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A Complex Power approach to characterise Joints in Experimental Dynamic

Substructuring

E. Barten, M.V. van der Seijs and D. de Klerk

Delft University of Technology, Faculty of Mechanical, Maritime and Materials Engineering

Department of Precision and Microsystem Engineering, section Engineering Dynamics

Mekelweg 2, 2628CD, Delft, The Netherlands

E-mail: e.barten@gmail.nl,m.v.vanderseijs@tudelft.nl

ABSTRACT

The dynamic response of coupled structures is inﬂuenced by the joints connecting the individual substructures.

The friction induced by the interfaces causes non-linear and damping-like eﬀects, which need to be taken into ac-

count when applying Experimental Dynamic Substructuring techniques. This paper proposes a compliant interface

model in the framework of substructuring, in order to account for the inﬂuence of jointed connections. Rather than

modelling damping as a separate phenomenon, the proposed compliant interface model characterises (non-linear)

damping as a function of the interface force and motion directly. As such the model ﬁts into the Lagrange-Multiplier

FBS method. In addition, the concept of complex power is adopted to characterise the eﬀect of damping and isolate

the contribution of the interface from the overall dissipation. The theory is illustrated with a test-case on a dedicated

test structure. A successful attempt was made to identify damping parameters based on power dissipation of the

structure.

Keywords: frequency based substructuring, experimental dynamic substructuring, compliant interface, complex power, damp-

ing

Nomenclature:

u,f,g– Vector of displacements, external forces, internal forces

∆uc,λ– Vector of relative displacement at the interface, interface force intensity

M,C,K– Mass-, Damping- and Stiﬀness matrix of unassembled structure

Mc,Cc,Kc– Mass-, Damping- and Stiﬀness matrix of the compliant interface

B,L– Signed Boolean matrix, Boolean Localisation matrix

Z,Y– Dynamic Stiﬀness, Receptance

Zc,Yc– Dynamic Stiﬀness, Receptance of compliant interface

Pe,Pi– Excitation power, Interface Power

Pd,Pk– Dissipative power, Exchange Power

DS – Dynamic Substructuring

LM-FBS – Lagrange Multiplier Frequency Based Substructuring

FRF – Frequency Response Function

DoF – Degree of Freedom

FEM – Finite Element Method

1 INTRODUCTION

It is known that joints used for assembly of substructures have a signiﬁcant inﬂuence on the dynamic response of coupled

structures[11]. At the interface between two individual substructures, non-linear and damping-like eﬀects are a result of friction

which in turn highly inﬂuences the energy dissipation of the entire structure. Diﬀerent strategies to implement the eﬀects of

mechanical joints have been proposed in literature. One approach tries to implement the eﬀects from a physical understanding

of friction, such as the Iwan model which uses four parameters to characterise the behaviour of a jointed interface [9] . With

this same purpose, Bograd et al. [3] proposed a so called non-linear Jenkins model from which equivalent linear stiﬀness and

damping coeﬃcients can be derived. Alternatively, methods have been proposed to characterise the inﬂuence of the interface

experimentally by means of Dynamic Substructuring (DS) coupling and decoupling techniques (see for instance[6,16]). In [2]

and[14] procedures are proposed that employ substitute parts to keep the interfaces connected during measurement, such that the

interface eﬀects are indeed present. The added dynamics associated with these substitutes is then either removed or interchanged

in a later stage.

The interface model as proposed in this work can be positioned in between both approaches. Essentially, the substructures are

coupled following the procedures of DS, but without the strict requirement of rigid compatibility at the interface. Instead, a

certain compliance can be implemented based upon physical (damping and stiﬀness) parameters, possibly derived from above-

mentioned physical models. In[17] a method was already described to implement a compliant interface in the DS framework. This

description is however limited to the time domain. This paper proposes an extension of this method to make it also applicable

for Lagrange-Multiplier Frequency Based Substructuring (LM-FBS) [5].

A useful tool to analyse a structure is to consider the ﬂow of power when subjected to a harmonic excitation[8,18]. This complex

valued quantity allows for identiﬁcation of dissipative power which is associated with damping present in a structure. The

compliant interface as discussed before allows to distinguish between the power dissipated due to intrinsic damping and the

power dissipated at the interface.

This paper starts in Section 2 with theory of DS and how a compliant interface ﬁts in its framework. The concept of complex

power is explained in Section 3 and how it can be used to obtain the dissipation due to a compliant interface. Section 4 continues

with an academic example which illustrates the use of complex power to identify the compliance of an interface. The paper is

concluded in Section 5 and 6 which includes an outlook for future research.

2 INTERFACE COUPLING IN THE FRAMEWORK OF DS

The key concept of Dynamic Substructuring is to divide rather complex structures into smaller, less complex structures[6]. When

the dynamic responses of the individual substructures are known, the dynamic behaviour of the full structure can be constructed

by numerically coupling its subcomponents at the interfaces. Let us write the equation of motion for nunassembled subsystems

in the physical domain:

M¨

u+C˙

u+Ku =f+g(1)

with system matrices (mass, damping and stiﬀness)

M,diag(M(1), . . . , M(n)),C,diag(C(1) , . . . , C(n)),K,diag(K(1), . . . , K(n))

and sets of displacements, external forces and interface forces1

u,

u(1)

.

.

.

u(n)

,f,

f(1)

.

.

.

f(n)

,g,

g(1)

.

.

.

g(n)

For the deﬁnitions the reader is referred to Figure 1(a).

1The explicit time dependencies are omitted for simplicity.

A

fA

i

uA

b

B

uB

b

gA

bgB

b

(a) Assembly of two substructures A and B. u

are displacements, ginternal forces and fap-

plied forces.

A

fA

i

uA

b

B

uB

b

(b) Assembly using a rigid con-

nection, i.e. no relative displace-

ment at the interface.

A

fA

i

uA

b

B

uB

b

Cc

Kc

(c) Assembly using a compliant connection, i.e.

relative displacement at the interface. The inter-

faces are connected by a linear spring and dashpot

in parrallel.

Figure 1: Assembly of two substructures A and B using one node at each interface.

Starting with a recap of the DS framework, a rigid connection is considered ﬁrst, i.e. no relative displacement between sub-

structures at the interface is allowed. This is followed by the introduction of the compliant interface model. In both cases, an

expression in terms of system matrices as well as the system’s receptance is considered.

2.1 Rigid Connection

Traditional substructuring techniques couple substructures as if they are rigidly connected, i.e. no relative displacement of the

interface nodes between the substructures is allowed, see Figure 1(b). This results in two conditions that need to be satisﬁed

when coupling substructures:

1. Compatibility: this condition states that the Degrees of Freedom (DoFs) associated with coinciding interface nodes of the

respective substructures are equal. The compatibility condition can be expressed by

Bu =0(2)

where as Bis a signed boolean matrix operating on the interface DoFs. This expression states that any pair of matching

interface DoFs uA

band uB

bhave the same displacement, i.e. uA

b−uB

b= 0.

2. Equilibrium: this condition requires force equilibrium between the interface DoFs. The equilibrium condition can be

expressed by

LTg=0(3)

where Lis a boolean matrix localising the interface DoFs from the global set (for more details on constructing Band L,

the reader is referred to[6]). This expression states that the sum of a matching pair of interface DoFs gA

band gB

bshould

be equal to zero, i.e. gA

b+gB

b= 0.

The coupled system can be obtained by using either a primal or a dual formulation approach, as discussed in [6] . When the primal

formulation is used, a unique set of interface DoFs is deﬁned while the interface forces are eliminated as unknowns. In the dual

formulation however, the full set of DoFs is retained. Additionally, the dual assembled system is obtained by satisfying the

interface equilibrium a priori. For an interface model which allows for relative displacement between the two substructures, the

dual formulation is therefore the only feasible formulation. Consequently, the equilibrium condition is implemented by deﬁning

g=−BTλ(4)

with λbeing a Lagrange Multiplier representing the interface force intensity. It can be veriﬁed that Equation (4) automatically

satisﬁes the equilibrium condition[6].

In terms of System Matrices

Substituting this relation in Equation (1), and moving λto the left-hand side together with the unknown degrees of freedom, the

dual formulation is found:

(M¨

u+C˙

u+Ku +BTλ=f(5a)

Bu =0(5b)

In the frequency domain one can conveniently write the dynamic stiﬀness for u(ω)as Z(ω) = −ω2M+jωC+K. The dually

assembled system in the frequency domain reads:

(Zu +BTλ=f(6a)

Bu =0(6b)

In terms of Receptance

Typically, the dynamics of a system is characterised by measuring a limited number of input and output points. Therefore,

no explicit mass, damping and stiﬀness matrix is obtained experimentally. Instead a receptance matrix Y(ω)is measured,

comprising the Frequency Response Functions (FRFs) of the tested structure. It can be shown that the receptance matrix is the

inverse of the dynamic stiﬀness Z(ω)of the system constrained at the measurement points, such that displacements and forces

are related by u=Yf. Similarly, the mobility matrix relates the excitation with velocities, i.e. ˙

u=jωu=jωYf .

To be able to do the assembly, Equation (6a) can be rewritten by eliminating the interface force intensity λ. This formulation

is suitable for receptance matrices obtained by experiment. This method is called the Lagrange Multiplier Frequency Based

Substructuring (LM-FBS) and has been well established in the experimental substructuring community[4,5,10].

u=Yf −YBT(BYBT)−1BYf (7)

The separate terms can be distinguished to give a mechanical interpretation:

u=Yf −YBTλ(8)

λ=Zint∆uint (9)

∆uint =BYf,Zint = (BYBT)−1(10)

The response of the individual, unassembled, substructures, resulting from a force input, is given by u=Yf. This force input

results in a ∆uint which represents the distance between the substructures at the interface in the uncoupled situation. In order

to close this distance, an interface force λis applied. Finally, Zint gives the dynamic stiﬀness of the substructures condensed

at the interface, which relates the force needed to obtain a unit gap at the interface. When this term is multiplied with the initial

gap ∆uint, the intensity of the force required to close the gap is obtained.

2.2 Compliant Interface

As previously discussed, regular substructuring techniques require compatibility and equilibrium conditions to be satisﬁed at

all time. Indeed, a rigid connection is assumed where no relative displacement between the substructures is allowed. However

in reality, due to some interface compliance, relative displacement between two subcomponents can occur. In other words,

the compatibility conditions no longer hold. What will follow is a description of a method by which this compliance can be

accounted for, based on [17] , see Figure 1(c). Again, a description is given in terms of system matrices as well as the system’s

receptance.

In terms of System Matrices

Friction at the interface will be modelled by a stiﬀness and damping term only. Mass corresponding to for instance a bolt can

either be added as a separate substructure or by a distributed contribution to the diagonal of the mass matrices. Since the interface

is modelled without an interface mass, i.e. the equilibrium is not aﬀected by internal dynamics, the force equilibrium condition

still holds. On the contrary, the compatibility condition can be reformulated when relative displacement between the structures

is allowed:

(M¨

u+C˙

u+Ku +BTλ=f(11a)

Bu = ∆uc(11b)

where the subscript cindicates a correspondence with the incompatibility at the interface. Please note the diﬀerence between

∆uint and ∆ucin Equation (8); ∆uint represents the distance between the uncoupled substructures, whereas ∆ucrepresents

the “play” at the interface between the coupled structures in case of a compliant interface.

Now instead of leaving the set of interface intensities λas unknowns, they can be written in terms of the relative displacement

and relative velocity between the substructures interfaces. By assuming a linear dependency, a stiﬀness and damping term can

be distinguished.

λ=Cc∆˙

uc+Kc∆uc(12)

This is equivalent with modelling a linear spring and viscous damper between the coupling DoFs of the substructures. Note that

Ccand Kcreside in the space of the interface problem, which is dimensionally diﬀerent than the physical domain of the system

matrices. More discussion of Ccand Kcis found at the end of this section.

Substituting Equation (11b) in Equation (12) results in the following expression for the interface force intensities.

λ=CcB˙

u+KcBu (13)

When ﬁnally Equation (5a) is combined with Equation (13), the interface model can be included in the equation of motion.

M¨

u+ (C+BTCcB)˙

u+ (K+BTKcB)u=f(14)

In terms of Receptance

In order to implement the compliant interface in the frequency domain, one starts with the dual assembled matrix notation of

the equation of motion as given in Equation (6a). Recall that the compatibility condition is reformulated as Bu = ∆uc.

(Zu +BTλ=f(15a)

Bu = ∆uc(15b)

Rewriting the ﬁrst equation in Equation (15a) yields uexplicitly and after subsequently pre-multiplying with Bone obtains:2

Bu =BYf −BYBTλ(16)

Recall the compliant interface model of Equation (12) and write it as frequency dependent compliance:

λ=ZcBu (17)

with Zc,jωCc+Kc(18)

Zcrepresents the dynamic interface stiﬀness. Inverting Equation (18) to obtain the dynamic interface receptance Yc, one can

write for the interface incompatibility of Equation (17):

Z−1

cλ=Ycλ=Bu (19)

2Note that the inverse of the dynamic stiﬀness equals the receptance, i.e. Y,Z−1.

Substituting this result in Equation (16), one now ﬁnds for the interface force intensity of the compliant connection:

λ=BYBT+Yc−1BYf (20)

Finally substitution of this expression in Equation (15a) results in an extension of the LM-FBS method where a compliant

interface is accounted for.

u=Yf −YBTBYBT+Yc−1BYf (21)

This allows for a clear mechanical interpretation. The result is similar to the rigid connection expression, except for the dynamic

stiﬀness of the interface.

Zint =BYBT+Yc−1(22)

The total dynamic interface stiﬀness is extended by the Yccorresponding to the compliance of the interface. Clearly, when no

compliance is present, the expression is the same as in Equation (7), i.e. the connection is considered rigid. The more compliant

the interface, the bigger the contribution to the dynamic interface stiﬀness.

The following remarks on the construction of the compliant interface model are noteworthy:

-Ccand Kccan by default be constructed as diagonal matrices. As a result Zcand Ycare also diagonal. It is possible

however to introduce coupling between certain degrees of freedom by introducing oﬀ-diagonal terms.

- Besides viscous damping, it is possible to have hysteretic damping at the interface by writing Kc=Kr

c+jKi

c.

- The proposed linear compliant interface model ﬁts well in the DS framework. On top of that it is possible to implement

non-linear models such as the Jenkins model[3] or Iwan model[9]. For instance in [3] , an approach is proposed that results

in equivalent interface stiﬀness and damping values that are possibly frequency dependent.

3 COMPLEX POWER

A useful tool to analyse a system is to consider the complex power ﬂow of a structure. The strength of complex power is the

possibility to distinguish between the real and the imaginary component, both having a clear physical interpretation. It will be

shown later that the real part of the power is associated with dissipative power, later referred to as Pd. The imaginary part is

associated with exchange power, later referred to as Pk. The latter one can be interpreted as power being exchanged between the

inertia and elastic forces.

Two types of complex power are deﬁned. The ﬁrst one is the so called excitation power, later referred to as Pe. Excitation power

gives information about the power ﬂow in the structure as a result of a forced excitation. It gives for instance insight in how

much of the input power is dissipated. However, since it is a global measure it does not give any information about the source

of the power dissipation. If one is interested in the power ﬂow over the interface, one can examine the so called interface power,

later referred to as Pi. This helps to identify the amount of power dissipated over the interface due to its compliance. Both terms

will be discussed in this section separately.

To be able to demonstrate phase diﬀerences in the complex power, we assume a forced harmonic vibration u(ω) = ˆ

uejωt , where

the amplitude shape ˆ

ucan be complex-valued.

3.1 Excitation Power

As mentioned earlier, excitation power gives information about the power in the system as a whole. It can be used to examine

by which extend power is dissipated in the structure as a result of the input power. What will follow is the derivation of the

excitation power, both in terms of the system matrices and in terms of the receptance matrix.

In terms of System Matrices

The excitation power can be deﬁned as the projection of the applied force fonto the corresponding structure’s velocities ˙

uin

the complex plane. Information about a possible phase diﬀerence between force and velocity is given by the complex valued

amplitude shapes.

Pe=˙

uHf=˙

uH(M¨

u+C˙

u+Ku)(23)

When oscillatory excitation and motion is assumed, it follows that

Pe(ω) = −jωe−jω t ˆ

uH−ω2M+jωC+Kˆ

uejωt

=−jω ˆ

uH−ω2M+Kˆ

u+ˆ

uH(jωC)ˆ

u

=jPk+Pd(24)

In Equation (24) the total power is decomposed in its real and its imaginary part. The imaginary part of the complex power of a

structure corresponds to the exchanged power (Pk). The remaining part is dissipated (Pd), which is the real part of the complex

power. This term exists only due to the presence of damping.

Pk(ω) = Im {Pe(ω)}=−ωˆ

uH−ω2M+Kˆ

uPd(ω) = Re {Pe(ω)}=−ωˆ

uH(ωC)ˆ

u(25)

In terms of Receptance

When the dynamics of a system are represented by its FRFs, one might want to write the power of the system in terms of its

receptance Yor its mobility jωY. Recall that the excitation power of the system is the complex projection of the velocity ˙

u

onto the excitation force f.

Pe(ω) = ˙

uHf

= (jωYˆ

fejω )Hˆ

fejωt

=−jωˆ

fHYHˆ

f(26)

Observe that if the excitation is given at a single degree of freedom l, the excitation power is simply determined by an element

of the conjugated mobility matrix:

Pe(ω) = −jωY ∗

ll f2

l

To give an example, Figure 2(a) shows the power for a limited frequency range, being illustrative for an entire frequency range.

Clearly, dissipative power dominates at resonances as well as anti-resonances. The drop of exchange power at the resonance

frequency is a result of the elastic and inertia forces being in equilibrium. At the anti-resonance frequencies, due to the very

nature of the anti-resonance, the system is not receptive for motion. Hence, exchange power will not be present meaning that all

power is directly dissipated.

Figure 2(b) shows the ratio of the dissipative power over the excitation power, giving a measure for the relative power dissipated

compared to the power input. Clearly, maxima can be observed at both anti-resonance and resonance frequencies.

900 950 1000 1050 1100 1150 1200 1250

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Frequency [Hz]

Power [W]

Absolute Power

Exchange Power

Dissipative Power

(a) Zoom of excitation power being typical for the full frequency

range.

0 500 1000 1500 2000 2500 3000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Power ratio [−]

(b) Ratio of dissipative power over total power.

Figure 2: Example of complex power plots to illustrate its characteristics.

3.2 Interface Power

Consider a compliant interface as described in Section 2.2. Damping-like behaviour can be present due to local friction behaviour

at the interface. Therefore, it can be interesting to examine the inﬂuence of a compliant interface onto the power in a structure.

To that extend consider the complex projection of the relative velocity between two interface DoFs ∆˙

uconto the interface force

intensity λin order to obtain the power over the interface.

Pi= ∆ ˙

uH

cλ(27)

Assume again oscillatory excitation and response, both with a possibly complex amplitude. Recall the following expressions for

the relative displacement and velocity at the interface.

∆uc=Bu =BYˆ

fejωt ∆˙

uc=B˙

u=jωBYˆ

fejωt (28)

In Section 2.2 it was proposed to decompose the interface force λinto a linear stiﬀness and damping term.

λ=Cc∆˙

uc+Kc∆uc

=CcB˙

u+KcBu

=jωCcBYˆ

fejωt +KcBuYˆ

fejωt (29)

Substituting Equation (28) and (29) into Equation (27) leads to the following expression for the power over the interface.

Pe= ∆ ˙

uH

cλ

= (jωBYˆ

fejωt )HjωCcBYˆ

fejωt +KcBYˆ

fejωt

=−jωˆ

fHYHBTe−jωt (j ωCcBY +KcBY)ˆ

fejωt

=ω2ˆ

fHYHBTCcBYˆ

f

| {z }

real, dissipative power

+−jωˆ

fHYHBTKcBYˆ

f

|{z }

imaginary, exchange power

(30)

Notice the two diﬀerent terms in Equation (30) denoting power corresponding to interface damping and interface stiﬀness, both

coeﬃcients being real valued. Since damping can be the only source of dissipation, real power is considered to be dissipative.

On the contrary, the power associated to the stiﬀness is imaginary and therefore represents the exchange power over the interface.

Note that both Kand Kccan be complex valued in case of hysteretic damping, i.e. K=Kr+jKi, resulting in the dissipation

term in the equations from (24) to be extended by the imaginary component of the stiﬀness.

4 TEST CASE

This section addresses an application of the compliant coupling of two substructures, based on the theory discussed in Section

2. The complex power method of Section 3 is used to estimate proper compliant interface parameters.

Figure 3 shows the considered assembled structures being a U-shaped beam and a straight beam. The substructures are attached

at two locations, which is realised by two M10 bolts. The simplicity in geometry of the structures makes it easy to numerically

model. Although having two interface connections, the high level of symmetry of the structure allows for good interpretation

of the contribution of a single connection. Moreover, the structures are made out of a single piece of aluminium which ensures

homogeneous material properties, unaﬀected by welds or any other form of connection. Finally, the physical attachment is

realised by a bolted connection typically used in industry.

(a) Picture of the considered assembled structure (20 ×200 ×

300mm). The structure is made out of a single piece of alu-

minium.

x

y

z

(b) CAD model of the considered assembled structure. The

contribution of the bolts is added in a later stage.

Figure 3: A picture and a CAD model of the considered assembly. The in-plane coordinates are xand y. The out of plane

coordinate is z.

The dynamics of the two unassembled structures and the assembled structure is experimentally obtained by combining the

following experiments: 1) The eigenmodes are obtained by hammer impact and acceleration measurement. Unfortunately,

the wire of the acceleration sensor used for the impact measurement contributed signiﬁcantly to the damping. 2) A second

measurement was therefore done where the response was obtained acoustically using a microphone. This resulted in a much

more accurate estimation of the eigenfrequencies and damping values. For the interested reader, the methods used for the

damping estimation is discussed in Appendix A.

To avoid the use of measured response data which might be contaminated with measurement noise, the assembly is realised based

on the FEM modes (obtained by ANSYS) of both substructures. A comparison of the FEM modes with the measured modes

based on the Modal Assurance Criteria (MAC)[1] results in values between 0.9 and 1 in the considered frequency range of 3kHz.

Moreover, the FEM eigenfrequencies are within 1%agreement with respect to the experimentally obtained eigenfrequencies.

Even beyond the considered bandwidth up to 6kHz, still a 1.5% agreement is obtained. Both results proof good resemblance

between the FEM model and the experimentally obtained modal data.

In the substructuring results that follow, the contribution of the bolts and sensor is accounted for by means of adding lumped

masses with accompanying inertia to the assembled structure.

A good comparison with experimental data is realised by re-synthesis of the measured response, rather than using the measured

response directly. The re-synthesis is based on the measured eigenfrequencies and modal damping values.

4.1 Classical rigid interface connection results

First, a comparison is made using diﬀerent sets of coupling DoFs based on a rigid connection, i.e. where no relative displacement

is allowed between the coupling degrees of freedom. Distinction is made between the following sets of coupling DoFs.

1. Direct node coupling considering only translation information. A selection of six DoFs is taken from three nodes spanning

a plane, being three translations in node 1, two translations in node 2and one translation in node 3.

2. A Virtual Point Transformation [13] using the 6-DoF translational and rotational rigid Interface Deformation Modes (IDMs),

i.e. no ﬂexible IDMs are considered. The nodes closest to the virtual point are used for the transformation.

3. A rigidiﬁed interface condensed in one point having both translational and rotational information. All nodes at the interface

are connected by rigid links to a single point which is used for assembly. This results in a truly rigid interface.

Direct node coupling did not give satisfactory results. Neither eigenmodes nor eigenfrequencies did match the measurement. The

other two options on the other hand showed better similarity with the experimentally obtained eigenmodes. The eigenfrequencies

however were signiﬁcantly higher than the measured eigenfrequencies. Clearly, the considered connection is stiﬀer than reality,

which shows its largest inﬂuence for the in-plane response.

Moreover, the rigidiﬁed eigenfrequencies were slightly higher than the virtual point eigenfrequencies. This is due to the fact

that, for the virtual point transformation, compatibility is enforced only for the rigid part of the interface behaviour while the

elastic part remains uncoupled. For a rigidiﬁed interface on the other hand, the interface is fully rigid and assembled as such.

So it is expected that the virtual point connection underestimates the stiﬀness of the interface, whereas the rigidiﬁed interface

yields an overestimation. The true interface stiﬀness is therefore most likely somewhere in between.

Based on these results and the fact that a rigidiﬁed interface showed better out-of-plane results when compliantly connected,

compared to the same connection with a virtual point, it is chosen to continue this test case with the rigidiﬁed interface. Recall

that only the inertial contribution of the bolt was accounted for. The rigidiﬁed interface therefore accounts for the added stiﬀness

of the bolt, making it the best possible option for this test case.

For the considered frequency band the observed modes are, due to the geometry of the structures, either purely in-plane or out-

of-plane. Therefore, the response of the structure is considered accordingly. Figure 4 and 5 show the results of a rigid connection

by means of a MAC table and FRFs. The ﬁgures show that the out-of-plane result is already quite promising in contrast with

the in-plane result, which indicates that the interface has its largest inﬂuence on the in-plane dynamics.

598 868 1757 1931 2495

415

802

1183

1653

2056

2757

FEM Eigenfrequencies [Hz]

Measured Eigenfrequencies [Hz]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) In-plane eigenmodes.

402 703 1123 1832 2210 2495 2586

372

716

1101

1802

2045

2447

2537

FEM Eigenfrequencies [Hz]

Measured Eigenfrequencies [Hz]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Out-of-plane eigenmodes.

Figure 4: MAC table of the measured eigenmodes and substructured eigenmodes with a rigid interface connection. Resonance

at 2495 Hz showed both in-plane and out-of-plane displacements.

0 500 1000 1500 2000 2500 3000

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency [Hz]

Receptance Y [m/N]

Experimental data

Rigid connection

(a) In-plane response.

0 500 1000 1500 2000 2500 3000

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency [Hz]

Receptance Y [m/N]

Experimental data

Rigid connection

(b) Out-of-plane response.

Figure 5: In-plane and out-of-plane response of the assembled structure, using 1) rigid connection, 2) compliant connection

case 1. The compliant connection (1) only aﬀects the in-plane dynamics.

4.2 Compliant interface connection results

Figure 6: CAD model showing the compliant

directions of the interface connection.

The aim is to improve the results by implementing a compliant interface for both

connections. Consider the following three cases.

1. Solely the in-plane rotation, i.e. the rotations around the bolts, are compli-

antly connected. Only ﬂexibility is added, no damping (kθz).

2. Both in-plane and out-of-plane rotation is compliant in stiﬀness (kθx, kθz).

3. Both in-plane and out-of-plane rotation is compliant in stiﬀness (kr

θx, kr

θz)

and damping (ki

θx, ki

θz, cθx, cθz).

It holds for all cases that the remaining DoFs are rigidly connected and that the

compliance is symmetrically applied, i.e. both sides of the structure have the same

compliance. Each subsequent case uses the same compliant stiﬀness values as

used in its preceding case.

In Figure 7 the measured (resynthesised) response is compared with both a rigid connection as well as compliant connection

case 1. The in-plane response shows the inﬂuence of adding compliance in a single DoF. Clearly, due to the compliance in

θz, only the in-plane resonances have shifted in frequency. The out-of-plane response given in Figure 7(b) is not aﬀected by

the in-plane compliance. This allows for independent shifting of eigenfrequencies for either in-plane or out-of-plane vibration

modes.

The results are improved (both eigenfrequency and eigenmode) when the compliance of the interface connection is extended by

a compliance for the out-of-plane motion (case 2), not shown in the ﬁgure.

Next, a MAC comparison is made between the measured modeshapes and the substructured modeshapes of case 2, see Figure

8. The majority of both in-plane and the out-of-plane eigenmodes have MAC values close to one, prooﬁng a good resemblance

with the measured data. Note that a MAC value for the ﬁrst in-plane mode is missing; this mode was not identiﬁed correctly.

0 500 1000 1500 2000 2500 3000

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency [Hz]

Receptance Y [m/N]

Experimental data

Rigid connection

Compliant connection (1)

(a) In-plane response.

0 500 1000 1500 2000 2500 3000

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency [Hz]

Receptance Y [m/N]

Experimental data

Rigid connection

Compliant connection (1)

(b) Out-of-plane response.

Figure 7: In-plane and out-of-plane response of the assembled structure, using 1) rigid connection, 2) compliant connection

case 1. The compliant connection (1) only aﬀects the in-plane dynamics.

419 790 1166 1792 2018 2698

415

802

1183

1653

2056

2757

FEM Eigenfrequencies [Hz]

Measured Eigenfrequencies [Hz]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) In-plane eigenmodes.

379 702 1120 1809 2189 2474 2549

372

716

1101

1802

2045

2447

2537

FEM Eigenfrequencies [Hz]

Measured Eigenfrequencies [Hz]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Out-of-plane eigenmodes.

Figure 8: MAC table of the measured eigenmodes and substructured eigenmodes with compliant interface case 2.

Estimation of interface damping based on complex power

The excitation power for the assembled structure is computed from the resynthesised receptance using Equation (26). Focussing

on the dissipative part of the power helps one to identify the dissipative phenomena on the interface and make an estimate of the

involved interface damping. Figure 9 gives the dissipative power of the resynthesised measurement and the compliant interface

cases 2 and 3. The essential diﬀerence between the two compliant interfaces is the presence of interface damping in case 3.

When no interface damping is present, obviously the only damping of the system is due to the contribution of the damping

present in the individual structures. The corresponding dissipation level is clearly underestimated. It is therefore not suﬃcient to

describe the interface connection solely with stiﬀness terms. Adding damping, both in viscous and hysteretic form, improves the

result signiﬁcantly. By tuning the damping values, the overall dissipation level increases and the dissipation close to resonance

frequencies approximate the measurement much better.

0 500 1000 1500 2000 2500 3000

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency [Hz]

Power [W]

Resynthesised Measurement

Compliant interface (2)

Compliant interface (3)

Figure 9: Dissipative part of the excitation power of the resynthesised measurement, compliant interface case 2 and case 3.

Unfortunately, due to measurement limitations it is not possible to directly obtain the interface power. Also, the way the measured

response is resynthesised results in a global level of the interface damping, hence no information about the interface dissipation

can be obtained. Nonetheless the eﬀect of the interface for the overall damping can indeed be indicated by comparing the total

amount of dissipation of the assembled structure with the dissipation of the substructures being rigidly coupled.

We have already shown that the substructured eigenfrequencies and eigenmodes show good resemblance with the measurement.

On top of that we now have a good estimate of the interface damping since the dissipation levels match. Now ﬁnally the

substructured response based on compliant interface case 3 can be compared with the resynthesised measurement, see Figure

10. For the majority of eigenmodes, both the in-plane as well as the out-of-plane substructured response approximate the

measured response very well.

0 500 1000 1500 2000 2500 3000

10−10

10−8

10−6

10−4

10−2

100

Frequency [Hz]

Receptance Y [m/N]

Experimental data

Compliant connection (3)

(a) The in-plane response.

0 500 1000 1500 2000 2500 3000

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency [Hz]

Receptance Y [m/N]

Experimental data

Compliant connection (3)

(b) The Out-of-plane response.

Figure 10: Response of assembled structure. The resynthesised measurement is compared with compliant interface case 3.

5 CONCLUSIONS

This paper introduces a method to implement interface compliance in the context of Experimental Dynamic Substructuring.

The method is validated by means of a test case in which it is shown that one can approximate the eigenfrequencies by choosing

the proper compliance of the interface in a certain degree of freedom.

Complex power was used to judge the damping of the interface compliance. By looking at the dissipation level, one can tune

the interface damping such that it approaches the measured dissipation. It was shown that by adding both hysteretic and viscous

damping for a small set of interface degrees of freedom, the power level showed good resemblance with the resynthesised power

dissipation.

6 OUTLOOK

As for now, the excitation power is used to tune the damping values of the compliant interface. For future work, however, it can

be interesting to examine the possibilities to use interface power to estimate the compliant interface parameters.

Yet to be done is to categorise stiﬀness and damping parameters of typical joints in order to implement jointed connections

directly from the shelf into the compliant interface model.

A DAMPING ESTIMATION

Two methods are adopted to determine modal damping: a frequency domain approach and a time domain approach. The Circle

Fit[7] method is a well known frequency domain approach and relies on the fact that a response in the vicinity of a resonance

follows a circular shape when plotted in the complex plane. Using a circle to ﬁt the response data, one can determine the damping

as a function of the sweep rate of the response over the circle, see Figure 11(a).

−1 −0.5 0 0.5 1

x 10−3

−1

−0.5

0

0.5

1x 10−3

FRF Data

Circle Fit

Eigenfrequency

(a) Circle ﬁt of accelerance response data.

0 5 10 15 20

−0.02

0

0.02

Time (s)

Amplitude

0 5 10 15 20

10−10

10−5

100

Time (s)

Amplitude

(b) Response in time domain (top) and linear ﬁt of response envelope on a

logarithmic scale up to a certain noise level (bottom).

Figure 11: A frequency and time domain approach to determine modal damping.

The time domain approach relies on the assumption that any signal in time can be modelled as a superposition of periodic

components with its speciﬁc frequency and amplitude (the deterministic part) plus uncorrelated noise (the stochastic part). When

an eigenfrequency is considered, the amplitude of the corresponding wave has the analytical expression of x(t) = Ae−ζrωrt.

When plotted on a logarithmic scale, a straight line can be ﬁtted up to a certain noise level, see Figure 11(b). The damping

associated to that eigenfrequency is a function of the slope of this line3.

3Rather that using a conventional Least-Squares Complex Exponential (LSCE) algorithm to ﬁt modal parameters to the time data, a Vold-Kalman ﬁlter

For both methods it is assumed that damping in the structure described in the test case of Section 4 is well described by modal

damping. The order of magnitude of the damping ratio ranging from 10−3to 10−5justiﬁes this assumption. For eigenfrequencies

for which both methods were successful, the obtained damping values are within 10% agreement.

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