Dynamics of Very High Dimensional Systems

The following sections are included:

• Preface

• Contents

To begin, consider a prototypical DISCRETE system that has played an important role in the development of the subject of dynamics from the time of Galileo, i.e. the pendulum. The basis for a then new revolutionary technology, the mechanical clock, the pendulum is now familiar to every undergraduate who takes a course in dynamics. Here it is used as a starting point in our journey to consider more complex dynamical systems…

In some sense the counterpart of the pendula (or a collection of point masses) as a prototypical model of a DISCRETE system is the string for a CONTINUUM system. Indeed within the framework of linear dynamical models all the essential elements appear in the string. For nonlinear continuum systems there is no corresponding generality given the diverse and fascinating array of possible physical mechanisms that may give rise to nonlinearities. Even so the string example is a useful way to introduce some of the complexities that may arise due to nonlinearities…

This is another opportunity to illustrate the power of Hamilton's Principle in deriving dynamical equations of motion. Hamilton's Principle of course is

The answer to the question of convergence, i.e. how rapidly the terms of a modal series decrease, is critical to the success of a modal representation for either a continuous or a discrete system. Here such matters are considered for the first time. However this question will continue to be addressed throughout this book in various contexts. Indeed the number of terms one needs to retain in a modal series to achieve a desired accuracy is a crucial issue for almost any application…

The following sections are included:

• Eigenvalues and Eigenvectors

• Proper Orthogonal Decomposition

• Second Order (in Time Differentiation) Nonlinear Systems

One can, of course, demonstrate orthogonality by direct calculation of using the eigenfunctions of a continuous system or the eigenvectors of a discrete system. However orthogonality may also be demonstrated using the governing differential equations for the eigenfunctions along with the corresponding boundary conditions or the governing matrix equations for the eigenvectors as will be discussed here.

In our earlier discussion of the string example, the concept of nonlinear normal modes was introduced. This arose inter alia because of the happy coincidence of the linear eigenmodes and the nonlinear normal modes. Here we briefly consider again nonlinear normal modes in a more formal and systematic context. Much of the recent work on this topic is nicely summarized by [Shaw, Pierre and Pescheck, 1999] who themselves have made notable contributions to the advance of this subject. There is a now a burgeoning literature and a recent issue of a prestigious journal is entirely devoted to this topic [Many Authors, Nonlinear Dynamics, 2001]…

This derivation is well known of course, but we particularly want to emphasize the case when the system (and the generalized coordinates) are constrained. For this case the introduction of a Lagrange multiplier has certain advantages and this methodology will prove particularly advantageous when the discussion turns to what is often called “component mode analysis”…

If one expands say the deflection of an elastic structure or an acoustic system in terms of global modes then the kinetic and potential energy take certain characteristic or normal forms within a linear, small amplitude approximation. For definiteness again think of a string, although a beam, membrane or plate as well as an closed acoustic space give rise to similar forms as well…

For simple geometrical shapes of strings, membranes, beams and plates of constant mass and stiffness properties it is often possible to develop analytical formulae for the eigenvalues and eigenmodes. And when the mass and stiffness properties are variable, an efficient solution can be obtained by expanding the deflection of the structure in terms of the eigenmodes of the corresponding uniform property structure. Using Galerkin's method, Lagrange's equations or Hamilton's Principle one can construct a discrete model from this eigenmode expansion and determine a good approximation to the eigenmodes of the continuous, variable property structure. For structures with complex geometries, however, this approach is not as readily applied. Thus finite element, finite difference or finite volume methods are often employed that more readily account for the complex geometrical shapes of automobiles, aerospacecraft, ships, skyscrapers or long span bridges, for example. As has been discussed previously such models may be of very high dimension and determining the eigenvalues may be a challenging task. Nevertheless algorithm developments continue to address this challenge including the POD method discussed in an earlier section of this text…

In our earlier discussion of the convergence of a modal series, we saw that the convergence is rapid for modes whose resonant frequencies lie beyond the range of the frequencies of the exciting forces. Thus often only a relative small number of terms in the modal series is required to give an accurate representation of the dynamical system. However if the exciting forces are random in time and "broadband", i.e. they extend over a wide range of frequencies, then many resonant modes may be excited. Here we consider this case in more depth; in particular we consider what happens as the number of resonant modes within the bandwidth of excitation becomes large. It turns out that an asymptotic limit can be found that has some very interesting and perhaps some unexpected properties. Much of the interest in this subject derives from a theoretical framework that is usually called Statistical Energy Analysis (SEA). The standard reference for SEA is the classic book by [Lyon, 1975] and now in a revised edition [Lyon and De Jong, 1995]. The first author became interested in this subject because of some questions about the underlying basis for SEA and thus considered the case of many resonant modes being excited as an asymptotic limit of classical modal analysis. An early publication is that of [Dowell and Kubota, 1985]. A recent publication is that of [Dowell and Tang,1998] which contains references to the earlier theoretical and experimental literature since 1985.

The interaction between the dynamics of a fluid field and a structure arises in a rich variety of physical contexts. Aerodynamic flows over high performance aircraft may give rise to oscillating forces on the structure and the resulting dynamic deformation of the structure may in turn modify those aerodynamic forces. This feedback between fluid force field and structural deformation may be favorable, e.g. the fluid may increase the damping of the structural motion, or unfavorable, e.g. the fluid-structural system may become dynamically unstable. Both linear and nonlinear oscilllations may arise. Similar issues are confronted in the fluid-structure interaction that occurs in turbomachinery (jet engines and ground based turbine power systems), in heating-cooling devices in nuclear reactors and in long span flexible bridges and tall buildings that sway under wind loadings. Fluid-structure issues also arise in biological systems such as the vibration of a flexible tongue due to air flow in the throat that is relevant to snoring (for some people this vibration can be so severe as to be a medically dangerous condition) and perhaps most famously in blood flow through arteries and fluid flows in the oscillating heart…

The following sections are included:

• Introduction

• Coupled Fluid-Structural Motion of an Acoustic Cavity with a Flexible and/or Absorbing Wall: General Theory
  • Acoustical Problem
  • Structural Modal Expansion
  • Structured Considerations
  • A Variational Formulation

• Acoustic Natural Modes in Multiply Connected Cavities

• Numerical Results for Acoustic Modes and Comparisons with Experiments
  • Acoustic Natural Modes in Multiply Connected Cavities

• Forced Response of a Cavity with a Flexible and/or Absorbing Wall
  • Basic Model
  • Simplified Model
    • Exciting frequency=structural wall resonant frequency
    • Structural resonant frequency changed by coupling with cavity modes
    • Exciting frequency=cavity resonant frequency
    • Effect of cavity wall absorption (damping)
    • Exciting frequency=structural wall and cavity resonant frequencies in very close proximity
    • Convection effects on external sound sources and internal sources
    • Summary of key relations from simplified models to determine cavity pressure

• Numerical Results for Forced Response of a Single Cavity with a Flexible Wall and Comparisons with Experiment
  • Experimental Arrangement
  • Cavity Pressure Measurement
  • Effect of Cavity on Panel Resonant Frequency
  • Panel Damping
  • Cavity Pressure and Damping Effects

• Concluding Remarks

• References

• Appendix 1: Further Data on the Experimental Apparatus of Section 12.6
  • Plate (in Vacuo) Modes
  • Cavity (Rigid Wall) Modes
  • Change in Wall Mode Shapes with Cavity Depth

• Appendix 2: List of Symbols

The following sections are included:

• Introduction

• The Range of Physical Models
  • The Classical Models
  • The Distinction Between Linear and Nonlinear Models
  • Computational Fluid Dynamics Models
  • The Computational Challenge of Fluid Structure Interaction Modeling

• Time-Linearized Models
  • Classical Aerodynamic Theory
  • Classical Hydrodynamic Stability Theory
  • Parallel Shear Flow with an Inviscid Dynamic Perturbation
  • General Time-Linearized Analysis
  • Some Numerical Examples

• Nonlinear Dynamical Models
  • Harmonic Balance Method
  • System Identification Methods
  • Nonlinear Reduced-Order Models

• Reduced-Order Models
  • Constructing Reduced-Order Models
  • Linear and Nonlinear Fluid Models
  • Eigenmode Computational Methodology
  • Proper Orthogonal Decomposition Modes
  • Balanced Modes
  • Synergy Among the Modal Methods
  • Input/Output Models
  • Structured, Aerodynamic, and Aeroelastic Modes
  • Representative Results
  • The Effects of Spatial Discretization and a Finite Computational Domain
  • The Effects of Mach Number and Steady Angle of Attack: Subsonic and Transonic Flows
  • The Effects of Viscosity
  • Nonlinear Aeroelastic Reduced-Order Models

• Concluding Remarks and Directions for Future Research

• References

• Appendix: Singular-Value Decomposition, Proper Orthogonal Decomposition, and Balanced Modes

The following sections are included:

• Introduction

• Scope of this Chapter
  • Airfoil Plus a Control Surface with Freeplay
  • Low Aspect Ratio, Plate-Like Wing
  • High Aspect Ratio, Beam-Like Wing
  • Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence Flutter and Limit Cycle Oscillations

• Concluding Remarks

• Future Work

• References

• Appendix 1: Modeling for Nonlinear Aeroelastic Analysis
  • Modeling of the Structure
  • Modeling of the Fluid
  • Solving the Aeroelastic Equations

• Appendix 2: List of Symbols

The following sections are included:

• Introduction

• Theoretical Model
  • Static Equilibrium Position
  • Small Dynamic Perturbations
  • Reduced-Order (Dynamic) Model
    • Small perturbation, linear reduced-order model with quasi-static correction
    • Nonlinear reduced-order model

• Numerical Analysis
  • Results for I = 100
    • Evaluating the Linearized Perturbation Approach
    • Results from the linear reduced-order model
    • Results from the nonlinear reduced-order model
  • Results for I = 1000
    • Evaluating the Linearized Perturbation Approach
    • Results from the linear reduced-order model
    • Results from the nonlinear reduced-order model

• Concluding Remarks

• References

• Appendix A: Dynamic Perturbation Variables

• Appendix B: Derivation of Dynamic Perturbation Equations

The following sections are included:

• Mathematical Methods

• Physical Media and Models

The following sections are included:

• Epilogue

• Index