Conference PaperPDF Available

Application of dynamic substructuring in NVH design of electric drivetrains


Abstract and Figures

This paper shows the advantages of frequency-based substructuring (FBS). It shows that FBS can bridgethe gap in full-vehicle responsibility between the conception phase and full-vehicle hardware testing in theV-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 PathAnalysis (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 simulationdata from FE-models, will be shown. The excitation by equivalent forces, the mount models, the couplingof the subsystems and the full-vehicle synthesis showing acceleration and sound pressure level results arevalidated one by one. As a last step, a parameter variation as a powerful application example is mentioned.
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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
Am Technologiepark 10, D-82229 Seefeld bei M ¨
unchen, Germany
2BMW Group,
Knorrstrasse 147, D-80788, M¨
unchen, Germany
Molengraaffsingel 14, 2629 JD Delft, The Netherlands
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
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
EDU mounts rear (2x)
RAC mounts (4x) EDU mounts front (2x)
(a) Schematic of the subystem topology.
(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
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
61 f1;u6=YDS
62 feq
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(IBTBYBT1BY)(2)
where we have for the uncoupled subsystem FRFs, response DOFs and force DOFs:
11 YA
21 YA
22#0 0
33 YC
43 YC
0 0 "YE
55 YE
65 YE
and for the Boolean compatibility matrix:
B,0 I
0 0I 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= 0
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= Υiiλi
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(IBTBYBT+Υ1BY)(7)
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:
2= (YA
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, 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
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.
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
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.
(a) Model of mount characterization.
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
(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
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.
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
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
3.2.2, we assume that the free accelerations behave like the operational accelerations uA,free
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:
22 1
22 1
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.
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.
(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
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
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
(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
(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.
(a) E-modulus variation for the inner driver ear posi-
(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
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.
[1] 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 & Signal Processing, vol. 68–69, pp. 217–
244, 2016.
[2] D. de Klerk, D. J. Rixen, and S. N. Voormeeren, “General framework for dynamic substructuring: His-
tory, review and classifcation of techniques,AIAA Journal, vol. 46, no. 8, pp. 1169–1181, 2008.
[3] A. T. Moorhouse, A. S. Elliott, and T. A. Evans, “In situ measurement of the blocked force of
structure-borne sound sources,Journal of Sound & Vibration, vol. 325, no. 4–5, pp. 679–685, 2009.
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bench dynamics,Mechanical Systems & Signal Processing, vol. 24, no. 6, pp. 1693–1710, 2010.
[5] M. V. van der Seijs, “Experimental dynamic substructuring: Analysis and design strategies for vehicle
development,” Ph.D. dissertation, Delft University of Technology, 2016.
[6] J. W. R. Meggitt, A. S. Elliott, A. T. Moorhouse, and H. K. Lai, “In situ determination of dynamic
stiffness for resilient elements,Proceedings of the institution of mechanical engineers, Part C: Journal
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[7] M. Haeussler, S. W. B. Klaassen, and D. J. Rixen, “Experimental twelve degree of freedom rubber
isolator models for use in substructuring assemblies,Journal of Sound & Vibration, p. 115253, 2020.
[8] M. V. van der Seijs, D. D. van den Bosch, D. J. Rixen, and D. de Klerk, “An improved methodology for
the virtual point transformation of measured frequency response functions in dynamic substructuring,
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Engineering Dynamics, ser. CISM Course and Lectures. Springer, 2019, vol. 594.
... This work focuses on the vibrations generated by the EDU and uses sound pressure level in the vehicle cabin as well as accelerations on the driver's seat-rail as the targets for evaluation. This project builds upon learnings from previous work which was carried out in 2018 on a BMW iX3 with a comparable drivetrain [2]. The considered subsystems are shown schematically in figure 2 and listed below: ...
Conference Paper
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This paper shows a modular NVH engineering process based on Dynamic Substructuring and component TPA techniques, using experimental data obtained on a fully electric BMW i4 vehicle. Following the component TPA approach, the electric drive unit (EDU) of the BMW i4 is considered as the vibration source and is described by equivalent forces on the EDU. To describe the presence of a second vibration source, originating from the wheels running on the drums, a set of equivalent forces at the rear wheel hubs is included. The quality of the equivalent forces is evaluated using criteria as defined in a recent ISO standard on the topic [1]. Transfer paths from the EDU up to the targets in the cabin, i.e. sound pressure at the driver's ear and vibrations at the seat rail, are obtained through Dynamic Substructuring of the individual subsystem models using the Lagrange Multiplier Frequency Based Substructuring (LM-FBS) method. The subsystem models include multiple sets of rubber bushings, a rear axle carrier and the vehicle trimmed body. Transfer paths from the rear wheel hubs up to the targets in the vehicle are obtained from FRF measurements. The individual subsystem models are obtained through measurements using the Virtual Point Transformation in DIRAC, a software application specifically designed to generate subsystem models from FRF measurements using 3-or 6-DoF Virtual Points. The rubber bushings are modeled using the inverse-substructuring approach, which is also available in DIRAC. A second application, COUPLE, is then used to generate NVH predictions based on the modular subsystem models and the equivalent force descriptions.
... The blocked force independently describes the operational activity of a vibration source; it does not depend on what the source is connected to. This is advantageous as it means that the blocked force can be used in conjunction with structural modification techniques, where the receiver structure is modified in some way, for example in a design optimisation context [10][11][12] . ...
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In-situ Transfer Path Analysis is a diagnostic method used to analyse the propagation of noise and vibration through complex built-up structures. Its defining feature is the invariant characterisation of an assembly's active components (i.e. vibration sources) by their blocked forces. This invariant characterisation enables the downstream structural modification of an assembly without affecting the sources' operational characteristics. In practical engineering structures, however, there is often a need to alter or replace components that reside within a vibration source, for example resilient mounts. An upstream structural modification of this sort would alter the blocked force and thus invalidate any response predictions made thereafter. Hence, an alternative approach is required. In the present paper a transmissibility-based structural modification method is introduced. We derive a set of equations that relate the blocked force and forward transfer functions obtained from an initial assembly, to those of an upstream modified assembly. Exact formulations are provided, together with first and zeroth order approximations for resiliently coupled structures. These component replacement expressions are verified by numerical examples.
Thanks to the recent advances in digital vision systems, a question might arise about where the full-field optical and contactless methods can bring modern design procedures. An answer lies on the methodologically detailed comparison of the results, processed from different full-field optical techniques, in exploring consistent and high-resolution maps of rotational and strain FRFs that is done through this paper. Great exertions were first put in thoroughly testing a thin aluminium plate, in its real dynamics as a lightweight structure with broad frequency band dynamics and high modal density, in a unique comparative set-up, to obtain Receptance FRF maps of displacements-over-force by means of 3 different full-field optical techniques (SLDV, DIC, ESPI). There resulted superior quality Receptance maps in a broad and dense frequency domain, with high-resolution and continuity-wise consistent mapping at each frequency line. This paper exploits the here detailed robust numerical differentiation and signal processing in order to calculate the Impedance-based models of rotational and strain FRFs. Especially for DIC and for ESPI, both rarely used on many frequency lines, the fully populated FRFs for rotations and strains are a clear novelty, with rotational Coherence functions as added quality assessment features. The systematic comparison of the results obtained in the same location of the sample, by means of spatial and frequency domain metrics, is possible because of the proposed pointwise procedure, permitting the first numerical matching assessment of the 3 optical technologies on these awaited experiment-based quantities, as the full-field rotational and strain FRFs. Accepted on September 3rd 2021, available online since September 25th 2021, printed version March 1st 2022.
Conference Paper
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Dynamic Substructuring methods play a significant role in the analysis of today’s complex systems. Crucial in Dynamic Substructuring is the correct definition of the interfaces of the subsystems and the connectivity between them. Although this is straightforward practice for numerical finite element models, the experimental equivalent remains challenging. One of the issues is the coupling of the rotations at the interface points that cannot be measured directly. This work presents a further extension of the virtual point transformation that is based on the Equivalent Multi-Point Connection (EMPC) method and Interface Deformation Mode (IDM) filtering. The Dynamics Substructuring equations are derived for the weakened interface problem. Different ways to minimise the residuals caused by the IDM filtering will be introduced, resulting in a controllable weighting of measured Frequency Response Functions (FRFs). Also some practical issues are discussed related to the measurement preparation and post-processing. Special attention is given to sensor and impact positioning. New coherence-like indicators are introduced to quantify the consistency of the transformation procedures: sensor consistency, impact consistency and reciprocity.
Full-text available
Sound and vibration have a defining influence on our perception of product quality. They are especially well-known aspects in the automotive industry; a branch which sees, besides safety and driving comfort, ever-increasing expectations of the acoustic experience. After all, a smooth and silent driving experience appeals to a feeling of premiumness, a connotation no longer reserved to the top segment in the industry. While traditional combustion engines are gradually getting replaced by hybrid or full-electric drive-lines, other electromechanical (so-called mechatronic) systems make their entrance. As a consequence, the sound experience shifts from low-frequent engine roar to high-frequent humming and whining – a yet unfamiliar experience that calls for redefinition of the soundscape. To support such change, it is necessary that sound and vibration aspects can be considered in an early phase of development by means of simulations. This poses a true challenge: although state-of-art numerical modelling techniques can simulate the low-frequent dynamics fairly well, they often fail to provide reliable answers for the higher acoustic frequency range.This thesis presents techniques that aim to implement measurements of structural dynamics and active vibration sources into development processes. By characterising the passive and active dynamics of yet available components by means of measurements and combining those with numerical models, a hybrid simulation emerges that may provide answers to high-frequent problems in an early phase of development. This hybrid simulation is facilitated by use of Experimental Dynamic Substructuring: a methodology that determines structural dynamic aspects of complete products based on individually measured components.Part one of this thesis presents a variety of methods for simulation and substructuring that form the basic toolbox for generation, analysis, coupling and decoupling of dynamic models. Pivotal is the experimental approach, which means that dynamic models are obtained from measurements rather than numerical modelling efforts. To transform such measurements into a model that is compatible for coupling with other (numerical) models, the virtual point transformation is proposed. This method considers measured responses and applied forces around (user-chosen) points as locally rigid displacements and forces. Doing so, every connection point of a component can be described by three translations and three rotations with respect to a global reference frame, perfectly suited for substructuring. At the same time, the quality of the measurement and transformed frequency response functions can be quantified objectively using the proposed consistency functions. Altogether, the virtual point method bridges the gap between experimental and numerical modelling activities and enables us to exploit substructuring effectively for complex high-frequency systems.Part two presents a comprehensive study of Transfer Path Analysis (TPA); a collection of methods that contemplate a vibration problem as a source, transmission and receiver. A general framework for TPA is presented by re-interpreting eleven methods from the perspective of substructuring. It is shown that these methods can be categorised into three families, that in turn differ in the nature of characterisation of the source. The component-based TPA is regarded the most promising family, which allows to characterise a source independent of the environment in which it has been measured. The vibrations of the active source can be replaced by equivalent force spectra that, multiplied with the (simulated) FRFs of the assembled vehicle, predict what this source would sound like in the vehicle. Several practical methods are discussed to determine such equivalent forces: from forces measured against a blocked boundary, using free velocities, based on measurements on a compliant test bench or using the so-called in-situ and pseudo-forces methods. For further generalisation, a notation is presented that governs the abovementioned principles and facilitates the application and comparison of component-based TPA methods. In particular, it is shown that controllability and observability – concepts adopted from control theory – are strongly related to TPA; proper understanding of these principles yields interesting opportunities for analysis and simulation.The developed methods have been applied to analyse the vibrations of the electric power-assisted steering (EPS) system, which is reported on in part three. It is demonstrated that the virtual point transformation is able to determine accurate FRFs in a frequency range up to 6000 Hertz. Substructuring is applied to simulate the FRFs of a vehicle by applying the principle of substitute coupling, which employs a substitute beam during measurement in the vehicle to represent the dynamic effects of the steering system to couple. For the purpose of characterisation of the steering system’s excitations, several testing environments are discussed: a stiff test bench, more compliant test benches and the vehicle itself. Each configuration is accompanied by a specific method for source characterisation, for which it is demonstrated that the equivalent forces are indeed an environment-independent description of the active excitations of the steering system. It is shown that these forces can be used for the prediction of sound and vibrations in the vehicle. The presented applications offer, with understanding of substructuring and TPA theory, insights in the practical aspects of the methodology. This opens interesting opportunities for early-phase development of sound and vibration.
Full-text available
An in situ method for the measurement of a resilient elements dynamic transfer stiffness is outlined and validated. Unlike current methods, the proposed in situ approach allows for the characterisation of a resilient element whilst incorporated into an assembly, and therefore under representative mounting conditions. Potential advantages of the proposed method include the simultaneous attainment of both translational and rotational transfer stiffness components over a broad frequency range without the need for any cumbersome test rigs. These rotational components are obtained via the application of a finite difference approximation. A further advantage is provided via an extension to the method allowing for the use of remote measurement positions. Such an extension allows for the possible characterisation of hard-to- reach elements, as well as the over-determination of the problem. The proposed method can thus be broken into two sub-methods: direct and remote. Preliminary results are shown for the direct method on a simple mass-isolator-mass laboratory test rig along with a more realistic beam-isolator-plate system. Validation of this method is provided for by a transmissibility prediction, in which an obtained dynamic stiffness value is used to predict the transmissibility of a separate system. Further results are presented for the remote case using a beam-isolator-plate system. In all cases the results are obtained over a substantial frequency range and are of a sufficient quality to be used as part of structure borne sound and vibration predictions. (Pre-print copy uploaded)
Full-text available
Transfer Path Analysis (TPA) designates the family of test-based methodologies to study the transmission of mechanical vibrations. Since the first adaptation of electric network analogies in the field of mechanical engineering a century ago, a multitude of TPA methods have emerged and found their way into industrial development processes. Nowadays the TPA paradigm is largely commercialised into out-of-the-box testing products, making it difficult to articulate the differences and underlying concepts that are paramount to understanding the vibration transmission problem. The aim of this paper is to derive and review a wide repertoire of TPA techniques from their conceptual basics, liberating them from their typical field of application. A selection of historical references is provided to align methodological developments with particular milestones in science. Eleven variants of TPA are derived from a unified framework and classified into three categories, namely classical, component-based and transmissibility-based TPA. Current challenges and practical aspects are discussed and reference is made to related fields of research.
Commercial off-the-shelf rubber isolators often come with no additional information other than the static stiffness in three translational directions. Hydraulic testing machines can be used to obtain frequency dependent dynamic stiffnesses of rubber isolators in translational degrees of freedom (DoF). Alternatively, dynamic substructuring based methods can be used, which can additionally identify the dynamic stiffness in rotational DoF while requiring only standard vibration testing equipment. Results of two substructuring methods will be compared to those from a hydraulic machine. Both of the presented methods use locally rigid fixtures, mounted to the bottom and top of the isolators. Frequency based substructuring (FBS) requires knowing the fixtures dynamics to decouple them. Inverse substructuring, also called in-situ decoupling, does not require knowing the fixtures dynamics, but is assuming negligible mass and a special stiffness matrix topology of the rubber isolator. Both methods produce accurate results for translational DoF up to the kilo Hertz range, which is confirmed by comparison to measurements on the hydraulic machine. However, FBS does not rely on specific assumptions about the isolator, like inverse substructuring. The limits of inverse substructuring's underlying assumptions are shown theoretically and in the measurements presented here. We propose two extensions to compensate for the assumptions and present their results. Nevertheless, the rubber model obtained with the FBS decoupling can provide better results when used in an assembly. This is illustrated by testing the experimental rubber element models, obtained with either method, in a substructuring prediction of coupled frequency response functions (FRFs) and comparing that to reference measurements.
In this article a component transfer path analysis (TPA) procedure is proposed. The method allows one to calculate the total system response resulting from a subcomponent's source excitation. It is based on the knowledge of the frequency response functions (FRFs) of the total system and on a measurement of the stand-alone subcomponent on a test bench. As the true source excitation, for example an engines combustion, is not measurable, equivalent forces at the subcomponent interface are found. The equivalent forces are multiplied with the total system FRFs from the subcomponent interface to response nodes of interest. The resulting responses at and in front of the subcomponent interface are shown to be physically exact for linear, time invariant and stationary operating systems.However, for the method to succeed, the source forces will have to be independent of the global dynamics. In addition, the test bench needs to be rigid in the frequency range of interest. This is typically hard to achieve for analysis in the mid frequency range (100–1000Hz in vehicle acoustics). Therefore, a way to compensate for the test bench dynamics is also discussed. It is shown that one needs the receptance matrix of the free component at its interfaces and the operational motions of the interface on the test bench. Knowledge of the test bench dynamics is not needed.Measuring excitation and response at the source interface may not be feasible in practice due to space restrictions. In this case, the proposed TPA method can be extended with substitute nodes on the subsystem which are reachable on the test setup and the total system. With the knowledge of the free subcomponent FRFs, physically exact responses at and in front of the gearbox interface can also be calculated.
It is shown that the blocked force of a structure-borne sound source can be obtained from measurements made in situ, i.e. when the source is connected to a receiver structure. This potentially removes the need for special test rigs employing blocked terminations. A corollary of this relationship is that a theoretically exact ‘in situ transfer path analysis’ is possible with a fully assembled structure, such as a vehicle, without at any stage needing to separate the substructures. The results are validated by numerical simulation and measurement on beam-like sources and receivers.
Substructuring in Engineering Dynamics, ser. CISM Course and Lectures
  • M S Allen
  • D Rixen
  • M Van Der Seijs
  • P Tiso
  • T Abrahamsson
  • R L Mayes
M. S. Allen, D. Rixen, M. van der Seijs, P. Tiso, T. Abrahamsson, and R. L. Mayes, Substructuring in Engineering Dynamics, ser. CISM Course and Lectures. Springer, 2019, vol. 594.