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Component TPA: benefit of including rotational degrees of freedom and over-determination

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Before performing a transfer path analysis (TPA), the engineer needs to think about the right modeling of the source's interface with the receiver. In practice, the vibration transfer from the source to the receiver is often modeled with three translational forces in each connection point. Mechanically this corresponds to a ball joint connection, which cannot transfer any moments. Our goal is to compare different complexities of interface descriptions on the industrial example of an electromagnetic roll control (ERC) in a passenger car. Therefore, different variants of interface degrees of freedom and matrix over-determination are compared: 1. Three hammer impact points in x,y,z-direction (no sensor over-determination). 2. Multiple impacts, transformed with the virtual point transformation (VPT) to 3 forces. 3. Multiple impacts, transformed with the VPT to 3 forces and 3 moments. These interface descriptions are compared in terms of an on-board validation, the interface-completeness-criterion and by evaluating the transferability to a modified vehicle design. It was found that the over-determination of the matrix inverse should be used in any case to avoid spurious noise artifacts. For best quality TPA results at higher frequencies, it was found necessary to include rotational moments in the interface description.
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Component TPA: benefit of including rotational degrees of
freedom and over-determination
M. Haeussler 1, T. Mueller 1, E.A. Pasma 1, J. Freund 2, O. Westphal 2, T. Voehringer 2
1VIBES.technology
Lichtenbergstr. 8, 85748, Garching, Germany
e-mail: mhaeussler@vibestechnology.com
2ZF Friedrichshafen AG
Graf-von-Soden-Platz 1, 88046, Friedrichshafen, Germany
Abstract
Before performing a transfer path analysis (TPA), the engineer needs to think about the right modeling of
the source’s interface with the receiver. In practice, the vibration transfer from the source to the receiver is
often modeled with three translational forces in each connection point. Mechanically this corresponds to a
ball joint connection, which cannot transfer any moments. Our goal is to compare different complexities of
interface descriptions on the industrial example of an electro-magnetic roll control (ERC) in a passenger car.
Therefore, different variants of interface degrees of freedom and matrix over-determination are compared:
1. Three hammer impact points in x,y,z - direction (no sensor over-determination).
2. Multiple impacts, transformed with the virtual point transformation (VPT) to 3 forces.
3. Multiple impacts, transformed with the VPT to 3 forces and 3 moments.
These interface descriptions are compared in terms of an on-board validation, the interface-completeness-
criterion and by evaluating the transferability to a modified vehicle design. It was found that the over-
determination of the matrix inverse should be used in any case to avoid spurious noise artifacts. For best
quality TPA results at higher frequencies, it was found necessary to include rotational moments in the inter-
face description.
1 Introduction & outline of the paper
Transfer path analysis (TPA) has established in industry as a tool for noise vibration harshness (NVH) engi-
neering1. A broad review and comparison of methods in a unified notation can be found in [2]. In general,
a TPA studies machines which actively excite a final assembly and thereby cause noise and vibrations. As
one of the first applications, Verheij described the transmission of vibrations, from a ship engine to the hull,
by interface forces transmitted over the rubber isolators [3]. In 1982 this was mainly driven by the desire to
make military ships more stealthy. Nowadays, TPA is commonly applied in NVH engineering of vehicles [4,
5]. Classically, TPA has been used as trouble-shooting tool, using interface forces to understand the trans-
mission of vibrations from the source to the receiver. A current trend is to use approaches which describe the
source independently from a specific receiver, e.g. via blocked forces [6–8]. A popular method for obtaining
the blocked forces is the in-situ method [9], which will also be used in this paper. It yields results comparable
to classical TPA, with some improvements if also rotational degrees of freedom (DoF) are included in the
source description (see [8]).
1Some contents of the introduction and theory sections were published in similar form in [1]. The text hereafter has been
modified and adapted to the paper.
(a) photo of the ERC in the vehicle
(b) Setup of the ERC in DIRAC
Figure 1: Overview of electric roll control (ERC) in the vehicle. In figure 1(b) the four connection points
with the vehicle are shown: Two drop link connections to the wheel hub (WH) on driver and co-driver side.
Two connection points of the ERC to the subframe (with rubber isolation). These two connection points will
be modelled with so-called virtual points (VP).
This paper investigates the benefits of including rotational DoF in the interface description on the industrial
example of a electro-magnetic roll controll (ERC) in a passenger car, see figure 1. The ERC contains an
electric motor and a gear transmission to control the wheel hubs and compensate the vehicle roll motion.
It allows dynamic driving with increased safety and improved comfort. One important performance aspect
of this system is its NVH behavior in the vehicle. The ERC introduces vibrations due to the pole-pairs of
the e-motor and the gear meshing of the transmission. Additionally, the impulsive loading from road bumps
introduces impact-like excitations into the vehicle. Both excitation phenomena contain higher frequency
content (>400Hz). The ERC is connected to the vehicle at four points, see figure 1(b): The drop links are
connected to the wheel hub (WH) on both sides. The ERC is connected to the front subframe with two
connection points (with a rubber bushing).
In industrial practice, the NVH development is a modular and collaborative process. The supplier component
shall be integrated into many different vehicles which are developed independently by the OEMs. Therefore,
it is advantageous to describe the excitation of the component with a common quantity that is independent of
the final vehicle. This will be done with blocked forces, but the proper modeling of the ERC interface with
the vehicle needs to be investigated for achieving optimal results. This paper will show the following:
Blocked forces determined on one vehicle variant can be transferred to another vehicle configuration.
6 DoF on the interface, i.e. including rotational DoF, yield a better on-board validation, interface
completeness and transfer validation for higher frequencies (>400Hz).
In TPA it is important to predict the right vibration magnitude. We propose to evaluate the interface
completeness with a coherence like measure, since MAC like criterion is not sensitive to magnitude
differences.
The concept underlying the blocked forces will be briefly explained in section 2. Their computation with
the in-situ method is explained in section 3. In section 4, the virtual point transformation (VPT) will be
explained for the blocked forces. The results for different interface complexities will be shown in section 5.
Figure 2: Overview of the source receiver problem and the equivalent modeling of interface vibration trans-
mission by blocked forces fbl
2.
2 Component TPA with blocked forces
The general problem studied with TPA can be described by the situation shown in the upper left part of
figure 2. An assembly AB contains a vibration source A, which is subject to internal loads fA
1. The exact
mechanisms creating the internal forces fA
1and the location of their DoF might be unknown or cumber-
some to model. It is therefore desirable for an NVH engineer to find another more abstract, yet complete
description of the source. It is assumed that the receiver Bis a purely passive structure with no external
forces.
The following explanation treats the underlying concepts of a component TPA, with strongly reduced math-
ematical detail (see [2] for a derivation), but a hopefully intuitive explanation.
1. Situation in vehicle: The source’s internal forces fA
1are transferred to vibrations uB
3or sound pressures
pB
3in the receiver, via the frequency response function (FRF) matrix YAB
31 :
pB
3=YAB
31 fA
1,(1)
where subscript (?)31 indicates that the FRF matrix describes the vibration transfer from the internal source
DoF (subscript (?)1) to the final receiver DoF (subscript (?)3). Superscript (?)AB indicates that the FRF
matrix is a property of the coupled system, source Aand receiver B.
2. Blocked interface: Now consider the following thought experiment: The operating source is rigidly
clamped on its interface so that the interface vibration uA
2is zero, see figure 2. The subscript (?)2denotes
forces and vibrations on the interface. The reaction forces in the clamped support are called ’blocked forces’
fbl
2and ensure that:
0!
=uA
2=YA
21fA
1+YA
22fbl
2.(2)
3. Noise cancelation: If fbl
2could be applied as an external load in the interface between source and receiver
(remember this is just a thought experiment) then they would act on the source, just like before, as a perfect
clamping support. The motion on the interface of the assembly AB would thus also be zero:
0!
=uAB
2=YAB
21 fA
1+YAB
22 fbl
2.(3)
(a) In-situ TPA
mx
my
mθz
rhehfh
(b) Virtual Point
Figure 3: (a) In-Situ determination of blocked forces. (b) General interface connection point. Exemplary
quantities for one force input h.
However, if the assembly AB has no motion on the interface and there is no other vibration source on the
receiver B, then also the sound and vibration at all other points in the receiver would be zero:
0!
=pB
3=YAB
31 fA
1+YAB
32 fbl
2.(4)
The blocked forces act like a noise cancellation on the source. This is the theoretical basis for the blocked
force concept (or in fact all component TPA concepts, see [2]).
4. Force superposition: Of course, the discussion so far was just a thought experiment (artificially applying
the blocked forces at the interface DoF of assembly AB is usually not possible). However, since the assembly
AB is modelled as a linear time invariant system, it is allowed to add and subtract the effect of the blocked
forces from the original problem in equation (1) without modifying the outcome (superposition principle):
pB
3=YAB
31 fA
1+
=0
z }| {
YAB
32 fbl
2YAB
32 fbl
2.(5)
5. Equivalent source description: Using the blocking effect on the original exciation fA
1from equation (4),
one finds that:
pB
3=YAB
32 fbl
2.(6)
Notice that the derivation did not specify which particular receiver structure Bis used. The blocked forces
are thus a valid source description for any receiver B. Also note that the blocked forces are a property of the
source alone, see equation (2). The minus sign in equation 6 will be neglected in the rest of this paper for
simplicity.
A thorough derivation of the concept, as well as different methods for obtaining the blocked forces in practice
are described in [2]. A theoretical comparison of these methods is given in [10]. An important assumption
for the derivation of the blocked force concept, is that the internal source excitation fA
1is independent of
the source mounting, i.e. the receiver B. This is (to the authors experience) a good assumption for climate
compressors, electric motors, rear axle differentials and many other components that are usually mounted
with rubber isolators, like the ERC. However, the concepts applicability needs to be thoroughly investigated
per component.
3 In-Situ determination of blocked forces
A popular method for determining the blocked forces in pracitice is the in-situ method [9]. For identifying
the blocked forces, the system is equipped with indicator sensors, denoted as u4, which have to be at or
downstream of the interface (see figure 3(a)). As discussed in the previous section, when artificially applying
the blocked forces fbl
2to the interface, they would have to cancel out all vibration at these points:
0!
=YAB
41 fA
1
| {z }
u4
+YAB
42 fbl
2,(7)
The responses in the indicator sensors u4can be recorded for different operational conditions of the source.
The blocked forces for this operational condition can then be computed by:
fbl
2=YAB
42 +u4,(8)
where (?)+indicates the least squares pseudo inverse, and the minus sign has again been dropped for clarity.
A pseudo inverse has to be used if the system of equations is over-determined, i.e. the vector u4contains
more channels than the actual number of blocked forces to be computed in fbl
2. The pseudo inverse can either
be built with least squares, or with a regularized inverse to suppress the detrimental effects of measurement
noise even more than with least squares [1].
4 Virtual point transformation
In the previous sections, it was implicitly assumed that the blocked forces fbl
2contain enough DoF to control
the full interface motion of the source, such that they can block all vibrations at and downstream of the in-
terface. In industry practice, the interface is often modeled with three translational forces in each connection
point, e.g. by performing an impact measurement at three points in x,y,z-direction. The blocked forces are
then represented by these impact points, which is reasonable as long as the full interface can be controlled
and numerical issues due to matrix inversion are not prominent. Obviously, the rotational DoF are neglegted
by this approach. Especially towards higher frequencies, with increasing complexity of the vibration modes,
the rotational DoF can become relevant for an accurate description of the interface. This is also important
for obtaining a set of blocked forces that can actually be transferred to a different vehicle design. Despite the
challenges of measuring rotational DoF, it has been shown in dynamic substructuring applications that they
are crucial for accurate results [11–15].
The method we employed for computing forces and moments in each connection point, is called the ’virtual
point transformation’ (VPT) [16]. It is using kinematic assumptions of the local displacement field directly
at the interface. In most cases, it is assumed that the interface is behaving rigid in a small area around the
connection point. A reference point for the computation of forces and moments is chosen. This point is
called the virtual point. Multiple impacts, contained in the vector f, are performed around the interface, see
figure 3(b). Their linear combinations shall be used to represent 3translational forces and 3moments around
the virtual point, which are contained in the vector m. Each force input hcan be written as a 3×1vector in
space fh, which is composed from its unit direction vector ehand scalar magnitude fh. The vector from the
virtual point to the impact position will be denoted as rh. Thereby, the translational forces mtand moments
mθthat the impact creates around the virtual point can be computed by:
m="mt
mθ#=eh
rh×ehfh=Rfhfh,(9)
where Rfhdenotes the 6×1matrix representing the virtual point load mresulting from a unit force input
in fh. The virtual point loads resulting from all other force inputs can be found equivalently. Placing each
Rfhin a column of the matrix Rf, one can write:
(a) Left: full virtual point DoF (b) Right: full virtual point DoF
(c) Left: reduced impacts & sensors (d) Right: red. impacts & sensors
Figure 4: Overview interface DoF used for modeling the interface. Impacts (blue arrows) are either directly
used as blocked force DoF (reduced setup in (c) & (d)) or transformed to VP DoF indicated as orange axis
in (a) & (b). Not all of the attached sensors and performed impacts are shown.
m=Rffand f= (Rf)+m,(10)
where the left part of equation (10) computes the resulting virtual point loads mfrom a fixed combination
of force inputs f. Inversely, the right part of equation (10) computes the minimal set of forces fnecessary
for creating a fixed virutal point load m[17]. Since the number of force inputs is typically higher than the
number of virtual point loads, a pseudo-inverse must be used. The impact positions must be carefully chosen,
so that all moments are excited [16].
5 Comparison of different interface complexities for the ERC
The ERC is connected at four points to the vehicle (see figure 1(b)). The two drop links connect to the wheel
hubs (WH) with a ball joint. Therefore, each WH interface is modeled with three forces in x, y, z direction,
and one triaxial sensor is used as indicator. This is common for each of the interface complexity variants
studied in this paper. The ERC is also connected at two points to the vehicle subframe. These connection
points were equipped with fixtures for applying 10 impacts and 3triaxial sensors on each fixture (not shown
in figure 4 (a) and (b)).
The effect of different interface complexities will be studied by using only a subset of impacts, or applying
a VP transformation (including or neglecting rotational DoF) for the inverse estimation of fbl
2at these con-
nection points. The FRFs from hammer impacts at the interface to indicator sensors u4, i.e. FRF matrices
YAB
42 , are transformed to virtual point loads by:
YAB
42m=YAB
42f(Rf)+.(11)
For including the rotational DoF in the interface description, the full matrix Rfhfrom equation (9) is used.
Table 1: Overview of different interface complexities and over-determinations used.
Force DoF Indicator DoF u4size YAB
42
3 DoF active (fig 4 (c) and (d))
untransformed f2
(fig 4 (c) and (d))
triax sensor active
12 ×12
3 DoF passive (fig 4 (c) and (d))
untransformed f2
(fig 4 (c) and (d))
triax sensor passive
12 ×12
VP 3 DoF only translations in Rf
VP transformed,
(fig 4 (a) and (b))
all sensors
42 ×12
VP 6 DoF transl. and rot. in Rf
VP transformed,
(fig 4 (a) and (b))
all sensors
42 ×18
(a) Condition Number VPT (b) Condition Number 3DoF
Figure 5: Condition numbers for the different interface complexities and over-determinations (see table 1).
The y-axis scale is the same in both plots.
For including only the translational DoF, the lower part of Rfhin equation (9), which computes the virtual
point moments mθfor each impact h, is neglected. The geometry preparation, FRF measurement and the
VP transformation in equation (11) is performed in the software DIRAC. The indicator channels used for
the inversion will also vary. A summary of the different variants of the blocked force evaluations is given
in table 1. The condition number of the inverted matrices YAB
42 , resulting from the different interface force
DoF and sets of indicator sensors, is shown in figure 5.
5.1 On-board validation
An initial check that can be done with the computed blocked forces is often called on-board validation (see
e.g. [18]). For computing the blocked forces in this paper a standard, least-squares pseudo-inverse without
any regularization was used. The blocked forces, computed with the signals u4via equation (8), can be
used to predict the vibration at other sensors uB
3which were also applied to the vehicle. As a load case, an
artificial excitation of the ERC with an impulse hammer on its housing will be used, see the impact shown in
figure 6. This can be seen as a repeatable internal load case fA
1as described in section 2. This single column
of the FRF matrix to the on-board validation and indicator sensors can then be used for the blocked force
estimation and on-board validation, i.e.:
uB
3=YAB
31 and u4=YAB
41 .(12)
The signals uB
3are responses to the same impact as u4, but are not used for the calculation of the blocked
Figure 6: Impacts on ERC housing which are used as artificial load cases.
(a) Subframe (b) Seatrail (c) Microphones
Figure 7: Validation Points
forces in (8). As validation points, a sensor on the vehicle subframe, the seatrail and two microphones in the
driver’s cabin were used, see figure 7. The response in these channels is predicted with the blocked forces,
as in equation (6):
˜uB
3=YAB
32 fbl
2.(13)
If the description of the interface is complete, this should yield equivalent vibration levels to the ones actually
recorded during the measurement (see explanation in section 2). Comparing the measured uB
3and the TPA
prediction ˜uB
3, i.e.:
uB
3
?
=˜uB
3,(14)
then serves as an initial validity check of the computed blocked forces. In case the description of the interface
loads is inappropriate, e.g. since an important transfer path on the interface was neglected, this would
manifest in a bad predictability of the measured uB
3. Figure 8 shows the on-board validation for the y-
channel of the subframe sensor (see figure 7(a)) and the z-channel of the seatrail sensor (see figure 7(b)).
It can be observed that for lower frequencies (<400Hz) all variants for computing the blocked forces can
reproduce the reference measurement well. The untransformed 3DoF variant with the indicator sensors on
the active side shows some deviations from the reference also in the lower frequency region. For higher
frequencies, especially in the region from 600-800Hz, the results computed with the untransformed 3DoF
variants are showing large spurious peaks. The results with the VP transformation, including the full 6DoF
per coupling, point generally yields the best on-board validation over the full frequency range.
5.2 Interface Completeness criterion
Another valuable check for testing the ’completeness’ of the interface description is the interface complete-
ness criterion (ICC) [19]. With the ICC, the blocked force prediction in the validation sensors ˜uB
3is compared
to the reference measurement uB
3on the full response vector at once. This is done with a modal assurance
criterion (MAC) like evaluation:
ICC =||uH
3˜u3||2
||uH
3u3|| ||˜uH
3˜u3||.(15)
(a) Subframe +YVPT (b) Subframe +Y3DoF
(c) Seatrail +ZVPT (d) Seatrail +Z3DoF
Figure 8: On-Board validation of different blocked force descriptions (same y-axis scale in all plots), for
artificial load-case 1in figure 6. The +Ydirection for the Subframe channel corresponds to the sensor axis,
wich can be seen on the sensor housing in figure 7(a). Likewise for the +Zdirection of the seatrail sensor in
figure 7(b).
(a) ICC VPT (b) ICC 3DoF
Figure 9: Interface completeness criterion for different interface complexities. Artificial load-case 1in
figure 6.
The ICC is bounded between zero and one. If reference and prediction are equal, the ICC will have a value
of 1. The ICC is evaluated for all channels of the subframe and seatrail validation sensor in figure 9, i.e. ˜u3
and u3are both 6×1vectors. As before, it can be observed that the blocked force determination with the
full 6DoF on each connection point yields the highest and most stable ICC over the full frequency range.
However, evaluating the ICC with a MAC like criterion makes it insensitive to amplitude differences in the
responses. As an example:
u3= [10 10]T,˜u3= [1 1]T,ICC = 1.(16)
For the comparison of mode shape vectors, an amplitude difference is insignificant, but for blocked force pre-
dictions it is relevant. We propose to evaluate the ICC in each sensor channel individually with a coherence
like criterion [16]. This compares the response in each channel iof ˜u3and u3at each frequency:
cohu3,i, u3,i ) = u3,i +u3,i)(˜u
3,i +u
3,i)
2(˜u3,i ˜u
3,i +u3,iu
3,i).(17)
This criterion is also bounded between zero and one, but sensitive to phase and amplitude differences in the
complex numbers aand b. A comparison of the MAC definition of the ICC (equation (15)) and the coherence
definition (equation (17)) is shown in figure 10. Both ICCs were evaluated for the 6DoF VP variant of the
interface description and use the response at the subframe and seatrail validation sensors. The coherence
definition of the ICC was then averaged over the results in all 6 entries of the vectors ˜u3and u3. It can be
seen that the results are in general very comparable.
Another advantage of evaluating the ICC with the criterion in equation (17) is that the quality of the pre-
diction can be evaluated for different load cases and validation channels individually, whereas the MAC
definition requires multiple channels to be evaluated at once. Averaging the coherence evaluation of the
ICC over multiple frequencies allows depicting the ICC for different load cases and validation channels in a
matrix plot, see figure 11.
5.3 Transfer validation
As mentioned in section 2, an advantage of the blocked forces is that they describe the source independently
of the final receiver (provided the interface modeling and computation of blocked forces is done right). Com-
puting the blocked forces in one vehicle configuration (or on a testrig), and using them to predict the response
in a modified vehicle configuration, allows to virtually predict the vehicle sound after a modification. Testing
this capability is called transfer validation. The blocked forces were computed in the normal vehicle, like
Figure 10: Comparison of MAC and coherence evaluation of the ICC criterion for the 6DoF VP interface
description.
(a) VP 6DoF (b) VP 3DoF (c) 3DoF active (d) 3DoF passive
Figure 11: ICC with coherence evaluation (17) averaged over the frequency range 20-1500Hz. Rows repre-
sent signals in the channels of validation sensors (see figure 7). Columns represent the 4impacts on the ERC
housing which can be seen in figure 6.
Figure 12: Difference in normal and modified vehicle FRF, for impact 1on ERC (figure 6) to subframe
y-channel (figure 13(a)).
(a) Subframe VPT (b) Subframe 3DoF
Figure 13: Transfer validation for artificial load-case 1in figure 6: computing blocked forces in normal
vehicle for predicting response in modified vehicle configuration.
in section 5.1. Subsequently, the system was modified by gluing a steel slap of ca. 7kg weight with dental
cement to the subframe, see figure 7(a). In the modified system, the same FRFs on the interface were newly
measured (see figure 4 (a) and (b))2. The same impact on the ERC housing (see figure 6) was also measured
to serve as a validation for the artificial loadcase. The FRF from this ERC housing impact to the subframe
y-channel is significantly altered by the additional weight compared to the normal vehicle, see their plots in
figure 12. Figure 13 shows a comparison of the validation with the blocked force predictions (see table 1).
The blocked forces were determined in the unmodified vehicle.
6 Conclusion
This paper investigates the applicability of the blocked force concept for different complexities of the in-
terface description and varying degrees of over-determination in the matrix inverse. For lower frequencies
(<400Hz) an over-determination in the sensors, and averaging of multiple impacts helps to improve the pre-
dictions and avoid spurious peaks in the results (compare BF VP 3DoF prediction in figure 13(a) to the
predictions with no over-determination and averaging in figure 13(b)). For higher frequencies (>400Hz), it
was found necessary to include also rotational DoF in the interface description (see the ICC in figure 11
and transfer validation in figure 13(a)). This result was expected physically, since at higher frequencies the
rotational DoF become more important due to the more complex mode shapes. The description of the com-
ponent interface with 6 DoF in the connection points was done with the VP transformation. This comes
2In principle the FRFs of the modified system can also be obtained virtually by substructuring [20], but this is beyond the scope
of this paper, so they were newly measured.
with the practical advantage of not having to measure exactly the same impacts during the blocked force
characterization as in the final receiver, which might often not be possible due to physical space restrictions.
The DoF in the VP provide the required common interface between the different experiments. The software
DIRAC was used for placing impacts and sensors on the 3D geometry, thereby taking care of the geometric
position and orientation so the virtual point transformed FRFs can be directly exported.
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... Combining the concepts of TPA with the principles of Dynamic Substructuring (DS) 20 has led to an approach in which the source is characterized using forces and moments in a virtual point (VP) [6]. The virtual point, typically used in frequency based substructuring (FBS) applications [7], has the advantage of taking into account moments in the transfer paths that are otherwise not measurable with conventional force transducers [8]. ...
... In the following, the use of VPT in a Y AB 42 measurement is assumed due to the advantages of including moments in the interface description [8] and non-rigid motion filtering at the low frequencies [23]. The VPT also filters the measurement errors to some extent by reduction of the forces in Eq. (7). ...
... (3) and compared to the measured response u 3 . Various criteria can be applied to estimate the responses' agreement [8,24]. In this study, the coherence criterion is used [8] as it is sensitive to both 170 phase and amplitude differences. ...
Article
Full-text available
When determining the critical paths for the transmission of sound and vibration in assembly products, transfer path analysis (TPA) is a reliable and effective tool. TPA represents a source with a set of forces that replicate the operational responses. However, admittance-based TPA methods are prone to experimental errors, as small measurement inaccuracies can lead to large discrepancies in the source characterization. The admittance of the transfer paths is preferably obtained through impact testing. Thus, poor repeatability in the position of the successive impacts affects the consistency of the interface forces. In this study, uncontrolled location variations in a structure’s excitation are characterized by a sensitivity analysis based solely on an experimental model. The functional dependency of a frequency response function on the impact location is deduced from the measured data. This makes it possible to reconstruct numerous responses for variations in the impact location and provides an appropriate sample size for the global sensitivity analysis. The influence of a random error at an individual impact location is quantified on the basis of variations in the response prediction. The approach is useful for cases where the source characterization is affected by location variations of the force input, e.g., lightly damped or complex structures where the impact locations are not easily accessed. An experimental study on an electric motor demonstrates that controlling the impact location’s repeatability in a TPA is important and can lead to a more consistent source characterization.
... Note that the joint accelerance of Fig. 12 is coupled as such without any fitting to the respective substructures due to the abovecited difficulties. A similar approach can be found in other works in literature [45][46][47]. In fact, the authors in [45] during identification of their known mass by substructure decoupling method noted that it was quite straightforward to detect errors in case of the known mass; however, it would not be the case if the system to be identified is unknown. ...
... They went on to assert that the only check that can be performed is to couple the predicted FRFs of the unknown subsystem with those of the known subsystem (to form the mathematically coupled system) and to compare it with the reference measured FRF on the assembly. This type of validation is called on-board validation [46,47] i.e. the reference FRFs (not included in the identification) in the same measurement campaign should be predicted by recoupling of the identified joint with the respective substructure models. ...
Preprint
Full-text available
In mechanical systems coupled with joints, accurate prediction of the joint characteristics is extremely important. Despite years of research, a lot is yet to be learnt about the joints' interface dynamics. The problem becomes even more difficult when the interface Degrees-of-Freedom (DoF) are inaccessible for Frequency Response Function (FRF) measurements. This is, for example, the case of bladed-disk systems with dove-tail or fir-tree type joints. Therefore, an FRF based expansion method called System Equivalent Model Mixing (SEMM) is used to obtain expanded interface dynamics. The method uses numerical and experimental sub-models of each component and their assembly to produce the respective expanded or hybrid sub-models. By applying substructure decoupling to these sub-models, the joint can be identified. However, the joint can be noisy due to expansion and measurement errors which propagate to the hybrid sub-models. In this paper, a correlation based approach is proposed in the SEMM method wherein the quality of the expanded sub-models is improved. In this new approach, several expanded models are generated systematically using different combinations of the experimental FRFs and computing a parameter, Frequency Response Assurance Criteria (FRAC), to evaluate quality of the contribution of the different measurements. The lowest correlated channels or FRFs can be filtered out based on a certain threshold value of FRAC. Using the improved hybrid sub-models, the joint identification also shows a remarkable improvement. The test object for the method is an assembly of disk and one blade with a dove-tail joint.
... Note that the joint accelerance of Fig. 12 is coupled as such without any fitting to the respective substructures due to the above-cited difficulties. A similar approach can be found in other works in literature [45][46][47] . In fact, the authors in [45] during identification of their known mass by substructure decoupling method noted that it was quite straightforward to detect errors in case of the known mass; however, it would not be the case if the system to be identified is unknown. ...
... They went on to assert that the only check that can be performed is to couple the predicted FRFs of the unknown subsystem with those of the known subsystem (to form the mathematically coupled system) and to compare it with the reference measured FRF on the assembly. This type of validation is called on-board validation [46,47] i.e. the reference FRFs (not included in the identification) in the same measurement campaign should be predicted by recoupling of the identified joint with the respective substructure models. ...
Article
In mechanical systems coupled with joints, accurate prediction of the joint characteristics is extremely important. Despite years of research, a lot is yet to be learnt about the joints’ interface dynamics. The problem becomes even more difficult when the interface Degrees-of-Freedom (DoF) are inaccessible for Frequency Response Function (FRF) measurements. This is, for example, the case of bladed-disk systems with dove-tail or fir-tree type joints. Therefore, an FRF based expansion method called System Equivalent Model Mixing (SEMM) is used to obtain expanded interface dynamics. The method uses numerical and experimental sub-models of each component and their assembly to produce the respective expanded or hybrid sub-models. By applying substructure decoupling to these sub-models, the joint can be identified. However, the joint can be noisy due to expansion and measurement errors which propagate to the hybrid sub-models. In this paper, a correlation based approach is proposed in the SEMM method wherein the quality of the expanded sub-models is improved. In this new approach, several expanded models are generated systematically using different combinations of the experimental FRFs and computing a parameter, Frequency Response Assurance Criteria (FRAC), to evaluate quality of the contribution of the different measurements. The lowest correlated channels or FRFs can be filtered out based on a certain threshold value of FRAC. Using the improved hybrid sub-models, the joint identification also shows a remarkable improvement. The test object for the method is an assembly of disk and one blade with a dove-tail joint.
... The FRAC [20][21][22][23] is a frequency-dependent normalized single-value metric that estimates the correlation between two FRFs with the same excitation and response points. We computed it to assess the accuracy of the transfer functions found with the reference FE model. ...
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Full-text available
Structural engineering simulations have required increasingly complex computational models to replace physical tests accurately. This work focuses on the numerical analysis of vehicle body structures, whose size and complexity make the use of very accurate nonlinear models unfeasible due to the prohibitive computational costs involved. The purpose of this study is to find a new approach to model spot-welded joints in linear finite element models of thin-wall vehicle body structures, improving the accuracy of the model without increasing its complexity. Using a set of simplified nonlinear models, we fitted the stiffness and damping properties of these welded joints and used those adjusted values into a linear model of the entire vehicle body structure. The results were compared with experimental tests, showing a clear improvement in the accuracy of the modal and frequency responses provided by the linear finite element model, but keeping its initial complexity level. The adjusted model was then used in an optimization analysis to reduce the structure's weight, leading to interesting cost savings and important reductions in the use of natural resources and carbon emissions.
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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.
Thesis
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[ Link to PhD defense video: https://www.youtube.com/watch?v=IEVuF2rJOYs&t=6s ] This thesis is the result of a 4-year collaboration between the Technical University of Munich and the BMW Group. The goal was to apply substructuring methods to the Noise Vibration Harshness (NVH) engineering needed for integrating electric climate compressors in upcoming vehicles. The compressor is one of the major contributors to the cabin noise in battery electric vehicles (BEVs). An accurate yet practical development process for its vehicle integration is crucial for industry. Specifically, the aim was to simulate the compressor noise in the cabin for different, virtual design variants of the isolation concept. Therefore, the methods from two broader fields were applied: First, the excitation of the compressor was modeled with component transfer path analysis (TPA) methods. Second, the full transfer path from the compressor to the driver’s ear is assembled from multiple subcomponent models, via dynamic substructuring (DS). For accomplishing the above mentioned goals, different gaps in the current technology have been identified, which will be addressed in this thesis. With frequency based substructuring (FBS), a subclass of DS, it is possible to couple experimental and numerical substructure models in a virtual assembly. For the compressor, it was found that including rigid body models in the transfer path is a valuable addition. The proper formulation and integration of rigid body models in the framework of FBS will be presented. Another bottleneck at the onset of this project, was the proper modeling of rubber bushings in the transfer path. A novel method for experimentally identifying accurate substructure models of rubber isolators was developed. The rotating components in the compressor introduce gyroscopic effects that influence its dynamics. A novel substructuring method for virtually coupling gyroscopic terms to a component could prove that these effects are not relevant for the compressor case. The compressors excitation is described by blocked forces. Applying the blocked forces to the substructured transfer path of the assembly allows to simulate the sound in a virtual prototype. One goal was to make the simulated results audible to non-acoustic experts, which required the creation of sound files. This allowed for a subjective comparison of different designs at an early development stage. Since the noise predictions with TPA are typically in the frequency domain, some signal processing is required to create sound files in the time domain. Different methods for auralization will be compared, which could not be found in the existing TPA literature. Due to the inverse process for identifying the blocked forces, measurement noise can be amplified to unacceptably high levels, which are audible in the sound predictions. Regularization methods have the potential to significantly suppress the noise amplification, which is explained and exemplified for blocked force TPA. Additionally, it was found that only the structure-borne sound transmission was not sufficient to describe the compressor noise in the cabin. The compressor is also directly radiating air-borne sound from its housing, which will be included in the NVH model by means of equivalent monopoles. The application examples at the thesis’ end are extending the current state-of-the-art, by showing how the modular vehicle models can be used for early phase, parametric design optimizations on a complex NVH problem.
Article
Full-text available
Blocked forces can be used to describe, independently, the operational characteristics of a vibratory source. Their use within a computational model avoids the need to represent explicitly the complex mechanisms that lead to vibratory excitation. To obtain and apply an experimental blocked force with confidence it is important that likely sources of error are known, and measures of their severity are available. In this paper we introduce the notions of completeness and consistency, and detail their role in the introduction of systematic errors in a blocked force characterisation. Their mathematical origins are described and criteria to quantify their severity are proposed; the Interface Completeness Criterion (ICC), and the Measurement Consistency Criterion (MCC). These are illustrated through numerical and experimental examples. Completeness is related to the interface description adopted in a source characterisation (i.e. the number of degrees of freedom used). The ICC represents the quality of an interface description and can be quantified from in-situ measurements, i.e without having to remove the source from its assembly. Consistency is related to the underlying dynamics shared by active and passive quantities (whether measured or modelled). The issue of consistency is more general, completeness being a special case, and so a single criterion is hard to formulate. When an inconsistency arises between the blocked force of a vibration source and its corresponding free interface frequency response function matrix, the MCC provides a quantitative indication of its severity. Importantly, many of the concepts discussed apply equally in the context of experimental dynamic sub-structuring.
Conference Paper
Full-text available
A detailed case study on using in-situ blocked forces for advanced NVH development of automotive structure-borne sound sources is presented. The discussed approach provides a realistic auralization of a steering system virtually operated in a vehicle using in-situ blocked forces from a bench with vehicle transfer paths allowing reliable subjective and objective design evaluations. In-situ Transfer Path Analysis is used to validate the obtained Virtual Acoustic Prototype from operational measurements with low-noise Driver Simulators (DS) yielding highly repeatable steering conditions on the bench and in the vehicle. Potential impact of the DS on the measured and predicted cabin sound is examined. The presented method is believed to be a promising approach towards enabling OEMs to accurately source vibrating components and suppliers to develop robust components on test benches due to the invariant property of the blocked force.
Conference Paper
Full-text available
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.
Conference Paper
Full-text available
Frequency based dynamic substructuring (FBS) allows to predict the dynamic behavior of a complex system where neither building a physical prototype of the assembled system, nor possessing a detailed numerical model of all substructures is required. A task that frequently arises in engineering practice when developing a product containing many supplier parts. However, in the experimental realm, modeling the interface connection between two substructures is not as straightforward as in numerical analysis. The consideration of rotational degrees of freedom (rdof) on the interface seems to be crucial for accurate results, but no common procedure has been established yet. By projecting measured sensor data on interface deformation modes (IDMs) it is possible to consider rdof as well as filtering out uncorrelated measurement noise. The transformation of a measured frequency response function (FRF) matrix on some generalized IDMs has recently been derived by directly using Moore-Penrose pseudoinverses. The transformation process can also be seen as a minimization procedure, e.g. as simple least squares for the displacements and a convex optimization for the forces. This contribution derives the pseudoinverses starting from this minimization point of view, where the engineer is free to choose the quantity to be minimized. From this interpretation, some suggestions for including more engineering judgment in the transformation are made (either gained during testing practice, from measurement variances or mechanical energy minimization principles). We also show that the coupling of transformed FRF matrices effectively corresponds to a weakening of the interface compatibility conditions. Thereby, we hope to give some insight in the meaning of the weighting matrices involved in the transformation, and provide a framework for deriving improved coupling methods in the future.
Thesis
Full-text available
Summary The sound transfer from resiliently mounted shipboard machinery to the ship structure is fundamentally of a multi-path nature. It occurs simultaneously via the resilient mountings, via the surrounding air and via mechanical links such as pipes, propeller shaft etc. At the present stage it is usually unknown which factors limit the effectiveness of a resilient mounting system as a noise reduction measure. This hampers a cost-effective improvement. Complete theoretical analysis of a multi-path system.is too complex. On the other hand experimental evaluation requires measuring methods which can be applied under the very restrictive conditions on board ships. For most sound transfer paths such methods are lacking. In the Chapters 2-5 of this thesis new experimental methods have been developed and tested for quantifying the sound transfer respectively via the resilient mountings underneath machinery, via shallow air cavities below machinery and via pipes. All these methods can be applied on board without disturbing seriously normal ship programs. The Chapters 6 and 7 are concerned with a case study of the multi-path noise reduction properties in a representative shipboard mounting system and with the development of a simple experimental method for aiding the design of improved multi-directional structureborne sound isolation. In Chapter 1 an overview is given of the knowledge with respect to the effectiveness of resilient mounting systems on board ships. Different approaches for the in-situ analysis of multi-path mounting systems are compared and an outline of the thesis is presented. In Chapter 2 a method is described for the experimental analysis of the multi-directional structureborne sound transfer through the resilient mountings and through the ship structure. Basic elements are a newly developed technique for measuring multi-directional sound transfer properties of mountings in a laboratory test rig and previously published reciprocity techniques for measuring ship transfer functions. The feasibility of the measurements on resilient mountings is illustrated with some test results. In Chapter 3, the mounting path analysis procedure is investigated in a scale model for the complete path from a diesel engine-like vibration source, via resilient mountings and ship structure, to an accommodation deck. Because of the multi-directional vibrations the complete analysis for a multi—mounting system requires the measurement of an enormous amount of data. Investigations were carried through to what extent the accuracy of the analysis is affected when simplified procedures are applied. Chapter 4 describes two experimental methods for determining the airborne sound transfer through shallow reverberant cavities below resiliently mounted machinery, in cases where these cavities are inaccessible for loudspeakers as substitution sources. Basic elements are the introduction of hypothetical acoustic point sources in a cavity and reciprocal transfer function measurements for such point sources. One of the methods is tested and validated by scale model experiments. The theoretical analysis leads also to improvements in theoretical models for sound transfer through shallow or narrow cavities published previously. In Chapter 5, experimental methods are investigated for the assessment of structureborne sound transfer along pipes. Laboratory tests show that direct determination of the sound transfer using energy flow measurements is feasible at frequencies below the initiation of 2nd order circumferential waves. Two substitution source methods for indirect determination of the sound transfer appear also feasible. One of the methods uses energy flow measurements on the pipe, whereas the other method uses squared radial accelerations averaged over a certain pipe length. The latter method is also usable at frequencies above the cut-off frequency of 2nd order circumferential waves. Of great practical interest is the use of reciprocal measurement of the sound transfer from the substitution sources, when the signal to noise ratio for direct measurements is low. The sound transfer from a resiliently mounted medium-speed propulsion diesel engine to the accommodation is analysed in Chapter 6. It concerns a mounting system representative of several modern passenger and car-ferries. The multi-path system insertion loss is some 12-17 dB for octave bands with centre frequencies 63 Hz - 1 kHz, which is typical for similar systems in other ships too. For octave bands with centre frequencies up to 250 Hz the contributions of the resilient mounting path and the airborne paths appear to be much smaller than the total sound transfer. On the basis of both shipboard and scale model measurements, system parameters which are important for the sound transfer through the resilient mountings and through the air, are discussed for the system investigated. Estimates are given for the upper‘ limit of the insertion loss for similar single stage mounting systems without acoustic enclosure. Compared to the present situation an improvement can be obtained of maximally some 20 dB for the octave bands with centre frequencies 63-250 Hz and of some 10 dB for the 500 Hz and 1 kHz octave bands. In Chapter 7, a simple experimental method is described and tested for estimating frequency bandwidth averages of real parts of point admittances for each of 6 degrees of freedom. Again, use is made of a substitution source principle and of reciprocity relations for transfer functions. The method is very useful for collecting multi-directional admittance data at resilient isolator locations on board ships. Moreover, it is of great practical use as a tool for designing seating structures, taking into account the multi-directionality of machinery vibrations and the multi-directional sound transfer properties of flexible isolators. Finally, in Chapter 8, an attempt is made to evaluate to what extent problems of the experimental analysis of multi-path resilient mounting systems has been solved in the present thesis and what type of work has still to be done. Moreover, some factors are indicated that may form either a practical or a fundamental limitation for mounting system improvement.
Article
Full-text available
Vibro-acoustic source characterization is an essential task in vehicle development to enable prediction of receiver response. For structure-borne noise, the interface forces in multiple degrees of freedom due to internal loads are often quantified for root cause analyses in a single system assembly, as in transfer path analysis (TPA). However, for a reliable prognosis of the acoustic performance of a known component such as a motor or pump, a receiver-independent source characterization is required, and the method of acquiring blocked forces from in-situ measurements has been shown to be a preferred technique for such purposes. The benefits of the method are the characterization of the intrinsic properties of the source and the possibilities of measuring the component attached to receivers with varying dynamic properties. There is to date a limited number of validation cases where blocked forces from in-situ measurements are acquired for automotive source–receiver assemblies. In this study the blocked forces of a vacuum pump in nine degrees of freedom were determined when connected to a bracket whose boundary conditions were modified in order to achieve four assemblies with different source/receiver dynamic properties. The results show that the blocked forces are transferable, i.e. the receiver response in one assembly was predicted in a wide frequency range by combining source–receiver transfer functions of that assembly with blocked forces estimated in another assembly. Furthermore, an in-situ blocked force TPA was applied to a double-isolated complete vehicle source–receiver case of an electric rear axle drive with interior compartment sound pressure as target. The reconstructed magnetic tonal harmonics agreed with the measured target response in the frequency range 50–500 Hz, which further motivates the use of the blocked force principles for TPA and source requirements specifications.
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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.
Article
15 years of NVH applications make Transfer Path Analysis appear a commodity tool. This is however not the case. Required insight in the application constraints makes TPA remain an expert approach. This paper reviews past progress in TPA methodology and its limitations. It then introduces a number of innovative approaches addressing these, opening new application fields. This includes speed improvement (Fast TPA), structural modeling integration (Modal Contribution Analysis), CAE integration (Hybrid TPA), sound quality interpretation (TPA-sound synthesis) and supporting better exploitation of operational data (Operational Path Analysis). An outlook is given to the next challenge, the application to transient problems.