Approximate description of structure-borne sound sources

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AIA-DAGA 2013 Merano
Approximate description of structure-borne sound sources with
continuous line-shaped contact interfaces
Sebastian Mathiowetz1 , Hannes A. Bonhoff
Institute of Fluid Mechanics and Engineering Acoustics, Technische Universität Berlin, Germany
1
Email: [email protected]
Introduction
A majority of research concerning the characterization
of structure-borne sound sources focuses on sources with
multiple discrete contact points. Large efforts have
been made in finding simplified methods for source
characterization and the calculation of the power injected
into adjacent structures. However, there is a lack of
studies that investigate the applicability of such methods
to source structures with continuous line-shaped contact
interfaces by means of discretization. In this work the
accuracy of the single equivalent approximation [1] and
the interface mobility method [2] to be expected with
regard to power transmission at continuous line-shaped
contact interfaces is examined. Both methods have
their advantages with respect to simplicity or physical
insight, making each approach inherently better suited
for different use. The study is based on simulated as
well as experimentally collected data from two sources
with generic line-shaped contact interfaces connected
continuously to plate-like receiver structures. The study
is confined to the vertical component of motion.
of the interface and with the maximum frequency to be
resolved, the problem is generally more severe than in the
case of point-connected contact interfaces. Additionally,
the identification of dominating contributions to the
power transmission is hindered as only one single power
spectrum is obtained. Particularly for practitioners
like consultant engineers or manufacturers, a matrix
description is hardly to manage and a description in
terms of singular values is favored.
Recently, the approach of single equivalent approximation was presented [1]. Here, the concept of effective
mobilities [4] was generalized by applying a spatial
averaging over the contact points to both the effective
mobilities and the activity of the source. Source and
receiver mobility matrices are represented by singular
equivalent source and receiver mobilities YS,eq and YR,eq ,
respectively, whereas the source activity can be given
as the sum of squared
free velocity magnitudes at the
N
2
|v
contact points
n=1 SF,n | . The calculation of the
transmitted power thereby is rigorously simplified and
given by
Theory
The assessment of the structure-borne sound transmission from an active source into a passive receiver connected continuously to each other generally requires an
integral formulation. An integral formulation, however,
would assume that integral descriptions of mobilities and
free velocities are available which is usually not the case
in practice. To circumvent this problem, the contact
interface can be discretized. Discretizing the interface at
N sampling points and introducing source and receiver
mobility matrices YS and YR as well as the source
activity in terms of a free velocity vector vSF leads to
the well known matrix formulation for the transmitted
power, as shown in [3], for instance,
W =
1
T
T
∗ −1 ∗
Re(vSF
(YS +YR )−T YR
(YS∗ +YR
) vSF ). (1)
2
In Eq. (1), superscripts −1 indicate a matrix inversion
and ∗ denotes the complex conjugate. Provided that the
spatial Nyquist criterion is fulfilled, that is the distance
between sampling points has to be smaller than half the
minimum wavelength of interest, the complex power at
the discretized interface equals that of the continuous
interface. The matrix formulation can be problematic
as it involves matrix inversions. As the size of mobility
matrices of discretized interfaces increases with the size
Weq
N
Re(YR,eq )
1
.
=
|vSF,n |2 2 n=1
|YR,eq + YS,eq |
(2)
In order to derive the effective mobilities, assumptions
about the force ratios between the contact points have
to be made. At low frequencies where the source is
supposed to be acting in bouncing mode, force ratios
with unit magnitude and a zero phase difference can
be assumed [4], [5]. At higher frequencies where the
structure is controlled by resonant behavior, force ratios
with unit magnitude and random phase distribution are
likely to be applicable [4], [5]. Thus, the approximate
character of the method is due to the spatial averaging
of data over the contact points as well as due to the
applied force ratio assumption. The method is tested
for both force ratio assumptions in this study.
The interface mobility method [2] offers a complementary
approach where the vibrational behavior at the interface
is split up into a sum of orders p. Each order can
be associated with a certain component of motion as
it reveals the phase difference between two adjacent
contact points such that Δφ = 2π · p/N . For p = 0 all
contact points move in phase, so it can be associated
with the bouncing mode of the source, for instance.
Interface mobilities of source and receiver are derived
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AIA-DAGA 2013 Merano
as the Fourier coefficients of the respective ordinary
mobility matrices. From a two-dimensional Fourier
transformation of mobility matrices, equal- and crossorder interface mobilities are derived [2]. By neglecting
the latter, only equal-order interface mobilities Ŷp−p,S
and Ŷp−p,R remain. Along with the Fourier transform
of the free velocity, v̂p,SF , a singular value description
is obtained for each power order Wp and the complete
power is retained by the sum of all power orders:
installations, source and receiver mobilities show the
same order of magnitude, i.e. matched conditions.
N/2−1
WIF =
WIF,p
(3)
p=−N/2
with
WIF,p =
1 |v̂p,SF |2 Re(Ŷp−p,R )
.
2 |Ŷp−p,S + Ŷp−p,R |2
Cross-order interface mobilities account for a variation of
ordinary point mobilities along the main diagonal of the
mobility matrix and for variations of transfer mobilities
with an equidistant ”spacing” in the mobility matrix, e.g.
Y12 , Y23 , Y31 in a three-point interface [2]. With regard
to continuous line-shaped contact interfaces, cross-order
interface mobilities account for a non-circular interface
contour, an interrupted interface contour, as well as
for discontinuities of source and receiver structures.
Their importance with regard to continuous line-shaped
contact interfaces is tested in this study.
Test setup
In this work, structure-borne sound power transmission
from two source-receiver installations is examined. First,
a laboratory source structure with a circular interface
(r = 0.16 m) that is driven by a shaker with pink
noise is considered (see Fig. 1 (a), top). Initially, the
source is assumed to be connected to a rectangular
simply supported chipboard plate (h = 8 mm) at a nonsymmetric, off-diagonal position. Source and receiver
mobilities as well as the free velocities were measured
in the dissembled state at N = 24 sampling points
(see Fig. 1 (a), bottom). This allows the vibrational
behavior at the interface to be discretized up to about
5, 000 Hz. Subsequently, the circular source is assumed
to be connected to a corresponding infinite plate.
In a second setup, the power transmission from a fan unit
source is considered (see Fig. 1 (b), top). The flangelike footing structure of the source is made from three
thin bent metal sheets that are arranged perpendicular
to each other. The source mobility and free velocity were
measured at N = 27 sampling points (see Fig. 1 (b),
bottom). In conjunction with a thin infinite steel plate
(h = 10 mm) assumed as receiver, the interface can be
discretized up to about 3, 500 Hz. However, due to some
yet unsolved problems in the higher frequency region,
results are analyzed up to 1, 000 Hz and calculated from
N = 15 sampling points. Mobilities of the infinite
plate receivers were derived by calculation. In both
Figure 1: (a) Laboratory source with circular interface,
(b) Fan unit source with flange-like rectangular interface
•: sampling points
Results
Laboratory source with circular interface
In order to study the accuracy of the approximate
methods with respect to the circular interface source
(see Fig. 1 (a)), the transmitted power was calculated
by means of Eqs. (2) and (3) and compared with the
transmitted power calculated by Eq. (1). For the case
that the source is connected to a simply supported plate,
results are shown in Fig. 2. In the mass-like region, that
is up to about 90 Hz, the single equivalent approximation
with zero phase assumption Weq,zero is in good agreement
with the matrix calculation. Above, the deviations become larger. From 160 Hz onwards, the single equivalent
approximation with random phase assumption Weq,rand
yields an overall agreement of ±5 dB or less. The
interface mobility method WIF yields a good agreement
throughout the frequency range, but deviations of about
7 dB can be observed at two distinct frequencies. In
summary, for both methods the largest deviations are
observed in the transition region between mass-controlled
and resonance-controlled frequency region of the source,
that is from 100 Hz to 250 Hz. Here, pronounced
variations of point and transfer mobilities appear due to
the onset of resonances and anti-resonances in the source
structure that are not accounted for when mobilities are
averaged over the interface or when cross-order interface
mobilities are neglected.
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10
10
10
W / WMatrix(dB)
(b)
-6
Weq, zero
WIF
Weq, rand
(a)
10
10
W (Nm/s)
10
WMatrix
-7
-8
10
10
10
-9
63
125
250
500
f (Hz)
1k
2k
10
4k
(b)
15
10
W / WMatrix(dB)
W (Nm/s)
(a) 10-5
5
0
-5
-10
-15
63
125
250
500
f (Hz)
1k
2k
-3
WMatrix
Weq, zero
-4
WIF
Weq, rand
-5
-6
-7
-8
125
250
500
f (Hz)
1k
2k
63
125
250
500
f (Hz)
1k
2k
15
10
5
0
-5
-10
-15
4k
63
4k
Figure 2: (a) Transmitted power from circular source on
simply supported plate, (b) Normalized transmitted power
Figure 4: (a) Transmitted power from circular source on
infinite plate, (b) Normalized transmitted power
The vibrational behavior at the interface can be analyzed
in more detail by using the interface mobility method.
In Fig. 3, the first three power orders WIF,0 , WIF,1 and
WIF,2 are compared with the sum of all power orders
WIF for the frequency range from 50 Hz to 1, 000 Hz.
Here, corresponding positive and negative power orders
have been summed up, that is W1 = W−1 + W1 and
W2 = W−2 + W2 .
contact points becomes important. Accordingly, the
structural wavelength matches half the circumference of
the source in this case. From about 250 Hz onwards, the
motion components occur less distinctive.
10
W (Nm/s)
10
10
10
10
10
WIF
-5
-6
WIF, 0
WIF, 1
WIF, 2
-7
-8
-9
-10
10
2
10
3
f (Hz)
Figure 3: Power orders for circular source on simply
supported plate
The zero order that represents the bouncing motion
component where all sampling points move in phase is
seen to be dominant mainly in the mass-like region of
the source. The first orders describe motion components
where adjacent sampling points exhibit a phase difference
of Δφ = ±2π · 1/24. Here, the structural wavelength
matches the circumference of the source. Positive and
negative first order can superimpose and result in a
rocking motion. The first order motion is seen to be
most dominant where the source behaves mass-like and
in the transition region between mass-like and resonancecontrolled frequency region of the source. Above, the
second order motion component that describes a phase
difference of Δφ = ±2π · 2/24 between two adjacent
When the circular source is assumed to be connected
to a corresponding infinite plate for which the receiver
mobilities were derived by calculation, the deviations
shown in Fig. 2 (b) are expected to be reduced. As the
interface is rotationally symmetric, there is no variation
in point and transfer mobility of the receiver when
moving along the interface with a constant distance
between the sampling points. In this case, cross-order
interface mobilities of the receiver are equal to zero.
Equivalently, spatial averaging of mobilities over the
contact points does not introduce errors. Results are
shown in Fig. 4. It can be seen that the deviations in the
transition region between mass-controlled and resonancecontrolled frequency region of the source remain, even if
reduced. The zero phase assumption, Weq,zero , leads to
large errors when applied to frequencies above the masslike region. As the infinite plate mobility is independent
of location, the vibrational behavior at the interface
and thus the force distribution can be assumed to be
predominantly determined by the source [1].
Fan unit source
In this section, the installation of a fan unit source (see
Fig. 1 (b)) connected to an infinite steel plate receiver is
considered. In Fig. 5, the transmitted power calculated
by means of Eqs. (2) and (3) is compared with the
transmitted power calculated by Eq. (1). Below the
main excitation frequency of 50 Hz, the deviation of
the single equivalent approximation is largest. The
zero phase assumption Weq,zero yields deviations in a
range of ±5 dB throughout the rest of the spectrum.
The random phase assumption Weq,rand yields larger
deviations in the higher frequency region. The interface
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mobility method overestimates the transmitted power at
most parts of the spectrum with a maximum deviation
of about 8 dB at 63 Hz.
W (Nm/s)
(a) 102
10
10
10
10
W / WMatrix(dB)
(b)
WMatrix
0
Weq, rand
Weq, zero
-2
WIF
-4
-6
63
125
250
f (Hz)
500
1k
63
125
250
f (Hz)
500
1k
15
10
5
0
-5
-10
-15
Figure 5: (a) Transmitted power from fan unit source on
infinite plate, (b) Normalized transmitted power
In Fig. 6, the first three power orders are compared
with the sum of all power orders WIF for the frequency
range from 40 Hz to 200 Hz. Except for the main
excitation frequency of the electric motor at 50 Hz and its
harmonics, the single power orders cannot be identified
as dominant in certain frequency regions. This might
be due to the structure of the source whose dynamic
behavior is less distinctive with regard to resonances and
mode shapes compared to that of the circular source. On
the other hand, the frequency region between the main
excitation frequency and its harmonics can be seen as a
more or less random ”noise floor” to the peaks in the
spectrum, thus it is of minor importance with regard to
power transmission.
10
10
W (Nm/s)
interfaces was tested in the example of two sourcereceiver installations. In summary, both methods reveal
a similar degree of accuracy on average. The input data
have been collected by discretizing source and receiver
structures which results in large mobility matrices. The
calculation of reference values involves matrix inversions,
thus the shown deviations between approximate methods
and reference values might also result from uncertainties
in the reference values to some extent. With regard
to prediction, the approach of the single equivalent
approximation is generally more appealing as it allows
the transmission process to be described by singular
values of source and receiver. The interface mobility
method, in contrast, allows to obtain some physical
insight in the transmission process, but requires larger
data sets to be analyzed. The results presented are based
on small datasets compared to the variety of conditions
that can be found in practice. The study should be
extended to other generic interface geometries in order
to yield results on a more general basis.
10
10
10
10
10
2
WIF
0
WIF, 0
-2
WIF, 1
-4
WIF, 2
-6
-8
-10
10
f (Hz)
2
Figure 6: Power orders for fan unit source on infinite plate
Concluding remarks
Acknowledgments
The authors would like to thank Dr. Andreas Mayr from
the University of Applied Sciences Rosenheim, Germany,
for collaborative discussions on the work presented in this
paper. The financial support received from the German
Federal Ministry of Economics and Technology - BMWi
through grant 03SX305G is gratefully acknowledged.
References
[1] Mayr, A.R., Gibbs, B.M.:
Single equivalent
approximation for multiple contact structure-borne
sound sources in buildings. Acta Acustica united with
Acustica 98 (2012), 402-410
[2] Bonhoff, H.A., Eslami, A.: Interface mobilities for
characterization of structure-borne sound sources
with multi-point coupling. Acta Acustica united with
Acustica 98 (2012), 384-391
[3] Bonhoff, H.A.: The influence and significance of
cross-order terms in interface mobilities for structureborne sound source characterization (Ph.D.). Universitätsverlag der TU Berlin, Berlin, 2010
[4] Petersson, B.A.T., Plunt, J.: On effective mobilities
in the prediction of structure-borne sound transmission between a source structure and a receiver
structure, part 1: Theoretical background and basic
experimental studies. Journal of Sound and Vibration
82 (1982), 517-529
[5] Fulford, R.A., Gibbs, B.M.: Structure-borne sound
power and source characterization of structure-borne
sound sources in multi-point-connected systems, part
1: Case studies for assumed force distributions.
Journal of Sound and Vibration 204 (1997) 659-677
In this paper, the applicability of the single equivalent approximation as well as the interface mobility
method with regard to continuous line-shaped contact
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