Substructuring in Engineering Dynamics

Matt Allen Daniel Rixen Maarten van der Seijs Paolo Tiso Thomas Abrahamsson Randy Mayes

– Emerging Numerical and Experimental Techniques –

July 2018

To be Published by Springer in 2019

Springer Preface

One fundamental paradigm in engineering is to break a structure into simpler components in order to sim- plify test and analysis. In the numerical world, this concept is the basis for Finite Element discretization and is also used in model reduction through substructuring. In experimental dynamics, substructuring approaches such as Transfer Path Analysis (TPA) are commonly used, although the subtleties involved are perhaps not always adequately appreciated. Recently, there has been renewed interest in using measurements alone to create dynamic models for certain components and then assembling them with numerical models to predict the behavior of an assembly. Substructured models are also highly versatile; when only one component is modified, it can be readily assembled with the unchanged parts to predict the global dynamical behavior. Substructuring concepts are critical to engineering practice in many disciplines, and they hold the potential to solve pressing problems in testing and modeling structures where nonlinearities cannot be neglected. This book originates from lecture notes created for a short course in Udine, Italy in July 2018. We will review a general framework which can be used to describe a multitude of methods and the fundamental concepts underlying substructuring. The course was aimed at explaining the main concepts as well as specific techniques needed to successfully apply substructuring both numerically (i.e. using finite element models) and experimentally. The course centered around the following topics, which range from classical substructuring methods to topics of current research such as substructuring for nonlinear systems. 1. Introduction and motivation 2. Primal and dual assembly of structures 3. Model reduction and substructuring for linear systems including Guyan and Hurty/Craig-Bamtpon reduction, McNeal, Rubin, Craig-Chang, interface reduction methods and model reduction in the state space. 4. Experimental-Analytical substructuring including frequency based substructuring or impedance coupling, substructure decoupling methods including the transmission simulator method, measurement methods for substructuring, the virtual point transformation, state space substructuring and an overview of finite element model updating (a common alternative to experimental substructuring). 5. Industrial applications and related concepts including Transfer Path Analysis and finite element model updating. 6. Model reduction and substructuring methods for non-linear systems are highlighted with a focus on geometrically non-linear structures and nonlinearities due to bolted interfaces. This text was designed to provide practicing engineers or researchers such as PhD students with a firm grasp of the fundamentals as well as a thorough review of current research in emerging areas. The reader is expected to have a solid foundation in structural dynamics and some exposure to finite element analysis. The material will be of interest to those who primarily perform finite element simulations of dynamic structures, to those who primarily focus on modal test, and to those who work at the interface between test and analysis.

Udine, Matt Allen July 2018 Daniel Rixen Maarten van der Seijs Paolo Tiso Thomas Abrahamsson Randall Mayes

v Contents

Preface ...... v

Contents ...... vii

1 Introduction and Motivation ...... 1 1.1 Divide and conquer approaches in the history of engineering mechanics ...... 1 1.2 Advantages of substructuring in mechanical engineering ...... 2

2 Preliminaries: primal and dual assembly of dynamic models ...... 5 2.1 The dynamics of a substructure: domains of representation ...... 5 2.1.1 Spatial descriptions ...... 6 2.1.2 Spectral representation ...... 8 2.1.3 State representation ...... 9 2.1.4 Summary of representation domains ...... 10 2.2 Interface conditions for coupled substructures ...... 10 2.2.1 Interface equilibrium ...... 11 2.2.2 Interface compatibility ...... 13 2.3 Primal and dual assembly ...... 15 2.3.1 Primal assembly ...... 15 2.3.2 Dual assembly...... 17 2.3.3 Usefulness of different assembly formulations ...... 18

3 Model Reduction Concepts and Substructuring Approaches for Linear Systems ...... 21 3.1 Model reduction – general concepts (reduced basis) ...... 21 3.1.1 Reduction by projection ...... 21 3.1.2 The Guyan-Irons method ...... 22 3.1.3 Model reduction through substructuring ...... 24 3.2 Numerical techniques for model reduction of substructures ...... 24 3.2.1 The Hurty/Craig-Bampton Method ...... 24 3.2.2 Substructure reduction using free interface modes ...... 27 3.2.3 Numerical examples of different substructure reduction techniques ...... 32 3.2.4 Other reduction techniques for substructures ...... 35 3.3 Interface Reduction with the Hurty/Craig-Bampton method: Characteristic Constraint Modes ...... 35 3.3.1 System-Level Characteristic Constraint (S-CC) Modes ...... 36 3.3.2 Local-Level Characteristic Constraint Modes ...... 37 3.4 State-space model reduction...... 39 3.4.1 Structural dynamics equations in state-space ...... 43 3.4.2 State observability and state controllability ...... 44 3.4.3 State-space realizations ...... 48 3.4.4 State-space reduction based on transfer strength...... 52

vii viii Contents

4 Experimental Substructuring ...... 57 4.1 Introduction ...... 57 4.2 Why is experimental substructuring so difﬁcult? ...... 58 4.3 Basics of Frequency Based Substructuring (FBS) ...... 60 4.3.1 Lagrange Multiplier FBS - the Dual Interface Problem ...... 61 4.3.2 Lagrange Multiplier FBS - the dually assembled admittance ...... 63 4.3.3 On the usefulness of dual assembly of admittance in experimental substructuring . . . 65 4.3.4 Weakening the interface compatibility ...... 67 4.3.5 Dual assembly in the modal domain...... 69 4.3.6 A special case: substructures coupled through bushings ...... 71 4.4 Substructure decoupling ...... 76 4.4.1 Theory of decoupling ...... 76 4.4.2 Inverse substructuring ...... 80 4.4.3 The Transmission Simulator Method for Substructuring and Substructure Decoupling ...... 83 4.5 Measurement methods for substructuring ...... 91 4.6 Virtual Point Transformation ...... 97 4.6.1 Interface modelling ...... 97 4.6.2 Virtual point transformation ...... 99 4.6.3 Interface displacement modes ...... 104 4.6.4 Measurement quality indicators ...... 106 4.6.5 Instrumentation in practice ...... 110 4.7 Real applications ...... 111 4.7.1 Experimental modelling of a substructure using the virtual point transformation . . . . 111 4.7.2 Experimental dynamic substructuring of two substructures ...... 113 4.8 Finite element model updating for substructuring ...... 114 4.8.1 Finite element model parameter statistics and cross-validation ...... 115 4.8.2 Finite element model calibration ...... 119 4.9 Substructuring with state-space models ...... 125 4.9.1 State-space system identiﬁcation ...... 126 4.9.2 Physically consistent state-space models ...... 127 4.9.3 State-space realization on substructuring form ...... 130 4.9.4 State-space model coupling ...... 131

5 Industrial Applications Related Concepts ...... 133 5.1 Introduction to Transfer Path Analysis ...... 133 5.2 The transfer path problem ...... 134 5.2.1 Transfer path from assembled admittance ...... 135 5.2.2 Transfer path from subsystem admittance ...... 135 5.3 Classical TPA...... 136 5.3.1 Classical TPA: direct force ...... 137 5.3.2 Classical TPA: mount stiffness ...... 137 5.3.3 Classical TPA: matrix inverse ...... 138 5.4 Component-based TPA ...... 139 5.4.1 The equivalent source concept ...... 140 5.4.2 Component TPA: blocked force ...... 140 5.4.3 Component TPA: free velocity ...... 141 5.4.4 Component TPA: hybrid interface ...... 141 5.4.5 Component TPA: in-situ ...... 142 5.4.6 Component TPA: pseudo-forces ...... 143 5.5 Transmissibility-based TPA ...... 145 5.5.1 The transmissibility concept ...... 146 5.6 Substructuring to estimate ﬁxed-interface modes ...... 149 5.6.1 Modal Constraints ...... 149 5.6.2 Singular Vector Constraints ...... 150 Contents ix

5.7 Payload and Component Simulations using Effective Mass and Modal Craig Bampton Form Modal Models ...... 150 5.8 Calibration of an SUV rear subframe ...... 151 5.9 Coupling of two major SUV components ...... 159

6 Model Reduction Concepts and Substructuring Approaches for Non-Linear systems...... 165 6.1 Geometric Nonlinearities ...... 165 6.1.1 2D Beam ...... 165 6.1.2 Von-Karman Model ...... 167 6.2 Finite Element Discretization ...... 169 6.3 Galerkin Projection ...... 170 6.4 Nonlinear Normal Modes ...... 172 6.4.1 Periodic Motions of an Undamped System: Nonlinear Normal Modes ...... 172 6.5 Non-intrusive Reduced Order Modeling (ROM) for geometrically nonlinear structures: ICE&ED ...... 175 6.5.1 Enforced Displacements Procedure ...... 176 6.5.2 Applied Loads Procedure or Implicit Condensation and Expansion ...... 176 6.6 Intrusive Methods ...... 177 6.6.1 Modal Derivatives ...... 177 6.6.2 Wilson Vectors ...... 179 6.7 Data driven methods ...... 181 6.7.1 Proper Orthogonal Decomposition ...... 181 6.8 Hyper-reduction ...... 182 6.8.1 Discrete Empirical Interpolation and variants ...... 182 6.8.2 Energy Conserving Sampling and Weighting ...... 186 6.9 Hurty/Craig-Bampton substructuring with ROMs and interface reduction...... 187 6.10 Modal Derivatives Enhanced Craig Bampton Method with Interface Reduction ...... 189 6.11 Substructuring Methods for Flexible Multibody Systems ...... 190

7 Weakly Nonlinear Systems: Modeling and Experimental Methods ...... 191 7.1 Modal models for weakly nonlinear substructures and application to bolted interfaces . . . . . 191 7.2 Test Methods for Identifying Nonlinear Parameters in Pseudo-Modal Models ...... 192

References ...... 197 Chapter 1 Introduction and Motivation

Abstract "Divide and Conquer" is a paradigm that helped Julius Cesar to dominate on the wide roman Empire. The power of dividing systems, then analyze them as parts before combining them in an assembly, is also an approach often followed in science and engineering. In this introductory chapter, we shortly discuss the main idea behind domain decomposition and substructuring applied to mechanical systems. — Chapter Author: Daniel Rixen

1.1 Divide and conquer approaches in the history of engineering mechanics

The work of engineers consists in understanding systems in order to build new solutions or optimize existing ones. To analyze systems, it is very common and efficient to decompose them into subcomponents. This reduces the complexity of the overall problem by considering first its parts and can provide very useful insight for optimizing or troubleshooting intricate structures. In this introductory chapter, the general idea of substructuring is discussed and an overview of the topics treated in these lecture notes is given. One often refers to Schwarz [254] who proved the existence and uniqueness of the solution of a Poisson problem on a domain that was a combination of a square and a circle. By considering the problem as two separate problems on simple overlapping square and circular domains for which existence and uniqueness of the solution was already known, and devising an iteration strategy between the two domains with guaranteed convergence, he could conclude his very important proof. The algorithm that he devised as a means for his proof is still used today, underlying some very popular solution strategies on multiprocessor computers (see for instance [292]). But in essence, approximation techniques such as the Rayleigh-Ritz method [82, 225, 229], where the solution space is approximated by a reduced number of admissible functions, can also be considered as divide an conquer methods: the solution space is decomposed into functional subspaces in which an approximate solution can be found. The Finite Element technique, pioneered by mathematicians like Courant [44, 45], can in fact be seen as a clever application of the Rayleigh-Ritz technique on small region of the computational domain, resulting therefore in not only a function decomposition of the problem within the element, but also a spatial decomposition of the problem as the physical domain is broken into elements. Today when we consider substructuring, most often we envison dividing a finite element model into subcomponents that each contain many elements. This substructuring approach was developed in the sixties, motivated by the need to solve structural problems in aerospace and aeronautics of several tens of nodes (considered to be very large problems at that time, given the fact that computers were just arising). Hence, the idea was to add another level of decomposition and reduction by subdividing the mesh of a finite element model into substructures and to represent the dynamics per substructure in a reduced and approximate way. Some of the substructuring techniques still commonly used today originate from that time where memory and CPU power were very limited. Later, in the eighties and especially in the nineties, solving very large problems using domain decomposition became a very active research topic. In order to efficiently use the computational power of new multi-processor machines, it was essential to minimize the communications between processing units and to ensure that each CPU was fed with enough work that could be done independently. Domain Decom- position is the paradigm to achieve this, by letting each CPU solve the local problems of each domain

1 2 1 Introduction and Motivation and searching for the global solution by iterating on the interface solutions [292]). Nowadays, high performance computing techniques solve engineering problems of close to a billion degrees of freedom on several hundreds of thousands of processors thanks to that paradigm [290]. The last wave of intensive research in substructuring came at the beginning of the 21st century, this time not directly powered by exponential grows of computing power, but by signiﬁcant progress in techniques to accurately measure the dynamics of components. Experimental substructuring aims to build models of assemblies from the data measured on components, possibly in combination with numerical models. Al- though the theory of experimental dynamic substructuring is essentially the same as its analytical counterpart, many novel techniques needed to be developed in order to alleviate the adverse effects of measurement errors (even if strongly reduced thanks to novel sensing and acquisition techniques) when building a model based on experimentally characterized components.

1.2 Advantages of substructuring in mechanical engineering

In mechanical engineering practice, substructuring techniques are useful for the following cases: When working on a large project (for example the design of an aircraft), different groups and departments • work on different components. In that case, it is essential that each team can concentrate on the modeling of their part, and then have an efficient means for assembling to other components in order to predict the overall dynamics and optimize their design accordingly. In industry, it is common that models of subsystems must be shared. A classical example is the coupled • load analysis of the assembly of a satellite and a launcher: in that case, the satellite company needs to share its model with the launcher operator in order to perform a global dynamic analysis and guarantee that vibration limits will not be crossed. In that case, exchanging information can be efficient if the different components have been modeled separately in such a way that their complexity (number of degrees of freedom) is low and such that they can be assembled on their interface. In many engineering problems, parts of the system to be analyzed are significantly more complex than • the rest. In structural dynamics for instance, parts of the model could be highly non-linear whereas the remainder can be considered as linear. In that case, it is very advantageous to substructure the problem in such a way that only the parts that are non-linear are treated with the appropriate techniques, whereas the linear parts are for instance condensed once and for all. Such a subdivision of the problem into substructures of different complexity can greatly improve the analysis speed and give better insight on the dynamic behavior. When optimizing the design during the development of a new product, or during an improvement of an • old design, one often wants to modify only a small number of parts (for instance change the bushings connecting the power-train of a car to the chassis in order to modify the noise and vibration harshness). When the parts that will not be modified are available as substructures and already precomputed (e.g. their dynamics on the interface is known), then recomputing the global behavior when only the components under optimization are changed requires a significantly reduced cost, enabling much faster design cycles. In experimental testing, it is not always possible to test the full system; often system level analysis is • needed before all of the components have been manufactured. In other cases it is impossible to test the full assembly because appropriately exciting a large structure would require forces that are beyond the capabilities of existing shakers. It is then very advantageous to experimentally characterize the structure part by part. This allows troubleshooting problems arising from the local dynamics in those parts, but also to build a full model by assembling the measured parts using experimental substructuring techniques. When parts of the system have not yet been built, it is possible to combine the measured dynamics of • hardware components with other parts that are modeled only numerically (hybrid substructuring). In some cases, the combination in real-time of the transient response of a hardware component (the physical substructure) with the dynamics of a numerical substructure is essential to predict the dynamics of the hardware part in realistic conditions. Such hardware-in-the-loop tests (also called Real-Time Sub- structuring, Hybrid Testing or Cyber-Physics) allow testing very complex systems by having only a part in the lab, the rest of the system being co-simulated in real time. Substructuring methods open new opportunities for efficient analyses and design, including the develop- • ment of digital twins, or a model that can be used to monitor the heath of a system. 1.2 Advantages of substructuring in mechanical engineering 3

These are only a few examples of cases in which substructuring can be advantageous. For these and other reasons, substructuring has attracted a lot of attention over the years and is yet today a very active research ﬁeld. Chapter 2 Preliminaries: primal and dual assembly of dynamic models

Abstract There are several ways to formulate the dynamics of a substructure. The different domains in which the dynamics can be described will be reviewed since the manner in which substructures are characterized will later determine the substructuring methodology that can be applied. In addition to how the substructures are formulated, the way in which the coupling/decoupling problem is expressed will allow us in the subsequent chapters to develop different numerical and experimental techniques1. Two conditions must be satisfied on the interface between substructures: a condition on the displacement field (compatibility) and on the interface stresses (force equilibrium). Those conditions can be accounted for following several different formulations, all mathematically equivalent, but each leading to different numerical methods, experimental approaches and approximation techniques, as will be explained in the following chapters. In this chapter we outline the basic concepts of the so-called three field formulation, dual and primal assembly. — Chapter Author: Daniel Rixen

2.1 The dynamics of a substructure: domains of representation

In these lecture notes, we assume that the problem has already been discretized (using for instance an appropriate Finite Element or Boundary Element formulation). The discretized problem describing the dynamics in a substructure Ω (s) can be written in the form

(s) (s) (s) (s) (s) (s) (s) M u¨ + fnl u˙ ,u = f + g (2.1) The superscript (s) indicates that the equation is written for a given substructure s. M denotes the discretized 2 mass matrix , fnl the discretized non-linear internal force function, which depends in general on the nodal velocities and displacements u˙ and u. Two types of forces are applied on the substructures: the externally applied forces, denoted f, and the forces due to interactions between substructures, denoted g. The latter are in fact internal forces when considering the entire structure, but are considered as applied forces when analyzing the substructure problem. In case of small displacements, the internal forces fnl can be linearized and the substructure dynamics are described by

M(s)u¨ (s) + C(s)u˙ (s) + K(s)u(s) = f(s) + g(s) (2.2)

(s) (s) (s) (s) (s) (s) where C = ∂fnl /∂u˙ and K = ∂fnl /∂u are the linearized (tangential) damping and stiffness matrices typically computed around equilibrium positions of the system [82]. The unknowns u in equations (2.1,2.2) express the behavior of a substructure in terms of its physical displacements (discretized at nodes) in the time domain. Therefore, we will refer to them as the equations in the physical and time domain. The ﬁrst qualiﬁcation refers to the spatial meaning of the unknowns whereas

1 A similar classiﬁcation can be found in a different form in [146]. 2 Here we assume that the mass matrix is constant. This is not the case for models in multibody dynamics for instance [80], but this will not be considered here.

5 6 2 Preliminaries: primal and dual assembly of dynamic models the second refers to the spectral description of the problem. Other spatial and spectral description of the dynamic problem will now be introduced since the way in which the problem is described has an impact on what method can be applied for reduction or experimental assembly, as will be shown throughout these lecture notes. A summary of the different domains in which the substructure dynamics can be considered is given in Figure 2.1. The substructure data in the different domains can be obtained either from numerical models, from experimentally measured data or from a mix of both. The different aspects are explained next.

Numerical Modelling Modal Representation

Physical Representation [ , , . . . ]

, , modal reduction FRF modal synthesis parameter dynamic estimation transformation sti ness to rst order form inversion Frequency (FRF) Representation time State Space Representation integration ( )

A B Time (IRF) Fourier Representation Transform C D impulse response Inverse ( ) Fourier Transform system identi cation Experimental Testing

Fig. 2.1: Schematic overview of substructuring domains [259]

2.1.1 Spatial descriptions

Physical domain (continuous and discrete)

The description outlined in (2.2) expresses the dynamics of the substructure using the displacement at speciﬁc nodal location and is referred to as a physical representation. Note that representation in the physical domain can be either discretized (as assumed in this text) or continuous. In the later case, the unknown is the physical response ﬁeld and the associated equations are partial differential equations describing the (non-linear) continuous dynamics in a body. This will not be considered here.

Modal domain (reduced and unreduced)

It is often handy and useful to consider the unknowns of a dynamic problem as a combination of vectors of a (sub)space. The most well-known subspace representation is probably the one obtained by mode superposition. The free vibration modes of a substructure deﬁned by the eigenvalue problem [82]

(s)2 (s) K(s) ω M(s) φ = 0 (2.3) − i i 2.1 The dynamics of a substructure: domains of representation 7

(s) (s) where ωi are the eigenfrequencies of the free-free substructure and φ i the associated eigenmodes that have the fundamental property of being mass and stiffness orthogonal, namely

T 2 (s) (s) (s) (s) φ i K φ j = ωi δi j (2.4) T (s) (s) (s) φ i M φ j = δi j (2.5) where we have assumed that the modes amplitudes have been chosen to be mass-normalized and where δi j is the Kronecker symbol such that

δi j = 1 if i = j δ = 0 if i = j i j 6 A modal representation of the substructure is then obtained by the change of variable

n(s) (s) (s) (s) (s) (s) u = ∑ φ i ηi = Φ η (2.6) i=1

(s) (s) where n is the number of degrees of freedom in substructure (s). Here, ηi are the amplitudes of the modal component of the response and are often called modal coordinate of substructure (s). Often we will use a matrix notation as in the second equality of (2.6), where Φ (s) is a matrix containing in its columns the vibration modes and η (s) is a uni-column matrix containing all modal coordinates. Usually only a subset of modes is considered in order to have an approximated but reduced representation of the substructure. This will be discussed in later chapters. In general, the response u(s) can be represented as a combination of n(s) independent vectors and we write

n(s) (s) (s) (s) (s) (s) u = ∑ vi qi = V q (2.7) i=1 where V(s) is a square matrix containing the basis vectors for the desired change of variables. Substituting T in the dynamic equation (2.2) and premultiplying by V(s) to project the equations onto the same space, we obtain T T T T T V(s) M(s)V(s)q¨ (s) + V(s) C(s)V(s)q˙ (s) + V(s) K(s)V(s)q(s) = V(s) f(s) + V(s) g(s) (2.8) which is usually written as

(s) (s) (s) (s) M˜ q¨ (s) + C˜ q˙ (s) + K˜ q(s) = f˜ + g˜(s) (2.9) where the tilde superscript indicates that the matrices and vectors pertain now to a representation in a transformed space. The representation vectors stored in V(s) can be any set of independent vectors, in particular, they can be chosen as the vibration modes Φ (s), in which case the transformed mass and stiffness (s) (s) matrices M˜ and K˜ will be diagonal. The representation (2.9) will often be referred to as the modal representation, even when the base vectors are not vibration modes Φ (s) but general representation modes V(s). The associated degrees of freedom q˙ (s) are then called generalized degrees of freedom or modal coordinates and do in general not represent the solution at a particular physical location. In case an incomplete basis is used for the representation, namely when fewer modes than the number n(s) of degrees of freedom in the substructure are used, the modal representation represents the dynamics in a reduced subspace and in general only in an approximate way (this will be discussed in detail in Chapter 3). We will then call the representation a reduced modal representation. 8 2 Preliminaries: primal and dual assembly of dynamic models 2.1.2 Spectral representation

Time domain (continuous and discrete)

In the form of Eq. (2.2), the unknown dynamic response is considered a function of time and we say that, from a spectral point of view, the equations are expressed in the time domain. The dynamic equation (2.2) considers the spatial unknowns to be continuous function of time and the dynamics are expressed as ordinary differential equations in time (the space having already be discretized in that equation). As was done for the spatial domain, time can also be discretized using methods related to ﬁnite differences (typically variants of the Newmark time integration scheme in structural dynamics [33, 41, 82, 196], but other approaches can also be applied such as Finite Elements in time or variational based approaches [158]). When discretized in time, we will still consider the resulting equations as being in the time domain since the unknowns are the responses at discrete time instances. The equations are then algebraic equations that are typically solved in a recursive form (time stepping), given the fact that the time problem is typically an initial value problem3.

Frequency domain (reduced and unreduced)

Similar to the decomposition of the spatial response in component modes, the time dependency of the response can also be decomposed into a combination of time contributions. The most classical one is the Fourier decomposition4 that writes the time function of the response in terms of harmonic functions. Using complex number notations, the Fourier decomposition can be written as

∞ (s) (s) iωt u (t) = u¯ (ω) e− dω (2.10) ∞ Z− where i is to be understood as the imaginary unit number. This decomposition is very suitable for linear systems since replacing in (2.2) and using the orthogonality properties of harmonic functions, the harmonic component u(ω) can be computed individually from the harmonic dynamic equation:

(s) ω2M(s) + iωC(s) + K(s) u¯ (s) = f¯ + g¯(s) ω ] ∞,+∞[ (2.11) − ∈ − (s) where f¯ and g¯(s) are the Fourier components of the forces, for instance

∞ (s) 1 f¯ = f(s)(t) eiωt dt (2.12) 2π ∞ Z− The dynamic equation in the frequency domain (2.13) is also often written as

(s) Z(s)u¯ (s) = f¯ + g¯(s) where Z(s)(ω) = ω2M(s) + iωC(s) + K(s) (2.13) − Z(s) is a dynamic stiffness matrix and is a function of the frequency ω. In this form, u¯ (s) is the complex amplitude of the harmonic displacement response (or equivalently the Fourier component of the transient response). A similar equation can be written for the amplitude of the harmonic velocity or accelerations in which case the operator Z(ω) is commonly called the mechanical impedance or the apparent mass respectively. The dynamic relation can also be inverted and written

1 (s) (s) (s) (s) (s) (s) 1 2 (s) (s) (s) − u¯ = Y f¯ + g¯ where Y (ω) = Z (ω)− = ω M + iωC + K (2.14) − Y(s) is a Frequency Response Function (FRF) matrix and is often called the admittance or dynamic ﬂexibil- ity, or more speciﬁcally receptance, mobility or accelerance/inertance if u¯ (s) are displacements, velocities or accelerations respectively.

3 For an interesting matrix description of time discretization see [241, 299]. 4 Note that other base functions in time can be used (such as wavelets), but this will not be discussed here. 2.1 The dynamics of a substructure: domains of representation 9

Obviously, in practice, the harmonic components are calculated only for a ﬁnite discrete number of frequencies ω, and (2.10) is approximated by the Discrete Fourier Decomposition

Nω (s) (s) iωk u (t) = ∑ u¯ k e (2.15) k= Nω − choosing a frequency range covering the spectral range of the excitation. It is noteworthy that the decomposition in (2.15) is comparable to the decomposition of the space function of the response in (2.6) and can also be seen as a reduction of the transient response in the time domain. It is an approximation unless the excitation can be exactly represented by a ﬁnite combination of harmonics.

Laplace domain

Another often used representation of the time evolution of the dynamic response is in terms of the Laplace components. The idea is to look for the dynamic repsonse when modulated with a decreasing exponential function, namely

∞ (s) (s) st (s) u¯ (s) = L u (t) = e− u (t) dt (2.16) 0 Z This transformation changes the differential equation in time into an algebraic equation in the Laplace variable s thanks to the fact that Laplace transforms of time derivatives of u¯ (s)(t) can be written in terms of u¯ (s)(s) using integration by parts. For instance

L u¨ (s)(t) = s2u¯ (s)(s) su(s)(t = 0) u˙ (s)(t = 0) − − Clearly, there is a similarity between Laplace and Fourier transforms since (2.16) becomes a Fourier transform if s is taken as imaginary. The main difference is that the inverse transform is trivial for the Fourier domain (leading to the frequency domain decomposition (2.10) or (2.15)) whereas finding the inverse Laplace transform is far more difficult and not general. In structural dynamics, Laplace transforms are used for highly transient problems that can not efficiently be represented by harmonic superposition, such as impact responses and shock propagations (see for instance section 4.3.2 in [82]).

2.1.3 State representation

Displacement space

In addition to represent to space and spectral behavior of the system in different domains as explains above, the very state of the system can be described in mainly two different manners: either one sees the displacements as the only independent unknowns (velocities and accelerations being dependent on the displacement through derivatives) or the velocities are seen as additional independent variables for which the derivative relation to the displacement is explicitly expressed in the formulation. For the ﬁrst approach, often used in structural dynamics, the dynamics of the system are described by a single set of second order equations as in (2.1) (for non-linear structures) or (2.2) (for linear ones).

State Space representation

In the second case, namely the velocities are seen as additional independent variables, the state of the system is described both by the displacements and the velocities:

u(s)(t) x(s)(t) = (2.17) u˙ (s)(t) 10 2 Preliminaries: primal and dual assembly of dynamic models and the associated linear dynamic equation can for instance be written as

I(s) 0 0 I(s) 0 x˙(s) = x(s)(t) + (2.18) 0 M(s) K(s) C(s) f(s) + g(s) − − In this state-space representation the number of equations has doubled, but the order of the differential is now reduced to one. This representation is often used especially in control. This form is also commonly used in structural dynamics when strong damping is present since the concept of modes of vibration properly generalizes only when writing the system in the State Space (see for instance section 3.3 in [82]).

2.1.4 Summary of representation domains

From the short summary of the formulation of the structural dynamics problem, it is clear that many variants to describe the problem, combining a spatial, spectral and state representation, can be constructed. In Fig. 2.1, the different aspects are shown graphically and relations between them are summarized. Depending on the representation chosen, numerical and experimental techniques in substructuring can signiﬁcantly differ as will be seen in these lecture notes.

2.2 Interface conditions for coupled substructures

Let us consider again the linearized dynamic equilibrium equation (2.11) of a substructure in the physical space and in the frequency domain5

(s) Z(s)u¯ (s) = f¯ + g¯(s) s = 1...Nsub (2.19) where Nsub is the total number of substructures in the system. It is common to write the equilibrium of all substructures in a block matrix form as

Zu¯ = f¯ + g¯ (2.20) with the deﬁnitions

Z(1) 0 .. Z =  .  (2.21) (Nsub)  0 Z    (1) u¯ (1) f¯ g¯(1) . ¯ . . u¯ =  .  f =  .  g¯ =  .  (Nsub) (Nsub) (Nsub) u¯ f¯  g¯             The dimension of these block matrices and block vectors is (∑ n(s)) (∑ n(s)) and (∑ n(s)) 1 respec- s × s s × tively. Since the substructures are part of a same assembly, two interface conditions need to be satisﬁed: interface equilibrium and compatibility. 6

5 Expressing the coupling of substructures in other domains (modal, time, state space ...) will be discussed in later chapters and use exactly the same approach. 6 These lecture notes deal with structural problems. Nevertheless, the general theory is also applicable to the coupling of other physical domains such as acoustics or thermal problems. 2.2 Interface conditions for coupled substructures 11 2.2.1 Interface equilibrium

The interface equilibrium requires that the interface forces, g¯(s), which are internal forces between the substructures, sum to zero when assembled. This is merely a manifestation of Newton’s “actio-reactio" principle. Considering for instance an interface Γ (sr) between two substructures s and r, one could express this condition as7

(s) (r) (sr) gb + gb = 0 on Γ (2.22) (s) (r) gi = 0 gi = 0 (2.23) where the subscript b indicates a restriction of the DOF to the boundary and where we assumed that the DOF are numbered in the same manner on both sides of the interface. The subscript i denotes DOF that are not on a boundary and are thus internal DOF. On internal DOF, no connecting forces should exist. In practice, the numbering of the DOF on the interface will not match across the interfaces and in addition more than two substructures can intersect on an interface (so-called cross-points in 2 and 3-D, and edges in 3-D). Hence in general, the interface equilibrium condition needs to be expressed using Boolean localization T matrices L(s) of dimension n ns that combine the forces on either side of the interface to satisfy force × equilibrium. Interestingly, these localization matrices also map the DOF of substructure s to a global and unique set of n global DOF, as will be elaborated later. In general, the interface equilibrium thus is written as

Nsub T ∑ L(s) g¯(s) = 0 (2.24) s=1

This equilibrium condition can also be written using the block matrix LT of dimension n ∑ ns acting on b × s the set of all substructure interface forces

T T T sub T L g¯ = 0 where L = L(1) L(N ) (2.25) ··· h i

I To illustrate the notation, consider the examples in Figure 2.2. The green numbers indicate the global nodes for which the interface force conditions are written (i.e. the row number in the localization matrices). For the beam example in Figure 2.2.a, the localization matrices are

1 1 0 0 0 T T T L(1) = 0 L(2) = 0 1 0 L(2) = 0       0 0 0 1 1       and the interface equilibrium condition can be written as

7 In general, we will assume that the decomposition in substructures generates an interface discretization that is conforming and matching, namely that the shape functions used on either side of the interface are identical and that the nodes coincide. In case of non-conforming or non-matching interfaces, the theory used in these lecture notes are still generally applicable, but the assembly operators are then no longer Boolean. Indeed, going back to the variational principle underlying the discretized problem, the part related to the compatibility condition over an interface Γ can be written as

µT vdΓ ZΓ where µ and v are the ﬁeld. More details about non-matching interfaces can be found, for instance, in [230] 12 2 Preliminaries: primal and dual assembly of dynamic models

(1) g¯2  h (2)i  1 1 0 0 0 g¯1 1 ALT generalg¯ = 0 0 1 framework 0 0  g¯(2)  = for0 dynamic(2.26) substruc-       2  0 0 0 1 1  (2)  1 A general1 framework A general for framework dynamic g3 for substruc- dynamic substruc- 1 A general frameworkturing   for dynamic   substruc-  (3) turing turing  g¯1  turing (s) (s)  (s) (s) (s) (s) M u¨ (t)+Cu˙ (t)+h Ki  u (t)=f (t)+g (t) (1) (s) (s) (s) (s) (s) (s) (s) MFor theu(¨s) second((st))+ exampleCu˙ (tM)+ in FigureKu¨(s 2.2.b,()us()t)+(s we()tC)= haveu˙ (ft two)+(s() degreestK)+(s)gu of((ss) freedom)((tt))=f per(s) node(t (1))+ andg( thes) ( Local-t) (1) izationM matrixu¨ can(t)+ be writtenCu˙ (t as)+K u (t)=f (t)+g (t) (1) u(s)u(s) (s) (s) 1 2 1(s 0) 0(s 0) 0 0 0 0 0 0 0(s 0) 0(s 0) 0 0 0 0 0 0 0 0 0 0 u1 u2 u u u01 1u 02 0 0 0 0 0 0 0 01 0 02 0 0 0 0 0 0 0 0 0 0 0       0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0  (1) (1) (1) (2) (2) (3) (3)  (1) (1) (1)0(2) 0 0(2) 0 1(3) 0(1) 0 0(3)u(1)10u(1) 02 0u 0(2)3 0u 0(2)1 0 0u(3)30u(3)1 0 0u 02 0 0 0 0 u (1)u (1)u (1)u (2)u (2)uu(3)uu(3)u u u u u  1u1 2u2 3u301u 01 03u 03 01 u1 0 02u203 0 0 01 0 03 0 01 02 0 0 0 0 0 0 0       0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0       0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 AL generalT = 1L(1) L( A2)1L framework(3 general) = A general framework for framework dynamic for  dynamic substruc- for dynamic substruc- substruc- 1 A general framework0 0 0 0 0for 0 0 0 dynamic0 0 1 0 0 0u 0I 0 substruc-0 0 0 0 0 0 0 0 uIu   uI    turing turing turing0 0I 0 0 0 0 0 0 0 0 0 1 0 0uII 0 0 0 0 0 0 0 0 0 0 turing uIIuII   uII2 3   20 03 0 0 0 0 0 0 0 02 0 0 0u3 0III 1 0 0 0 1 0 0 0 0 0 u2III 3   u    (s) (s) (s) 0u( 0IIIs) 0( 0s)( 0s) 0(s 0)(s 0) 0 0 0 0(III(ss) 0) 0( 0s) 1(s(s)0) 0(s 0)(s 1) 0 0 0 0(s()s) (s) (sM) (s)u¨ (t)+MCu˙(tu¨)+M(s(K)t)+(us¨)uCu(˙t()+t)=)+(Cs)Kfu˙ (4t)+(ut)+(sK)5(gt)= u(ft)(t()=t)+fg (1) ((t)+t)g (t) (1) (1) M u¨ (t)+Cu˙ (t)+440K 05 05 0u 0 0( 0t)= 0 f0 04 0(t 0)+ 05 0g 0 0(t)0 0 0 0 1 0 (1) 0 0       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0       0( 0s) 0 0((s 0)) 0(s 0) 0 0 0 0(s 0) 0( 0s) 0 0(s)0(s 0) 0 0 0 0 1 0 u u u   u u u u  01 0 0 021 0 02 0 0 0 0 01 0 02 0 0 01 02 0 0 0 0 0 0 1             (s) (s) (s)(s)(s)(s)(s)(s) (s) (s) (s) (s) (1) g(1)g g(1)g f(2)f (2) (3) (3)g gg1 fg2 f u u(1)1 u1 (1)2 u2 (1)u(2)uu(1)(2)uu(1)(3)u1(1)(1)(3)u2(2)(1)u(2)(1)u(3)(2)u(3)(2) (3) (3) 1 u21 u32 u13 u31 u113 u221 u312 u12 u33 u11 u23 u1 u2 2 1 2 3 1 (1) (1) (2) (2) (3) (3) (1) (1) (2) (2) (3) (3) (1) g g(1)g g(1)g g(2)g g(2)g g(3)g gg(2)(3)(1)gg(1)gg(2)gg(2)gg(3)gg(3) g (3) 1 1 2 2 1 1 3u3I 1 1 2 21 21 12 u 31 13 21 2 1 uI 2 I 3 uI a. A bar assembly uII u 2 3uII II uII uIII2 3 2 3 2 3 uIII(1)(1) uIII u(1) 1 2 1 4 2 5uu u III(1) 2 2 4 5 2 u2 4 (2)(2)5 4(2) (2)5 u1u1 u1 u 1 1 100100 0 0 212 (2)31003 00 02 231 30 (1) (2) u(2)u1 ==100 0u(2) (2)= 0 000100011 1 2 2 00 2 u2 = 600016(2)0001(2)77 1 1 (2) 0 0    u  6 67(2)7 6u63 377 6u3 67u 7  66(3)(3)77 6 (3)73 3 4 3 4 6u6u 77 6 67(3)7 (s) (s)66(s1) 177 6u1 67u 7 y g1 g2 f4 5 (s) (s) 6 1 7 1 2(s)4 (s) 5(s) (4s) 5 g g f(1)(1) g1 g2 f (s) 4((1)s) (s5) x (3) 1 2 g2g2 g1 gg22 f (1) (2)(2) (2) g2 110003 4 g g 0 (2) 1100022111000133 0 g1 0 b. A 2D assemblyT T (1) (1) (2)T (2) (3)(2)(2)(3)110002 (2)3g1 0 L Lg =g =0010000100g1 g2 g1L gg3 =g1g g00100g2 == 00 g 2 (2)=3 0 2 2 3L3T g22=2 (1)00100(1)22(2)33 (2)3(3) 2(3)g 2=3 0 0001100011(1) (1) (2)66(2)(2)00011g(2)7(3)7g(3)g00(1)g (1)g 6(2)g(2)(2)72(3) 0(3) Fig. 2.2: Examplesg ofg assemblies:g 6gg6 substructuregg37127g2 1 DOFs3 and16g interface32 7(2) forces.2 3 1 2 1 633 1 7000112 g1 g2 g1 3g36 g1 7g2 0 4 4 55664(3)(3)77 4455 5 6 (3)67g3 74 5 6g g17 6g 67(3)7 661(1)747 6 51 7 4 5 J 4 5 6g1 7 (1) 4 u2 5 4 65 7 (1)u (1) (2) (1) 4 5 These matricesu2 2 are in fact100100 identical tou the2 localization(1) u 100(1) matrices used inu Finite Element codes to assemble (2)(2) (2) u 2 1 3u 2 (1) elementary matricesu in1 the global100100 system, however 0u 2 here(2) the2100 localization0 is not(2) writtenu for one element but for u21 1 3 100 u1uI I(2) u 100(2)= uI 2 one substructure2 (that(2)3 one could2 consider23 as a3 super-element2 or macro-element).u1 Obviously,(2) it is not efﬁcient (2) 0001=2 0103 (2)u1u 2= Luu110003 2u 3 u u2 = 010uu2IIII11=6=3(2)010Lu10027assembledassembled3 0uII (2)I= Luassembledu01 to store these Boolean2 matrices1 as100 written2 above,(2)6 03u but3 7 rather2(2) one should03 store them=2 as3 sparse matrices, or 6(2)(2)7 0012 (2)u 3 1 1002 3u2 0 (2) 0 6 6u7 7 60016 767uIII7uIII00012 6 6(3)001=7u2010=7 u1III uII = Lu0 assembled 6u3 37 6000167u3 7 61 7 0 26 (2)37 u2 = 6 (3)7 6 0017 6 (2)7u616 0001(2)7 7 u 1  0 6 6(3)u7 7 60016 76744(3)67u5657600167u60017 7 4 7 6u5III3 7 6 (2)7 6u6 17 7 6 6 767u 73 4 6 563 7 7 6 (3)7 6u 7  6 1 7 4 65 1 67(3)7 6(1)6(3)0017 7 6 7 3 4 4 5 5 4 5 6u 74g6 7 5 4 u1 5 6 (3)7 4 651 7 2u61 7 6 7 6u 7 6(2)4 7 5 4 5 6 1 7 110004 5 g4 5 0 (1) 2 1 3(1) g 4 5 T (2) g 2 (1) L g = 00100 g 2= 0 (2) g 2 3 2 (2) 2 3 g 2 0001111000611000(2)7g 0 0 2 1 3 (2)0 T 6g3 271 3 (2) g T L g = 6001007(2) 11000g =2 10 3 0 L g4= 001001 5 6 T(3)7g 4=5 0 2 (2) 1 2Lg1g =2 001001 3 6 (2)7 2g 3 = 0 2 6000113 67(2)7 2 3 6g 7 20 000114 5 2 0 3 3 6 (2)7 2 3 (1) 6g3 7000111 6 (3)7 0 4 6 7 5 6 7 64g3 57 u2 1004 5 (3) 4 5 g1 6 7 (2) 6g1 47 6 75 (3) 4 5 u 6 7 4 5 6g1 7 1 100(1)uI 6 7 2 (2)3 2 3 4 5 (1) u2 4 5 u2 u = 010 uII (1)=100Luassembled (2) 2 1002(2) 3u 6 7(2) 60017u uIII 2100100u 6u3 7u 6 27 1 3 (2) I 6 (3)71 100(2) u2I 3 2 (2)360012 7u4 3 =5u1010100uII = LuuI assembled 6u1 7 6 7 2 2 3 6 7u2 = 010(2) uII(2) = Lu2 assembled2 33 4 5 4 65 72 u60013 = 0107 uIII uII = Luassembled 6 (2)7 6u3 7 2 6u 7 60016 77 6uIII6(2)7 7 2 3 3 6 6 (3)77 6u60017 60017 4 57 uIII 6 (3)7 6001u1 7 4 63 5 67 7 6u 7 6 7 6 (3)7 001 6 1 7 6 4 75 6u4 7 65 7 4 5 4 5 4 5 6 1 7 6 7 4 5 4 5 1

1 1 1 AB AB ABAB ua Yab Yac AB AB ua Yab Yac1 AB AB AB A AB AB 1 uc = Ycb u Ycc = YYcc AB Ycc Y ABYcb Y ABfb Y A Y AB f AB 2 3 02 AB3 c 2 AB3 cb cc 1cc cc cb b ub Y 2 3Y 02 AB3 2 AB3 1 bb ub bc ⇥ Ybb ⇤Ybc 4 5 @4 5 4 5 ⇥A ⇤ A 4 A5 @4A 5 4 5 A ua Y 1 A 2.2 Interface conditions for coupled substructuresac ABua A 13 Y AB ABAB A 1 AB AB = A Ycc Y=cc Yaccb Yfb Y Y f uc Y A cc cc cb b even better one should construct the mapping tables based on the connectivity cc of the substructuresuc over theYcc interfaces. ⇥  ⇤  1 ⇥ ⇤ AB AB AB ABA ABAB ABAB A 1 AB AB ub = Ybb Ybc uYcc= Y Ycc YYcb Y fb Y Y f b bb bc cc cc cb b ⇣ ⌘ 2.2.2 Interface compatibility ⇥ ⇣ ⇤ ⇥ ⇤ ⌘ 0 uAB f AB0 AB AB AB a a ua fa The second condition that needs to be satisﬁed on the interfaceZ is that DOF pertaining0 to theZAB someI structuralAB0 I AB AB uc 0 u 0 node have the some response on bothAB sides of the2 interface,AB orAB in other words that the DOF2 are3 compatible3 2 3 c ua YabAB YacAB 2 0 2 AB3 2 33 2 3 2 3 ua YabAB YacAB AB A 1 AB AB (sr) u 0 0 AB on the interface. Considering theu DOF of= twoY substructuresY s and Yr coupledAB Y onA the1Y interfaceAB f ABΓ , theb u 0 u c Y cbAB YccAB cc cc cb b = b compatibility condition becomes2 c 3 = 02 cb6AB3 2 ccAB3 Ycc Ycc Ycb4 1 f5b 7 A A = 2 ub3 02 Y AB3 2Y AB3 6 1I 6 u 7 640 757 u (s) (rY) bb6 Ybc A 7 a A I 6 ua 7 6 0 7 b u¯ u¯ bb= 0 on0 bcΓ (sr)⇥ Z ⇤6⇤ 0 6 AZ7 6 77 6 7 6 7 4 5 b @4Ab 6 5 4 5 6 A0 7 u 0 7 A 4 5 −@4A 5 A4 5 A 6 c 7 6 07 6 u 7 6 0 7 ua 6 YacA AB A 1 AB6 AB 7 7 c where, as before, the subscript b indicatesu thata the=6 compatibilityYac Y isAB writtenY A for the1Y boundaryABf AB DOF and7 where6 7 6 7 6 7 6 7 AB = A Ycc Ycc Ycb6 fb 6 7 6 0 77 we assumed that the DOF are numbered identicallyuuc AB6 on0Y bothccA IAB sidescc0 of the interface.cc I cb0 b 0 7 6 7 6 0 7 ua  c Yab  ccYac 6 0 I 0 6 I 70 6 0 77 6 7 6 7 AB  AB AB6  ABAB⇥ AB ⇤ A 61 AB AB 7 7 In general the numberingua onAB the interfaceY does not coincideY ⇥ and therefore⇤ the1 compatibility conditions4 are 5 4 5 uc = ABabYcbAB4 ABacABYcc ABYcc A Ycc1 ABYcb AB fb 5 4 5 4 5 ua uYab=(sAB) Y ABYacYABAB Y ABAB(s) Y AA Y1 AB4ABf ABAB 5 expressed using signed Booleanu2 matrices3 u b0B=Y2.Y WhenABbb 3 operatingYY2bc ABY on3cc u¯ , theseYcc1 operatorsYcb extractfb 1 the interfaces c ub = b ABcbYbbbb ⇥ABccbcYbc ABYcccc⇤⇥ A Ycccc ABYcbcb⇤⇥ABbfb ⇤⇥0 ⇤ 2uc 3 = 0Y2cb 3 Y2cc Y3cc Ycc Ycb fb 1 0 DOF and give them an opposite sign on eachAB side of theAB interface.⇥ The interface⇤ compatibility can thenT be AB AB AB AB 2 u4b3 5 02 @YAB4bb⇣3 25 ABABY4bc3 ⇥ 5 ⇤ 1 ⌘ A + u f +T ub YbbA⇣ Ybc ⇥⇥ ⇤ ⇤ AB⌘ A a Aa ua fa written in the following general form: Z A ⇥ ⇤ AB0 Z AB 0 4 5 @u4a 5 Y4ac 5AB 0 A u1 ABAB ABfAAB c AB c AB 4 5 @4A A2 5= 4A 5 Y Y 2uaY Af fa 2 33 3 u ABAB AY A cc cc1 ABa cb b a uc 2 0 3 u 0 ua a ucZ Y acY AB0 AB A AI1 AB ABABAB AB c Z= ac A cc0Ycc YccI Yucbc fb 0 0 2 AB3 2 0 3 u22 = Nsub AY Ycc Ycc2 3Y3cb 2fubc 3 2 0 3 2 AB3 2 3 uc c Ycc cc ⇥ u 0 ⇤2 ABAB3 2 3 u 0 u 0  6  (s) (s) a 0AB u 0 +T 7 b T b  ABB u¯ AB= 0 ABu2 A6u1bb AB= AB04 (2.27)5 4= + 57 = ∑ ⇥ ⇥ c ⇤ ⇤ A = A A A A u66b =6Ybb Ybc Ycc 4 I1Y5cc716uYAcb7 6fb 0 7 7 7 6 ABABs=1 ABAB ABAABu A 3AI 7AB6 uABaABA7 AB6 0 7 c 6A ua 7 6c 0 7 6 ua 7 6 0 7 uub b==6YY60 Y Y 0Y ZYA 4Yb Y Y76Y ZfAa 7f 60 7 7Z 7 66 bb bb0 bc bc ccZcc cc cc0= 7cb6 cbu Ab 77 b 66 0 77 6 A 7 6 7 6 A 7 6 7 6 6⇣ ⇥ 5 0 7⇤66uYcc 7 ⌘ 6 0 7 0 7 u 00 7 6⇣6⇣  7 6⌘ AB⌘77 66 77 6 c 7 6 7 6 uc 7 6 0 7 One can use a block matrix notation6 to write6 this condition⇥ ⇥ also in⇤0 the⇤ form760YABab 7 6 AB0 7 7  7 6 00 II 00 I 0 0 7u6 2AB 7 f6 0 7 6 7 6 7 6 7 6 7 66 AB6 + 0 0 AB 7AB66acb +AB77 AB66a+ 77 7+ 0 7 6 A ua u7@Y4ABAfa 5f 4 5 6 n 7 6 7 6 n 7 6 0 7 1 AAB4Z general6 0 frameworkuI 5suba 64bb for5 a A4 dynamic5 substruc-7A 7 1 A general1 A framework generalZ 4AB framework0 0 forI dynamic(10) a ABA( forN 5u4) c 0 dynamic substruc-30 0 0 substruc-0 6 0 7 60 7 6 7 6 7 1 A general frameworkBu¯2=Z0⇥ where for⇤⇥0 Bc dynamic= B⇤2Iu u3B3 AB substruc-c 0 cY AB (2.28)c 2 ⇥ 6 ⇤⇥2 3⇤ 3  c u206AB3 20 ac 3 7 7 2 2 ···32c 30AB3c 2 3ABY AB 4 5 4 5 4 5 4 5 6 0 2 +6uTT3 2 AB2AB30 AB 7 7 ABAB u 0 ub= AAB+ b 05 uua cc ffa turing turingZ 0 hb ubA=i A =Ya0 a turing turing62 6 Z 4h 40 =547ihI5A7c 4uh=3 ABABABi0 ih 5 i 5 These equations can be understood2 6 as compatibilityA constraintsA I imposed62ua776A ontoY6a3A37 the0 u7 independent4bc6 7 0 sets of DOF 66 0 Z 4 7I527 6 u6 7ac3 76ucc60 7 7Y3 0 6 60 (s) Z A Y6(AB7s) A770aY 6A 2 7AB3 2AB 30 0 6 0 Z 07 bb 0u6 6 u7A 7026 AB6 370 7cc2 3 in the substructures.(s) (s The) matrices66 (6s)B have(s( dimensions)) (s()s) nλ 7⇣n6(s7) ,c where77A cc(6csn()sλ) is(suu7) theb number(s)0 of0 interface+T(s) AB AB+T AB AB (s) (s) 6 (s)(s) (s)  (s) 06(s)6(su7(6)s)+TT76 7767(sb60) 75 7⇥ (s) M u¨ (t)+6M66Cu6˙ (u¨t)+M(Kt)+u¨Cu u˙2AB((t(t)+t)+)=CK×7fu˙ (47tu)+(t7)+AKc(+tg)=557uY(ftAB)(AtAB)=(t)+==f g (1)(t)+(At)g (t)u (1) (1) fA u f compatibilityM constraintsu¨ (t)+ thatCu6 need˙6(t)+ to beK imposed.u (Zt)=Z f  (t6)+6g76A (7t6)077706cc7Z6A 70 (1)7 Y 0 a a a a 660 6I0 0I 0 I 0 I AB0A0 7 077Y cAB 66 uua 77 66 0077A 66 0 Z 6 6 7bc6c 76 777⇥6760a 7 7 ccc,a c,a 66 06 I0 0 I6 0 Z 07 77 66 A 77 66 77 1 AB 2 3 AB 466 2 6 5 47 6 540274 775665u4A 77 526 0 7 33 u 0 3 u 0 6 4 6 45 0 5Y 7 64 ucc Y75A 6 0 7 c c 64⇥ ⇤⇥ ⇤ 0 5 AB76 7cc 6 70 Y AB 0 6 ⇥ ⇤⇥6 ⇤ 0 cc7 6 7 61 7 ⇤ cb 2 3 2 3 2 AB3 2 3 I Example: Boolean Compatibility6 (+s+) (s Matrix)6 (s) + (s)T (s) (ABs) 7(s6) n AB7 6 0 7 AB 6AB⇥ A ⇤⇥ ⇤A + 0 uT ⇥ 7 6ABfn 7 6AB0 7 Z6 0 ABuA u 060u 0 A 0uA1 u2 +0ua u7AB6u a 7 AB6fY 7 ub 0 ub 0 66 0ABZ cc16 0216 2 0cc 0 c 0 +TA 61ABu772 6aY 7f 46a AB 7+T 57 f1 4 +T 57 To illustrate this notation,2 consider again the4 examples 2 of FigureA3 2.2.3c u a 44ABA055 a⇤44cb 5A5 AB A = A A = 2 66Z 6 0 2c 6c3377AB cc 7 b 7 2 4h 2 ihI i0 2 3AB35Au2c 31 0 f AB 6A u 7 6c,a0 7 6 u 7 6 0 7 4h ⇥ ih 2i 03u6 0u5c 0 0 bc,a A a 7 a 6 0 0 T 0 b Z2 AB30 2 3 7Z 6 Z 4 + 57 0 22 AB=u3 2Y AB30 6 A 7 6 7 6 A 7 6 7 6 6 AB Z A 7 T6IA b ⇤ 7 6 6 A 4 6+ u+5TT7u7b 6ABAB0 7cb0 ABAB 7 0 u 0 66 06 ABAB Z 4 c +7TA5A7 a uuA= = "ff 0 # 6 uc 7" 6 0 7# 6 c 7 6 7 6 6 ZZ 6 A0 A6406A773Au6ABaa 7 0aa 7 7 66 (1)6 (1)(1) (1)(1) (1)(2) (2) (1)(3)(2)A⇤⇥(1)(3)I(3) (1)(3)(2)7c(2)c,a7 (3)u63(3)a 7 60 f AB7 6 7 6 7 6 6 62 0 Z (1)0 2c(1) 6(1)7u6c6(2)733736a (2)76a ABAB07(3)67(3)6 7 7 6 7 6 77 6 u61 2uu20 uu3 u61 u3 uZu1 uuu2uu u u7 I6 u677u3u2ABu76uAc 6+7 7⌘0b0 7+ 0 6 n 7 6 0 7 66 6 1 2 3 1 13 2+1 0u32 u1 7u30 07u15 +6uA2cu uc 7 u 6 0 7 6 n 7 6 7 66 + 6 + A 1 20 636n61A77u 22376cAB01A363720227 33 7A 7 66 06 A 0 0 A 0 0  70 6 7 770u6cABu6uAB7 7 6 00070 6 0 7 60 7 6 7 6 7 6 6 6c + 6 0c + 0 4 7 +6T6T7576b 7bb0c,a6 7 7c,a 7 606 6 + A + Ac,a 4 4A7+67c,a5u5676 74n 7365==f60AB70 7 4 5 4 5 664 6 6 A A 0 ⇤ 0 7 A7777An AA a 4 5 4 5 2 h0 60 c 0 601 c20 A0 3 0 06766a76617uu 776 66 70077 7 7 4h 6 6A +c ih c i 5 c,a 677 6 aa 7 2 6 7 6 c6 00 Z 7 u A 746 =7 57 046 57 6 6 6 4 0 77446ch77 657AA 47 65ih7 5 i 5 (1) 6 h (2) ih0 6 7667uuc i77 (3)66 00 77 4h4h66 ihih i i +T"1 5524AB5##377 4 cAB560 6 7 2 AB01 2uI3 4 5 6 A7 u 667 f 677 66 3 77 Z66 uZ0IA 2 uI0 0 a 777 a 0 a. A bar assembly Z ++ + c,a ABu 766nn677 66 00 77 2 AB 66 AA A 2 A +33T +T I77AB AB6 AB AB 6 ABABihu c T+ A0uc 5 66 0077 667 77 6 66 00Z c,ac,aII00u 0c,ac,a 40 0uIIA u77a ua 6 fa 7fa 6 Z A + II 0 c,ac,a2 AB3 4426 553 44 55 66 2 c 3 202 3 u 3uII77AB AB0 0 7 6 62 2 u 2 3 4 2 +T 57 3 3b 3 u 4 7 0 6 44hh III ihih iAuIII 0 23u A u55c=3 c 70 6 6 uIII0 A+ 0 7 AB 2 3 2 3 1122 33 44 55 66 77 A  c c,aT 6 uaua2III7AB36AB0 27 3 1 6 0 62 10 2 Z 57 6u AB u7 u6 57 0 0 6 66 6 4 5 4 4 +7T75+27T 5A7 b b 6 4 5 "0 40 A# 75A6c uc 7 A 6A=0 =7 1 0 66 6 0 i A 77u AB7 7u u 0 0 4 6 A c,a7c,a6b64 6 76a5 76a 767 6 7 7 2 h 6 6 0+ 0 + Z Z0 7 6 7n 773 f6AB0 7 A + 6 u76A 6 6A 7A 76 6 7 7 c,a 6 6 A A 7 6 6 7 77 6a 7 3 6 0 c,a 0 c,a 0 "0 "7 0 7a 0# #6 u6c u7c 76 60 70 7 (1) 6 6 (2) 7u6A 4 7=6775 0274 65 7 4 66 6c 7 77 6 7 6 7 0 6 + + 0 76 67n 7 6 0 70 46h Z A +ih 2 i+ 7 5 77 60n 7 6 7 5 6 A A AA A 4n 67 6 7 6 7 1 2630405 6 07 0 + 0 0 5 0 0 7 6 73 6 7 6 ih6 c,a c,a c,a c,aT 7 470 5 4 5 6 6 4 7 7 764 5 4 5 7 A + 0 6 7 4 c,a 4 5 5 60 3 4 h4h3 ihih 3i i 65 7 1 82 3 4 5 (6s)7(s) A 3u AB 7 1 2 3 4 5 6 "7 (+s8) 9 10 11 12 40 7 9 0 g g(s)c,af (sT) 5 a (s) (s) (s) y 1 2 (s) 7u2ABg g f 7 10 1 g2 0 g f 7 c 1 2 (s) (s) 7 i(3) 1 2 7u AB 5(s) x 11 6b g g f 0 # 7 6 3 f AB1 2 7 u A a 3 12 7 6a 4 7 u6A =7 02 7 6c 7 b. A 2D assembly (1) (1) (2) (2) 7(3)6 (3) 7 0 5 4n 7 6 3 g1 g(1)2 g1(1)g3(2)g1(2)g(1)2(3)7(1)(3) (2)6 (2) (3) (3) g 7g g60 g g g Fig. 2.3: Examplesg of assemblies:g g g interpretationg1 5 g2 61 of the3 7 Lagrange1 2 multipliers 1 2 1 3 1 2 60 (1) 7(1) (2) (2) (3) (3) 6 7 40 g1 7g2 g1 g3 g1 g2 7 7 5 u(1) 2 (1) (1) (2) u u2 u 2 (2) 1 100 02 1 3(2) 0 u (1) (2) u 2 1 3 u2 1 u21002 1=3 0 (2) 0 00011 1001 0 (2) 0 0 u = (2)  0001 6 (2)7u  =11 1 2 u10 6u3 72 11 6 (2)7 2 3 0001 61 671(2)7 1000 0 (2) 0  6 (3)7u  6u3 7 u = u1 6 00013 7 6 (3)71 2 0 6 67(3)7 6u 7 6 (2)7 4 65u11 7 6 17 6u 7  6(1)1 7 3 11 g 4 5 6 (3)7 11 42 5 (1) 6u 711 (2) (1) g2 6 1 7 11000 g1 g2 0 (2) 4 5 2 (2)311(2) LT g = 00100 11000g = 0 g1 (1)0 11000T 2 g1 11 0 2 (2)3 g2 2 L g =3 600100(2)27 32 3 = 0 LT g =00011001006g 7(2) 0= 0 g2 (2) 2 3 g2 110003 6 (2)7 g21 3 0 2 6000113(3)7 2 3 2 0 3 4 000115 6 T67(2)74 5 0 6g3 7 (2) Lg1 6gg= 7001006 7 = 0 46 673 7 5 (3) g42 5 4 4 5 5(3)2 4 56g1 73 (2) 2 3 (1) 6g 7000116 7 6 7 0 u 6 1 7 4 5 6g3 7 2 100(1) 4 45 5 6 (3)7 4 5 (2) (1) u 6g 7 u1 u 1002 uI 100 6 1 7 2 (2)32 2 1003(2) (2)= 010u u =100Lu u 4 5 u2 u 2 1 3II (1) assembledI (2) 1 100(2)2 3u2uI 3 6 2 7(2)360012 u7 u3III= 0102 100uII = Luassembled 6u3 7 2 u2 6= 0107(2) uII(2) = Luassembled2 3 6 (3)7 600167 47 2 56u0013 7100uIII u 6u6 7(2)7 6u3 7 2 1 3 I 6 16u7 76 600167 7 u6III(2) 27 3 4 53 4 6 65(3)7 6u001= 70104 5 uII = Luassembled 6 (3)7 6001u1 7 4 6 2 5 7 6u 7 6 7 6 (2)7 2 3 6 1 7 6 4 57 64u 7 650017 uIII 4 5 4 5 6 3 7 6 7 6 (3)7 60017 4 5 6u1 7 6 7 4 5 4 5 1 1 1

1 14 2 Preliminaries: primal and dual assembly of dynamic models

For the beam example, the compatibility condition can be written as

(1) u¯2  h i  u¯(2) 1 1 0 0 0  1  ¯ = (1) (2) (3) ¯ =  (2)  = Bu B B B u − u¯2  0 (2.29) 0 0 0 1 1   −  u(2)   3       (3)   u¯   1   h i  The side on which the entry in B is positive and negative can be chosen freely. The interpretation of B and its associated Lagrange multipliers is depicted in Figure 2.3.a. For the second example in Figure 2.2, the constraint matrix can be written as

0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0    −    0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 − 0 0 0 0 0 0 0 1  0 0 0 0 0 1 0 0  0 0 0 0 0 0 0 0    −    0 0 0 0 0 0 0 0  0 0 0 0 1 0 0 0  1 0 0 0 0 0 0 0 B = B(1) B(2) B(3) =     −  0 0 0 0 0 0 0 0  0 0 0 0 0 1 0 0  0 1 0 0 0 0 0 0      −  0 0 0 0 0 0 0 0  0 0 0 0 0 0 1 0  0 0 1 0 0 0 0 0      −  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 1  0 0 0 1 0 0 0 0      −  0 0 0 0 0 0 1 0  0 0 0 0 0 0 0 0  1 0 0 0 0 0 0 0     −  0 0 0 0 0 0 0 1  0 0 0 0 0 0 0 0  0 1 0 0 0 0 0 0      −        The interpretation of B and its associated Lagrange multipliers is depicted in Figure 2.3.b. Note that the last two constraints are redundant. The compatibility of node 4 in Ω (1) with node 3 in Ω (2), and of node 1 in Ω (3) with node 3 in Ω (2) is imposed in lines 3, 4, 5 and 6 of B, so there is no need to impose in addition the compatibility between node 4 in Ω (1) and node 1 in Ω (3). Adding this redundant constraint does usually not harm the computation and can even be helpful for instance in the parallel computing algorithms . J

As for the localization matrix L(s), the constraint matrices B(s) are, in practice, not stored as full but as sparse, or only connectivity information is used to apply them to a vector. To summarize this section, an assembly of substructures in the physical frequency domain (and similarly in other domains) is obtained by imposing compatibility and interface equilibrium conditions, leading to the set of equations

(s) Z(s)u¯ (s) = f¯ + g¯(s) s = 1...Nsub Nsub  B(s)u¯ (s) = 0  ∑ (2.30)  s=1  Nsub  T ∑ L(s) g¯(s) = 0  s=1   or in block matrix form 

Zu¯ = f¯ + g¯ Bu¯ = 0 (2.31)  T  L g¯ = 0  2.3 Primal and dual assembly 15 2.3 Primal and dual assembly

The form (2.30) (or equivalently (2.31)) of the coupled problem uses two interface ﬁelds, namely the primal unknowns u per substructure (i.e. on each side of the interfaces) and the substructure interface forces g called dual unknowns.8 Solving the dynamic problem of the assembly in the form (2.30) can be expensive since many interface unknowns need to be resolved. Hence, these equations can be rearranged in order to eliminate the interface forces and write the problem in terms of unique interface displacements (primal assembly) or by introducing interface forces satisfying the interface equilibrium (dual assembly).

2.3.1 Primal assembly

We can deﬁne as primal unknowns for the interface a set of DOF ug that are global and uniquely deﬁned for the entire structure. The DOF of each substructure are then obtained by mapping the global set ug to the local DOF of each substructure u(s). Such a mapping was already introduced in the previous section, (2.24), to map the local interface forces to a global set. The same mapping, but now from the global DOF to the local ones can be used to write

(s) (s) u = L ug or u = Lug (2.32)

If the substructure DOF are obtained from a unique set as described above, they automatically satisfy the compatibility conditions that matching interface DOF must be equal. Hence

BLu = 0 u (2.33) g ∀ g This relation, mathematically means that L represents the nullspace of the constraint matrix B:

L = null(B) (2.34)

I In the example of the beam in Figure 2.2.a for which L was found in (2.26), this leads to

(1) 1 0 0 u¯2  h (2)i    u¯1 1 0 0 u¯1 (2)  u¯  =  0 1 0  u¯2 (2.35)  2      (2)   0 0 1  u¯3  u¯3          (3)    u¯   0 0 1   1     h i    Clearly it can be seen that the substructure DOF are “drawn" from a global and unique set of DOF and thus automatically satisfy the compatibility constraint on the interface: with the compatibility constraint matrix B obtained in (2.29), one veriﬁes that

8 In these lecture notes we will introduce the coupling conditions using basically a two ﬁeld approach. A more general three ﬁeld approach can also be considered but will not be discussed here. See for instance [303] 16 2 Preliminaries: primal and dual assembly of dynamic models

1 0 0 1 0 0 1 1 0 0 0   Bu = BLug = − 0 1 0 ug = 0 0 0 0 1 1   −  0 0 1    0 0 1     J

When choosing a unique set of DOF as in (2.32), the coupled problem (2.31) can be simpliﬁed and written as

ZLu = f¯ + g¯ g (2.36) LT g¯ = 0 where the compatibility condition is not present anymore since it is automatically verified. This expression is sometimes denoted as the Neumann-Dirichlet form of the coupled problem. The interface forces g can be eliminated from this relation by premultiplying the dynamic equilibrium (first line of (2.36)) by LT , which builds the sum of the equilbrium equations on matching nodes. Con- sidering the equilibrium of the internal forces (second line of (2.36)), the coupled problem can finally be expressed as

T T Zgug = fg with Zg = L ZL and fg = L f¯ (2.37)

This form is generally referred to as the primal assembly of the coupled problem and matrix LT ZL is the primal assembled impedance of the global system. This assembly is similar to the assembly of ﬁnite elements.

I To illustrate this process, consider again the ﬁrst example in Figure 2.2.a, with the matrix L given in (2.26). The assembled impedance matrix for this example then writes

(1) 1 0 0 Z22 0 0 h i (2) (2) (2)   1 1 0 0 0  Z11 Z12 Z13  1 0 0 T (2) (2) (2) Zg = L ZL = 0 0 1 0 0 0 Z Z Z 0  0 1 0         21 22 23    0 0 0 1 1  (2) (2) (2)  0 0 1   Z31 Z32 Z33              0 0  (3)    Z11  0 0 1     (1) (2) (2) (2)  h i  Z22 + Z11 Z12 Z13 = (2) (2) (2)  Z21 Z22 Z23  (2.38) Z(2) Z(2) Z(2) + Z(3)  31 32 11 33    and the dynamics of the beam are expressed by

(1) (2) (2) (2) ¯(1) ¯(2) Z22 + Z11 Z12 Z13 u¯1 f2 + f1 (2) (2) (2) (2) Zgug = = fg = Z Z Z u¯2 = f¯ (2.39) ⇒  21 22 23    2  Z(2) Z(2) Z(2) + Z(3) u¯3 f¯(2) + f¯(3)  31 32 11 33   3 1       J 2.3 Primal and dual assembly 17 2.3.2 Dual assembly

Let us start again from the coupled form (2.30) (or equivalently (2.31)), but instead of satisfying a priori the interface compatibility as in the primal assembly, we satisfy a priori the interface equilibrium.9 This can be achieved by choosing coupling forces as

g = BT λ (2.40) − Note that, since BL = 0 and thus LT BT = 0, such coupling forces automatically satisfy the interface equilibrium:

LT BT λ = 0 λ (2.41) ∀ The minus in this deﬁnition is a choice to obtain a symmetric form later. The expression (2.40) can also be written substructure-wise as

g(1) B(1) . . . = . λ (2.42)  .  − .  (Nsub) (Nsub) g  B      showing clearly that the coupling forces are generated from a unique set of interface forces, whose intensities are given by λ and are attributed, with the appropriate positive or negative sign, to the substructure interface. The interface force intensities λ are unknowns pertaining to the interface. The size nλ of λ is equal to the number of interface compatibility conditions on the interface. Substituting the coupling forces by their form (2.40), the coupled problem can be written as

(s) T Z(s)u¯ (s) = f¯ + B(s) λ s = 1...Nsub Nsub  (s) (s) (2.43)  ∑ B u¯ = 0  s=1  where the interface equilibrium no longer appears since it is automatically satisﬁed given our choice (2.40). In block matrix form, we can write

Zu¯ + BT λ = f¯ (2.44) Bu¯ = 0 which is often put in the symmetric form

ZBT u¯ f¯ = . (2.45) B 0 λ 0 One observes that λ can be interpreted as the Lagrange multipliers associated with the compatibility constraint (see the general theory of constrained systems in dynamics, for instance in [82]).

I To illustrate this form, consider again the ﬁrst example in Figure 2.2.a, with the matrix B given in (2.29). The dually assembled form is then written as

9 For the description of the problem, it does obviously not manner if one first satisfies the equilibrium or the compatibility. This only determines which equation in the three-field formulation (2.30) is eliminated. The end result will be identical, but the mathematical form and the approximations that can be applied will be depending on the form that will be solved. 18 2 Preliminaries: primal and dual assembly of dynamic models

(1) (1) ¯(1) Z22 0 0 1 0 u¯2 f2 h i (2) (2) (2) h (2)i h ¯(2)i  Z11 Z12 Z13 1 0  u¯1   f1  (2) (2) (2) − (2) ¯(2)  0 Z21 Z22 Z23  0 0 0 u¯2   f2         Z(2) Z(2) Z(2) 0 1  u¯(2)   f (2)    31 32 33   3  =  3  (2.46)    (3)   (3)  (3)  0 0 Z 0 1  u¯   f¯   11 −  1   1        h i  h i   h i   1 1 0 0 0  λ   0   − 0  1     0 0 0 1 1  λ   0   −  2         Notice that this representation of the system is larger than the primally assembled counterpart, but in this representation the interface forces, λ are explicitly available. In future sections we shall see that this can be useful. J

2.3.3 Usefulness of different assembly formulations

In this section, we have expressed the coupled problem in three different forms that are recalled in their block matrix form below for clarity:

Zu¯ = f¯ + g¯ General form Bu¯ = 0 (2.31)  T  L g¯ = 0  T T Primal Assembly Zgug = fg with Zg = L ZL and fg = L f¯ (2.37)

ZBT u¯ f¯ Dual Assembly = (2.45) B 0 λ 0 To illustrate the primal and dual assembly, Figure 2.4 shows the assembly of components with forces or with displacements, and the corresponding matrices.

Primal Dual

Fig. 2.4: Primal and dual assemble illustration

Each of these representations is mathematically equivalent and, hence, one might wonder why considering one form or the other might be advantageous. From a mechanical point of view the interpretation of these forms is different, and so each provides different insights. The general form considers each substructure as independent (the DOF and coupling forces are deﬁned separately per substructure) and therefore 2.3 Primal and dual assembly 19 requires writing all of the interface conditions explicitly. In the primal form, coupling forces are no longer part of the problem: the only interface unknowns are the interface DOF, which are unique for the entire structure. In the dual form, it is the interface force amplitudes λ that are unique for the entire structure and represent the interface unknowns (the interface DOF remaining deﬁned per substructure). The different mechanical interpretations of the assembly can be used to develop different numerical and experimental techniques in substructuring. This will hopefully become clear in the other chapters of these lecture notes, but here we provide a brief preview to the various uses of these forms.

Model Reduction (Chapter 3) In model reduction by substructuring, the dynamics in a substructure are approximated by representing the behavior of its physical DOF in a subspace of representative modes. – When the primal assembled form (2.37) is considered, a substructure is seen as receiving information from its neighboring substructures over the interface DOF they share. Hence, a natural set of modes for the reduction might correspond to the static response of the substructure to unit displacements at the interface, and these could be augmented by vibration modes in which the interface is ﬁxed (i.e. the Hurty/Craig-Bampton family of methods naturally arise). – When the dual assembly form (2.45) is considered, the substructure receives excitation from the neighbors as imposed interface forces. A natural set of representation modes in that case would consist of the static response of the substructure to unit interface forces and the additional dynamics would naturally be represented by vibration modes computed with the interface free. The reduction methods will thus be based on different approximated representations depending on which mechanical interpretation of the coupling is considered.

Experimental Substructuring (Chapter 4) In experimental substructuring, the dynamics of (some) substructures are obtained be measurements. Since in experimental dynamics one typically measures a component with free interface, applying excitations and measuring the dynamic response, one typically measures dynamic flexibilities (FRFs), namely the response of the component to excitations. Therefore, considering the dually assembled form seems more natural. Obviously, the measured dynamic flexibility could be inverted and assembled in a primal way (impedance coupling) which would be mathematically equivalent. In practice however, since the measured dynamic flexibility of the substructure always contains imperfections or are not complete, an assembly should usually be performed only in a weak sense : the degrees of freedom on the interface should not be forced to match exactly, but rather only some simple deformation modes on the interface should be made compatible, allowing for some local incompatibility. This alleviates the effect of measurement errors in the assembly as will be explained later (Section 4.3). References

1. Abaqus analysis user’s guide, User Manual, Simulia (2014) 2. Abad, J., Franco, J.M., Celorrio, R., Lezáun, L.: Design of experiments and energy dissipation analysis for a contact mechanics 3D model of frictional bolted lap joints. Advances in Engineering Software 45(1), 42–53 (2012). DOI: 10.1016/j.advengsoft.2011.09.021 3. Abrahamsson, T., Kammer, D.C.: Finite element model calibration using frequency responses with damping equalization. Mechanical Systems and Signal Processing 62-63, 218–234 (2015) 4. Abrahamsson, T.J.S.: FEMcali (Finite Element Model CALibration). URL mathworks.com/matlabcentral/ fileexchange/ 5. Aguirre, L.A.: Quantitative measure of modal dominance continuous systems. In: Anon (ed.) Proceedings of the 32nd IEEE conference on decision and control, vol. 3. IEEE (1993). DOI: 10.1109/CDC.1993.325629 6. Allemang, R., Brown, D.L.: A correlation coefficient for modal vector analysis (1982) 7. Allen, M.S., Kammer, D.C., Mayes, R.L.: Uncertainty in experimental/analytical substructuring predictions: a review with illustrative examples. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, pp. 1833–1850 (2010) 8. Allen, M.S., Mayes, R.L.: Comparison of frf and modal methods for combining experimental and analytical substructures. In: 25th International Modal Analysis Conference (IMAC XXV) (2007) 9. Allen, M.S., Mayes, R.L.: Comparison of frf and modal methods for combining experimental and analytical substructures (2007) 10. Allen, M.S., Mayes, R.L., Bergman, E.J.: Experimental modal substructuring to couple and uncouple substructures with flexible fixtures and multi-point connections. Journal of Sound and Vibration 329(23), 4891–4906 (2010) 11. Anderson, B.D.O.: A system theory criterion for positive real matrices. SIAM Journal on Control and Optimization 5(2), 171–182 (1967) 12. Antoulas, A.C.: Approximation of large-scale dynamical systems. Society for Industrial and Applied Mathematics, Philadelphia (2005) 13. Aoyama, Y., Yagawa, G.: Component mode synthesis for large-scale structural eigenanalysis. Computers & Structures 79(6), 605–615 (2001) 14. Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quart. Appl. Math. 9, 17–29 (1951) 15. van der Auweraer, H., Mas, P., Dom, S., Vecchio, A., Janssens, K., van de Ponseele, P.: Transfer path analysis in the critical path of vehicle refinement: the role of fast, hybrid and operational path analysis. Tech. Rep. 2007-01-2352, SAE Technical Paper (2007). DOI: 10.4271/2007-01-2352 16. Avitabile, P.: Twenty years of structural dynamic modification – a review. Sound & Vibration Magazine January, 14–25 (2003) 17. Balmès, E.: Use of generalized interface degrees of freedom in component mode synthesis. Proceedings of the 14th International Modal Analysis Conference 1 (1996) 18. Balmès, E.: Modèles expérimentaux complets et modèles analytiques réduits en dynamique des structures. Ph.D. thesis, Université Pierre et Marie Curie, Paris VI, ONERA, Directions des Structure, DRET/AC (1997) 19. Bampton, M.C.C., Craig, J.R.R.: Coupling of substructures for dynamic analyses. AIAA Journal 6(7), 1313–1319 (1968) 20. Barten, E., van der Seijs, M.V., de Klerk, D.: A complex power approach to characterise joints in experimental dynamic substructuring. In: Dynamics of Coupled Structures, Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, vol. 1, chap. 27, pp. 281–296. Springer New York (2014). DOI: 10.1007/978-3-319-04501-6_27 21. Bathe, K.J., Dong, J.: Component mode synthesis with subspace iterations for controlled accuracy of frequency and mode shape solutions. Computers & Structures 139, 28–32 (2014)

197 198 References

22. Beck, J.L., Yuen, K.V.: Model selection using response measurements: Bayesian probabilistic approach. Journal of Engineering Mechanics 130, 192–203 (2004). DOI: 10.1061/(ASCE)0733-9399(2004)130:2(192) 23. Bello, M., Sestieri, A., D’Ambrogio, W., La Gala, F.: Development of a rotation transducer based on bimorph PZTs. Mechanical Systems & Signal Processing 17(5), 1069–1081 (2003). DOI: 10.1006/mssp.2003.1607 24. Bendat, J.S., Piersol, A.G.: Engineering Applications of Correlation and Spectral Analysis. John Wiley & Sons, Inc., New York, London, Sydney (1980) 25. Benfield, W., Hruda, R.: Vibration analysis of structures by component mode substitution. AIAA journal 9(7), 1255– 1261 (1971) 26. Bhushan, B.: Concepts and Applications of Tribology, 2nd edn. Wiley (2013) 27. Bisplinghoff, R.L., Ashley, H., Halfman, R.L.: Aeroelasticity. Addison-Wesley Publishing Company, Cambridge, MA (1955) 28. Bograd, S., Reuss, P., Schmidt, A., Gaul, L., Mayer, M.: Modeling the dynamics of mechanical joints. Mechanical Systems and Signal Processing 25(8), 2801–2826 (2011). DOI: 10.1016/j.ymssp.2011.01.010 29. Bonhoff, H.A.: The influence and significance of cross-order terms in interface mobilities for structure-borne sound source characterization. Ph.D. thesis, Technische Universität Berlin (2010) 30. van den Bosch, D.D., van der Seijs, M.V., de Klerk, D.: Validation of blocked-force Transfer Path Analysis with compensation for test bench dynamics. In: Dynamics of Coupled Structures, Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, vol. 1, chap. 4, pp. 37–49. Springer New York (2014). DOI: 10.1007/978-3- 319-04501-6_4 31. Brake, M.R.: The Mechanics of Jointed Structures: Recent Research and Open Challenges for Developing Predictive Models for Structural Dynamics, 1st edn. Springer (2018) 32. Bregant, L., Sandersons, M.: Rotational degrees of freedom: an historical overview on techniques and methods. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, pp. 973– 980 (2000) 33. Brüls, O., Arnold, M.: The generalized-α scheme as a linear multistep integrator: Towards a general mechatronic simulator. Trans. ASME 3, 041,007–1/041,007–10 (2008) 34. Bylin, A.: Dynamic substructuring using experimental-analytical state-space models of automotive components. Mas- ter’s thesis, Chalmers University of Technology, Dept Mechanics and Maritime Sciences, report 2018:69 (2018) 35. Bylin, A., Gibanica, M., Abrahamsson, T.J.S.: Experimental-analytical state-space synthesis of a body-in-white and rear subframe (2018) 36. Carne, T.G., Dohrmann, C.R.: Improving experimental frequency response function matrices for admittance modeling (2006) 37. Castanier, M.P., Tan, Y.C., Pierre, C.: Characteristic constraint modes for component mode synthesis. AIAA journal 39(6), 1182–1187 (2001) 38. Cerrato, G.: Automotive sound quality – powertrain, road and wind noise. Sound & vibration 43(4), 16–24 (2009). URL http://www.sandv.com/downloads/0904cerr.pdf 39. Chaturantabut, S., Sorensen, D.: Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing 32(5), 2737–2764 (2010). DOI: 10.1137/090766498. URL http://epubs.siam.org/doi/abs/ 10.1137/090766498 40. Choi, H.G., Thite, A.N., Thompson, D.J.: Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination. Journal of Sound & Vibration 304(3), 894–917 (2007). DOI: 10.1016/j.jsv.2007.03.040 41. Chung, J., Hulbert, G.M.: A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method. ASME J. Appl. Mech. 60, 371–375 (1993) 42. Cloutier, D., Avitabile, P.: Dynamic uncoupling of a system model for component identification. In: IMAC-XXIX: International Modal Analysis Conference, Jacksonville, FL. Society for Experimental Mechanics, Bethel, CT (2011) 43. Coppolino, R.N.: Consolidation of weakly coupled experimental system modes. In: Proceedings of IMAC-XXXVI: A Conference on Structural Dynamics, Society for Experimental Mechanics, Bethel, CT (2018) 44. Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations Bull. Amer. Math. Soc 49(1), 23 (1943) 45. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 1. Interscience Publishers, New york (1953) 46. Craig, R.R.: Coupling of substructures for dynamic analyses: an overview. In: Proceedings of AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference and exhibit, pp. 1573– 1584 (2000) 47. Craig, R.R., Chang, C.J.: On the use of attachement modes in substructure coupling for dynamics analysis. In: 18st Structures, Structural Dynamics and Material Conference, pp. 89–99. AIAA/ASME, San Diego (1977) 48. Craig Jr, R.R., Chang, C.J.: Substructure coupling for dynamic analysis and testing. Report, National Aeronautics and Space Administration (1977) References 199

49. Crowley, J.R., Klosterman, A.L., Rocklin, G.T., Vold, H.: Direct structural modification using frequency response functions. Proceedings of IMAC II (1984) 50. D’Ambrogio, W., Fregalent, A.: Decoupling procedures in the general framework of frequency based substructuring. In: IMAC-XXVII: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2009) 51. D’Ambrogio, W., Fregalent, A.: Are rotational dofs essential in substructure decoupling? In: IMAC-XXXII: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2014) 52. D’Ambrogio, W., Fregolent, A.: The role of interface DoFs in decoupling of substructures based on the dual domain decomposition. Mechanical Systems & Signal Processing 24(7), 2035–2048 (2010). DOI: 10.1016/j.ymssp.2010.05.007 53. D’Ambrogio, W., Fregolent, A.: Ignoring rotational dofs in decoupling structures connected through flexotorsional joints. In: IMAC-XXXIII: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2015) 54. Davison, E.: A method for simplifying linear dynamic systems. Auto Control IEEE Trans 11(1), 93–101 (1966). DOI: 10.1109/tac.1966.1098264 55. Deaner, B.J., Allen, M.S., Starr, M.J., Segalman, D.J., Sumali, H.: Application of viscous and Iwan modal damping models to experimental measurements from bolted structures. Journal of Vibration and Acoustics 137(2), 021,012 (2015). DOI: 10.1115/1.4029074 56. Di Maio, D., Schwingshackl, C., Sever, I.A.: Development of a test planning methodology for performing experimental model validation of bolted flanges. Nonlinear Dynamics 83, 983–1002 (2016). DOI: 10.1007/s11071-015-2382-9 57. Dickens, J.M., Stroeve, A.: Modal truncation vectors for reduced dynamic substructure models. In: Structures, Struc- tural Dynamics and Material Conference. 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, Atlanta (2000) 58. Dobson, B.J., Rider, E.: A review of the indirect calculation of excitation forces from measured structural response data. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 204(2), 69–75 (1990). DOI: 10.1243/PIME_PROC_1990_204_080_02 59. Dossogne, T., Jerome, T.W., Lancereau, D.P.T., Smith, S.A., Brake, M.R.W., Pacini, B., Reuß, P., Schwingshackl, C.W.: Experimental assessment of the influence of interface geometries on structural dynamic response. In: M. Allen, R. Mayes, D. Rixen (eds.) Dynamics of Coupled Structures 4, Conference Proceedings of the Society of Experimental Mechanics Series, pp. 255–261. Springer (2017). DOI: 10.1007/978-3-319-54930-9_22 60. Duarte, M.L.M., Ewins, D.J.: Rotational degrees of freedom for structural coupling analysis via finite-difference technique with residual compensation. Mechanical Systems & Signal Processing 14(2), 205–227 (2000). DOI: 10.1006/mssp.1999.1241 61. Elliott, A.S.: Characterisation of structure borne sound source in-situ. Ph.D. thesis, University of Salford (2009). URL http://usir.salford.ac.uk/id/eprint/26661 62. Elliott, A.S., Moorhouse, A.T.: Characterisation of structure borne sound sources from measurement in-situ. Journal of the Acoustical Society of America 123(5), 3176–3176 (2008). DOI: 10.1121/1.2933261 63. Elliott, A.S., Moorhouse, A.T., Huntley, T., Tate, S.: In-situ source path contribution analysis of structure borne road noise. Journal of Sound & Vibration 332(24), 6276–6295 (2013). DOI: 10.1016/j.jsv.2013.05.031 64. Elliott, A.S., Moorhouse, A.T., Pavic,´ G.: Moment excitation and the measurement of moment mobilities. Journal of Sound & Vibration 331(11), 2499–2519 (2012). DOI: 10.1016/j.jsv.2012.01.022 65. Enns, D.F.: Model reduction with balanced realizations: An error bound and frequency weighted generalization. Pro- ceedings of the IEEE Conference on Decision and Control pp. 127–132 (1984) 66. Epp, D.S., Ozdoganlar, O.B., Chaplya, P.M., Hansche, B.D., Carne, T.G.: A base excitation test facility for dynamic testing of microsystems. In: IMAC-XXII: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2004) 67. Ersal, T., Fathy, H.K., Rideout, D.G., Louca, L.S., Stein, J.L.: A review of proper modeling techniques. J Dyn Syst Measur Control Trans ASME 130(6) (2008). DOI: 10.1115/1.2977484 68. Ewins, D.J.: Modal Testing: Theory, Practice and Application, 2nd edn. John Wiley & Sons, Inc., New York, London, Sydney (2000) 69. Ewins, D.J., Inman, D.J.: Structural dynamics @ 2000: current status and future directions. Research Studies Press, Baldock, Hertfordshire, England (2001) 70. Fahy, F.J.: The vibro-acoustic reciprocity principle and applications to noise control. Acta Acustica united with Acustica 81(6), 544–558 (1995) 71. Fahy, F.J.: Some applications of the reciprocity principle in experimental vibroacoustics. Acoustical Physics 49(2), 217–229 (2003). DOI: 10.1134/1.1560385 72. Farhat, C., Avery, P., Chapman, T., Cortial, J.: Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency. International Journal for Numerical Methods in Engineering 98(9), 625–662 (2014). DOI: 10.1002/nme.4668. URL http://onlinelibrary. wiley.com/doi/10.1002/nme.4668/abstract 200 References

73. Farhat, C., Chapman, T., Avery, P.: Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models. International Journal for Numerical Methods in Engineering 102, 1077–1110 (2015). DOI: 10.1002/nme.4820. URL http: //onlinelibrary.wiley.com/doi/10.1002/nme.4820/abstract 74. Ferri, A.A.: Friction damping and isolation systems. Journal of Mechanical Design 117, 196–206 (1995). DOI: 10.1115/1.2836456 75. Festjens, H., Chevallier, G., Dion, J.L.: A numerical tool for the design of assembled structures under dynamic loads. International Journal of Mechanical Sciences 75, 170–177 (2013). DOI: 10.1016j.ijmecsci.2013.06.013 76. Gajdatsy, P., Janssens, K., Gielen, L., Mas, P., Van der Auweraer, H.: Critical assessment of operational path analysis: effect of neglected paths. In: Proceedings of the XV International Congress on Sound and Vibration (ICSV), Daejeon, Korea, pp. 1090–1097 (2008) 77. Gajdatsy, P., Janssens, K., Gielen, L., Mas, P., Van Der Auweraer, H.: Critical assessment of operational path analysis: effect of coupling between path inputs. Journal of the Acoustical Society of America 123(5), 3876 (2008). DOI: 10.1121/1.2935777 78. Gaudin, A., Beniguel, J.F.: Low frequency road noise decomposition at wheel center on an roller bench. In: Acous- tics 2012. Société Française d’Acoustique, Nantes, France (2012). URL https://hal.archives-ouvertes.fr/ hal-00811236 79. Gaul, L., Lenz, J.: Nonlinear dynamics of structures assembled by bolted joints. Acta Mechanica 125(1), 169–181 (1997). DOI: 10.1007/BF01177306 80. Géradin, M., Cardona, A.: Flexible multibody dynamics: a finite element approach. Wiley (2001) 81. G’eradin, M., Rixen, D.: Mechanical Vibrations: Theory and Application to Structural Dynamics, 2nd Edition. 0-471- 97524-9. Wiley (1997) 82. Géradin, M., Rixen, D.: Mechanical Vibrations. Theory and Application to Structural Dynamics, 3d edn. Wiley & Sons, Chichester (2015) 83. Géradin, M., Rixen, D.J.: Impulse-based substructuring in a floating frame to simulate high frequency dynamics in flexible multibody dynamics. Multibody System Dynamics pp. 1–31 (2017) 84. Gialamas, T., Tsahalis, D., Otte, D., Van der Auwaraer, H., Manolas, D.: Substructuring technique: improvement by means of singular value decomposition (svd). Applied Acoustics 62(10), 1211–1219 (2001) 85. Gibanica, M., Abrahamsson, T.J.S.: Parameter estimation and uncertainty quantification of a subframe with mass loaded bushings. In: R.B. (ed.) Proceedings of 32nd IEEE conference on decision and control, vol. 3, pp. 61–76. Springer International Publishing (2017) 86. Gibanica, M., Abrahamsson, T.J.S., j. Allemang, R.: Residual states for modal models indentified from accelerance data. In: M. Maines, B. Dilworth (eds.) Topics in Modal Analysis & Testing, vol. 9. Springer International Publishing (2018) 87. Gibanica, M., Abrahamsson, T.J.S., Kammer, D.C.: Redundant information rejection in sensor localisation using system gramians. In: M. Mains (ed.) Topics in Modal Analysis and Testing, vol. 10. Springer International Publishing (2016) 88. Ginsberg, J.H.: Mechanical and Structural Vibrations, first edn. John Wiley and Sons, New York (2001)

89. Glover, K.: All optimal hankel-norm approximations of linear multivariable system and their L∞-error bounds. Interna- tional Journal of Control 39, 1115–1193 (1984) 90. Golub, G.H., van Loan, C.F.: Matrix computations, vol. 3. John Hopkins University Press (2012) 91. Gordis, J.H.: Integral equation formulation for transient structural synthesis. AIAA Journal 33(2), 320–324 (1995) 92. Gordon, R.W., Hollkamp, J.J.: Reduced-order models for acoustic response prediction. AFRL-RB-WP-TR-2011-3040 (July 2011) 93. Groper, M.: Microslip and macroslip in bolted joints. Experimental Mechanics 25(2), 171–174 (1985). DOI: 10.1007/BF02328808 94. Groult, B.: Extension d’une mèthode de modiﬁcation structurale pour la conception de dispositifs dissipatifs intègrant des matèriaux viscoèlastiques. Ph.D. thesis, Ecole Centrale Paris, Laboratoire de MÈcanique des Sols, Structures et MatÈriaux (2008) 95. Gruber, F., Rixen, D.: A comparison of craig-bampton approaches for systems with arbitrary viscous damping in dynamic substructuring. In: IMAC-XXXVI: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2018) 96. Gruber, F.M., Rixen, D.J.: Evaluation of substructure reduction techniques with ﬁxed and free interfaces. Strojniški vestnik - Journal of Mechanical Engineering 62(7-8), 452–462 (2016) 97. Gruber, F.M., Rixen, D.J.: Dual craig-bampton component mode synthesis method for model order reduction of nonclas- sically damped linear systems. Mechanical Systems and Signal Processing 111, 678 – 698 (2018) 98. Guasch, O.: Direct transfer functions and path blocking in a discrete mechanical system. Journal of Sound & Vibration 321(3), 854–874 (2009). DOI: 10.1016/j.jsv.2008.10.006 99. Guasch, O., García, C., Jové, J., Artís, P.: Experimental validation of the direct transmissibility approach to classical transfer path analysis on a mechanical setup. Mechanical Systems & Signal Processing 37(1), 353–369 (2013). DOI: 10.1016/j.ymssp.2013.01.006 References 201

100. Guasch, O., Magrans, F.X.: The global transfer direct transfer method applied to a finite simply supported elastic beam. Journal of Sound & Vibration 276(1), 335–359 (2004). DOI: 10.1016/j.jsv.2003.07.032 101. Guyan, R.: Reduction of mass and stiffness matrices. AIAA Journal 3(2), 380 (1965) 102. Haeussler, M., Klaassen, S., Rixen, D.: Comparison of substructuring techniques for experimental identification of rubber isolators dynamic properties. In: P.S. et al. (ed.) International Conference on Noise and Vibration Engineering, ISMA. KUL, Leuven, Belgium (2018) 103. Halmos, P.: Finite Dimensional Vector Spaces. Van Nostrand, New York (1958) 104. Harrison, M., Sykes, A.O., Martin, M.: Wave effects in isolation mounts. Journal of the Acoustical Society of America 24(1), 62–71 (1952). DOI: 10.1121/1.1906850 105. Hartwigsen, C.J., Song, Y., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Experimental study of non-linear effects in a typical shear lap joint configuration. Journal of Sound and Vibration 277(1–2), 327–351 (2004). DOI: 10.1016/j.jsv.2003.09.018 106. Harvie, J., Avitabile, P.: Effects of precise FRF measurements for frequency based substructuring. In: Topics in Experi- mental Dynamic Substructuring, Proceedings of the 31st IMAC, A Conference on Structural Dynamics, vol. 2, chap. 22, pp. 277–285. Springer New York (2013). DOI: 10.1007/978-1-4614-6540-9_22 107. Helderweirt, S., Van der Auweraer, H., Mas, P., Bregant, L., Casagrande, D.: Application of accelerometer-based rotational degree of freedom measurements for engine subframe modelling. In: Proceedings of the XIX International Modal Analysis Conference (IMAC), St. Louis, MO, vol. 2, pp. 1298–1304. Society for Experimental Mechanics, Bethel, CT (2001) 108. Herting, D.: A general purpose, multi-stage, component modal synthesis method. Finite elements in analysis and design 1(2), 153–164 (1985) 109. Hixson, E.L.: Mechanical impedance and mobility. In: C.M. Harris, C.E. Crede (eds.) Shock and Vibration Handbook, 1st edn., chap. 10, pp. 1–59. McGraw-Hill (1961) 110. Höller, C., Gibbs, B.M.: Indirect determination of the mobility of structure-borne sound sources. Journal of Sound and Vibration 344, 38–58 (2015) 111. Hollkamp J. J., G.R.W., Spottswood, S.M.: Nonlinear modal models for sonic fatigue response prediction: a comparison of methods. Journal of Sound and Vibration 284, 1145–63 (2005) 112. Holzwarth, P., Eberhard, P.: Svd-based improvements for component mode synthesis in elastic multibody systems. Eu- ropean Journal of Mechanics-A/Solids 49, 408–418 (2015) 113. Hong, S.K., Epureanu, B.I., Castanier, M.P.: Next-generation parametric reduced-order models. Mechanical Systems and Signal Processing 37(1–2), 403–421 (2013) 114. Hurty, W.C.: Dynamic analysis of structural systems using component modes. AIAA journal 3(4), 678–685 (1965) 115. Idelsohn, S.R., Cardona, A.: A load-dependent basis for reduced nonlinear structural dynamics. Computers & Struc- tures 20, 203–210 (1985). DOI: 10.1016/0045-7949(85)90069-0. URL http://www.sciencedirect.com/science/ article/pii/0045794985900690# 116. Idelsohn, S.R., Cardona, A.: A load-dependent basis for reduced nonlinear structural dynamics. Computers & Structures 20(1), 203 – 210 (1985). DOI: https://doi.org/10.1016/0045-7949(85)90069-0. URL http://www.sciencedirect. com/science/article/pii/0045794985900690. Special Issue: Advances and Trends in Structures and Dynamics 117. Idelsohn, S.R., Cardona, A.: A reduction method for nonlinear structural dynamic analysis. Computer Methods in Applied Mechanics and Engineering 49(3), 253–279 (1985). DOI: 10.1016/0045-7825(85)90125-2. URL http://www. sciencedirect.com/science/article/pii/0045782585901252 118. Imregun, M., Robb, D., Ewins, D.: Structural modification and coupling dynamic analysis using measured frf data. In: IMAC-V: International Modal Analysis Conference, London, England, pp. 1136–1141. Society for Experimental Mechanics, Bethel, CT (1987) 119. Ind, P., Ewins, D.: Impedance based decoupling and its application to indirect modal testing and component measurement: A numerical investigation. In: IMAC-XXI: International Modal Analysis Conference, Kissimmee, FL. Society for Experimental Mechanics, Bethel, CT (2003) 120. Irons, B.: Structural eigenvalue problems-elimination of unwanted variables. AIAA journal 3(5), 961–962 (1965) 121. Ismail, M., Ikhouane, F., Rodellar, J.: The hysteresis Bouc-Wen model, a survey. Archives of Computational Methods in Engineering 16(2), 161–188 (2009). DOI: 10.1007/s11831-009-9031-8 122. ISO Technical Committee 43 / Subcommittee 1 / Workgroup 22 (ISO/TC43/SC1/WG22): Acoustics – Characterization of sources of structure-borne sound with respect to sound radiation from connected structures – Measurement of velocity at the contact points of machinery when resiliently mounted. ISO 9611, International Standards Organisation (1996) 123. ISO Technical Committee 43 / Subcommittee 1 / Workgroup 22 (ISO/TC43/SC1/WG22): Measurement of noise emitted by accelerating road vehicles – Engineering method – Part 1: M and N categories. ISO 362-1, International Standards Organisation (2015) 124. Iwan, W.D.: A distributed-element model for hysteresis and its steady-state dynamic response. Journal of Applied Mechanics 33(4), 893–900 (1966). DOI: 10.1115/1.3625199 202 References

125. Jakobsson, H., Bengzon, F., Larson, M.G.: Adaptive component mode synthesis in linear elasticity. International J. Numer. Methods Engineering 86, 829–844 (2011) 126. Janssens, M.H.A., Verheij, J.W.: A pseudo-forces methodology to be used in characterization of structure-borne sound sources. Applied Acoustics 61(3), 285–308 (2000). DOI: 10.1016/S0003-682X(00)00035-9 127. Janssens, M.H.A., Verheij, J.W., Loyau, T.: Experimental example of the pseudo-forces method used in characterisation of a structure-borne sound source. Applied Acoustics 63(1), 9–34 (2002). DOI: 10.1016/S0003-682X(01)00023-8 128. Janssens, M.H.A., Verheij, J.W., Thompson, D.J.: The use of an equivalent forces method for the experimental quantification of structural sound transmission in ships. Journal of Sound & Vibration 226(2), 305–328 (1999). DOI: 10.1006/jsvi.1999.2303 129. Jetmundsen, B., Bielawa, R.L., Flannelly, W.G.: Generalized frequency domain substructure synthesis. Jnl. American Helicopter Society 33(1), 55–64 (1988) 130. Jewell, E., Allen, M.S., Lacayo, R.: Predicting damping of a cantilever beam with a bolted joint using quasi-static modal analysis. In: Proceedings of the ASME 2017 International Design Engineering Technical Conference IDETC 2017, Cleveland, OH, USA (2017) 131. Jones, D.I.G.: Handbook of Viscoelastic Vibration Damping. Wiley, Chichester (2001) 132. Kailath, T.: Linear Systems. John Wiley and Sons (1980) 133. Kalaycıoglu,˘ T., Özgüven, H.N.: Experimental verification of a recently developed frf decoupling method for nonlinear systems. In: IMAC-XXXVI: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechan- ics, Bethel, CT (2018) 134. Kammer, D.C.: Input force reconstruction using a time domain technique. Journal of Vibration and Acoustics 120(4), 868–874 (1998). DOI: 10.1115/1.2893913 135. Kammer, D.C.: Sensor set expansion for modal vibration testing. Mechanical Systems and Signal Processing 19, 700– 713 (2005) 136. Kay, S.M.: Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory, 1st edn. Prentice Hall, Engle- wood Cliffs (1993) 137. Keersmaekers, L., Mertens, L., Penne, R., Guillaume, P., Steenackers, G.: Decoupling of mechanical systems based on in- situ frequency response functions: The link-preserving, decoupling method. Mechanical Systems and Signal Processing (2015) 138. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes. part i. a useful framework for the structural dynamicist. Mechanical Systems and Signal Processing 23(1), 170–94 (2009). Copyright 2008, The Institution of Engineering and Technology 10330862 0888-3270 nonlinear normal modes NNMs large-scale engineering structures time-frequency analysis structural dynamics nonlinear vibrating structures wavelet transform modal analysis 139. Kim, J.H., Kim, J., Lee, P.S.: Improving the accuracy of the dual craig-bampton method. Computers & Structures 191, 22–32 (2017) 140. de Klerk, D.: Dynamic response characterization of complex systems through operational identification and dynamic substructuring. Ph.D. thesis, Delft University of Technology, The Netherlands (2009). URL http://resolver.tudelft. nl/uuid:2e4dd63e-68d5-41a4-8201-1e784f2f752c 141. de Klerk, D., Lohrmann, M., Quickert, M., Foken, W.: Application of operational transfer path analysis on a classic car. In: DAGA. Rotterdam (2009) 142. de Klerk, D., Ossipov, A.: Operational transfer path analysis: Theory, guidelines and tire noise application. Mechanical Systems & Signal Processing 24(7), 1950–1962 (2010). DOI: 10.1016/j.ymssp.2010.05.009. Special Issue: ISMA 2010 143. de Klerk, D., Rixen, D., Voormeeren, S., Pasteuning, F.: Solving the RDoF problem in experimental dynamic substructuring. In: Proceedings of the XXVI International Modal Analysis Conference (IMAC), Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2008) 144. de Klerk, D., Rixen, D.J.: Component transfer path analysis method with compensation for test bench dynamics. Me- chanical Systems & Signal Processing 24(6), 1693–1710 (2010). DOI: 10.1016/j.ymssp.2010.01.006 145. de Klerk, D., Rixen, D.J., de Jong, J.: The frequency based substructuring (FBS) method reformulated according to the dual domain decomposition method. In: International Modal Analysis Conference, IMAC-XXV. SEM, St Louis, MO (2006) 146. de Klerk, D., Rixen, D.J., Voormeeren, S.N.: General framework for dynamic substructuring: History, review and clas- sifcation of techniques. AIAA Journal 46(8), 1169–1181 (2008). DOI: 10.2514/1.33274 147. de Klerk, D., Voormeeren, S.: Uncertainty propagation in experimental dynamic substructuring. In: Proceedings of the XXVI International Modal Analysis Conference (IMAC), Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2008) 148. Klerk, D.D., Rixen, D.J., Voormeeren, S.N.: General framework for dynamic substructuring: History, review and classification of techniques. AIAA Journal 46(5), 1169–1181 (2008) 149. Klosterman, A.L.: A combined experimental and analytical procedure for improving automotive system dynamics. Ph.D. thesis, University of Cincinnati, Department of Mechanical Engineering (1971) References 203

150. Krattiger, D., Wu, L., Zacharczuk, M., Buck, M., Kuether, R.J., Allen, M.S., Tiso, P., Brake, M.R.W.: Interface reduction for hurty/craig-bampton substructured models: Review and improvement. Mechanical Systems and Signal Processing 114, 579–603 (2019) 151. Kuether, R.J., Allen, M.S., Hollkamp, J.J.: Modal Substructuring for Geometrically Nonlinear Finite Element Models. AIAA Journal 54(2), 691–702 (2016). DOI: 10.2514/1.J054036. URL http://dx.doi.org/10.2514/1.J054036 152. Kuether, R.J., Allen, M.S., Hollkamp, J.J.: Modal substructuring of geometrically nonlinear finite element models with interface reduction. AIAA Journal (accepted) (2017) 153. Kuether, R.J., Renson, L., Detroux, T., Grappasonni, C., Kerschen, G., Allen, M.S.: Nonlinear Normal Modes, Modal Interactions and Isolated Resonance Curves. Journal of Sound and Vibration 351(299-310) (2015). DOI: 10.1016/j.jsv.2015.04.035. URL http://dx.doi.org/10.1016/j.jsv.2015.04.035 154. L., W.E., Ming-Wu, Y., M., D.J.: Dynamic analysis by direct superposition of ritz vectors. Earthquake Engineering & Structural Dynamics 10(6), 813–821 (1982). DOI: 10.1002/eqe.4290100606. URL https://onlinelibrary.wiley. com/doi/abs/10.1002/eqe.4290100606 155. Lacayo, R.M., Allen, M.S.: Updating structural models containing nonlinear iwan joints using quasi-static modal analysis. Mechanical Systems and Signal Processing Submitted Aug 2017 (2018) 156. Lacayo, R.M., Deaner, B.J., Allen, M.S.: A numerical study on the limitations of modal Iwan models for impulsive excitations. Journal of Sound and Vibration 390, 118–140 (2017). DOI: 10.1016/j.jsv.2016.11.038 157. Leclère, Q., Roozen, N.B., Sandier, C.: Experimental estimation of transmissibility matrices. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium (2012) 158. Lew, A., Marsden, J., Ortiz, M., West, M.: Variational time integrators. International Journal for Numerical Methods in Engineering 60(1), 153–212 (2004) 159. Liljerehn, A.: Machine tool dynamics - a constrained state-space substructuring approach. Ph.D. thesis, Chalmers Uni- versity of Technology, Dept Applied Mechanics (2016) 160. Liljerehn, A., Abrahamsson, T.: Dynamic sub-structuring with passive state-space components. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, pp. 3879–3890 (2014) 161. Lim, T.C., Steyer, G.C.: An improved numerical procedure for the coupling of dynamic components using frequency response functions. In: IMAC-IX: International Modal Analysis Conference, Florence,Italy, pp. 355–355. SEM SOCIETY FOR EXPERIMENTAL MECHANICS INC (1992) 162. Lindberg, E., Hörlin, N.E., Göransson, P.: Component mode synthesis using undeformed interface coupling modes to connect soft and stiff substructures. Shock and Vibration 20, 157–170 (2013) 163. Liu, J., Li, M., Qin, L., Liu, J.: Principle research on a single mass piezoelectric six-degrees-of-freedom accelerometer. Sensors 13(8), 10,844–10,855 (2013). DOI: 10.3390/s130810844 164. Liu, W., Ewins, D.: The importance assessment of RDOF in FRF coupling analysis. In: Proceedings of the XVII In- ternational Modal Analysis Conference (IMAC), Kissimmee, FL, pp. 1481–1487. Society for Experimental Mechanics, Bethel, CT (1999) 165. Ljung, L.: System Identification: Theory for the User. Pearson Education (1998) 166. MacNeal, R.H.: Special issue on structural dynamics a hybrid method of component mode synthesis. Computers & Structures 1(4), 581–601 (1971) 167. Madsen, M.B.: Electrical power assisted steering – dynamic source strength characteristic and vehicle NVH prediction. Master’s thesis, University of Southern Denmark (2014) 168. Magrans, F.X.: Method of measuring transmission paths. Journal of Sound & Vibration 74(3), 321–330 (1981). DOI: 10.1016/0022-460X(81)90302-3 169. Mahmoudi, A.E., Meyer, C.H., Rixen, D.J.: Comparison of different approaches to include rubber bushings into frequency-based substructuring coupling process. Experimental Techniques submitted (2019) 170. Maia, N.M.M., Silva, J.M.M., Ribeiro, A.M.R.: The transmissibility concept in multi-degree systems. Mechanical Sys- tems & Signal Processing 15(1), 129–137 (2001). DOI: 10.1006/mssp.2000.1356 171. Martinez, D.R., Carne, T.G., Gregory, D.L., Miller, A.K.: Combined experimental/analytical modeling using component mode synthesis (1984). Compilation and indexing terms, Copyright 2007 Elsevier Inc. All rights reserved 84060103006 COMPUTED MODAL FREQUENCY ACCURACY MODE SHAPES SIMPLE BEAM STRUCTURE FREE-FREE MODES EXPERIMENTAL MODAL DATA BASE FINITE ELEMENT MODEL 172. Martinez, D.R., Miller, A.K., Carne, T.G.: Combined experimental and analytical modeling of shell/payload structures (1985) 173. Mas, P., Sas, P.: Indirect force identification based upon impedance matrix inversion: a study on statistical and determin- istical accuracy. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium (1994) 174. Mayes, R.: A modal craig-bampton substructure for experiments, analysis, control and specifications (2015) 175. Mayes, R., Linehan, D.: Measuring effective mass of a circuit board (2014) 176. Mayes, R., Ross, M.: Advancements in hybrid dynamic models combining experimental and finite element substructures. Mechanical Systems and Signal Processing 31, 56–66 (2012) 204 References

177. Mayes, R., Schoenherr, T., Blecke, J., Rohe, D.: Efficient method of measuring effective mass of a system (2013) 178. Mayes, R.L., Allen, M.S., Kammer, D.C.: Eliminating indefinite mass matrices with the transmission simulator method of substructuring (2012) 179. Mayes, R.L., Arviso, M.: Design studies for the transmission simulator method of experimental dynamic substructuring. In: International seminar on modal analysis, ISMA. KUL, Leuven (2010) 180. Mayes, R.L., Johansen, D.D.: A modal parameter extraction algorithm using best-fit reciprocal vectors (1998). Requires guessing initial frequencies, mode vectors and damping ratios and using optimization to improve the result. Sounds pretty weak, although it allows one to play around with the result. Check future papers. Interesting links to papers on "Modal Filters" 181. Mayes, R.L., Pacini, B.R., Roettgen, D.R.: A modal model to simulate typical structural dynamic nonlinearity (2016) 182. Mayes, R.L., Stasiunas, E.C.: Lightly damped experimental substructures for combining with analytical substructures (2007) 183. Meggitt, J., A.T.Moorhousea, A.S.Elliott: On the problem of describing the coupling interface between sub-structures: an experimental test for ‘completeness’. In: IMAC-XXXVI: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2018) 184. Meggitt, J., Moorhouse, A.: In-situ dynamic sub-structure decoupling of resiliently coupled assemblies. Mechanical Systems and Signal Processing (in press) (2018) 185. Meggitt, J.W.R., Elliott, A.S., Moorhouse, A.T., Lai, H.K.: In situ determination of dynamic stiffness for resilient elements. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science p. 0954406215618986 (2015) 186. Mignolet M. P., P.A.R.S.A., Spottswood, S.M.: A review of indirect/non-intrusive reduced order modeling of nonlinear geometric structures. Journal of Sound and Vibration 332(10), 2437–2460 (2013) 187. Mondot, J.M., Petersson, B.A.T.: Characterization of structure-borne sound sources: The source descriptor and the coupling function. Journal of Sound & Vibration 114(3), 507–518 (1987). DOI: 10.1016/S0022-460X(87)80020-2 188. Montalvao, D., Ribeiro, A., Maia, N., Silva, J.: Estimation of the rotational terms of the dynamic response matrix. Shock and Vibration 11(3–4), 333–350 (2004). DOI: 10.1155/2004/780837 189. Moore, B.: Principal component analysis in linear systems: Controllability, observability and model reduction. IEEE Transaction of Automatic Control AC-26, 17–32 (1981) 190. Moorhouse, A., Elliot, A.S., Heo, Y.: Intrinsic characterisation of structure-borne sound sources and isolators from in-situ measurements. In: Proceedings of Meetings on Acoustics ICA2013, vol. 19, p. 065053. ASA (2013) 191. Moorhouse, A.T.: On the characteristic power of structure-borne sound sources. Journal of Sound & Vibration 248(3), 441–459 (2001). DOI: 10.1006/jsvi.2001.3797 192. Moorhouse, A.T.: Compensation for discarded singular values in vibro-acoustic inverse methods. Journal of Sound & Vibration 267(2), 245–252 (2003). DOI: 10.1016/S0022-460X(02)01432-3 193. Moorhouse, A.T., Elliott, A.S., Evans, T.A.: In situ measurement of the blocked force of structure-borne sound sources. Journal of Sound & Vibration 325(4–5), 679–685 (2009). DOI: 10.1016/j.jsv.2009.04.035. URL http: //www.sciencedirect.com/science/article/pii/S0022460X09003794 194. Muravyov, A.A., Rizzi, S.A.: Determination of nonlinear stiffness with application to random vibration of geometrically nonlinear structures. Computers & Structures 81(15), 1513–1523 (2003) 195. Nelson, R.B.: Simplified calculation of eigenvector derivatives. AIAA Journal 14(9), 1201–1205 (1976). URL http: //arc.aiaa.org/doi/abs/10.2514/3.7211?journalCode=aiaaj 196. Newmark, N.: Method of computation for structural dynamics. J. Eng. Mech. 85, 67–94 (1959) 197. Nicgorski, D.: Investigation on experimental issues related to frequency response function measurements for frequency based substructuring. Master’s thesis, University of Massachusetts Lowell (2008) 198. Nicgorski, D., Avitabile, P.: Conditioning of FRF measurements for use with frequency based substructuring. Mechanical Systems & Signal Processing 24(2), 340–351 (2010). DOI: 10.1016/j.ymssp.2009.07.013 199. Nicgorski, D., Avitabile, P.: Experimental issues related to frequency response function measurements for frequency- based substructuring. Mechanical Systems and Signal Processing 24(5), 1324 – 1337 (2010) 200. Nicgorski, D., Avitabile, P.: Experimental issues related to frequency response function measurements for frequency-based substructuring. Mechanical Systems & Signal Processing 24(5), 1324–1337 (2010). DOI: 10.1016/j.ymssp.2009.09.006 201. Noor, A.K.: Recent advances and applications of reduction methods. App Mech Rev 47(5), 125–146 (1994) 202. Norton, E.L.: Design of finite networks for uniform frequency characteristic. Tech. rep., Technical Report TM26–0– 1860, Bell Laboratories (1926) 203. Noumura, K., Yoshida, J.: A method of transfer path analysis for vehicle interior sound with no excitation experiment. In: Proceedings of FISITA 2006 World Automotive Congress, F2006D183. JSAE (2006). URL http://www.fisita. com/publications/papers?id=4748 204. O’Callahan, J.C., Avitabile, P., Riemer, R.: System equivalent reduction expansion process (SEREP). In: Proceedings of the VII International Modal Analysis Conference (IMAC), Boston, MA, vol. 1, pp. 29–37. Society for Experimental References 205

Mechanics, Bethel, CT (1989) 205. On, F.J.: Mechanical impedance analysis for lumped parameter multi-degree of freedom/multi-dimensional systems. Tech. rep., NASA Technical Note D-3865 (1967) 206. Otte, D., Leuridan, J., Grangier, H., Aquilina, R.: Prediction of the dynamics of structural assemblies using measured frf-data: some improved data enhancement techniques. In: IMAC-IX: International Modal Analysis Conference, Flo- rence,Italy, pp. 909–918. Society for Experimental Mechanics, Bethel, CT (1991) 207. van Overschee, P., de Moor, B.: Subspace Identification for Linear Systems. Kluwer Academic Publishers (1996) 208. Pacini, B.R., Mayes, R.L., Owens, B.C., Schultz, R.: Nonlinear finite element model updating, part i: Experimental techniques and nonlinear modal model parameter extraction (2017) 209. Pasma, E.A., van der Seijs, M.V., Klaassen, S.W.B., van der Kooij, M.W.: Frequency Based Substructuring with the Virtual Point Transformation, Flexible Interface Modes and the Transmission Simulator (2018) 210. Pasteuning, F.: Experimental dynamic substructuring and its application in automotive research. Master’s thesis, Delft University of Technology, The Netherlands (2007) 211. Peharz, R., Pernkopf, F.: Sparse nonnegative matrix factorization with `0-constraints. Neurocomputing 80, 38–46 (2012) 212. Penne, F.: Shaping the sound of the next-generation BMW. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, pp. 25–39 (2004) 213. Perez, R.A.: Multiscale reduced order models for the geometrically nonlinear response of complex structures (2012) 214. Petersson, B.: On the use of giant magnetostrictive devices for moment excitation. Journal of Sound & Vibration 116(1), 191–194 (1987). DOI: http://dx.doi.org/10.1016/S0022-460X(87)81332-9 215. Petersson, B.A.T., Gibbs, B.M.: Use of the source descriptor concept in studies of multi-point and multi-directional vibrational sources. Journal of Sound & Vibration 168(1), 157–176 (1993). DOI: 10.1006/jsvi.1993.1367 216. Petersson, B.A.T., Gibbs, B.M.: Towards a structure-borne sound source characterization. Applied Acoustics 61(3), 325–343 (2000) 217. Petrov, E.P., Ewins, D.J.: Analytical formulation of friction interface elements for analysis of nonlinear multi-harmonic vibrations of bladed disks. Journal of Turbomachinery 125(2), 364–371 (2003). DOI: 10.1115/1.1539868 218. Plunkett, R.: Interaction between a vibratory machine and its foundation. Noise Control 4(1), 18–22 (1958) 219. Plunt, J.: Finding and fixing vehicle NVH problems with transfer path analysis. Sound & vibration 39(11), 12–16 (2005) 220. Powell, R.E., Seering, W.: Multichannel structural inverse filtering. Journal of Vibration and Acoustics 106(1), 22–28 (1984). DOI: 10.1115/1.3269147 221. Putner, J., Fastl, H., Lohrmann, M., Kaltenhauser, A., Ullrich, F.: Operational transfer path analysis predicting contributions to the vehicle interior noise for different excitations from the same sound source. In: InterNoise12, New York City, NY, no. 3 in INTER-NOISE and NOISE-CON Congress and Conference Proceedings, pp. 2336–2347 (2012). URL http://www.ingentaconnect.com/content/ince/incecp/2012/00002012/00000009/art00034 222. Putner, J., Lohrmann, M., Fastl, H.: Contribution analysis of vehicle exterior noise with operational transfer path analysis. Journal of the Acoustical Society of America 133(5), 3323–3323 (2013). DOI: 10.1121/1.4805558 223. Qu, Z.Q.: Model order reduction techniques: with application in finite element analysis. Springer, New York (2004) 224. Rahrovani, S., Vakilzadeh, M.K., Abrahamsson, T.J.S.: Modal dominancy analysis based on modal contribution to frequency response function 2-norm. Mechanical Systems and Signal Processing 48(1-2), 213–231 (2014) 225. Rayleigh, J.: The Theory of Sound, second edition. Dover (1896) 226. Reuss, P., Zeumer, B., Herrmann, J., Gaul, L.: Consideration of interface damping in dynamic substructuring. In: Topics in Experimental Dynamics Substructuring and Wind Turbine Dynamics, Proceedings of the 30th IMAC, A Conference on Structural Dynamics, vol. 2, chap. 10, pp. 81–88. Springer New York (2012). DOI: 10.1007/978-1-4614-2422-2_10 227. Ribeiro, A.M.R., Maia, N.M.M., Silva, J.M.M.: Experimental evaluation of the transmissibility matrix. In: Proceedings of the XVII International Modal Analysis Conference (IMAC), Kissimmee, FL (1999) 228. Ribeiro, A.M.R., Silva, J.M.M., Maia, N.M.M.: On the generalisation of the transmissibility concept. Mechanical Sys- tems & Signal Processing 14(1), 29–35 (2000). DOI: 10.1006/mssp.1999.1268 229. Ritz, W.: Uber eine neue methode zur lösung gewisser variations probleme der mathematishen physik Journal für die Reine und Angewandte Mathematik 135, 1–61 (1909) 230. Rixen, D.: Substructuring and dual methods in structural analysis. Ph.D. thesis, Université de Liège, Belgium, Collection des Publications de la Faculté des Sciences appliquées, n◦ 175 (1997) 231. Rixen, D.: High order static correction modes for component mode synthesis. In: Fifth World Congress on Computational Mechanics. Vienna, Austria (2002) 232. Rixen, D.: A dual Craig-Bampton method for dynamic substructuring. Journal of Computational and Applied Mathe- matics 168(1-2), 383–391 (2004) 233. Rixen, D.: How measurement inaccuracies induce spurious peaks in frequency based substructuring. In: Proceedings of the XXVI International Modal Analysis Conference (IMAC), Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2008) 206 References

234. Rixen, D., Godeby, T., Pagnacco, E.: Dual assembly of substructures and the fbs method: Application to the dynamic testing of a guitar. In: International Conference on Noise and Vibration Engineering, ISMA. KUL, Leuven, Belgium (2006) 235. Rixen, D.J.: How measurement inaccuracies induce spurious peaks in frequency based substructuring. In: IMAC-XXVII: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2008) 236. Rixen, D.J.: Dual craig-bampton with enrichment to avoid spurious modes. In: IMAC-XXVII: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2009) 237. Rixen, D.J.: Substructuring technique based on measured and computed impulse response functions of components. In: P.S. et al. (ed.) ISMA. K.U. Leuven (2010) 238. Rixen, D.J.: Substructuring using impulse response functions for impact analysis. In: IMAC-XXVIII: International Modal Analysis Conference, Jacksonville, FL. Society for Experimental Mechanics, Bethel, CT (2010) 239. Rixen, D.J.: Interface reduction in the dual craig-bampton method based on dual interface modes. In: IMAC-XXIX: International Modal Analysis Conference, Jacksonville, FL: International Modal Analysis Conference, Jacksonville, FL. Society for Experimental Mechanics, Bethel, CT (2011) 240. Rixen, D.J., Boogaard, A., van der Seijs, M.V., van Schothorst, G., van der Poel, T.: Vibration source description in substructuring: A theoretical depiction. Mechanical Systems & Signal Processing 60–61, 498–511 (2015). DOI: 10.1016/j.ymssp.2015.01.024 241. Rixen, D.J., van der Valk, P.L.: An impulse based substructuring approach for impact analysis and load case simulations. Journal of Sound and Vibration 332, 7174–7190 (2013) 242. Roettgen, D.R., Allen, M.S.: Nonlinear characterization of a bolted, industrial structure using a modal framework. Me- chanical Systems and Signal Processing 84, Part B, 152–170 (2017). DOI: 10.1016/j.ymssp.2015.11.010 243. Roettgen, D.R., Allen, M.S., Kammer, D.C., Mayes, R.L.: Substructuring of a nonlinear beam using a modal iwan framework: Part i: Nonlinear modal model identification (2017) 244. Roozen, N.B., Leclère, Q.: On the use of artificial excitation in operational transfer path analysis. Applied Acoustics 74(10), 1167–1174 (2013). DOI: 10.1016/j.apacoust.2013.04.011 245. Rosenberg, R.M.: Normal modes of nonlinear dual-mode systems. Journal of Applied Mechanics 27, 263–268 (1960) 246. Rubin, S.: Mechanical immittance- and transmission-matrix concepts. Journal of the Acoustical Society of America 41(5), 1171–1179 (1967). DOI: 10.1121/1.1910455 247. Rubin, S.: Improved component-mode representation for structural dynamic analaysis. AIAA Journal 13(8), 995–1006 (1975) 248. Rubin, S.: Improved component-mode representation for structural dynamic analysis. AIAA Journal 13(8), 995–1006 (1975) 249. Sachse, D., Geluk, T., van Wayenberge, T.: Accuracy of inverse load identification techniques for transfer path analysis. In: Automotive Acoustics Conference 2013, 2nd International ATZ Conference. ATZlive (2013) 250. Sanderson, M.A., Fredö, C.R.: Direct measurement of moment mobility: Part I: A theoretical study. Journal of Sound & Vibration 179(4), 669–684 (1995). DOI: 10.1006/jsvi.1995.0043 251. dos Santos, F.L.M., Peeters, B., Lau, J., Desmet, W., Góes, L.C.S.: An overview of experimental strain-based modal analysis methods. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium (2014) 252. Scheuren, J., Lohrmann, M.: Transfer path analysis – experiences, expectations and perspectives. In: International Noise and Vibration Colloquium. SAE Brazil (2014) 253. van Schothorst, G., Boogaard, M.A., van der Poel, G.W., Rixen, D.J.: Analysis of ground vibration transmission in high precision equipment by Frequency Based Substructuring. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, pp. 3501–3514 (2012) 254. Schwarz, H.A.: Gesammelte Mathematische Abhandlungen, vol. 2, pp. 133–143. Springer Verlag, Berlin (1890). First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, volume 15, 1870, pp. 272–286. 255. Sedaghati, R., Soucy, Y., Etienne, N.: Experimental estimation of effective mass for structural dynamics and vibration applications (2003) 256. Segalman, D.J.: A four-parameter Iwan model for lap-type joints. Journal of Applied Mechanics 72(5), 752–760 (2005). DOI: 10.1115/1.1989354 257. Segalman, D.J.: Modelling joint friction in structural dynamics. Structural Control and Health Monitoring 13(1), 430– 453 (2006). DOI: 10.1002/stc.119 258. van der Seijs, M., de Klerk, D., Rixen, D., Rahimi, S.: Validation of current state frequency based substructuring technology for the characterisation of steering gear – vehicle interaction. In: IMAC-XXXI: International Modal Analysis Conference, Garden Grove, California USA. Society for Experimental Mechanics, Bethel, CT (Feb. 11-14, 2013) 259. van der Seijs, M.V.: Experimental dynamic substructuring: Analysis and design strategies for vehicle development. Ph.D. thesis, Delft University of Technology (2016). DOI: 10.4233/uuid:28b31294-8d53-49eb-b108-284b63edf670 260. van der Seijs, M.V., van den Bosch, D.D., Rixen, D.J., de Klerk, D.: An improved methodology for the virtual point transformation of measured frequency response functions in dynamic substructuring. In: M. Papadrakakis, V. Papadopoulos, References 207

V. Plevris (eds.) 4th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earth- quake Engineering (COMPDYN), pp. 4334–4347. Kos Island, Greece (2013). DOI: 10.13140/RG.2.1.2715.3126 261. van der Seijs, M.V., de Klerk, D., Rixen, D.J.: General framework for transfer path analysis: History, theory and classification of techniques. Mechanical Systems & Signal Processing 68–69, 217–244 (2016). DOI: 10.1016/j.ymssp.2015.08.004 262. van der Seijs, M.V., de Klerk, D., Rixen, D.J., Rahimi, S.: Validation of current state frequency based substructuring technology for the characterisation of steering gear – vehicle interaction. In: Topics in Experimental Dynamic Substruc- turing, Proceedings of the 31st IMAC, A Conference on Structural Dynamics, vol. 2, chap. 20, pp. 253–266. Springer New York (2013). DOI: 10.1007/978-1-4614-6540-9_20 263. van der Seijs, M.V., Pasma, E.A., van den Bosch, D.D., Wernsen, M.W.M.: A benchmark structure of validation of experimental substructuring, transfer path analysis and source characterisation techniques. In: Dynamics of Coupled Structures, Proceedings of the 35th IMAC, A Conference and Exhibition on Structural Dynamics, vol. 1, chap. 6, pp. 73–82. Springer New York (2017) 264. van der Seijs, M.V., Pasma, E.A., de Klerk, D., Rixen, D.J.: A robust transfer path analysis method for steering gear vibrations on a test bench. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium (2014). DOI: 10.13140/RG.2.1.4297.8647 265. van der Seijs, M.V., Pasma, E.A., de Klerk, D., Rixen, D.J.: A comparison of two component TPA approaches for steering gear noise prediction. In: Dynamics of Coupled Structures, Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, vol. 4, chap. 7, pp. 71–79. Springer New York (2015). DOI: 10.1007/978-3-319-15209-7_7 266. Shyu, W.H., Ma, Z.D., Hulbert, G.M.: A new component mode synthesis method: Quasi-static mode compensation. Finite Elements Anal. Des. 24(4), 197–298 (1997) 267. Siddhi, V.: A generalized approach for calculation of the eigenvector ensitivity for various eigenvector normaliza- tions. Master’s thesis, University of Missouri - Columbia (2005). URL https://mospace.umsystem.edu/xmlui/ bitstream/handle/10355/4244/research.pdf?sequence=3 268. Sjövall, P., Abrahamsson, T.: Component system identification and state-space model synthesis. Mechanical Systems & Signal Processing 21(7), 2697–2714 (2007). DOI: 10.1016/j.ymssp.2007.03.002 269. Sjövall, P., Abrahamsson, T.: Substructure system identification from coupled system test data. Mechanical Systems and Signal Processing 22, 1–260 (2007) 270. Sjövall, P., Abrahamsson, T.: Substructure system identification from coupled system test data. Mechanical Systems & Signal Processing 22(1), 15–33 (2008). DOI: 10.1016/j.ymssp.2007.06.003 271. Sjövall, P., McKelvey, T., Abrahamsson, T.: Constrained state-space system identification with application to structural dynamics. Automatica 42, 1539–1546 (2006) 272. Skelton, R.E.: Cost decomposition of linear systems with application to model reduction. Int J Control 32(6), 1031–1055 (1980) 273. Skelton, R.E., Yousuff, A.: Component cost analysis of large scale systems. Int J Control 37(2), 285–304 (1983) 274. Smallwood, D.O., Gregory, D.L., Coleman, R.G.: Damping investigations of a simplified frictional shear joint. In: 71st Shock and Vibration Symposium. Arlington, Virginia (2000) 275. Snowdon, J.C.: Vibration isolation: use and characterization. Journal of the Acoustical Society of America 66(5), 1245– 1274 (1979). DOI: 10.1121/1.383546 276. Soliman, J., Hallam, M.: Vibration isolation between non-rigid machines and non-rigid foundations. Journal of Sound & Vibration 8(2), 329–351 (1968). DOI: 10.1016/0022-460X(68)90236-8 277. Song, Y., Hartwigsen, C., McFarland, D., Vakakis, A., Bergman, L.: Simulation of dynamics of beam structures with bolted joints using adjusted Iwan beam elements. Journal of Sound and Vibration 273, 249–276 (2004). DOI: 10.1016/S0022-460X(03)00499-1 278. Stoev, J., Möhring, J.: Hybrid reduction of non-symmetric mimo systems. In: P.S. et al. (ed.) ISMA. K.U. Leuven (2010) 279. Sturm, M., Moorhouse, A.T., Alber, T., Li, F.F.: Force reconstruction using an adaptive algorithm in time domain. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, pp. 17–19 (2012) 280. Sturm, M., Moorhouse, A.T., Kropp, W., Alber, T.: Robust calculation of simultaneous multi-channel blocked force signatures from measurements made in-situ using an adaptive algorithm in time domain. In: Proceedings of the XX International Congress on Sound and Vibration (ICSV), Bangkok, Thailand, vol. 2, pp. 1610–1617 (2013) 281. Su, T.J., Craig, R.: Model reduction and control of flexible structures using krylov vectors. Journal of Guidance, Control, and Dynamics 14(2), 260–267 (1991) 282. Thévenin, M.L.: Sur un nouveau théoreme d’électricité dynamique [On a new theorem of dynamic electricity]. Comptes Rendus hebdomadaires des des Séances de l’Académie des Sciences 97, 159–161 (1883) 283. Thite, A.N., Thompson, D.J.: The quantification of structure-borne transmission paths by inverse methods. part 1: Im- proved singular value rejection methods. Journal of Sound & Vibration 264(2), 411–431 (2003). DOI: 10.1016/S0022- 460X(02)01202-6 284. Thite, A.N., Thompson, D.J.: The quantification of structure-borne transmission paths by inverse methods. part 2: Use of regularization techniques. Journal of Sound & Vibration 264(2), 433–451 (2003). DOI: 10.1016/S0022-460X(02)01203- 208 References

8 285. Thite, A.N., Thompson, D.J.: Selection of response measurement locations to improve inverse force determination. Applied Acoustics 67(8), 797–818 (2006). DOI: 10.1016/j.apacoust.2006.01.001 286. Thompson, D.J., van Vliet, W.J., Verheij, J.W.: Developments of the indirect method for measuring the high frequency dynamic stiffness of resilient elements. Journal of Sound & Vibration 213(1), 169–188 (1998). DOI: 10.1006/jsvi.1998.1492 287. Tiso, P., Rixen, D.J.: Discrete empirical interpolation method for finite element structural dynamics. In: G. Kerschen, D. Adams, A. Carrella (eds.) Topics in Nonlinear Dynamics, Volume 1, pp. 203–212. Springer New York, New York, NY (2013) 288. T.L. Hill A. Cammarano, S.N., Wagg, D.: An analytical method for the optimisation of weakly nonlinear systems. Proceedings of the 9th International Conference on Structural Dynamics - EURODYN 2014 Porto, Portugal (2014) 289. T.L. Hill A. Cammarano, S.N., Wagg, D.: Interpreting the forced response of a two-degree-of-freedom nonlinear oscil- lator using backbone curves. Journal of Sound and Vibration 349, 276–288 (2015) 290. Toivanen, J., Avery, P., Farhat, C.: A multilevel feti-dp method and its performance for problems with billions of degrees of freedom. International Journal for Numerical Methods in Engineering 116(10-11), 661–682 (2018) 291. Tol, ¸S.,Özgüven, H.N.: Dynamic characterization of bolted joints using frf decoupling and optimization. Mechanical Systems and Signal Processing 54, 124–138 (2015) 292. Toselli, A., Widlund, O.: Domain decomposition methods-algorithms and theory, vol. 34. Springer Science & Business Media (2006) 293. Touzé, C., Vidrascu, M., Chapelle, D.: Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models. Computational Mechanics 54(2), 567–580 (2014). DOI: 10.1007/s00466-014-1006-4. URL https://doi.org/10.1007/s00466-014-1006-4 294. Ungar, E., Dietrich, C.: High-frequency vibration isolation. Journal of Sound & Vibration 4(2), 224–241 (1966). DOI: 10.1016/0022-460X(66)90123-4 295. Urgueira, A.P.V.: Dynamic analysis of coupled structures using experimental data. Ph.D. thesis, Imperial College, Lon- don (1989) 296. Vakakis, A.F.: Non-linear normal modes (nnms) and their applications in vibration theory: an overview. Mechanical Systems and Signal Processing 11(1), 3–22 (1997). Compilation and indexing terms, Copyright 2009 Elsevier Inc. Compendex 1997163536841 08883270 Nonlinear normal mode Shock isolation designs Vibrational energy 297. Vakilzadeh, M.K., Rahrovani, S., Abrahamsson, T.J.S.: Modal reduction based on accurate input-output relation preser- vation. In: R. Allemang, J.D. Clerck, C. Niezrecki, A. Wicks (eds.) Topics in Modal Analysis, vol. 7, pp. 333–342. Springer, New York (2014) 298. Vakilzadeh, M.K., Yaghoubi, V., McKelvey, T., Abrahamsson, T.J., Ljung, L.: Experiment design for improved frequency domain subspace system identification of continuous-time systems. pp. 886–891. IFAC-PapersOnLine (2015) 299. van der Valk, P.L., Rixen, D.J.: An impulse based substructuring method for coupling impulse response functions and finite element models. Computer Methods in Applied Mechanics and Engineering 275(0), 113 – 137 (2014) 300. Varga, A.: Enhanced modal approach for model reduction. Math Model Syst 1(2), 91–105 (1995). DOI: 10.1080/13873959508837010 301. Varoto, P.S., McConnell, K.G.: Single point vs. multi point acceleration transmissibility concepts in vibration testing. In: Proceedings of the XVI International Modal Analysis Conference (IMAC), Santa Barbara, CA, vol. 1, pp. 83–90. Society for Experimental Mechanics, Bethel, CT (1998) 302. Verheij, J.W.: Multi-path sound transfer from resiliently mounted shipboard machinery: Experimental methods for analyzing and improving noise control. Ph.D. thesis, Delft University of Technology, The Netherlands (1982). URL http://resolver.tudelft.nl/uuid:30d02e94-2fce-4b88-ab8f-6a58a08f0c06 303. Voormeeren, S.: Dynamic substructuring methodologies for integrated dynamic analysis of wind turbines. Ph.D. thesis, Delft University of Technology (2012) 304. Voormeeren, S., de Klerk, D., Rixen, D.: Uncertainty quantification in experimental frequency based substructuring. Mechanical Systems and Signal Processing 24(1), 106 – 118 (2010) 305. Voormeeren, S., Nortier, B., Rixen, D.: Error estimation and adaptive model reduction applied to offshore wind turbine modeling. In: IMAC-XXXI: International Modal Analysis Conference, Error Estimation and Adaptive Model Reduction Applied to Offshore Wind Turbine Modeling. Society for Experimental Mechanics, Bethel, CT (2013) 306. Voormeeren, S., Rixen, D.: Substructure decoupling techniques - a review and uncertainty propagation analysis. In: IMAC-XXVII: International Modal Analysis Conference, Orlando, FL. Society for Experimental Mechanics, Bethel, CT (2009) 307. Voormeeren, S., Rixen, D.: A family of substructure decoupling techniques based on a dual assembly approach. Me- chanical Systems and Signal Processing 27, 379–396 (2012) 308. Voormeeren, S., Rixen, D.: A family of substructure decoupling techniques based on a dual assembly approach. Me- chanical Systems & Signal Processing 27(18), 379 (2012). DOI: 10.1016/j.ymssp.2011.07.028 References 209

309. Voormeeren, S.N.: Dynamic substructuring methodologies for integrated dynamic analysis of wind turbines. Ph.D. thesis, Delft University of Technology, The Netherlands (2012). DOI: 10.4233/uuid:f45f0548-d5ec-46aa-be7e- 7f1c2b57590d 310. Voormeeren, S.N., van der Valk, P.L.C., Rixen, D.J.: Generalized Methodology for Assembly and Reduction of Compo- nent Models for Dynamic Substructuring. AIAA Journal 49, 1010–1020 (2011) 311. Vorländer, M., Dietrich, P.: Transfer path analysis and synthesis for auralization. In: 39th Congreso Espanol de Acústica 2008, Coimbra, Portugal (2008) 312. Wang, Z., Zhu, P., Liu, Z.: Relationships between the decoupled and coupled transfer functions: Theoretical studies and experimental validation. Mechanical Systems and Signal Processing 98, 936–950 (2018) 313. Wang, Z.W., Wang, J.: Inverse substructure method of three-substructures coupled system and its application in product- transport-system. Journal of Vibration and Control 17(6), 943–951 (2011) 314. Weeger, O., Wever, U., Simeon, B.: On the use of modal derivatives for nonlinear model order reduction. International Journal for Numerical Methods in Engineering pp. n/a–n/a (2016). DOI: 10.1002/nme.5267. URL http://dx.doi. org/10.1002/nme.5267. Nme.5267 315. Weijtjens, W., De Sitter, G., Devriendt, C., Guillaume, P.: Operational modal parameter estimation of mimo systems using transmissibility functions. Automatica 50(2), 559–564 (2014). DOI: 10.1016/j.automatica.2013.11.021 316. Wenneker, F.: Component mode synthesis for geometrically nonlinear structures (2013) 317. Witteveen, W., Pichler, F.: Efﬁcient model order reduction for the nonlinear dynamics of jointed structures by the use of trial vector derivatives. In: IMAC-XXXII: International Modal Analysis Conference, Orlando, FL. Society for Experi- mental Mechanics, Bethel, CT (2014) 318. Woinowski-Krieger, S.: The effect of axial force on the vibration of hinged bars. Journal of Applied Mechanics 17 (1950) 319. ten Wolde, T.: Reciprocity experiments on the transmission of sound in ships. Ph.D. thesis, Delft University of Technology, The Netherlands (1973). URL http://resolver.tudelft.nl/uuid: 6d3d9f26-e631-4043-a3ab-04bd45edde98 320. ten Wolde, T.: Reciprocity measurements in acoustical and mechano-acoustical systems. Review of theory and applications. Acta Acustica united with Acustica 96(1), 1–13 (2010). DOI: 10.3813/AAA.918250 321. ten Wolde, T., Verheij, J.W., Steenhoek, H.F.: Reciprocity method for the measurement of mechano-acoustical transfer functions. Journal of Sound & Vibration 42(1), 49–55 (1975). DOI: 10.1016/0022-460X(75)90301-6 322. Wu, L.: Model order reduction and substructuring methods for nonlinear structural dynamics. dissertation, Delft Univer- sity of Technology (2018) 323. Wu, L., Tiso, P.: Nonlinear model order reduction for ﬂexible multibody dynamics: a modal derivatives approach. Multi- body System Dynamics 36(4), 405–425 (2016). DOI: 10.1007/s11044-015-9476-5. URL https://doi.org/10.1007/ s11044-015-9476-5 324. Zeller, P.: Handbuch Fahrzeugakustik. Springer (2009). DOI: 10.1007/978-3-8348-8657-6 325. Zhen, J., Lim, T.C., Lu, G.: Determination of system vibratory response characteristics applying a spectral-based inverse sub-structuring approach. part i: analytical formulation. International journal of vehicle noise and vibration 1(1), 1–30 (2004) 326. Zhen, J., Lim, T.C., Lu, G.: Determination of system vibratory response characteristics applying a spectral-based inverse substructuring approach. part ii: motor vehicle structures. Int. J. Vehicle Noise and Vibration 1(1/2), 31–67 (2004)