identifying joints from measured reflection coefficients in beam-like structures with application to...

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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 24 (2010) 784–795

0888-32

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Identifying joints from measured reflection coefficients in beam-likestructures with application to a pipe support

Bing Zhang, Timothy P. Waters �, Brian R. Mace

Institute of Sound and Vibration Research (ISVR), University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom

a r t i c l e i n f o

Article history:

Received 3 October 2008

Received in revised form

16 September 2009

Accepted 31 October 2009Available online 10 November 2009

Keywords:

Joints

Pipe

Model updating

Waves

Inverse problems

70/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ymssp.2009.10.023

responding author. Tel.: +44 23 8059 4979; f

ail address: [email protected] (T.P. Waters

a b s t r a c t

The properties of joints in mechanical systems are notoriously uncertain causing

corresponding uncertainty in the systems’ dynamic responses. A piping system is one

such example where an accurate knowledge of joint properties is useful for the

purposes of structure-borne sound transmission, fatigue considerations and structural

health monitoring. This paper presents an inverse technique that is applicable to joint

estimation in one-dimensional structures such as a pipe. Measured wave reflection

coefficients are used which have several advantages over modal information. First, they

characterise just the joint and adjacent pipes and are independent of the rest of the

built-up system. Second, they are potentially more sensitive to the joint parameters in

question than are modal parameters.

The method is illustrated by means of an experimental case study featuring a

straight pipe suspended by a cantilevered hanger. The stiffness and inertia of the hanger

are accurately identified from measured data at frequencies significantly higher than

the fundamental modes of the structure.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Vibration of engineering structures is of concern for reasons of noise propagation and radiation, discomfort and structuralintegrity. Numerical tools such as finite element (FE) modelling can in principle compute the response of structures todynamic forces but require complete and accurate input data concerning the properties, including boundary conditions, of thewhole system being modelled. In practice this information is often not available. Joints are a particularly common source ofuncertainty in built-up structures, and inaccuracies in their properties can affect the predicted responses profoundly.

Even when a structure has been designed and built it is not usually possible to measure joint properties directly.Instead, inverse methods can be employed to infer the joint properties from measured response quantities. Sometechniques arrive at a frequency domain description of the joint without making any a priori assumptions about thephysical form of the joint [1,2]. Other techniques presuppose a parametric joint model (e.g. masses, springs and dampers)and tune the parameter values so as to best match experimental data [3–5]. In the case of FE models, modal or frequencyresponse data are frequently used as they are often readily measurable [6]. The problem frequently encountered is thatwhilst measured data are plentiful much of it contains redundant information regarding the net effect on the response ofthe individual joint parameters. Consequently, the inverse problem of parameter estimation is over-determined but can behighly ill-conditioned.

ll rights reserved.

ax: +44 23 8059 3190.

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B. Zhang et al. / Mechanical Systems and Signal Processing 24 (2010) 784–795 785

Some structures, particularly uniform one-dimensional structures, are more appropriately described in terms of thewaves they carry than their modes of vibration. Piping systems are one such example. Joints are then characterised not bytheir effect on global response quantities such as natural frequencies and mode shapes, but locally in terms of how theyreflect and transmit incident waves [7]. The potential advantage here is that the task of updating a number of joints in alarge built-up system can in principle be decoupled into a number of smaller and therefore better conditioned parameterestimation problems. An additional benefit of the wave approach is that measured data can be exploited at higherfrequencies where modal overlap becomes large and renders modal methods unsuitable [8].

This paper investigates the feasibility of updating joint models where the adjoining structures are modelled as one-dimensional waveguides. The general approach is not dissimilar to that proposed in [9] which was applied to a rectangularbeam with attached blocking masses. First, a technique for calculating the reflection and transmission coefficients of adiscontinuity between two arbitrary waveguides [10] is briefly reviewed. The procedure is then illustrated for the specificcase of two beams connected by a mechanical joint comprising mass, moment of inertia and grounded throughtranslational and rotational springs. Second, an experiment is conducted on a straight in vacuo pipe supported by a hanger.Given the simplicity of the system it is valid to assume that over a limited bandwidth only flexural waves arise fromtransverse excitation and so it is straightforward to estimate reflection and transmission coefficients of the hanger. Anestablished technique used elsewhere [11] is briefly described in the context and notation of this paper. In the case of jointasymmetry [12] or pipe bends [13] wave mode conversion occurs and the process is more complicated but still tractable.

Finally, a procedure is described whereby the model of the joint between the waveguides can be updated basedon experimentally determined reflection and transmission coefficients. The procedure is applied to the case of thepipe hanger. The estimated parameter values are shown to be plausible and the updated model closely predicts themeasured behaviour.

2. Theory: wave reflection and transmission in waveguides

For many one-dimensional systems, such as beams, rods and ducts, the motion can be described concisely by the wavesthey carry. Whilst the wave types present are system and frequency dependent, the analytical formulation of the wavefield is essentially the same. In this section, a general procedure for determining reflection and transmission coefficients atdiscontinuities in waveguides which is similar to that presented in [10] is briefly described. This procedure is then appliedto the case of a mechanical joint connecting two identical Euler–Bernoulli beams.

2.1. Discontinuity between two arbitrary waveguides

Consider time harmonic excitation of a waveguide in which an arbitrary number of wave types propagate. The waveamplitudes associated with positive and negative going waves can be grouped into vectors a+ and a�, respectively.

At any point on the waveguide, both the vector of generalised displacements, W, and the vector of internal forces, F, canbe expressed as a linear combination of positive and negative going wave amplitudes as

W

F

� �¼

Wþ W�

Uþ U�

" #aþ

a�

� �ð1Þ

The coefficient matrices W+ , W�, U+ and U� are functions of the wavenumbers of the various wave types.Now consider two such waveguides a and b which are connected so as to form a single waveguide with a discontinuity

at the connection. Such a system comprising two beams is shown in Fig. 1. Immediately either side of the discontinuity, thedisplacement and force vectors can be similarly described, i.e.

Wa

Fa

( )¼

Wþa W�aUþa U�a

" #aþ

a�

� � Wb

Fb

( )¼

Wþb W�bUþb U�b

" #bþ

b�

( )ð2Þ

where b+ and b� are positive and negative going waves in waveguide b, and subscripts a and b denote terms pertaining towaveguides a and b, respectively.

The displacement and force vectors on one side of the discontinuity WaFa

n oare related to those on the other side Wb

Fb

n oby

equilibrium and continuity conditions imposed by the connection. Sufficient equations can always be written and solved to

x

ba

b-a-

a+ b+

Fig. 1. Two beams connected by an arbitrary mechanical joint, and the wave field in the assembly.

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B. Zhang et al. / Mechanical Systems and Signal Processing 24 (2010) 784–795786

relate the outgoing and incoming waves in terms of a matrix of transmission and reflection coefficients, i.e.

a�

bþ

� �¼

Raa Tba

Tab Rbb

" #aþ

b�

� �ð3Þ

Submatrices R and T are reflection and transmission matrices whose sizes correspond to the number of wave types.Superscript ab indicates transmission from waveguide a to waveguide b; superscript aa denotes reflection of incidentwaves in waveguide a, and so on.

The diagonal elements of R (and T) correspond to where the reflected (and transmitted) wave type is the same as that ofthe incident wave. Off-diagonal elements correspond to wave mode conversion.

In many cases the equilibrium and continuity equations can be written in terms of a transfer matrix X, i.e. in the form

Wa

Fa

( )¼

X11 X12

X21 X22

" #Wb

Fb

( )ð4Þ

Substituting for Wa Fa� �T

and Wb Fb� �T

from (2) into (4), and rearranging, gives

Raa Tba

Tab Rbb

" #¼

W�a �ðX11Wþ

b þX12Uþ

b Þ

U�a �ðX21Wþ

b þX22Uþ

b Þ

" #�1�Wþa ðX11W

�

b þX12U�

b Þ

�Uþa ðX21W�

b þX22U�

b Þ

" #ð5Þ

In cases where the transfer matrix does not exist the reflection and transmission matrices can still be determined, butthe details depend on the exact nature of the discontinuity.

2.2. Two identical beams connected by a mechanical joint

In this section, the general formulation presented above is applied to the specific case of two uniform Euler–Bernoullibeams, as shown in Fig. 1.

The equation of motion for free vibration of either beam is

EI@4w

@x4þrA

@2w

@t2¼ 0 ð6Þ

where EI and rA are the bending rigidity and mass per unit length of the beam, respectively, and w is the transversedisplacement. The response is time harmonic, i.e.

wðx; tÞ ¼WðxÞeiot ð7Þ

and the four solutions for W(x) are of the form eikx where

k¼7

ffiffiffiffiffiffiffiffiffiffiffiffiffio2rA

EI

4

r; k¼7i

ffiffiffiffiffiffiffiffiffiffiffiffiffio2rA

EI

4

rð8a;bÞ

are the wavenumbers. Eqs. (8a,b) correspond to propagating and evanescent waves in each direction, so the waveamplitude vectors in beams a and b are all two-element vectors, i.e.

aþ ¼aþpaþe

( ); a� ¼

a�pa�e

( ); bþ ¼

bþp

bþe

( ); b� ¼

b�p

b�e

( )ð9Þ

where subscripts p and e denote propagating and evanescent waves, respectively. The generalised displacement and forcevectors have as elements the displacement, slope, bending moment and shear force, and are given by

W¼W

@W=@x

( ); F¼

�EI@3W

@x3

EI@2W

@x2

8>>><>>>:

9>>>=>>>;

ð10Þ

and the coefficient matrices are

Wþa ¼Wþb ¼1 1

�ik �k

� �; W�a ¼W�b ¼

1 1

ik k

� �ð11Þ

and

Uþa ¼Uþb ¼�iEIk3 EIk3

�EIk2 EIk2

" #; U�a ¼U�b ¼

iEIk3 �EIk3

�EIk2 EIk2

" #ð12Þ

In the specific example considered in detail, the joint is represented by translational and rotational springs(KT, FR) to ground and mass/second moment of inertia (m, J), as shown in Fig. 2. The transfer matrices of the joint are

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KT

KR

m,J

Fig. 2. Lumped parameter model of a joint between two beams.


given by

X11 ¼X22 ¼1 0

0 1

� �; X12 ¼

0 0

0 0

� �; X21 ¼

�ðKT �o2mÞ 0

0 �ðKR �o2JÞ

" #ð13Þ

The full 4�4 matrix of reflection and transmission coefficients can be readily computed numerically from Eqs. (5),(11)–(13). Alternatively, algebraic manipulation can yield more direct equations for the individual scattering coefficients.For example, it is shown in the Appendix that

rpp

tpp

rpe

tpe

8>>>><>>>>:

9>>>>=>>>>;¼

1 �1 1 �1

ik ik k k

iEIk3 KT �o2mþ iEIk3 �EIk3 KT �o2m� EIk3

�EIk2 �ikðKR �o2JÞþEIk2 EIk2 �kðKR �o2JÞ � EIk2

266664

377775

�1�1

ik

iEIk3

EIk2

8>>><>>>:

9>>>=>>>;

ð14Þ

where r and t denote individual reflection and transmission coefficients relating to waves impinging on the joint fromwaveguide a. Subscripts p and e denote propagating and evanescent wave types and the first and second subscript refer tothe incident and reflected/transmitted wave, respectively.

It should also be noted that Raa=Rbb and Tab=Tba in this case since waveguides a and b are identical and the joint issymmetric.

The purpose of this paper is to update the joint parameters, KT, KR, m and J of a practical jointed structure from measuredcounterparts of one or more of these coefficients.

3. Experiment on a straight pipe

A practical application of a beam-like structure with a discontinuity is a pipe connection. At low frequencies, i.e. belowthe cut-on frequency for the n=2 ovalling mode, it is assumed that only n=1 flexural waves exist and that these are in theplane of the point excitation. Consequently, the motion of the pipe can be adequately represented by positive- andnegative-going propagating and evanescent flexural waves, as outlined in Section 2.2. This section describes an experimentperformed on a long straight copper pipe suspended by an idealised hanger which can be regarded as a joint between twoidentical sections of pipe. Some selected results are presented here for illustrative purposes; further results are shown inSection 4 where they are compared with theory.

3.1. The rig

Fig. 3(a) shows a schematic of the experimental rig featuring a 6 m long straight thin-walled cylindrical copper pipe ofouter diameter 0.028 m and wall thickness 0.9 mm of the type used extensively in domestic central heating systems. Thepipe is held at its mid-length from above and below by a symmetric support arrangement, detailed in Fig. 3(b). The supportcomprises two cantilever rods which are connected at one end to a collar on the pipe and clamped to a large seismic massat the other end. In bending, the bars add translational stiffness to the pipe and in torsion they provide rotational stiffness.The rods and collar can also be expected to contribute significant inertia to the joint. The support bars are interchangeableallowing both aluminium and steel bars to be tested. Their effective lengths can be altered by moving the clamps todifferent positions. Two lengths of support were chosen, such that the total separation between the clamps was either125 mm or 185 mm.

Each end of the pipe was buried in a sandbox to provide partially anechoic conditions; this is not implicitly assumed butimproves conditioning in the data post-processing.

3.2. Experimental procedure

The pipe was excited by an electrodynamic shaker to inject flexural waves into the system. The shaker was placed at asufficient distance from the joint such that the incidence of evanescent waves on the joint could be neglected. The stingerconnecting shaker and pipe was orientated in the radial direction and in the horizontal plane, i.e. perpendicular to theflexible supporting rods. A random excitation signal was used with a bandwidth of 2.5 kHz.

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rod

sand box pipe collar shaker’s

stinger (into page)

da db

accelerometers accelerometers

pipe collar

rod

adjustableheight clamp

seismic mass

Δ = 0.1m

Fig. 3. Schematic showing experimental set-up: (a) side view and (b) detail of hanger arrangement viewed along axis of pipe.


First, the dispersion relation for flexural waves in the pipe was measured. Three equally spaced accelerometers (PCBmodel 352C22) were placed with a spacing of D=0.1 m far from the joint, shaker and sandboxes. Data from all transducerswere acquired using a Data Physics Mobilyzer II frequency analyser, and transmissibility frequency response functionsformed between the signals from the outer accelerometers and the middle reference accelerometer.

If the propagating wave amplitudes are defined at the middle sensor then the displacement here is simply given by

W2 ¼ aþp þa�p ð15Þ

whereas the displacements at the other two sensors are given by

W1 ¼ aþp eikDþa�p e�ikD; W3 ¼ aþp e�ikDþa�p eikD ð16a;bÞ

The wavenumber k is obtained from Eqs. (15) and (16a,b) by summing W1 and W3, and dividing by W2 to give

coskD¼1

2

W1

W2þ

W3

W2

ð17Þ

where kD is the phase difference between adjacent sensors. The displacement transmissibility functions in Eq. (17)are of course identical to the acceleration transmissibility functions available from accelerometer measurements. Fig. 4shows a graph of coskD against frequency for the bandwidth of the measurement. The results are anomalous at2250 Hz. This is consistent with the cut-on of the n=2 wave mode which, assuming a nominal axial wave speed incopper of 3750 m/s, is predicted at about 2 kHz. At frequencies lower than this, but high enough to neglect evanescentwaves at the sensor locations (about 100 Hz in this case) it can be assumed that only propagating flexural wavesare observed. If it is further assumed that Euler–Bernoulli theory holds true then the dispersion relation can be expected tobe of the form kp

ffiffiffiffiffiop

. A least-squares fit of this model to the experimental data over an appropriate frequency rangeresults in

k¼ 0:170ffiffiffiffiffiop

ð18Þ

which is depicted in Fig. 4. A close agreement between the model and the experimental data is apparent.Next, wave amplitudes were measured in the far field either side of the joint, from which reflection and transmission

coefficients of propagating waves could be estimated. A pair of accelerometers was placed on either side of the joint, asshown in Fig. 3(a). The spacing of each pair was again chosen as D=0.10 m which corresponds to less than half a

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0 500 1000 1500 2000 2500-1

-0.5

0

0.5

1

cos

(kΔ)

Frequency, Hz

Fig. 4. Cosine of phase angle between adjacent sensors, kD, as a function of frequency. (— measured, - - - - fit).


wavelength for the whole bandwidth of interest. A standard method for wave amplitude decomposition was employed[11] to estimate the wave amplitudes at the sensor locations in terms of the spatial displacements W1, y, W4, viz.

aþpa�p

( )¼

eikD=2 e�ikD=2

e�ikD=2 eikD=2

" #�1W1

W2

( );

bþp

b�p

( )¼

eikD=2 e�ikD=2

e�ikD=2 eikD=2

" #�1W3

W4

( )ð19Þ

Taking into account the distances da and db of the sensor pairs from the joint, the reflection and transmissioncoefficients at the joint can be estimated from

rp

tp

( )¼

aþp e�ikda b�p e�ikdb

b�p e�ikdb aþp e�ikda

24

35�1

a�p eikda

bþp eikdb

8<:

9=; ð20Þ

For the purposes of practical implementation it is convenient to employ transfer functions of accelerations with respect toa reference such as the input force. The wave amplitudes above are then accelerations per unit force; this does not affectthe reflection and transmission coefficients.

At high frequencies, the sensors are many wavelengths from the joint so that kda and kdb are potentially large comparedwith 2p. Consequently, the phases of the exponential terms in Eq. (20) are sensitive to small errors in estimations ofwavelength and distances da and db. For this reason, only the power coefficients are presented and used in this paper,although these too are not immune to errors in kda and kdb. For flexural to flexural waves between two identicalwaveguides the power reflection and transmission coefficients are given by

r¼ rp

�� 2; t¼ tp

�� 2 ð21Þ

On inverting the matrix in Eq. (20) algebraically it can be shown that in this case

r¼aþp a�p � bþp b�p

L

��2

; t¼aþp bþp e�2ikðda�dbÞ � aþp b�p

L

��2

ð22Þ

where

L¼ ðaþp Þ2e�2ikda � ðb�p Þ

2e�2ikdb ð23Þ

is the determinant of the matrix. It is apparent that the sensitivity of L to small errors in wavenumber k and/or distances da

and db causes r and t to be under- or overestimated by the same factor 91=L29. Consequently r+t, which should be unityfor a conservative system, will be similarly erroneously scaled. Provided that damping can be neglected then r/(r+t) andt/(r+t) can be expected to provide more robust estimates for the power coefficients.

It is also clear from Eq. (23) that ill-conditioning can occur at some frequencies if the reflected wave b�p is comparable inmagnitude to the incident wave aþp . This problem is circumvented to some extent by the presence of the sandboxes at theends of the pipes. In the extreme case when reflections from the termination of beam b are negligible then the matrix in Eq.(20) becomes diagonal.

3.3. Illustrative results

Fig. 5 shows the magnitude of a typical accelerance measurement at one of the sensor locations. The structure exhibitsnumerous modestly damped resonances below 200 Hz (corresponding to a flexural wavelength of about 1 m) after whichthe sandboxes become progressively more effective. The lightly damped resonances above 2 kHz are due to cut-on of then=2 wave mode which provides an upper bound on the frequency range of interest.

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101 102 103 10410-3

10-2

10-1

100

101

102

Frequency (Hz)

Mag

nitu

de (m

s-2/N

)

Fig. 5. Magnitude of typical accelerance FRF measured on pipe supported by hanger and terminated at both ends by sand boxes.


Figs. 6(a),(b) show the power reflection and transmission coefficients as given by Eq. (22) for the case when longaluminium rods are used in the pipe hanger arrangement. Unlike the FRFs from which they derive, the reflection andtransmission coefficients are smoothly varying functions of frequency. This is an important property in model updatingwhere the response is often linearised with respect to the model parameters. However, it should be noted that reflectionand transmission coefficients can become rapidly varying when the joint itself is resonant [14].

In theory one might expect total reflection as frequency tends to zero. However, at low frequencies (below 100 Hz, say)this effect is masked by noise because the wavelength is too long to resolve the wave amplitudes reliably with the smallsensor spacing used.

At high frequencies, where the impedances of the springs are negligible compared with those of the pipe, flexural wavesare mostly transmitted through the joint. The reflection coefficient is notably less noisy than the transmission coefficient.For a weakly reflecting boundary b�p is small and, from Eq. (22), r is dependent largely on the wave amplitudes, and hencethe sensor outputs, on just the directly excited side of the joint.

At intermediate frequencies the coefficients vary with frequency, indicating total reflection at 400 Hz and totaltransmission at 800 Hz. The particular shape of the curve is determined by the exact joint parameter values. This frequencyrange is potentially the most useful for the subsequent purpose of accurately estimating the joint parameters.

Fig. 6(c) shows the sum of the power coefficients which should be unity for a conservative system. Some deviation fromunity is apparent for reasons explained in Section 3.2, and values exceeding unity are clearly unphysical. In subsequentresults, both r and t are normalised by r+t to enforce conservation of power.

4. Model updating using power reflection and transmission coefficients

In Section 2, a general procedure for calculating reflection and transmission coefficients was applied to the case of ajointed beam. In Section 3, an experiment was reported in which reflection and transmission coefficients were measuredfor flexural waves in a pipe supported by a hanger. The purpose of this experiment was two-fold. First, it provided evidencethat the dynamic model adopted for the joint is representative of the real structure in the frequency range of interest. Thisis apparent from the comparisons presented later in this section. Second, the measured response data can be usedsubsequently to update the parameters of the model. The importance of the former step should not be underestimatedsince a physically unrepresentative model, albeit updated to match a set of measured data, will not in general be able topredict unmeasured responses. This is a particular issue for many practical welded, bonded or bolted joints where it is notat all obvious how they should be modelled and how many joint parameters are necessary. For the structure underconsideration in this paper the form that the model should take is reasonably obvious and comparative results betweenexperiment and theory presented in this section validate the choice of model. First, the model updating method adoptedfor estimating the joint parameters is described.

4.1. Application of nonlinear least squares

The objective of model updating is to choose parameter values for a model that achieve a best match between themodel and the experimental data. In the case of data in the wave domain this can be achieved in a least-squares sense byminimising an objective function of the form

JðpÞ ¼ 12 ðqm � qÞT Wrðqm � qÞþ1

2ðsm � sÞT Wtðsm � sÞ ð24Þ

where qm and sm are vectors of measured power reflection and transmission coefficients of length N, and N is the numberof frequencies selected from the available measurements. The vectors q and s correspond to predicted values at the same

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0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

Frequency, Hz

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

Frequency, Hz

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

Frequency, Hz

��

�+�

Fig. 6. Measured power coefficients for hanger with long aluminium rods: (a) power reflection coefficient, (b) power transmission coefficient, (c) sum of

power reflection and transmission coefficients.


frequencies. These are functions of the vector of model parameters, p, and in this case can be calculated by squaring themagnitudes of rpp and tpp obtained from Eq. (14).

Weighting matrices Wr and Wt are positive definite and usually diagonal. In the results presented in this paper Wr andWt are taken to be the identity matrix and the null matrix, respectively. That is, only the measured reflection coefficientsare used, and all frequencies are given equal weighting in the fit. Other choices have been explored in [8] and found tomake little difference in this study.

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Minimising J(p) in Eq. (24) is a nonlinear least-squares problem which is solved here by the standard Gauss–Newtonmethod [15]. The parameter values at the (j+1)th iteration are given by

pjþ1 ¼ pjþDpj ð25Þ

where

Dpj ¼ � ðSTj WrSjÞ

�1STj Wrqj ð26Þ

and Sj is the Jacobian of qj, i.e.

Sj ¼

@r1

@p1

@r1

@p2� � �

@r1

@pq

@r2

@p1

@r2

@p2� � �

@r2

@pq

^ ^ & ^@rN

@p1

@rN

@p2� � �

@rN

@pq

266666666664

377777777775

j

ð27Þ

The Jacobian matrix can in general be computed numerically by a finite difference approach, for example. However, forthe joint model used here closed form solutions can be found for the power reflection coefficient which can bedifferentiated analytically. These are rather cumbersome and are reported in [8]. The closed form solutions are alsoinsightful when considering asymptotic behaviour at both low and high frequencies.

In Eq. (26) it is assumed that STj WrSj is a well conditioned matrix such that it can be inverted reliably. For this, it is necessary

but not sufficient for the number of frequencies to be equal to or exceed the number of updating parameters. Convergence canbe deemed to have occurred when, for example, the differences in either the estimated parameters Dpj or predicted responsesDqj become suitably small from one iteration to the next. In this work a criterion based on the norm of Dqj was used.

4.2. Results

The model updating method was applied to estimate the mass, inertia and translational/rotational stiffness of the pipehanger, i.e.

p¼ KT KR m J� �T

ð28Þ

from the power reflection coefficients obtained by the method reported in Section 3. The most suitable frequency range ofdata to include in the model updating process was chosen on a case by case basis by visual inspection. Typically data below200 Hz or so were discarded due to a combination of measurement noise and expected evanescent wave effects, and dataabove 2 kHz were excluded due to the presence of higher order wave types.

Figs. 7(a)–(d) show the measured power reflection coefficients for the four permutations of support rod (long/short andsteel/aluminium) together with their predicted counterparts using the updated joint parameter values. Close agreement isachieved in all cases except for the stiffest (short steel) support which, at high frequencies, exhibits dynamic behaviour notincluded in the joint model. The joint parameter values estimated by the model updating process are listed in Table 1.

The translational stiffness and mass are reasonably straightforward to measure directly from a driving point dynamicstiffness measurement of the hanger with the pipe removed. Dynamic stiffness measurements were undertaken for all foursupports, the procedure and results for which are reported fully in [8]. Fig. 8 shows the real part of one such measurement,for the long aluminium support, which can be seen to exhibit a natural frequency at 800 Hz and to resemble the dynamicstiffness of an undamped single-degree-of-freedom (SDOF) system,

D¼ KT �o2m ð29Þ

Fitting equation (29) to the experimental data in a least-squares sense yields the effective translational stiffness of thehanger, and also its effective mass (once account is duly taken of mass loading by the force gauge). The ‘directly’ measuredeffective mass and stiffness values are also reported in Table 1. The translational stiffness values are in very goodagreement with the estimates obtained through model updating from power reflection coefficients. The estimated massvalues are 30–40% lower than the directly measured values but the rank ordering is consistent, i.e. the long steel support isthe heaviest in both cases, and so on. Note that one would not necessarily expect the parameter estimates from themeasured dynamic stiffness and from the measured wave amplitudes to be equal because the in situ conditions of thehanger alone and the hanger with the pipe present are likely to affect the effective mass and stiffness somewhat.

The rotational stiffness is more difficult to measure directly. Instead, a comparison is made in Table 1 with the theoreticalclosed form expression for the torsional stiffness of two perfectly constrained shafts of length L acting in parallel, i.e.

KR ¼2GJ

Lð30Þ

where G is the shear modulus and J is the polar second moment of area. The simple expression over-predicts the rotationalstiffness by 50% or so compared with estimates from model updating except for the short steel support which is in worse

ARTICLE IN PRESS

Table 1Joint parameter values obtained from model updating estimation compared with directly measured values and simple theory (Eq. (30)).

KT (106 N m�1) m (kg) KR (103 N m rad�1) J (10�6 kg m2)

Support Updated Direct Updated Direct Updated Theory Updated

Long aluminium 2.32 2.60 0.056 0.077 1.57 1.98 32.8

Long steel 4.47 5.13 0.077 0.106 3.79 5.95 34.4

Short aluminium 7.93 9.02 0.046 0.076 2.37 3.56 17.9

Short steel 14.7 14.6 0.065 0.091 4.72 10.7 64.7

200 400 600 800 100012001400160018002000-15

-10

-5

0

5x 106

D (N

/m)

Frequency, Hz

Fig. 8. Real part of driving point dynamic stiffness for long aluminium pipe support.

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

Frequency, Hz0 500 1000 1500 2000

0

0.2

0.4

0.6

0.8

1

Frequency, Hz

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

Frequency, Hz0 500 1000 1500 2000

0

0.2

0.4

0.6

0.8

1

Frequency, Hz

��

��

Fig. 7. Power reflection coefficients for four support configurations: measured estimate r/(r+t); - - - - predicted from identified parameter values.

(a) long aluminium support, (b) long steel support, (c) short aluminium support, (d) short steel support.


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agreement. This is to be expected since in practice there is compliance in the assembly which affects the stiffest support to thelargest extent.

The estimates for the second moment of inertia in Table 1 imply radii of gyration of about 20 mm which is the correctorder of magnitude given the dimensions of the pipe collar assembly.

5. Conclusions

This paper has considered the widely acknowledged problem of joint identification. In the case of one-dimensionalstructures it has been argued that the problem may be more tractable when adopting a wave rather than a modalapproach. First, the wave field depends only on the local properties of the joint and the waveguide and not the globalproperties such as boundary conditions. In principle, this enables joints to be identified individually, i.e. a potentially largeill-conditioned inverse problem can instead be cast as a number of small, better conditioned inverse problems. Second, theresponse at significantly higher frequencies can be used to inform the joint parameter estimation process without theassociated problems of mode pairing and high modal overlap.

This paper has presented an overall strategy for identifying discontinuities in waveguides in which a parametric model ofthe discontinuity is updated to match observed wave reflection and transmission behaviour. The approach has beendemonstrated by means of an experimental case study on a straight pipe suspended by a hanger. Flexural wave amplitudes andhence reflection and transmission coefficients have been measured using established experimental techniques. A correspondingwave model of the structure featuring a lumped inertia-spring representation of the joint gave qualitatively similar reflectionand transmission coefficients. Standard nonlinear least-squares estimation was employed to estimate the joint parametervalues from measured power reflection coefficients. The updated wave model accurately predicts the measured reflection andtransmission coefficients. Estimates for the effective mass and translational stiffness of the hanger are in reasonably closeagreement with directly measured values. Estimates for the rotational stiffness and moment of inertia are plausible.

Application of the proposed approach to in situ piping systems may be complicated by potentially large reflections from otherboundaries and by wave mode conversion arising from asymmetric joints and pipe bends. This is the subject of further work.

Acknowledgments

The authors are grateful for the financial support of the Engineering and Physical Science Research Council (EPSRC).

Appendix

The reflection and transmission matrices for a discontinuity in a waveguide in which an arbitrary number of wave typespropagate is given by (see Eq. (5))

W�a �ðX11Wþ

b þX12Uþ

b Þ

U�a �ðX21Wþ

b þX22Uþ

b Þ

" #Raa Tba

Tab Rbb

" #¼�Wþa ðX11W

�

b þX12U�

b Þ

�Uþa ðX21W�

b þX22U�

b Þ

" #ðA:1Þ

where the submatrices of X define the transfer matrix of the joint and matrices w and U relate to the waveguide, asdescribed in Section 2.1. Taking the first column only,

W�a Raa� ðX11W

þ

b þX12Uþ

b ÞTab

U�a Raa� ðX21W

þ

b þX22Uþ

b ÞTab

24

35¼ �Wþa

�Uþa

" #ðA:2Þ

In the case of the specific mechanical joint considered in Section 2.2, Raa and Tab are 2�2 matrices

Raa¼

rpp rep

rpe ree

" #; Tab

¼tpp tep

tpe tee

" #ðA:3Þ

U�a , Uþb , W�a and Wþb are as given in Eqs. (11) and (12) and X11, X12, X21, X22 are given in Eq. (13). Substituting these intoEq. (A.2) gives

1 1

ik k

� � rpp rep

rpe ree

" #�

1 1

�ik �k

� � tpp tep

tpe tee

" #

iEIk3 �EIk3

�EIk2 EIk2

" #rpp rep

rpe ree

" #�

�ðKT �o2mÞ 0

0 �ðKR �o2JÞ

" #1 1

�ik �k

� �þ�iEIk3 EIk3

�EIk2 EIk2

" # !tpp tep

tpe tee

" #2666664

3777775

¼

�1 1

�ik �k

� �

��iEIk3 EIk3

�EIk2 EIk2

" #266664

377775

ðA:4Þ

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Expanding Eq. (A.4) and taking the first column only gives

rppþrpe � tpp � tpe

ikrppþkrpeþ iktppþktpe

iEIk3rpp � EIk3rpeþðKT �o2mþ iEIk3ÞtppþðKT �o2m� EIk3Þtpe

�EIk2rppþEIk2rpe � ðikðKR �o2JÞ � EIk2Þtpp � ðkðKR �o2JÞþEIk2Þtpe

266664

377775¼

�1

ik

iEIk3

EIk2

26664

37775 ðA:5Þ

Eq. (A.5) can be rewritten in the following form:

1 �1 1 �1

ik ik k k

iEIk3 KT �o2mþ iEIk3 �EIk3 KT �o2m� EIk3

�EIk2 �ikðKR �o2JÞþEIk2 EIk2 �kðKR �o2JÞ � EIk2

266664

377775

rpp

tpp

rpe

tpe

8>>>><>>>>:

9>>>>=>>>>;¼

�1

ik

iEIk3

EIk2

8>>><>>>:

9>>>=>>>;

ðA:6Þ

The reflection and transmission coefficients can be computed at each frequency by matrix inversion or obtainedalgebraically by further analysis.

References

[1] J.-S. Tsai, Y.-F. Chou, The identification of dynamic characteristics of a single bolt joint, Journal of Sound and Vibration 125 (3) (1988) 487–502.[2] Y. Ren, C.F. Beards, Identification of joint properties of a structure using FRF data, Journal of Sound and Vibration 186 (4) (1995) 567–587.[3] S.-W. Hong, C.-W. Lee, Identification of linearised joint structural parameters by combined use of measured and computed frequency responses,

Mechanical Systems and Signal Processing 5 (4) (1991) 267–277.[4] W.L. Li, A new method for structural model updating and joint stiffness identification, Mechanical Systems and Signal Processing 16 (1) (2002)

155–167.[5] S. Frikha, G. Coffignal, J.L. Trolle, Boundary condition error for parametric updating of in-operation systems—application to operating piping systems,

Journal of Acoustical Society of America 241 (3) (2001) 373–399.[6] M.I. Friswell, J.E. Mottershead, in: Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, 1995.[7] L. Cremer, M. Heckl, E.E. Ungar, in: Structure-borne Sound, second ed, Springer-Verlag, Berlin, 1988.[8] B. Zhang, Joint identification in structural waveguides using wave reflection and transmission coefficients, Ph.D. Thesis, University of Southampton,

2007.[9] M.-H. Moulet, F. Gautier, Reflection, transmission and coupling of longitudinal and flexural waves at beam junctions. Part II: experimental and

theoretical results, Acta Acustica 93 (1) (2007) 37–47.[10] N.R. Harland, B.R. Mace, R.W. Jones, Wave propagation, reflection and transmission in tunable fluid-filled beams, Journal of Sound and Vibration 241

(5) (2001) 735–753.[11] C.R. Halkyard, B.R. Mace, Structural intensity in beams—waves, transducer systems and the conditioning problem, Journal of Sound and Vibration

185 (2) (1995) 279–298.[12] M.-H. Moulet, F. Gautier, Reflection, transmission and coupling of longitudinal and flexural waves at beam junctions. Part I: Measurement methods,

Acta Acustica 92 (6) (2006) 982–997.[13] J.M. Muggleton, T.P. Waters, B.R. Mace, B. Zhang, Approaches to estimating the reflection and transmission coefficients of dicontinuities in

waveguides from measured data, Journal of Sound and Vibration 307 (1–2) (2007) 280–294.[14] D.J. Allwright, M. Blakemore, P.R. Brazier-Smith, J. Woodhouse, Philosophical Transactions of the Royal Society of London A 346 (1994) 511–524.[15] P.E. Gill, W. Murray, M.H. Wright, in: Practical Optimization, London Academic Press, 1993.

identifying joints from measured reflection coefficients in beam-like structures with application to...

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