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A NOTE ON THE USE OF VIBRO-ACOUSTIC TRANSMISSI-
BILITY TO ESTIMATE VIBRO-ACOUSTIC RESPONSES
Neves, M.M.1, Policarpo, H.1,2, Maia, N.M.M.1, Tcherniak, D.3
1IDMEC, Instituto Superior Técnico, Universidade de Lisboa
Av. Rovisco Pais, 1049-001, Lisbon, Portugal 2Universidade Nova de Lisboa, Campus da Caparica, Caparica, Portugal
3Brüel & Kjær Sound & Vibration Measurement A/S,
Nærum, Denmark
ABSTRACT
Authors present part of their research on the implementation of vibro-acoustic analysis using transmissibility
functions to relate excitations from some parts with their counterpart in interior acoustic environments. Simple
models are presented with basic cases of vibration transmissibility and acoustic transmissibility. The extension to
vibroacoustic problem is introduced and briefly discussed with respect to applications like e.g. source identification
and operational transfer path analysis (OTPA). Essentially, the model involves accelerometers at a few position
and a microphone at another position to define the related transmissibilities (in the frequency domain). Authors
conclude that in many practical cases an Hs estimator should produce a better transmissibility matrix estimation,
and discuss on which conditions the contributions from classical TPA and OTPA are expected to be practically the same.
KEYWORDS: Displacement transmissibility, acoustic transmissibility, vibro-acoustics transmissibility,
operational transfer path analysis.
1. INTRODUCTION
The concept of transmissibility is well known in vibration and acoustic fields but mainly in what concerns to single
degree-of-freedom systems. One question that aroused naturally was how to extend that idea of transmissibility to
a system with N degrees-of-freedom. Initial attempts to achieve that were done by Snowdon [1], Vakakis et al. [2-4] and Sciulli and Inman [5], Liu and Ewins [6] and Varoto and McConnell [7]. First generalizations involving
multiple degrees-of-freedom can be found e.g. in [8-10], where the corresponding transmissibility matrix between
two sets of response positions is obtained from mobility matrices of the structure.
However, the concept of multipoint transmissibility is a relatively new concept in the acoustics and vibro-acoustic
fields. Among the works published, one can find the work from Curling and Païdoussis [11] that proposed a
discrete acoustic transmissibility applied to a pressure transducer-in-capsule to measure wall pressure spectra. In
Devriendt and Guillaume [12], “scalar transmissibilities” are considered of high potential in diverse fields, as e.g.
in operational modal analysis [12–14], model updating [15] and structural health monitoring [16].
Such a transmissibility concept seems of interest for situations where one has technical difficulties in measuring
the responses at some co-ordinates of the structure. For example, if the transmissibility matrix of the original
system can be evaluated beforehand, it is to expect in some conditions that by measuring a few responses, one would be able to estimate other responses, e.g. at inaccessible co-ordinates as described in Maia et al. [17].
In the single degree-of-freedom (SDOF) case, the conclusion is that the transmissibility expressions for both
displacement and force are identical. Some efforts can be found for the indirect measurement of vibration
excitation forces [18-22]. A more general approach for the force transmissibility is given in [23]. As explained in
[24], transmissibility expressions for both force and displacement are not strictly identical for multiple degree-of-
freedom (MDOF) systems. However, and naturally, they are very closely related, as shown in [25], where the force
transmissibility in a structure with fixed supports is extracted directly from the displacement transmissibility
obtained with the same structure but in free displacement boundary conditions.
Various applications of transmissibility approach have been developed, such as structural response estimation [26],
damage detection [27, 28], operational modal analysis [29], evaluation of unmeasured frequency response
functions (FRFs) [30, 31], and force identification [32, 33]. Another example of application is the problem of
transfer path analysis in vibro-acoustics, where classical techniques based on measured transfer functions and
estimated source strengths, may often be time consuming and error-prone. Tcherniak [34-35] and Tcherniak et al.
[36] sought for easier and more reliable ways to address those types of problems through the use of the
transmissibility matrix, extracted from operating measurements and therefore not requiring the measurement of
transfer functions.
It is not easy to find works in acoustic transmissibility, as few are fully dedicated to it, but instead some appear in
its applications like e.g. operational transfer path analysis. Let us first mention a few ones focused on acoustic
transmissibility. Devriendt et al [37] used a scalar pressure ratio FE-based transmissibility for an acoustic cavity. Beside the works in [12-16], one can mention here the works from Kletschkowski [38] and Weber et al [39] to
obtain unknown pressures via measured pressures and to identify noise sources. Inspired on these works, Guedes
and Neves [40] proposed an Acoustic Transmissibility Matrix for 1D acoustic problems, which Guedes [41]
extended to the 2D case for noise source identification and reconstruction, and finally performed some
experimental work to validate the method as described in [42].
There are several vibro-acoustic studies, as e.g. [43,44], and from some is concluded that the FEM can more easily
treat complicated boundary conditions compared to BEM. An analysis of the contribution to the acoustic pressure
at node i due to a unit velocity at node j is presented in [45] (using the Helmholtz integral) as well as the proposed
by Maressa et al. [46] which uses an Acoustic Transfer Vector approach [47] for efficient pressure predictions
after structural modifications.
Finally, let us mention a few works in applications of acoustic transmissibility like e.g. Operational Transfer Path Analyses (OTPA or OPA) beside the ones already mentioned [34,35]. Indeed, in recent years, we observed a
renewed interest and need for faster Transfer Path Analysis (TPA) methods [48], and one way of doing that is
mainly achieved by using operational data in conjunction with the transmissibility concept. TPA methods have
various family members (presenting similarities or are even identical) named as Operational TPA [49], Operational
Path Analysis with exogenous inputs (OPAX), blocked-force TPA, Gear Noise Propagation, in situ Source Path
Characterization and Virtual Acoustic Prototyping. However, literature soon described limitations and drawbacks
of these methods, see e.g. [50], which includes problems related to the transmissibility’s estimation, or errors due
to coupling between path inputs, among others.
In this work, authors review the fundamentals from part of their current research – on vibration, acoustic as well
as vibro-acoustic transmissibility – which is being developed to relate (in the frequency domain) excitations from
some parts with their response counterpart in interior acoustic environments. The main concepts for vibration and
acoustic transmissibility are briefly described in section 2. Some aspects of the acoustic transmissibility are discussed at section 3. The extension to vibro-acoustic problem is briefly discussed in the section 4. In section 5,
authors elaborate on the operational transfer path analysis (OTPA) and explain why in many practical cases an Hs
estimator should produce a better transmissibility matrix estimation, and which are the conditions where the
contributions from classical TPA and OTPA are expected to be practically the same.
2. FUNDAMENTALS
In this section, we briefly review the concept of transmissibility in its more general form. It is a normalized function
strongly dependent of its zeros (antiresonances) and independent of the poles (resonances). Due to it,
transmissibility shows “flat spot” in the plot instead of a peak in the frequency range where a resonance peak occurs. For a more detailed description we recommend e.g. [25, 30, and 42].
2.1. Vibration transmissibility in MDOF dynamic systems
The structural response of linear viscoelastic structure assuming n DOFs, is given by
[M]{y(t)} + [C]{y(t)} + [K]{y(t)} = {f(t)}, (1)
where [M], [C] and [K] are the n×n mass, damping and stiffness matrices, respectively; {y} the nodal displacement
vector; the one and two dots are for its first and second time derivatives; and {f} is the excitation load vector.
When excitation load is harmonic, the separation of space and time dependent variables allows to replace {y} and
{f} by their respective amplitudes {Y(ω)} and { F(ω)}, and (1) is rewritten to
[[K] + 𝐢ω[C] − ω2[M]]{Y(ω)} = {F(ω)}. (2)
The dynamic stiffness matrix [Z()] is in this case given by
[Z(ω)] = [[K] + 𝐢ω[C] − ω2[M]], (3)
and the receptance or FRF matrix [H(ω)] may be obtained as the inverse of [Z(ω)] , which gives an equation
equivalent to (2) that is
{Y(ω)} = [Z(ω)]−1{F(ω)} = [H(ω)]{F(ω)}. (4)
In classical books, the dynamical displacement transmissibility is introduced as the ratio between the amplitudes
(in absolute value) of the applied and transmitted displacements, for single DOF systems. This generalization is
useful when done through the definition of sets of DOFs as illustrated in Fig. 1. The sets U and K are composed
of coordinate points where the responses can be measured and the set A by coordinate points where loads are
applied (which may have DOFs from other sets).
Figure 1 – One possible choice of sets: K is a set of coordinates of interest, U is other set with points of interest, and C is a set
of coordinates that includes all other coordinates. The set A includes points where excitation forces are applied.
At equation (4) the amplitudes {YU} and {YK} and the applied force amplitude {FA} can be related by
{ {𝑌𝑈} = [𝐻𝑈𝐴]{𝐹𝐴}
{𝑌𝐾} = [𝐻𝐾𝐴]{𝐹𝐴} (5)
where [HUA] is the truncated FRF matrix to the sets of DOFs U and A. [HKA] is obtained similarly. From (5) one
can conclude that transmissibility matrix involves in general the pseudo-inverse of [HKA], i.e.
{YU} = [TUKA (d)
] {YK} = [HUA] ∙ ([HKA]+){YK}. (6)
This pseudoinversion requires that set K has larger or equal dimension (i.e. number of coordinate points) than the
set A (i.e. #K≥#A). The expression (6) depends on the DOFs A (where the loads are applied) but not on their amplitudes. The transmissibility matrix that relates amplitudes {YU} and {YK} is obtained by
[TUKA (d)
] = [HUA]([HKA]+). (7)
2.2. Acoustic pressure transmissibility
A linear dynamic and weakly damped fluid (upper index ‘f’) acoustic system may be modelled in the frequency-domain (i.e. after separation from time) using the following equation
[[𝐾𝑓] + 𝐢ω[𝐶𝑓] − ω2[𝑀𝑓]] {P(ω)} = {Q(ω)}. (8)
whereis the angular frequency, [Kf] and [Mf] are respectively the global stiffness and mass matrices of the acoustic medium, P is the sound pressure amplitude vector in steady-state, and Q is the volume accelerator vector
(or source vector). For harmonic type sources, Q is of the form A, where A is the particle velocity in the sound
field and is the fluid density. The steady-state pressure field P can mathematically be given by
{P(ω)} = [Z(ω)]−1{Q(ω)} = [𝐻𝑓]{Q(ω)}. (9)
where [Hf] is the global frequency-response matrix. In practice, obtaining [Hf] is numerically expensive and
therefore its calculation is usually done with efficient numerical methods like the modal superposition method.
Let us consider the domain in Fig. 2. One can divide the domain in the following four sets of coordinates named
as: 1) the set of known coordinates K where pressure can be measured; 2) the set of known coordinates U where
pressure amplitude is unknown (or to be estimated); 3) the unknown set of coordinates S were sources are located
and 4) the set C representing the remaining coordinates. The sets could be unions of subsets separated in the space.
Figure 2 – Schematic of domain discretized in coordinate sets K, U, S and C.
Assuming that there are no other sources than in set S, i.e. QK ,QU and QC are zero, then (9) can be written as
{
𝑃𝐾
𝑃𝑈
𝑃𝑆
𝑃𝐶
} = [
𝐻𝐾𝑆
𝐻𝑈𝑆
𝐻𝑆𝑆
𝐻𝐶𝑆
] {𝑄𝑆}
(10)
From (10) one can obtain an expression relating PU as function of Pk given by the following product which is
independent of the source amplitudes QS
{P(ω)} = [Z(ω)]−1{Q(ω)} = [𝐻𝑓]{Q(ω)} (11)
The acoustic pressure transmissibility matrix between sets U and S, and assuming that the sources are acting in set S only (even if some or all with zero amplitude), is given by
[𝑇UK𝑆 (𝑝)
] = [HUS] ∙ [HKS]+ (12)
and has dimensions nu x nk , respectively the number of coordinates in sets U and K. It requires that nk must be
greater or equal than nS (the latter being the case where the pseudo-inverse takes the form of an inverse).
A particular case is when the two coordinate sets U and K have dimension one and S is equal or greater than one
for which the transmissibility is given by
𝑇𝑈𝐾 (𝜔) =𝑝𝑈(𝜔)
𝑝𝐾(𝜔) =
∑ 𝐻𝑈𝑆𝑖(𝜔)𝑛𝑠𝑖=1 𝑞𝑖(𝜔)
∑ 𝐻𝐾𝑆𝑖(𝜔)𝑛𝑠𝑖=1 𝑞𝑖(𝜔)
(13)
which depends on the acoustic source amplitudes qi, with i=1,2,…,nS i.e. the number of coordinates in S.
The transmissibility matrix allows to calculate the pressure amplitudes in coordinates U, with the sole requirement
of knowing the pressures in another set K. However, this matrix may have ill-conditioning problems that must be
considered when measurement errors are taken into account.
Finally, the great interest of transmissibility is that HUS and HKS may be obtained experimentally.
3. SOME CONSIDERATIONS ON THE ACOUSTIC TRANSMISSIBILITY
Let one consider the following configuration of the two dimensional interior domain given at Fig. 3.
Figure 3 – Discretized domain representing each set of coordinates to be measured.
The following sets of coordinates are defined K={1,2}; U={3,4} and S={5,6}. In the particular case the
transmissibility is one-dimensional (scalar), and the calculation of the transmissibility is considered as
straightforward. Applying a source in a specific set of coordinates S, the transmissibility is obtained by (13) taking
the ratio of the measured pressures in one of coordinates (points) of set K and another of set U.
The transmissibility matrix between the complete sets U and K is a 2x2 matrix, with the two sources applied at
coordinates 5 and 6. Using equation (11) it yields
𝑇𝑈𝐾 = [𝐻35 𝐻36
𝐻45 𝐻46
] [𝐻15 𝐻16
𝐻25 𝐻26
]
−1
(14)
Theoretically, it is possible to obtain each entry exciting the domain by applying a given source in each coordinate
S at a time. For that, one only has to take the ratio of the measured pressures i=1,2… in sets K and U, with the
source amplitude applied in coordinates j=1,2,… of set K, i.e.
𝐻𝑖𝑗(𝜔) =𝑝𝑖(𝜔)
𝑞𝑗(𝜔)
(15)
In practice, the sources used are not point sources, but speakers with a given area. This raises an important question:
how to treat distributed sources? In a FE analysis, a distributed source reduces to a line source when 2D domains
are considered and it is modelled by a succession of punctual sources. Let one consider the theoretical situation where one is measuring a transmissibility (scalar) in an interior pressure
field due to a source S1 of unknown location and unitary amplitude. That transmissibility is given experimentally by
𝑇𝑈𝐾𝑆𝑖 (𝜔) =
𝑃𝑈𝑆𝑖(𝜔)
𝑃𝐾
𝑆𝑖(𝜔) =
𝐻𝑈𝑆𝑖(𝜔)
𝐻𝐾𝑆𝑖(𝜔)
(16)
If another source S2 with amplitude “a” is introduced, the transmissibility will take the form
𝑇𝑈𝐾𝑆1+𝑆2(𝜔) =
𝑃𝑈𝑆1+𝑆2(𝜔)
𝑃𝐾𝑆1+𝑆2(𝜔)
=𝐻𝑈𝑆1(𝜔) + 𝑎𝐻𝑈𝑆2(𝜔)
𝐻𝐾𝑆1(𝜔) + 𝑎𝐻𝑈𝑆2(𝜔)
(17)
One can observe that when the introduced source S2 location tends to the location of the source S1 then
𝐻𝑈𝑆2→ 𝐻𝑈𝑆1
; 𝐻𝐾𝑆2→ 𝐻𝐾𝑆1
𝑎𝑛𝑑 𝑇𝑈𝐾𝑆1+𝑆2 → 𝑇𝑈𝐾
𝑆1 (18)
Pioneer work on operational acoustic modal analysis (OAMA) using transmissibility measurements was published
by C. Devrient et al. [37]. In their work, they discuss difficulties observed at acoustic parameters identification.
Indeed, experimental Acoustic Modal Analysis techniques use volume acceleration sources while OAMA not. A
disadvantage of existing OAMA is that the non-measured acoustic sources must be pure white noise excitation but
in operation it cannot always be the case. One advantage of the transmissibility functions is that they can be
measured without the knowledge of the excitation forces or volume accelerations q.
In what concerns to the noise source identification one can mention here the works from Kletschkowski [38] and
Weber et al [39] to obtain unknown pressures via measured pressures and to identify noise sources. Guedes [41]
used acoustic transmissibility matrix for noise source identification and reconstruction in very simple cases. For more than one source it requires to address the cross-talk effect.
4. VIBRO-ACOUSTIC TRANSMISSIBILITY
Consider now a Fluid-Structure Interaction (FSI) case. It combines its structural model
[[K] + 𝐢ω[C] − ω2[M]]{Y(ω)} = −[A]{P(ω)} + {F(ω)}, (19)
with the acoustic fluid model (compressible fluid)
∇2p =��
𝑐2 ,𝜕𝑝
𝜕𝑛= −𝜌��𝑛, (20)
which is expressed by the following system of equations
[[𝐾𝑓] + 𝐢ω[𝐶𝑓] − ω2[𝑀𝑓]] {P(ω)} = ω2𝜌[A]{Y(ω)} + {Q(ω)}, (21)
Considering (19) coupled to (21) one may write the following system of equations
([K 𝐴
−𝜌𝐴 𝐾𝑓] + 𝐢ω [C 00 𝐶𝑓] − ω2 [M 0
0 𝑀𝑓]) {Y(ω)
P(ω)} = {
F(ω)
Q(ω)},
(22)
The coupled system may be expressed by
([𝐾𝐹𝑆𝐼] + 𝐢ω[𝐶𝐹𝑆𝐼] − ω2[𝑀𝐹𝑆𝐼]){X(ω)} = {𝐹𝐹𝑆𝐼}, (23)
which can alternatively be expressed as
{X(ω)} = [𝐻𝐹𝑆𝐼]{𝐹𝐹𝑆𝐼}, (24)
From (24), it is possible to set relations between some displacement responses and excitation loads
{𝑌𝐴(ω)} = [𝐻𝐴𝐴𝐹𝑆𝐼]{𝐹𝐴(ω)}, (25)
as well as to set relations between some pressure responses and the same excitation loads
{𝑃𝐵 (ω)} = [𝐻𝐵𝐴𝐹𝑆𝐼]{𝐹𝐴(ω)}, (26)
Relating these pressure responses with displacement responses through the excitation forces one gets
{𝑃𝐵 (ω)} = [𝐻𝐵𝐴𝐹𝑆𝐼]{𝐹𝐴(ω)}, (27)
i.e. the corresponding vibro-acoustic transmissibility
[𝑇𝐵𝐴𝐹𝑆𝐼] = [𝐻𝐵𝐴
𝐹𝑆𝐼][𝐻𝐴𝐴𝐹𝑆𝐼]−1. (28)
The transmissibility relates displacements to pressure, which in a linear system do not depend on the excitation
amplitudes. Note also that one important aspect is that sub-matrices HAA and HBA may be obtained experimentally
if necessary.
5. SOME CONSIDERATIONS ON CLASSIC TPA AND OPERATIONAL TPA
In many industries, especially in automotive NVH, characterizing noise sources and noise propagation paths is
an important problem. Fig. 4 presents a very typical, though simplified NVH source-path-receiver problem.
Here, the noise and vibration at sensitive locations defining driving comfort are under concern. Such locations are
driver’s ears, steering wheel, heel point, etc., are typically instrumented by microphones and accelerometer and
according to the NVH terminology are called receivers. The noise and vibration come from noise sources, in the
case shown in the figure, there are two uncorrelated sources, the engine and the ventilation fan. The noise and
vibration from the engine propagate to the cabin by noise propagation paths, in the illustrated case, there are two
structure borne path, namely two engine mounts, and the air-borne path. The noise from the fan takes the air-borne path to reach the receivers. The transfer path analysis (TPA) attempts to estimate the contributions from
each identified noise sources and propagation paths.
Figure 4 – Typical NVH source-path-receiver problem
5.1. Classic TPA A classical approach to the problem is illustrated in Fig. 5, using a simple example.
Figure 5 – Source-receiver model with the source-indicator sensor matrix.
Let 𝑢𝑛 , 𝑛 = 1, … , 𝑁 are noise (vibration) sources (or paths) to be characterized. Then the contributions of n-th
source, 𝐶𝑛 to the total noise 𝑌 picked up at the receiver position (y),
𝑌 = ∑ 𝐶𝑛𝑁𝑛=1 , (28)
is estimated as
𝐶𝑛 = 𝐻𝑛𝑌𝑈𝑛. (29)
In the following notation, the capital letters denote the complex Fourier spectra of the corresponding physical
quantities in time domain, the latter are denoted by small letters.
In (29), the contribution is estimated as a product of the frequency response function (FRF) 𝐻𝑛𝑌 and the operational
source strength 𝑈𝑛. Whilst the FRF can be measured in the lab (when the machine is not under operation) using some measurement technique, it is not possible or very difficult to directly measure the source strength while the
machine is in operation. Thus, some indirect methods have to be involved. The most popular one is the matrix
inversion method. Following this method, several indicator sensors are placed near the sources, and the FRFs
between the sources and the indicators are measured (in the lab). These FRFs are combined into the source-to-
indicator matrix [𝐻𝑈𝑉]. When the machine is under operation, the signals from the indicator sensors {𝑉} are
recorded. Since
{𝑉} = [𝐻𝑈𝑉]{𝑈} (30)
where {𝑉} = {𝑉1, 𝑉2, … , 𝑉𝑁}𝑇 and {𝑈} = {𝑈1, 𝑈2, … , 𝑈𝑁}𝑇, the operational source strength can be estimated as
{��} = [𝐻𝑈𝑉]−1{𝑉} (31)
Due to it, the number of the indicator sensors should be greater or equal than the number of sources, in practical
cases the over-determination is advantageous, making [𝐻𝑈𝑉] rectangular; then matrix pseudo-inverse shall be used
instead of the inverse.
The classical TPA method, described above, requires a set of FRFs to be measured, which makes the method time consuming. For the last few decades, NVH engineers are looking for a faster technique. Perhaps the most
significant contribution to this was the introduction of so-called Operational TPA (introduced by Honda and
commercialized by M-BBM).
5.2. Application of transmissibility to transfer path analysis
In contrast to classic TPA, OTPA is based solely on operational measurements (conducted when the machine is in
operation), not requiring any FRFs, thus significantly reducing the complexity of the measurement campaign and
reducing the measurement time. However, the method has been criticized as it may provide incorrect results. Let
us consider the method in detail to identify its drawbacks and possibly find a solution to circumvent them.
OTPA method is based on using a transmissibility matrix. It is easy to derive the method from the classic TPA.
Rewriting (28) and (29) in matrix form
𝑌 = [𝐻1𝑌 𝐻2𝑌 … 𝐻𝑁𝑌]{𝑈1, 𝑈2, … 𝑈𝑁}𝑇 = [𝐻𝑈𝑌]{𝑈}, (32)
and substituting (31) into (32) yields
𝑌 = [𝐻𝑈𝑌][𝐻𝑈𝑉]−1{𝑉}. (33)
Expression (33) links two responses: 𝑌 measured at the receiver position and {𝑉}, measured at the indicator
positions. The matrix linking them together is a transmissibility matrix
[𝑇𝑉𝑌,𝑈] = [𝐻𝑈𝑌][𝐻𝑈𝑉]−1. (34)
Notation [𝑇𝑉𝑌,𝑈] denotes that the matrix links the response 𝑌 with the responses {𝑉} under the condition that the
forces are acting at the locations (DOFs) {𝑈}
𝑌 = [𝑇𝑉𝑌,𝑈]{𝑉}. (35)
Thus, the response 𝑌 can be expressed as a linear combination of the indicator responses {𝑉}. For simplicity,
further in the text, the indices of [𝑇𝑉𝑌,𝑈] will be omitted, and the transmissibility matrix will be denoted as [𝑇]. OTPA method postulates that, if the indicator sensors are placed in a vicinity of noise sources, contributions (29)
can be approximated by
𝐶𝑛 ≈ 𝑆𝑛 = 𝑇𝑌𝑛𝑉𝑛. (36)
where 𝑆𝑛 is OTPA contribution from n-th source/path, and 𝑇𝑌𝑛 is the corresponding element of the transmissibility
matrix.
The method is called operational because the transmissibility matrix [𝑇] can be estimated from operating
measurements. This is a real beauty of the transmissibility matrix. Indeed, instead of calculating it from FRFs,
following the definition (34), one can estimate it without measuring FRFs. It can be done in different ways, one of
them is outlined below.
One can notice that transmissibility expression (35) closely resembles force-to-response relation of multiple input
multiple output (MIMO) known from modal analysis. Therefore, it is sensible to apply one of the well-known
MIMO FRF estimators to assess [𝑇]. Below, the application of 𝐻1 estimator is described.
Let’s post-multiply expression (35) by {𝑉}𝐻
𝑌{𝑉}𝐻 = [𝑇]{𝑉}{𝑉}𝐻. (37)
Averaging over time,
𝐸(𝑌{𝑉}𝐻) = 𝐸([𝑇]{𝑉}{𝑉}𝐻) = [𝑇]𝐸({𝑉}{𝑉}𝐻). (38)
we readily recognize that 𝐸(𝑌{𝑉}𝐻) = [𝐶𝑌𝑉] is the cross-spectra matrix between the receiver and indicator signals
and 𝐸({𝑉}{𝑉}𝐻) = [𝐶𝑉𝑉] is the cross-spectra matrix between the indicator signals
[𝐶𝑌𝑉] = [𝑇][𝐶𝑉𝑉]. (39)
Now the transmissibility matrix can be expressed as
[��] = [𝐶𝑌𝑉][𝐶𝑉𝑉]−1 = [𝑇][𝐶𝑉𝑉]. (40)
In many practical cases, so-called 𝐻𝑆 estimator provides a better transmissibility matrix estimation [51].
From practical viewpoint, OTPA steps are:
1. Place the indicator sensors close to the noise sources and the receiver sensors at the comfort-sensitive
locations;
2. Operate the machine at the regimes of interest, recording receiver(s) and indicators signals 𝑌 and {𝑉}, to
collect sufficient amount of averages to properly estimate [𝐶𝑌𝑉] and [𝐶𝑉𝑉];
3. Estimate [��] using (40);
4. Estimate the contributions from the sources / paths using (36);
This could be done for operating measurements performed at step 1, but also for any other operating
measurements conducted for the given instrumentation and same regimes of the machine.
5.3. Conditions where TPA and OTPA produce practically the same result
After OTPA was introduced, there were many publications pinpointing the drawbacks of the method, where the
main critics was put on the assumption that OTPA contributions are approximately equal to TPA ones: 𝑆𝑖 ≈ 𝐶𝑖.
Let us consider an application examples to illustrate when this assumption is valid, and where one shall pay a
special attention when applying the OTPA method.
Let’s use a very simple example to introduce the methodology, as the same analysis template will be used when
considering other examples. Let’s consider two omnidirectional noise sources with source strength 𝑄1 and 𝑄2, in
the free field, and one receiver microphone (Fig. 6). To characterize the sources using OTPA, we use two indicator
microphones, which we place very close to the sources.
Figure 6 – Measurement setup: two omni-directional noise sources, one receiver microphone and two indicator microphones.
The model in Fig. 7 is described by the following (unknown) FRF matrices
[𝐻𝑄𝑌] = [𝐻𝑄1𝑌 𝐻𝑄2𝑌],
[𝐻𝑄𝑉] = [𝐻11 𝜀 𝐻12
𝜀 𝐻21 𝐻22].
(41)
Here 𝜀 is introduced as a bookkeeping device to characterize the level of cross-talk between the sources and
indicator sensors. Referring to the general case considered above, [𝐻𝑈𝑌] ≡ [𝐻𝑄𝑌] and [𝐻𝑈𝑉] ≡ [𝐻𝑄𝑉].
The assumed (unknown) operating sources {𝑈} ≡ {𝑄1 𝑄2}𝑇 are characterized by the following (unknown) cross-
spectra matrix
[𝐶𝑄𝑄] = [𝐶11 𝛾 𝐶12
𝛾 𝐶12∗ 𝐶22
]. (42)
where symbol ‘*’ denotes the complex conjugate and 𝛾 is a bookkeeper to characterize the degree of correlation between the two sources.
The cross-spectra matrices [𝐶𝑌𝑉] and [𝐶𝑉𝑉] can be expressed as
[𝐶𝑌𝑉] = [𝐻𝑆𝑌][𝐶𝑄𝑄][𝐻𝑆𝑉]𝐻, and [𝐶𝑉𝑉] = [𝐻𝑆𝑉][𝐶𝑄𝑄][𝐻𝑆𝑉]𝐻 (43)
where [ ]𝐻 denotes conjugate transpose. Note, in practice, we will be measuring these two matrices.
From matrices (43), the transmissibility matrix can be computed using [𝑇] = [𝐶𝑌𝑉][𝐶𝑉𝑉]−1, (40). For this particular
example,
[𝑇] =1
det([𝐻𝑄𝑉])[𝐻𝑄1𝑌 𝐻22 − 𝜀 𝐻𝑄2𝑌 𝐻21 𝐻𝑄2𝑌𝐻11 − 𝜀 𝐻𝑄1𝑌𝐻12]
(44)
Note, all 𝐶𝑖𝑗 got cancelled out, as well as 𝛾. Also, note that exactly same expression can be obtained from definition
(34).
Now the OTPA contributions can be readily obtained using (36), (44) and (30)
𝑆1 =1
det([𝐻𝑄𝑉])(𝑄1𝐻11 + 𝜀 𝑄2𝐻12)(𝐻𝑄1𝑌𝐻22 − 𝜀 𝐻𝑄2𝑌 𝐻21),
𝑆2 =1
det([𝐻𝑄𝑉])(𝑄2𝐻22 + 𝜀 𝑄1𝐻21)(𝐻𝑄2𝑌𝐻11 − 𝜀 𝐻𝑄1𝑌 𝐻12), (45)
and here det([𝐻𝑄𝑉]) = 𝐻11𝐻22 − 𝜀2𝐻12𝐻21.
Expressions (45) can be expanded into power series about 𝜀 = 0 as
𝑆1 = 𝐻𝑄1𝑌 𝑄1 + (−𝐻21𝐻𝑄2𝑌 𝑄1
𝐻22⁄ +
𝐻12𝐻𝑄1𝑌 𝑄2𝐻11
⁄ ) 𝜀 + 𝑂(𝜀2),
𝑆2 = 𝐻𝑄2𝑌 𝑄2 + (𝐻21𝐻𝑄2𝑌 𝑄1
𝐻22⁄ −
𝐻12𝐻𝑄1𝑌 𝑄2𝐻11
⁄ ) 𝜀 + 𝑂(𝜀2). (46)
The baseline TPA contributions, in this case, are
𝐶1 = 𝐻𝑄1𝑌𝑄1, and
𝐶2 = 𝐻𝑄2𝑌𝑄2, (47)
reason why comparing with the obtained OTPA contributions, one can observe that:
1. The OTPA contributions differ from the baseline contributions by terms of order of 𝜀:
𝑆1 − 𝐶1 = (−𝐻21𝐻𝑄2𝑌 𝑄1
𝐻22⁄ +
𝐻12𝐻𝑄1𝑌 𝑄2𝐻11
⁄ ) 𝜀 + 𝑂(𝜀2),
𝑆2 − 𝐶2 = (−𝐻21𝐻𝑄2𝑌 𝑄1
𝐻22⁄ +
𝐻12𝐻𝑄1𝑌 𝑄2𝐻11
⁄ ) 𝜀 + 𝑂(𝜀2), (48)
2. The OTPA contributions will coincide with the baseline if 𝜀 = 0, i.e. if there is no cross-talk, namely,
no energy transfer between the first source and the second indicator sensor and the second source and
the first indicator sensor.
3. The contributions (47) don’t depend on 𝛾, i.e. the expressions are valid independently of the correlation
between the sources;
4. The sum of the OTPA contributions differs from the sum of TPA contributions by the term of the order
of 𝜀2:
𝑆1 + 𝑆2 = 𝐶1 + 𝐶2 + 𝑂(𝜀2). (49)
Finally, to quantify the example, imagine that the two sources are placed 50 cm from each other, and the indicator
microphones are placed such way that the distance to the nearest source is 10 cm and to the furthest is 50 cm. The
energy arriving to the furthest microphone is (10 cm / 50 cm)2 = 0.04 of the energy picked by the nearest
microphone, and thus the cross-talk factor 𝜀 = √0.04 = 0.2. Assuming ythat the setup is symmetric, then 𝐻12 =𝐻21 and 𝐻𝑆1𝑌 = 𝐻𝑆2𝑌.
Let source 𝑄2 = 𝛼 𝑄1. Then the OTPA contributions 𝑆1 = 𝐶1 + 𝜀 (𝛼 − 1)𝐶1 + 𝑂(𝜀2) and 𝑆2 = 𝐶2 +𝜀 (1/𝛼 − 1)𝐶2 + 𝑂(𝜀2). If the sources radiate same amount of noise, 𝛼 = 1, the OTPA contributions are correct.
If, for example, 𝛼 = 2, then 𝑆1 = 1.2 𝐶1 + 𝑂(𝜀2), i.e. 20% overestimates the correct value and 𝑆2 = 0.9𝐶2 +𝑂(𝜀2), that is 10% underestimate the baseline. Interesting enough, their sum is almost correct: 𝑆1 + 𝑆2 = 𝐶1 +𝐶2 + 𝑂(𝜀2).
For a recent publication with a discussion on some aspects of the application of vibro-acoustic operational transfer path analysis see e.g. [52].
6. CONCLUSIONS
The formulations for vibration transmissibility and acoustic transmissibility, as well as its extension to vibro-
acoustic problem, were briefly reviewed. Some considerations were done to the application of these concepts with
respect to OAMA, noise source identification, TPA and OTPA. Because transmissibility expression closely
resembles force-to-response relation of multiple input multiple output case, authors consider that makes sense to
apply the known H1 estimator. It is also observed that the sum of the OTPA contributions differs from the sum of
TPA contributions by the term of the order of 𝜀2 where 𝜀 characterize the level of cross-talk between the sources and indicator sensors.
ACKNOWLEDGEMENTS
This work was supported by FCT, through IDMEC, under LAETA, project UID/EMS/50022/2019.
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