PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

In-situ transfer path analysis of multiple vibration sources in acomplex source- receiver assembly
Lucy BARTON1; Andrew ELLIOTT1; Andy MOORHOUSE1; John SMITH2
1

University of Salford, England
2

DSTL, England

ABSTRACT
Depending on where the source-receiver boundary is defined within an assembly, an airborne sound source
can also be considered as a structure borne sound source. For example, if the internal forcing mechanisms
within a machine are regarded as the sources, the rest of the assembly including the surrounding air can beconsidered the receiver. If the structure borne noise resulting from these internal forcing mechanisms can bepredicted in real-time, the radiated noise can potentially be predicted and managed- either actively or
passively- using numerous approaches. The primary aim of the work presented in the paper is thedevelopment of an accurate method for the real-time prediction of structure borne noise, whilst continuallymonitoring the reliability of these predictions. To facilitate this process, it is necessary to also monitor thestability of the vibro-acoustic frequency response functions that relate the internal forcing mechanisms to thereceiver position, and to eliminate the influence of background noise resulting from noise or vibration sources
other than those of interest. Presented in the paper are the results of a laboratory Transfer Path Analysis (TPA)
case study of a source-receiver assembly composed of multiple sources attached to a single receiver structure.
Keywords: Noise, TPA, Transmission

1. INTRODUCTION
This paper documents a case study carried out as part of a wider PhD project investigating the useof In-Situ TPA methods (1) for the prediction of radiated noise. Considering an arbitrary source andreceiver structure, predictions are made for a remote response point on the receiver by using a matrix
of collected FRFs from the coupling points of the source and receiver as a reference point to validate
the accuracy of the predictions. The wider scope of the project aims to evaluate methods for theprediction of noise created by the structure, consisting of multiple sources, with a focus on methods
which can be implemented in real time. Also considered are elements of structural health monitoring,such as unmeasured background noise, extra connection points, and source de -activation.
The paper outlines the background theory of transfer path analysis and in-situ transfer path analysis
using the matrix inversion method (section 2.1) together with alternative formulations using crossspectral matrices and principal component analysis (section 2.2). Following that an experim ent is
described which consists of two vibration sources incorporated into an assembly and it is shown thatin-situ transfer path analysis can be used to identify the contributions from each source to the overall
response of the receiver structure when both sources are operating simultaneously. Finally, a fault(bridging of the vibration isolation) on one of the vibration sources is introduced and it is shown thatthe sum of the contributions of the two sources no longer equals that observed in the receiver structure
due to the additional path. A future aim of the study is to investigate methods that not only identify
the presence of a fault but also to locate its origin, i.e. which item of equipment is affected, ideallyusing operational data measured simultaneously on the source and receiver structures.

435

2. THEORY
Following is a description of the TPA approach and Principal Component Analysis. The theory ispresented for a simple source- receiver structure, where one or more sources (A) are coupled to anarbitrary receiver structure (B) through multiple contacts at an interface (c) . The target quantity to
be predicted can be either a remote point away from the structure itself (sound pressure at d), or aremote point on the receiver (e.g. acceleration at b).
2.1 Conventional Transfer Path Analysis
The conventional matrix inversion TPA method, or source path contribution method, uses inverse
force synthesis to find the operational forces at work on the interface between the vibrating source
and the receiver of an assembly.

Figure 1: Schematic for source- receiver structure
Figure 1 shows a schematic diagram of an arbitrary vibration source A coupled to an arbitraryreceiver structure B. The vibration source has multiple generalised forces at (a) during operation,
which may then apply forces to the receiver structure B through the interface (c).
The sound pressures at position (d) are a combination of the airborne sound radiated by thesource A and the structure borne sound radiated by the receiver B. The points (b) shown on
subsystem B of the assembly C may be considered as a set of vibration velocities due to the internalmechanisms of A. Inverse force synthesis (2)A set of complex operational forces, fC,c, at interface (c) can be found using,܎ሶ ۱ǡ‫ ܋‬ൌ ‫܇‬െ૚۰ǡ‫ܞ ܋܋‬ሶ ۱ǡ‫܋‬(1)܎ሶ ۱ǡ‫ ܋‬ൌ ‫܇‬൅۰ǡ‫ܞ ܋܊‬ሶ ۱ǡ‫܊‬(2)‫ܡ‬۰ǡ‫܋܋‬൅܎ሶ ۱ǡ‫ ܋‬ൌ ൤ ‫ ܂‬൨ ൜‫܇‬۰ǡ‫܊܋‬‫ܞ‬ሶ ۱ǡ‫܋‬‫ܞ‬ሶ ۱ǡ‫܊‬ൠ

(3)

Where: upper-case subscripts are the structure A, B, C; lower case are measurement location(a), (b), (c), (d); and a dot denotes that the source is operational. ‫ܞ‬ሶ ۱ǡ‫ ܋‬and ‫ܞ‬ሶ ۱ǡ‫ ܊‬are the vectors ofcomplex velocities at measurement locations (c) and (b), on assembly C, whilst the source A is‫܂‬
operational. Subscripts denote row (response) then column (excitation): ‫ܡ‬۰ǡ‫ ܋܋‬and ‫ܡ‬۰ǡ‫ ܋܊‬ൌ  ‫܇‬۰ǡ‫܊܋‬arematrices of frequency response functions of velocities due to force/ mobility (3); ‫ܡ‬۰ǡ‫ ܋܋‬is a squaresymmetric mobility matrix for the receiver B, where the rows are the responses and columns are
excitations on the interface (c).
‫܂‬. It is not symmetric and may not be square, so is pseudo‫ܡ‬۰ǡ‫  ܋܊‬is usually measured as ‫܇‬۰ǡ‫܊܋‬
inverted. Equation 3 includes a square symmetric matrix, allowing optimization by over436determination and regularization, which is desirable due to the fact that i nverting the often large
FRF matrix for finding the operational forces at (c) can be highly sensitive to noise and otherinconsistencies. In most cases equation 1 is sufficient and Equation 2 might be used in the case theinterface (c) is not physically accessible, which can be a problem when manually exciting thestructure.
The sound pressure at point (d) is then found using:ሶ‫۾‬ሶ۱ǡ‫ ܌‬ൌ ۶۰ǡ‫܎ ܋܌‬۱ǡ‫܋‬(4)

Where ‫۾‬ሶ۱ǡ‫ ܌‬is a complex vector of radiated sound pressures from the receiver B, caused by theoperational source A. The matrix ۶۰ǡ‫ ܋܌‬is made up of sound pressure frequency response functions
due to force, or through reciprocal measurements, velocity due to volume velocity excitation (4).
2.2

In-situ Transfer Path Analysis

In-situ transfer path analysis is an alternative to conventional TPA that does not require the source
receiver-assembly to be dismantled with the use of blocked forces rather than interface forces, where
the blocked force for the source A in Figure 2 is defined by:ሶ ҧ ൌ െ܎ሶ ห܎‫ۯ‬ǡ‫܋‬‫ۯ‬ǡ‫ܞ ܋‬ሶ(5)‫ۯ‬ǡ‫ ܋‬ୀ૙where the bar denotes that the force is blocked.
The mobilities of the subsystems which make up the assembly C provide the relationship between
the operational blocked forces of the source A and the operational forces of the assembly C.
ሶҧሶ ൌ ൫‫ۯ܇‬ǡ‫ ܋܋‬൅ ‫܇‬۰ǡ‫ ܋܋‬൯ି૚ ‫ۯ܇‬ǡ‫ۯ܎ ܋܋‬ǡ‫܋‬܎۱ǡ‫܋‬(6)

Which after substitution and rearrangement of equations 1-3, gives the blocked force version ofthese equations.
ሶ ҧ ൌ ‫ି ܇‬૚ ‫ܞ‬ሶ(7)܎‫ۯ‬ǡ‫܋‬۱ǡ‫܋‬۱ǡ‫܋܋‬ሶ ҧ ൌ ‫ ܇‬ା ‫ܞ‬ሶ܎‫ۯ‬ǡ‫܋‬۱ǡ‫ ܋܊‬۱ǡ‫܊‬ା(8)ሶ ҧ ൌ ൤ ۱ǡ‫ ܋܋‬൨ ൜ ۱ǡ‫ ܋‬ൠ܎‫ۯ‬ǡ‫܋‬
‫܂܇‬
‫ܞ‬ሶ
‫܇‬

‫ܞ‬ሶ

۱ǡ‫܋܊‬(9)۱ǡ‫܊‬Finally, assuming that the sound pressure at (d) caused by structure A during the measurement of H isnegligible, the structure borne noise at (d) is:ሶҧ‫۾‬ሶ۱ǡ‫ ܌‬ൌ ۶۱ǡ‫ۯ܎ ܋܌‬ǡ‫܋‬

(10)

437

where ۶۱ǡ‫ ܋܌‬is the measured transfer function for the entire assembly C, as opposed to ۶۰ǡ‫܋܌‬, usedin equation 4, which is only for the receiver structure. Thus, the method does not require the separationof source and receiver saving time and effort. Another advantage of the in-situ approach is that theoperational blocked forces are independent of the receiver structure, meaning that they remain valid
for different assemblies by re-measuring or modelling ۶۱ǡ‫ ܋܌‬for the new assembly. This property is
not shared by the conventional approach as the contact forces measured can vary significantly between
receiver structures (5).
2.3. Principal Component Analysis
When a system has multiple uncorrelated system inputs, the measured operational data issometimes conditioned using principal component analysis to avoid issues resulting from averaging
in the frequency domain (6,7). Considering the matrix inversion TPA approach outlined in section
2.1, figure 1 shows the interfaces of a coupled system. If there are multiple uncorrelated operational
forces (vibration source mechanisms) acting at the (a), the velocities at (c) may not be fully
correlated, meaning there may not be a steady phase relationship between the (c) positions. The
relative phases of the forces at (c) must be known to ensure the phase information is correct for thepartial pressure contribution predictions (1). By referencing the operational data to virtual sources,
or principle components, PCA aims to alleviate this issue.
As opposed to the vector form of complex operational data shown in the matrix inversion method,
PCA uses a cross spectral matrix, for which a definition for the sets of points and degrees of
freedom on interface (c) is shown by:۶۵ሶ۱ǡ‫ ܋܋‬ൌ ۳උ‫ܞ‬ሶ ۱ǡ‫ܞ ܋‬ሶ ۱ǡ‫܋‬ඏ

(11)

Where ۵ሶ is the averaged cross spectral matrix of the velocities, ‫ܞ‬ሶ ۱ǡ‫ ܋‬is instantaneous velocity
vector, ‫ ܧ‬denotes ‘expectation of’, and ‫ ܪ‬is the Hermitian transpose. Equations 1-3 and 7-9 can all
be converted to this form by post-multiplying both sides by its own complex conjugate transpose. Forି૚ሶ ൌ ‫܇‬۰ǡ‫܋܋‬‫ܞ‬ሶ ۱ǡ‫ ܋‬becomes:
example, equation 1 ܎۱ǡ‫܋‬ି૚ ሶି۶۴ሶ۱ǡ‫ ܋܋‬ൌ ‫܇‬۰ǡ‫܋܋‬۵۱ǡ‫܇ ܋܋‬۰ǡ‫܋܋‬(12)ሶ ҧ ൌ ‫ି ܇‬૚ ‫ܞ‬ሶ becomes:
And equation 7 ܎‫ۯ‬ǡ‫܋‬۱ǡ‫ ܋܋‬۱ǡ‫܋‬ି૚ ሶି۶۴തሶ‫ۯ‬ǡ‫ ܋܋‬ൌ ‫܇‬۰ǡ‫܋܋‬۵۱ǡ‫܇ ܋܋‬۰ǡ‫܋܋‬(13)

ሶ
Where ۴തሶ is the force cross spectral matrix, instead of ݂ (force vector). Equation 4 ‫۾‬ሶ۱ǡ‫ ܌‬ൌ ۶۰ǡ‫܎ ܋܌‬۱ǡ‫܋‬
now becomes:۶۵ሶ۱ǡ‫ ܌܌‬ൌ ۶۰ǡ‫ ܋܌‬۴ሶ۱ǡ‫ ܋܋‬۶۰ǡ‫܋܌‬(14)

ሶ ҧ becomes:
And equation 10 ‫۾‬ሶ۱ǡ‫ ܌‬ൌ ۶۱ǡ‫ۯ܎ ܋܌‬ǡ‫܋‬۶۵ሶ۱ǡ‫ ܌܌‬ൌ ۶۰ǡ‫܋܌‬۴തሶ‫ۯ‬ǡ‫ ܋܋‬۶۱ǡ‫܋܌‬(15)

Here, ۵ሶ۱ǡ‫ ܌܌‬is a cross-spectral matrix of sound pressure at (d) - the auto-spectra are on the diagonal.
A principle component analysis can now be conducted using a set of slave/ reference sensors fordetermination of the ‘virtual sources’. If these sensors are placed at (a), the cross -spectral matrix۶۵ሶ۱ǡ‫ ܉܉‬ൌ ۳උ‫ܞ‬ሶ ۱ǡ‫ܞ ܉‬ሶ ۱ǡ‫܉‬
ඏof size n × n is defined (where n is the number of sensors at (a)). The auto-spectraof the principle components can then be found from the eigenvalue or singular value decomposition438

of the reference cross spectral matrix:۶۵ሶ۱ǡ‫ ܉܉‬ൌ ۳උ‫ܞ‬ሶ ۱ǡ‫ܞ ܉‬ሶ ۱ǡ‫܉‬ඏ ൌ ‫܁܃‬ሶ۱ǡ‫ ܃ ܉܉‬۶(16)

‫ ܃‬is a unitary matrix and the singular values ‫܁‬ሶ۱ǡ‫ ܉܉‬are the principle components. Velocity spectra
referenced to the principle components (6) are then found for the degrees of freedom at (c) and (b)using:
ᇱ(17)‫ܞ‬ሶ ۱ǡ‫܋‬۵ሶ۱ǡ‫܋܉‬ି૚Ȁ૛ቈ ᇱ ቉ൌቈ቉ ሾ‫܃‬ሿൣ‫܁‬ሶ۱ǡ‫ ܉܉‬൧ሶ‫ܞ‬ሶ ۱ǡ‫܊‬۵۱ǡ‫܋܊‬ᇱᇱare the referenced velocity spectra with rows relating to the velocities at (c), ‫ܞ‬ሶ ۱ǡ‫܊‬areWhere ‫ܞ‬ሶ ۱ǡ‫܋‬
the referenced velocity spectra with rows relating to the velocities at (b) and the columns of each areᇱ
canreferenced to principle components according to the references at (a). The reference spectra ‫۾‬ሶ۱ǡ‫܌‬
be converted from the sound pressures at (d) using:
ି૚Ȁ૛
ᇱ
(18)
ൣ‫۾‬ሶ۱ǡ‫܌‬൧ ൌ ൣ۵ሶ۱ǡ‫ ܌܉‬൧ሾ‫܃‬ሿൣ‫܁‬ሶ۱ǡ‫ ܉܉‬൧The matrix inversion method can then be conducted using the referenced spectra in (7.7) to f ind apartial pressure contribution prediction for each position of (d), with reference to a virtual source. The
conventional method would take the form:ା ‫ܞ‬ሶ ᇱ
(19)
‫܇‬
۱ǡ‫܋‬
ᇱ
ൣ‫۾‬ሶ۱ǡ‫܌‬൧ ൌ ۶۰ǡ‫ ܋܌‬൤ ۰ǡ‫܋܋‬‫ ܂‬൨ ቈ ᇱ ቉‫܇‬۰ǡ‫܊܋‬‫ܞ‬ሶ ۱ǡ‫܊‬With the in-situ method taking the form:ା ‫ܞ‬ሶ ᇱ
‫܇‬
۱ǡ‫܋‬
ᇱ
ൣ‫۾‬ሶ۱ǡ‫܌‬൧ ൌ ۶۱ǡ‫ ܋܌‬൤‫܇‬۱ǡ‫܋܋‬‫ ܂‬൨ ቈ ᇱ ቉‫ܞ‬ሶ۱ǡ‫܊܋‬
۱ǡ‫܊‬

(20)

ᇱ
‫ܞ‬ሶ ۱ǡ‫܋‬
Where equation 17 is used to obtain ቈ ᇱ ቉ (1).
‫ܞ‬ሶ ۱ǡ‫܊‬

3. EXPERIMENTThe experiment was carried out on a bespoke test rig, consisting of an aluminium box with a steel‘lid’ plate. Two smaller rafts are attached to the lid via vibration isolators, with one isolator at thecorner of each raft.

Figure 2: Test Rig
In order to test as many positions as possible, the testing was carried out in stages . Firstly, themotor was placed in situ and the coupling points were instrumented with six accelerometers each inorder to obtain accelerations in 5 degrees of freedom (three translations and two rotations) . To create
a better surface for impacts, the accelerometers were mounted on b locks at each coupling point.
FRF’s for five degrees of freedom at each coupling point were then obtained using an instrumeted439

hammer. A triaxial was mounted on the motor itself as a phase reference during operational
measurements.

Figure 3&4: Motor test and receiver test
Following the FRF measurements, testing with motor running was conducted. The motor was run
at speeds of 2kHz, 3kHz, and 4kHz.
The accelerometers were then transferred to the shaker coupling points and the testing repeated .
The accelerometers were then moved to twenty-one remoted positions on the receiver box, with atriaxial accelerometer on each of the sources.

4. RESULTSି૚ ሶି۶
Equation 13 ۴തሶ‫ۯ‬ǡ‫ ܋܋‬ൌ ‫܇‬۰ǡ‫܋܋‬۵۱ǡ‫܇ ܋܋‬۰ǡ‫܋܋‬, was used to determine the forces required as the input to۶Equation 14, ۵ሶ۱ǡ‫ ܌܌‬ൌ ۶۰ǡ‫ ܋܌‬۴ሶ۱ǡ‫ ܋܋‬۶۰ǡ‫܋܌‬, Where H is a set of transfer FRF’s relating the source receivercoupling points to the response point on the box. These values were calculated for both the shaker andmotor, as seen in the lower plot of figure 5.

Figure 5: Plot of predictions using Equations 14 and 15 (motor 4kHz speed)This plot shows the individual contributions of source to the overall measured vibration level. It
can be observed that the shaker contributes most of the broadband signal, whereas the tonal peak at~500Hz is contributed by the motor. The upper plot shows the two predictions added together tomake an overall combined prediction. Good agreement is observed at this operational speed (4kHz).

440

Figure 6: (motor 3kHz)

Figure 7: (motor 2kHz)

Figures 6&7 show that the prediction works for the varying speeds of motor; the peaks in themeasured response and in the calculated prediction appear to have very good agreement, and areaccurate from ~20Hz, up to approximately 1kHz. This restriction may be due to the degrees of freedom
used. A more detailed instrumentation of the coupling points may lead to better accuracy at higher
frequencies. The frequency range is also dependent on the accelerometers used and the level ofvibration isolation provided by the vibration isolators under each raft structure ; for example,
accelerometers with a higher sensitivity could potentially give better predictions at both ends of thefrequency range shown.

5. FAULT CONDITION
As mentioned in the introduction one of the aims of the project is to identify system faults using
the measured operational data in real time. For this reason a known fault was introduced to the source
receiver assembly by bridging the isolation system on one of the sources.
The introduced fault condition consisted of a small magnetic structure which created a bridge
between the motor raft and the lid of the box, which created an unaccounted connection point
between the source and the receiver.

Figure 8: Fault condition

Figure 9: Fault condition results (motor 4kHz)

441

The upper plot of Figure 9 shows the measured response against the calculated response, as
shown in Figures 5. It can be seen that additional tones are now present in the receiver response.
The lower plot shows the individual contributions of each source. The motor measurement
corresponds to these additional tones in the overall response, but the overall level calculated fromthe blocked forces of the two sources is now no longer in agreement at these frequencies ,highlighting the presence of the fault.

6. Conclusions
In conclusion, the contributions from each source have been accurately identified with simultaneous
source operation using cross-spectral matrices. It is also shown that following the introduction of a
fault, the sum of the contributing sources no longer corresponds completely with the observed
vibration. The next stages implied are to further investigate the possibilities of source separation using
the PCA methods described, and to carry these out in the presence of background noise.

REFERENCES
1.

Elliott AS, Moorhouse AT, Huntley T, Tate S. In-situ source path contribution analysis of structure borne
road noise. J Sound Vib. 2013;

2.

Dobson BJ, Rider E. A Review of the Indirect Calculation of Excitation Forces from Measured Structural
Response Data. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of MechanicalEngineering Science. 1990.

3.

Ewins DJ. Modal testing : theory, practice, and application. 2nd ed. Philadelphia, PA: Research Studies
Press, Baldock, Hertfordshire, England; 2000.

4.

Verheij JW. Inverse and Reciprocity Methods for Machinery Noise Source Characterization and Sound
Path Quantification Part 1: Sources. Int J Acoust Vib [Internet]. 1997;2(1):11–20. Available from:http://s3.amazonaws.com/academia.edu.documents/38269668/Jan_Verheij_Inverse_and_Reciprocity_Methods_for_Machinery_Noise__Part_1_Sources.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1494226668&Sign
ature=nopyeQWx1toaVcDhxLgfPDXQ5L4%3D&response-content-

5.

Elliott A. Characterisation Of Structure Borne Sound Sources In-Situ. PhD Thesis. University Of Salford,
Greater Manchester; 2009.

6.

D. Hendricx WSFV. Interior road noise optimization in a multiple input environment. Inst Mech Eng Conf
Publ 3. 1994;3:53–8.

7.

Otte, D. Development and Evaluation of Singular Value Analysis Methodologies for Studying Multivariate
Noise and Vibration Problems [Internet]. PhD K. U. Leuven. 1994 [cited 2018 Aug 23]. Available from:https://ci.nii.ac.jp/naid/10003083914/

442