Joseph P. Cusumano, Ph.D. - US grants

This Research Initiation Award is aimed at developing better techniques for obtaining finite degree of freedom nonlinear models of flexible structures. Overall, the project has three phases: experiments; signal analysis; and modeling. The modeling phase of the project represents an application of the theory or chaotic vibrations to the problem of nonlinear modeling.

9418674 Duffy This research will test the hypothesis that the rainfall-runoff process can be represented as a low-dimensional dynamical system, forced by topographic, soil, geologic and climatic variability. By "low-dimensional " we mean the minimum number of state variables required to approximate the processes as a system of nonlinear ordinary differential equations. We examine the case where rainfall-runoff is controlled by porous soils and shallow groundwater circulation, and where the ability of the catchment to store water and yield runoff depends on the nature of the storage-flux relationships of the system. The role of evapotranspiration as a parametic function of soil-moisture storage will also be examined. The "low-dimensional" model will serve as a physically-based alternative to the nonlinear partial differential equations (Richard's equation) representing the local processes. In constructing the dynamical model, the essential problem is to separate or distinguish among spatial and temporal components of watershed dynamics, such that the important mechanics of the processes involved are elucidated, without loss of critical nonlinear structure. The research has the following elements: (1) Comprehensive numerical experiments based on finite element solutions to the partial differential equations for saturated-unsaturated flow will be performed To establish terrain-integrated constitutive relations (e.g. storage-flux relations). the proposed research will build on a series of steady-state numerical experiments (Lee, 1993; Duffy, 1994) for two dimensional hillslope geometry with uniform soil properties. This previous work found that recharge to the water table and subsurface flow to the stream were nonlinear functions of at least two state variables: the integrated soil moisture and integrated saturated storage. The present objective is to extend these experiments to the case of fully three dimensional and time varying flow, and test the role of soil stratification and variability on nonlinear storage-flux relations and runoff response. (2) Develop and test procedures for scaling and spatial integration of the state variables and fluxes for the Shale Hills watershed, 8 hectare, forested, catchment in central PA (J. Lynch et al, 1976). Shale Hills was the site of a unique experiment in the 1970's to evaluate the effects of antecedent soil moisture on stormflow volume and timing. Rainfall was artificially applied for 8 events with initial moisture ranging from dry to very wet. a comprehensive accounting of soil moisture and saturated storage at multiple depths was made over the entire watershed. Although if may seem to be a straight forward problem, spatial integration of scattered field observations requires an appropriate weighting function. Duffy (1994) has proposed weighting function derived from the hypsometric distribution, and a local rescaling of hillslope trajectories for each hillslope or zero-order basin. This scaling and averaging method will be carried out for the Shale Hills data base. Field-estimated storage-flux relations will be compared with the numerical experiments in (1). (3) An independent method known as proper orthogonal decomposition (POD), allows the essential spatial structure of the dynamics to be reconstructed directly from random field data or from the governing pde's. The method is widely applied to detecting coherent structures in hydrodynamics turbulence (Lumley, 1967), the evolution of climatic fields (North et al, 1982), and nonlinear vibration (Cusumano, and Bai, 1993, Cusumano et al, 1993, Lin and Cusumano, 1993, Cusumano et al, 1994). The POD method will be applied to the field data of the Shale Hills experiment and the Richard's equation to provide a measure of the dimensionality, or number of state variables required to model the system.

This interdisciplinary research program is focused on the development and application of a new class of nonlinear dynamical system inspired by the study of repeated precision human movements. The fundamental qualitative dynamics of these systems will be examined. Our approach is based on a novel definition of movement tasks in terms of goal functions, which encode the interaction between the body and the environment needed for perfect execution. The resulting task dynamical systems model the trial-to-trial performance of precision goal-directed actions, such as, for example, the repeated throwing of a ball at a target or the hammering of a nail. Using these models, one can examine how performance arises from the interaction between the geometry of task-specific goal equivalent manifolds, passive sensitivity, control and active stability, and intrinsic noise.

New concepts and methods will be developed for modeling and characterizing trial-to-trial variability in a range of different tasks. The resulting theoretical models will be used to study the mechanisms of variability generation in repeated movements, to determine the simplest model features that are capable of exhibiting observed phenomena, and to examine the dynamical implications of specific control assumptions. The theoretical results will be used to develop new experimental methods and to make experimentally testable qualitative predictions. The resulting conceptual framework and analysis methods will enable researchers to, for the first time, untangle the passive mechanical aspects of movement from the perceptual and neurological aspects believed to be related to active control.

This project involves collaboration between researchers in nonlinear dynamics, movement science (kinesiology), and robotics. In addition to addressing fundamental questions in movement science, our effort will lead to: new noninvasive approaches to studying, monitoring, and diagnosing neurological movement disorders; monitoring progress in physical therapy after surgery or injury; and characterizing motor learning and performance in subjects involved in sports and other precision tasks. In the area of technology, our work will find application in the design of biologically-inspired precision machines and advanced man-machine interfaces needed for such applications as remote
tele-surgery. Such machines will be designed to exploit task redundancy and passive stability properties so that they inherently respond to changes in their operating environment, or internal changes caused, for example, by component wear. This, in turn, could lead to improved repeatability, reliability and service life, even for entirely open loop devices.

This grant provides travel support for students and young investigators to attend and present their research at the 16th US National Congress of Theoretical and Applied Mechanics (USNCTAM). The congress will be held June 27 - July 2, 2010 at The Pennsylvania State University in University Park, PA. The funding will be used to help defray registration, hotel, food, and transportation costs of roughly 20 students and young investigators presenting their research at the Congress. Recipients will be selected by the organizers based on an application and review process that considers need and contribution. By encouraging the participation of the next generation of researchers, who typically have limited financial resources for conference travel, the long range interests of NSF in maintaining an engaged research workforce are served.

Mechanics is a dynamic and evolving area of research with impacts in broad areas such as advanced transportation materials, energy storage and generation, and human health care. Fifteen professional societies belong to the US National Committee on Theoretical and Applied Mechanics (USNC/TAM)?the nonfinancial underwriting agency of the Congress. USNC/TAM represents the interests of the United States in international scientific activities relating to the field of mechanics, specifically to the International Union of Theoretical and Applied Mechanics (IUTAM). The committee is the focal point for the U.S. engineering, scientific, and mathematical communities working in mechanics and serves as the national forum for defining major issues in mechanics research, technology, and education. The proposed travel support provides the financial means for young researchers to attend the Congress, intermingle with leading researchers, and advance their professional and intellectual development in areas of national importance.