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Chapter 14

On the Problem of Describing the Coupling Interface Between

Sub-structures: An Experimental Test for ‘Completeness’

J. W. R. Meggitt, A. T. Moorhouse, and A. S. Elliott

Abstract The implementation of both vibration source characterisation and sub-structure coupling/decoupling procedures

rely on the complete description of a coupling interface, that is, the inclusion of coupling forces in all signiﬁcant degrees

of freedom (DoFs). However, it is not straight-forward to establish which DoFs are required in the description. E.g. is it

necessary to include moments and/or in-plane forces? This is an important question as an incomplete description will lead

to an erroneous representation of the dynamics. However, there are currently no methods of quantifying the completeness of

an interface description. In this paper an experimental procedure is described for the assessment of interface completeness.

Based on the theoretical blocking of DoF subsets, a relation is presented that allows for the contribution of an unknown DoF

to be established. Further, a coherence style criterion is proposed to estimate the completeness of a given interface description.

This criterion may be used to check whether sufﬁcient coupling DoFs have been included in both source characterisation and

sub-structure coupling/decoupling procedures. Numerical and experimental examples are provided to illustrate the concept.

Keywords Interface · Completeness · Characterisation · Sub-structure · Coupling

14.1 Introduction

Interface dynamics play an essential role in the coupling of a built up structures. The complete description of an interface

(i.e. the correct number, position and orientation of coupling DoFs) is key when attempting to experimentally characterise

a structural assembly and/or its constituent components. This paper is concerned with the development of an ‘Interface

Completeness Criterion’ that quantitatively assesses the degree to which an interface has been correctly described.1

In general, the interface DoFs of a point connected structure are able to move in 6 coordinate DoFs. In theory, a

complete interface description must account for all of these. Experimentally, however, this is typically not possible, nor

is it often warranted.2In any practical scenario access to the coupling interface is likely to be restricted. This complicates

the measurement of coupling DoFs. Furthermore, the experimental challenges associated with the measurement of rotational

and in-plane DoFs has led to their near constant neglect. For this reason, incomplete interface descriptions are routinely

encountered in practice. Whilst considerable research has focused on the importance of rotational DoFs [1–5], in-plane

DoFs have received far less attention. According to Moorhouse and Elliott [6], however, their importance in the coupling

of structural elements should not be understated. With this in mind, the ability to assess the importance of a particular DoF,

perhaps unknown, would clearly be advantageous.

In the analysis of a continuous interface it is standard practice to approximate the interface in terms of a ﬁnite number of

point-like DoFs so as to acquire an approximate description of its dynamics [7,8]. Such an approximation is often based on

1Although we will refer to the proposed quantity as a criterion, given its deﬁnition, the authors are undecided as to whether ‘coefﬁcient’ may be a

more appropriate term.

2Whilst a complete interface description is required from a theoretical basis, it is often the case that their exists a subset of DoFs that are of

particular importance, and themselves provide a satisfactory description of the interface. In such a case the remaining DoFs may not need to be

accounted for.

J. W. R. Meggitt () · A. T. Moorhouse · A. S. Elliott

Acoustics Research Centre, University of Salford, Greater Manchester, UK

e-mail: j.w.r.meggitt1@salford.ac.uk

© The Society for Experimental Mechanics, Inc. 2018

A. Linderholt et al. (eds.), Dynamics of Coupled Structures, Volume 4, Conference Proceedings of the Society

for Experimental Mechanics Series, https://doi.org/10.1007/978-3-319-74654- 8_14

171

172 J. W. R. Meggitt et al.

a points per wavelength argument, although the completeness of this description remains unclear. Alternatively, a continuous

interface may be approximated in terms of a truncated orthonormal series expansion [9]. Similarly, however, it is unclear the

degree to which this expansion suitably describes the interface.

In the analysis of both discrete and continuous interfaces, an indication as to whether sufﬁcient DoFs have been accounted

for would clearly beneﬁt the experimenter, and aid in avoiding the potential problems associated with an incomplete

interface description. Based on the mathematical blocking of known interface DoFs, theory is developed that allows for

the contribution of unknown interface DoFs to be established. This contribution is used as part of a coherence style criterion

to assess the completeness of an interface description. The proposed criterion may be used to indicate missing DoFs, weigh

up the potential beneﬁts of including additional DoFs, or provide a level of uncertainty in a given interface description.

The remainder of this paper will be organised as follows: In Sect. 14.2 the interface completeness problem will be

demonstrated mathematically by considering the independent characterisation of a vibration source. Having introduced the

problem, Sect. 14.3 will describe the relevant theory, before Sect. 14.4 outlines the proposed Interface Completeness Criterion

(ICC). Section 14.5 will provide a numerical and experimental case study. Lastly, Sect. 14.6 will present some concluding

remarks.

14.2 The Interface Completeness Problem

The interface completeness problem is illustrated here with reference to the inverse determination of blocked forces using

the in-situ approach presented in [10–12]. A similar problem is encountered in the coupling (or decoupling) of sub-structures

through dynamic sub-structure coupling procedures.

It was shown by Moorhouse et al. [13] that the blocked force, which represents an independent property of a structural

source, may be acquired through an inverse procedure similar to that used in operational force identiﬁcation and standard

transfer path analysis (TPA) procedures [14]. The relation of note is given by,

vbDYbcN

fc(14.1)

where Ybc 2CNNis the measured mobility matrix of a coupled assembly, vb2CNis a measured operational velocity vector,

and N

fc2CNis the vector of unknown blocked forces. Here subscripts fbgand fcgrepresent complete sets of remote receiver

and coupling interface DoFs, respectively. Variables are given in the frequency domain, with explicit notation omitted for

clarity.

To illustrate the interface problem we begin by considering the in-situ blocked force relation in the form,

vbDYbciN

fciCYbcjN

fcj:(14.2)

where the velocity contributions from known and unknown DoFs are separated. Here, fcicgrepresents the set of coupling

interface DoFs that are known and considered measurable, whilst fcjcjcj6 cigrepresents the set of interface DoFs that

are unknown, or known but can not be measured. An illustrative example of such an assembly is given in Fig.14.1a.

In the determination of the blocked force one pre-multiplies the operational velocity vector by the inverse of a mobility

matrix pertaining to the known DoFs, Ybci. Pre-multiplication of Eq. 14.2 by Y1

bcithus yields,

Y1

bcivbDQ

N

fciDN

fciCY1

bciYbcjN

fcj:(14.3)

The acquired blocked force, Q

N

fci, although correct in its own respect,3is not the true blocked force at the DoFs ci,N

fci.The

neglect of the DoFs cjhas resulted in the additional term, Y1

bciYbcjN

fcj. This term is a property of the coupled assembly and thus

the acquired blocked force is no longer an independent property of the source. This loss of independence may have severe

implications when the acquired blocked force is transferred to a secondary assembly and used to predict the operational

response, for example in the construction of a Virtual Acoustic Prototype [15]. This is demonstrated below.

3The blocked force Q

N

fcirepresents the reaction forces due to a source that is blocked in ciDoFs but unrestrained in the remaining cjDoFs. It is

therefore by deﬁnition a blocked force, albeit not the true blocked force.

14 On the Problem of Describing the Coupling Interface Between Sub-structures: An Experimental Test for ‘Completeness’ 173

ab

Fig. 14.1 Diagrammatic illustration of the constrained and unconstrained assemblies corresponding to Eqs. 14.7 and 14.8, respectively. (a)

Unconstrained assembly. (b) Constrained assembly

We are interested in the prediction of the remote receiver velocity, v0

b, on a secondary assembly (denoted by the prime

symbol, 0). The true velocity, that we are aiming to predict, may be expressed in terms of the true blocked force as,

v0

bDY0

bci

N

fciCY0

bcj

N

fcj:(14.4)

To predict the remote velocity the acquired blocked force, Q

N

fci, is pre-multiplied by the coupled transfer mobility of the

secondary assembly that corresponds to the known and measurable DoFs, Y0

bci.

Q

v0

bDY0

bci

Q

N

fciDY0

bci.N

fciCY1

bciYbcjN

fcj/(14.5)

Expanding the above we arrive at,

Q

v0

bDY0

bci

N

f0

ciCY0

bciY1

bciYbcjN

fcj:(14.6)

From the above it is clear that the true velocity, v0

b, and the predicted velocity, Q

v0

b, are not equal. Hence the blocked force is

no longer transferable. Here, the unknown blocked force, N

fcj, contributes to the predicted velocity response through the

propagating transfer function, Y0

bciY1

bciYbcj. So, whilst the response prediction still accounts for the unknown blocked

force, its contribution to the response of a secondary assembly is incorrect. This discrepancy is clearly a consequence of

an incomplete interface description. Had all coupling DoFs been accounted for the true blocked force would have been

acquired, and subsequently transferred to the secondary assembly with no problems.4The uncertainty introduced as a result

of an incomplete interface description is referred to here as model uncertainty.

In this paper we are interested in establishing an experimental procedure to minimize, or identify the severity/cause of,

this source of model uncertainty through the development of an Interface Completeness Criterion (ICC).

14.3 Theoretical Development

In this section we will develop the theory required to formulate the proposed Interface Completeness Criterion.

Consider the assembly depicted in Fig. 14.1a, where a source and receiver sub-structure are coupled via two sets of

interface DoFs. Denoted ciand cj, these DoFs form subsets of the complete coupling interface c, such that fcicgand

4A similar problem is encountered in the application of sub-structure coupling and decoupling procedures. The neglect of coupling interface DoFs

results in an erroneous representation of the assembly/substructure dynamics.

174 J. W. R. Meggitt et al.

fcjcjcj6 cig. Also included are two sets of remote measurement DoFs on the source and receiver. These are referred

to as aand b, respectively. The equations that govern the behaviour of the coupled source-receiver assembly (as depicted in

Fig. 14.1a) are given generally by,

0

B

B

@

va

vci

vcj

vb

1

C

C

A

D

2

6

6

4

Yaa YaciYacjYab

YciaYciciYccjYcib

YcjaYcjciYcjcjYcjb

Yba YbciYbcjYbb

3

7

7

5

0

B

B

@

fa

fci

fcj

fb

1

C

C

A

:(14.7)

We begin by considering the case where only two external forces are applied, i.e. fcjDfbD0. The ﬁrst is an arbitrary force

at the remote source DoFs a,fa. The second is a constraint force at the coupling interface DoFs ci,N

fci. This second force is

the blocking force required to constrain the velocity vcito 0. The constrained assembly may be represented by the following

set of equations,

0

B

B

@

va

0

vcj

vb

1

C

C

A

D

2

6

6

4

Yaa YaciYacjYab

YciaYciciYcicjYcib

YcjaYcjciYcjcjYcjb

Yba YbciYbcjYbb

3

7

7

5

0

B

B

@

fa

N

fci

0

0

1

C

C

A

:(14.8)

The above describes an assembly that is excited by an arbitrary force at a, whilst the coupling interface, c,ispartially

constrained. This constrained assembly is shown diagrammatically in Fig. 14.1b. Taking the second line of Eq.14.8,

0DYciafaCYciciN

fci:(14.9)

and solving for the blocked force, we arrive at,

N

fciDY1

ciciYciafa:(14.10)

The above relation is a force transmissibility that relates the externally applied force at a, to the resultant blocked force at ci.

Taking the 4th line of Eq. 14.8, whilst introducing the notation v.cj/

bto describe the velocity of the constrained assembly, i.e.

the contribution through the coupling DoFs cjonly,

v.cj/

bDYbafaCYbciN

fci:(14.11)

The ﬁrst RHS term describes the velocity at bdue an applied force at aon the unconstrained assembly, i.e. v.c/

bDYbafa.

The second RHS term describes an added contribution to this velocity due to the blocking force required to constrain the

interface ci. The blocked force may be eliminated by expressing it in terms fa, as per Eq. 14.10,

v.cj/

bDYbafaYbciY1

ciciYciafa(14.12)

or equivalently,

v.cj/

bDYba YbciY1

ciciYciafa:(14.13)

Equation 14.13 relates an externally applied force at ato the resultant velocity at bon the constrained assembly depicted

in Fig. 14.1b.5The bracketed mobility term therefore represents the transfer mobility through the constrained assembly.

Consequently, let us deﬁne,

Y.cj/

ba ,Yba YbciY1

ciciYcia(14.14)

as the transfer mobility from ato b, through the DoFs cj, whilst the DoFs ciare blocked.

5Although not discussed in any further detail here, for jcjjD1,Eq.14.13 may be considered the velocity at bdue to the contribution of a

single path, i.e. whilst all other paths are blocked. This is a similar concept to that used by Margans [17] in the formulation of the GTDT TPA

procedure [18].

14 On the Problem of Describing the Coupling Interface Between Sub-structures: An Experimental Test for ‘Completeness’ 175

Noting that the left hand term in the bracket of Eq. 14.13 is the transfer mobility of the unconstrained assembly, let us

redeﬁne this as,

Y.c/

ba ,Yba (14.15)

that is, the transfer mobility from ato bthrough all coupling DoFs (ciand cj). Lastly, the right hand term in the bracket of

Eq. 14.13 can be seen to form a round trip identity [16] for a mobility similar to Yba. It is interesting to note that for the case

whereby only a single set of coupling DoFs exist, i.e. jcjjD0, this round trip identity is equal exactly to the unconstrained

mobility, Yba [16].6For the constrained interface considered, however, the mobility product YbciY1

ciciYciacorresponds to a

transfer mobility between aand bwhilst accounting for only the coupling DoFs ci. As such, let us deﬁne,

Y.ci/

ba ,YbciY1

ciciYcia:(14.16)

Using the above deﬁnitions Eq. 14.14 may be rewritten as,

Y.cj/

ba DY.c/

ba Y.ci/

ba (14.17)

or alternatively,

Y.c/

ba DY.ci/

ba CY.cj/

ba :(14.18)

Equation 14.18 states that the transfer mobility from ato bthrough the unconstrained interface, Y.c/

ba , may be represented as

the sum of two mobility terms. These correspond to; the transfer mobility from ato bwhilst the DoFs ciare constrained,

Y.cj/

ba ; and transfer mobility from ato bwhilst the DoFs cjare in some way neglected, Y.ci/

ba .

Experimentally we are able to measure Y.c/

ba directly, simply through the excitation of a, and response measurement of b,

since the physical assembly is not constrained. Similarly, we are able to compute Y.ci/

ba through the direct measurement of its

associated mobilities, Ybci,Y1

cici, and Ycia. We are therefore able to calculate the contribution of the DoFs cjwithout having

access to them directly.

We note here that if the DoF subset cicontains all DoFs, ciDc, then jcjjD0and Y.c/

ba DY.ci/

ba . Experimentally this

would mean that all of the interface DoFs have been included and the interface description is complete.7This ﬁnal remark

will be used in the formulation of an Interface Completeness Criterion. This criteria may be used to provide a quantitative

assessment of an interface’s ‘completeness’, that is, whether sufﬁcient DoFs have been accounted for.

14.4 Interface Completeness Criterion

In the ﬁeld of modal analysis there exist a number of criteria to assess conﬁdence in acquired modal parameters. For example,

the modal assurance criterion (MAC), the coordinate modal assurance criterion (COMAC), and the frequency response

assurance criterion (FRAC), to name but a few [19].

The aim here is to establish a similar criterion for the ‘completeness’ of an interface description. Here ‘completeness’

refers to the degree to which the coupling interface DoFs are accounted for by the coupling interface mobility matrix, Ycc.A

complete interface is deﬁned as one whose DoFs are completely described by Ycc.

A suitable criteria should be bound between 0 and 1 such that the degree of completeness may be assessed with ease. An

assessment is proposed in the same form as the well established modal assurance criterion (MAC). The proposed criteria will

be referred to as the Interface Completeness Criterion (ICC).

6Consequently, Y.cj/

ba D0since the DoF subset cjdoes not exist.

7Alternatively, if one considers Eq. 14.14, a complete interface description would block entirely the coupling interface, resulting in a velocity of 0

at the remote source DoFs b.

176 J. W. R. Meggitt et al.

Using an expression identical in form to the MAC, the ICC yields a frequency dependent scalar value bound between 0

and 1, and is deﬁned as,

ICCbaDˇ

ˇ

ˇYc

ba Yci

baHˇ

ˇ

ˇ

2

Yc

ba.Yc

ba/HYci

ba.Yci

ba/H(14.19)

where, Y.ci/

ba DYbciY1

ciciYcia.

Recalling the closing remark of Sect. 14.3, we note that if all coupling DoFs are accounted for, that is, Y.c/

ba DY.ci/

ba ,the

ICCbais equal to one. Similarly, if none of the coupling DoFs are accounted for, that is, Y.ci/

ba D0,theICC

bais equal to zero.

The ICCbais thus related to the degree of completeness, and consequently, the degree of model uncertainty present in a given

interface description.

We note that the ICCbadescribes the degree to which an interface is complete through the blocking of a transfer function

measured between a set of source side DoFs, a, and a single receiver side DoF, b. As such, a different ICCbais obtained for

each remote receiver DoF considered. Furthermore, it should be noted that whilst the above criterion, in theory, yields 1for a

complete interface description this is likely unachievable in an experimental scenario. The measured and predicted mobilities,

Y.c/

ba and Y.ci/

ba , respectively, will be contaminated to some extent by measurement uncertainty and perfect agreement is

unlikely. The question of experimental error is further addressed in the case study of Sect. 14.5.2, although a more thorough

investigation regarding the sensitivity of the ICC is considered beyond the scope of this work.

14.5 Case Studies

In this section the application of the ICC will be demonstrated as part of two case studies. In the ﬁrst, the discretisation of a

continuous interface between two coupled plates will be considered numerically. In the second, an experimental study will

investigate the completeness of an interface between two sections of steel rod.

14.5.1 Numerical: Continuous Interface

In this study the Interface Completeness Criterion (ICC) is used too assess the completeness of a discrete approximation

to a continuous interface. The interface is considered to be an arbitrary line that separates two sides of a simply supported

rectangular plate, as illustrated in Fig. 14.2. Each side of the interface is considered a separate plate, and are referred as source

(plate 1) and receiver (plate 2). The coupled plate assembly is modelled analytically using a truncated modal summation.

Both translational z, and xyrotational DoFs, ˛and ˇ, are included. The geometric and material properties of the model are

presented in Table 14.1. Five source side DoFs (a) are considered on plate 1 and a single receiver side DoF (b)onplate2.

The coupling interface is represented by an increasing number of positional DoFs, as illustrated in Fig. 14.3. All positional

DoFs, unless otherwise speciﬁed, are made up of three coordinate DoFs, relating to z,˛and ˇ. The aim of this study is to

illustrate the use of the ICC in quantitatively assessing the discretisation of a continuous interface. Interfaces of this type are

often encountered in practical scenarios and are typically represented by a series of discrete point like DoFs. The degree to

which this approximation accounts for the continuous nature of the interface is of interest. The ICC between the source side

DoFs and each of the receiver side DoFs are calculated (these are referred to as ICCz,ICC

˛and ICCˇ).

Five interface descriptions are considered here, corresponding to 1, 2, 3, 4 and 5 positional DoFs, evenly spread across

the breadth of the plate, as illustrated through Fig. 14.3a–e. The corresponding ICCs are shown in Fig. 14.4, from top to

bottom, respectively. The ICCs pertaining to translational and rotational (˛and ˇ) receiver DoFs are shown in blue, orange

Table 14.1 Geometry of the coupled plate assembly, plate 1 and plate 2. The material properties of the three plates were; Young’s modulus

ED200 109,densityD9000, Poison’s ratio D0:3, and loss factor D0:1

Plate Dimensions

Coupled 10:8 0:005

Plate 1 0:35 0:8 0:005

Plate 2 0:65 0:8 0:005

14 On the Problem of Describing the Coupling Interface Between Sub-structures: An Experimental Test for ‘Completeness’ 177

Fig. 14.2 Diagrammatic representation of the continuous interface numerical study. Two plates are coupled via a continuous interface (dashed red

line) in the translational zand x=yrotational DoFs, ˛and ˇ

ab

de

c

Fig. 14.3 Diagrammatic representation of the continuous interface numerical study. Two plates are coupled via a continuous interface which is

approximated through varying degrees of discretisation. (a) 1 positional DoF. (b) 2 positional DoFs. (c) 3 positional DoFs. (d) 4 positional DoFs.

(e) 5 positional DoFs

and yellow, respectively. Figure 14.4 clearly illustrates that the ICC converges to one as the number of DoFs used to describe

the interface is increased. Furthermore, the results highlight the difference between ICCz,ICC

˛and ICCˇ. Although sharing

an overall trend, clear differences can be observed. This suggests that a single ICC may not sufﬁciently describe the interface

completeness, and that multiple ICCs may be required for a more thorough insight. That said, the experimental effort required

to include additional response DoFs is minimal as no further excitations are required.

Shown in Fig. 14.5 are the ICCs corresponding to an interface description made up of 10 positional DoFs. In Fig. 14.5athe

coupling interface description includes both translational and rotational coordinate DoFs. In Fig. 14.5b the coupling interface

description includes only translational coordinate DoFs, i.e. the rotational DoFs were neglected. As one might expect, the

neglect of rotational coupling DoFs has resulted in a worsening of the ICC. Whilst this is an intuitively obvious result, it

highlights the importance of rotational DoFs in the description of a continuous interface and, furthermore, the ability of the

ICC to quantify it.

178 J. W. R. Meggitt et al.

102103

0

0.2

0.4

0.6

0.8

1

ICC

ICCabzICCabaICCabb

102103

0

0.2

0.4

0.6

0.8

1

ICC

102103

0

0.2

0.4

0.6

0.8

1

ICC

102103

0

0.2

0.4

0.6

0.8

1

ICC

102103

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

ICC

Fig. 14.4 Interface Completeness Criteria (ICC) for a continuous interface approximated using ﬁve different levels of discretisation (see Fig. 14.3).

From top to bottom, the interface is represented by 1, 2, 3, 4 and 5 positional DoFs, each of which includes translational zand x=yrotational DoFs,

˛and ˇ. The ICC is calculated using 15 source side positional DoFs (including z,˛and ˇat each) and a single receiver side DoF (including z,˛

and ˇ). An ICC is presented for each receiver DoF

14 On the Problem of Describing the Coupling Interface Between Sub-structures: An Experimental Test for ‘Completeness’ 179

102103

0

0.2

0.4

0.6

0.8

1

ICC

ICCabzICCaba

a

b

ICCabb

102103

0

0.2

0.4

0.6

0.8

1

ICC

Fig. 14.5 Interface Completeness Criteria (ICC) for acontinuous interface approximated using 10 positional DoFs, both with (a) and without (b)

rotational coupling. The ICC is calculated using 15 source side positional DoFs (including z,˛and ˇat each) and a single receiver side DoF

(including z,˛and ˇ). An ICC is presented for each receiver DoF. (a) 10 positional DoFs. (b) 10 positional DoFs (translation zonly, neglecting ˛

and ˇ)

Fig. 14.6 Diagrammatic representation of the experimental study. Two rods are coupled via a single discrete interface (red line) in translational y

and z,andy=zrotational DoFs, ˇand

Lastly, it is perhaps worth noting the improvement in the ICC between Figs. 14.4 (bottom) and 14.5a due to the extra 5

positional DoFs.

14.5.2 Experimental: Point Connected Interface

The experimental case study presented here concerns the description of an interface between two rigidly coupled rods. A

diagrammatic illustration of the study is given in Fig. 14.6. The coupling interface, shown in red, was considered as a single

positional DoF, and therefore described, generally, by the 6 coordinate DoFs, x,y,z,˛,ˇand .

The aim of this study was to use the ICC to access various interface descriptions consisting of the DoFs; y,z,ˇand .

The in-plane DoF x(along the length of the rod) and its associated rotation ˛were neglected due to the difﬁculties associated

with their measurement.

The translational, rotational and cross mobilities associated with the retained DoFs considered were approximated using

the ﬁnite difference method [20], which itself imposes a limit on the validity of an interface description over a frequency

range related to the sensor spacing (taken here as 2 cm).

Twelve source side DoFs (a) were excited whilst simultaneously measuring the response at the coupling interface (ci) and

the single receiver side DoF (b). This enabled the calculation of Y.c/

ba and Ycia. The interface DoFs were subsequently excited

and the mobilities Yciciand Ybcimeasured. Together, the above measurements allowed for the calculation of the mobility

180 J. W. R. Meggitt et al.

100 1000 5000

0

0.2

0.4

0.6

0.8

1

Frequency

ICC

4 DoFs 2 DoFs

a

b

c

100 1000 5000

0

0.2

0.4

0.6

0.8

1

ICC

100 1000 5000

0

0.2

0.4

0.6

0.8

1

ICC

Fig. 14.7 Interface Completeness Criteria (ICC) obtained from the experimental study using 4 different interface descriptions. (a)AnICC

comparison based on all 4 interface DoFs (z,y,,ˇ), and the reduced interface DoFs (z,). (b) An ICC based on the translational zDoF,

i.e. neglecting the rotational DoF, .(c) An ICC based on the translational and rotational DoFs, yand ˇ

Y.ci/

ba DYbciY1

ciciYcia. The ICC was then calculated as per Eq. 14.19. The importance of a particular DoF to the interface

description may be assessed by simply neglecting said DoF in the calculation of the ICC.

Shown in Fig. 14.7 are the ICCsobtained from the experimental study illustrated in Fig. 14.6 for a number of different

interface descriptions.

Presented in Fig. 14.7a are the ICCs associated with a 4 DoF description (blue) and a reduced 2 DoF description (orange),

where the DoFs yand ˇhave been neglected. It is interesting to note that the two ICCs are in near perfect agreement.

This suggests that the interface DoFs, yand ˇ, have a negligible effect on the coupling of the structure. It should be noted,

however, that this result was obtained from an ICC in which the source side excitation was made in the vertical zdirection.

Had additional excitations been applied in the ydirection, coupling in yand ˇwould likely have been of greater importance.

It is important to note that the calculation of the ICC, as per Eq. 14.19, is particularly sensitive to experimental error,

notably the error associated with inconsistent excitations in the measurement of the required mobility matrices. Errors of

this sort can lead to inconsistencies within the interface mobility matrix, Ycici, which may result in large errors following

its inversion. This effect is particularly relevant for under-damped systems, where structural anti-resonances are sensitive to

14 On the Problem of Describing the Coupling Interface Between Sub-structures: An Experimental Test for ‘Completeness’ 181

excitation position. As an example, consider the sharp notch in the ICCs of Fig. 14.7a in the region of 200 Hz. This is likely

a result of inconsistencies in the measured mobilities due to the small amount of damping present in the rod, and not due to

an incomplete interface description.

Shown in Fig. 14.7b is the ICC associated with the single translational coupling DoF, z. A comparison against the reduced

description in Fig. 14.7a (orange) clearly illustrates a worsening of the ICC due to the now neglected rotational DoF, .The

effect is particularly pronounced in the region of 150 and 450 Hz. This suggests that in these regions the rotational coupling

of the interface is of greater importance.

Lastly, shown in Fig. 14.7c is the ICC associated with the coupling DoFs yand ˇ. As one would expect considering

Fig. 14.7a, the ICC suggests that, by themselves, the DoFs yand ˇprovide a very poor interface description.

14.6 Conclusion

This paper has been concerned with the development of an Interface Completeness Criterion (ICC) suitable for the

quantitative assessment of the degree to which a coupling interface has been described. The ICC is based on the mathematical

blocking of a subset of the coupling interface DoFs. This blocking allows for the separation of known and unknown DoFs,

and consequently the formulation of a bound completeness criterion, referred to here as the ICC. The application of the ICC

was illustrated by two case studies; the ﬁrst a numerical simulation concerning the discretisation of a continuous interface,

and the second, and experimental study concerning a discrete interface.

Acknowledgements This work was funded through the EPSRC Research Grant EP/P005489/1, Design by Science.

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