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A benchmark structure for validation of Experimental Substructuring, Transfer Path Analysis and Source Characterisation techniques

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This paper presents a practical study on popular Experimental Dynamic Substructuring topics. A series of substructures is designed of such complexity to fit in right between " real life " structures as often found in industrial applications and " academic " structures which are typically the simplest models to identify a particular phenomenon. The designed benchmark structure comprises an active side with a vibration source, a passive side and a test rig for source characterisation. The connectivity is scalable in complexity, meaning that a single-point, two-point and continuous interface can be established. Substructuring-compatible component models are obtained from impact measurements using the Virtual Point Transformation. The vibration source on the active structure is characterised on the test rig using the in-situ TPA concept. Hereafter the component TPA method is applied to simulate the response on the passive side of the coupled structure, in turn obtained using dynamic substructuring.
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A benchmark structure for validation of Experimental Substructuring, Transfer
Path Analysis and Source Characterisation techniques
M. V. van der Seijsa, E. A. Pasmaa, D. D. van den Boscha, M. W. F. Wernsena,b
aVIBES.technology
Molengraaffsingel 14, 2629 JD, Delft, The Netherlands
bDelft University of Technology, Department of Precision and Microsystems Engineering
Mekelweg 2, 2628CD, Delft, The Netherlands
ABSTRACT
This paper presents a practical study on popular Experimental Dynamic Substructuring topics. A series of
substructures is designed of such complexity to fit in right between “real life” structures as often found in industrial
applications and “academic” structures which are typically the simplest models to identify a particular phenomenon.
The designed benchmark structure comprises an active side with a vibration source, a passive side and a test rig
for source characterisation. The connectivity is scalable in complexity, meaning that a single-point, two-point and
continuous interface can be established. Substructuring-compatible component models are obtained from impact
measurements using the Virtual Point Transformation. The vibration source on the active structure is characterised
on the test rig using the in-situ TPA concept. Hereafter the component TPA method is applied to simulate the
response on the passive side of the coupled structure, in turn obtained using dynamic substructuring.
Keywords: dynamic substructuring, virtual point transformation, transfer path analysis, blocked force, in-situ.
1 INTRODUCTION
Dynamic Substructuring (DS), Transfer Path Analysis (TPA) and Source Characterisation (SC) are three research fields that have
received tremendous attention from both science and industry. All three aim to provide practical solutions for engineering of
structural vibrations and sound, with applications stretching from the automotive and aerospace industry to high-tech precision
machinery and sustainable energy solutions. However, it is fair to say that the similarities between the three fields have not always
been well understood [1]. This is perhaps motivated by their different origins, for instance: substructuring finds its oldest roots in
numerical modelling and reduction of aerospace structures [2–5], transfer path analysis evolved hand-in-hand with automotive
NVH engineering [6–10] and source characterisation emerged from fields such as vibration isolation and structure-borne sound
engineering [11–15]. Only in recent years, some studies [9, 16–18] have appeared that incorporate various aspects of the three
fields, while [19, 20] extend to yet other fields such as feedback control theory.
At the same time, many methods within DS, TPA and SC prove to be rather challenging to validate in the context of an indus-
trial application. This is often due to a myriad of experimental uncertainties (signal-to-noise problems, incomparable opera-
tional/boundary conditions, presence of secondary excitation sources, etcetera) on top of the applications’ inherent complexities.
To avoid such uncertainties and reduce the overall complexity of a problem, studies on “academic” structures are often conducted
first, such that the method’s key properties present themselves as clearly identifiable and unambiguous properties. And although
such studies provide an excellent basis for theory development, it remains difficult to transpose a proof of a theoretical concept
to application on an industrial problem.
Figure 1: The three benchmark substructures: active source A (left), passive receiver B (centre) and test rig parts R (right).
Paper goal & outline
The goal of this paper is twofold. First, a benchmark structure is proposed of such complexity, that it fits in right between the
“real-life” industrial structures on the one hand, and the limited-DoF academic examples on the other hand. Section 2 introduces
the three benchmark substructures: an active, passive and test rig substructure. The benchmark substructures have been designed
to allow for three different coupling configurations with increasing interface complexity.
The second goal of this paper is to apply concepts of DS, TPA and SC using measurements on the constructed benchmark struc-
tures. Section 3 presents a high-level overview of a selection of possible applications, including a virtual point transformation,
coupling of substructures A and B, source characterisation of A in test assembly AR and transfer path analysis for prediction of
vibrations in assembly AB.
2 BENCHMARK DESIGN
The benchmark construction has been motivated by the desire to validate methods within the fields of experimental DS and
TPA. More specifically, the aim was to experiment with methods as covered in the general framework articles on the two topics,
respectively [5] and [1]. In the latter one, three types of substructures are used for theory development: an active source structure,
a passive receiving structure and a test rig for source characterisation. These three substructures have formed the basis for the
benchmark. Also, as the interest is in validating methods in a frequency range of 0 to 5000 Hz, the benchmark substructures are
supposed to display sufficient dynamics (i.e. vibration modes) in this range.
2.1 Substructure design
Figure 1 shows the three substructures. Let us introduce them one by one and briefly touch upon some design considerations:
- Substructure A is welded together from 3 pieces of solid aluminium (30×30 mm). It forms an evenly sided triangle and
loosely resembles the character ‘A, but was made asymmetric to avoid double resonance modes. It comprises a number
of 10 mm diameter holes, at the corner points and along the length of the members, evenly spaced at distances of 75 mm.
It hosts a vibration source (further discussed below) and can therefore represent the active source system in a TPA or SC
problem. The combined weight is circa 2.5 kg.
- Substructure B is constructed from two plates of stainless steel with a solid piece of steel welded in between. The plates
are produced using precise laser-cutting. Five holes are placed spanning a total distance of 300 mm, again with 75 mm
spacing in between. A honeycomb-like pattern of cuts was introduced to reduce weight, as well as to provide a pattern to
align sensors for an observability investigation1. As such, substructure B represents a receiving side into which the source
vibrations of substructure A may propagate. The total weight is circa 10 kg.
- Substructure R is a collection of small identical support structures, together forming a test rig for testing of substructure A.
The supports are machined from solid aluminium blocks and can be mounted on a wooden base plate. An opening in the
centre was made to reduce the stiffness of the top with respect to the fixed base. The test rig R can be used to characterise
1See the paper in the proceedings of SEM IMAC 2017: M. Wernsen et al. An indicator sensor criterion for in-situ characterisation of source vibrations.
Figure 2: Various configurations of the benchmark structures. Upper row: assemblies of A and B in 1/2/5-point configuration.
Lower row: assemblies of A and R in 1/2-point configuration.
the source vibrations of substructure A, for prediction of vibrations in an assembly with substructure B. The weight of
each support is 670 gramme.
2.2 Vibration source
For application of SC and TPA, substructure A needs to have an active vibration source. Many applied studies in this field
report on difficulties with maintaining identical operational conditions throughout the various testing environments [9, 21–23].
In the definition of the benchmark, it has therefore been a key requirement to have a source that generates perfectly reproducible
vibrations. Other requirements were to have a source with distinct orders, i.e. a very constant speed with a stationary excitation
profile, and an excitation spectrum that renders sufficient signal-to-noise in the frequency range of interest.
The chosen vibration source is a NEMA 17 stepper motor: a typical electric motor used in 3D printing hardware. The stepper is
controlled using an Arduino DUE with a Pololu A4988 stepper motor driver. It uses a PWM protocol for speed control, which
can be programmed to run through various speeds. In contrast to e.g. an AC electric motor, a stepper motor has a rather ‘rippled’
rotational speed profile, generating a lot of motor orders, i.e. harmonics proportional to the fundamental frequency. If desired,
an unbalance mass can be connected to the shaft of the motor to further amplify the vibration amplitude. The PWM signal can be
connected to a tacho-pulse channel of a DAQ system, allowing for accurate speed monitoring during operational measurement.
2.3 Assembly variants
The individual substructures have been designed to allow for multiple assembly configurations. Figure 2 shows these assemblies
for AB (top) and AR (bottom). The configurations respectively represent a single coupling point, two coupling points and five
coupling points, the latter resembling a continuous interface. In all cases, standard M10 bolts and nuts can be inserted to fasten
the structures.
- The single-point configuration is meant to be the simplest assembly to comprehend from a structural-dynamic point of
view. Although both structures comprise dozens of modes in a bandwidth of 5 kHz, one might reason that only 6 vibration
modes can be transferred over the interface. Substructure coupling of A and B would thus imply writing an interface
condition for the 3 translational and 3 rotational DoFs. The single-point configuration is well suited to investigate e.g.
experimental substructure coupling and decoupling [24, 25].
(a) Assembly AB in the single-point coupling configuration, free-
floating suspended by soft springs.
(b) Assembly AR in the two-point coupling configuration, mounted
to a wooden base plate resting on a test table.
Figure 3: Two test configurations.
- The two-point configuration roughly doubles the complexity of the assembly. Following the same reasoning, a maximum
of 12 vibration modes would now be present in the vibration transfer between A and B. However, it is evident that there is
interplay between the two coupling points, which deserves special attention in any application of substructuring or force
identification. Indeed, the two-point coupling configuration forms a perfect basis to study phenomena related to interface
conditioning, such as matrix regularisation and observability of the interface vibrations.
- The five-point configuration resembles a continuous interface. In this case, it is likely that the effect of each coupling
point can no longer be distinguished. Instead, the combined interface effect would probably be best considered in terms
of modes. Modal substructuring techniques such as [26] can be tested on this assembly, as well the transmission simulator
method [27].
The entire benchmark collection provides an abundance of data for investigation of many DS, SC and TPA topics, which shall
be the topic of the next section.
3 APPLICATIONS
The coming sections provide a high-level overview of some experimental applications. Examples are shown of experimental
modelling using virtual point transformation, dynamic substructuring, source characterisation and transfer path analysis. In
order not to dwell in theory, the derivations and equations have been kept to a minimum. Reference is made to original literature
for the interested reader.
For the purpose of this study, an extended range of assemblies has been subjected to impact hammer and operational measure-
ments. Figure 3 shows two of those assemblies: AB in single-point and AR in two-point coupling configuration. All separate
substructures and the assemblies AB have been measured in free-floating conditions, i.e. suspended by soft springs. The assem-
blies AR have been mounted onto a wooden base plate, in turn resting on a test table on air springs.
3.1 Experimental modelling
Experimental modelling can be understood as the art of obtaining a structural-dynamic model (such as FRFs) from measurements
[18, 28, 29]. It constitutes a fundamental step in experimental substructuring, but also finds application in component transfer
path analysis. This section briefly discusses how a nodal FRF model can be obtained from impact hammer measurements,
demonstrated for experimental modelling of substructures A and B.
VP2
VP1
VP3
(a) Substructure A. The three coupling points are each instrumented
by 3 tri-axial accelerometers and 16 impact points.
VP3
VP1
VP2
(b) Substructure B. The three coupling points are each instrumented
by 3 tri-axial accelerometers and 16 impact points; 2 additional sen-
sors register target responses in the structure.
Figure 4: Acceleration sensors (indicated by grey cubes) and impact locations (red arrows) visualised on the substructures.
3.1.1 Short theory of the Virtual Point Transformation
Typically, experimentally obtained models lack a common interface which allows for substructure coupling. In the numerical
domain, nodes provide this common interface, as a direct result of FE modelling (sometimes after remeshing or a node colloca-
tion technique [30]). The Virtual Point Transformation [29] introduces such nodes in experimentally obtained models. The main
idea is to choose a point on or near a physical interface of a substructure that can be made compatible with the other (experimen-
tal or numerical) substructure to couple. All measured displacements uand forces faround the interface can be transformed to
this virtual point, resulting in a 6-DoF ‘nodal’ description consisting of virtual translations/rotations qand forces/moments m:
Displacements: u=Ruq=q= (Ru)+u RuRn×6(1a)
Forces: m=RT
ff=f=RT
f+m RfRm×6(1b)
The two transformations allow to compute a 6×6virtual point FRF matrix Yqm (ω)from a measured n×mmatrix Y(ω).
This can easily be set up for each coupling point, building a experimental ‘super-element’ that is compatible for substructuring
with other models:
Measured FRFs: u=Yf (2a)
Virtual point FRFs: q= (Ru)+YRT
f+m=q=Yqmm(2b)
One underlying assumption of this transformation is that the measured substructures behave rigidly in the vicinity of this interface
in the frequency range of interest [29]. This assumption and other criteria will be discussed next.
3.1.2 FRF measurement
All substructure FRFs have been obtained by impact hammer testing. Figure 4 depicts how hammer impact points (red arrows)
and tri-axial accelerometers (grey cubes) have been positioned and oriented on substructures A and B. Besides some internal
points, the main interest for both substructures are the three coupling points. Each coupling point has been instrumented by 3
tri-axial accelerometers of type PCB 356B21. To determine forces and moments, 16 impact hammer positions are chosen per
coupling point. Altogether, this results in sufficient overdetermination of the virtual point transformations.
Overall sensor consistency of VP2 w.r.t. VP1
0 1000 2000 3000 4000 5000
Frequency (Hz)
0%
50%
100%
Consistency (-)
(a) Overall sensor consistency of 9 sensor channels around VP2 with
respect to excitations around VP1.
(b) Overall impact consistency of 16 (light blue) and 13 (blue) im-
pacts out of 16 around VP1 with respect to responses around VP2.
Figure 5: Sensor and impact consistency checks for substructure A.
3.1.3 FRF consistency
In order to evaluate the above assumption on rigidity and obtain insight in the contribution of single force impacts or displace-
ments to the VP dynamics, several consistency checks can be done. With a consistency check, the experimentally obtained
results are first transformed to the virtual point and then expanded (or projected) back on the original measured DoFs [18, 29].
The difference in the original response and the projected response provides inside on how much residual dynamics (interface
flexibility) has been neglected with the virtual point transformation. More practically, this technique is used to evaluate the
contribution of single measured DoFs to the transformed VP dynamics. This can be used to find erroneous definitions of sensor
and impact positions or directions in the transformation, or discard ‘bad impacts’2from the transformation.
Let us illustrate the various consistency checks for substructure A. Figure 5a shows the overall sensor consistency of VP2 for
excitations around VP1. This operation takes the accelerances of all 9 sensors channels (FRF matrix rows) around VP2 (u) for
a combination of hammer impacts (FRF matrix columns) around VP1, transforms these to the virtual point qand expands the
accelerances back to their original sensor channels (˜
u). The score of 100% over the full bandwidth of 5000 Hz indicates that all
sensor channels are perfectly consistent, i.e. ˜
u=u. This is obvious as the region between the three sensors is very stiff; values
below 100% would probably indicate incorrect placement of a sensor.
Figure 5b shows the overall impact consistency for VP1 with respect to responses around VP2. The light-blue area was computed
for all 16 impact points, which is clearly not optimal. Looking into the specific impact consistency for each 16 impacts, three
impacts had significant lower score than average. By discarding these 3 from the set of 16, the full 6-DoF set of virtual point
forces/moments can still be determined. The dark-blue area was computed for the optimised set, clearly showing an improved
overall impact consistency.
3.1.4 FRF reciprocity
The VP transformation allows to validate reciprocity of the obtained virtual point FRFs, as computed by equation (2b). Note
that this is possible as the VP displacements (i.e. linear and rotational accelerations) are perfectly ‘vectorially associated’ with
the corresponding VP loads (i.e. forces and moments). In other words, the virtual point FRFs behave as if they were computed
for nodes of an FE model.
Figure 6 shows two typical virtual point FRFs: response VP2Y over force VP3Y of substructure A (left) and response VP2Z
over force VP3Z of substructure B (right). The FRFs reciprocal FRFs are displayed in red. It can be observed that reciprocity is
indeed satisfied, especially up to 2 kHz.
2Bad impacts can for instance be caused by a low impact energy in the frequency range of interest, low signal-to noise ratio, poor reachability with an impact
hammer due to geometric constraints, double pulses, etcetera.
0 1000 2000 3000 4000 5000
10-3
10-2
10-1
100
101
102
103
Accelerance (m/s 2/N)
FRF reciprocity of substructure A
VP2 Y / VP3 Y
VP3 Y / VP2 Y
0 1000 2000 3000 4000 5000
Frequency (Hz)
-180
-90
0
90
180
Phase (deg)
(a) Substructure A: Y-direction of VP2 to VP3 and its reciprocal.
0 1000 2000 3000 4000 5000
10-3
10-2
10-1
100
101
102
103
Accelerance (m/s 2/N)
FRF reciprocity of substructure B
VP2 Z / VP3 Z
VP3 Z / VP2 Z
0 1000 2000 3000 4000 5000
Frequency (Hz)
-180
-90
0
90
180
Phase (deg)
(b) Substructure B: Z-direction of VP2 to VP3 and its reciprocal.
Figure 6: Reciprocity of the virtual point FRFs of the experimental models of substructure A and B.
3.2 Dynamic substructuring
Now that VP transformed FRFs are available for substructure A and B, both structures are coupled using the LM-FBS algorithm
[5, 31]. To do so, the substructure FRF matrices of A and B are put in block-diagonal form and an appropriate Boolean matrix
Bis written (not discussed here):
˜
Y=YYBTBYBT1BY Y ,YA0
0 YB(3)
The two-point coupling configuration is considered, which means that coupling is performed by requiring strict coordinate
compatibility and force equilibrium for the FRFs of virtual points 2 and 3. We now focus on the frequency range of 0 to 1600
Hz.
Some results of the substructured FRFs of AB are depicted in figure 7. First in 7a, a driving point FRF on the coupling interface
is shown, namely for VP2 in Z-direction. The phase is shown as well to assess the passivity3of the FRF. Figure 7b shows a
transfer FRF from an internal force impact point on structure A to an acceleration response internally on structure B. Both points
are not part of a coupling VP, hence the transfer FRF is realised by coupling over the interface. The substructured FRFs (blue)
are compared the measured FRFs of the coupled structure AB.
The first substructuring results, although not perfect yet, are by all means encouraging. It can be seen how resonance frequencies
are created at roughly the right frequencies. The phase around anti-resonances is not fully stable, however the overall amplitude
of the FRFs match quite well. Note that no filtering or processing has been applied to the measured FRF data, except for
transformation to virtual points.
3.3 In-situ source characterisation
To characterise the active vibrations of source structure A, the in-situ characterisation method is used [15]. This method describes
the source structure using ‘blocked forces’ on its interfaces (as if the component were connected to a fully rigid boundary) by
measuring operational responses on a connected receiver structure. More specifically, this method is able to characterise a source
structure in an assembly, with the resulting characterisation being a property of the source structure only rather than being a
property of the combined assembly. Because this characterisation is a source-inherent property, the obtained blocked forces are
3For an accelerance driving point FRF, the phase should be bounded by 0 and +180 degrees.
0 200 400 600 800 1000 1200 1400 1600
10-3
10-2
10-1
100
101
102
103
Accelerance ((m/s 2)/N)
Measured results
DS results
0 200 400 600 800 1000 1200 1400 1600
-180
-90
0
90
180
Phase (deg)
DS coupling - Driving point FRF VP2Z / VP2Z
(a) Driving-point FRF for VP3 in Z-direction.
0 200 400 600 800 1000 1200 1400 1600
10-3
10-2
10-1
100
101
102
103
Accelerance ((m/s 2)/N)
DS coupling - Transfer FRF 10Z / 3Z
Measured results
DS results
(b) Transfer FRF for an internal impact point on A to an internal
acceleration response on B, both in Z-direction.
Figure 7: Application of dynamic substructuring: assembled FRFs of AB (blue) in two-point coupling configuration, compared
against the validation measurement (red).
transferable to other receiving structures. Therefore, in theory, the source may be characterised in the original assembly (e.g.
AB) or on a test rig with different dynamic properties (e.g. AR). This paper shows examples of both variants.
The in-situ source characterisation method comprises three steps. Here, it is discussed for the test rig variant; for more explana-
tion of the notation and terminology used, see [1].
1. Operational measurement of the source structure A mounted to a test rig R where indicator responses u4on test rig R are
measured (see the test setup in figure 3b);
2. FRF measurement of the combined structure AR, more specifically from force inputs at the interface f2to the indicator
responses on the test rig u4. Here it is key that the DoFs u4are the same set as with the operational measurement;
3. Characterisation of the active source by means of a matrix-inverse operation, resulting in blocked forces for each opera-
tional measurement cycle:
feq
2=YAR
42 +u4(4)
where feq
2denotes the blocked forces representing the source structure, u4the measured operational responses of step 1
and YAR
42 the FRFs of the source on test rig measured in step 2.
Note that if a virtual point transformation to a 6-DoF description is done on the force input side of the FRF matrix (i.e. the columns
of Y42 relate to forces and moments in virtual point format), the resulting blocked forces feq
2will also present themselves in this
form, making them easily transferable to other structures. In other words, one would obtain a source characterisation comprising
3 forces and 3 moments per coupling point, instead of a series of only translational forces.
The source vibrations of the active structure A have been characterised in the original ‘target’ assembly with passive side B
and on the test rig structure R. Hence, the two in-situ characterisations yield two sets of 12 blocked forces/moments: 6 for each
coupling point. These sets are used for vibration prediction in target assembly AB, which is presented in the next section.
3.4 Component-based Transfer path analysis
For the purpose of virtual noise and vibration prediction (sometimes called Virtual Acoustic Prototyping, [32]), component
Transfer Path Analysis is applied on the benchmark data. The advantage of component TPA is the ability to predict target
response levels u3in/on a passive structure B using an independent source characterisation (i.e. blocked forces) of a source A.
0 500 1000 1500 2000 2500 3000
Frequency (Hz)
10-5
10-4
10-3
10-2
10-1
100
101
Acceleration (m/s 2)
Validation on AB
In-situ on AB, TPA using measured AB
(a) In-situ on AB, TPA prediction using measured FRFs of AB.
0 500 1000 1500 2000 2500 3000
Frequency (Hz)
10-5
10-4
10-3
10-2
10-1
100
101
Acceleration (m/s 2)
Validation on AB
In-situ on AB, TPA using substructured AB
(b) In-situ on AB, TPA prediction using substructured FRFs of AB.
0 500 1000 1500 2000 2500 3000
Frequency (Hz)
10-5
10-4
10-3
10-2
10-1
100
101
Acceleration (m/s 2)
Validation on AB
In-situ on AR, TPA using measured AB
(c) In-situ on AR, TPA prediction using measured FRFs of AB.
0 500 1000 1500 2000 2500 3000
Frequency (Hz)
10-5
10-4
10-3
10-2
10-1
100
101
Acceleration (m/s 2)
Validation on AB
In-situ on AR, TPA using substructured AB
(d) In-situ on AR, TPA prediction using substructured FRFs of AB.
Figure 8: Four results of in-situ source characterisation and TPA prediction (blue/red/yellow/green), all compared to a validation
measurement in structure AB (black).
The governing equation of component TPA is as follows:
u3=YAB
32 feq
2(5)
where feq
2is a set of equivalent/blocked forces for instance obtained by equation (4) and YAB
32 the FRF matrix of assembly AB
from the interfaces to the responses of interest u3(sometimes called Noise Transfer Functions). Combining this with Dynamic
Substructuring, one can predict the vibration levels as described above without ever physically assembling structures A and B
[16]. In practice, this allows for a separation in development process, since two parties are able to work on their own structures
A and B and an interface is provided in terms of blocked forces, possibly in the format of Virtual Point forces and moments.
Four results of component TPA are shown in figure 8, namely for source characterisation in AB (top) and AR (bottom) and for
component TPA using a measured set of FRFs of AB (left) and substructured FRFs obtained by coupling of A and B (right). A
validation response measurement is added for comparison, which is identical for all four plots. Remind that all characterisations
are expressed in virtual point blocked forces and moments, i.e. 12 DoFs in total.
The first result in figure 8a constitutes the most literal application of in-situ TPA: the source is characterised in the original
assembly AB, after which the virtual point blocked forces and moments are applied to the same assembly. Near identical
results are obtained, especially at the peaks corresponding to the actual signal (source vibration orders) of the stepper motor.
Figure 8b shows an application of the same blocked forces to the substructured FRFs of assembly AB. Most peaks are well
approximated, which may be considered a very good result considering the various substructure FRF measurements involved
(A, B and AB). Note that this approach goes into the direction of virtual vibration prototyping, which heavily relies on the virtual
point transformation to provide common interfaces between the various measurements.
The results of figure 8c and 8d present similar results as above, yet for source characterisations calculated from operational
measurements on the test rig. The imposed challenge here is that the test rig structure R possesses very different dynamic
properties than B, resulting in totally different operational indicator responses (uAR
4) than in the original assembly (uAB
4). It is
thus interesting to investigate if a source characterisation can be determined that renders similar responses on another passive
side, i.e. is transferable to arbitrary assemblies.
Figure 8c on the left depicts the test rig characterisation applied to the measured FRF of AB. The results are encouraging, as
many peaks that exceed the signal noise floor find roughly the right order of magnitude. The noise floor, indeed, has been
a limiting factor in this measurement, as the signal on the test rig sensors hardly exceeded the noise level. Finally, figure 8d
shows what might be considered the holy grail of component TPA: a characterisation of source vibrations on a test rig, applied
to an experimental model of the target assembly obtained using dynamic substructuring. This indeed constitutes a novelty in
experimental DS and TPA, and shall be a direction for further investigation.
4 CONCLUSIONS & OUTLOOK
In this paper, a series of benchmark substructures has been presented for method development and validation in the field of
DS, TPA and SC. The three benchmark substructures can be connected in several ways, which makes the complexity of the
interface problem adjustable to a one-point, two-point or continuous connection. Several applications have been shown to
validate methods of experimental modelling, virtual point transformation, dynamic substructuring and source characterisation.
Many more validations can be done, which is topic of further research at VIBES.technology.
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... The Virtual Point Transformation is extended in various ways to account for additional local flexibility not captured by a single 6-DOF node. The usage and effectiveness of these techniques is compared and validated using a well-known benchmark structure [5]. In section 2 the theory of the standard Virtual Point Transformation is covered and an extension to flexible IDMs is proposed. ...
... This section demonstrates practical applications of frequency based substructuring (FBS) using the the virtual point transformation with six degrees of freedom, the virtual point transformation extended with flexible modes and making use of a transmission simulator [4]. For the test-cases, we will make use of structures that have been previously used for experimental substructuring and source characterisation method validation [5]. In summary, there are three structures: substructure A, substructure B and the transmission simulator TS; see figure 2. In this paper, focus lies on comparing different strategies for modelling the interface. ...
Conference Paper
Full-text available
In this contribution, Frequency Based Substructuring is applied to a well-known benchmark structure with special attention to correct modelling of the interface problem. Various approaches with the Virtual Point Transformation are used to incorporate the effect of interface flexibility that is present in the clamped connections. An extension of the rigid displacement basis with flexible interface displacement modes is proposed and evaluated. Furthermore, a practical approach using a Transmission Simulator is used in order to improve the quality of the interface description.
... The Virtual Point Transformation is extended in various ways to account for additional local flexibility not captured by a single 6-DOF node. The usage and effectiveness of these techniques is compared and validated using a well-known benchmark structure [5]. In section 2 the theory of the standard Virtual Point Transformation is covered and an extension to flexible IDMs is proposed. ...
... This section demonstrates practical applications of frequency based substructuring (FBS) using the the virtual point transformation with six degrees of freedom, the virtual point transformation extended with flexible modes and making use of a transmission simulator [4]. For the test-cases, we will make use of structures that have been previously used for experimental substructuring and source characterisation method validation [5]. In summary, there are three structures: substructure A, substructure B and the transmission simulator TS; see figure 2. In this paper, focus lies on comparing different strategies for modelling the interface. ...
... This consists in two subsystems: an 'A'-shaped component and an 'S'-shaped component. The outer dimensions are 307 × 186 × 40 mm for A and 258 × 368 × 20 mm for B. The design concept is similar to what presented in [32]. The two components are machined from a single aluminum block and provided with several circular holes to guarantee a large variety of connectivity and boundary conditions. ...
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Substructure decoupling is the process of identifying the dynamic behavior of one component by removing the dynamic influence of the second component from the assembled system. In experimental practice, several techniques have been developed to address the decoupling problem. In this context, measurements errors of random and systematic nature remain a major hindrance to a successful implementation of the methodology. For this reason, approaches such as extended interface, Virtual Point Transformation and truncated Singular Value Decomposition are commonly adopted on top of a standard interface decoupling procedure. This paper introduces the Singular Vector Transformation. The idea is to weaken the interface problem by using the Singular Value Decomposition to extract reduction spaces directly from the measured dynamics. A least square smoothing minimizes random errors and outliers, thereby improving the conditioning of the interface matrix inversion. No geometrical or analytical model is required. The reduction basis are frequency-dependent and can include flexible interface behavior, if properly controlled and observed. Further understanding and interpretation of the interface problem in frequency-based decoupling is provided. Numerical and experimental examples show the potential of the proposed technique in comparison with state-of-the-art approaches.
Chapter
There are several ways to formulate the dynamics of a substructure. The different domains in which the dynamics can be described will be reviewed since the manner in which substructures are characterized will later determine the substructuring methodology that can be applied. In addition to how the substructures are formulated, the way in which the coupling/decoupling problem is expressed will allow us in the subsequent chapters to develop different numerical and experimental techniques. Two conditions must be satisfied on the interface between substructures: a condition on the displacement field (compatibility) and on the interface stresses (force equilibrium). Those conditions can be accounted for following several different formulations, all mathematically equivalent, but each leading to different numerical methods, experimental approaches, and approximation techniques, as will be explained in the following chapters. In this chapter, we outline the basic concepts of the so-called three-field formulation, dual and primal assembly.—Chapter Author: Daniel Rixen
Chapter
“Divide and Conquer” is a paradigm that helped Julius Cesar to dominate on the wide Roman Empire. The power of dividing systems, then analyze them as parts before combining them in an assembly, is also an approach often followed in science and engineering. In this introductory chapter, we shortly discuss the main idea behind domain decomposition and substructuring applied to mechanical systems. — Chapter Author: Daniel Rixen
Chapter
This chapter reviews common nonlinearities that are encountered in engineering structures, with a particular emphasis on geometric nonlinearity. Popular ways to construct reduced order models for geometrically nonlinear problems are discussed. The concept of nonlinear normal modes is presented to help understand the dynamics of these structures, and some recently presented substructuring methods are reviewed. —Chapter Authors: Matt Allen & Paolo Tiso.
Thesis
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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. 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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.
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This paper presents a reformulation of dynamic substructuring for vibrating structural systems as a feedback control problem. Frequency based substructuring (FBS) using admittance coupling of two substructures is shown to be mathematically equivalent to a feedback control system with the primary substructure acting as the controller and the secondary substructure acting as the plant. This formulation can be used to perform time-domain simulations of dynamic substructuring problems using MATLAB's Simulink environment, whereby the primary substructure can be modeled by three possible approaches: 1) Laplace domain transfer functions, 2) state-space models, or 3) finite impulse response functions. The secondary substructure can be represented by a variety of Simulink blocks including nonlinear elements. By inserting an actuator transfer system between the two substructures, this formulation also provides the basis for real-time hybrid substructuring (RTHS) for coupling numerical and physical substructures as a cyber-physical system similar to hardware-in-the-loop testing. In typical RTHS the numerical substructure acts as the controller and the actuator with physical substructure acts as the plant. This feedback control formulation will lead to further advancements for both dynamic substructuring and RTHS by adapting methods from classical and modern feedback control theory.
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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.
Chapter
In the design of mechanical systems, there are constraints imposed on the vibration of mechanical equipment to limit the vibration transmission into its support structure. To accurately predict the coupled system response, it is important to capture the coupled interaction of the two portions, i.e., the mechanical equipment and the support structure, of the mechanical system. Typically during a design, the analysis of the full mechanical system is not possible because a large part of the system may be non-existent. Existing methods known as Transfer Path Analysis and Frequency Based Substructuring are techniques for predicting the coupled response of vibrating mechanical systems. In this paper, a control based hybrid substructuring approach to Transfer Path Analysis is proposed. By recognizing the similarities between feedback control and dynamic substructuring, this paper demonstrates that this approach can accurately predict the coupled dynamic system response of multiple substructured systems including operating mechanical equipment with a complex vibration source. The main advantage of this method is that it uses blocked force measurements in the form of a power spectral density matrix measured uncoupled from the rest of the system. This substructuring method is demonstrated using a simplified case study comprised of a two-stage vibration isolation system and excited by operating mechanical equipment.
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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.
Article
The transmission simulator method of experimental dynamic substructuring has the capability to couple substructures with continuous connections. A hardware example with continuous connections is presented in which the method is used to couple an experimental substructure with a finite element substructure to predict full system response. The predicted response is compared with frequency response functions measured on the full system hardware. The experimental substructure captures the motion of a component packed in foam. This is coupled to a finite element model of a cylindrical metal case which contains the foam and is attached through a flange to a plate and beam structure.