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Interval-based identification of response-critical joints in complex built-up structures: A tool for model refinement

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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.
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 eort 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 eorts 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 eort 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 eorts 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 eort 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 diculty 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 oer 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 stiness 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 diculties 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 dierent 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 stiness 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),xy=(xryr)+i(xiyi),x×y=(xryrxiyi)+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
+ixiyrxryi
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 stiness/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
xy, b) subtraction xy, c) multiplication xy, 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 stiness matrices of the ith component, uis a vector
of displacements with ˙
denoting dierentiation 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 stiness matrix,
Zi=ω2Mi+iωRi+Ki.(16)
Alternatively, the dynamic stiness matrix can be obtained through the inversion of a measured or modelled admit-
5
tance matrix,
Zi=Y1
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 stiness, 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 stiness 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=
Y1
A
ZJ
Y1
B
(20)
where Y1
Aand Y1
Bare the inverse admittance matrices of components Aand B, and ZJis the joint dynamic stiness
matrix. For the spring-damper joint considered,
ZJ=
k+ir kir
kir k +ir !1...
k+ir kir
kir k +ir !N
(21)
where kis the joint stiness, and rits damping coecient. 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
Y1
A
ZJ
Y1
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 stiness matrix. Whilst in
this form we are able to accept arbitrarily complex joints, including inertial eects, an ideal rigid coupling would
require the joint stiness 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=YfBTλ.(32)
where Y=Z1has 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 fBTλ=Γλ. (33)
Solving for λthen yields,
λ=BYBT+Γ1BYf.(34)
Finally, substituting equation (34) into equation (32) leads to,
u=YCf(35)
where,
YC=YYBTBYBT+Γ1BY (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 Γ0(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=YYBTBYBT+Γ1BY (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=A11
1+tr XA1A1XA1,(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=A11
1+tr Γ(n)A1A1Γ(n)A1(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=A1wnA1PnA1(42)
where,
wn=γn
1+tr PnA1γn
=1
1
γn+tr PnA1=1
Zn+tr PnA1(43)
is a complex scalar value, and Zn=kn+irnis the dynamic stiness 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 eect on the assembled
structure.
Herein we propose that the interface stiness Znbe treated as a complex interval {Zn}, such that its bounds are
supposed to contain all possible values of joint stiness 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 stiness as a complex interval {Zn}, equation (43) now describes a complex interval recipro-
cal,
{wn}=1h{Zn} ⊕ tr PnA1i=1 {zn}(44)
where {zn}={Zn} ⊕ tr PnA1={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}=YYBTBYBT+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)
CnY(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, nY(n)
Corepresents an interval matrix that describes the eect, 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)
ChnY(n)
Co¯
fSi=u(n)
Cnu(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 nu(n)
Co.
5. Identification of response-critical joints
The principal aim of this paper is to provide an ecient 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 eort 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 eciency, 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 sucient samples are
drawn, for a system with many joints the computational eort 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 eort
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 oered
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
dierential of the assembly admittance,
dYC=d(Y)dYBTBYBT+Γ1BY.(49)
Noting that interest lies in the dierential interface flexibility d(Γ), the above reduces to,
dYC=dYBTBYBT+Γ1BY.(50)
Application of the product rule, d (ABC)=d(A)BC +Ad(B)C+ABd(C), then leads to,
dYC=YBTdBYBT+Γ1BY.(51)
Recalling the dierential of a matrix inverse, d(A)1=A1d(A)A1[56], yields
dYC=YBTBYBT+Γ1dBYBT+ΓBYBT+Γ1BY (52)
which reduces to,
dYC=YBTBYBT+Γ1d(Γ)BYBT+Γ1BY.(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+Γ1PnBYBT+Γ1BY dγn=Jndγn(54)
where dγnis a scalar dierential 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 suciency small interface relaxation, the coupled admittance can be estimated according to,
YC=Y(n)
C+ ∆Y(n)
C=Y(n)
C+Jnγn(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(EXi(Y|Xi)) (59)
Vi,j=VXij (EXi j (Y|Xi,Xj)) ViVj(60)
and so on, where E() denotes the expectation operator. The term EXn(Y|Xn)) is read as, the expected value of the
model output Ytaken over all possible values of Xn, 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 eects 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(EXn(Y|Xn))
V(Y).(62)
Note that Snis a normalised metric, as VXn(EXn(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 Xi.
12
Another related metric is that of the total eect index, given by,
ST n =1VXn(EXn(Y|Xn))
V(Y).(63)
Here, the term VXn(EXn(Y|Xn)) describes the first order eect due to all input parameters Xnbar Xi. Hence, V(Y)
VXn(EXn(Y|Xn)) describes the contribution of all terms in the variance decomposition which include Xi. The total
eect index thus describes the influence of the input parameter Xi, along with the eect 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 eect 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 stiness 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, oset by the computational eort
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 eciency. 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 stiness 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 Jnγn, 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)
Creal ¯
Y(n)
CE2+Dimag ¯
Y(n)
Cimag ¯
Y(n)
CE2
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 suciently large lower stiness bound ¯
Zn, we would expect S(i)
nto follow the same trend as S(g)
n, as {zn}=
{Zn} ⊕ tr PnA1would be dominated by the joint stiness {Zn}, and so {w} ≈ 1 {Zn}={γn}would appear simply
as a scaling factor. Treating the interface stiness 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 stiness 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 stiness 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)
Creal ¯
Y(n)
CEDimag ¯
Y(n)
Cimag ¯
Y(n)
CE/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}=YYBTBYBT+Γ(n)1BY "{wn} ⊗ YBTBYBT+Γ(n)1PnBYBT+Γ(n)1BY#
where
{wn}=1h{Zn} ⊕ tr PnA1i=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 stiness 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 eort 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
·103
b)
Real (w= 1/z) = u
100101102
1
0.5
0
·102
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 stiness, {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
·103
a)
Real(YC)
Sample-based estimate Interval computation
100101102
2
0
·103
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
1012
1010
108
106
104
102
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 stiness 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 nY(n)
Coalso contains the
origin (since complex multiplication can be interpreted as a rotation followed by a scaling). The subsequent addition,
Y(n)
CnY(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
1012
1010
108
106
104
102
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
1012
1010
108
106
104
102
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 stiness 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 stiness 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 dierences
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 stiness 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 stiness/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 eort. 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 stiness/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
dierences 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 eort, 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. stiness 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 dierence 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 dierent 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 dierent 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 stiness 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.
108
107
106
105
|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)
108
107
106
105
|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 stiness 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 stiness 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 dierences can be seen. This again illustrates the importance of considering the entire range
of possible stiness 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 eorts 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 eort 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 stiness/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 stiness 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 dier from the gradient-based; the relative importance of a joint depends on the range of permissible
stiness 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 eorts, 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 eciency), the interval-based metric provides a convenient and computationally ecient 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 eort 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 =uiv
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+v2u
¯
x= u1
2¯
x!2
+(v0)2 1
2¯
x!2
,(A.4)
or equivalently,
1
2¯
x!2
> u1
2¯
x!2
+(v0)2.(A.5)
Recalling that a circle of radius rwith centre point (h,k), is given by,
r2=(uh)2+(vk)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
< u1
2¯
x!2
+(v0)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=(u0)2+
v1
2
¯
y
2
1
2
¯
y
2
(A.8)
or equivalently,
1
2
¯
y
2
>(u0)2+
v1
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
<(u0)2+
v1
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 r2r1then
// 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 r2r1then
¯
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 r2r1then
¯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,
(uu1)2+(vv1)2=r2
1,(uu2)2+(vv2)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(u1u2)2v(v1v2)=r2
1r2
2u2
1u2
2v2
1v2
2(B.2)
Isolating v, whilst noting that for all cases u2=0 and v1=0, yields,
v=1
2(v1v2)1r2
1r2
2u2
1u2
2v2
1v2
2+2u(u1u2)=1
2v2r2
1r2
2u2
1+v2
2+2uu1.(B.3)
Further, noting that u2
1=r2
1and v2
2=r2
2we have that,
v=1
2v2r2
1r2
2r2
1+r2
2+2uu1=u1u
v2
.(B.4)
Substituting the above into equation (B.1a) we get,
(uu1)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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... This Jacobian matrix would form the basis of a local/ gradient-based SA (Meggitt, 2022). From the above, it is clear that fΔY ðnÞ C g is proportional to the system Jacobian, but includes the interval contribution of the uncertain joint through the interval term {z}. ...
... The sensitivity with respect to a particular joint depends on the size of the input interval; the sensitivity metric captures the nonlinear contribution of the uncertain joint's dynamics towards the system response (which arises due to equation (30)). Though only a minor effect in the present case study, more complex systems can exhibit much greater effects (Meggitt, 2022). This ability to capture non-linear behaviour is a principal advantage of interval-based SA over conventional gradient-based methods, which rely on a linearisation of the system dynamics. ...
... For more complex systems, the sensitivities of several joints can be combined to form a single sensitivity value. For example, the stiffness and damping in each coordinate direction (x, y, z) for a single connection point might be combined to yield a sensitivity value for the connection as a whole (Meggitt, 2022). ...
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