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An indicator sensor criterion for in-situ characterisation of source vibrations

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Component-based Transfer Path Analysis allows us to analyse and predict vibration propagation between an active source and passive receiver structures. The forces that characterise the active source are determined using sensors placed on the connected passive substructure. These source characterisation forces, often called blocked or equivalent forces, are an inherent and unique property of the source, allowing to predict vibration levels in assemblies with different connected passive structures. In order to obtain a unique and accurate characterisation, accurate measurements are of key importance. The success of the characterisation is not only dependent on the hammer skill of the experimentalist, but also relates to sensor placement, overdetermination and matrix conditioning. In this paper the effects of each of these influences are studied using theoretical approaches, numerical studies and measurements on a benchmark structure designed for in-situ source characterisation. An assembly of two substructures is tested, representing an active substructure with a source and a passive substructure. In order to determine a criterion for the placement of indicator sensors, the effect of the various influences on the in-situ characterisation is compared. Using the results, a structured approach for the use of indicator sensors for in-situ blocked force TPA is proposed.
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An indicator sensor criterion for in-situ characterisation of source
vibrations
M. W. F. Wernsena,b, M. V. van der Seijsb, D. de Klerka,b
aDelft University of Technology, Department of Precision and Microsystems Engineering
Mekelweg 2, 2628CD, Delft, The Netherlands
bVIBES.technology
Molengraaffsingel 14, 2629 JD, Delft, The Netherlands
ABSTRACT
Component-based Transfer Path Analysis allows us to analyse and predict vibration propagation
between an active source and passive receiver structures. The forces that characterise the active source
are determined using sensors placed on the connected passive substructure. These source characterisation
forces, often called blocked or equivalent forces, are an inherent and unique property of the source, allowing
to predict vibration levels in assemblies with different connected passive structures. In order to obtain a
unique and accurate characterisation, accurate measurements are of key importance. The success of the
characterisation is not only dependent on the hammer skill of the experimentalist, but also relates to sensor
placement, overdetermination and matrix conditioning. In this paper the effects of each of these influences
are studied using theoretical approaches, numerical studies and measurements on a benchmark structure
designed for in-situ source characterisation. An assembly of two substructures is tested, representing
an active substructure with a source and a passive substructure. In order to determine a criterion for
the placement of indicator sensors, the effect of the various influences on the in-situ characterisation is
compared. Using the results, a structured approach for the use of indicator sensors for in-situ blocked
force TPA is proposed.
Keywords: transfer path analysis, dynamic substructuring, source characterisation, blocked force, in-situ, indicator
sensors.
1 INTRODUCTION
Knowledge of the vibrations of a system is essential in designing mechanically and acoustically sound products.
With the current trend of modular design, in which different companies design specific parts for the final product,
predicting the dynamic and acoustic qualities of the final product in an early design stage becomes more difficult.
A useful tool to assess these qualities is Transfer Path Analysis (TPA). Using TPA, a vibrationally active source
can be represented by a set of forces, relating to a certain dynamic load case. This characterisation can be used by
engineers to calculate the dynamic response of the assembled system and track down the critical paths of vibration
transmission.
There are many different flavours of TPA [1] and the ones most suited to solve source characterisation problems are
from the category component-based TPA [2, 3, 4]. The useful property of component-based TPA methods is that
the source can be characterised by forces that are inherent to the source structure only. Any assembly of this source
structure with a passive structure can be simulated for, without the need to do a new operational measurement on
the source. In practice this means that a company can characterise their active source structure on an in-house test
bench and use this characterisation to predict sound and vibrations in the assembled product, or put more general:
a characterisation of source A done in an assembly AB is also valid for use with any other passive side B.
The forces that characterise a source are often called equiva lent or blocked forces, as they represent blocking forces
that would be generated if the source structure was mounted to a rigid boundary. Yet a more practical method to
obtain them is by measuring vibration responses in an assembly using acceleration sensors on the passive side. In a
second step, Frequency Response Functions (FRFs) are obtained using for instance impact hammer measurements
on the assembly. A matrix-inverse procedure is performed afterwards to determine the forces that exactly represent
those vibrations. This approach is popularly known as in-situ characterisation [5], and is particularly effective for
characterisation in the original assembly (hence the name ‘in-situ’).
Combining concepts of TPA with the knowledge of Dynamic Substructuring (DS) [6] has led to a component TPA
approach in which the source is characterised using forces and moments in a virtual point (VP) [7]. The virtual point
has the advantage that it allows for easy coupling between two substructures, taking into account rotational coupling
(and thus moments in the coupling points) as well. Implementing DS and VP technology gives rise to a modern
approach for TPA [1], in which substructures can easily be coupled, source characterisations can be exchanged and
in which numerical and experimental models can be merged to create hybrid simulations.
To ensure that a characterisation truly represents the source excitation and is property of solely the active substruc-
ture, the experimental procedures should meet certain requirements. Apart from the impact hammer skills of the
experimentalist (which should be outstanding), the results of the characterisation highly depend on choices regarding
sensor placement, overdetermination and matrix conditioning.
Some theoretical methods have been proposed that give an indication of the ideal location to place sensors for
force identification. Examples are methods based on conditioning such as the composite conditioning number [8], or
methods based on energy and system modes [9]. However, almost all of these methods require an accurate numerical
model of the structure. Small errors of the model will have a large influence on the results and the results are specific
for a certain structure and load case. As a numerical model often does not suffice to truly represent the experimental
structure for moderate to high frequencies (typically above 500 Hz), such methods will not provide a suitable solution.
Hence, a structured approach is needed for the use of indicator sensors that does not require a numerical model.
Instead, by answering rudimentary questions such as where to place the sensors with respect to the excitation, one
should be able to choose a set of indicator sensor locations that will yield the best possible characterisation.
Paper outline
In this paper the influence of sensor placement, overdetermination and matrix conditioning is investigated using
experimental measurements on a benchmark structure, substantiated by insights from numerical models. Section
2 of the paper will present the theory behind TPA and the methods used to perform source characterisation. In
section 3 the problems are identified that influence the quality of a characterisation using experimental data. Section
4 introduces the experimental structure used for this paper and presents the results of the measurements on this
structure, identifying the effects of varying sensor locations and overdetermination on the quality of the source
characterisation.
2 THEORY
2.1 Component-based TPA
Figure 1 shows the presence of two forces: the unknown and unmeasurable source excitation, represented by f1,
and the equivalent forces that are used to characterise the source, represented by feq
2. Furthermore, two response
f1
u2
u3
u4
YAB
42
YAB
32
feq
2
Figure 1: An overview of the nodes and DoFs in assembly AB.
locations are shown, namely the responses of interest at the structure B, u3, and the indicator sensor responses u4
used for the in-situ characterisation of source f1, also located on structure B.
The fundamental difference between component-based TPA and other types of TPA is that the forces feq
2obtained
from the characterisation are a property of substructure A only, and thus not depending on structure B. A physical
interpretation can therefore be that when the equivalent forces are applied in the opposite direction with the source
in operation, no responses should be present onward from u2. If the source is now deactivated, the response u3for
the assembled system AB, due to application of feq
2, should be equal to the original responses caused by the active
source. Using the assembled FRF matrix YAB, with elements YAB
ij representing the response at degree of freedom
(DoF) idue to a unit force excitation at DoF j, this relation can be expressed as follows:
u3(ω) = YAB
32 (ω)feq
2(ω)(1)
For the sake of notation all equations may be assumed to be in the frequency domain, unless otherwise stated, and
ωis therefore emitted from the coming equations.
Expanding the assembled admittance using the substructures’ admittances gives:
u3=YAB
32 feq
2=hYB
32(YA
22 +YB
22)1YA
22ifeq
2(2)
As the response at u3due to application of feq
2and f1should be equal, the following relation for the forces can be
deduced:
YB
32(YA
22 +YB
22)1YA
22feq
2=YB
32(YA
22 +YB
22)YA
21f1
=feq
2= (YA
22)1YA
21f1(3)
Equation (3) shows that equivalent forces are indeed a property of substructure A.
The most practical method to determine the equivalent forces is by means of in-situ characterisation [5, 10]. The
in-situ method makes use of indicator sensors on substructure B in order to determine the equivalent forces:
feq
2= (YAB
42 )1u4(4)
This method is easy to apply, as one only has to place sensors on structure B, determine YAB
42 , and activate the source.
Depending on the chosen amount of indicator sensors, the equivalent forces are solved either using a standard inverse
or a pseudo-inverse. In other words, if the inverse problem is overdetermined (more u4than f2) the equivalent forces
are solved by minimising the the sum of the squared error in u4:
feq
2= (YAB
42 )+u4=feq
2=arg min ||u4YAB
42 f2|| (5)
2.2 Equivalent vs. blocked force
In the theory above, the term equivalent forces is used. In literature, one often finds the denotation ‘blocked’ force,
which refers to the blocking effect that these forces have on the active source when applied in opposite direction:
u4=YAB
41 f1YAB
42 feq
2=0
In other words, feq
2represent the reaction forces if the source structure A were connected to a rigid boundary
[3, 11, 12]. This indeed explains why the word “blocked” is commonly used, however it still leaves some room for
interpretation. As stated before, the number of indicator DoFs is typically larger than the number of DoFs of the
interface. This number is in turn bounded by a maximum of 6 per coupling point (3 translations and 3 rotations),
assuming that the structures are rather stiff in the area where they interconnect. It is known that different set
of forces can be found that represent the vibrations of the source that also function as ‘equivalent’ forces (see for
instance [13]). In this study we use 6-DoF virtual point forces, which ensures that all translational forces as well
as rotational moments are available for the characterisation. This way, if the equivalent forces are defined on the
basis of these 6-DoF-per-point sets, it is fair to say that these forces are indeed blocking the interface in all possible
directions. Therefore, we will continue with referring to feq
2by blocked forces.
3 SOURCE CHARACTERISATION
Although equation (5) gives the impression that determining blocked forces is a straight-forward job, many difficulties
arise when trying to correctly solve the inverse problem when using experimental data. Many of the problems
concerning the calculation of blocked forces comes down to the effect of measurement noise and the inversion of
YAB
42 , as will be explained further in the coming section.
3.1 Numerical vs. experimental characterisation
In the simplified world of numerical models no noise will be present, and thus solving for the blocked forces will
be easy and correct. Linear algebra tells us that the only requirement for solving equation (5) is that the matrix
YAB
42 is full rank. In order to achieve a full-rank YAB
42 , the amount of indicator sensors u4should be larger or equal
to the number of forces that have to be identified and the columns of YAB
42 should be linearly independent. Linear
independent columns mean that each force of feq
2gives a linearly independent response at u4. When looking at a
single frequency bin for which equation (5) is solved, this means that in order to get a full-rank matrix YAB
42 , ‘enough’
dynamics should be present. Theoretically, this means that the amount of eigenmodes of the system participating
at this frequency bin should be higher than the amount of forces feq
2used for the characterisation. If the amount of
modes participating at a frequency bin is too small, a linear dependence will exist between the columns of YAB
42 , as
simply not enough dynamic information is available to create linearly independent responses.
For a numerical model this criterion should not pose a problem as each mode of the system will have some (albeit
very small) contribution at each frequency bin. This means that having a numerical model with neigenmodes
allows you to solve equation (5) for nblocked forces, as long as the amount of sensors is equal or larger than n. As
numerical models often consist of thousands of DoFs, more than enough dynamic information is available and solving
for blocked forces is a straightforward process.
A real-life structure theoretically has an infinite amount of dynamics, but the presence of noise in the measurement
results in much of the dynamics being unmeasurable. In order to show the effect that noise has on the resulting
characterisation, a theoretical expansion of the measured data in a noise and signal part is analysed in the following
section.
3.2 Blocked force noise
Let us consider the measurements and computational steps used to perform a source characterisation. Firstly an FRF
measurement is required to determine the FRF matrix of transfer paths YAB
42 . Secondly an operational measurement
is required in which the responses at the indicator DoFs u4is measured for a certain load case of the source. Noise
has an effect on both of these measurements, however the effect of noise on YAB
42 is outside the scope of this paper1.
Let us consider a set of operational data umeas
4that is corrupted with sensor noise. This means that the measured
umeas
4is a combination of both the true usignal
4belonging to the blocked forces feq
2and the sensor noise unoise
4.
umeas
4=usignal
4+unoise
4(6)
Using these measured responses umeas
4, the blocked forces fmeas
2are calculated:
fmeas
2= (YAB
42 )+umeas
4= (YAB
42 )+(usignal
4+unoise
4)(7)
By assuming the physical world acts as a linear system, the following expansion can be made:
fmeas
2= (YAB
42 )+usignal
4+ (YAB
42 )+unoise
4(8)
fmeas
2=feq
2+fnoise
2(9)
Equation (9) shows how the calculated blocked forces fmeas
2using a noisy measurement are simply built up from a
part corresponding to the true blocked forces feq
2and a noise part fnoise
2which we shall refer to as blocked force noise.
The blocked force noise corresponds to the blocked forces that are calculated using a noise measurement on the
sensors u4, i. e. with the source deactivated. Using this noise measurement, the blocked force noise can be calculated
by solving equation (3).
To achieve that the calculated blocked forces are equal to the true blocked forces (fmeas
2=feq
2), two possibilities with
a different physical interpretation are considered:
1. Increase the blocked force feq
2with respect to the blocked force noise fnoise
2
2. Reduce the blocked force noise fnoise
2with respect to the blocked force feq
2
The first case describes a situation in which the blocked forces are much larger than the blocked force noise. A
physical interpretation for this is that the source (f1) is exciting the structure in such a manner that high blocked
forces are required, thus minimising the influence of the blocked force noise on the results. The excitation levels of
the source are however not tunable, as it is simply a property of the load case. As maximising feq
2is not possible (and
probably the thing you are trying to prevent when doing TPA measurements), an alternative is to instead minimise
the blocked force noise, as is defined by case two. The minimisation of these blocked forces will be dealt with in the
next section.
3.3 Minimising blocked force noise
Minimising the blocked force noise comes down to minimising the solution of (YAB
42 )+unoise
4. The outcome of this
inverse operation can be minimised by either minimising the amount of sensor noise unoise
4or by wisely choosing
your measurement setup that defines the FRF matrix YAB
42 . As the sensor noise relates to the total noise picked up
by the sensor, cabling, DAQ system etcetera, proper sensor selection in combination with minimising electrostatic
interference is the main influence an experimentalist has on this noise level. However, the experimentalist does have
a certain influence on YAB
42 . The DoFs for the blocked forces are prescribed by the virtual point (i. e. the 3 forces
and 3 moments centred in the coupling point), but depending on the selection of the amount of sensors and their
location, YAB
42 will have certain properties which will be discussed next.
1The effect of noise on the measurement of YAB
42 is probably of minor importance, due to the fact that higher responses at the sensors
can be generated using an impact or shaker measurement. This renders the influence of noise on the FRF measurement negligible.
3.3.1 Matrix conditioning and singular values
Looking in more detail at how YAB
42 is built up, reveals which parameters mostly influence the magnitude of the
blocked force noise. Two aspects that are often used to analyse the properties of a matrix, especially when dealing
with inverse problems, are the condition number and the singular value decomposition (SVD).
The condition number shows the amount of linear dependence that exists between the columns of matrix YAB
42 . A
high condition number indicates that there is a high linear dependence between the columns of YAB
42 . It can thus be
understood as some of the blocked forces showing a similar response at the indicator sensors u4, making it difficult to
observe the difference between an excitation by feq
2,i and a second excitation feq
2,j . As a result, if one wants to identify
these blocked forces using responses at u4, a small error in u4may lead to a large amplification in the blocked forces,
as will be explained next. A high condition number is a good indication that a problem exists, but it does not explain
precisely what is happening.
To get a better understanding, one can use a singular value decomposition of YAB
42 . The singular value decomposition
can be written as follows:
YAB
42 =UΣVT(10)
with Ubeing the matrix with left singular vectors, Vthe matrix with right singular vectors and Σthe matrix with
the singular values on the diagonal. The pseudo-inverse of YAB
42 can now be written as follows:
(YAB
42 )+=VΣ+UT(11)
Expressing fnoise
2using this SVD expansion gives:
fnoise
2= (YAB
42 )+unoise
4=VΣ+UTunoise
4(12)
To see the effect that a single singular value has on the level of blocked force noise, unoise
4is assumed to be equal
for all sensors DoF’s of u4. Using equation (12) it can be reasoned that the magnitude of a singular value σi,Σii
determines how much of the noise measured on the space spanned by Uiis amplified to the forces that excite this
displacement space. If σihas a very low value, its inverse in equation (12) will be large, meaning that a high blocked
force noise can be expected for the forces that excite the displacement space belonging to σi. The smallest singular
value of Y42 belongs to the displacement space that is least measured by the indicator sensors, and for the largest
the opposite is true. A low singular value can be expected when two elements from feq
2have a very similar response
at u4or when a force from feq
2has a very low contribution to the excitation of the structure, in both cases rendering
the forces badly observable using u4.
As a result of this, a good indication of the magnitude of the blocked force noise are the magnitudes of the smallest
singular values of YAB
42 . In general, a higher condition number indicates the presence of a low singular value, resulting
in an increase of the total blocked force noise. Therefore, reducing the condition number also reduces the magnitude
of the blocked force noise.
3.3.2 Blocked force signal-to-noise ratio
In order to get an indication of the quality of the calculated blocked forces, use can be made of a blocked force noise
ratio. Rewriting equation (9) into a ratio gives:
feq
2,i
fnoise
2,i
=
fmeas
2,i
fnoise
2,i
1(13)
This calculation will show for each element iof the blocked forces a ratio between the useful information and noise.
This ratio can easily be calculated using the calculated blocked forces and the blocked forces obtained from a noise
measurement. A fraction of 0 will mean that the calculated blocked forces are solely the blocked force noise, and a
value of infinite will tell you that the calculated blocked forces are the true blocked forces. Everything in between
shows the ratio of blocked force signal-to-noise present in calculating the blocked forces.
3.4 Practical implementation
Concluding from the previous section there are two rudimentary choices that influence the linear dependence and
are a direct choice of the experimentalist: the sensor location with respect to the blocked forces and the amount of
sensors used. The influence of both of these will be discussed in the following section.
3.4.1 Sensor distance
An empirically determined rule of thumb is that placing sensors closer to the forces that one wants to identify increases
the linear independence, and thus decreases the condition number of the matrix YAB
42 that has to be inverted [14].
The linear dependence increases as the sensors are moved further away from the location where the forces are applied
due to a "blurring" effect present in the FRF. Intuitively this make sense: If you want to measure a difference between
two sources that are placed close together, gut feeling tells you to measure close to these sources. When you are
measuring further away, both sources will be much more difficult to distinguish. Theoretically this effect of blurring
can also be explained by the reduced amount of anti-resonances when one moves further away from the excitation
point [15, 16].
The physical reason behind this reduction in anti-resonances can be understood by looking at the two extreme cases,
namely the driving point FRF (Yii), and a transfer point FRF (Yji) far away from the excitation point. To calculate
the admittance Yji , the participation of each mode is added to the total, and the modal sign of each added mode will
influence the final result. An example of the admittance calculated from two modes of a 2-DoF mass-spring system
without damping is shown in equation (14) and visualized by figure 2.
Magnitude
180
90
0
90
180
Frequency (Hz)
Angle (deg)
Magnitude
180
90
0
90
180
Frequency (Hz)
Figure 2: Accelerance plot of a) Driving point FRF Yii, in which an antiresonance is visble at the point where both
modal contributions (dashed) have the same magnitude b) Transfer FRF Yji, in which an minimum is visible at the
location where both modal contribution have the same magnitude.
Yii =xi,1xi,1
ω2
1ω2+xi,2xi,2
ω2
2ω2(14a)
Yji =xj,1xi,1
ω2
1ω2xj,2xi,2
ω2
2ω2(14b)
In these equations xi,n is the eigenvector of mode nat index iand ωnis the eigenfrequency of mode n. The
modal sign of a mode is determined by the sign of the eigenmode at both the receiving and exciting DoF. When
two consecutive modes have the same modal sign, an antiresonance will be present as can be seen in the left plot
of figure 2. The yellow and red dotted lines represent the modal participation of mode 1 and 2, and at the point
where they have the same magnitude they exactly cancel each other out, as can be reasoned using equation (14a).
When the two modes have a different modal sign there will just be a minimum, and no antiresonance, as the two
modal participations are simply added. The sign of the modes is determined by the value of the eigenvector at both
the exciting and receiving node. This means that at a driving point there will always be an anti-resonance visible
(figure 2a) as the square of the eigenvector value will always be positive. For a transfer FRF this is however not
necessarily true. Depending on the location, modal contributions may be positive or negative, resulting in an FRF
with sometimes an antiresonance and sometimes a minimum (figure 2b). Statistically it can be shown that when one
moves the receiving and exciting point further apart, the chances of two consecutive modes differing in sign increases
[16], resulting in what we observe as a ’blurred’ FRF.
In practice this means that when the indicator sensors u4are moved further away from the blocked forces, less
difference can be observed between the different blocked forces. As a result, the indicator responses due to different
components of feq
2will show a higher resemblance. Due to the higher resemblance, the value of the lowest singular
value will decrease, increasing the conditioning of the matrix YAB
42 , in turn increasing the blocked force noise level.
3.4.2 Numerical example
Using a numerical model of a beam consisting of 1600 nodes the effect that sensor distance has on the conditioning of
the matrix YAB
42 is visualised. The conditioning of YAB
42 is evaluated for in total 6 different sensor groups, consisting
of 12 DoF each. For this numerical example five forces have to be determined, placed on two nodes in the x, y and
z-direction. The results of this numerical study are shown in figure 3. The left figure shows the condition number of
YAB
42 for the different sensor-sets. An increase in the condition number of a factor 1×103can be observed for the
position at the far end of the beam compared to the position close to the forces. The norm of the blocked force noise
for the different sensor-sets is also calculated for a noise input of 1 m s1, and plotted in the right figure of figure 3.
The norm of the blocked force noise varies with a factor 1×104, with the sensors closest to the forces performing the
best. This numerical example indeed confirms the theory that sensors closer tot the blocked forces show the least
amount of blocked force noise.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
100
101
102
103
104
105
106
107
0 200 400 600 800 1000 1200 1400 1600 1800 2000
102
103
104
105
106
107
108
109
Figure 3: Conditioning of Y42 (left) and norm of the blocked force noise (right) for a numerical beam model for
different sensor sets (colours are linked between graph and diagram).
An interesting observation is that this effect is only visible when a significant linear dependence exists between the
columns of the matrix YAB
42 , i. e. when some of the components of feq
2show a similar response at u4. When the same
numerical case is evaluated with a subset of the forces of feq
2, no difference in conditioning and blocked force noise
norm is found for the different sensor distances. This logically makes sense, as forces that excite the structure in a
similar manner will look more and more similar when moving further away from them, due to the blurring effect.
Forces that show a distinctly different response will be much less influenced by the blurring effect.
3.4.3 Sensor quantity
If an experimentalist is lucky enough to have plenty of sensors available for his/her measurement, it is a possibility
to use a set of indicator sensors that is larger than the amount of forces that have to be identified, resulting in a
overdetermined problem. The effect of overdetermination however depends on whether new dynamic information
can be measured by the extra sensors, thus decreasing the condition number of the matrix YAB
42 , and minimising the
blocked force noise. One would expect to observe more and more information by the addition of each sensor, until a
certain point at which all possible information is observed. This is further addressed in section 4.
3.4.4 Matrix regularisation
Matrix regularisation is an often used tool to improve the results of the inversion problem. Many types of regu-
larisation methods exists, with the two most common being truncated singular value decomposition and Tikhonov
regularisation [9]. All regularisation methods however come down to rejecting or minimising the solution belonging
to the smallest singular value or a set of smallest singular values. In an optimal case, the error due to the discarding
of the lowest singular value will be much smaller than the error due to the sensitivity to the other errors [17].
As discussed in the theory, the smallest singular value of Y42 belongs to the displacement space that is least measured
by the indicator sensors. It can be reasoned that rejecting this displacement space from the solution, will result in a
characterisation that is build up without the part of the blocked forces that is responsible for the excitation of this
displacement mode. If one is purely interested in the blocked forces that are of the greatest influence on the assembly
used for the characterisation, regularisation could help out to get a better conditioning of the inverse problem, but
due to the fact that a certain set of blocked forces is being left out of the solution, it may render the blocked forces
useless to be used as a unique set to describe the source for any other combination than AB.
4 EXPERIMENTAL CASE STUDY
In the coming section the theory that has been discussed in the previous section will be applied to an experimental
case study. A graphical render of the structure that was constructed for this specific task is visible in figure 4.
Figure 4: Artist impression of substructure A and B used for the experimental measurements, including the stepper
motor that functions as the vibration source.
4.1 Measurement setup
The structure used for the measurements consists of a substructure A, having the shape of an A-frame, and a
structure B being a plate-like structure. Structure A and B can be coupled using one or two coupling points, by
simply rotating the A-frame (the single point coupling is shown in the figure). A vibration source is used in the
form of a NEMA 17 stepper motor, often used in the 3D printing industry. This stepper motor is controlled by an
Arduino DUE with a Pololu A4988 stepper motor driver using the PWM protocol. A Müller-BBM PAK MKII was
used for data acquisition in combination with 11 tri-axial PCB 356B21 sensors. The stepper motor PWM signal was
connected to a tacho-pulse input channel of the DAQ system, allowing for easy speed monitoring and order tracking
of the stepper motor. All measurements were done in a free-floating environment.
Two types of measurements have been performed on structure AB: 1) impact hammer measurements to determine
the admittance FRFs YAB
42 from the virtual point to the receiving indicator sensors and 2) operational measurements
with the stepper motor running for various load cases to determine u4. The operational measurements consisted
of the stepper motor rotating at various constant speeds, doing a continuous sweep and lastly a discrete sweep,
in which the motor speed is increased with 5% every 3 seconds. The discrete sweep is especially useful for source
characterisation, as it provides many loadcases with a constant source excitation.
4.2 Bottleneck effect
In order to show that indeed a virtual point can be used to describe the coupling between the substructures, the
presence of the bottleneck effect is analysed using the complex mode indicator function (CMIF) [18] of the FRF
matrix. The bottleneck effect is a consequence of the rigidity assumption of the interface, which is what the virtual
point concept is based on [1]. It can be understood that a maximum of 6 different modes can be visible in the
connecting substructure B, independent of the amount of modes being excited in substructure A, due to the fact
that they are connected with a 6-DoF interface. If this bottleneck effect is indeed present, the choice for a virtual
point with 6 DoFs for the blocked forces is made legitimate.
0k0.5k1k1.5k2k2.5k3k3.5k4k4.5k5k
103
100
103
Frequency (Hz)
Figure 5: CMIF of YAB
42 for 15 indicator DoFs u4as a result of 16 excitations f2at the A-side.
A CMIF is calculated for 24 response DoFs on substructure B to 16 excitations on substructure A. Figure 5 shows
that up to approximately 1000 Hz the responses on substructure B are dominated by a maximum of 6 modes per
frequency bin, clearly confirming the presence of the bottleneck effect. Onwards from 1000 Hz the amount of modes
present at each frequency bin in the system increases as the interface starts to show more flexibility, and thus acts
less as a bottleneck. Theses results thus show that a 6 DoF virtual point can be used for the blocked forces.
4.3 Typical BF-TPA application
4.3.1 Dominance of motor orders
A first look at the operational results show that the stepper motor excites the structure at specific orders, and
is therefore not a broadband source. Using the continuous sweep of the stepper motor, a waterfall and Campbell
diagram (including order tracking) can be constructed. A typical waterfall and Campbell diagram of a sensor on the
B-side of the assembly AB is shown in figure 6.
Both diagrams show that the response of the sensor is almost completely built up using the orders of the stepper
motor. As the stepper orders are dominant in the sensor responses, the same would be expected for the blocked
forces.
0k1k2k3k4k5k
50
100
150
Frequency [Hz]
Time [s]
0k1k2k3k4k5k
2
4
6
Frequency [Hz]
Stepper speed [Rps]
103
102
101
Acceleration m/s2
Figure 6: Waterfall diagram (left) and Campbell diagram (right) of u3response in z-direction
4.3.2 Blocked force characterisation
Using the FRF measurement of YAB
42 and an operational measurement, a source characterisation for one of the many
load cases of the stepper motor can be performed using the theory described. Using the measured u4responses
and the FRF matrix YAB
42 , equation 5 is solved, giving a set of blocked forces that characterise the loadcase. In
addition, using a noise measurement in which the source is turned off, the blocked force noise can be determined by
again solving equation (5) with u4being unoise
4. A typical result of such a characterisation and blocked force noise
calculation that were done using a set of 5 sensors to determine 6 blocked forces is shown in figure 7.
104
102
100
Force [N]
0.5k1k1.5k2k2.5k3k3.5k4k4.5k5k
180
200
220
240
260
Frequency [Hz]
Time [s]
Figure 7: Typical characterisation result of a load case. Top plot: calculated feq
2component in z-direction for a
load case (blue) and measured blocked force noise (red), including stepper orders (dashed), bottom plot: waterfall
diagram of feq
2in z-direction for different loadcases with increasing stepper speeds over time.
The top plot of figure 7 shows the results for a component of the calculated blocked forces (fmeas
2), namely the blocked
force acting in the z-direction of the virtual point (blue). In addition, also the blocked force noise that was calculated
for this component is visualized in the same plot (red). The plot shows that indeed the characterisation can be
interpreted as being a combination of the blocked force noise and a blocked force part, as was stated by equation
(9). The bottom plot shows a waterfall diagram of the same blocked force component as was portrayed in the top
plot. The waterfall diagram is build up of in total 32 loadcases over time. The waterfall diagram clearly visualizes
how the blocked force noise dominates the characterisation at all of the characterisations and thus minimizing the
effect of the blocked force noise would certainly be an advantage for the quality of a characterisation.
4.3.3 Response reconstruction
Using a characterisation such as the one that was shown in the previous chapter, one can reproduce the response at a
reference DoF u3using the transfer path YAB
32 . Performing this reconstruction and comparing it with the measured
u3will give an indication if the characterisation is indeed capable of predicting the dynamic response at a location
at structure B. Figure 8 shows a typical result for such a u3reconstruction in combination with a measured u3.
0k0.5k1k1.5k2k2.5k3k3.5k4k4.5k5k
104
102
Frequency [Hz]
Acceleration [m/s2]
Figure 8: Typical reconstructed u3(blue) and measured u3(red)
The reconstructed and measured response are almost identical, which shows that the characterisation is capable of
predicting the response at points in the structure that were not used to calculate the characterisation.
4.4 Sensor placement
To study the effect that sensor placement has on the magnitude of the blocked force noise and thus overall quality
of the characterisation, two configurations of the experimental structure are analysed. The first configuration that
is analysed is the single point coupling as can be seen in the left image of figure 9. Secondly also a two-coupling
configuration is analysed which is shown in the right image of figure 9.
u4sensor set 1 u4sensor set 2
u4sensor set 3 u4sensor set 4
u4sensor set 5 u3sensors
Figure 9: Sensor location groups. Left: single-point coupling configuration; right: two-point coupling configuration.
4.4.1 Single-point coupling
To evaluate the influence of the distance of the sensors on the quality of the characterisation using the single-
point coupling configuration, five sensor-sets are evaluated at different locations on the structure B. Each sensor-set
consists of five tri-axial sensors, which should result in a thoroughly overdetermined matrix YAB
42 (15 ×6). The
different sensor-set locations are shown in the left image of 9.
In order to compare the different sensor-sets, a noise and impact measurement are performed for each sensor-set,
allowing one to calculate the blocked force noise in a similar manner as was done in the previous section. In order
to compare the calculated blocked force noise of the different sensor-sets, use is made of the norm of blocked force
noise, as this will give a clear indication of the total magnitude of the noise level. As the source is characterised
using forces and moments, they will be compared separately.
104
102
100
Frequency [Hz
||feq
2,noise,F ||
0k0.5k1k1.5k2k2.5k3k3.5k4k4.5k5k
104
101
102
||feq
2,noise,M ||
u4sensor set 1
u4sensor set 2
u4sensor set 3
u4sensor set 4
u4sensor set 5
Figure 10: Comparison of blocked force noise norm (top) and moment norm (bottom) for different sensor distances
in the single-point coupling configuration.
Figure 10 shows a comparison of the norm of the blocked force noise, done separately for the forces and moments.
Sensor-set 1 corresponds to the sensors placed closest to the blocked forces and sensor-set 5 to the set placed furthest
away. Due to the fact that no clear difference is observed between the different sensor locations it can not be
concluded whether placing sensor closer to the blocked forces is of any advantage, and that indeed the blurring effect
described in section 3 is applicable to experimental data. A minor improvement of the blocked force noise moments
can be seen for sensors placed close to the blocked forces, but for the forces shown in the top figure the opposite can
be concluded. A possible reason for the absence of any difference between the sensor location can be found in the
results of the numerical beam analysis. The numerical analysis already showed that characterisation problems that
have easily distinguishable blocked forces are almost not affected by sensor distance, as even at a large distance the
forces are still distinguishable. As use is made of a virtual point for the blocked forces, the matrix Y42 shows a high
amount of linear independence, as the six forces and moments excite the structure in linear independent manners.
In order to show that indeed placing sensors close to the blocked forces is of an advantage, the characterisation has
to be made more interesting by making use of a two-point coupling of structure AB as was shown in the right image
of figure 9.
4.4.2 Two-point coupling
In order to analyse the two-point coupling, a similar measurement and calculation procedure was followed as for the
single-point coupling. Again the norm of all the blocked force noise forces and moments is used as an indication of
the quality of the characterisation.
Figure 11 shows the results for this analysis. Both the results for the forces and moments show that sensor set
1, which is placed closest to the virtual points, has a blocked force noise norm which is a factor 10 smaller than
the other sensor sets. This means that indeed the sensor set placed closest to the blocked forces has the lowest
amount of blocked force noise acting on the results, as was predicted in section 3. Even more, all of the sensor-sets
except set 1 generate a blocked force noise level that is in the same order of magnitude as the blocked force itself,
rendering the characterisation useless. An example of this is shown in figure 12, which shows the blocked forces of
103
101
Frequency [Hz
||feq
2,noise,F ||
0k0.5k1k1.5k2k2.5k3k3.5k4k4.5k5k
101
101
||feq
2,noise,M ||
u4sensor set 1
u4sensor set 2
u4sensor set 3
u4sensor set 4
u4sensor set 5
Figure 11: Comparison of blocked force noise (top) and moment (bottom) norm for different sensor distances in the
two-point coupling configuration.
the right coupling point in the z-direction determined with sensor-set 1 and 5. In the waterfall diagram shown in
the middle plot (sensor-set 1), clear peaks are visible for each of the loadcases, corresponding to the orders of the
stepper. The waterfall diagram of the bottom plot corresponds to the same blocked force component, but this time
the characterisation is done using sensor-set 5. This waterfall diagram shows that the order peaks that were clearly
visible using sensor-set 1 are now masked by the blocked force noise. A zoomed-in load case from the waterfall
diagrams is shown in the top plot, and shows the characterisation for the same specific loadcase done by both sensor-
set 1 and sensor-set 5. This plot clearly visualizes that the blocked force noise of sensor-set 5 is higher than the
blocked forces that represent the characterisation of the source.
104
102
100
Force [N]
180
200
220
240
260
Time [s]
104
102
100
Force [N]
0.2k0.4k0.6k0.8k1k1.2k1.4k1.6k1.8k2k2.2k2.4k2.6k2.8k3k
180
200
220
240
260
Frequency [Hz]
Time [s]
104
102
100
Force [N]
Figure 12: Comparison of characterisation feq
2in z-direction. Waterfall diagram of sensor set 1 is shown in the middle
plot, and for sensor set 5 in the bottom plot. A zoomed in loadcase is presented in the top plot, showing the result
for sensor-set 1 (red) and sensor-set 5 (red).
4.5 Sensor overdetermination
4.5.1 Blocked forces
In order to study the effect of overdetermination of YAB
42 , the characterisation with sensor-set 1 is done with five,
four, three and two sensors per virtual point for the single-point coupling. To eliminate the effect of the chosen sensor
combinations, an average is taken of the n
5possibilities that exist.
0k0.5k1k1.5k2k2.5k3k3.5k4k4.5k5k
102
101
100
101
Frequency [Hz]
||fbfn,f||
2u4sensors
3u4sensors
4u4sensors
5u4sensors
0k0.5k1k1.5k2k2.5k3k3.5k4k4.5k5k
102
101
100
101
Frequency [Hz]
||fbfn,m||
Figure 13: Effect of overdetermination on the norm of the blocked force noise for the forces (top) and moments
(bottom), normalized to the results using two sensors
Figure 13 shows the norm of the blocked force/moment noise for different sensor amounts, normalised to the results
for a set with two sensors. The first observation is that indeed, increasing the amount of sensors increases the quality
of the result. The largest difference for the level of the blocked force noise is seen for the addition of one extra
sensor, and every other additional sensor does improve the result, but not as significant as the first sensor. The
question however remains if the sensor combinations are also able to correctly characterise a source when the number
of sensors decrease, which is analysed in the next section.
4.5.2 Response reconstruction
A comparison of the reconstructed u3using different amounts of sensor numbers is shown in figure 14. Firstly it can
be observed that indeed the peaks are of equal magnitude of all the reconstructed u3responses, meaning that indeed
the stepper orders are equally reconstructed by each amount of sensors. The second observation is that a lot more
noise is present in the reconstruction using the minimal amount of 2 sensors, which is what the results of figure 13
already showed us. Comparing the magnitude of the source characterisation with the noise level shows that a valid
characterisation is not possible using just two sensors.
0k0.5k1k1.5k2k2.5k3k
104
101
102
Accelerance [m/s2]
0k0.5k1k1.5k2k2.5k3k
104
101
102
Accelerance [m/s2]
2u4sensors
3u4sensors
4u4sensors
5u4sensors
Figure 14: Typical reconstructed u3result with different number of sensors used.
5 CONCLUSION
This paper introduces a structured approach for the use of indicator sensors for the characterisation of a dynamic
source using in-situ blocked force TPA. Using a combination of both theory, numerical testing,experimental mea-
surements and the introduction of the blocked force noise, results have been found that show that sensor placement
and sensor quantity have a large influence on the quality of the characterisation.
To minimise the blocked force noise and maximise the quality of the characterisation, the indicator sensors must
be placed close to the forces that one wants to identify, and at least one additional sensor must be used for the
overdetermination of the inverse problem. Although this paper mainly focusses on in-situ blocked force TPA, the
insights obtained can also be applied to the matrix inverse method or any other type of force identification methods,
in which the same type of inverse problem is solved.
6 OUTLOOK
As for now, a structured approach is proposed which helps an experimentalist in deciding where to place the indicator
sensors. Many future possibilities however exist, which can further improve the source characterisation procedure.
Quantifying the quality of a characterisation using a ratio such as the blocked forces signal to noise ratio that was
proposed in the theory section, would make for a good first step.
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