Transfer Path Analysis

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Transfer path analysis

This document describes how a Transfer path analysis is done.It covers:

� the theory of the different methodologies provided

� the type of test data required for the different methodologies

� the overall procedure and parameters used in the processing

� the organization and storage of results

� the results and the post processing facilities

Transfer Path AnalysisTransfer Path Analysis

page 2 LMS CADA−X Manuals

Transfer Path Analysis Transfer Path Analysis

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1 Introduction

The CADA−X TRANSFER PATH ANALYSIS module allows you to assess the possible waysof energy transfer from the various sources of excitation in an assembly to a given tar−get location. It supplies the tools required to locate the most important energy trans−fer paths for a specific problem, and to evaluate their individual effects on the target,thus providing valuable insight into the mechanisms responsible for the problem.

The essential elements in the analysis of the problem are illustrated in the schematicdiagram and listed below.

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

ÏÏÏÏ

ÏÏÏÏ

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

ÉÉÉÉÉÉ

ÏÏÏÏÏÏexcitation sources target locations transmission paths

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

ÏÏ ÏÏÏÏ

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

Figure 1 Schematic representation of the essential elements of an acoustic−vibrationassembly

� The excitation sourcesThese can be structure and/or airborne, acoustical or vibrational. Typicalsources include the vibration of a car engine, road induced vibrations or ra−diated noise from a vibrating panel.

� The target locationsThese are typically acoustical such as the acoustic pressure perceived by thepassengers in a car during engine run−up, but can be vibrational too such asthe vibrations of the steering wheel. It could also be the vibration of a spacestation caused by an instrument in operation on the carrier platform.

� The transfer pathsStructural transfer paths are represented by the physical mounts and rigidconnections whereby the noise and vibration are transferred from the sourceto the target location. Examples of airborne transfer paths are vibrating pan−els, intake or exhaust noise.

This document follows the format of a Transfer Path Analysis as carried out using theCADA−X TRANSFER PATH ANALYSIS software.



It starts with the basic theory used in relating the source of noise or vibration and thetarget location via the transfer paths, which are described in section 2. This includesthe processing of measurements to obtain volume accelerations and velocities whichcan be done in the panel contribution analysis module. Such data is required for anal−ysis of airborne sources from panels.

Section 3 describes the type of source data required for the analysis and important as−pects regarding its acquisition.

Section 4 explains the overall stages and parameters involved in transfer path analysis,including the data and memory structures used, while section 5 describes the resultsthat are obtained and how to visualize them.


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2 Theory

The basic methodology uses an FRF model description of the vibro−acoustical sys−tem which relates a loading or excitation vector {s(�)} to the target response vector{t(��} by an FRF matrix [H(�)]

{t(�)} � [H(�)]� .�{s(�)} Eqn 1

where s (or the source) can be a force or a volume velocity depending onwhether you are considering a structural or airborne path respective−ly.

The various targets and sources are considered as two different subsystems as illus−trated below. These subsystems are connected to each other by means of a numberof more or less stiff connections (or mounts), forming the so−called transfer paths.

ÉÉÉÉÉÉ

excitationsubsystem

targetsubsystem

transfer paths

ÉÉÉÉÉÉÉÉÉÉ

ÉÉÉÉÉÉÉÉ

ÉÉÉÉÉÉÉÉ

ÉÉÉÉÉÉÉÉ

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

ÉÉÉÉÉ

vibro acousticassembly

Figure 2 Schematic representation of a vibro−acoustic assembly with structure bornetransfer paths

If the system is composed of N transfer paths, then the total target response can bewritten as the sum of the partial responses from the individual paths.

t � (�) ��N

i�1

T(�)

Si(�)�si(�) Eqn 2

t�(�) is the target response which can be a function of frequency or rpm.

T(�)

Si(�) is the frequency response function between the target and the appliedsource (force or volume velocity) for transfer path idiscussed in section 2.1.



si� (�) is the operational force or volume velocity at transfer path i.Operational forces would be used for a structural path and are dis−cussed in section 2.2. Volume velocities are needed for airborne pathanalysis and are discussed in section 2.3.

2.1 The frequency response functions

Frequency response functions are required for all transfer paths at the target locationsin order for the location to be considered as a target. These would normally be mea−sured after disassembling the sources from the assembly structure to eliminate sourcecoupling of the FRFs.

Each direction at a given location is considered as one structural transfer path. It ispreferable that these FRFs should be measured with the source side disconnectedfrom the target side. Either hammer impact, or shaker excitation can be used. Theresponse can be acoustical or a mechanical.Airborne transfer paths are typically measured in a reciprocal way. The excitation isdone with a loudspeaker at the receiver location while the response is measured witha microphone at the source location.

The targets are described by means of the primary identifications of the frequency re−sponse functions.The secondary identification of the FRF in a path corresponds to the primary identifica−tion of the operational data at the target side.

2.2 The operating forces

These are required for a structural transfer path analysis. For details on the actualdata required see section 3.1. When performing an airborne source analysis operatingvolume velocities and accelerations are required which are discussed in section 2.3.Operating forces can be available directly as experimental data or can be derived indi−rectly from other physical quantities using the following analytical procedures.

The complex dynamic stiffness method

In cases where mounts are used to insulate the target from the source, operationalforces can be determined indirectly using the complex dynamic stiffness of the mountsand the operational displacements at either side of the mount during operation. Themethod is based on the following equation:


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fi(�) � K(�)�(�Xs(�)�Xt(�)�) Eqn 3

fi(�) is the operational force at transfer path i

K�(�) is the complex dynamic stiffness as function of frequency �

Xt�(�) is the displacement at the mount connection at the target side

Xs�(�) is the displacement at the mount connection at the source side

Further information on the data requirements for this methodology is given in section3.1.2.

The matrix inversion method

This method must be used when the transfer paths are formed by rigid connectionssince the mount method in inappropriate for this type of connection. This is also thecase for a mount whose stiffness is very large compared with the rest of the structureso that only a minimal relative displacement is achieved under normal operatingconditions.Under these conditions, you need to assemble FRFs in a matrix. These FRFs aremeasured between the structural (accelerations) or acoustical responses due to forceexcitation at all transfer paths. The matrix is then inverted and combined with opera−tional measurements of the structural responses in order to obtain force estimates.

{f (�)} � [H(�)]�1� .�{t�(�)} Eqn 4

or

Eqn 5��

f1....fN��

��

�

T1�F1

.

.

.

.TM�F1

�T1�F2

T2�F2....

��....

T1�FN

.

.

.

.TN�FN

��

�

�

�1

�x��

t1....tM��

�

In order to avoid numerical problems in the matrix inversion, singular value decom−position methods are used. The number of responses (M) should at least be equal tothe number of input forces to be estimated (N), in order to have a unique solution forthe operational forces. However, taking more response measurements at the targetside (M > N), the measurement locations not being restricted to the transfer pathlocations, allows you to overdetermine the set of equations. This allows you to obtaina more accurate least square estimate results, for the operational forces.



Further details on the measurement of this matrix are given in section 3.1.3.

The driving point inversion method

This method is the fastest, but the most inaccurate one. It is only applicable whencross−coupling effects can be neglected.To use this method only the diagonal elements of the matrix are required since thecross−elements are considered zero.

2.3 Operating volume velocities/accelerations

When performing an analysis of airborne transfer paths, operating volume velocitiesor accelerations are required. The radiating surface (source) is subdivided into small−er areas called patches and airborne source quantification assesses the volume accel−eration or volume velocity of different source patches.

By quantifying the airborne source by their volume acceleration or volume velocity,rather than their power, a measure for the source strength is obtained which is sup−posed to be independent of the (acoustic) boundary. The subdivision of the radiatingsurface into patches is crucial, but a useful guideline is that the dimension of thepatches should be smaller than the smallest acoustic wavelength of interest (depen−dent on frequency range) divided by 6.

Volume acceleration Q.

will be used to specify the load vector, but it is always possibleto use volume velocity instead Q. These quantities can be computed using a numberof different methodologies discussed in this document. Further details on the inputdata required can be seen in section 3.2. The processing of this data is done in thePanel contribution analysis module described in the ‘Using Panel contribution analy−sis’ document in the �Panel contribution analysis" book.

Operational volume velocities are available either as experimental data, analyticalsimulations, or more commonly, have to be determined indirectly.

Point to point surface sampling

This method allows you to derive the equivalent source volume acceleration from asampled scanning of the radiating surfaces. Different types of measurements can bemade.


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Acceleration measurements

The assumption is that the acceleration, measured at a given point in the normaldirection on the surface S, represents the (constant) acceleration profile that canbe used in a given surface area, for quantifying the volume acceleration of that sur−face. This requires the global radiating surface to be divided into individual sur−faces.

S1

S2

S3

Figure 3 Acceleration measurements on panel patches

The volume acceleration Q.

j of each of these m sampled surfaces Sj is then calcu−

lated as:

Eqn 6Q

.

j � Aj � x..

jm

where Q.

j is the volume acceleration for surface Sj

Aj is the area of surface Sjx..

jm is the normal surface acceleration at surface Sj

The volume velocity can of course be calculated as the integral of the volume accel−eration.

Laser scanning measurements

With laser scanning measurements velocity is measured, rather than acceleration.In principle however, there is no real difference between laser scanning and accel−eration measurements.

Both measurement techniques can be used for interior and exterior radiation prob−lems.



The matrix inversion techniques

Matrix inversion can be used to estimate indirectly the volume velocity of an acousticsource. To do this a number of indicator pressure responses (pj) are measured closeto the source under operating conditions. In combination with near−field transferfunctions [H] between pressure at those indicator positions and volume velocity by theradiating surface, the operating volume velocity of the radiating acoustic source canbe calculated.

ÄÄ

ÄÄÄÄ

ÄÄ

S1S2

S3

p1

p2

p3

q1

q3q2

source

{q.} � [H]�1{p}

Figure 4 Measurement of indicator pressure responses

The transfer function matrix is measured in a reciprocal way by putting loudspeakersat the location of the indicator pressure microphones, and microphones placed at theradiating surface as illustrated below.


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S1S2

S3

p1

p2p3

Q3

[Hi� j] �pi

Qj

ÄÄÄÄÄÄ

ÄÄ

Q1

Q2

Figure 5 Measurement of inverse transfer functions

Eqn 7��

q1....qn��

��

�

P1�Q1

.

.

.

.Pm�Q1

�P1�Q2

P2�Q2....

��....

P1�Qn

.

.

.

.Pn�Qn

��

�

�

�1

� ��

p1....pm��

�

The matrix inversion can become ill−conditioned. By overdetermination, i.e. by tak−en the number of indicator pressures (m) greater than the number of equivalent vol−ume velocity sources (n) the condition of the matrix can be improved.

Intensity measurements

Under steady−state conditions near−field intensity measurements on the source canbe performed in an anechoic room (not in situ). This results in a sound power estimateof the source under anechoic conditions. From this sound power, an estimate can bederived for the (phase−less) volume velocity of the source. The volume velocity ofthe source is assumed to be invariable with changing acoustic boundary conditions.

From intensity measurements the mean intensity of a measurement surface S aroundthe source can be obtained. The mean intensity is given by the time averaged product of sound pressure and par−ticle velocity :

Eqn 8I�� 1

T�

T

0

p�(t)�v�(t)�dt



where p� is the pressure and v� is the particle velocity.

I1

S1

S2

S3

I2

I3

Figure 6 Measurements of sound intensity

The surface S is divided into m subareas Sj which are considered as partial sources.

Integrating the scalar product of the mean intensity vector I�

and the normal vector n�

over a subarea Sj leads to the radiated sound power from that subarea Sj :

Eqn 9Powj � �Sj

I�� n

��ds

The sound power Powj can be obtained from the Acoustic monitor.

If you assume that the radiating source can be represented by a monopole sourceplaced against an acoustically hard and infinite surface, then the following relation−ship between the volume velocity and the sound power can be derived.

Eqn 10Q2j � n(j) � Q2

jeq � Powj�2�c��2

where c= the particle velocity of air and � = the density of air.

Note! In this method no phase information is available of the source strength (Q2 is calcu−lated). For low frequencies, however the assumption that the equivalent sources areuncorrelated will no longer hold and as a result phase information does become im−portant.

When the assumption of a monopole radiating into half space no longer holds, a factorC can be introduced. This factor has to be entered by the user and is defaulted to 2(monopole radiating into half space).



Eqn 11Q2j � Powj �

4�c��2

� 1C

Comparison of methods

Point to point surface sampling is only possible when the surface can be divided intoa number of well defined subareas for which the normal acceleration and area can becalculated or measured. This method can be used for instance to qualify the contribu−tions from the different panels in a car cavity.

Sound intensity measurements have less restrictions on the complexity of the radiat−ing surface. This method has proved to give good results, except for the low frequencyrange. A disadvantage is that the sound intensity measurements have to be performedin anechoic conditions and that in practice they are limited to stationary conditionsmeaning that no run−up/run−down measurements are possible.

The inverse method on the contrary is not restricted to stationary conditions and themeasurements are also performed under realistic conditions (source in situ). Thismethod however requires a large number of transfer function measurements and thecondition of the matrix inversion can be a problem. The inverse method could alsobecome ill−conditioned when the pressure indicator microphones and the target mi−crophones are situated in the same cavity (as for instance would be the case for panelcontribution applications).

2.4 The Multiple reference problem

For multiple reference problems, such as road induced noise, another approach isnecessary. A characteristic of such problems is that several mutually incoherent orpartially coherent inputs are acting simultaneously on the system. Depending on thetime history of the sources, the momentary amplitude and phase relation of the differ−ent contributions can vary widely. The transfer path analysis therefore is preceded bya principal component decomposition in which the multi−reference problem is splitup into several orthogonal single reference problems. Each of these individual prob−lems are then treated as a single reference transfer path analysis. The total contribu−tion of a single transfer path to the overall noise or vibration level is then calculatedas the RMS sum of each of the individual principal components’ contributions. Moreinformation on this subject can be found in the ‘Theoretical considerations’ documentof the �Principle component analysis" book. This overall approach is illustrated inFigure 7.



x1 x2 . . . xr

x11 ... x21 ... xr1 x12 ... x22 ... xr2 x1m ... x2m ... xrm

PCA

x11 ... x21 ... xr1 x12 ... x22 ... xr2 x1m ... x2m ... xrm

Forceid.

f11 ... f21 ... fn1 f12 ... f22 ... fn2 f1m ... f2m ... fnm

phenomenon 1

TPA

phenomenon 2 phenomenon m

p1 ��i

Hai .fi 1 p2 ��

i

Hai .fi 2 pm ��

i

Hai .fim

p � � |pi|2�

Figure 7 Multi−reference Transfer path analysis



3 Data requirements

Depending on the analysis to be performed Transfer Path Analysis requires differenttypes of input data.

Section 3.1 describes the data requirements for a structural transfer path analysis. Insections 3.2 and 3.3 the details about an airborne transfer path analysis are given. Itis also possible to combine structural and airborne transfer paths in one analysis bysupplying the program with the appropriate input data.

Section 3.4 describes which types of operational data is supported in Transfer PathAnalysis and how it can be obtained.

3.1 Structural transfer path analysis

Consider a simple application with an engine which is flexibly connected to the carbody in two points body:1 and body:2. When the three translation degrees of free−dom are taken into account, this results in 6 transfer paths :

body:1:+X, body:1:+Y, body:1:+Z body:2:+X, body:2:+Y, body:2:+Z.

As a target a microphone response in the car cavity mic:1:S is taken.

engi:1 engi:2

body:1 body:2 body:1002body:1001

K2K1

mic:1:S



3.1.1 Direct method

If it is possible to measure the force at the 6 transfer paths, the direct methodologycan be used (no indirect force estimation required). In this case the following datais required to perform a transfer path analysis.

Operational data

6 force records measured in the transfer path locations.

Primary identifier Secondary identifier Function class Unit labelbody:1:+X - frequency_spectr Nbody:1:+Y - frequency_spectr Nbody:1:+Z - frequency_spectr Nbody:2:+X - frequency_spectr Nbody:2:+Y - frequency_spectr Nbody:2:+Z - frequency_spectr N

Transfer functions

6 vibro−acoustical FRFs between the transfer paths and the acoustical target. Theprimary identifier of the FRFs corresponds to the target location, the secondary iden−tifiers refer to the transfer paths.

Primary identifier Secondary identifier Function class Unit labelmic:1:S body:1:+X FRF Pa/Nmic:1:S body:1:+Y FRF Pa/Nmic:1:S body:1:+Z FRF Pa/Nmic:1:S body:2:+X FRF Pa/Nmic:1:S body:2:+Y FRF Pa/Nmic:1:S body:2:+Z FRF Pa/N

3.1.2 Complex dynamic stiffness method

If the connections between source and target subsystem are flexible the complex dy−namic stiffness method can be used. The following data sets are required:

Operational data

12 accelerations measured at the mounts: 6 on the source side (engi:1 and engi:2) and6 on the body side (which are the transfer path locations).



Primary identifier Secondary identifier Function class Unit labelengi:1:+X - frequency_spectr m/s2

engi:1:+Y - frequency_spectr m/s2

engi:1:+Z - frequency_spectr m/s2

engi:2:+X - frequency_spectr m/s2

engi:2:+Y - frequency_spectr m/s2

engi:2:+Z - frequency_spectr m/s2

body:1:+X - frequency_spectr m/s2

body:1:+Y - frequency_spectr m/s2

body:1:+Z - frequency_spectr m/s2




Transfer functions



Complex dynamic stiffness data

6 FRFs describing the complex dynamic stiffness of the mount. There is no strict nam−ing convention in existence, but but it is advisable to set the primary identifiers to cor−respond to the transfer paths and the secondary identifiers to correspond to the pointsdescribing the source side of the mounts. The unit of the mount can be ‘force overdisplacement’, ‘force over velocity’ or ‘force over acceleration’.

Ref Primary identifier Secondary identifier Function class Unit labelK1 body:1:+X engi:1:+X FRF N/mK1 body:1:+Y engi:1:+Y FRF N/mK1 body:1:+Z engi:1:+Z FRF N/m



K2 body:2:+X engi:2:+X FRF N/mK2 body:2:+Y engi:2:+Y FRF N/mK2 body:2:+Z engi:2:+Z FRF N/m

3.1.3 Matrix inversion method

If the connections between source and target subsystem are rigid or very stiff, then ma−trix inversion can be used. In our example we are applying an overdetermination fac−tor of 2 which means that 6 extra accelerations have to be acquired at the target subsys−tem. This results in the following data :

Operational data

6 accelerations measured at the the transfer path locations and 6 extra accelerationsat the car body (body:1001 and body:1002).

Primary identifier Secondary identifier Function class Unit labelbody:1:+X - frequency_spectr m/s2












Target transfer functions





Accelerance transfer functions

72 accelerance FRFs between the transfer paths and the 12 acceleration responses (6responses at the transfer paths and 6 extra acceleration responses). The primary iden−tifier of the FRFs corresponds to the response location, the secondary identifiers referto the transfer paths. In the following table only the first 13 of 72 data records arelisted.

Primary identifier Secondary identifier Function class Unit labelbody:1:+X body:1:+X FRF m/s2/Nbody:1:+Y body:1:+X FRF m/s2/Nbody:1:+Z body:1:+X FRF m/s2/Nbody:2:+X body:1:+X FRF m/s2/Nbody:2:+Y body:1:+X FRF m/s2/Nbody:2:+Z body:1:+X FRF m/s2/Nbody:1001:+X body:1:+X FRF m/s2/Nbody:1001:+Y body:1:+X FRF m/s2/Nbody:1001:+Z body:1:+X FRF m/s2/Nbody:1002:+X body:1:+X FRF m/s2/Nbody:1002:+Y body:1:+X FRF m/s2/Nbody:1002:+Z body:1:+X FRF m/s2/Nbody:1:+X body:1:+Y FRF m/s2/N… … … …

3.1.4 Driving point method

By means of the driving point method you can reduce the amount of measurementwork with respect to the other methods. The following data sets are required :



Operational data

6 accelerations measured at the transfer paths.

Primary identifi-er

Secondary identifier Function class Unit label










Accelerance transfer functions

6 driving point FRFs at the transfer paths. The primary and secondary identifier ofthe FRFs corresponds to the transfer paths.

Primary identifier Secondary identifier Function class Unit labelbody:1:+X body:1:+X FRF m/s2/Nbody:1:+Y body:1:+Y FRF m/s2/Nbody:1:+Z body:1:+Z FRF m/s2/Nbody:2:+X body:2:+X FRF m/s2/Nbody:2:+Y body:2:+Y FRF m/s2/Nbody:2:+Z body:2:+Z FRF m/s2/N



3.2 Airborne transfer path analysis

The aim is to quantify the noise contribution from the exhaust exh:1:S. As a targeta microphone response in the car cavity mic:1:S is taken.

exh:1:S

mic:1:S

exh:1002:S

exh:1001:S

3.2.1 Direct method

If it is possible to obtain the volume velocity at the airborne transfer paths, the directmethodology can be used (no indirect volume velocity estimation required in TPA).These volume velocities can be calculated easily from acceleration, sound intensityor sound power measurements by means of the Panel Contribution Analysis module(see section 3.3). In this case the following data is required to perform a transfer pathanalysis.

Operational data

1 volume velocity record at the airborne transfer path exh:1:S.

Primary identifier Secondary identifier Function class Unit labelexh:1:S - frequency_spectr m3/s

Transfer functions

1 acoustical FRF between the airborne transfer path and the acoustical target. Theprimary identifier of the FRF corresponds to the target location, the secondary identi−fier refers to the airborne transfer path. In practice these FRFs are measured in a re−ciprocal way by switching the location of source and microphone.



Primary identifier Secondary identifier Function class Unit labelmic:1:S exh:1:S FRF Pa/m3/s

3.2.2 Matrix inversion method

Intake and exhaust noise can be determined in a indirect way by means of matrix in−version. It combines pressure measurements around the source with acoustical FRFsbetween the source and those pressure points.

Operational data

2 pressure measurements at position exh:1001:S and exh:1002:S.

Primary identifier Secondary identi-fier

Function class Unit label

exh:1001:S - frequency_spectr Paexh:1002:S - frequency_spectr Pa


1 acoustical FRF between the airborne transfer path and the acoustical target. Theprimary identifier of the FRF corresponds to the target location, the secondary identi−fier refers to the transfer path. In practice these FRFs are measured in a reciprocalway by switching the location of source and microphone.

Primary identifier Secondary identifier Function class Unit labelmic:1:S exh:1:S FRF Pa/m3/s

Extra transfer functions

2 acoustical FRFs between the airborne transfer path and the extra pressure points.The primary identifier of the FRF corresponds to the extra pressure points, the secon−dary identifier refers to the airborne transfer path. In practice these FRFs are mea−sured in a reciprocal way by switching the location of source and microphone.



Primary identifier Secondary identifier Function class Unit labelexh:1001:S exh:1:S FRF Pa/m3/sexh:1002:S exh:1:S FRF Pa/m3/s

3.3 Panel contribution analysisVolume velocities can be calculated easily from acceleration, sound intensity or soundpower measurements by means of the Panel Contribution Analysis module. Thesevolume velocities are required for the direct method in Transfer Path Analysis.

3.3.1 Panel acceleration

Suppose that you want to estimate the volume velocity wind:1000:S of a part of a wind−screen based on three normal accelerations on that part (wind:1001:+Z towind:1003:+Z).

ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ

wind:1001:+Z

wind:1002:+S

wind:1003:+Z

windscreen

wind:1002:+Z

Operational data

Acceleration measurements on the vibrating panels.

Primary identifier Secondary identifier Function class Unit labelwind:1001:+Z - frequency_spectr m/s2

wind:1002:+Z - frequency_spectr m/s2

wind:1003:+Z - frequency_spectr m/s2

The area of the three patches must to be known.



3.3.2 Sound intensity or sound power

A similar processing can be done based on sound intensities or sound powers. Theoperational data in that case becomes :

Primary identifier Secondary identifier Function class Unit labelwind:1001:+Z - acoustic_intens W/m2

wind:1002:+Z - acoustic_intens W/m2

wind:1003:+Z - acoustic_intens W/m2

or

Primary identifier Secondary identifier Function class Unit labelwind:1001:+Z - sound_power Wwind:1002:+Z - sound_power Wwind:1003:+Z - sound_power W

The particle velocity, air density and C factor intensity measurements have to suppliedto the Panel Contribution Analysis module (see page 11). The area of the threepatches only needs to be known in case of sound intensities.

3.4 Operational data types

All data has to be available in test sections of Cada−x project databases. TPA doesnot allow importing data directly from throughput or TRDS file.

Single reference problems

For steady state applications frequency spectra or autopowers can be used as operation−al data. They can be measured in CADA−X FOURIER MONITOR. In the above tables thefunction class frequency_spectr of operational data records can always be replaced byautopower_spectr.

Ordercuts have to be used when studying the behavior of an engine in run−up or run−down. These records can be obtained in the CADA−X SIGNATURE MONITOR.



Multi-reference problems

For multi−reference problems only frequency spectra can be used. During operation−al conditions crosspowers must be measured. These crosspowers are processed in thePrincipal Component Analysis module in order to obtain different sets of virtual spec−tra. This is explained in the Principal Component Analysis manual.



4 Implementation of TPA

This section describes the terminology, the overall steps and the data structures usedin performing a transfer path analysis using the CADA−X TRANSFER PATH ANALYSIS mod−ule once your test data has been acquired.

4.1 The TPA model

The analysis is based on the use of a TPA model. This is a means of grouping a setof data on which to perform the analysis and outcome of that analysis. It is a conve−nient mechanism for accessing particular sets of data and the corresponding results.This is illustrated in Figure 8.

Project1

Testspectra

Testorders1

Project2

TestFRFs

Testspectra

Teststiffness

Testorders2

Model 1

Calc 1 Calc 2

Project3

Testresults

Model 2

Calc 1 Calc 2

TestResults

Figure 8 The TPA model

Test data can be retrieved from different projects and gathered into one TPA model.A model once opened by one user can not be accessed by any others.

Note! The source data is just referenced by the model so moving a project data base ormoving the model in such a way as to make the referencing link invalid will renderthe model unusable. You can restore this link using the Model � Change project

paths menu option.



A number of models can be defined, referencing different aspects of the total amountof data on a structure. These models will be archived in the file TPAm_MODEL_IDon a defined directory. A model is characterized by the source data it contains, andwithin one model different data cases can be established.

Cases

One set of operational data corresponds to one case and the first case of operationaldata and will be defined as the reference case. All other cases must contain the opera−tional data for the same measurement points, though they can relate to different op−erational parameters such as frequency or rpm ranges. A case is defined by a name(identification), a specific frequency or rpm range, an increment value for the calcula−tions, and possibly an order number. If a second case is found to be inconsistent inthis respect to the reference case, then although you can choose to define the secondcase to be the reference case, you must however adjust the data in one or other caseto so that both sets match.

When computing the results for the data in a model then all cases are included in thecalculations. It is in the analysis and display of these results that distinctions can bemade between the different cases.

Calculation tables

Based on the type of data that is available in the model, a number of different calcula−tion schemes can be executed, using a variety of computational methods, paths andtargets. These schemes are known as ‘‘Calculation tables" and a series of tables canbe defined for each model.

Each calculation procedure will result in a set of functions being generated. Thesefunctions can be loaded into Block Data Memory from where they can be visualizedin a set of dedicated Static display layouts and be stored to a separate test section ina database from where they can be accessed elsewhere in the Cada−x software. Thisaspect is further discussed in section 5 of this document and section 2.5 of the ‘Transferpath menus’ document in the �Transfer path analysis" book.

A number of functions can be performed on models using the Model menu in theCADA−X TRANSFER PATH ANALYSIS module. This allows you to create, select, copy anddelete complete models − which includes results as well as the references to thesource data.

4.2 Model data

A transfer path analysis requires the definition of paths, targets and possibly mounts.These parameters are determined by the selection of data that is included in the TPAmodel. This data falls into three categories, namely; operational data, transfer func−tions and mount connection data.



Operational data

The selection of the operational data determines the paths and is the first stage increating a model. It determines the framework of the model. Operational data is de−scribed in section 3. It can be considered as either ‘Spectra’ or ‘Order cuts’. One mod−el can contain only spectra, ordercuts or autopowers. As mentioned above the firstset of data defines the reference case and all other cases must contain operational datawith the same primary identifications as the reference case.

Rms summation

For multiple references different cases will contain sets of operational data belongingto different principal components: referenced virtual spectra. To assess the overalleffect of the individual principal components, rms summation over the contributionsfrom the different principal components must be done. In order to do so a special casecan be defined as the rms summation over a number of cases.

The procedure for defining cases of operational data for a model is given in section1.2 of the ‘Using transfer path analysis’ document in the �Transfer path analysis" book.

Transfer functions

The targets in a transfer path analysis are defined by transfer functions and as mini−mum the model must contain a set of FRFs between a target location and all pathscontained in the model. Such functions are termed ‘‘target FRFs’’. If you wish to usean indirect methodology then you need to enter the requisite measured FRFs, bothacoustical and acceleration, into the model. They can only be selected once the opera−tional data has been defined.

When autopowers are used, as operational data, then the FRFs are automaticallymultiplied by their complex conjugate. This is not done when FRFs have the functionsqualifier �rms_power". In this case the software assumes that this multiplication hasalready been performed. In Panel contribution analysis it is possible to average FRFswith a quadratic method and end up with squared FRFs with the function qualifier�rms_power".

Mount connection data

These are necessary if you wish to use the complex dynamic stiffness methodology.These can only be selected once the operational data and the transfer functions havebeen defined. Mount connection data in the model defines the characteristics of amount and the physical locations where they are placed. The data describing thephysical characteristics of the mount must be available in a test section.

When the relevant test section has been selected, three file lists will be generated.

mount pid mount sid Path id Input id

body:1:X souc:1:X body:1:X souc:1:X



body:1:Y souc:1:Y body:1:Y souc:1:Y

body:1:Z souc:1:Z body:1:Z souc:1:Z

body:2:X souc:2:X body:2:X souc:2:X

body:2:Y souc:2:Y body:2:Y souc:2:Y

body:2:Z souc:2:Z body:2:Z souc:2:Z

body:3:Z

body:4:Z

body:5:Z

body:6:Z

The first list contains the characteristic records of the mounts represented by the pri−mary and secondary ids. The CADA−X TRANSFER PATH ANALYSIS program does not ac−cept empty primary and/or secondary identification header fields for the mount stiff−ness data.

The second list contains the possible connection points on the target side (path identi−fication) of the mounts.

The third list is created as the difference between the list of points where operationalaccelerations are available and the list of paths (second list). It is used to define thecorresponding mount connection point on the source side (input identification).

A mount connection is created by selecting a mount identification from the first list,a path identification from the second list, and a source (input) identification from thethird list. Once a connection has been defined the concerned path and input identifi−cation are removed from the file lists, because it is impossible to connect multiplemounts in one point.

Normally the software expects the mounts to be measured under tension. If this is notthe case, it is necessary to reverse the sign of the phase of the FRF with the [Reverse]button. Further connections can be added or deleted from the model.

4.3 The Calculation table

After the model has been created, then all the possible calculation methodologies, allthe possible targets and all the paths are known. This information is presented to youwhen you execute Process � Calculation table.

The number of paths

is listed on the left hand side. This list can be divided into structural and airbornepaths depending on the type of operational data you have selected. The paths avail−able depends on the operational data that has been included in the model.



The methods

These are again dependent on the data in the model.

The ‘Direct’ method will be available for those paths for which force spectra orvolume velocities have been defined.

The ‘Mount’ method refers to the ‘complex dynamic stiffness’ method and re−quires that a mount connection has been defined for the path.

The ‘Point inversion’ method will be available for a path if driving point FRFsfor a particular path have been included in the model.

The ‘Matrix inversion’ method will be available for those paths for which the re−sponse functions have been included. A further column is also displayed headed‘‘Nr of FRFs". which is of relevance when the Matrix inversion method is se−lected for a calculation table as described below.

The availability of a method is indicated by the letter ‘‘A" and its non−availabilityfor a path by ‘‘−’’.

The targets

The number of targets depends on the FRF data that has been included in the mod−el. A target is defined as a primary id of an FRF for which the secondary id corre−sponds to a path id. The number of targets is the same for each table.

The user is able to select any combination of targets, paths and methodology for whichresults are to be calculated from those available. Each resulting set of calculationsare held in a calculation table. It is possible to create several calculation tables withina model as illustrated below.

Table MountDirectMatr

invPointinv

Targets(3)

Paths(10)

S

S

S

S

1

2

3

A calculation table is identified by a name and the procedure for creating a calculationtable is given in section 1.3 of the ‘Using transfer path analysis’ document in the�Transfer path analysis" book.

The paths to be analyzed are selected from the list and a methodology can be selectedfor each. Paths for which no methodology is selected will not be included in the cal−culations, except for airborne paths, which if present, are always taken into account.



The matrix inversion methodology

If the matrix inversion methodology is selected, then the inversion parameters needto be checked. The maximum dimension of the inversion problem is indicated by thenumber of paths and the number of responses.

For each path two numbers separated by a slash are shown. The first one is the num−ber of measured frequency response functions in that path, the second is the numberof responses taken into account for the matrix inversion. When the first number issmaller than the second (meaning not all FRF’s were measured), then the missingFRF’s are replaced by zeroes in the matrix.

The number of inversion points to be taken into account for the matrix inversion canbe adjusted through the [Edit matrix] button. Deselecting responses will reduce thenumber displayed in the list while the overall number of responses displayed at thetop of the calculation table remains unchanged.

Matrix inversion is done by using singular value decomposition. By overdetermina−tion and omission of some singular values, the influence of noise on the measurementscan be reduced. Therefore three methods are provided:

Absolute criteriaall singular values smaller than a specified value (x>0) are put to zero.

Relative criteriaall singular values that are less than a specified percentage (0<x<100) of thehighest singular value are put to zero.

Number criteriaa specified number (0<x< number of paths) of singular values, starting from thesmallest, will be put to zero.

The condition number of the matrix which indicates the influence of the noise can becalculated and visualized in a Static display. By comparing several condition functionsthe influence of the different inversion methods can be investigated.

4.4 The Calculations

These are initiated by the [Calculate] button which starts the processing of the re−quested results. Checks are made that a suitable number of inversion points are de−fined.

Upon completion of the calculations, information on the results is displayed. If theresults were successful the results are stored in the calculation table, which in its turnis stored in the model. The results are now available for viewing and/or storing as de−scribed in the following section.



5 Results and post-processing

The calculations procedure can give rise to a large number of results, since one modelcan contain multiple calculation tables and each calculation table can contain the re−sults for a number of cases, paths and targets. Because of the large quantity of results,they are archived into files linked to their calculation table and their model. Two op−tions exist however whereby you can access particular results in a logical fashion andload them into BDM for either viewing in the Static display or selectively transferringthem to a database test section for storage. Two separate sections of BDM are desig−nated for these two purposes.

The menu entry Results � Load enables you to load selected functions from differentmodels and different calculation tables into BDM. The selected functions will be ac−cumulated in memory and can then be stored to a test section of the currently openproject for access, viewing and comparison from within other modules of the Cada−xsoftware. The section of BDM used for this purpose is known as the user space andby default consists of the first 50 blocks in memory. You can adjust the size of the userspace in BDM through the Options � Set user space menu entry. You can also clearthis specific area in memory through the Test data � Delete block � User blocks menuentry. Loading results into memory is described in section 2.5 of the ‘Transfer pathmenus’ document in the �Transfer path analysis" book.

The menu entry Results � Display enables you to select functions which will be loadedinto BDM and immediately displayed in an appropriate dedicated Static display lay−out. Functions selected for display are loaded into a section of memory termed proces−sing space which follows the user space. The user has no control over the destinationblocks used for display and each new function requested will overwrite the existingcontents of the processing blocks. The processing blocks are kept completely separatefrom the user blocks which can only be cleared or overwritten on instruction from theuser.

User blocks Processing blocks

user space

Multi-reference Transfer path analysis

For the multiple reference Transfer path analysis it is possible to define an RMS loadcase, which consists of the RMS summation of all the results from the different princi−pal components. Due to this summation, the phase information disappears. The fol−lowing table gives an overview of the contents of such an RMS case for k transfer pathsand m references.



Loadingcases(principal

components)

PCA1 PCA2 ... PCAm RMS1

Sourcesin path1, ... k

f(1,PCA1)

f(2,PCA1)

...

f(k,PCA1)

f(1,PCA2)

f(2,PCA2)

...

f(k,PCA2)

f(1,PCAm)

f(2,PCAm)

...

f(k,PCAm)

f(1,RMS)

f(2,RMS)

...

f(k,RMS)

Calculatedpartialtargets

tc(1,PCA1)

tc(2,PCA1)

...

tc(k,PCA1)

tc(1,PCA2)

tc(2,PCA2)

...

tc(k,PCA2)

tc(1,PCAm)

tc(2,PCAm)

...

tc(k,PCAm)

tc(1,RMS)

tc(2,RMS)

...

tc(k,RMS)

Calculatedtotaltargets tc(PCA1) tc(PCA2) tc(PCAm)

tc(RMS)

tc(complex)

Measuredtotaltargets

tm(PCA1) tm(PCA2) tm(PCAm) tm(RMS)

This RMS case is also accessible for postprocessing in the same way as the individualprincipal component contributions.

5.1 Loading results

A range of functions are available depending on the model and the methods used eachof which are described below. In addition, you can make choices as to the format ofthe results, e.g., whether they are expressed as a function of frequency or rpm, whetherweighting functions will be applied. These are described in section 2.5 of the ‘Transferpath menus’ document in the �Transfer path analysis" book, as is the procedure forloading data.

Data can be loaded into memory by executing Results � Load

Partial contribution

This is partial contribution to the response at the selected target dueto the selected individual path(s).

RMS calc over subset of PCAi

Complex sum



XYZ contribution

This is contribution to the response at the selected target due to theeffect of each selected path summed over all the measurement direc−tions.

Group contribution

This is the contribution grouped according to the specifications fromthe user.

Target FRF The frequency response function(s) between the selected target rel−ative to the selected path(s).

Calculated response

The calculated response of the selected target(s) due to all paths.

Measured response

The measured response of the selected target(s) due to all paths.

Operating inputs

These are the calculated (or measured) forces or volume velocitiesfor the selected path(s). Measured forces will only be available forthose paths where the ‘Direct’ method was used.

Operational data

The measured data (accelerations) for each of the selected path(s).Two functions (source and target side of the mount) are available forthose paths where the ‘Mount’ method was used.

Direct FRF The driving point frequency response function(s) for the selectedpath(s). These functions are only available for those paths for whichthe matrix inversion or point inversion methods were used.

Mount FRF The mount characteristic function for the selected path(s). Thesefunctions are only available for those paths for which the ‘mount’(complex dynamic stiffness method) was used.

5.2 Displaying results

The results arising from an analysis are listed above. The Results � Display optionprovides you with a range of graphical formats to aid the assessment and the compari−son of the data. As for the loading of results, additional choices as to the format ofthe results can be applied. These are described in section 2.5 of the ‘Transfer pathmenus’ document in the �Transfer path analysis" book.



The View options are Path related dataPartial sumsComparison of XYZ sums3D display and Color displayVector displaysHistogram displaysPercentual path contribution

Path related data

This option displays a range of functions related to a specified target due to one se−lected path in a specific layout. Details on the information and the windows in whichthey are displayed in this layout is given in Figure 9.

Target FRF

Measured response in that target (ifavailable) − Trace 3

Partial contribution of the selected pathto the specified target − Trace 1

No entry (Direct)

Operational acceleration on target side(Pnt inv and Mat inv)

No entry (Direct)

Body FRF (Pnt inv and Matinv)

The mount stiffness FRF(Mount)

Total calculated sum of all path con−tributions to the target − Trace 2

Measured operating inputs(Direct)Calculated operating inputs

Operational accelerations on target and sourceside (Mount)

Operating inputs

Display dialog

Figure 9 Display of ‘Path related data’ layout

Partial sums

The partial contributions to a specified target are summed over the selected paths.The summation can be RMS or complex calculated. This partial sum (Trace 1) isshown together with the total calculated sum (Trace 2) and the measured response inthe target (Trace 3) in an appropriate Static display layout as shown in Figure 10.



Figure 10 Example of a ‘Partial sum’ layout

Comparison of XYZ sums

With this option you can see the relative contributions of the different six measure−ment directions of a path on a selected target. Six Static display windows are sched−uled, and in each case the total calculated response at the target is shown. In additioneach window shows the response due to the summed partial contributions in the X,Y, Z, Rx, Ry and Rz direction of the path. In most cases the Rx, Ry and Rz contribu−tions are not taken into account and are therefore not displayed, which means thatonly three displays are shown.

The sum of the measurement directions related to the selected path (Trace 1) is showntogether with the total calculated sum (Trace 2). In Figure 11 the third trace (Trace3) is the partial contribution due to the X direction for the upper window, the Y direc−tion for the middle window and the Z direction for the lower window.



Figure 11 Display of ‘XYZ sums’

3D display and Color display

These option allows you to compare the partial contributions to a specified target ofa number of selected paths. The response of each path as a function of rpm or fre−quency are displayed in either in a color map or waterfall display. It therefore pro−vides a very easy way to compare the contributions of each path at any particular valueof rpm or frequency.



The first selected path is shown as the lowest color bar

Figure 12 Color display

The first selected path is shown as the front trace

Figure 13 3D display

Vector displays

Two options are provided whereby the partial contributions of the selected paths toa specified target at a specific frequency or rpm value, are represented as vectors. Thelength of the vector represents the amplitude of the response and the angle its phase.



Linear vector

measuredresponse

calculatedresponse

selected path

The individual partial contributions of all thepaths are displayed as a series of vectors in onecolor (shown in the figure as solid lines). Thecontribution of the selected path is shown in adifferent color (represented in the figure as aheavy line) and as different paths are selectedfrom the list, the display is refreshed. In additionthe total calculated response is represented aswell as the measured response. The color codingcan be seen by executing Options � Display an−

notation This vector representation can be cal−culated for any frequency or rpm value and themechanism for scrolling and selecting frequen−cies are described in section 2.5 of the ‘Transferpath menus’ document in the �Transfer pathanalysis" book.

Logarithmic vector

measuredresponse

calculatedresponse

selected path

In this case the same functions are displayed us−ing the same color coding conventions but sincelogarithmic vector addition is not possible, allthe vectors are portrayed starting from a centralpoint which by default is 0 dB. Mechanical tar−gets can result in logarithmic values smaller than0 dB in which case no vectors would be visual−ized. The ‘Minimum amplitude’ field allows youto change the starting value to an appropriatevalue.

The vector is shown in an upper window while the lower window contains the partialcontribution of the selected path (Trace 1) together with the total calculated sum(Trace 2) and the measured response in the target (Trace 3).

Histogram displays

With these options, the amplitude and phase of the vectors discussed above are shownas histogram bars.



Absolute histogram

�1

�3

�2

A1

A2

A3

A1

A2

A3

�1

�2

�3

The absolute amplitude and phase of the vectors of each path are displayed as a histo−gram in the order in which they appear in the path list. The selected one is highlightedin a different color.

Relative histogram

��

��

For the amplitude, the value is calculated as a percent−age of the amplitude of the total calculated responsevector, and the phase as the smallest angle between thepartial contribution vector and a perpendicular on thereference vector. This phase value can be interpretedas following : the smaller the angle, the less this pathcan contribute to the reference vector.

Projected histogram

A third option projects the partial contributions on the vectorof the total calculated sum.

The vector is shown in an upper window while the lower win−dow contains the partial contribution of the selected path(Trace 1) together with the total calculated sum (Trace 2) andthe measured response in the target (Trace 3).



Percentual path contribution

In this view the relative importance of a number of the transfer paths to the total ener−gy contents is calculated and visualized. The automatic option will search for the fivemost significant paths. This allows you to detect problem frequency ranges and thepaths responsible.

B

A

C

D

path 1

path 2

overall rms level

The length |AB| is the rms sum of the total energy contribution of all the transferpaths in dB. The length |BC| represents the contribution of the most significant path.The second length |CD| represents the second most significant path. The lowestlength represents the sum of all other remaining paths. The view can be scaled to apercentage as shown in the example below.

Documents

Transfer Path Analysis