Nonlinear Modal Substructuring of Geometrically Nonlinear ... Nonlinear Modal Substructuring of Geometrically

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  • Nonlinear Modal Substructuring of Geometrically Nonlinear Finite

    Element Models


    Robert John Kuether

    A dissertation submitted in partial fulfillment of the requirements for the degree of


    (Engineering Mechanics)

    at the



    Date of final oral examination: 12/11/2014 The dissertation is approved by the following members of the Final Oral Committee:

    Matthew S. Allen, Associate Professor, Engineering Physics

    Daniel C. Kammer, Professor, Engineering Physics

    Melih Eriten, Assistant Professor, Mechanical Engineering

    Krishnan Suresh, Associate Professor, Mechanical Engineering

    Joseph J. Hollkamp, Senior Aerospace Engineer, Air Force Research Laboratory

  • © Copyright by Robert J. Kuether 2015

    All Rights Reserved

  • i


    In the past few decades reduced order modeling (ROM) strategies have been developed

    to create low order modal models of geometrically nonlinear structures from detailed finite

    element models built in commercial software packages. These models are capable of accurately

    predicting responses at a dramatically reduced computational cost, but it is often not

    straightforward to determine which modes must be included in the reduction basis. Furthermore,

    much of the upfront cost associated with these reduced models comes from the static load cases

    applied to the full order model that are used to estimate the nonlinear stiffness coefficients, and

    this cost grows in proportion to the number of component modes used to capture the kinematics.

    Hence, there is strong motivation to include only those modes that are absolutely necessary to

    predict the response. The accuracy of the ROMs and their identified nonlinear stiffness

    coefficients are also sensitive to the amplitude of these static loads. Typically, these issues are

    addressed by directly integrating the full model subject to a small number of representative

    loading scenarios and its response is then compared with that computed by the candidate ROMs

    to evaluate their accuracy. These strategies have been successfully used for many engineering

    applications, but the truth data can be prohibitively expensive to compute and it is difficult to

    ascertain whether the model would work in different loading scenarios. This dissertation

    proposes contributions to the existing ROM strategies in two ways.

    The first contribution is the use of the nonlinear normal mode (NNM) as a metric to

    gauge the convergence of candidate ROMs and to observe similarities and differences between

    them. These may be compared with the NNMs computed from the full order model if it is

    computationally feasible, although these are not necessary. The undamped NNM is defined as a

  • ii

    not necessarily synchronous periodic response to the undamped nonlinear equations of motion.

    Since geometric nonlinearities depend only on displacements, the undamped NNM framework

    serves as an ideal metric for comparison since they are load independent properties of the system

    and capture a wide range of response amplitudes experienced by the structure. While

    superposition and orthogonality do not hold for NNMs, many existing works have shown that the

    NNMs are intimately connected to the damped, forced response of the system. If the NNMs of

    the ROMs converge and coincide with the true NNMs of the full order model over a range of

    frequency and energy, then the model is at least guaranteed to represent the full order model in a

    number of important conditions. This validation framework provides the tools and guidelines to

    create a reduced order model that is orders of magnitude less expensive to integrate for response


    The second contribution of this work is the development of a modal substructuring

    framework that utilizes the existing ROM approaches for geometrically nonlinear finite element

    models. Using the proposed approach, one creates a reduced order model of a large, complicated

    structure by first dividing it into smaller subcomponents. The efficiency of a modal

    substructuring procedure strongly depends on the type of subcomponent modes used to create

    each ROM, and this dissertation explores the use of linear free-interface modes, fixed-interface

    plus constraint modes, and fixed-interface plus characteristic constraint modes. The Implicit

    Condensation and Expansion method is used to fit each subcomponent model by applying a

    series of static forces in the shapes of the subcomponent modes and computing the response

    using the built-in nonlinear static solver in the finite element code. These forces and responses

    are used to estimate the nonlinear stiffness coefficients in the subcomponent modal model. Then

    the geometrically nonlinear, reduced modal equations for each subcomponent are coupled by

  • iii

    satisfying compatibility and force equilibrium. Creating a reduced order model with

    substructuring allows one to build ROMs of simpler subcomponent models that may require

    fewer modes, and hence fewer static load cases needed to fit the nonlinear stiffness terms. This

    procedure is readily applied to geometrically nonlinear models built directly within commercial

    finite element packages, and the nonlinear normal modes of the assembled ROMs serve as a

    convergence metric to evaluate the sufficiency of the basis used to create them.

  • iv


    First off, I would like to give thanks to my advisor, Prof. Matt Allen, for his guidance and

    patience during my time as his Ph.D student. He has taught me many valuable lessons in

    engineering and structural dynamics, and he has been a great example of an honest, hardworking

    and kind person.

    Thank you to my Ph.D committee for taking the time to review the material in this

    document and for providing feedback. I'd like to extend a special thanks to Dr. Joe Hollkamp for

    his helpful insight into reduced order modeling and for sharing his computer codes. His guidance

    was instrumental in the development of this work, and it was a great pleasure to work with him

    during my time as a student. I would also like to thank Dr. Matt Brake and Dr. Gaëtan Kerschen

    for arranging for me to work in their laboratories while I pursued my doctorate.

    I am grateful for receiving support from an Air Force Office of Scientific Research grant

    under grant number FA9550-11-1-0035. I am also thankful for the funding I received from the

    National Physical Science Consortium (NPSC) Fellowship, as well as the J. Gordon Baker


    I have had the great pleasure of meeting so many great and talented colleagues

    throughout the years. These people have helped make my time as a student much more

    enjoyable, so thanks to all for our informative conversations about research, and the great


    Finally I would like to thank my family and loved ones for all their love and support

    during my time as a student. I am grateful for the countless times we've shared together and I

    will keep these memories with me for the rest of my life. This work was made possible because

    of the support from those close to me so I dedicate this dissertation to them with love.

  • v

    Abbreviations and Nomenclature


    AMF Applied Modal Force

    CB Craig-Bampton

    CB-NLROM Craig-Bampton Nonlinear Reduced Order Model

    CC Characteristic Constraint

    CC-NLROM Characteristic Constraint Nonlinear Reduced Order Model

    CPU Central Processing Unit

    DOF Degree-of-freedom

    ED Enforced Displacement

    EMD Enforced Modal Displacement

    EOM Equation of Motion

    FEA Finite Element Analysis

    FEP Frequency-Energy Plot

    ICE Implicit Condensation and Expansion

    MC Milman-Chu

    MIF Mode Indicator Function

    NNM Nonlinear Normal Mode

    NLROM Nonlinear Reduced Order Model

    ROM Reduced Order Model


    rA , rB quadratic and cubic nonlinear stiffness coefficient, respectively

    B modal transformation matrix

    rCD constant modal displacement scaling factor (ED)

    rCLD constant linear displacement scaling factor (ICE)

    rCS constant modal scaling factor (ED)

     tf vector of external forces

  • vi

    cF , staticF vector of static forces

     xf NL vector of nonlinear restoring forces

    rf̂ scaling for the rth mode using (ICE)

     tg half-sine pulse

    H shooting function

    I identity matrix

    K linear stiffness matrix

    K̂ reduced stiffness matrix

    L connectivity matrix

    M linear mass matrix

    M̂ reduced mass matrix

     qN1 quadratic nonlinear modal stiffness matrix

     qN 2 cubic nonlinear modal stiffness matrix P tangent prediction vector

    p vector of generalized membrane coordinates

    q vector of modal coordinates

    uq vec