Direct decoupling of substructures using primal and dual formulation

Direct decoupling of substructures using primaland dual formulation

Walter D’Ambrogio and Annalisa Fregolent

Abstract The paper considers the decoupling problem, i.e. the identification of thedynamic behaviour of a structural subsystem, starting from the known dynamic be-haviour of the coupled system, and from information about the remaining part of thestructural system (residual subsystem). Substructure decoupling techniques can beclassified as inverse coupling techniques or direct decoupling techniques. In inversecoupling, the equations written for the coupling problem are rearranged to isolate(as unknown) one of the substructures instead of the coupled structure. Examplesof inverse coupling are impedance and mobility approaches. Direct decoupling con-sists in adding to the coupled system a fictitious subsystem which is the negativeof the residual subsystem. Starting from the 3-field formulation (dynamic balance,compatibility and equilibrium at the interface), the problem can be solved in a pri-mal or in a dual manner. Compatibility and equilibrium can be required either atcoupling DoFs only, or at additional internal DoFs of the residual subsystem. Fur-thermore DoFs used to enforce equilibrium might be not the same as DoFs usedfor compatibility: this generates the so called non collocated approach. In this pa-per, direct decoupling techniques are considered: primal and dual formulation arecompared in combination with collocated and non collocated interface.

1 Introduction

Dynamic substructuring consists in building the structural dynamic model of a com-plex system by assembling the dynamic models of simpler components (substruc-

Walter D’AmbrogioDipartimento di Ingegneria Meccanica, Energetica e Gestionale, Universita dell’Aquila, Via G.Gronchi, 18 - I-67100, L’Aquila (AQ), Italy, e-mail: [email protected]

Annalisa FregolentDipartimento di Ingegneria Meccanica e Aerospaziale, Universita di Roma La Sapienza, Via Eu-dossiana 18, I 00184 Rome, Italy e-mail: [email protected]

1

2 Walter D’Ambrogio and Annalisa Fregolent

tures or subsystems). This kind of problem is also known in the literature as cou-pling problem or subsystem addition. Many well established techniques exist, whenall substructures are modeled theoretically. However, a very important issue is thepossibility of combining different kind of models, some obtained from a theoret-ical or numerical analysis such as a Finite Element Model (FEM) and some oth-ers derived from experimental tests (Frequency Response Functions: FRFs). Whenthe model of at least one subsystem derives from experimental tests, the processis called experimental dynamic substructuring. The subject is particularly relevantin virtual prototyping of complex systems and responds to actual industrial needs.Due to modal truncation problems, in experimental dynamic substructuring, the useof FRFs (Frequency Based Substructuring) is preferred with respect to the use ofmodal parameters. The main algorithm for frequency based substructuring is theimproved impedance coupling [4] that involves just one matrix inversion with re-spect to the classical impedance coupling technique that requires three inversions.A general framework for dynamic substructuring is provided in [6, 5], where primaland dual formulation are introduced.

Sometimes the opposite need arises, namely how to extract a substructure modelfrom the assembled system. In this case one speaks of decoupling problem or sub-system subtraction. A trivial application of decoupling is mass cancellation, to getrid of the effect of the accelerometer mass on FRF measurements. Another applica-tion is joint identification. More generally, decoupling is a relevant issue for subsys-tems that cannot be measured separately, but only when coupled to their neighboringsubstructure(s) (e.g. a fixture needed for testing or subsystems that are very delicateor in operational conditions). To be more precise, the decoupling problem is definedas the identification of the dynamic behaviour of a structural subsystem, startingfrom the known dynamic behaviour of the assembled system, and from informationabout the remaining part of the structural system (residual subsystem).

Substructure decoupling techniques can be classified as inverse coupling tech-niques or direct decoupling techniques. In inverse coupling, the equations written forthe coupling problem are rearranged to isolate (as unknown) one of the substructuresinstead of the assembled structure. Examples of inverse coupling are impedance andmobility approaches [1, 7].

Direct decoupling consists in adding to the assembled system a fictitious sub-system, which is the negative of the residual subsystem. The technique starts fromthe 3-field formulation: one set of equations expressing the dynamic balance of theassembled system and, separately, of the fictitious subsystem; one set of equationsenforcing compatibility at interface DoFs, one set of equations enforcing equilib-rium of constraint forces at interface DoFs. To solve the problem, a primal approachor a dual approach can be used. Compatibility and equilibrium can be required ei-ther at coupling DoFs only (standard interface), or at additional internal DoFs ofthe residual subsystem (extended interface): as shown in [3] for the dual approach,the choice of interface DoFs determines a set of frequencies at which the decou-pling problem is ill conditioned. Apparently, when using an extended interface, theproblem is singular at all frequencies, although this singularity is easily removedby using standard smart inversion techniques. To circumvent this problem, in [8, 9]

Direct decoupling of substructures using primal and dual formulation 3

it is pointed out that DoFs used to enforce equilibrium need not to be the same asDoFs used to enforce compatibility: this gives rise to the so called non collocatedapproach, as opposite to the traditional approach in which such DoFs are the same,which is called collocated.

In this paper, direct decoupling techniques are considered. Specifically, primalformulation for decoupling is developed and compared with dual formulation: it isshown that both formulation provide the same result when the number of DoFs usedto enforce compatibility is equal to the number of DoFs used to enforce equilibrium,i.e. in the collocated approach and in some special case of non collocated approach.On the contrary, when the number of DoFs used to enforce compatibility is differentfrom the number of DoFs used to enforce equilibrium, the two approaches providedifferent results. The techniques are applied using simulated data from a torsionalsystem describing a two-speed transmission.

2 Direct decoupling techniques

The coupled structural system AB (NAB DoFs) is assumed to be made by an unknownsubsystem A (NA DoFs) and a residual subsystem B (NB DoFs) joined through anumber of couplings (see fig. 1). The residual subsystem (B) can be made by one ormore substructures. The degrees of freedom (DoFs) of the coupled system can bepartitioned into internal DoFs (not belonging to the couplings) of subsystem A (a),internal DoFs of subsystem B (b), and coupling DoFs (c).

COUPLED SYSTEM

INTERNAL

DOFS

UNKNOWN

SUBSYSTEM

COUPLING

DOFS

RESIDUAL

SUBSYSTEM

INTERNAL

DOFS

FRFS AT COUPLING DOFS +

FRFS AT SOME INTERNAL DOFS

→ NOISE + IDENTIFICATION ERRORS

PHYSICAL (FE) MODEL

A B

Fig. 1 Scheme of the decoupling problem

It is required to find the FRF of the unknown substructure A starting from theFRF of the coupled system AB. The subsystem A can be extracted from the cou-pled system AB by cancelling the dynamic effect of the residual subsystem B. This


can be accomplished by adding to the coupled system AB a fictitious subsystemwith a dynamic stiffness opposite to that of the residual subsystem B and satisfyingcompatibility and equilibrium conditions. The dynamic equilibrium of the coupledsystem AB and of the fictitious subsystem can be expressed in block diagonal formatas: [[

ZAB]

[0][0] −

[ZB

]]{{uAB

}{uB

} }=

{{f AB

}{f B} }

+

{{gAB

}{gB

} }(1)

where:

• [ZAB], [ZB] are the dynamic stiffness matrices of the coupled system AB and ofthe residual subsystem B, respectively;

• {uAB)}, {uB)} are the vectors of degrees of freedom of the coupled system ABand of the residual subsystem B, respectively;

• { f AB}, { f B} are the external force vectors on the coupled system AB and on thefictitious subsystem, respectively;

• {gAB}, {gB} are the vectors of connecting forces between the coupled system andthe fictitious subsystem, and viceversa (constraint forces associated with compat-ibility conditions).

According to this point of view, the interface between the coupled system ABand the fictitious subsystem should not only include all the coupling DoFs betweensubsystems A and B, but should as well include all the internal DoFs of subsystemB. However, by taking into account that the problem can be solved by consideringjust coupling DoFs, the number of interface DoFs should be greater than or equal tothe number of coupling DoFs nc. Therefore, three options for interface DoFs can beconsidered:

• standard interface, including only the coupling DoFs (c) between subsystems Aand B;

• extended interface, including also a subset of internal DoFs (i ⊆ b) of the residualsubstructure;

• mixed interface, including a subset of coupling DoFs (d ⊆ c) and a subset ofinternal DoFs (i ⊆ b) of the residual substructure.

The compatibility condition at the (standard, extended, mixed) interface DoFsimplies that any pair of matching DoFs uAB

l and uBm, i.e. DoF l on the coupled

system AB and DoF m on subsystem B must have the same displacement, that isuAB

l −uBm = 0. Let the number of interface DoFs on which compatibility is enforced

be denoted as NC.The compatibility condition can be generally expressed as:

[[BAB

C

] [BB

C

]]{{uAB

}{uB

} }= {0} (2)

where each row of [BC] =[[BAB

C ] [BBC]]

corresponds to a pair of matching DoFs. Notethat [BC] has size NC × (NAB +NB) and is, in most cases, a signed Boolean matrix.


Before deriving the equilibrium condition, it should be noted that the interfaceDoFs involved in the equilibrium condition are not necessarily the same used to en-force the compatibility condition, as long as controllability between equilibrium andcompatibility DoFs is ensured. If the compatibility and the equilibrium DoFs are notthe same, the approach is called non-collocated [8]. Note that a non-collocated ap-proach requires an extended or mixed interface and therefore it is only possible inthe decoupling problems (in coupling problems only standard interface can be de-fined). Obviously, the traditional approach, in which compatibility and equilibriumDoFs are the same, is called collocated.

Let NE denote the number of interface DoFs on which equilibrium is enforced.The equilibrium condition for constraint forces implies that, when the connect-ing forces are added for a pair of matching DoFs, their sum must be zero, i.e.gAB

r +gBs = 0: this holds for any pair of matching DoFs. Furthermore, if DoF k on

the coupled system AB (or to the residual subsystem B) does not belong to the equi-librium interface, it must be gAB

k = 0 (gBk = 0): this holds for any DoF not involved

in the equilibrium condition.Overall, the above conditions can be expressed as:[[

LABE][

LBE] ]T {{

gAB}{

gB} }

= {0} (3)

where the matrix [LE ] =[[LAB

E ] [LBE ]]

is a Boolean localisation matrix. Note that thenumber of columns of [LE ] is equal to the number NE of equilibrium interface DoFsplus the number NNE of DoFs not belonging to the equilibrium interface. Note thatNNE = NAB +NB − 2NE : in fact, the number of DoFs belonging to the equilibriuminterface must be subtracted once from NAB and once from NB. Therefore, the sizeof [LE ] is (NAB +NB)× (NAB +NB −NE).

Eqs. (1-3) can be put together to obtain the so-called 3-field formulation:

[[ZAB

][0]

[0] −[ZB

]]{{uAB

}{uB

} }=

{{f AB

}{f B} }

+

{{gAB

}{gB

} }(4)

[[BAB

C] [

BBC]]{{

uAB}{

uB} }

= {0} (5)

[[LAB

E][

LBE] ]T {{

gAB}{

gB} }

= {0} (6)


2.1 Primal formulation

In the primal formulation, a unique set of DoFs is defined:{{uAB

}{uB

} }=

[[LAB

C][

LBC

] ]{q} (7)

where {q} is the unique set of DoFs, and [LC] is a localisation matrix similar to [LE ]introduced previously. Note that [LC] is a (NAB +NB)× (NAB +NB −NC) matrix.Since there is a unique set of DoFs, {q}, the compatibility condition is satisfiedautomatically for any set {q}, i.e.

[[BAB

C] [

BBC]]{{

uAB}{

uB} }

=[[

BABC] [

BBC]][[LAB

C

][LB

C

] ]{q}= {0} ∀{q} (8)

Hence, [LC] is the nullspace of [BC] and, viceversa, [BC]T is the nullspace of [LC]

T :[[

BABC

] [BB

C

]][[LABC][

LBC

] ]= {0}[[LAB

C][

LBC

] ]T [[BAB

C

] [BB

C

]]T= {0}

(9)

Since the compatibility condition, Eq. (5), is satisfied (see Eq. (8)) by the choiceof the unique set {q}, the 3-field formulation reduces to:

[[ZAB

][0]

[0] −[ZB

]][[LABC

][LB

C] ]{q}=

{{f AB

}{f B} }

+

{{gAB

}{gB

} }[[

LABE][

LBE] ]T {{

gAB}{

gB} }

= {0}

(4∗)

(6)

Pre-multiplying the equation (4∗) by [LE ]T and noting that [LE ]

T{g} = {0}, theformulation reduces to:[[

LABE][

LBE] ]T [[ZAB

][0]

[0] −[ZB

]][[LABC

][LB

C

] ]{q}=

[[LAB

E][

LBE] ]T {{

f AB}{

f B} }

(10)

from which:

{q}=

[[LAB

E][

LBE] ]T [[ZAB

][0]

[0] −[ZB

]][[LABC

][LB

C] ]+[[

LABE][

LBE] ]T {{

f AB}{

f B} }

(11)

where the superscript + denotes the generalized inverse.


To obtain a determined or overdetermined matrix for the generalized inversionoperation, the following condition must be satisfied:

number of columns of [LE ]≥ number of columns of [LC]

i.e.(NAB +NB −NE)≥ (NAB +NB −NC) ⇒ NC ≥ NE ≥ nc (12)

where it is worth to recall that nc ia the number of coupling DoFs.Note that, if NC > NE , Eq. (10) is not satisfied exactly by vector {q} given by

Eq. (11), but only in the minimum square sense. This implies that also Eq. (6) is notsatisfied exactly, i.e. equilibrium conditions at interface are approximately satisfied.On the contrary, compatibility is satisfied exactly due to the unique choice of {q}.

From Eq. (11), the FRF of the unknown subsystem A can be written as:

[HA]=

[[LAB

E][

LBE] ]T [[ZAB

][0]

[0] −[ZB

]][[LABC

][LB

C] ]+[[

LABE][

LBE] ]T

(13)

[HA]= ([

LABE]T [

ZAB][LABC]−[LB

E]T [

ZB][LBC])+ [[

LABE]T [LB

E]T]

(14)

With the primal formulation, the columns of [HA] corresponding to the equilib-rium interface DoFs appear twice. Furthermore, when using an extended interface,[HA] contains some meaningless rows and columns: those corresponding to the in-ternal DoFs of the residual substructure B. Obviously, only meaningful and inde-pendent entries are retained.

2.2 Dual formulation

In the dual formulation, the total set of DoFs is retained, i.e. each interface DoFis present as many times as there are substructures connected through that DoF.The equilibrium condition g(r)l +g(s)m = 0 at a pair of equilibrium interface DoFs is

ensured by choosing, for instance, g(r)l =−λ and g(s)m = λ . Due to the constructionof [BE ], the overall interface equilibrium can be ensured by writing the connectingforces in the form: {{

gAB}{

gB} }

=−

[BABE]T[

BBE]T

{λ} (15)

where {λ} are Lagrange multipliers corresponding to connecting force intensities.The interface equilibrium condition (6) is thus written:

8 Walter D’Ambrogio and Annalisa Fregolent[[LAB

E][

LBE] ]T {{

gAB}{

gB} }

=−

[[LAB

E][

LBE] ]T

[BABE]T[

BBE]T

{λ}= {0} (16)

Then [BE ]T is the nullspace of [LE ]

T , and viceversa [LE ] is the nullspace of [BE ]:[[

BABE] [

BBE]][[LAB

E][

LBE] ]= {0}[[

LABE][

LBE] ]T [[

BABE] [

BBE]]T

= {0}(17)

Since Eq. (16) is always satisfied, the 3-field formulation reduces to:

[[ZAB

][0]

[0] −[ZB

]]{{uAB

}{uB

} }+

[BABE]T[

BBE]T

{λ}=

{{f AB

}{f B} }

[[BAB

C

] [BB

C

]]{{uAB

}{uB

} }= {0}

(4∗∗)

(5)

To eliminate {λ}, Eq. (4∗∗) can be written:

{{uAB

}{uB

}}=−

[[ZAB

][0]

[0] −[ZB

]]−1[BAB

E]T[

BBE]T

{λ}+

[[ZAB

][0]

[0] −[ZB

]]−1{{ f AB}{

f B}} (18)

which substituted in Eq. (5) gives:

[[BAB

C

] [BB

C

]][[ZAB]

[0][0] −

[ZB

]]−1[BAB

E]T[

BBE]T

{λ}=

=[[

BABC] [

BBC]][[ZAB

][0]

[0] −[ZB

]]−1{{f AB

}{f B} } (19)

from which {λ} is obtained:

{λ}=

[[BAB

C

] [BB

C

]][[ZAB]

[0][0] −

[ZB

]]−1[BAB

E]T[

BBE]T

+

×

×[[

BABC] [

BBC]][[ZAB

][0]

[0] −[ZB

]]−1{{f AB

}{f B} } (20)


To obtain a determined or overdetermined matrix for the generalized inversionoperation, the following condition must be satisfied:

number of rows of [BC]≥ number of rows of [BE ]

i.e.NC ≥ NE ≥ nc (21)

which is the same as (12).Note that, if NC > NE , Eq. (19) is not satisfied exactly by vector {λ} given by

Eq. (20), but only in the minimum square sense. This implies that also Eq. (5) is notsatisfied exactly, i.e. compatibility conditions at interface are approximately satis-fied. On the contrary, equilibrium is satisfied exactly due to the introduction of theconnecting force intensities {λ} as in Eq. (15).

Substituting {λ} in Eq. (4∗∗), it is obtained:

[[ZAB

][0]

[0] −[ZB

]]{{uAB

}{uB

} }+

[BABE]T[

BBE]T

×

×

[[BAB

C] [

BBC]][[ZAB

][0]

[0] −[ZB

]]−1[BAB

E]T[

BBE]T

+

×

×[[

BABC

] [BB

C

]][[ZAB]

[0][0] −

[ZB

]]−1{{f AB

}{f B} }

=

{{f AB

}{f B} }

(22)

Finally, {u} can be written in the form {u} = [H]{ f}, which provides the FRFof the unknown subsystem A:

{{uAB

}{uB

} }=

[[ZAB

][0]

[0] −[ZB

]]−1

−

[[ZAB

][0]

[0] −[ZB

]]−1[BAB

E]T[

BBE]T

×

×

[[BAB

C

] [BB

C

]][[ZAB]

[0][0] −

[ZB

]]−1[BAB

E]T[

BBE]T

+

×

×[[

BABC

] [BB

C

]][[ZAB]

[0][0] −

[ZB

]]−1{{f AB

}{f B} }

(23)

i.e., by noting that the inverted dynamic stiffness matrices [ZAB]−1 and [ZB]−1 areequal to the FRF matrices [HAB] and [HB] at the full set of DoFs:


[HA]=[[

HAB]

[0][0] −

[HB

]]−[[HAB

][0]

[0] −[HB

]][BABE]T[

BBE]T

×

×

[[BAB

C] [

BBC]][[HAB

][0]

[0] −[HB

]][BABE]T[

BBE]T

+

×

×[[

BABC

] [BB

C

]][[HAB]

[0][0] −

[HB

]](24)

With the dual formulation, the rows and the columns of [HA] corresponding toall the interface DoFs appear twice. Furthermore, when using an extended inter-face, [HA] contains some meaningless rows and columns: those corresponding tothe internal DoFs of the residual substructure B. Obviously, only meaningful andindependent entries are retained.

In Eq. (24), the product of the three matrices to be inverted can be defined asinterface flexibility matrix. The interface flexibility matrix can be rewritten as:

[[BAB

C] [

BBC]][[HAB

][0]

[0] −[HB

]][BABE]T[

BBE]T

=

=[BAB

C][

HAB][

BABE]T −

[BB

C][

HB][

BBE]T

(25)

It can be noticed that [BAB

C][

HAB][

BABE]T

=[HAB

]where [HAB] is a subset of the FRF matrix of the coupled structure: pre-multiplicationby [BAB

C ] extracts rows at compatibility DoFs, and post-multiplication by [BABE ] ex-

tracts columns at the equilibrium DoFs. Similarly,[BB

C

][HB

][BB

E]T

=[HB

]where [HB] is the FRF of the residual structure at the same DoFs as above.

Therefore, the interface flexibility matrix becomes:[BAB

C

][HAB

][BAB

E]T −

[BB

C

][HB

][BB

E]T

=[HAB

]−[HB

](26)

Note that, whenever compatibility and equilibrium DoFs are the same, [HAB] and[HB] can be seen as the inverse of the condensed dynamic stiffness matrices of thecoupled structure [ZAB] and the residual structure [ZB], respectively. In this case, theinterface flexibility matrix can be rewritten as:

Direct decoupling of substructures using primal and dual formulation 11[HAB]− [

HB]= [ZAB]−1 [

ZAB]([HAB]− [HB])[ZB][ZB]−1

=

=[ZAB]−1 ([

ZB]− [ZAB])[ZB]−1

=[HAB]([ZB]− [

ZAB])[HB] (27)

In the following, the influence of the choice of compatibility and equilibriumDoFs on singularity and ill-conditioning of ([HAB]− [HB]) will be analysed.

2.3 Singularity of the interface flexibility matrix (dual formulation)

As shown in [3], and recalled in the Appendix,[ZB

]and

[ZAB

]differ only in the up-

per left cc block, i.e. that relative to the coupling DoFs, and they can be convenientlywritten in block matrix form as:

[ZAB]=

[ZAB]

cc

[ZB

]ci[

ZB]

ic

[ZB

]ii

[ZB]=

[ZB]

cc

[ZB

]ci[

ZB]

ic

[ZB

]ii

(28)

where subscripts c and i represent coupling DoFs and internal DoFs, respectively.Therefore:

[ZB]− [

ZAB]= [[ZB

]cc −

[ZAB

]cc [0]ci

[0]ic [0]ii

](29)

Therefore, the interface flexibility matrix in Eq. (27) can be expanded as:

[HAB]

cc

[HAB

]ci[

HAB]

ic

[HAB

]ii

−

[HB]

cc

[HB

]ci[

HB]

ic

[HB

]ii

=

=

[HAB]

cc

[HAB

]ci[

HAB]

ic

[HAB

]ii

[ZB]

cc −[ZAB

]cc [0]ci

[0]ic [0]ii

[HB]

cc

[HB

]ci[

HB]

ic

[HB

]ii

(30)

Note that the interface flexibility matrix is expressed as a product of three matri-ces, and it is singular if just one of these matrices is singular. In fact, the determinantof a matrix product equals the product of the determinants.

2.3.1 Singularity using collocated extended interface

When using collocated approach with extended interface, the interface flexibilitymatrix can be singular for several reasons.

1. Because ([ZB]− [ZAB]) is singular at all frequencies: this is true if i is not anempty set as assumed for the extended interface. Note that the matrix is truly


singular only when all data are known without errors or noise. In any case, theuse of smart inversion techniques, e.g. the truncated SVD, allows to deal with theproblem.

2. Because, as shown in [3] and recalled in appendix, [HAB] is singular at the reso-nant frequencies of the coupled structure AB with interface DoFs grounded.

3. Because [HB] is singular at the resonant frequencies of the residual substructureB with interface DoFs grounded.Note that the resonant frequencies of the residual substructure B, with interfaceDoFs grounded are a subset of the resonant frequencies of the coupled structureAB with interface DoFs grounded. (This is also apparent from Fig. 2, showingthat the unknown subsystem A and the residual subsystem B are independent oneof another when interface DoFs are grounded.) Therefore, the interface flexibilitymatrix is twice singular at the resonant frequencies of the residual subsystem Bwith coupling DoFs grounded.It should be noted that, differently from what could be expected from previousstatements, the interface flexibility matrix is not singular at the resonant frequen-cies of the unknown subsystem A with interface DoFs grounded. In fact these arecancelled by the frequencies at which the determinant of [ZB]− [ZAB] = −[ZA]tends to infinity, that (see Appendix) are the resonant frequencies of the unknownsubsystem A with interface DoFs grounded.In cases 2 and 3, if noise is present as usual, the problem becomes ill conditionedbut smart inversion techniques are not able to remove ill conditioning.

INTERNAL

DOFS

UNKNOWN

SUBSYSTEM

COUPLING

DOFS

RESIDUAL

SUBSYSTEM

INTERNAL

DOFS

A B

Fig. 2 Structure with (extended) interface DoFs grounded

A particular case is that of standard interface where Eq. (30) reduces to:[HAB]

cc −[HB]

cc =[HAB]

cc

([ZB]

cc −[ZAB]

cc

)[HB]

cc (31)

and singularity occurs only, but twice, at the resonant frequencies of the residualsubstructure B with coupling DoFs grounded.


2.3.2 Singularity using non collocated extended interface

For the sake of simplicity, a special case of non collocated approach is consideredwhere compatibility is enforced at DoFs c and i and equilibrium only at DoFs c. Inthis case, Eq. (30) becomes:[HAB

]cc[

HAB]

ic

−

[HB]

cc[HB

]ic

=

[HAB]

cc[HAB

]ic

([ZB]

cc −[ZAB]

cc

)[HB]

cc

It can be noticed that [HB]cc is singular at resonances of the residual substructure Bwith coupling DoFs grounded. Therefore, the use of non collocated approach doesnot prevent from this kind of singularity.

3 Primal vs dual formulation

In this section, the expressions of the FRF of the unknown subsystem A providedby primal formulation and dual formulation are compared to establish whether theyare the same or not, and under which conditions. The FRF of subsystem A providedby the primal formulation, Eq. (13), can be rewritten in compact form as:

[HA]P =([LE ]

T [Z] [LC])+

[LE ]T (13)

where the subscript P stands for primal. If NC = NE , Eq. (13) becomes:

[HA]P =([LE ]

T [Z] [LC])−1

[LE ]T (32)

Premultiplying by [LE ]T [Z] [LC], one obtains:

[LE ]T [Z] [LC] [HA]P = [LE ]

T (33)

The FRF of the unknown subsystem A provided by dual formulation, Eq. (24),can be rewritten in compact form for NC = NE :

[HA]D = [H]− [H] [BE ]T([BC] [H] [BE ]

T)−1

[BC] [H] (34)

where the subscript D stands for dual. Premultiplying by [Z], one obtains:

[Z] [HA]D = [I]− [BE ]T([BC] [H] [BE ]

T)−1

[BC] [H] (35)

Premultiplying by LTE , one obtains:

[LE ]T [Z] [HA]D = [LE ]

T − [LE ]T [BE ]

T([BC] [H] [BE ]

T)−1

[BC] [H] (36)


from which, being [LE ]T [BE ]

T = 0 since BTE is the nullspace of LT

E (see Eq. 17):

[LE ]T [Z] [HA]D = [LE ]

T (37)

By comparing Eq. (37) and Eq. (33), it is found that:

[HA]D = [LC] [HA]P (38)

The former equation shows that, if NC = NE , primal and dual formulations pro-vide the same result. In fact, by considering the primal formulation, the relationbetween the unique set of DoFs {q} and the redundant set of DoFs {u} is given by{u}= [LC]{q}.

On the contrary, if NC > NE , Eq. (33) can not be obtained from Eq. (13) since:([LE ]

T [Z] [LC])(

[LE ]T [Z] [LC]

)+= [I]

due to the definition of pseudoinverse.In fact in case of overdetermined system, the pseudoinverse of a matrix [A] (num-

ber of rows > number of columns) is:

[A]+ =([A]T [A]

)−1[A]T

from which:

[A]+ [A] =([A]T [A]

)−1[A]T [A] = [I]

[A] [A]+ = [A]([A]T [A]

)−1[A]T = [I]

Therefore, if NC > NE , the primal and the dual formulation provide differentresults.

4 Application

A relatively simple application is considered on a torsional system that represents amodel of a two-speed transmission. The complete system consists in three shafts: aninput shaft, a layshaft and an output shaft (see Fig. 3). The layshaft is coupled bothto the input shaft and to the output shaft by helical gears. Power flows through thegear that is locked to the output shaft by the shift collar (e.g. through gears 5 and 7in Fig. 3). Input and output shafts are assumed to be fixed at the outer ends. Such aboundary condition is a good approximation whenever the mass moments of inertiaupstream and downstream the transmission are very large compared to those withinthe transmission.


Fig. 3 Sketch of the test system: a two speed transmission in first gear

Rotational inertias and torsional stiffnesses, as well as torsional dampings, areshown in Table 1 together with the number of teeth z of each gear.

Table 1 Physical parameter values and number of teeth

Item J [kg m2] c [Nm s/rad ] k [Nm/rad] z

1 3 ·10−3 8.25 ·10−4 82.5 –2 1.98 ·10−4 8.25 ·10−4 82.5 303 7.62 ·10−4 1.65 ·10−3 165 424 8.13 ·10−5 1.65 ·10−3 165 245 2.57 ·10−5 6.875 ·10−3 687.5 186 1.3 ·10−3 3.435 ·10−3 343.5 487 2.08 ·10−3 3.435 ·10−3 343.5 548 1 ·10−2 – – –

The rotations of meshing gear pairs are related by their gear ratio, e.g. the rota-tions θ2, θ6 and θ7 are related to rotations θ3, θ4 and θ5, respectively. The wholeset of rotations θ1 · · ·θ8 can be expressed through a reduced set of five independentrotations, as shown by Eq. (39):

θ1θ2θ3θ4θ5θ6θ7θ8

=

1 0 0 0 00 z3/z2 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 z4/z6 0 00 0 0 z5/z7 00 0 0 0 1

θ1θ3θ4θ5θ8

(39)

Therefore, the coupled system can be modelled as a 5 DoFs lumped parametersystem.


As outlined in Fig. 3, the unknown subsystem A is a 2-DoFs system made bythe output shaft to which gear 7 and flywheel 8 are locked. Note that gear 6 is notlocked to the output shaft so that it can be considered as belonging to the residualsubsystem B.

For the residual subsystem B, rotations θ B1 · · ·θ B

6 can be expressed through a re-duced set of four independent rotations, as shown by Eq. (40):

θ B1

θ B2

θ B3

θ B4

θ B5

θ B6

=

1 0 0 00 z3/z2 0 00 1 0 00 0 1 00 0 0 10 0 z4/z6 0

θ B1

θ B3

θ B4

θ B5

(40)

Therefore, subsystem B can be modelled as a 4 DoFs lumped parameter system.To have an idea of the dynamic behaviour of the torsional system, the natural

frequencies of the subsystems A and B, and of the coupled system AB are shown inTable 2.

Table 2 Natural frequencies of the systems [Hz]

ModeSystem 1 2 3 4 5

A 26.3881 72.2981 – – –B 22.7742 56.3484 120.4927 416.6248 –AB 23.3477 37.5007 59.7304 106.7501 184.4634

4.1 Decoupling

It is assumed that the rotational FRFs (mobilities) describing the angular veloc-ity/torque relationship of the coupled system AB, and the mechanical impedance ofthe residual subsystem B are known. It is desired to determine the rotational mobilityof subsystem A. The exact FRFs Hi j of the coupled system AB and the impedancesof subsystem B are computed starting from the physical parameters shown in Ta-ble 1. To simulate the effect of noise on the FRFs of the coupled system, a complexrandom perturbation is added to the true FRFs:

Hi j(ωk) = Hi j(ωk)+mi j,k + ini j,k (41)

where mi j,k and ni j,k are independent random variables with gaussian distribution,zero mean and a standard deviation of 0.1 rad/sNm. The effect of such perturbation


on the drive point rotational mobility at the coupling DoF is shown in Fig. 4 togetherwith the FRF obtained after curve-fitting.

Fig. 4 Drive point rotationalmobility of the completesystem at the coupling DoF:true (—), perturbed by noise(∗∗∗ ) and fitted (—)

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]0 20 40 60 80 100 120 140 160 180 200

−4

−2

0

2

4

Frequency [Hz]P

hase

[rad

]

In the sequel, only FRFs perturbed by simulated noise will be considered. Infact, if noise-free FRFs of the coupled system are used, the FRF of the unknownsubsystem is always predicted without errors (although the problem may be singularfor several reasons, as stated in section 2.3, the use of smart inversion techniquescompletely removes the singularity). Furthermore, perturbed FRFs are not used inraw form but are smoothed through a curve fitting procedure.

The rotational mobility of subsystem A can be determined by using the proceduredescribed in section 2. The compatibility condition is written generally as:

[BC]{u}=[BAB

C]{

uAB}+ [BB

C]{

uB}= {0}

where {uAB}T

=[θ1 θ3 θ4 θ5 θ8

] {uB}T

=[θ B

1 θ B3 θ B

4 θ B5]

and [BABC ] and [BB

C] built as detailed afterwards. In non collocated approach, thecompatibility and the equilibrium DoFs are not the same. Therefore, a localizationmatrix [LE ] = [LC], where [LC] is the nullspace of [BC], is defined to enforce equi-librium of constraint forces. A matrix [BE ] can be defined accordingly, such that[LE ]

T [BE ]T = 0.

4.1.1 Compatibility

Compatibility is alternatively enforced:

• only at the coupling DoFs (standard interface). In this case, it is:

[BC] =[θ1 θ3 θ4 θ5 θ8

0 0 0 1 0[BAB

C

] |θ B

1 θ B3 θ B

4 θ B5

0 0 0 −1[BB

C] ]


• also at some internal DoFs of the residual subsystem B (extended interface). Byadding all internal DoF, it is:

[BC] =

θ1 θ3 θ4 θ5 θ80 0 0 1 01 0 0 0 00 1 0 0 00 0 1 0 0

∣∣∣∣∣∣∣∣∣∣[BAB

C]

θ B1 θ B

3 θ B4 θ B

5

0 0 0 −1−1 0 0 00 −1 0 00 0 −1 0

[BB

C]

By adding only one internal DoF, it is:

– using θ1 as additional internal DoF:

[BC] =

[θ1 θ3 θ4 θ5 θ8

0 0 0 1 01 0 0 0 0[

BABC

]∣∣∣∣

θ B1 θ B

3 θ B4 θ B

5

0 0 0 −1−1 0 0 0[

BBC]

]


[BC] =

[θ1 θ3 θ4 θ5 θ8

0 0 0 1 00 1 0 0 0[

BABC

]∣∣∣∣θ B

1 θ B3 θ B

4 θ B5

0 0 0 −10 −1 0 0[

BBC]

]


[BC] =

[θ1 θ3 θ4 θ5 θ8

0 0 0 1 00 0 1 0 0[

BABC

]∣∣∣∣θ B

1 θ B3 θ B

4 θ B5

0 0 0 −10 0 −1 0[

BBC]

]

4.1.2 Equilibrium

In collocated approach, [BC] = [BE ] in all the above cases.Except for three cases whose results will be shown afterwards, in non collocated

approach equilibrium is enforced only at the coupling Dofs. Therefore:

[BE ] =[θ1 θ3 θ4 θ5 θ8

0 0 0 1 0[BAB

E] |

θ B1 θ B

3 θ B4 θ B

5

0 0 0 −1[BB

E] ]


4.2 Results

First of all, the case of standard interface (rotational mobility only at the couplingDoF θ5) is considered. In Fig. 5, the true drive point rotational mobility at the cou-pling DoF of subsystem A is compared with the corresponding FRF computed usingthe dual formulation, Eq. (24), starting from the fitted perturbed FRF of the coupledsystem. The same result is obtained using the primal formulation, Eq. (14).

Fig. 5 Rotational mobility atthe coupling DoF of subsys-tem A: true (—), computedfrom fitted perturbed FRF(∗∗∗ ) without additional inter-nal DoFs

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]M

agni

tude

[rad

s−

1 /(N

m)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]

As discussed in previous papers [2, 3], the predicted rotational mobility of sub-system A is badly identified at frequencies around 30, 70 and 150 Hz. This dependson ill conditioning due to unmeasured internal DoFs, as explained in section 2.3.In fact, the coupled system AB and the residual subsystem B, with the ”measured”coupling DoF θ5 grounded, share three resonance frequencies, namely fn1 = 28.94Hz, fn2 = 72.23 Hz and fn3 = 151.5 Hz. Around these frequencies, [HAB]− [HB] isill-conditioned and noise is greatly amplified.

A way to circumvent this problem is to use an extended interface, i.e. to assumethat the FRF matrix of the coupled system is known not only at the coupling DoFbut also at a subset of the three internal DoFs (θ1, θ3, θ4) of the residual subsystemB. The predicted rotational mobility of the unknown subsystem A, obtained usingcollocated approach with all the internal DoFs, is shown in Fig. 6: in this case, theresidual subsystem is fully grounded and no ill-conditioned frequencies appear. Thepredicted rotational mobility of the unknown subsystem A, obtained using non col-located approach with compatibility at all DoFs and equilibrium at the coupling DoFθ5, is shown in Fig. 7 for primal formulation and in Fig. 8 for dual formulation. Asdiscussed in section 2.3.2, ill conditioned frequencies are the same as for standardinterface but they affect the predicted FRFs to a lower extent if compared with thestandard interface.


Fig. 6 Rotational mobility atthe coupling DoF of subsys-tem A: true (—), computedfrom fitted perturbed FRF(∗∗∗ ) using collocated ap-proach with all the internalDoFs 0 20 40 60 80 100 120 140 160 180 200

10−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

Frequency [Hz]

Pha

se [r

ad]

Fig. 7 Rotational mobility atthe coupling DoF of subsys-tem A: true (—), computedfrom fitted perturbed FRF(∗∗∗ ) using primal formulationwith compatibility at all DoFsand equilibrium at the cou-pling DoF θ5

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]

Fig. 8 Rotational mobility atthe coupling DoF of subsys-tem A: true (—), computedfrom fitted perturbed FRF(∗∗∗ ) using dual formulationwith compatibility at all DoFsand equilibrium at the cou-pling DoF θ5

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


Since the full set of internal DoFs can be difficult to measure in practice, resultsobtained by adding only one internal DoF are considered. The resonant frequenciesof the residual subsystem B, with the measured DoFs grounded (the coupling DoFand one additional internal DoF), are shown in Table 3 versus the added internalDoF.

Table 3 Resonant frequencies of the residual subsystem B with extended interface DoFs grounded(the coupling DoF θ5 and one additional internal DoF)

Added DoF fn1[Hz] fn2[Hz]

θ1 69.4 151.5θ3 37.3 143.4θ4 31.5 87.15

The predicted rotational mobility of the unknown subsystem A obtained usingcollocated approach with the additional internal DoF θ1 is shown in Fig. 9: in thiscase, troubles arise at frequencies close to 70 and 150 Hz, as shown in Table 3,around which the inversion of an ill conditioned matrix is performed. The predictedrotational mobility of the unknown subsystem A obtained using non collocated ap-proach with compatibility at DoFs θ1 and θ5 and equilibrium at the coupling DoFθ5 is shown in Fig. 10 for primal formulation and in Fig. 11 for dual formulation. Illconditioned frequencies are the same as for standard interface.

Fig. 12 shows the predicted FRF of subsystem A using collocated approach withthe additional internal DoF θ3: in this case, troubles could arise at frequencies closeto 37 and 143 Hz as shown in Table 3, but they do not appear. Using non collocatedapproach with compatibility at DoFs θ3 and θ5 and equilibrium at the coupling DoFθ5, results are shown in Fig. 13 for primal formulation and in Fig. 14 for dual for-mulation. Again, ill conditioned frequencies are the same as for standard interface.

Fig. 15 shows the predicted FRF of subsystem A using collocated approach withthe additional internal DoF θ4: in this case, troubles could arise at frequencies closeto 31 and 87 Hz as shown in Table 3, but only very slight effects can be noticed. Us-ing non collocated approach with compatibility at DoFs θ4 and θ5 and equilibriumat the coupling DoF θ5, results are shown in Fig. 16 for primal formulation and inFig. 17 for dual formulation. Again, ill conditioned frequencies are the same as forstandard interface.

Generally, collocated approach seems to provide the best results, because in noncollocated approach, with equilibrium enforced only at coupling DoFs, ill condi-tioning occurs at the same frequencies that are troublesome with standard interface.


Fig. 9 Rotational mobility atthe coupling DoF of subsys-tem A: true (—), computedfrom fitted perturbed FRF(∗∗∗ ) using collocated ap-proach with internal DoF θ1

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]

Fig. 10 Rotational mobility atthe coupling DoF of subsys-tem A: true (—), computedfrom fitted perturbed FRF(∗∗∗ ) using primal formulationwith compatibility at DoFs θ1and θ5 and equilibrium at thecoupling DoF θ5

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]

Fig. 11 Rotational mobility atthe coupling DoF of subsys-tem A: true (—), computedfrom fitted perturbed FRF(∗∗∗ ) using dual formulationwith compatibility at DoFs θ1and θ5 and equilibrium at thecoupling DoF θ5

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


Fig. 12 Rotational mobil-ity at the coupling DoF ofsubsystem A: true (—), com-puted from fitted perturbedFRF (∗∗∗ ) using collocated ap-proach with internal DoF θ3

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

Frequency [Hz]

Pha

se [r

ad]


0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


Fig. 15 Rotational mobil-ity at the coupling DoF ofsubsystem A: true (—), com-puted from fitted perturbedFRF (∗∗∗ ) using collocated ap-proach with internal DoF θ4

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


In the sequel, non collocated approach is used with NC = NE . The coupling DoFθ5 is always included among both compatibility and equilibrium DoFs. As demon-strated in section 3, primal and dual formulation give the same results when thenumber of equilibrium DoFs is the same as the number of compatibility DoFs.

Fig. 18 shows the predicted rotational mobility of unknown subsystem A ob-tained using non collocated approach with compatibility at DoFs θ1 and θ5 andequilibrium at DoFs θ3 and θ5. Figure 19 is obtained by exchanging compatibilityand equilibrium DoFs. The second option seems to give better results.

Fig. 18 Rotational mobility atthe coupling DoF of subsys-tem A: true (—), computedfrom fitted perturbed FRF(∗∗∗ ) with compatibility atDoFs θ1 and θ5 and equilib-rium at DoFs θ3 and θ5

0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

Frequency [Hz]

Pha

se [r

ad]


Fig. 20 shows the predicted rotational mobility of unknown subsystem A ob-tained using non collocated approach with compatibility at DoFs θ1 and θ5 andequilibrium at DoFs θ4 and θ5. Figure 21 is obtained by exchanging compatibilityand equilibrium DoFs. The second option seems to give slightly better results.


0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]

Fig. 22 shows the predicted rotational mobility of unknown subsystem A ob-tained using non collocated approach with compatibility at DoFs θ3 and θ5 andequilibrium at DoFs θ4 and θ5. Figure 23 is obtained by exchanging compatibilityand equilibrium DoFs. The second option seems to give the best results.



0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]


0 20 40 60 80 100 120 140 160 180 20010

−5

100

105

Frequency [Hz]

Mag

nitu

de [r

ad s

−1 /(

N m

)]

0 20 40 60 80 100 120 140 160 180 200−4

−2

0

2

4

Frequency [Hz]

Pha

se [r

ad]

5 Summary and discussion

The paper considers the decoupling problem, i.e. the identification of the dynamicbehaviour of a structural subsystem, starting from the known dynamic behaviour ofthe assembled system, and from information about the residual subsystem, i.e. theremaining part of the structural system. Direct decoupling techniques are consid-ered, that consist in adding to the assembled system a fictitious subsystem, whichis the negative of the residual subsystem. Starting from the 3-field formulation (dy-namic balance, compatibility and equilibrium), a primal approach for decoupling isdeveloped and compared with a dual approach.

Compatibility and equilibrium can be required either at coupling DoFs only(standard interface), or at additional internal DoFs of the residual subsystem (ex-tended interface): for the dual approach, the choice of interface DoFs determines aset of frequencies at which the decoupling problem is ill conditioned. Apparently,when using an extended interface, the problem is singular at all frequencies, al-though this singularity is easily removed by smart inversion techniques. To avoidthis problem, the non collocated approach, in which DoFs used to enforce equilib-rium are not the same as DoFs used to enforce compatibility, can be considered: inthis case the number of DoFs used to enforce compatibility must be less than the


number of DoFs used to enforce equilibrium. However, it is shown that, when equi-librium is enforced at coupling DoFs only, the use of non collocated approach givesrise to an interface flexibility matrix that is ill conditioned at the same frequenciesas for standard interface.

Furthermore, it is shown that primal formulation and dual formulation providethe same result when the number of DoFs used to enforce compatibility is equal tothe number of DoFs used to enforce equilibrium, i.e. in the collocated approach andin some special case of non collocated approach. On the contrary, when the numberof DoFs used to enforce compatibility is greater than the number of DoFs used toenforce equilibrium, the two approaches provide different results: in fact, using theprimal approach, compatibility is enforced exactly whilst equilibrium of constraintforces is only approximate; the opposite occurs using the dual approach, i.e. com-patibility is only approximate whilst equilibrium of constraint forces is enforcedexactly.

The techniques are applied using simulated data from a torsional system describ-ing a two-speed transmission. Results agree with theory.

Acknowledgements This research is supported by grants from University of Rome La Sapienza.

Appendix: Format and properties of dynamic stiffness matrices

To obtain the condensed dynamic stiffness matrices of the coupled structure AB andof the residual substructure B, master DoFs M (interface DoFs to be measured andto be reatined after condensation) and slave DoFs S (unmeasured and to be elim-inated after condensation) must be defined. According to which option is selectedfor interface DoFs, master DoFs and slave DoFs are:

a) When standard interface is considered:

• M ≡ c, that is interface DoFs are just the coupling DoFs;• for the residual subsystem B, S ≡ b, that is the slave DoFs are all its internal

DoFs;• for the coupled system AB, S ≡ a∪b, that is the slave DoFs are all its internal

DoFs.

b) When extended interface is considered, also some internal DoFs i of the residualsubstructure are included among master DoFs:

• M ≡ c∪ i, as stated previously;• for the residual subsystem B, S ≡ b\i ≡ u, that is the slave DoFs are given by

the set difference between its internal DoFs and the internal DoFs included inthe interface;

• for the coupled system AB, S ≡ a∪u.

b1) A special case occurs when i ≡ b. In this case, u is an empty set.


The dynamic stiffness matrices of the residual subsystem B and of the coupledsystem AB can be partitioned as follows:

[ZB]= [[

ZB]

MM

[ZB

]MS[

ZB]

SM

[ZB

]SS

]=

[ZB

]cc

[ZB

]ci

[ZB

]cu[

ZB]

ic

[ZB

]ii

[ZB

]iu[

ZB]

uc

[ZB

]ui

[ZB

]uu

(42)

[ZAB]= [[

ZAB]

MM

[ZAB

]MS[

ZAB]

SM

[ZAB

]SS

]=

[ZA

]cc +

[ZB

]cc

[ZB

]ci

[ZB

]cu

[ZA

]ca[

ZB]

ic

[ZB

]ii

[ZB

]iu [0]ia[

ZB]

uc

[ZB

]ui

[ZB

]uu [0]ua[

ZA]

ac [0]ai [0]au[ZA

]aa

(43)

where the zero blocks arise from the fact that the unknown subsystem A is joinedto the residual subsystem B only through the coupling DoFs c. Note that, whenstandard interface is considered, i is an empty set, and u ≡ b.

Using dynamic condensation, the condensed dynamic stiffness matrix of theresidual subsystem B is:

[ZB]= [[

ZB]

cc

[ZB

]ci[

ZB]

ic

[ZB

]ii

]−

[[ZB

]cu[

ZB]

iu

][ZB]−1

uu

[[ZB

]uc

[ZB

]ui

]=

=

[ZB]

cc

[ZB

]ci[

ZB]

ic

[ZB

]ii

−

[ZB]

cu

[ZB

]−1uu

[ZB

]uc

[ZB

]cu

[ZB

]−1uu

[ZB

]ui[

ZB]

iu

[ZB

]−1uu

[ZB

]uc

[ZB

]iu

[ZB

]−1uu

[ZB

]ui

(44)

and the condensed dynamic stiffness matrix of the coupled subsystem AB is:

[ZAB]= [[

ZA]

cc +[ZB

]cc

[ZB

]ci[

ZB]

ic

[ZB

]ii

]−

−

[[ZB

]cu

[ZA

]ca[

ZB]

iu [0]ia

][[ZB

]uu [0]ua

[0]au[ZA

]aa

]−1[[ZB]

uc

[ZB

]ui[

ZA]

ac [0]ai

]=

=

[[ZA

]cc +

[ZB

]cc

[ZB

]ci[

ZB]

ic

[ZB

]ii

]−

−

[ZB]

cu

[ZB

]−1uu

[ZB

]uc +

[ZA

]ca

[ZA

]−1aa

[ZA

]ac

[ZB

]cu

[ZB

]−1uu

[ZB

]ui[

ZB]

iu

[ZB

]−1uu

[ZB

]uc

[ZB

]iu

[ZB

]−1uu

[ZB

]ui

(45)


It can be observed that[ZB

]and

[ZAB

]differ in only the upper left cc block, i.e.

that relative to the coupling DoFs, and they are conveniently written as:

[ZAB]=

[ZAB]

cc

[ZB

]ci[

ZB]

ic

[ZB

]ii

[ZB]=

[ZB]

cc

[ZB

]ci[

ZB]

ic

[ZB

]ii

(28)

Note that, when standard interface is considered, only the upper left cc block existsbecause i is an empty set.

By looking at Eq. (44), it can be noticed that [ZB]uu must be inverted. [ZB]uu is thedynamic stiffness matrix of the residual subsystem B with master (interface) DoFsgrounded, and it is singular at its own resonant frequencies. Therefore, det([ZB])tends to infinity at the resonant frequencies of [ZB]uu: at those frequencies, det([HB])tends to zero and [HB] is singular. Similarly, det([ZAB]) tends to infinity at the reso-nant frequencies of the coupled structure AB with master (interface) DoFs grounded:at those frequencies, [HAB] is singular. By looking at Eq. (45), where the matrix tobe inverted is a block diagonal matrix including [ZB]uu and [ZA]aa, it can be noticedthat the resonant frequencies of the residual substructure B, with interface DoFsgrounded [ZB]uu, are a subset of the resonant frequencies of the coupled structureAB with interface DoFs grounded.

References

1. D’Ambrogio, W., Fregolent, A.: Promises and pitfalls of decoupling procedures. In: Proceedingof 26th IMAC. Orlando (U.S.A.) (2008)

2. D’Ambrogio, W., Fregolent, A.: Decoupling procedures in the general framework of frequencybased substructuring. In: Proceedings of 27th IMAC. Orlando (U.S.A.) (2009)

3. D’Ambrogio, W., Fregolent, A.: The role of interface dofs in decoupling of substructures basedon the dual domain decomposition. Mechanical Systems and Signal Processing 24(7), 2035–2048 (2010). Doi:10.1016/j.ymssp.2010.05.007, also in Proceedings of ISMA 2010, pp. 1863-1880, Leuven (Belgium)

4. Jetmundsen, B., Bielawa, R., Flannelly, W.: Generalised frequency domain substructure syn-thesis. Journal of the American Helicopter Society 33(1), 55–64 (1988)

5. de Klerk, D.: Dynamic response characterization of complex systems through operational iden-tification and dynamic substructuring. Ph.D. thesis, TU Delft (2009)

6. de Klerk, D., Rixen, D.J., Voormeeren, S.: General framework for dynamic substructuring:History, review, and classification of techniques. AIAA Journal 46(5), 1169–1181 (2008)

7. Sjovall, P., Abrahamsson, T.: Substructure system identification from coupled system test data.Mechanical Systems and Signal Processing 22(1), 15–33 (2008)

8. Vormeeren, S.N., Rixen, D.J.: A dual approach to substructure decoupling techniques. In:Proceeding of 28th IMAC. Jacksonville (U.S.A.) (2010)

9. Vormeeren, S.N., Rixen, D.J.: A family of substructure decoupling techniques based on a dualassembly approach. In: P. Sas, B. Bergen (eds.) Proceedings of ISMA 2010 - InternationalConference on Noise and Vibration Engineering, pp. 1955–1968. Leuven (Belgium) (2010)

Direct decoupling of substructures using primal and dual formulation

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