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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
2−YAB
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:
coh(˜u3,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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