THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

A method, called the Eigensystem Realization Algorithm (ERA), is developed for modal parameter identification and model reduction of dynamic systems from test data. A new approach is introduced in conjunction with the singular value decomposition technique to derive the basic formulation of minimum order realization which is an extended version of the Ho-Kalman algorithm. The basic formulation is then transformed into modal space for modal parameter identification. Two accuracy indicators are developed to quantitatively identify the system modes and noise modes. For illustration of the algorithm, examples are shown using simulation data and experimental data for a rectangular grid structure.

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An eigensystem realization algorithm for modal parameter identification and model reduction

Aircraft wings are a compromise that allows the aircraft to fly at a range of flight conditions, but the performance at each condition is sub-optimal. The ability of a wing surface to change its geometry during flight has interested researchers and designers over the years as this reduces the design compromises required. Morphing is the short form for metamorphose; however, there is neither an exact definition nor an agreement between the researchers about the type or the extent of the geometrical changes necessary to qualify an aircraft for the title ‘shape morphing.’ Geometrical parameters that can be affected by morphing solutions can be categorized into: planform alteration (span, sweep, and chord), out-of-plane transformation (twist, dihedral/gull, and span-wise bending), and airfoil adjustment (camber and thickness). Changing the wing shape or geometry is not new. Historically, morphing solutions always led to penalties in terms of cost, complexity, or weight, although in certain circumstances, thes...

A Review of Morphing Aircraft

Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature, but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.

On Dynamic Mode Decomposition: Theory and Applications

We develop a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, dynamic mode decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations---only snapshots in time of observables and actuation data from historical, experimental, or black-box simulations. W...

https://epubs.siam.org/doi/pdf/10.1137/15M1013857

Dynamic Mode Decomposition with Control

Uncertainty quantification using interval modeling with performance sensitivity

NASA and its contractors are working on structural concepts for absorbing impact energy of aerospace vehicles. Recently, concepts in the form of multi-cell honeycomb-like structures designed to crush under load have been investigated for both space and aeronautics applications. Efforts to understand these concepts are progressing from tests of individual cells to tests of systems with hundreds of cells. Because of fabrication irregularities, geometry irregularities, and material properties uncertainties, the problem of reconciling analytical models, in particular LS-DYNA models, with experimental data is a challenge. A first look at the correlation results between single cell load/deflection data with LS-DYNA predictions showed problems which prompted additional work in this area. This paper describes a computational approach that uses analysis of variance, deterministic sampling techniques, response surface modeling, and genetic optimization to reconcile test with analysis results. Analysis of variance provides a screening technique for selection of critical parameters used when reconciling test with analysis. In this study, complete ignorance of the parameter distribution is assumed and, therefore, the value of any parameter within the range that is computed using the optimization procedure is considered to be equally likely. Mean values from tests are matched against LS-DYNA solutions by minimizing the square error using a genetic optimization. The paper presents the computational methodology along with results obtained using this approach.

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A Computational Approach for Model Update of an LS-DYNA Energy Absorbing Cell

A flight flutter experiment at the National Aeronautics and Space Administration (NASA) Dryden Flight Research Center, Edwards, California, used an 18-inch half-span composite model called the Aerostructures Test Wing (ATW). The ATW was mounted on a centerline flight test fixture on the NASA F-15B and used distributed piezoelectric strain actuators for in-flight structural excitation. The main focus of this paper is to investigate the performance of the piezoelectric actuators and test their ability to excite the first-bending and first-torsion modes of the ATW on the ground and in-flight. On the ground, wing response resulting from piezoelectric and impact excitation was recorded and compared. The comparison shows less than a 1-percent difference in modal frequency and a 3-percent increase in damping. A comparison of in-flight response resulting from piezoelectric excitation and atmospheric turbulence shows that the piezoelectric excitation consistently created an increased response in the wing throughout the flight envelope tested. The data also showed that to obtain a good correlation between the piezoelectric input and the wing accelerometer response, the input had to be nearly 3.5 times greater than the turbulence excitation on the wing.

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Ground and Flight Test Structural Excitation Using Piezoelectric Actuators

This paper addresses the identification (realization) of approximate linear models from response data for certain nonlinear dynamic systems. Response characteristics for several typical nonlinear joints are analyzed mathematically and represented by series expansions. The parameters of the series expansion are then compared with the modal parameters of a linear model identified by the Eigensystem Realization Algorithm. The agreement of the identified model and the analytically derived representation is excellent for the cases studied. Also laboratory data from a model which exhibited stiffening behavior was analyzed using the Eigensystem Realization algorithm and Fast Fourier Transform. The laboratory experiment demonstrated the ability of the technique to recover the model characteristics using real data.

Identifying approximate linear models for simple nonlinear systems

Various concepts of the degree of controllability (DOC) have been previously developed for the purpose of choosing actuator locations in the control of large flexible spacecraft. A perturbation technique is presented to efficiently account for the actuator mass in the computation of the fuel-optimal, time-optimal, and energy-optimal DOC definitions. Methods are developed for computing each of the three DOCs, when the mass of the actuator is nonnegligible. The energy-optimal DOC is employed to find optimal actuator locations for a large spacecraft under study by the NASA Langley Research Center. The mass of the actuator is seen to have a significant effect on the optimal actuator locations and is shown to spread the actuator locations more evenly for large mass actuators. Results indicate that when the actuator mass is large when compared to the structural mass it should be included in selecting actuator locations.

Lucas G. Horta

Papers

Uncertainty quantification using interval modeling with performance sensitivity

A Computational Approach for Model Update of an LS-DYNA Energy Absorbing Cell

Ground and Flight Test Structural Excitation Using Piezoelectric Actuators

Identifying approximate linear models for simple nonlinear systems

Actuator placement by degree of controllability including the effect of actuator mass