

4 Validation and design modiﬁcation possibilities

In the previous section we discussed how to setup the models for the component-based TPA and how to

improve the quality of the measurements. Here different steps towards the full synthesis will be discussed

and validated.

4.1 Comparison of excitation from measurement and simulation

As stated in section 2.2, we calculated the equivalent forces based on free accelerations. However, it is

practice not possible to measure under purely free boundary conditions. Therefore, like discussed in section

3374 PROCEEDINGS OF ISMA2020 AND USD2020

3.2.2, we assume that the free accelerations behave like the operational accelerations uA,free

2≈uA,meas

2at

the interface points on the drivetrain housing. In further investigations this assumption was validated by

comparing the FRFs and operational accelerations at the roller test bench in full vehicle condition with the

drivetrain test bench as well as in free condition. This comparison showed us that the assumption is valid in

this case, but it is important to keep in mind the remarks in section 3.2.2.

Hence it is possible to compute the forces with (8). With the explanations from the previous section it is

possible to calculate the forces based on either the pure measured model or the hybrid model. In a ﬁrst step

it was decided to use the same operational data in both cases. That means for the two cases:

feq

2=

YA,meas

22 −1

uA,meas

2

YA,FE

22 −1

uA,meas

2

(9)

The following ﬁgure illustrates the forces exemplary for the rear left mount position and the ﬁrst order of the

ﬁrst pinion wheel of the gearbox. The operational condition is a certain middle engine torque.

(a) X-direction. (b) Y-direction. (c) Z-direction.

Figure 7: Comparison of equivalent forces derived from measured free accelerations, in combination with

either measured (blue) or simulated (red) FRF data of the EDU.

The results of the two approaches show partially higher discrepancies between the curves, caused by the

differences in the FRFs of the models of the EDU subsystem. Therefore these EDU FRFs are analyzed in

more detail.

(a) X-direction. (b) Y-direction. (c) Z-direction.

Figure 8: Comparison of virtual point accelerance for the rear left engine mount.

VEHICLE NOISE AND VIBRATION (N VH) 3375

Looking at ﬁgure 8 it can be seen that the trend ﬁts quite well in phase and at lower frequencies. But over

400 Hz, the curves starts to deviate, especially in Z-direction. Furthermore it can be seen that the level of

the simulated curves is higher than the measured ones. The levels of the simulated equivalent forces are

lower, because the higher simulated FRFs are inverted to calculate these force levels. It is an advantage of

FBS and component-based TPA that improved measurement or model results for the different inputs can be

interchanged, i.e. updated, very easily.

4.2 Coupling of the subsystems

After identifying all subsystems and making sure that all virtual points line up, the subsystems can be coupled

to get the assembled system. As described in section 2.2, a compliant dual coupling is used. For validation

purposes, an extra measurement with physically connected engine (EDU) and rear axle carrier (RAC) was

done. This was used to validate substructure coupling of the ﬁrst stage as shown in ﬁgure 9(a).

(a) Coupling of the ﬁrst stage of EDU to RAC. (b) Overview of the 24 matched DOFs in

each subsystem.

Figure 9: Overview of the ﬁrst stage coupling.

Figure 9(b) shows the Boolean connectivity matrix for the ﬁrst stage, coupling 24 virtual point DOFs. Hence

each DOF at the engine side matches with the corresponding DOF with the rear axle carrier side. In this

process all 6 DOFs (3 translations and 3 rotations) are included. This shown Boolean matrix, together

with the associated dynamic stiffness values for every connection, is all that is needed to establish the DS

coupling using (7). As stated in section 3.1.2, the mount stiffness is incorporated using the diagonal parts of

the stiffness matrices.

Figure 10 compares the coupled FRF through substructuring (blue) with the measured FRF of the coupled

system (red). The left side shows an FRF at the engine side; the right side for a position after the engine

mount; both for an impact at the engine in Z-direction.

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(a) FRF comparison of an excitation at the engine

in Z-direction to the rear-right engine measurement

point in Z-direction.

(b) FRF comparison of an excitation at the engine in

Z-direction to the rear axle carrier after the mount in

Z-direction.

Figure 10: Validation of the coupling for two complex structures.

Looking at ﬁgure 10 it can be seen that the comparison of the transfer function at the engine side ﬁts very

well up to one kHz. A similar result can be seen after the engine mount but with an increase deviation to

higher frequency. Reasons can be small positioning errors in the measurement or some numerical effects

during the coupling. Overall, the results show that the compliant coupling with the simpliﬁed mount model

performs well for the presented situation. As a conclusion from both diagrams it can be seen that the system

reaches a resonance at about 50 Hz, where the phase in the right diagram, i.e. after the mounts, shows the

corresponding phase shift. Above that frequency there is a transmission loss over the ﬁrst stage of one up to

two orders of magnitude.

This exercise gives a good understanding of the coupling process and will be a key enabler for the further

substructuring process.

4.3 Synthesis of the FBS-model

After validating the different steps of the frequency-based substructuring it is possible to calculate the syn-

thesis of the full FBS-model. As described in section 2.2, the synthesis results by applying the equivalent

forces to the corresponding DOFs of the assembled system. At ﬁrst, the resulting synthesis for the structure

borne noise of the hybrid simulation will be compared with the measurement at different coupling stages in

the ﬁrst order of the ﬁrst pinion wheel of the gearbox.

In ﬁgure 11 the hybrid simulation results are shown in the upper row of the ﬁgure and the validation mea-

surement underneath. This is for the ﬁrst order of the ﬁrst pinion wheel of the gearbox and for a certain

middle engine torque. The solid red line is the acceleration at the engine side. Subsequently the dashed red

line corresponds to the positions at the rear axle carrier, directly after the engine mounts. The dashed blue

line is the acceleration before and the solid blue line the acceleration after the mounts between the rear axle

carrier and the car body. It can be seen, that the match is quit good overall, only the accelerations at the car

body in the rear (i.e. the solid blue lines in the upper right two diagrams) are lower than the values of the

validation measurement.

Figure 12 illustrates the sound pressure in the cabin of the car for the same manoeuvre as in ﬁgure 11.

As mentioned above the airborne contribution is not yet included, this will be a future step. As can be

seen, the measurement and hybrid-based FBS model match well over the frequency range. Furthermore

there is a signiﬁcant gab between both synthesis and the measured curve. The reason for this is that we

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Figure 11: Validation of the structure borne noise before/after ﬁrst mounting level (red curves) and be-

fore/after second mounting level (blue curves) in the ﬁrst order of the ﬁrst pinion wheel of the gearbox in

vertical direction. The (dashed) black lines with constant amplitudes are for orientation purposes only. The

numbers in the legend refer to the numbers used in ﬁgure 2(b).

analyzed the drivetrain contribution only, but the frequency range below 500 Hz is typically dominated by

the rolling noise. By adding the airborne as well as the suspension contributions, the match of measurement

and synthesis can be improved. The next ﬁgure shows this for the interior positions. From the ﬁgure above it

is possible to see that the suspension path dominates below 500 Hz. Above that frequency the drivetrain path

becomes more important. The additional paths where determined by the classic matrix inversion method.

The theory and different processing steps of getting the results are beyond the scope of the paper and can be

found in [1].

The now completely described and validated simulation environment can be used to do various component-

based TPAs, which can be evaluated very easily. In addition, response points can be evaluated at all stages

on both the active and passive side. Furthermore it is now possible to change and to interchange single

substructures and to use hybrid models like presented in this paper. An example of such an application is

shown in the next section.

4.4 Parameter variation

One of the advantages of Dynamic Substructuring is the fact, that we can exchange substructures and simulate

the dynamics of the new system, leaving all other subsystems unchanged. In this synthesis project of the

electric car, we exchanged the ﬁnite element model of the rear axle carrier using different values for the

elasticity modulus and for the density. These two parameters are scaled respectively up and down with the

amount of 30% of the nominal value of the original model. This was more done to show the possibilities of

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(a) Airborne synthesis at the inner driver ear position. (b) Airborne synthesis at the inner ear position at the

back seat.

Figure 12: Resulting synthesis at different interior positions based on the two approaches.

(a) Different path contributions at the inner driver ear

position.

(b) Different path contributions at the inner ear posi-

tion at the back seat.

Figure 13: Visualisation of the different contributions regarding to the interior positions.

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(a) E-modulus variation for the inner driver ear posi-

tion.

(b) Density variation for the inner driver ear position.

Figure 14: Parameter variation of the elasticity modulus and the density of the rear axle carrier. The curves

can directly be compared with those in ﬁgure 12.

the method and out of curiosity than for physically based analytical reasons. The results can be seen in ﬁgure

14.

5 Conclusion and outlook

In this paper two FBS approaches have been presented: one based on measured models and a second from

a hybrid combination of models. Each processing step towards the component-based TPA was addressed

and validated separately. For mount characterization, an inverse substructuring approach was used which

ﬁt very well with common mount measurements. The coupling of subsystems is discussed and validated

against an assembled system in hardware. The FBS compliant coupling showed very comparable results. In

addition, the equivalent forces of the two approaches have been compared, where the differences between the

measurement-based and the hybrid-based approach result from the differences in accelerances of the drive-

train housing. In a last step the sound pressure level in the cabin caused by the structure-borne contribution

was discussed. Therefore it can be shown that the differences between measurement and synthesis is based

on the absence of the contribution over the suspension and airborne path of the engine and wheels.

It can be concluded that FBS can generate reliable results in a full-vehicle context. As presented in this paper,

the method allows to use hybrid modeling for sound prediction and enables to interchange substructures or

do parameter variations very easily. Therefore FBS is a powerful method in NVH-design at different stages

of the vehicle development process.

As next steps on the way to a comprehensive simulation environment for the acoustics of an electrical driv-

etrain of a passenger car, it is necessary to expand the frequency range of the hybrid simulation model as a

whole to higher frequencies. That means, the ﬁnite element models of the subsystems should be enabled to

describe the dynamics of these subsystems in this higher frequency ranges. At the moment, in our case, the

measured input is not restrictive here.

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