Matt Allen Daniel Rixen Maarten van der Seijs Paolo Tiso Thomas Abrahamsson Randy Mayes Substructuring in Engineering Dynamics – 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 ap- proaches 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 measure- ments 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 dy- namical 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 fundamen- tal 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 re- duction, McNeal, Rubin, Craig-Chang, interface reduction methods and model reduction in the state space. 4. Experimental-Analytical substructuring including frequency based substructuring or impedance cou- pling, 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 . ..