Content uploaded by Maarten Van der Seijs

Author content

All content in this area was uploaded by Maarten Van der Seijs on Apr 24, 2017

Content may be subject to copyright.

COMPDYN 2013

4th ECCOMAS Thematic Conference on

Computational Methods in Structural Dynamics and Earthquake Engineering

M. Papadrakakis, V. Papadopoulos, V. Plevris (eds.)

Kos Island, Greece, 12–14 June 2013

AN IMPROVED METHODOLOGY FOR THE VIRTUAL POINT

TRANSFORMATION OF MEASURED FREQUENCY RESPONSE

FUNCTIONS IN DYNAMIC SUBSTRUCTURING

Maarten V. van der Seijs1, Dani¨

el D. van den Bosch1, Daniel J. Rixen2and Dennis de

Klerk1

1Delft University of Technology

Faculty of Mechanical, Maritime and Materials Engineering, Section of Engineering Dynamics

Mekelweg 2, 2628CD, Delft, The Netherlands

e-mail: m.v.vanderseijs@tudelft.nl

2Technische Universit¨

at M¨

unchen

Faculty of Mechanical Engineering, Institute of Applied Mechanics

Boltzmannstr. 15, D - 85748 Garching, Germany

e-mail: rixen@tum.de

Keywords: Frequency Based Substructuring, Experimental Substructuring, NVH, Virtual Point

Transformation

Abstract. Dynamic Substructuring methods play a signiﬁcant role in the analysis of todays

complex systems. Crucial in Dynamic Substructuring is the correct deﬁnition of the interfaces

of the subsystems and the connectivity between them. Although this is straightforward practice

for numerical ﬁnite element models, the experimental equivalent remains challenging. One of

the issues is the coupling of the rotations at the interface points that cannot be measured di-

rectly. This work presents a further extension of the virtual point transformation that is based

on the Equivalent Multi-Point Connection (EMPC) method and Interface Deformation Mode

(IDM) ﬁltering. The Dynamics Substructuring equations are derived for the weakened inter-

face problem. Different ways to minimise the residuals caused by the IDM ﬁltering will be

introduced, resulting in a controllable weighting of measured Frequency Response Functions

(FRFs). Also some practical issues are discussed related to the measurement preparation and

post-processing. Special attention is given to sensor and impact positioning. New coherence-

like indicators are introduced to quantify the consistency of the transformation procedures:

sensor consistency, impact consistency and reciprocity.

4334

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

1 INTRODUCTION

Dynamic Substructuring (DS) has proven to be a powerful engineering tool for a wide vari-

ety of industries. It allows a complex dynamic system to be modelled and analysed as separate

components or “substructures”, building a bridge between the efforts of different suppliers and

design groups. Also, components may be coupled from either numerical or experimental na-

ture. The ﬁrst class is often characterised by ﬁnite element models or reduced-order models,

that tend to be relatively easy to assemble and highly controllable in terms of numerical accu-

racy. Substructuring methods associated with these models are numerous and have been well

accepted over the last decades [2]. However, with increasing product complexity, the question

rises how truthful some numerical models represent the actual behaviour of the components.

Experimental identiﬁcation of components is therefore often desired for the purpose of model

validation.

In recent years though, an increasing amount of research has been devoted to substructur-

ing with the experimental models themselves, leading to hybrid and very powerful modelling

methodologies [4, 11]. Although the substructuring techniques themselves are formulated in

a rather straightforward way (see for instance [5]), the largest challenges exist in the coupling

of the measured components. To couple substructures properly, one requires a complete and

accurate model of the dynamics at the interface for both translational and rotational degrees of

freedom [8, 12, 13].

1.1 Modelling the interface problem

The DS algorithm requires that displacement compatibility and force equilibrium is explicitly

satisﬁed on the interfaces of the connected substructures. In a discretised world, coupling could

be fully determined by imposing this two conditions on the respective coupling nodes of the

substructures.

However in industrial practice components are often connected by bolts or welds, that phys-

ically show more resemblance to a line or surface connection than to a single point. Modelling

a continuous contact surface or line is of course not feasible as one would theoretically need an

inﬁnite amount of DoFs. It is therefore common practice to reduce the interface problem to one

or more connecting nodes.

Let us for example consider a substructure connection as illustrated in ﬁgure 1a. If both

structures would be represented by an FE model, the substructuring task amounts to coupling the

displacements and interface forces of the coinciding nodes (ﬁgure 1b). By coupling a sufﬁcient

number of nodes over a larger area, any rotational coupling is implicitly accounted for.

However in connecting experimental substructures very few guidelines exist. A simple tri-

axial translational Single-Point Connection (SPC) is simply inadequate to couple the rotational

degrees of freedom (RDoFs). To overcome the RDoF problem, it was suggested by [6] and [14]

to introduce an Equivalent Multi-Point Connection (EMPC) as illustrated by ﬁgure 1c. This

method couples the translational directions of multiple points in the proximity of the interface

which implicitly accounts for the rotations. In practice, a minimum of 3 triaxial accelerome-

ters (not in line) is sufﬁcient to fully determine the 6 DoF coupling, i.e. 3 translational and 3

rotational directions.

1.2 Weakening the interface conditions

Unfortunately it was observed that the 9 DoFs resulting from a 3-point coupling may even

overdetermine the coupling problem. As the structure between the 3 connections point is typ-

4335

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

(a) The full contact area. (b) Discrete coupling with FEM nodes.

(c) Standard EMPC: multiple points, only

translational directions (9 DoFs).

(d) Extended EMPC: One virtual point, 3

translations and 3 rotations (9 →6 DoFs).

Figure 1: A bolted connection: three ways to model the interface problem.

ically very stiff, any discrepancy in motion (often due to measurement errors) will be over-

compensated for in the DS coupling equations, resulting in unwanted “stiffening” and spurious

peaks in the coupled FRFs [15].

In fact the local rigidness of the interface is the essential observation that leads to the concept

of Interface Deformation Mode (IDM) ﬁltering, as introduced in the same work. By deﬁning

6 rigid IDMs per interface and projecting the 9-DoF (or more) receptance matrix onto this

subspace, one only retains the dynamics that load the interface area in a rigid manner. When

substructuring is performed with this “ﬁltered” receptance, one only imposes the compatibility

and equilibrium conditions on the motion of those 9 DoFs that obey local rigid behaviour, while

the residual ﬂexible motion is left uncoupled. In effect the interface problem is “weakened”

and, due to the least-square reduction step, measurement errors are averaged out.

1.3 Virtual point description

More generally, one can look at this concept from a modal reduction point of view and simply

describe the rigid interface dynamics using the generalised set of coordinates associated with

the rigid IDMs. Doing so, one has in fact obtained a 6-DoF per node receptance model that is

directly compatible for coupling with FE models. This is illustrated by ﬁgure 1d. Indeed, the

nodal 6×6receptance matrices describe the dynamic responses between the so-called virtual

point displacements and virtual point forces: the receptance of the interface concentrated in a

virtual point1.

Interesting is now how to set up an experiment that leads to a complete and reciprocal 6×6

virtual point FRF matrix, including proper “driving-point” receptance on the diagonal. Vari-

ous studies have dealt with problems related to driving-point measurements [10, 16] and the

propagation of measurement errors into the coupled FRFs [7, 15].

1The point is said to be virtual, since no actual measurements have been done on this point and it can in fact be

chosen anywhere in the proximity of the interface.

4336

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

1.4 Paper outline

This paper discusses the virtual point transformation that translates the measured (impact)

forces and displacements to virtual point receptance. The theoretical background on dynamic

substructuring and the virtual point transformation is provided in section 2. Next, section 3 aims

at providing some practical guidelines to set up an experiment and means to evaluate the results

of the measurements in terms of consistency of the transformation and resulting reciprocity.

The paper is concluded with a short summary in section 4.

2 THEORY

The Virtual Point Transformation is based on the theory of the Equivalent Multi-Point Con-

nection (EMPC) method. This method was ﬁrst presented by [6, 14, 17]. A physical derivation

of the transformation and ways to construct IDM matrices for individual subsystems were sug-

gested by [16]. In this work however, the virtual point transformation is approached slightly

differently, namely from a Dynamic Substructuring point of view. Thereafter the physical

derivation of the IDM matrices is discussed.

2.1 Dynamic Substructuring with weakened interface conditions

Dynamic substructuring connects substructures by imposing two conditions: compatibility

of interface displacements and equilibrium of interface forces. For the derivation of the virtual

point transformation, let us ﬁrst consider a simple coupling problem of two substructures A

and B, characterised by their FRF matrices YA(ω)and YB(ω). In summary, one can start

with writing the dynamic equations of the uncoupled system using a block-diagonal receptance

matrix Y= diag(YA,YB), such that the uncoupled responses u= [uA;uB]to some force

loading ftot are found by2:

u=Yftot (1)

The total forces ftot are typically a combination of external loads fand interface forces g. Let

us for now imagine that the interface nodes of the two substructures coincide perfectly, such

that the compatibility condition can be stated by equating the translational DoFs of the two

substructure boundaries uA

b=uB

bor Bu =0, using the signed boolean matrix notation (see

[5]). The following coupled system of equations is then obtained:

u=Y(f+g) = Y(f−BTλ)(2a)

Bu =0(2b)

where the interface forces are expressed using the same matrix Band a set of Lagrange multi-

pliers λthat are yet unknown.

2.1.1 Interface reduction

Let us now imagine that one is interested in considering the interface displacements as rigid,

or at least represented by some more general Interface Deformation Modes (IDMs) contained

in the N×Mmatrix Rwith modal coordinates q. As the number of IDMs is smaller than the

number of interface DoFs (M < N), a residual on the displacements µis added:

u=Rq +µ(3)

2In this derivation the explicit frequency dependency of Y(ω),u(ω)and f(ω)is omitted for clarity.

4337

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

To ﬁnd qin a minimal sense, one could directly apply the (Moore-Penrose) pseudo-inverse of

R, minimising the norm of the residuals on the displacements. Instead one can also introduce a

symmetric weighting matrix Wthat can be chosen to have the residual satisfying

RTWµ=0(4)

such that the coordinate transformation and ﬁltering process becomes:

RTWu =RTWRq (5a)

q=RTWR−1RTWu (5b)

˜

u=RRTWR−1RTWu (5c)

or short:

q=Tu with T∆

=RTWR−1RTW(6a)

˜

u=RTu =Pu with P∆

=RRTWR−1RTW(6b)

Vector ˜

udenotes the physical displacements projected onto the IDM subspace. Pis the N×N

projection operator of reduced rank M < N.

Note that the above procedure is in fact a least-square projection minimising the squared

error µTWµ. If Wis chosen to be identity, the pseudo-inverse is found, T= (RTR)−1RT,

which leads to a minimisation of the error on the displacements. In general if Wis a diagonal

matrix, it gives weight to the individual displacements in the error minimisation such that one

can control the importance of a certain DoF for the transformation. However if Wis chosen to

represent a (dynamic) stiffness matrix, one is nullifying some local residual energy. Alternative

choices for Ware beyond the scope of this paper.

2.1.2 Substructure assembly

The interface reduction as derived above allows us to assemble only the rigid behaviour of

the interface (˜

u=Rq) that was properly measured. The rest (µ) is left free, being either mea-

surement noise or higher-order deformations that one could probably not measure correctly due

to inaccuracies in impact and sensor locations.

Rewriting the dual DS equations of (2a) and (2b) for rigid interface coupling:

u=Y(f+g)(7a)

Bq =BRTWR−1RTWu =0(7b)

The boolean matrix Bis different to the one of (2b) as it now couples the DoFs of the rigid

IDMs, hence the rigid behaviour of u. The forces gare still coupling forces in the unreduced

domain but conjugate to the IDM conditions, namely:

g=−WTRRTWR−1BTλ=−TTBTλ(8)

and the coupled system becomes:

u=Y(f−TTBTλ)(9a)

Bq =BTu =0(9b)

4338

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

Writing now the ﬁrst equation in the second and solving for λ:

BTYf −BTYTTBTλ=0

λ=BTYTTBT−1BTYf (10)

Putting everything together in (9a) gives us the dynamic equation for the reduced-interface

coupling:

u=Yf −YTTBTBTYTTBT−1BTYf (11)

The receptance matrix of the coupled system is given by

Yass =Y−YTTBTBTYTTBT−1BTY (12)

such that we can simplify to u=Yassf.

2.2 Dynamic Substructuring with virtual point DoFs

Equation (11) gives the coupled reduced-interface dynamic equations for the physical dis-

placements and forces uand f. However when coupling the experimental models to FE models

or even to each other, it is probably more convenient to set up the substructuring equations using

the virtual point DoFs directly. Let us therefore state the relation between measured and virtual

displacements and forces once more:

q=RTWR−1RTWu =Tu (13a)

f=WTRRTWR−1m=TTm(13b)

The ﬁrst equation was already deﬁned by (5b). The second equation to obtain the virtual forces

fuses the same Tas the virtual displacements and therefore the same IDMs3. However a dif-

ferent Tcould easily be constructed, as long as the IDM coordinates match: the components in

mare indeed a force or torque in the direction of the components of q. More detail on how to

obtain the IDMs for the sensors and impacts is found in section 2.3.

Combining equations (11), (13a) and (13b) gives us the coupled system equations in terms

of the virtual point displacements qand forces m:

q=TYTTm−TYTTBTBTYTTBT−1BTYTTm(14)

Observes that compared to equation (11) the applied forces are now also ﬁltered, so that only

the resulting virtual forces corresponding to the IDM subspace are implied in the dynamic

computations.

Substituting H=TYTTas a virtual point receptance matrix, we get:

q=Hm −HBTBHBT−1BHm (15)

and as a result

Hass =H−HBTBHBT−1BH (16)

which has exactly the same form as the well-known Lagrange Multiplier FBS formula [5] but

now for the virtual point dynamics. Indeed H(ω)comprises the virtual point receptance relating

the virtual point forces m(ω)to virtual point displacements q(ω). Due to the block-diagonal

construction of the subsystem IDM matrices, we can also conclude that H= diag(HA,HB),

showing that the virtual point receptance can be built up for the subsystems independently.

3This implies that the locations of the applied forces (hammer impacts or shaker excitations) should completely

coincide with the corresponding measured responses (sensors).

4339

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

X

Y

uk

x

uk

y

ek

x

ek

y

rkrhfh

eh

qv

X

qv

Y

qv

θ

Z

Figure 2: The IDMs associated with the virtual point (green) can be constructed from the positions and directions

of the measured displacements (red) and impacts (blue).

2.3 Construction of the Interface Displacement Modes

In the derivation above it was shown that the virtual point transformation relies on a re-

duction in a modal sense, namely reducing the interface connectivity using a set of Interface

Deformation Modes (IDMs) speciﬁed by R. Let us now elaborate on how to obtain these IDMs

for a single interface indicated in ﬁgure 2. Virtual point vis surrounded by Nk= 3 triaxial

acceleration sensors, registering a total of 9 translational displacements4in the local (x, y, z)

frame of the sensors. An impact is indicated by a thick blue arrow.

2.3.1 Displacements

Following equation (3) the sensor IDM matrix Rtranslates the local frame displacements

to 6 virtual point displacements and rotations plus a residual. Let us write out this equation

for a triaxial sensor k. The orientation of the sensor is determined by its three measurement

directions speciﬁed as the unit vectors [ek

xek

yek

z] = Ekwhile the distance from the sensor to

the virtual point is given by vector rk. The respective local displacements along these directions

are denoted by uk. The 6 DoFs of virtual point vare contained in the set qv.

uk

x

uk

y

uk

z

=

ek

x,X ek

x,Y ek

x,Z

ek

y,X ek

y,Y ek

y,Z

ek

z,X ek

z,Y ek

z,Z

1 0 0 0 rk

Z−rk

Y

0 1 0 −rk

Z0rk

X

0 0 1 rk

Y−rk

X0

qv

X

qv

Y

qv

Z

qv

θ

X

qv

θY

qv

θ

Z

+

µk

x

µk

y

µk

z

(17)

Introducing ¯

Rkv as the 3×6global frame matrix associated with sensor kand virtual point v,

we can write:

uk=EkT¯

Rkv qv+µk(18)

uk=Rkv qv+µkwith Rkv =EkT¯

Rkv (19)

4In fact the sensors measure accelerations but for simplicity of notation displacements are considered here.

4340

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

For a typical problem having 3 sensors per virtual point and sorted in ascending virtual point

order, the Rkv matrices can be stacked to form the following equations5:

u=Rq +µwith R=

R1,1

R2,1

R3,1

R4,2

R5,2

R6,2

...

RNk,Nv

(20)

Alternatively one may prefer to describe the virtual point relation per measurement direction

individually, such that a single displacement DoF uk

iin the local direction ek

ican be related to

the associated set of virtual point coordinates in qv:

uk

i=hek

i

Trk

i×ek

iTiqv+µk

i=Rkv

iqv+µk

i(21)

from which the full matrix Rcan be assembled row-wise per sensor channel iand column-wise

for the different virtual points.

Regardless of the construction method the resulting IDM matrix Rshall be block diagonal,

or at least have IDMs that are uncoupled over the various virtual points and sensor groups (if

a different ordering was used). Although not treated here, Rmay be augmented with ﬂexible

IDMs if desired. Note that in any case one should take care that Ris at least full rank (but

preferably overdetermined) such that a (generalised) inverse Tcan be computed by applying

equation (13a):

q=RTWR−1RTWu =Tu

2.3.2 Forces

The transformation of the forces can be performed in a similar step-wise way. Unlike dis-

placements, forces are not uniquely deﬁned by virtual point forces and moments (from here

on simply called virtual forces). In fact the other way around is true, such that the following

relation can be written for the set of virtual forces mvas a result of an impact fhin the direction

of ehat distance rh:

mv

X

mv

Y

mv

Z

mv

θ

X

mv

θY

mv

θ

Z

=

1 0 0

0 1 0

0 0 1

0−rh

Zrh

Y

rh

Z0−rh

X

−rh

Yrh

X0

eh

X

eh

Y

eh

Z

fh(22)

Now we like to use similar notation as (21), but to do so we need the transpose of Rhv:

mv="eh

rh×eh#fh=RhvTfh(23)

5it is easy to observe that if only translational information is desired for a certain virtual point (e.g. if one only

used one sensor), the sensor rotation matrix can be used directly: Rk v =EkT.

4341

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

Every impact adds a single column to the IDM matrix RT. Assuming 9 impacts per virtual

point, the system for the complete set of DoFs is constructed as follows:

m=RTfwith R=

R1,1

.

.

.

R9,1

R10,2

.

.

.

R18,2

...

RNh,Nv

(24)

Note that the transpose of the IDM matrix of the forces takes the same form as the IDM matrix

of the displacements from equation (20). In fact if one decides to excite only on the sensor faces

(which is generally not advised, see section 3.2), one has the exact same IDM matrix apart from

some sign changes.

With more than independent 6 excitations per virtual point, equation (24) becomes under-

determined, which means that there is no unique combination of the impacts ffrom which a

certain mcan be constructed. Again the solution is found by using a generalised inverse, only

this time a weighted “right inverse”:

f=WTRRTWR−1m=TTm(25)

that can be shown to minimise the error on the residual impact forces weighted by W. We can

now see that the underdetermination of (24) in fact reﬂects an overdetermination of the inversed

problem of (25), which again has advantageous properties for interface weakening and error

suppression as discussed in section 2.1.

3 PRACTICE

This section will elaborate on some more practical issues related to virtual point modelling.

Proper positioning of the sensors and impacts has to be taken into consideration. Much care

should be taken to ensure that all 6 DoFs per virtual point can be described independently. This

has implications for both sensor placement and impact positions. After measuring some careful

post-processing needs to be done to come up with good results. The following preparation and

post-processing steps are presented in the order as they may normally appear in time.

3.1 Sensor placement

The thought that two triaxial sensors (measuring two times 3 translations) can completely

describe the 6 DoF virtual point, can be deceiving. When the responses of two triaxial sensors

are transformed to a 6 DoF virtual point description, one linear dependence will always appear

in the virtual points DoFs. This linear dependence is caused by the fact that the triaxial sensors

are unable to describe the rotation over the axis spanned between both sensors as illustrated by

ﬁgure 3a. This is regardless of the position of the sensors relative to the virtual point. Adding

a third triaxial sensor, such that the three sensors span a surface (ﬁgure 3b), enables the three

sensors with a total of 9 DoFs to describe all 6 DoFs of the virtual point.

The additional beneﬁt of the third triaxial sensor is of course the overdetermination of the

interface problem. Using the virtual point transformation as discussed in section 2 the effects of

4342

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

(a) Two sensors spanning a line: one de-

pendency between the rotational axes.

(b) Three sensors spanning a surface: all

rotations fully determined.

uncorrelated measurement noise as well as errors due to misalignment of the sensor are reduced,

which is generally considered a good practice [6, 9, 16]. The use of at least three triaxial sensors

per virtual point can therefore be held as rule of thumb.

Finally, considering the rigid interface assumption, it seems evident that it would be best

practice to place the sensors as close as possible to the virtual point. The smaller the distances

are, the smaller the effects of ﬂexible interface modes are compared to the rigid interface motion.

On the other hand, for smaller distances, the virtual point transformation will get more sensitive

to to absolute errors on the position. In general one should approach this matter with some

engineering judgement or foreknowledge about the system.

3.2 Impact positions

Unlike the 6 displacements measured by 2 triaxial sensors, 6 well-positioned force impacts

could in practice be sufﬁcient to fully determine the 6 generalised forces at the virtual point.

Still, for the same reason, it is wise to choose more excitation points that overdetermine the

force transformation.

Similar to the sensor placement, three impacts in each direction (X, Y, Z)can be used as a

rule of thumb. Also one should include excitations of which the direction normal (eh) does not

point straight to the virtual point, in order to create “moment” along the rotational axes.

In some previous studies the sensor faces were chosen as possible impact locations [3, 16].

Depending on the chosen sensors this can have its advantages and disadvantages. Since the

impact locations and directions are equal to the locations and orientations of the measured XYZ

responses in the sensor, the same transformation matrix can be used for both transformations.

However, practice shows that FRFs obtained at the driving point exhibit poor coherence, espe-

cially for the cross-directional FRFs of one sensor. Also, the sensor is easily driven in overload.

As the virtual point transformation does not require physical driving-point receptance, this type

of excitations should be avoided.

It seems logical to apply the impacts as close as possible to the virtual point in order to avoid

inducing local deformation around the virtual point, but an the other hand one wants to be far

enough to be less sensitive to errors in the impact location and orientation. The same reasoning

as for the sensor placement holds here.

3.3 Post-processing

After measuring, the obtained FRF data should be organised into a matrix Y(ω)such that

the virtual point transformation can be performed:

H(ω) = TY(ω)TT

43

4343

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

The transformation matrices Tleft and right from the receptance matrix denote respectively the

sensor and impact transformation matrices as constructed from the IDMs6. In practice these

steps involve accurate determination of the positions and orientations (directions) of both the

sensors and impacts. Besides ruler and calliper measurement, optical measurement systems as

well as CAD models of the components at hand may dramatically improve the accuracy of the

position determination and with that the quality of the transformation. The use of such assisting

systems is therefore strongly suggested.

3.4 Performance indicators

In section 2 the relation between measured and virtual displacements and forces was set up.

The residuals that result from the least-square projection of the measured receptance data can be

used to derive a meaningful value to evaluate the accuracy of the sensor and impacts positions.

3.4.1 Sensor consistency

In order to say something about the how the consistency of the sensor placement and the

constructed IDMs, let us get back to the the interface reduction equation of (3):

u=Rq +µ

We deﬁne fhas the “load case” resulting from a unit impact somewhere at the structure, reason-

ably far away from virtual point 1. Then the measured responses of the sensors associated with

virtual point 1 due to the given load case are:

u1(fh) = Y1fh=Y1h

which is simply the receptance column corresponding to that impact. For substructure coupling,

we are interested in the ﬁltered responses ˜

u1due to the transformation via q1:

˜

u1(fh) = R11q1(fh) = R11 T11u1(fh) = P11 Y1h

Now, we are able to compare u1with ˜

u1using a formulation similar to the Modal Assurance

Criterion [1]:

ρ2

u1= MAC ˜

u1(fh),u1(fh)= MAC P11Y1h,Y1h(26)

with

MAC(a,b)∆

=(aHb)(bHa)

(aHa)(bHb)(27)

The value ρ2

u1is called the sensor consistency (in previous work rigidness [6, 16]) and gives a

value between 0and 1that indicates how well the sensors of virtual point 1 can describe the

dynamics (caused by a certain excitation) through the IDM modes. It is a frequency dependent

indicator that can be evaluated for every virtual point and for each excitation separately7. Evalu-

ating the sensor coherence for several impacts provides meaningful insights in the consequences

of the transformation. Low sensor consistency can have the following causes:

6In general the two Tmatrices are different and depend on the sensor and impact locations, see section 2.3.

7An excitation f2“far away” from the virtual point of interest is suggested because then one observes the

global response to some force that brought the complete system in motion, rather then a response that may be

highly affected by impact position differences in the vicinity of the virtual point of interest.

4344

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

•The position and orientation of a sensor was not properly determined, such that the mea-

sured kinematics appear incompatible with the IDM kinematics. If for instance one sensor

is rotated 90 degrees along the Z-axis, a rigid interface motion in the X, Y -plane will ap-

pear as a partly ﬂexible motion. The effect will be visible as a low consistency over the

full frequency range.

•The interface area spanned between the sensors is ﬂexible, such that the rigid IDMs can-

not fully describe the dynamics. This will mostly reﬂect in reduced consistency for in-

creasing frequencies.

•If the rigidness is particularly low for a certain impact, one may question if this impact

was either to soft (low signal-to-noise-ratio) or to strong (driving on or more sensors in

overload). In such cases it is preferred to leave this impact (column of Y) out of the

transformation.

Note that the sensor consistency requires an overdetermination of the problem, otherwise it will

just indicate a straight value of 1.

3.4.2 Impact consistency

Similar to the sensor consistency, it is possible to come up with a consistency measure for the

applied forces. This value is especially valuable to assess the accuracy of the impact positions

and by that the consistency of the force IDMs.

This time we are interested in the responses of a single sensor channel uito all the impacts

of virtual point 1, which is equal to a row of the measured receptance matrix:

ui(f1) = ui(f1), ui(f2), . . . , ui(f9)=Yi1

The response to the ﬁltered set of impacts, i.e. the combination of impacts that only result in a

rigid load, is found by applying the projection matrix for the force IDMs:

ui(˜

f1) = Yi1P11

The impact consistency is then deﬁned by:

ρ2

f1= MAC ui(˜

f1), ui(f1)= MAC Yi1P11,Yi1(28)

Also, this value is limited between 0 and 1.

3.4.3 Reciprocity

Evaluating sensor and hammer impact consistency can be useful to ﬁnd errors in the mea-

surement setup, calibration, etc. However a high consistency does not guarantee reciprocity of

virtual point FRFs. Reciprocity of the raw measurement data is often not meaningful as the

location of sensors and impact positions do not coincide. Reciprocity of the virtual point re-

ceptance however should be reciprocal, and can therefore be used to assess the transformation

quality.

Let iand jdenote two different DoFs from the set of virtual point DoFs. Then a non-

dimensional frequency dependent reciprocity value between 0 and 1 is deﬁned by:

χij =(Hij +Hji )(Hji

∗+Hji

∗)

2(Hij

∗Hij +Hji

∗Hji )(29)

434

5

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

This formula originates from the FRF input-output coherence function for the two measure-

ments Hji and Hij . It shows the similarity in both amplitude and phase between the two virtual

point FRFs.

4 SUMMARY

In this paper the theory behind the virtual point transformation is presented from a modal re-

duction perspective. We have shown that by weakening the interface problem, one only couples

the rigid motion described by the IDMs, while the residual motion is left free. By assigning a

certain weighting matrix, one can control this residual and with that the transformation. Ways

to construct different IDMs from the sensor and impact positions are discussed.

Also some practical aspects were discussed that one may encounter when doing experimental

substructuring. It appeared that sensor and impact positioning should be thought out carefully

beforehand, but can still be evaluated afterwards on the basis of a number of non-dimensional

performance indicators.

REFERENCES

[1] Randall J. Allemang. The modal assurance criterion twenty years of use and abuse. Sound

and Vibration, pages 14–21, August 2003.

[2] R.R.J. Craig and M.C.C. Bampton. Coupling of substructures using Component Mode

Synthesis. AIAA Journal, 6:1313–1319, 1968.

[3] D. de Klerk. Dynamic Response Characterization of Complex Systems through Op-

erational Identiﬁcation and Dynamic Substructuring. Phd thesis, Delft University

of Technology, http://www.3me.tudelft.nl/live/pagina.jsp?id=73932050-a4ca-46b5-bd9b-

d98b8a52dbe6, 2009.

[4] D. de Klerk, D.J. Rixen, and J. de Jong. The Frequency Based Substructuring Method

reformulated according to the Dual Domain Decomposition Method. In Proceedings of

the XXIV International Modal Analysis Conference. Society for Experimental Mechanics,

2006.

[5] D. de Klerk, D.J. Rixen, and S.N. Voormeeren. A General Framework for Dynamic Sub-

structuring. History, review and classifcation of techniques. AIAA Journal, 46(8), 2008.

[6] D. de Klerk, D.J. Rixen, S.N. Voormeeren, and F. Pasteuning. Solving the RDoF prob-

lem in Experimental Dynamic Substructuring. In Proceedings of the XXVI International

Modal Analysis Conference, Orlando, FL, Bethel, CT, 2008. Society for Experimental

Mechanics.

[7] D. de Klerk and S.N. Voormeeren. Uncertainty Propagation in Experimental Dynamic

Substructuring. In Proceedings of the XXVI International Modal Analysis Conference,

Orlando, FL, Bethel, CT, 2008. Society for Experimental Mechanics.

[8] M.L.M Duarte and D.J. Ewins. Some Insights into the Importance of Rotational Degrees

of Freedom and Residual Terms in Coupled Structure Analysis. In Proceedings of the

XIII International Modal Analysis Conference, Nashvill, TN, pages 164–170, Bethel, CT,

1995. Society for Experimental Mechanics.

4346

M.V. van der Seijs, D.D. van den Bosch, D.J. Rixen and D. de Klerk

[9] T.S. Edwards. Implementation of Admittance Test Techniques for High-precision Mea-

surement of Frequency Response Functions. In Proceedings of the XXXI International

Modal Analysis Conference. Society for Experimental Mechanics, 2013.

[10] J. Harvie and P. Avitabile. Effects of precise FRF measurements for Frequency Based

Substructuring. In Proceedings of the XXXI International Modal Analysis Conference,

Los Angeles, CA, Bethel, CT, 2013. Society for Experimental Mechanics.

[11] B. Jetmundsen, R.L. Bielawa, and W. G. Flannelly. Generalized frequency domain sub-

structure synthesis. Journal of the American Helicopter Society, 33(1):55–64, 1988.

[12] W. Liu and D.J. Ewins. The Importance Assessment of RDOF in FRF Coupling Analysis.

In Proceedings of the XVII International Modal Analysis Conference, Orlando, FL, pages

1481–1487, Bethel, CT, 1999. Society for Experimental Mechanics.

[13] J.C. O’Callahan, I.W. Lieu, and C.M. Chou. Determination of Rotational Degrees of Free-

dom for Moment Transfers in Structural Modiﬁcations. In Proceedings of the III Inter-

national Modal Analysis Conference, Orlando, FL, pages 465–470, Bethel, CT, January

1985. Society for Experimental Mechanics.

[14] F. Pasteuning. Experimental Dynamic Substructuring and its Application in Automotive

Research. Master’s thesis, TU Delft, Department of Engineering Mechanics, 2007.

[15] D.J. Rixen. How Measurement Inaccuracies Induce Spurious Peaks in Frequency Based

Substructuring. In Proceedings of the XXVI International Modal Analysis Conference,

Orlando, FL, Bethel, CT, 2008. Society for Experimental Mechanics.

[16] M.V. van der Seijs, D. de Klerk, D.J. Rixen, and S. Rahimi. Validation of current state

Frequency Based Substructuring Technology for the characterisation of Steering Gear –

Vehicle Interaction. In Proceedings of the XXXI International Modal Analysis Conference,

Los Angeles, CA, Bethel, CT, 2013. Society for Experimental Mechanics.

[17] S.N. Voormeeren. Improvement of Coupling Procedures and Quantiﬁcation of Uncer-

tainty in Experimental Dynamic Substructuring Analysis; Application and Validation in

Automotive Research. Master’s thesis, TU Delft, 2007.

4347