PAMM · Proc. Appl. Math. Mech. 17, 3 – 6 (2017) / DOI 10.1002/pamm.201710002
Experimental-Numerical Substructuring: a Comparison of Assemblies in
Primal and Dual Forms
Morteza Karamooz Mahdiabadi
, Yongle Qi
, and Daniel Jean Rixen
Chair of Applied Mechanics, Technical University of Munich, 85747 Garching, Germany
A hybrid dynamic substructuring method which combines experimentally measured and ﬁnite element (FE) modeled sub-
structures in primal and dual form is presented. On the one hand, the model of the FE substructure is reduced to decrease the
computational costs in the analysis of the hybrid model. On the other hand, the experimental substructure is measured while
an instrumented ﬁxture (known as Transmission Simulator) is assembled to its interface, providing the required interface dy-
namics for a high quality experimental model. The model of the transmission simulator is decoupled after measurement from
the experimental substructure. Both the coupling and decoupling procedures are performed once with interface displacement
(primal form) and once with interface forces (dual form). The hybrid substructuring methods are applied on an example and
a comparison between both methods are presented.
2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
The rapid development of technologies has leaded to the production of large and complex structural system, which cannot be
measured, modeled or analyzed as a whole. The dynamic substructuring methodologies offer a simple solution: divide the
whole structure into smaller substructures such that the substructures are small enough to be analyzed. After the individual
analysis of each substructure, they are coupled together by applying the boundary conditions to construct the whole structure.
Besides, substructuring provides the possibility of parallel computation and analysis of several substructures simultaneously
which makes the design procedure faster and more efﬁcient.
For many companies producing large and complex structures, one often encounters the case where some components are
produced by themselves, whereas many others are supplied by third parties who do not want to share all the information of
their production for conﬁdentiality reasons. However, the structural analysis of a dynamic system is crucial after assembly of
all its components to avoid failure in the operational performance. In order to facilitate fast analysis of such structures and
simultaneously keep the commercial details for all parties, hybrid substructuring of experimentally measured and numerically
simulated substructures is proposed [1, 3, 5]. In other words, hybrid substructuring allows annalists to experimentally identify
the structural properties of the substructures in which their numerical models are not available and combine them with other
substructures which are derived from FE models.
In order to obtain high quality modal properties of experimentally measured substructure, Allen et al.  proposed to couple a
ﬁxture (also known as transmission simulator) to the interface of the experimental substructure while measuring, and remove
the model of the ﬁxture from the measured data. Additionally, The transmission simulator (TS) supplies many information
at the interface of measured substructure such as mass loading, stiffening and damping due to the joints, which cannot be
achieved when the experimental substructures are measured in a free condition.
In this paper, hybrid substructuring of experimentally measured components and numerically modeled ﬁnite elements is
investigated. In the ﬁrst step the FE model of the numerical substructure is reduced using free and ﬁxed interface model
reduction approaches [2, 7]. The transmission simulator method is used to extract the high quality modal properties of the
experimental substructure including the interface dynamics of the substructure. The reduced substructures are coupled to
the experimental substructure with interface displacements (primal form) and interface forces (dual forms) . A veriﬁcation
example is ﬁnally employed to check the performance of the presented method and a comparison of the results is presented.
2 Overview of the Coupling Techniques
In the dynamic substructuring technique two boundary conditions (BC) are to be satisﬁed in the system:
1. Compatibility BC: The displacement of the connecting points at the substructures’ interfaces are equal. Deﬁning B as a
signed Boolean matrix which speciﬁes the connecting points at the interface, the compatibility BC can be written as
Bu = 0 (1)
2. Equilibrium BC: The reaction forces at the interface connecting points of substructures are equal with opposite direction.
The equation for the equilibrium BC can be written as
g = 0 (2)
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2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim