ss32_2the influence of contact force distribution on structure

The influence of force distribution on structureborne sound power sources in buildings
Barry Gibbs, Carl Hopkins
Acoustics Research Unit, School of Architecture, University of Liverpool, UK
Summary
Structure-borne sound power injected by a mechanical installation with multiple contacts into a
structure is influenced by the force distribution and phase differences between these contacts. Simplified methods to determine the injected power usually involve band-average values (typically in
one-third octave bands), where the phase information is lost. For simplified methods, which involve
average values over the contacts, the magnitude of each contact force and the phase difference between them also are lost. The effects of these forms of simplification are investigated by comparing
the approximate estimates with directly measured values for a range of source-receiver systems typical of building installations.
1
Introduction
To assess the contribution of a vibrating mechanical installation, to the sound pressure in a building, the structure-borne sound power at the installation location is required [1]. For the general
case, three quantities are required in some form,
for the structure-borne power. The first is source
activity: either the measured free velocity of the
freely suspended source, under otherwise normal
operating conditions, or the measured blocked
force, obtained when attached to a rigid supporting structure. The second is the source mobility
(or the inverse, impedance). The third is the receiver mobility (or the inverse, impedance). The
three quantities can be obtained using precision or
engineering methods. In this paper, the term engineering method means that the three quantities are
real-valued and represent frequency band averages, e.g. in one-third octave bands, and each of the
three quantities is expressed as an equivalent single value.
In order to address these requirements for sources,
a reception plate method (RPM) has been proposed, where the machine under test is attached to
a plate system and operated under otherwise normal conditions [2]. The total power injected by
the source, through all contacts with the plate, is
calculated from the plate parameters as [3]:
Psource   M v 2
(1)
The mean-square plate velocity v 2 is obtained
using accelerometers distributed over the plate
surface. The loss factor  of the plate of mass M is
obtained by the decay method [4]. Alternatively,
the plate can be calibrated with a shaker and inline force transducer to give the ratio of the
known input power to mean-square plate velocity
[5]. Replacing the shaker with the source under
test, then gives:
Psource  vsource Pcal / vcal
2
2
(2)
If the reception plate is thick, such that the source
mobility is much higher than the plate mobility,
then the source can be characterised by a single
2
value Fbeq
, related to the sum of square blocked
N
forces over the contacts
F
2
bi
[6]. In this case,
i
the source power into a thick plate of known mobility is:
FORUM ACUSTICUM 2014
7–12 September, Krakow
2
Psource  Fbeq
Re(YRe q )
Gibbs, Hopkins:
Force distribution of structure-borne sound power sources in buildings
(3)
2
The equivalent value Fbeq
is useful for predicting
installed power into structures of low mobility,
such as in heavyweight buildings.
However, this approach to characterizing multiple
contact sources with single values pre-supposes
that the contact forces are independent of each
other. This is likely to be the case at high frequencies, and for contacts which are distant from each
other, with respect to the bending wavelength on
the receiver plate. At frequencies and plate thicknesses where the bending wavelength is large,
then the contact forces interact and the sumsquare values are not obtained indirectly by the
reception plate method. Indeed, the force relationships will be strongly dependent on the installation condition and therefore difficult to predict.
Therefore, an estimate is required of the lower
frequency limit to the independent force assumption and the expected uncertainties below this
frequency limit.
The approach, described in this paper, is to first
consider simple force distributions on idealized
plates. Then secondly, to use measured source
data (free velocity and source mobility) and
measured plate data (receiver mobility). The target quantity is the sum-square blocked force,
since this is an independent source descriptor.
2. Force distributions on idealized plates
Yc 
1
8 Bm
(4)
B’ is the plate bending stiffness and m’ is the mass
per unit area. Termed the characteristic mobility
[3], it is real-valued and frequency invariant.
(a)
50 cm
(b)
30 cm
40 cm
(c)
70 cm
130 cm
100 cm
Figure 1: Force distributions of source in buildings
In order to include transfer mobility terms, reference is made to the Hankel function of second
kind, which is the solution to the bending wave
equation of a thin plate of infinite extent [3]. The
transfer mobility, between points i and j, is given
in terms of the characteristic mobility, as:
Yij  Yc H 0( 2) kb rij 
(5)
Figure 1 shows force distributions corresponding
to a range of common installation geometries: (a)
two point contacts, found for example in water
pipe connections to walls; (b) four point contacts,
corresponding to, for example, mounts between
medium-size fan units and floors; (c) six point
contacts, corresponding to larger machines on
floors. The contact dimensions in Figure 1 are
arbitrary but typical.
kb is the plate bending wave number and rij is the
distance between the ith and jth contact points.
The complex power at the ith contact, due to the
force Fi is:
For idealized receiver plate structures, consider
infinite thin plates with point mobility, given by:
The plate velocity vi results from the product of
force Fi and mobility Yii at the contact, and the
product of forces Fj at the other contacts and the
associated transfer mobility terms Yij:
Pi  Fi*vi
(6)
FORUM ACUSTICUM 2014
7–12 September, Krakow
Gibbs, Hopkins:
Force distribution of structure-borne sound power sources in buildings
N
vi  FiYii   F jYij
(7)
j i
rapid conversion; the low mobility case (green
line) shows the slowest conversion.
8
Ptrans  Re( Ptotal )  N F Yc  F Yc  H 0( 2) kb rij 
2
2
N
(8)
j i
7
POWERHI2
POWERMID2
6
POWERLO2
5
NORMALISED POWER
If the forces are assumed to be of equal magnitude and the transfer mobility terms are given by
(5), then the total transmitted power, through N
contacts, is given by the real part of the total
complex power:
4
3
2
1
If the forces are independent, then the total power is given by the first term of the RHS of (8).
Using this to normalize the total transmitted
power, to include interaction between forces,
gives:
N

1
Pnorm  1  Re H 0( 2) kb rij 
N  j i

(9)
The normalized transmitted power therefore is a
function of the sum of the complex interaction
terms in (9). In turn, the interaction terms are a
function of distance between contacts and bending wavenumber of the plate.
0
-1
-2
1
10
100
1000
10000
FREQUENCY Hz
Figure 2: Normalized power for two equal in-phase
forces
In Figure 3 is shown the case of four forces, distributed as shown in Figure 1(b), where phase
effects are included by considering forces in
phase (F, F, F, F) and out-of-phase (F, -F, F, -F)
and (F, F, -F, -F).
8
Powerhicase1
In Figure 2 are shown the normalized transmitted
power for two equal in-phase forces (see Figure
1(a)) on high-, mid- and low-mobility receivers.
There is the expected low-frequency asymptotic
value of 2 (3 dB), with interference effects and
convergence to unity at high frequency, where
the forces can be assumed to be independent.
In order that equivalent single values of the
source can be used for different receiver mobility
conditions, we require fast convergence to unity.
The high mobility case (blue line), where the
bending wave number is high, shows the most
Powerhicase2
Powerhicase3
Powermidcase1
6
Powermidcase2
Powermidcase3
Powerlowcase1
5
Powerlowcase2
NORMALISED POWER
The wavenumbers can be assigned values corresponding to the characteristic mobility of representative building plate elements: high mobility
(10-3m/sN) for lightweight elements such as plasterboard panels; mid-mobility (10-4m/sN) for
lightweight elements, at the timber-frame connection with plasterboard panels; low mobility
(10-5m/sN) for heavyweight elements, such as
concrete floors and masonry walls.
7
Powerlowcase3
4
Unity
3
2
1
0
-1
-2
1
10
100
1000
10000
FREQUENCY Hz
Figure 3: Normalized power for four forces of
equal magnitude, in and out of phase
The in-phase forces have a low-frequency asymptote of 4 (6 dB), with convergence with increased frequency to unity. The out-of-phase
forces have a low-frequency asymptote of zero.
In general, there is more rapid convergence than
for the two-force case, due to more complicated
geometry and thus increased randomizing effects. The asymptotes indicate the likely discrepancy at low frequency, between RPM estimates
FORUM ACUSTICUM 2014
7–12 September, Krakow
Gibbs, Hopkins:
Force distribution of structure-borne sound power sources in buildings
2
of single equivalent values Fbeq
, and the sumN
square blocked force
F
2
bi
recorded and the reception plate powers calculated
according to (2).
. Figure 4 shows
i
results for six equal in-phase forces distributed
as shown in Figure 1(c). There is greater complexity and greater distances between contacts
than for the previous cases and interaction effects
can be neglected above 100 Hz, for all receiver
mobilities.
8
7
POWER HI MOBILITY
POWER MID MOBILITY
6
POWER LOW MOBILITY
NORMALISED POWER
5
4
3
2
1
0
-1
-2
1
10
100
1000
10000
FREQUENCY Hz
Figure 4: Normalized power for six in-phase forces of
equal magnitude
The implication of these results is that equivalent
single quantities, obtained by reception plate
methods, will result in uncertainties in predictions
of transmitted power into other plate structures
with different governing bending wavelengths.
The loss of phase information is significant at low
frequencies and for low mobility plates, and introduces errors into the estimates of the sumsquare blocked force, which is the important independent source quantity, required for prediction
of installed power. This problem was explored
further by measurements of real machines on a
real plate.
3. Real sources on a low mobility plate
The sources were a compact air pump and a small
fan unit which in turn were attached to an aluminium plate of dimensions 2.12m x 1.50m and
thickness 20mm (Figure 5). The plate was supported on visco-elastic pads at corners and midedge which approximated a free boundary condition for the plate. With the sources in operation,
the mean square plate response velocities were
Figure 5: Pump (above) and fan (below) on free 20mm
aluminium plate
In Figure 6 are the point mobility at the contacts
at the sources and receiver plate. At frequencies
between 200 Hz and 400 Hz, the pump mobility
equals or is less than the plate mobility, but otherwise the plate provides a low value of receiver
mobility, relative to that of the sources, over the
frequency range of interest.
2
Re-arranging equation (3) gives Fbeq
in terms of
the RPM power Ptrans . Ideally, this value should
approximate the sum-square blocked force over
N
the contacts
F
2
bi
. The equivalent single receiv-
i
er plate mobility YR eq can be calculated several
ways [7]. The simplest is to average the measured
FORUM ACUSTICUM 2014
7–12 September, Krakow
Gibbs, Hopkins:
Force distribution of structure-borne sound power sources in buildings
point mobility of the plate over the contacts and
this was used in this study.
and point mobility at the four pump contacts,
where:
Fbi 
1.0E+0
Y11mag
Y22mag
v fi
(10)
Yi
Y33mag
1.0E-1
Y44mag
1.0E+2
1.0E-2
1.0E+1
1.0E-3
1.0E+0
1.0E-1
FORCE SQUARED Nsq
1.0E-4
1.0E-5
1.0E-2
1.0E-3
1.0E-4
1.0E-6
10
100
1000
10000
1.0E-5
1.0E+0
1.0E-6
Y11mag
Y22mag
Y33mag
1.0E-1
1.0E-7
Y44mag
40
400
4000
FREQUENCY Hz
1.0E-2
Figure 7: Sum-square blocked force of pump:
solid line, calculated from free velocity and source
mobility; dashed lines, RPM estimates at two positions on 20mm plate
1.0E-3
1.0E-4
1.0E-5
The RPM estimate is unreliable below 100 Hz
1.0E-6
10
100
1000
N
10000
and the calculated value of
1.0E+0
pmp1Y2
pmp1Y4
pmp2Y1
pmp2Y2
pmp2Y4
1.0E-3
1.0E-4
1.0E-5
1.0E-6
10
100
1000
10000
Figure 6: Point mobility of pump (top), fan (middle)
and 20mm plate (bottom). Horizontal log-frequency
scale 10 -10,000 Hz; Vertical log-mobility scale 10-6 –
100 m/sN.
Figure 7 shows the calculated sum-square blocked
N
force
F
2
bi
2
beq for
and RPM estimate F
exceeds the
nerated by the pump in rigid-body rocking motion. The agreement is more promising above
400 Hz.
pmp2Y3
1.0E-2
2
bi
2
estimates of Fbeq
due to out-of-phase forces, ge-
pmp1Y3
1.0E-1
F
i
pmp1Y1
two pump
i
locations on the plate. The blocked forces were
calculated from previously measured free velocity
The RPM estimates, for the two source locations,
differ by 5 dB between 80 Hz and 200 Hz. This
points to the need for averaging of estimates for
several source locations on the test plate. Above
1000 Hz, the RPM gives an overestimate, which
is likely to be caused by signal-noise problems.
However, at 1000 Hz, the force squared has reduced by 40 dB, with respect to the maximum
value. Hence in many installations the pump will
only cause noise problems as a structure-borne
source below this frequency.
In Figure 8 are the calculated sum-square
N
blocked
force
F
2
bi
and
RPM
estimates
i
of Fb2eq for two fan locations. Again, the RPM
FORUM ACUSTICUM 2014
7–12 September, Krakow
Gibbs, Hopkins:
Force distribution of structure-borne sound power sources in buildings
By considering idealized force distributions and
thin plates of infinite extent, it is demonstrated
that an increase in the number, spacing and complexity of contacts, increases the accuracy of the
RPM estimate of sum-square blocked forces and
the frequency range of application.
N
underestimates
F
2
bi
at low frequencies.
i
1.0E-1
1.0E-2
This is confirmed using real sources on a real
low-mobility plate. The RPM gives underestimates at low frequencies or is not applicable,
particularly for compact sources in rigid body
motion.
FORCE SQUARED Nsq
1.0E-3
1.0E-4
1.0E-5
1.0E-6
Acknowledgements
1.0E-7
1.0E-8
40
400
4000
FREQUENCY Hz
Figure 8: Sum-square blocked force of fan: solid
line, calculated from free velocity and source mobility; dashed lines, RPM estimates at two positions
on 20mm plate
The agreement is within 10 dB above 100 Hz,
although the RPM gives an overestimate above
800 Hz, due to signal-noise problems.
The force squared has reduced 40 dB, with respect to the maximum value, at 2000 Hz. Therefore, the fan will only cause noise problems as a
structure-borne source below this frequency.
4. Concluding remarks
This paper considers the requirements of the
reception plate method (RPM), used to obtain
single values of the blocked force of tested
vibration sources.
For this, the plate must provide a value of low
receiver mobility, relative to that of the tested
source.
However, in achieving this, the plate then has a
long governing bending wavelength, which
increases interaction effects between the contacts between the source and plate, particularly
at low frequencies.
This means that the single equivalent value of
the blocked force squared only approximates
the target value, the sum-square of the blocked
forces over the contacts.
The authors wish to thank Dr. Kevin Lai and his
colleagues at Boeing Commercial Airlines, and
also colleagues at ITT Enedine Inc., for their
financial support and helpful comments throughout the collaborative programme, a part of which
is the work reported here.
In addition, the measurement effort and skills of
Dr. Gary Seiffert, of the Acoustics Research
Unit, are gratefully acknowledged.
References
[1] EN 12354-5: 2009: Building acoustics- Estimation of acoustic performance of building from
the performance of elements – Part 5: Sound
levels due to service equipment.
[2] M. M. Späh and B. M. Gibbs, ‘Reception plate
method for characterisation of structure-borne
sound sources in buildings: Assumption and application’, Applied Acoustics 70, 361-368, 2008.
[3] L. Cremer, M. Heckl and B.A.T. Petersson, Structure-borne Sound, 3rd ed., 607 p., Springer Berlin
Heidelberg, New York, 2005.
[4] C. Hopkins, Sound Insulation, 622 p., Elsevier,
2007.
[5] M. Ohlrich, L. Fris, S.Aatola, A. Lehtovaara, M.
Martikainen and O. Nuutila, Round Robin test of
techniques for characterizing the structure-borne
sound source strength of vibrating machines, Proc.
Euronoise 2006, Tampere, Finland, 2006.
[6] B.M. Gibbs, N. Qi and A.T. Moorhouse, A
practical characterization for vibro-acoustic
sources in buildings, Acta Acust. Acust. 93, 8493, 2007.
[7] B.M. Gibbs, R. Cookson and N. Qi, Vibration
activity and mobility of structure-borne sources
by a reception plate method, J. Acoust. Soc.
Am. 123(6), 4199-4209, 2008.

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