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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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