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Interval-based identiﬁcation of response-critical joints in complex built-up

structures: A tool for model reﬁnement

J.W.R. Meggitt1

1Acoustics Research Centre, University of Salford, Greater Manchester, M5 4WT

Abstract

To accurately model the dynamics of a complex built-up structure it is often necessary to forgo the assumption of ideal

rigid coupling between components. The dynamic behaviour of inter-component connections, or joints, can have a

signiﬁcant impact on the performance, even survivability, of an assembled structure. For complex structures, where

many joints are present, the complete characterisation and/or modelling of all joints is often not feasible. Hence, in the

development or reﬁnement of a component-based model there is a need to identify the response-critical joints of an

assembly - those joints whose reﬁnement would beneﬁt the modelling eﬀort most. In the present paper we address this

issue by considering a ‘possibilistic’ interval-based assessment of joint variability, and propose a sensitivity metric

to rank order joints based on their inﬂuence towards the dynamics of the assembled structure. Results are compared

against local gradient-based and global sampling-based metrics as part of two numerical examples.

Keywords: joint dynamics, interval analysis, sensitivity analysis, possibilistic, model reﬁnement

1. Introduction

Complex engineering structures (such as vehicles, satellites, domestic products, buildings and more) are built-

up from individual components that are joined with connections that rely on frictional contact or other dissipative

mechanisms (such as elastomeric elements). To model such a structure it is conventional to follow a component-

based strategy, where individual components are measured, or modelled, independently of one another, before being

assembled to form the complete model. Typically, this assemblage is achieved by enforcing equilibrium and continuity

conditions (or energetic equivalents) at the connecting interfaces between components; a process referred to as sub-

structuring [1]. This is, however, an idealisation. The joints that connect components possess their own dynamics;

ideal rigid coupling is rarely achieved in practice. Importantly, these joint dynamics can have a signiﬁcant impact

on the performance, or even survivability, of an operational assembly [2]. Hence, when adopting a component-based

strategy, it is essential that, where necessary, jointed connections are characterised and modelled appropriately.

The types of joints encountered in practical engineering structures are wide ranging, and can include bolted con-

nections, rivets, welds, and resilient elements, among many others [3]. Likewise, their dynamics can vary greatly,

from linear spring or beam-like coupling, to non-linear hysteretic, perhaps discontinuous (e.g. frictional), connections

[4, 5]. Whether linear or non-linear, two principal issues arise when modelling complex jointed structures.

First, an experimental characterisation of each joint is required before they can be represented within a component-

based model. This characterisation is an experimentally demanding task [6, 7, 8, 9, 10, 11], often reinquiring supple-

mentary numerical models [12, 13, 14]. As an example, in [6, 7, 9] the experimental procedure involves measuring

the point and transfer admittance between the interface/joint and a set of remote DoFs, on both the assembled and

individual components. In [12] the individual component admittances are instead obtained by FE models. In [13, 14]

the need for interface-based measurements is avoided using a hybrid experimental/numerical ’System Equivalent

Model Mixing’ scheme, though several remote measurements are still required on the connected components. For a

non-linear characterisation experimental eﬀorts are multiplied, often requiring controlled shaker excitations with in-

creasing amplitude levels [11] or advanced time domain identiﬁcation methods [10]. As an alternative to experimental

characterisation, joint characteristics might be obtained indirectly by model updating [15, 16].

Preprint submitted to Journal of Sound and Vibration February 14, 2022

Second, the computational eﬀort required to incorporate a non-linear or 3D elastic joint model can be many times

greater than that of a linear or 1D joint [17, 18]. For complex built-up structures, where many joints are present,

these eﬀorts are multiplied, and a complete characterisation/modelling of all joints is often not feasible. Hence, in

the development or reﬁnement of a component-based model there is a need to identify the response-critical joints of

an assembly - those that have the greatest inﬂuence on the target response - so that experimental and computational

resources may be focused towards those joints which will beneﬁt the modelling eﬀort most. Importantly, the inﬂu-

ence of a speciﬁc joint will depend not only on the passive dynamics of its connected components, but also on the

operational activity of the assembly.

It is worth noting that the identiﬁcation of ‘critical joints’ is often considered in the context of structural analysis,

where joints that experience high static loads are identiﬁed and subject to local non-linear reanalysis [16]. In the

present paper we consider, instead, the dynamic response of an assembled structure, and the relative importance of

individual joints towards said response. We address the issue of response-critical joint identiﬁcation by considering

an interval-based sensitivity analysis (SA) of the assembled structure’s response, with respect to the relaxation of

individual interface connections.

When dealing with problems of uncertainty, such as sensitivity analysis (SA), are two common view points:

probabilistic or non-probabilistic. SA is typically treated probabilistically, describing uncertain parameters by their

probability distributions. A principal diﬃculty here is that to evaluate the uncertainty of a complex system by proba-

bilistic means, detailed statistical input data is required. If this data is not available, complex statistical analysis cannot

be justiﬁed. This issue can be overcome by non-probabilistic, or ‘possibilistic’, analysis methods, which include: in-

terval analysis [19, 20, 21, 22, 23, 24, 25], convex modelling [26] and fuzzy set theory [27, 28]. Possibilistic methods

do not consider the probability of an outcome occurring, rather whether or not is it possible.

Joint characterisation is an experimentally demanding task, and so it is often challenging to obtain the statistical

information required to conduct a conventional probabilistic analysis of uncertainty. Possibilistic, or interval analysis,

methods oﬀer a convenient alternative. Interval-based methods have found numerous applications in the analysis of

structural dynamic systems, including; interval Finite Element Analysis [20, 21], eigen frequency analysis [29, 30],

robustness evaluation [31], quantiﬁcation of experimental uncertainty [25], inverse force identiﬁcation [23], stochas-

tic response analysis [32, 33], among many others. In the present paper, we consider its application towards the

identiﬁcation of response critical joints.

Based on the rank ordering of a proposed sensitivity metric, we are able to infer the relative importance of indi-

vidual joints by their contribution towards the global dynamics of the assembled structure. The proposed metric is

formulated by considering the dual formulation of the sub-structuring problem in the presence of a ﬂexible interface

[34]. The Sherman-Morrison formula is used to separate out the contribution of an individual joint, at which point

a complex interval description is adopted for the interface stiﬀness and damping. Employing interval arithmetic, the

maximum and minimum bounds of the coupled assembly admittance (also operational response) are determined. Spa-

tial averaging of the interval bounds over selected degrees of freedom (DoFs) yields a metric that indicates the relative

importance of a particular joint towards the dynamics of the assembled structure, in a possibilisitic sense. Results are

compared against local gradient-based and global sampling-based (Sobol) metrics, also proposed herein.

Having introduced its context and principal aim, the remainder of this paper will be structured as follows. Section

2 will begin by introducing the notion of real and complex intervals, alongside their arithmetic operations. Primal and

dual approaches to modelling ﬂexible interface dynamics are then summarised in section 3, before section 4 details the

proposed interval-based assessment of joint variability. Section 5 then considers the identiﬁcation of response-critical

joints, before section 6 presents two numerical examples and section 7 draws some concluding remarks.

2. Interval arithmetic

Interval analysis falls within a family of non-probabilistic methods for evaluating uncertainty in complex systems

[35, 36]. One of the principal diﬃculties in evaluating the uncertainty of a complex system by probabilistic means

is the acquirement of detailed statistical input data. In the absence of such data, complex statistical analysis cannot

be justiﬁed. It is this issue that has lead to the development of non-probabilistic, or ‘possibilistic’, methods, to which

interval analysis belongs.

2

2.1. Real intervals

A real interval {x}=[ ¯x¯

x] (denoted by curly braces) is represented by two numbers forming the upper ( ¯x) and

lower (¯

x) bounds of the unknown variable x,

¯

x<x<¯x.(1)

Unlike a probability distribution, the interval bound provides no detailed information on the likeliness of a particular

value of x, only that its true value must occur within it.

With the deﬁnition of an interval, as per equation (1), it is the role of interval arithmetic to compute the upper and

low bounds of functions whose inputs are intervals. This is achieved by redeﬁning the basic arithmetic operations

+− × ÷ as interval operations ⊕ ⊗ .

For real valued xand y, the key interval arithmetic operations are deﬁned as [37]:

{x}⊕{y}=[¯

x¯x]⊕[

¯

y¯y]=[¯

x+

¯

y¯x+¯y] (2)

{x}{y}=[¯

x¯x][

¯

y¯y]=[¯

x−¯y¯x−

¯

y] (3)

{x}⊗{y}=[¯

x¯x]⊗[

¯

y¯y]=[min(¯

x

¯

y,¯

x¯y,¯x

¯

y,¯x¯y) max(¯

x

¯

y,¯

x¯y,¯x

¯

y,¯x¯y)] (4)

{x}{y}=[¯

x¯x]⊗[1/¯y1/

¯

y] if 0 <[

¯

y¯y] (5)

Care should be taken with interval division; if the interval {y}contains zero, its reciprocal will extend to ±∞.

Note that non-interval quantities can readily be incorporated within the interval arithmetic presented above by

considering an interval whose upper and lower bounds are of equal value, x={x}=[x x]. More complex interval

functions can be dealt with by recursively applying these arithmetic operations.

2.2. Complex intervals

There are two common ways in which interval arithmetic can be extended to complex variables. In one, the

complex interval is represented by a circular disk in the complex plane, characterised by its centre point and radius

[37, 38]. In the other, the real and imaginary parts of the complex number are represented by independent real

intervals. This leads to a rectangular interval in the complex plane [37, 39, 40]. The rectangular approach has the

advantage that the real and imaginary parts can be assigned diﬀerent intervals, which is not possible via the circular

disk representation, which assumes an inherent interdependency. In the present paper we will adopt the rectangular

complex interval approach as it will enable the independent assignment of joint stiﬀness and damping.

Taking xand ynow to be complex numbers, their rectangular intervals take the form,

{x}={xr}+i{xi},{y}={yr}+i{yi}(6)

where,

{xr}=[¯

xr¯xr],{xi}=[¯

xi¯xi],{yr}=[

¯

yr¯yr],{yi}=[

¯

yi¯yi].(7)

These intervals describe rectangular domains in the complex plane, as illustrated in ﬁgure 1 (orange and yellow

rectangles). To analyse complex interval functions we must extend the interval arithmetic deﬁned above to the complex

domain. For non-interval complex variables, addition, subtraction and multiplication can be deﬁned as,

x+y=(xr+yr)+i(xi+yi),x−y=(xr−yr)+i(xi−yi),x×y=(xryr−xiyi)+i(xryi+xiyr).(8)

3

Based on the above, an equivalent set of complex interval operations can be deﬁned as,

{x}⊕{y}={xr}⊕{yr}+i{xi}⊕{yi}=[¯

xr+

¯

yr¯xr+¯yr]+i[¯

xi+

¯

yi¯xi+¯yi] (9)

{x}{y}={xr}{yr}+i{xi}{yi}=[¯

xr−¯yr¯xr−

¯

yr]+i[¯

xi−¯yi¯xi−

¯

yi] (10)

{x}⊗{y}=({xr}⊗{yr}{xi}⊗{yi})+i({xr}⊗{yi}⊕{xi}⊗{yr})(11)

where ⊕,and ⊗are deﬁned as per the real interval operations.

These complex interval operations are optimal; they provide the smallest rectangular interval that includes the

set of all possible outcomes. Shown in ﬁgure 1abc are the complex intervals arising due to {x}⊕{y},{x}{y}and

{x} ⊗ {y}, respectively, where {x}and {y}are the complex intervals represented by the yellow and orange rectangles.

Results are compared against a sampling-based estimation (blue dots). Note that in using the rectangular interval

description described above, addition and subtraction do not alter the shape of the interval. Multiplication, on the

other hand, leads to a change in shape. Hence, the interval description over estimates the true region of possible

outcomes. Nevertheless, the resulting interval is still optimum in a rectangular sense.

Let us now consider the case of complex division, deﬁned as,

x÷y=xryr+xiyi

y2

r+y2

i

+ixiyr−xryi

y2

r+y2

i

.(12)

Applying the above interval operations yields the following,

{x}{y}={xr}⊗{yr}⊕{xi}⊗{yi}{y2

r}⊕{y2

i}+i{xi}⊗{yr}{xr}⊗{yi}{y2

r}⊕{y2

i}.(13)

The complex interval obtained as per equation (13) is not optimal. If standard interval arithmetic operations are applied

to complex division, the numerator and denominator intervals are treated independently (the so-called dependency

problem). For example, the instance of {yr}in the numerator is considered independent to that being squared in

the denominator {y2

r}. The same issues occurs with {yi}. This can lead to a huge over estimation of the interval, as

illustrated by the grey rectangle in ﬁgure 1d. Similar dependency issues are often encountered for more complex

functions of interval input parameters and represent one of the main limiting factors of conventional interval analysis.

For complex division, special algorithms are required to compute the optimal rectangular interval [39, 40, 41]. See

for example the black interval in ﬁgure 1d, which was obtained as per the algorithm presented in [41].

In the present work these algorithms are not well suited as the intervals used to describe a joint’s dynamics will

extend from some minimum stiﬀness/damping value, to inﬁnity (representing a rigid joint). Hence, we propose an

alternative algorithm to determine the rectangular bounds for the special case of complex division considered here

(see Appendix A).

3. Flexible interface dynamics in complex built-up structures

For the sake of clarity, we consider two arbitrary components Aand Bcoupled via Nspring-damper joints Jthat

connect the interface DoFs c(see ﬁgure 2). Though, the developments below are easily generalised to more complex

assemblies. Our intention is to consider the dynamics of this structure when a joint’s characteristics are described

in terms of a complex interval, whose upper bounds extend to inﬁnity (i.e. representing a rigid coupling). To this

end, we ﬁrst present the primal and dual sub-structuring formulations in the presence of ﬂexible interface dynamics.

The reader is referred to the many well established texts for further details on the primal and dual sub-structuring

formulations [1, 42].

For linear time invariant structures it is well known that the dynamics of a structural component can be described

4

0 2 4 6 8 10 12 14 16

2

4

6

8

10

12

a)

Imag( )

−10 −5 0 5 10

0

2

4

6

8

10

b)

−20 0 20 40

0

20

40

60

80

100

c)

Real( )

Imag( )

0 0.511.5 2

0

0.5

1

1.5

2

d)

Real( )

Figure 1: Interval operations applied to the complex intervals x=xr+ixiand y=yr+iyishown in yellow and orange, respectively: a) addition

x⊕y, b) subtraction xy, c) multiplication x⊗y, and d) division xy(grey box - naive implementation, black box - optimum rectangular bound

[41]). Note that the intervals xand yare omitted from b) to more clearly show the over estimation caused by a naive implementation of complex

division.

by the equation of motion (EoM),

Mi¨

u(t)+Ri˙

u(t)+Kiu(t)=f(t)+g(t) (14)

where M,Rand Krepresent the discretised mass, damping and stiﬀness matrices of the ith component, uis a vector

of displacements with ˙

denoting diﬀerentiation wrt. time, fis a vector of externally applied forces, and ga vector of

internal coupling forces [43].

Assuming time harmonic motion, equation 14 may be transformed into a more convenient frequency domain

representation,

−ω2Mi+iωRi+Kiu(ω)=f(ω)+g(ω) (15)

from which we can deﬁne the dynamic stiﬀness matrix,

Zi=−ω2Mi+iωRi+Ki.(16)

Alternatively, the dynamic stiﬀness matrix can be obtained through the inversion of a measured or modelled admit-

5

tance matrix,

Zi=Y−1

i.(17)

The component EoM then becomes,

Ziu(ω)=f(ω)+g(ω).(18)

The purpose of (frequency-based) sub-structuring is to obtain the equation of motion, dynamic stiﬀness, or admit-

tance matrix of a multi-component structure based on those of its individual components. There exist two common

formulations for doing so, primal and dual.

3.1. Primal

The EoM representing the uncoupled AJB system are obtained by block-diagonalising the component EoMs

(explicit frequency dependence has been omitted for clarity),

Zu =f+g(19)

where Zis a block diagonal dynamic stiﬀness matrix, u,f, and gare stacked response, external force and interface

force vectors, respectively. Importantly, the primal formulation considers the joint as an independent component such

that,

Z=

Y−1

A

ZJ

Y−1

B

(20)

where Y−1

Aand Y−1

Bare the inverse admittance matrices of components Aand B, and ZJis the joint dynamic stiﬀness

matrix. For the spring-damper joint considered,

ZJ=

k+ir −k−ir

−k−ir k +ir !1...

k+ir −k−ir

−k−ir k +ir !N

(21)

where kis the joint stiﬀness, and rits damping coeﬃcient. To assemble the AJ B system the conditions of continuity

and equilibrium must be satisﬁed. These are expressed concisely in matrix form, respectively, by,

Bu =0,(22)

and

LTg=0(23)

where Band Lrepresent signed and unsigned Boolean matrices [1]. The primal formulation continues by deﬁning

a new response vector uCthat belongs to the assembled structure. Continuity is then satisﬁed by relating the two

response vectors (coupled and uncoupled) as so,

u=LuC.(24)

By substituting equation (24) into equation (19), and pre-multiplying all terms by LT(hence satisfying equation (23))

we obtain,

LTZLuC=LTf(25)

6

J

A

a

o

δn=γnλn

c

B

b

Figure 2: General active-passive assembly (AJ B) with interface ﬂexibility. The relative interface displacement at the nth connection δn=uAn −uBn

is proportional to the interface ﬂexibility γnand the internal coupling force λn.aand brepresent remote source and receiver DoFs, care the

interface DoFs, and ois the internal region where operational forces develop if Awere an active component.

or equivalently,

uC=YCfC(26)

where fC=LTfis the external force applied to the coupled assembly, and

YC=

LT

Y−1

A

ZJ

Y−1

B

L

−1

(27)

is the coupled admittance of the C=AJB assembly.

Importantly, according to the primal procedure, a joint is represented by its dynamic stiﬀness matrix. Whilst in

this form we are able to accept arbitrarily complex joints, including inertial eﬀects, an ideal rigid coupling would

require the joint stiﬀness to tend to inﬁnity. This complicates the investigation of joint relaxation. Fortunately, a more

convenient (albeit less general) form can be derived following the dual sub-structuring formulation.

3.2. Dual

Unlike the primal formulation, which considers the joint as a separate component, the dual formulation treats

the joint as a relaxation of the continuity condition. In the presence of a ﬂexible interface, for example due to the

spring-damper joint, continuity is not satisﬁed. Instead,

Bu =δ=Γλ(28)

where the interface separation δis related to the unknown joint force λby an interface ﬂexibility matrix Γ. For the

spring-damper joint considered, this ﬂexibility matrix is given by,

Γ=

1

k+ir 1...

1

k+ir N

=

(γr+iγi)1

...

(γr+iγi)N

(29)

where,

γr=k

|Z|2, γi=−r

|Z|2,and Z=k+ir.(30)

7

To satisfy equilibrium, the interface force gis given the form,

g=−BTλ(31)

where λare the unknown joint forces and Bis a signed Boolean matrix. The presence of BTensures that the forces

applied to components Aand Bare equal and opposite, thus satisfying equilibrium.

The unknown joint forces λare found using the relaxed continuity condition as follows. The interface force gis

substituted into equation 19, to give,

u=Yf−BTλ.(32)

where Y=Z−1has been used. Note that Y, as per the dual approach, does not contain a joint admittance matrix

(ZJis non-invertible); instead the joint is accounted for by equation (28). Equation (32) is substituted into the relaxed

continuity condition (equation (28)),

BY f−BTλ=Γλ. (33)

Solving for λthen yields,

λ=BYBT+Γ−1BYf.(34)

Finally, substituting equation (34) into equation (32) leads to,

u=YCf(35)

where,

YC=Y−YBTBYBT+Γ−1BY (36)

is the coupled admittance of the C=AJB assembly. The advantage of this form of YCis that a rigidly connected

system is obtained by taking the limit as Γ→0(i.e. k,r→ ∞), as opposed to ZJ→ ±∞. This simpliﬁes the

derivation of an interface sensitivity metric which can be used to identify response-critical joints.

4. Interval-based assessment of joint dynamics

In this section we will consider an interval-based representation of joint dynamics, and its application in determin-

ing the maximum and minimum bounds of an assembled structure’s admittance matrix due to an interface relaxation.

As shown in section 3.2, the dual formulation, in the presence of a ﬂexible interface, leads to the following equation

for the assembled admittance matrix,

YC=Y−YBTBYBT+Γ−1BY (37)

where the interface ﬂexibility matrix Γis deﬁned as,

Γ=

γ1

γ2

...

γN

, γ =1

k+ir =γr+iγi, γr=k

|Z|2, γi=−r

|Z|2,and Z=k+ir.(38)

With attention focused towards the inﬂuence of a single joint, it is convenient to separate its contribution by means

8

of the Sherman–Morrison formula (rank-one update) [44],

(A+X)−1=A−1−1

1+tr XA−1A−1XA−1,(39)

where Ais any invertible matrix, and Xis a matrix of rank 1. To apply the Sherman–Morrison formula, the interface

ﬂexibility matrix Γis split into two parts; Γ(n), which contains the nth joint ﬂexibility of interest, and Γ(∼n), which

contains all remaining terms,

Γ(∼n)=

γ1

γ2

...

0

,Γ(n)=

0

0

...

γn

,Γ=Γ(∼n)+Γ(n).(40)

Note that Γ(n)is a rank one matrix, so can take on the role of Xin equation (39). Hence, the matrix inverse in equation

(37) can be rewritten in the form,

BYBT+Γ(∼n)+Γ(n)−1=A+Γ(n)−1=A−1−1

1+tr Γ(n)A−1A−1Γ(n)A−1(41)

where A=BYBT+Γ(∼n). Note that Γ(n)is a single entry matrix. Its only non-zero value, the interface ﬂexibility γn,

can be brought out as a scalar term, and Γ(n)replaced by Pn, which is a zero matrix bar the nth diagonal entry whose

value is 1. Equation (41) then reduces to,

BYBT+Γ(∼n)+Γ(n)−1=A−1−wnA−1PnA−1(42)

where,

wn=γn

1+tr PnA−1γn

=1

1

γn+tr PnA−1=1

Zn+tr PnA−1(43)

is a complex scalar value, and Zn=kn+irnis the dynamic stiﬀness of the nth interface connection. Note that wnis a

non-linear function of γn; the joint ﬂexibility, although linear in its dynamics, has a non-linear eﬀect on the assembled

structure.

Herein we propose that the interface stiﬀness Znbe treated as a complex interval {Zn}, such that its bounds are

supposed to contain all possible values of joint stiﬀness and damping, including those of a rigid connection. The

complex interval {Z}={k}+i{r}is characterised by a pair of real intervals, namely {k}=[¯

k¯

k] and {r}=[¯

r¯r], where

¯

and ¯

denote, respectively, lower and upper bounds. In particular, we are interested in the case that ¯

k=¯r=∞,

i.e. a rigid connection. That is, we are interested in the range of possible outcomes given the relaxation of a rigid

connection.

Treating the interface stiﬀness as a complex interval {Zn}, equation (43) now describes a complex interval recipro-

cal,

{wn}=1h{Zn} ⊕ tr PnA−1i=1 {zn}(44)

where {zn}={Zn} ⊕ tr PnA−1={x}+i{y}. In Appendix A we present a straightforward approach to determine the

bounds of the complex interval {wn}, given {x}=[¯

x∞] and {y}=[

¯

y∞].

Having determined the interval bounds of {wn}, substituting equation (44) and 42 into equation (37), leads to the

interval expression,

{YC}=Y−YBTBYBT+Γ(∼n)−1BY

| {z }

Coupled with rigid n

⊕"{wn} ⊗ YBTBYBT+Γ(∼n)−1PnBYBT+Γ(∼n)−1BY

| {z }

Interval matrix describing joint relaxation

#(45)

9

or

{YC}=Y(∼n)

C⊕n∆Y(n)

Co.(46)

where {YC}is a complex interval admittance matrix whose bounds enclose all possible values obtainable given the

range of permissible joint dynamics speciﬁed by the interval {Zn}.

Note that Y(∼n)

Ccontains no intervals, and represent the structure’s admittance when γn=0, i.e. when the nth

contact is made rigid. Hence, n∆Y(n)

Corepresents an interval matrix that describes the eﬀect, or inﬂuence, of the nth

joint’s relaxation on the coupled admittance.

4.1. Response interval

In the presence of an active component (i.e. a vibration source) the operational response uCis often of greater

importance than the assembly admittance. Hence, we look to extend the interval treatment above to account for an

assembly’s operational activity.

The operational response of a complex assembly can be expressed in the form,

uC=YC¯

fS(47)

where YCis the coupled admittance matrix, and ¯

fSis a vector of so called blocked forces [45]. The blocked force is

an independent source quantity that describes its operational activity. It has become an internationally standardised

method for experimentally characterising vibratory sources [46], and is used extensively across several engineering

sectors, including: automotive [47, 48], railway [49, 50], heavy machinery [51] and building acoustics [52].

As per equation (47), the complex interval of an operational response can be obtained by,

{uC}={YC}⊗¯

fS=u(∼n)

C⊕hn∆Y(n)

Co⊗¯

fSi=u(∼n)

C⊕n∆u(n)

Co.(48)

where it is noted that ⊗represents an interval matrix multiplication. Like the assembly admittance, the interval

response is made up of two parts; the ﬁrst is that of the assembly when the nth joint is considered rigid u(∼n)

C, and the

second is an interval contribution describing a relaxation of the nth joint n∆u(n)

Co.

5. Identiﬁcation of response-critical joints

The principal aim of this paper is to provide an eﬃcient identiﬁcation of the response-critical joints within a

complex built-up structure. Here, response-critical joints are deemed as those whose reﬁnement would beneﬁt the

modelling eﬀort most. In what follows we propose three sensitivity metrics for this purpose; a local gradient-based, a

global sampling-based, and a pseudo-global interval-based metric.

Sensitivity analysis (SA) is a tool used to allocate, or apportion, the uncertainty in a system’s output to that of

its input parameters [53]. Methods of SA can be broadly categorised as local or global. Local methods assume a

small variance on the input parameters and consider a linearisation of the system’s dynamics. Such methods are

typically based on partial derivatives (obtained analytically or by numerical approximation), much like a typical error

propagation [54]. Local methods are advantageous in terms of computational eﬃciency, but limited in terms of

accuracy, especially for non-linear systems with reasonable levels of input variance. Global methods overcome this

limitation by sampling the model output over the space of input parameters. Whilst computationally more expensive,

this approach ensures that model non-linearity and input parameter distributions are accounted for. Hence, more a

robust analysis is achieved.

In the context of response-critical joints, a naive identiﬁcation might consider the Jacobian matrix (containing

the partial derivatives of the coupled admittance with respect to each joint parameter) alone, which describes the

local sensitivity of the coupled assembly based on a inﬁnitesimal relaxation of the interface. Clearly, this does not

account for the non-linear contribution of a joint’s dynamics to those of the coupled structure (see equation (43)). An

alternative approach might be to consider a global sampling-based strategy, using metrics such as the Sobol indices

[55], to identify and rank order response-critical joints. Whilst a robust approach, providing suﬃcient samples are

drawn, for a system with many joints the computational eﬀort may become prohibitive. Furthermore, in the absence

10

of detailed statistical information, it is unclear how a joint’s dynamics should be represented. Herein, we propose

an interval-based sensitivity metric to identify the response-critical joints of an assembly. Its computational eﬀort

is not too dissimilar to that of the Jacobian-based approach, whilst its application extends beyond the assumption of

an inﬁnitesimal relaxation; it can be applied in the presence of an arbitrary relaxation. Whilst the interval approach

considers the entire input space (deﬁned by the joint interval {Z}n), it does not provide the detailed information oﬀered

by a global sampling-based approach, hence we term this approach a pseudo-global method.

In what follows we introduce three sensitivity metrics, based on the above strategies (local, global and pseudo-

global), to rank order the inﬂuence of individual joints towards the dynamics of the global assembly.

5.1. Gradient-based sensitivity

To develop a gradient-based sensitivity metric we ﬁrst derive the system’s Jacobian matrix, representing the deriva-

tive of the assembly admittance with respect to a speciﬁc joint’s ﬂexibility. We begin by considering the complex

diﬀerential of the assembly admittance,

dYC=d(Y)−dYBTBYBT+Γ−1BY.(49)

Noting that interest lies in the diﬀerential interface ﬂexibility d(Γ), the above reduces to,

dYC=−dYBTBYBT+Γ−1BY.(50)

Application of the product rule, d (ABC)=d(A)BC +Ad(B)C+ABd(C), then leads to,

dYC=−YBTdBYBT+Γ−1BY.(51)

Recalling the diﬀerential of a matrix inverse, d(A)−1=−A−1d(A)A−1[56], yields

dYC=YBTBYBT+Γ−1dBYBT+ΓBYBT+Γ−1BY (52)

which reduces to,

dYC=YBTBYBT+Γ−1d(Γ)BYBT+Γ−1BY.(53)

In equation (53), we can interpret dYCas the small (complex) change in admittance, due to a small (complex) change

in the interface ﬂexibility d (Γ). Importantly, d (Γ)represents a small change in each matrix entry. Focussing attention

on the nth interface connection in particular, such that d (Γ)→Pndγn, where Pnis a zero matrix bar the nth diagonal

entry, whose value is one, equation (53) becomes,

dYC=YBTBYBT+Γ−1PnBYBT+Γ−1BY dγn=Jndγn(54)

where dγnis a scalar diﬀerential of the nth interface ﬂexibility, and Jnis a complex Jacobian matrix containing the

partial derivatives of the coupled assembly admittance, with respect to the nth interface ﬂexibility,

Jn=

∂Y11

∂γn· · · ∂Y1N

∂γn

.

.

.....

.

.

∂YN1

∂γn· · · ∂YNN

∂γn

.(55)

Note that for a suﬃciency small interface relaxation, the coupled admittance can be estimated according to,

YC=Y(∼n)

C+ ∆Y(n)

C=Y(∼n)

C+Jn∆γn(56)

which represents a linear approximation of the assembly admittance, and is of similar form to equation (45).

11

To evaluate the Jacobian for a rigidly connected structure we set Γ=0. Its entries then describe the sensitivity

of each assembly admittance to an inﬁnitesimal relaxation of the nth interface. Note that equation (55) represents a

matrix of complex derivatives. To provide a convenient single number descriptor for the admittance sensitivity with

respect to the nth joint, it is convenient to consider the spatial average of the Jacobian’s squared magnitude. Hence,

we propose the following gradient-based sensitivity metric,

S(g)

n=D|Jn|2E

N(57)

where hidenotes the spatial average over a speciﬁed set of target DoFs, and Nis chosen such that S(g)

1+S(g)

2+· · · +

S(g)

N=1.

The greater the value of S(g)

n, the more sensitivity the admittance matrix is (on average) to an inﬁnitesimal relax-

ation of the nth interface.

5.2. Sample-based sensitivity

Among the global SA strategies available are a class of methods based on the use of a variance decomposition on

the system’s output [53]. A notable example being the Sobol indices [55].

Assuming a square integrable function Y=f(X), the total variance in the output Ycan be decomposed in the form

[55],

V(Y)=X

i

Vi+X

iX

i>j

Vi j +· · · +Vi,j,··· ,k(58)

where V() denotes the variance operator, and Vi,j,··· ,krepresents a contribution to the output variance due to the input

parameters Xi,j,··· ,k. The decomposed variance terms are given by,

Vi=VXi(EX∼i(Y|Xi)) (59)

Vi,j=VXij (EX∼i j (Y|Xi,Xj)) −Vi−Vj(60)

and so on, where E() denotes the expectation operator. The term EX∼n(Y|Xn)) is read as, the expected value of the

model output Ytaken over all possible values of X∼n, whilst keeping the input Xnﬁxed. The outer variance is then

taken over all possible values of Xn.

According to above decomposition, the variance in the output of a model can be attributed to terms related to each

input parameter (Vi), as well as the interaction eﬀects between them (Vi,j,··· ,k).

Normalising equation (58) by the total variance then yields,

X

i

Si+X

iX

i>j

Si j +· · · +Si,j,··· ,k=1.(61)

The factors Si,j,···,kdescribe sensitivity measures (Sobol indices) of increasing order. Of principal interest in most

cases are the ﬁrst order sensitivity measures, deﬁned as,

Sn=VXn(EX∼n(Y|Xn))

V(Y).(62)

Note that Snis a normalised metric, as VXn(EX∼n(Y|Xn)) varies between zero and V(Y). It provides a measure of the

sensitivity of the model output to the input Xialone; it neglects the inﬂuence of any interaction between Xiand the

remaining inputs X∼i.

12

Another related metric is that of the total eﬀect index, given by,

ST n =1−VX∼n(EXn(Y|X∼n))

V(Y).(63)

Here, the term VX∼n(EXn(Y|X∼n)) describes the ﬁrst order eﬀect due to all input parameters X∼nbar Xi. Hence, V(Y)−

VX∼n(EXn(Y|X∼n)) describes the contribution of all terms in the variance decomposition which include Xi. The total

eﬀect index thus describes the inﬂuence of the input parameter Xi, along with the eﬀect of its interaction with all other

inputs. For details on the numerical estimation of the Sobol indices the reader is referred to [57] (see Table 2 for

necessary formulas).

To develop a global identiﬁcation of response-critical joints we consider a model of the form,

Y=D|YC|2E=f(k1,r1, , k2,r2,· · · ,kN,rN)=f(X) (64)

and compute the total eﬀect index for each joint. To enable fair comparison against gradient and interval-based

metrics, each index is normalised,

S(s)

n=ST n

N(65)

such that S(s)

1+S(s)

2+· · · +S(s)

N=1, where the superscript (s)is used to indicate that this is a sample-based sensitivity

metric. These are then rank ordered to identify the response-critical joints.

The greater the value of S(s)

n, the greater the amount of output variance that can be attributed to the relaxation of

the nth interface.

To aid a fair comparison when using the sample-based metric S(s)

n, the range of permissible stiﬀness and damping

values (kn,rn) should be the same for each joint. A principal advantage of the global method is that the interaction

between several joints can be accounted for; the gradient-based metric presented above, and the interval-based metric

presented below, consider only a single joint at a time. This advantage is, however, oﬀset by the computational eﬀort

required to compute numerically the metrics.

5.3. Interval-based sensitivity

The local and global methods described above are limited, respectively, by the assumption of linearity and compu-

tational eﬃciency. Herein we propose an interval-based identiﬁcation that overcomes both these limitations. Whilst

the interval-based approach does not provide a detailed analysis like the global approach, it avoids the need to sample

the entire input space to identify the response-critical joints; instead of treating each input as a distribution, intervals

are considered and the bounds of the output are determined analytically. These bounds will form the basis of the

proposed interval sensitivity metric S(i)

n.

The general concept of interval-based sensitivity analysis was ﬁrst introduced in [58], where the authors considered

the change in absolute interval widths on the input (x) and the output (y) side of the problem. Their proposed sensitivity

metric takes the form,

S=∂(¯y−

¯

y)

∂( ¯x−¯

x).(66)

The authors present a numerical example whereby the frequency response function of a lumped parameter truck

model is subject to an interval sensitivity analysis with respect to its lumped mass values. In the present paper we are

interested in the sensitivity of a scalar interval output quantity (the spatially averaged magnitude squared admittance)

to a complex pair of input intervals (joint stiﬀness and damping) whose maxima extend to ∞. For this speciﬁc case

we propose an alternative metric, described below.

Recalling the interval matrix of equation (45), with consideration of equation (54), we have,

{∆Y(n)

C}={wn} ⊗ YBTBYBT+Γ(−n)−1PnBYBT+Γ(−n)−1BY ={wn} ⊗ Jn.(67)

13

Note that unlike Jn∆γn, which describes a linearised joint contribution, the interval {wn} ⊗ Jnrepresents the upper and

lower bounds for an arbitrary relaxation of the interface. Hence, it takes into account the non-linearities that arise due

to the matrix inversion (see equation (43)). Considering the upper and lower bounds of the complex interval {∆Y(n)

C},

we deﬁne the interval-based sensitivity metric,

S(i)

n=Dreal ∆¯

Y(n)

C−real ∆¯

Y(n)

CE2+Dimag ∆¯

Y(n)

C−imag ∆¯

Y(n)

CE2

N(68)

where, as before, Nis chosen such that S(i)

1+S(i)

2+· · ·+S(i)

N=1. The numerator of equation (68) may be interpreted as

an (average) measure of the size of the complex interval {∆Y(n)

C}; recalling that a complex interval describes a rectangle

in the complex plane, S(i)

nrepresents the normalised squared distance across the diagonal of said rectangle.1Hence,

the greater the value of S(i)

n, the greater the size of the output interval.

Unlike the metric proposed in [58], we consider only the interval bound of the output (as the input bound extends

to inﬁnity). To aid a fair comparison when using the interval-based metric S(i)

n, each {∆Y(n)

C}should be determined

subject to the same joint interval bounds {Zn}.

For a suﬃciently large lower stiﬀness bound ¯

Zn, we would expect S(i)

nto follow the same trend as S(g)

n, as {zn}=

{Zn} ⊕ tr PnA−1would be dominated by the joint stiﬀness {Zn}, and so {w} ≈ 1 {Zn}={γn}would appear simply

as a scaling factor. Treating the interface stiﬀness as a non-interval parameter, such that {γn} → ∆γn, this would be

equivalent to the linearised estimation of the coupled assembly admittance (equation (56)).

6. Numerical example

In this section we present two numerical examples demonstrating the interval assessment of joint variability and

the identiﬁcation of response-critical joints, the ﬁrst a simple mass-spring system and the latter a more representative

FE frame assembly.

6.1. A simple mass-spring system

The system considered in this initial example is illustrated in ﬁgure 3. It is an 11 DoF mass-spring-damper system.

It constitutes two sub-systems (green and blue) connected by 3 joint elements (red). System parameters are given in

table 1, and were chosen arbitrarily to obtain suitably spaced resonances between 1 to 200 Hz. Each joint element is

characterised by its ﬂexibility γn, or equivalently its dynamic stiﬀness Zn. We are interested in a) obtaining the interval

bounds of the coupled admittance matrix YC, and b) identifying the response-critical joints, i.e. determining, at each

frequency, which joint contributes most to the dynamics of the assembled system.

6.1.1. Interval-based joint variability

We begin by considering the interval bounds of the coupled admittance based on the interval description of a

particular joint; joints 2 and 3 are taken to be rigid (γ2=γ3=0) with joint 1 described by the complex interval {Z1}=

{k}+i{r}=[¯

k∞]+i[¯

r∞]. Note that the upper bounds of the joint interval represent a rigid connection. Hence,

the interval admittance matrix should describe the entire range of possible admittance values given any combination

of joint stiﬀness and damping that is made permissible by the interval {Z1}.

1An alternative metric could be formulated by instead considering the (average) area of the complex interval,

S(i)

n=Dreal ∆¯

Y(n)

C−real ∆¯

Y(n)

CEDimag ∆¯

Y(n)

C−imag ∆¯

Y(n)

CE/N

.

14

m1

m2

m3

m4

m5

k1

k2

k3

k4

k5

c1

c2

c3

c4

c5

m11

m9

m10

m8

m7

m6

k11

k9

k10

k7

k8

k6

c11

c10

c9

c6

c7

c8

γ3

γ2

γ1

Figure 3: Numerical mass-spring example. System parameters are given in table 1 along with the natural frequencies for the rigidly coupled case.

Table 1: System parameters used in numerical mass-spring example, and natural frequencies obtained for rigid coupling.

Index 1 2 3 4 5 6 7 8 9 10 11

m(kg) 1 0.5 1.2 0.65 0.9 0.56 1.2 0.52 1.1 0.25 1.5

k×104(N/m) 2 3 1 4 0.5 1 2 2 3 1 0.8

c(Ns/m) 1 1 1 1 1 1 1 1 1 1 1

f(Hz) 7.3 14.9 18.6 21.8 34.3 38.2 68.3 73.5 - - -

The complex admittance interval {YC}is determined as per the equation (45),

{YC}=Y−YBTBYBT+Γ(∼n)−1BY ⊕"{wn} ⊗ YBTBYBT+Γ(∼n)−1PnBYBT+Γ(∼n)−1BY#

where

{wn}=1h{Zn} ⊕ tr PnA−1i=1 {zn}(69)

is a complex interval to be determined. Using the approach outlined in Appendix A (see Algorithm 1), alongside the

interval arithmetic described in section 2, the complex reciprocal interval {wn}can be determined straightforwardly.

Shown in ﬁgure 4 are the upper and lower bounds of the real (b) and imaginary (c) parts of {wn}according to

the joint interval {Z1}=[100 ∞]+i[100 ∞]. The dashed blue curves indicate the interval computation, whilst

the orange curves show the max/min values obtained using a sampling-based estimation, where 40000 combinations

of kand rare considered. Shown in ﬁgure 4a are the lower bounds of the complex interval {zn}used to compute

{w}(note that the upper bounds are ∞, representing a rigid connection). It is clear from ﬁgure 4bc that a particular

combination of joint stiﬀness and damping will yield a wwith a particular set of peaks, and by uniform sampling we

obtain a series of such. Gaps between successive peaks are a result of the discrete sampling employed and represent

a limitation of such a sample-based estimation. That is not to say that the sample based estimate is inaccurate, rather

it requires a greater computational eﬀort to achieve a result similar to that of the interval computation, which, as

expected, encloses the max/min values of all samples drawn. The results of ﬁgure 4 demonstrate the application and

validity of the conformal mapping approach discussed in Appendix A.

Implementation of equation (45), taking into account the appropriate interval arithmetic, then yields a complex

interval for the coupled admittance. Shown in ﬁgure 5a and b are, respectively, the real and imaginary interval bounds

of the transfer admittance Y1,11. Dashed blue curves were obtained by the interval method described above, whilst

orange curves were obtained by sampling. Shown in ﬁgure 6 is the magnitude transfer admittance |Y1,11|of the rigidly

coupled assembly (black) alongside the maximum interval bound (blue) obtained from the real and imaginary intervals

presented in ﬁgure 5. Also shown in orange is the result obtained by sampling. The interval computation can be seen

15

to encapsulate entirely the sample estimate. This maximum bound represents the maximum attainable admittance

based on the relaxation of interface 1 in ﬁgure 6. It does not, however, provide any indication as to the most likely

outcome, as per the possibilistic paradigm.

100101102

−6

−4

−2

0

2

·105

a)

z

¯x=∞¯y=∞

¯

x¯

y

100101102

−4

−2

0

2

4

·10−3

b)

Real (w= 1/z) = u

100101102

−1

−0.5

0

·10−2

c)

Frequency (Hz)

Imag (w= 1/z) = v

Sample-based estimate

Interval computation

Figure 4: Interval computation for numerical example. a) Lower bounds of the complex interval {zn}={xn}+i{yn}. b/c) Upper and lower bounds

of the real/imaginary parts of the complex interval {wn}: blue dashed lines are interval computations, orange lines are sample-based estimates.

Note that the lower bound in ﬁgure 6 has been omitted. The reason for doing so is that in computing the interval

of the magnitude admittance, the real and imaginary parts must each be squared. If both parts have a lower bound less

than zero and an upper bound greater than zero (i.e. zero is enclosed within the bound), their squared intervals must

have a lower bound of zero. Hence, their sum also has a lower bound of 0, so it can not be presented on a logarithmic

scale. Nevertheless, ﬁgure 6 demonstrates that the proposed interval assessment of joint variability is able to provide

an upper limit on the possible admittance. This is, of course, for a single joint. The interval assessment of multiple

joints simultaneously would require a non-trivial extension of the method proposed herein.

Shown in ﬁgure 7 are the upper interval bounds obtained by relaxing each interface individually according to the

interval stiﬀness, {Zn}=[100 ∞]+i[100 ∞]. The upper bounds provide a visual indication of which interface joint

has the greatest possible inﬂuence on the coupled transfer admittance |Y1,11|. As expected, the relative importance of

each joint varies with frequency. A more intuitive presentation of this result is obtained by considering the sensitivity

metrics proposed in section 5. These will be shown in following sub-section.

16

100101102

−6

−4

−2

0

2

·10−3

a)

Real(YC)

Sample-based estimate Interval computation

100101102

−2

0

·10−3

b)

Frequency (Hz)

Imag(YC)

Figure 5: Real (a) and imaginary (b) interval bounds of the admittance Y1,11 due to relaxation of joint interface: blue lines are interval computations,

orange lines are sample-based estimates.

100101102

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency (Hz)

|YC|

Rigid coupling Sample-based estimate Interval computation

Figure 6: Upper bound of the magnitude admittance |Y1,11 |due to relaxation of the joint interface: blue lines are interval computations, orange

lines are sample-based estimates. Lower bounds are omitted for clarity.

Shown in ﬁgure 8 are the upper and lower bounds of the magnitude transfer admittance |Y1,11|obtained by suc-

cessively increasing the minimum bound of the joint stiﬀness and damping, such that {Zn}=[¯

Z∞]+i[¯

Z∞],

where ¯

Z=[100,1000,10000,100000]. As mentioned previously, if the complex interval {YC}contains the origin,

17

then the lower bound of its squared magnitude will always be zero. This occurs frequently for large joint intervals.

As the origin is always contained within the interval {wn}(see ﬁgure A.14), the bounds of n∆Y(n)

Coalso contains the

origin (since complex multiplication can be interpreted as a rotation followed by a scaling). The subsequent addition,

Y(∼n)

C⊕n∆Y(n)

Co, translates the bound. To avoid enclosing the origin, this translation must occur in the appropriate

direction. Had the upper bounds of {Dn}been considered ﬁnite, the origin would not be enclosed by {wn}, and a more

useful lower bound obtained.

100101102

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency (Hz)

|YC|

Rigid coupling {D1} {D2} {D3}

Figure 7: Comparison of the upper bound on |Y1,11 |due the relaxation of each interface joint such that {Zn}=[100 ∞]+i[100 ∞]. Rigid

coupling is shown in black. Lower bounds are omitted for clarity.

100101102

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency (Hz)

|YC|

Rigid coupling 100 1000 10000 100000

Figure 8: Comparison of the upper bound on |Y1,11 |due an increasing relaxation of the interface joint, {Zn}=[¯

Z∞]+i[¯

Z∞] where ¯

Z=

[100,1000,10000,100000].

18

From ﬁgure 8 it can be observed that the interval bounds are not simply scaled for increasing relaxation; new

characteristics emerge as the joint is made more ﬂexible. These ﬂexibility dependent characteristics are neglected

when considering a gradient-based estimation of the coupled admittance, as per equation (56).

6.1.2. Response critical joint identiﬁcation

To identify the response-critical joints of an assembly it is necessary to compare their inﬂuence on the dynamics

of the assembled structure. In section 5 we proposed three sensitivity metrics to do so. In this section we apply those

metrics to the numerical example of ﬁgure 3. We consider two cases: ¯

Zn=1×108(1 +i) and ¯

Zn=100(1 +i),

representing a minor and major relaxation of the interface, respectively.

The interval and gradient-based metrics, S(i)

nand S(g)

n, are computed analytically according to section 5.1 and 5.3

(making sure to adopt interval arithmetic for the former), taking mass 11 to be the target DoF. The sample-based

metric S(s)

nis computed numerically by sampling over the input parameter space. For the example considered, the

input space consists of 6 parameters, the stiﬀness and damping of each joint. At each frequency, 2000 samples were

drawn between ¯

Znand ¯

Zn×107(representing an approx. rigid connection). Samples were distributed such that each

parameter’s logarithm was uniformly distributed between the speciﬁed bounds.

0

0.2

0.4

0.6

0.8

1

S(i)

0

0.2

0.4

0.6

0.8

1

S(g)

100101102

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

S(s)

a)

0

0.2

0.4

0.6

0.8

1

S(i)

0

0.2

0.4

0.6

0.8

1

S(g)

100101102

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

S(s)

b)

Figure 9: Comparison of the interval (top), gradient (middle), and sample-based (bottom) sensitivity metrics for the minimum stiﬀness bounds

¯

Zn=1×108(1 +i) (a) and ¯

Zn=100(1 +i) (b): S()

1- blue, S()

2- yellow, S()

3- purple.

Shown in ﬁgure 9a and 9b are the sensitivity metrics obtained for the two cases considered (top - interval-based,

middle - gradient-based, bottom - sample-based). The results of ﬁgure 9a represent the minor relaxation of the

interface, and ﬁgure 9b the major. Considering ﬁrst ﬁgure 9a, it is clear that we get signiﬁcant agreement between

all three metrics (to be expected for a minor relaxation). We observe that at low frequencies, joints 1 (blue) and 2

19

(yellow) dominate the response, whilst at high frequencies, joint 3 (purple) becomes the key contributor. Across the

middle range joint 2 tends to dominate, with joints 1 and 3 contributing across narrow frequency ranges.

Let us now consider ﬁgure 9b, representing the major interface relaxation. Note that the gradient-based metric is

unchanged, as it does not depend on the level of relaxation. Whilst the interval and gradient-based metrics share some

similar trends, it is clear that across the middle frequency range some discrepancies have emerged. These diﬀerences

are due to the non-linear contribution of each joint’s ﬂexibility (as observed in ﬁgure 8). Importantly, this suggests

that the possible inﬂuence, or importance, of a joint depends on the range of permissible stiﬀness and damping values

that can be taken.

The sampling-based metric S(s)

nis harder to interpret. The issue is that given the range of stiﬀness/damping values

permissible, a large range of assembly responses are observed. This can cause numerical issues when estimating the

Sobol indices. Furthermore, as suggested by ﬁgure 6, at high frequencies a uniform sampling scheme leads to a series

of peaks in the response bound. These peaks cause rapid ﬂuctuations in the obtained Sobol indices, complicating their

interpretation. This issue could be alleviated by increasing the the number of samples drawn to more suitably cover

the entire input space. This would of course come at the cost of additional computational eﬀort. This is the principal

limitation of the sample-based approach. It is also worth noting that the interval-based metric considers the entire

range of possible assembly responses, no matter how unlikely the extreme values are. The sample-based metric,

on the other hand, considers the spread, or variance, of outcomes, so will be less inﬂuenced by unlikely extremum

values. For example, the sample-based metric does not indicate an increase in sensitivity due to joint 1 around 80-90

Hz, whilst the interval-based metric does. This is likely due to the small range of input stiﬀness/damping values

that cause this increased sensitivity. These particular values are likely missed by the sample-based approach, or their

inﬂuence is outweighed by regions of the output space that are more densely populated. Nevertheless, noticeable

diﬀerences are observed between ﬁgure 9a and b, inline with the results of the interval-based metric.

The results of ﬁgure 9b are supported by the upper interval bounds shown in ﬁgure 7, where it can be seen that, in

general, the greater the upper bound of the magnitude transfer admittance, the greater the sensitivity metric.

Whilst it may be argued that the sampling-based metric S(s)

nwill likely provide a more robust analysis, being less

sensitive to unlikely outcomes, this comes at the cost of considerable computational eﬀort, especially for large levels

of relaxation. For the mass-spring example considered (running on a standard desktop machine with an Intel(R)

Core(TM) i7-8700 CPU 3.20GHz and 16GB RAM, over 2000 frequency points), the metric computation times were

as follows: gradient-based - 0.155 seconds; interval-based - 0.424 seconds; sampling-based - 600.9 seconds. Note

that the sampling-based metric involves the repeated sampling of 2000 log-uniformly distributed values assigned to

each of the 6 joint parameters (i.e. stiﬀness and damping of each joint), and that is done at every frequency point,

hence its rather large computation time. Furthermore, without detailed statistical information of the joint, such a

method is somewhat unwarranted. Computation of the interval-based metric, on the other hand, is of the order of the

gradient-based metric, and does not require any detailed statistical information.

6.2. A more practical, albeit numerical, example

In this section we will demonstrate the identiﬁcation of response-critical joints on a more representative numerical

assembly. The chosen assembly consists of two frame-like components coupled together at four locations, as illus-

trated in ﬁgure 10. At each connection, coupling is enforced through 4 translational links in the vertical zdirection; for

simplicity in-plane coupling is neglected. Through an appropriate interface transformation, each 4 link connection can

be characterised by a single translational coupling, with two accompanying rotations, yielding a total of 12 interface

DoFs. A further remote DoF is included on each component.

The question to be answered is as follows: given limited experimental and/or numerical resources, which interface

connection (c1,c2,c3, or c4; starting from the connection closest to the origin and moving clockwise) should be treated

as a ﬂexible joint, characterised, and implemented numerically to aid model reﬁnement? In what follows we will use

the gradient and interval-based sensitivity metrics proposed through section 5 to identify the response-critical joints

and answer this question.

The assembly is modelled using the FE method, implemented in MATLAB using the PDE Toolbox [59]. Both

frames are speciﬁed as steel with structural loss factor η=0.025. Shown in ﬁgure 11 are some example mode

shapes of the rigidly coupled assembly. After solving the eigen-problem for each frame, the free interface admittance

matrices YAand YBare determined by modal summation, and a ﬁnite diﬀerence transformation is used to obtain the

20

sought after translational/rotational admittances at each connection [60]. These form the inputs to the gradient and

interval-based sensitivity analyses.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0.04

x (m)

y (m)

z (m)

Figure 10: Diagram of numerical example. Two frame structures coupled at 4 locations (c1,c2,c3, and c4, located clockwise from the origin),

each with 4 translational zDoFs.

Figure 11: Some example mode shapes of coupled structure.

As before, we consider two cases with diﬀerent levels of interface relaxation. In the ﬁrst each joint is described

in turn by the interval {Zn}=[1 ×108∞]+i[1 ×108∞], representing a minor relaxation of the interface. In the

second, we set {Zn}=[1×105∞]+i[1×105∞], representing a moderate relaxation of the interface. For each case

we employ the gradient and interval-based sensitivity metrics described in section 5.1 and 5.3, respectively. Owing it

its computational burden and poor performance, the sampling-based Sobol metric is not considered here. Results are

presented in ﬁgure 12 with left and right-hand side plots representing the minor and moderate relaxation, respectively.

Middle and bottom plots show, respectively, the interval and gradient-based sensitivity metrics for each connection.

Note that the metrics presented describe the contribution of the entire connection, including both translational and

rotational DoFs. To obtain these results the sensitivity metrics for each DoFs are summed together. For example, the

combined metric for connection 1 is given by,

S()

1=S()

11 +S()

12 +S()

13 (70)

where S()

11 ,S()

12 , and S()

13 are the sensitivity metrics for the translational and x/yrotational DoFs at connection 1.

Using this approach one could similarly compare the importance of diﬀerent DoFs (e.g. translational vs. rotational),

as opposed to connection points.

Shown in the top plots of ﬁgure 12 are the remote DoF transfer admittances for the rigidly coupled assembly

(black), and those obtained by independently relaxing (the DoFs at) each connection by an amount equal to the

21

minimal stiﬀness bound ¯

Zn. These plots provide an indication of the sensitivity of the transfer admittance given the

relaxation of each interface connection. They do not, however, provide an insight into the range of possible outcomes.

Nevertheless, we can use them to help interpret the sensitivity metrics displayed beneath.

10−8

10−7

10−6

10−5

|Y|

Rigid c1c2c3c4

0

0.2

0.4

0.6

0.8

1

S(i)

100 200 300 400 500

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

S(g)

a)

10−8

10−7

10−6

10−5

|Y|

0

0.2

0.4

0.6

0.8

1

S(i)

100 200 300 400 500

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

S(g)

b)

Figure 12: Comparison of the interval (middle) and gradient (bottom) based sensitivity metrics for minimum stiﬀness bounds ¯

Zn=1×108(1+i) (a)

and ¯

Zn=1×105(1+i) (b). Each metric describes the sensitivity of the transfer admittance with respect to a speciﬁc connection point, including both

its translational and rotational DoFs: S()

1- blue, S()

2- yellow, S()

3- purple, S()

4- green. Shown in the top ﬁgures are the transfer admittances

obtained by independently relaxing the DoF at each connection point by an amount equal to the minimal stiﬀness bound. In black is the response

obtained from the rigid assembly.

As before, for a minor relaxation the gradient and interval-based metrics display the same general trends. For the

moderate relaxation clear diﬀerences can be seen. This again illustrates the importance of considering the entire range

of possible stiﬀness and damping values when determining the inﬂuence of speciﬁc joint. For a minor relaxation,

in the low-mid frequency range (100-350 Hz), both metrics indicate that the connection 3 has the greatest inﬂuence,

with connection 4 contributing signiﬁcantly over narrower frequency ranges. At higher frequencies (>350 Hz) the

inﬂuence of connection 2 increases markedly. For the moderate relaxation, according to the interval-based metric,

the inﬂuence of connections 3 and 4 reduce considerably compared to ﬁgure 12a, particularity around 200 Hz, with

connection 2 becoming dominant. It is further observed that in the mid-high frequency range (>200 Hz) the inﬂuence

of connection 1 increases, and at some frequencies becomes dominant.

Comparing the interval-based metrics for the moderate relaxation (middle right plot) with their respective trans-

fer admittances (top right plot) it can be seen that regions of increased sensitivity generally correspond to greater

deviations from the rigid admittance (in black). Though, this is not always the case. It is important to recall that

the interval-based metric considers the entire range of all possible transfer admittances given the permissible joint

22

dynamics. The transfer admittances plotted are those obtained for Zn=1×108+i1×108alone, which is just one

realisation.

The information provided by the sensitivity metrics could be used to guide model reﬁnement. As an example,

from ﬁgure 12 it is clear that connection 1 has the least inﬂuence overall, hence eﬀorts should not be focused towards

reﬁnement of this particular connection. In the low frequency range connection 3 has the greatest inﬂuence and should

be prioritised. If attention is focused on the mid frequency range connection 4 might be prioritised instead.

7. Conclusions

Whilst there exists a large body of literature related to the characterisation and modelling of joint dynamics, it

appears few works have considered which joints to characterise and/or model; for a complex built-up structure the

number of joints present may prohibit the complete characterisation/modelling of all joints. With the development or

reﬁnement of a component-based model there is a need to identify the response-critical joints of an assembly - those

that have the greatest inﬂuence on the target response. Once identiﬁed, experimental and computational resources

may be focused towards those joints which will beneﬁt the modelling eﬀort most. In the present paper a possibilistic

interval-based sensitivity metric is proposed to rank order the inﬂuence of individual joints towards the dynamics of

an assembled structure.

The interval-based sensitivity metric was formulated by exploiting the Sherman-Morrison formula and the dual

sub-structuring formulation. A complex interval representation was adopted for the interface stiﬀness/damping, and

the complex interval bounds of the assembled admittance matrix (also operational response) were obtained. Spatial

averaging of the admittance interval bounds over selected DoFs yield the proposed metric. Scaled between 0 and 1 the

metric represents a normalised measure of the size of the complex output interval, and enables a clear identiﬁcation

of the assembly’s response-critical joints.

The advantages of an interval-based sensitivity metric, over a more conventional probabilistic one, are two fold: no

detailed statistical information is required to describe the joint parameters (only a minimum bound for the stiﬀness and

damping), and computationally expensive sampling methods are avoided. Furthermore, the interval-based approach,

whilst providing only the extremum outcomes, considers the entire input space and so implicitly takes into account

model non-linearity (hence we term the method ‘pseudo-global’). This is in contrast to typical gradient-based methods

which assume linearity and/or small input variance.

As part of a numerical example, the proposed metric was compared against local gradient-based and global

sampling-based (Sobol) metrics, also proposed herein. For low levels of interface relaxation, all metrics are in strong

agreement. For larger levels of relaxation, the inﬂuence of model non-linearity is observed and the interval/sampling-

based metrics diﬀer from the gradient-based; the relative importance of a joint depends on the range of permissible

stiﬀness and damping values that can be taken. Owing to numerical and ﬁnite sampling issues, the global met-

ric performs poorly at high frequencies where there is large variance in the model input/output. For this reason

the sample-based metric is considered the least beneﬁcial. In contrast, both gradient and interval-based metrics are

computed analytically, and so avoid such issues. Nevertheless, whilst the gradient-based metric avoids costly sam-

pling eﬀorts, it does not take into account the range dependent inﬂuence of a joint’s dynamics. For this reason, its

use should be limited to cases where joints are expected to be very nearly rigid. To account for a greater degree of

interface relaxation the interval-based metric should be used.

Overcoming the respective limitations of both the gradient and sampling-based strategies (linearity and computa-

tional eﬃciency), the interval-based metric provides a convenient and computationally eﬃcient identiﬁcation of the

response-critical joints in a complex built-up structure. The information gained though its application may be used to

focus model reﬁnement towards the connections or joints that will beneﬁt the modelling eﬀort most.

Appendix A. Conformal mapping

Equation (45) (48) yields the complex interval bounds of an assembly’s admittance matrix (response vector), based

on an interval relaxation of the nth joint. Its implementation requires ﬁrst establishing the interval bounds of {wn},

deﬁned as per equation (44). In what follows we consider equation (44) as a conformal map, and propose a simple

algorithm to establish said bounds.

23

¯

x

¯

y

z

a)

x

y

1

2¯

x

−1

2

¯

y

1/z

b)

u

v

Figure A.13: Conformal map f(z)=1/zapplied to the open regions x>¯

x(blue) and y>

¯

y(green).

The denominator of equation (44) represents a complex interval of the form {z}={x}+i{y}, where the real and

imaginary intervals are taken to be, {x}=[¯

x∞] and {y}=[

¯

y∞]. Taken together, these intervals describe an open

rectangular domain in the complex plane that extends from the point (¯

x,

¯

y) to inﬁnity, as shown in ﬁgure A.13a. This

domain contains all possible values of the complex number zgiven the real and imaginary intervals {x}and {y}. We are

interested in the mapping of this domain onto that of the complex variable w=1/z. That is, we are interested in the

conformal map f(z)=1/zapplied to the open rectangular domain {z}. In what follows we seek the smallest complex

rectangular interval {w}={u}+i{v}, where {u}=[¯

u¯u] and {v}=[¯

v¯v], that encloses the domain of f(z)=1/z.

We start by considering the complex relation,

w=u+iv =1

x+iy .(A.1)

Taking the reciprocal of the above,

x+iy =1

u+iv =u−iv

u2+v2(A.2)

we obtain expressions for the real and imaginary parts of zwith respect to uand v,

x=u

u2+v2>¯

x,y=−v

u2+v2>

¯

y(A.3)

where ¯

xand

¯

yrepresent the lower bounds of the intervals {x}and {y}. Assuming ¯

x>0, rearranging the left-hand

inequality above leads to,

0>u2+v2−u

¯

x= u−1

2¯

x!2

+(v−0)2− 1

2¯

x!2

,(A.4)

or equivalently,

1

2¯

x!2

> u−1

2¯

x!2

+(v−0)2.(A.5)

Recalling that a circle of radius rwith centre point (h,k), is given by,

r2=(u−h)2+(v−k)2.(A.6)

the conformal map f(z)=1/ztakes the region x>¯

xof the complex plane and maps it to a closed circular domain

24

of radius r=1/2¯

xcentred at the coordinates (1/2¯

x,0). See for example the blue domain in ﬁgure A.13. Note that if

¯

x<0, the above inequality ﬂips, such that,

1

2¯

x!2

< u−1

2¯

x!2

+(v−0)2.(A.7)

In this case, the conformal map f(z)=1/ztakes the region x>¯

xand maps it to the open domain surrounding a

circle of radius r=1/2|¯

x|centred at the coordinates (1/2¯

x,0) (with ¯

x<0 this circle is located in the negative real

half-plane). See for example the blue domain in ﬁgure A.14c.

Let us now consider the inequality for y. Following similar steps as above, assuming

¯

y>0, we arrive at the

inequality,

0>u2+v2+v

¯

y=(u−0)2+

v−−1

2

¯

y

2

−

1

2

¯

y

2

(A.8)

or equivalently,

1

2

¯

y

2

>(u−0)2+

v−−1

2

¯

y

2

(A.9)

which equally describes a closed circular domain in complex plane with radius r=1/2

¯

ycentred at the coordinates

(0,−1/2

¯

y). See for example the green domain in ﬁgure A.13. If

¯

y<0 the above inequality ﬂips, such that,

1

2

¯

y

2

<(u−0)2+

v−−1

2

¯

y

2

.(A.10)

In this case, the conformal map f(z)=1/ztakes the region y>

¯

yand maps it to the open domain surrounding a circle

of radius r=1/2|

¯

y|centred at the coordinates (0,−1/2

¯

y) (with

¯

y<0 this is in the positive imaginary half-plane). See

for example the green domain in ﬁgure A.14d.

The complex map f(z)=1/zis shown in ﬁgure A.13 for the two regions ¯

x>0 and

¯

y>0. The intersection of

x>¯

xand y>

¯

yin the x,yplane describes the domain of possible complex values z=x+iy that satisfy both x>¯

xand

y>

¯

y. This domain maps to a vesica piscis (i.e. the intersection of two circles) in the u,vplane, as shown in ﬁgures

A.13b and A.14ab, which describes the range of possible values the complex variable w=1/(x+iy) can take, given

the restrictions on xand y. For the case that ¯

x<0or

¯

y<0 the domain maps to a more complex region that is the

intersection of an open and closed circular region, as shown in ﬁgures A.14cd.

To adopted an interval-based approach we seek to describe the resulting intersection by means of the smallest

enclosing rectangle. To determine the interval bounds of 1 {z}we consider the four cases presented in ﬁgure A.14:

a) If 0 <¯

x<

¯

ythen ¯v=¯

u=0, ¯u=1/2

¯

y, and ¯

v<0 corresponds to the point of intersection between the two

domain’s circular boundaries.

b) If ¯

x>

¯

y>0 then ¯v=¯

u=0, ¯

v=−1/2¯

x, and ¯u>0 corresponds to the point of intersection between the two

domain’s circular boundaries.

c) If ¯

x<0<

¯

ythen ¯v=0, ¯

v=−1/

¯

y, ¯u=1/2

¯

y, and ¯

u<0 corresponds to the point of intersection between the two

domain’s circular boundaries if 1/2|

¯

y|<1/2|¯

x|, or −1/2

¯

yotherwise.

d) If

¯

y<0<¯

xthen ¯

u=0, ¯u=1/¯

x,¯

v=−1/2¯

x, and ¯v>0 corresponds to the point of intersection between the two

domain’s circular boundaries if 1/2|¯

x|<1/2|

¯

y|, or 1/2¯

xotherwise.

In the case that ¯

x,

¯

y<0, the domain of zencloses the origin and so the domain of wextends to ±∞.

25

¯u

¯

v

¯v=¯

u

a)

u

v

¯u

¯

v

¯v=¯

u

b)

u

v

¯

u

d)

¯u

¯

v

¯v

u

v

¯v

c)

¯

u¯u

¯

v

u

v

Figure A.14: Optimum complex interval enclosing w=1/(x+iy) for x>¯

x(blue) and y>

¯

y(green). Four scenarios are considered: a) 0 <¯

x<

¯

y,

b) 0 <

¯

y<¯

x, c) ¯

x<0<

¯

y, and d)

¯

y<0<¯

x.

An implementation of the above interval computation is summarised in Algorithm 1. Having established the

bounds of {wn}, as per the outlined procedure, equation (45) can be implemented by adopting the interval arithmetic

operations detailed in section 2, and maximum/minimum bounds of the assembly admittance determined.

26

Algorithm 1: Determine bounds of the complex interval {w}=[¯

u¯u]+i[¯

v¯v]=[¯

x∞]+i[

¯

y∞]−1

r1=1/(2|¯

x|);

r2=1/(2|

¯

y|);

if ¯

x>0∧

¯

y>0then

if r2≤r1then

// this is the case of figure A.14a

¯

u=0;

¯u=r1;

¯

v=−2r2

1/r21+(r1/r2)2;

¯v=0;

else

// this is the case of figure A.14b

¯

u=0;

¯u=2r1/1+(r1/r2)2;

¯

v=−r2;

¯v=0;

end

else if ¯

x<0∧

¯

y>0then

// this is the case of figure A.14c

if r2≤r1then

¯

u=2r1/1+(r1/r2)2

else

¯

u=−r2

end

¯u=r2;

¯

v=−2r2;

¯v=0;

else if

¯

y<0∧¯

x>0then

// this is the case of figure A.14d

¯

u=0;

¯u=2r1;

¯

v=−r1;

if r2≤r1then

¯v=−2r1/1+(r1/r2)2

else

¯v=r1

end

else

¯

u=−∞;

¯u=∞;

¯

v=−∞;

¯v=∞;

end

27

Appendix B. Intersection point

To determine the intersection point between the boundaries of two circular domains (i.e. the red markers in ﬁgure

A.14) we should solve the following pair of simultaneous equations,

(u−u1)2+(v−v1)2=r2

1,(u−u2)2+(v−v2)2=r2

2(B.1)

each representing a circle in the u,vplane. We take subscript 1 to denote the mapped x>¯

xdomain (blue), and

subscript 2 the mapped y>

¯

ydomain (green).

Subtracting equations B.1 from one another yields,

−2u(u1−u2)−2v(v1−v2)=r2

1−r2

2−u2

1−u2

2−v2

1−v2

2(B.2)

Isolating v, whilst noting that for all cases u2=0 and v1=0, yields,

v=−1

2(v1−v2)−1r2

1−r2

2−u2

1−u2

2−v2

1−v2

2+2u(u1−u2)=1

2v2r2

1−r2

2−u2

1+v2

2+2uu1.(B.3)

Further, noting that u2

1=r2

1and v2

2=r2

2we have that,

v=1

2v2r2

1−r2

2−r2

1+r2

2+2uu1=u1u

v2

.(B.4)

Substituting the above into equation (B.1a) we get,

(u−u1)2+ u1u

v2!2

=u2

1(B.5)

which, after expanding the above and grouping terms, yield a quadratic equation in u,

u2

1+ u1

v2!2

−2uu1=0,u=

2u1±q4u2

1

21+u1

v22=u1±u1

1+u1

v22.(B.6)

Taking the positive square-root leads to the following pair of equations for the intersection point,

u=2u1

1+u1

v22,v=−2u2

1

v21+u1

v22.(B.7)

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