Impulse response operators for complex structures
Authors: Maidanik G. , Dickey J. 
Published: October 2005

The response of a structure may be stated in terms of an impulse response function integrated over the external drive or equivalently in terms of an impulse response operator acting on the external drive. A complex structure usually consists of either several elemental structural components that are interconnected (e.g., two welded plates) each admitting to a single response type (e.g., either longitudinal or flexural), an elemental structural component that admits to several response types (e.g., longitudinal and flexural), or a combination of both. To derive the impulse response operator directly is not a simple task, even in these primitive examples. The boundaries between elemental structural components and/or the various response types in each component render the response of the complex structure orderly but multifaceted. Therefore, it is suggested that one may advantageously analyze the response, at a given spatial position in the structure and at a specific instant of time, by superposing the contributions to the response of a number of paths. A path defines a unique type of an impulse response operator. The impulse response operator of a path may be simpler to derive than that derived directly for the complex structure as a whole. In this multipaths analysis the complex structure is described by an impulse response vector operator, each element is an impulse response operator that is associated with a unique path. Correspondingly, the external drive is a vector, and the response is the scalar product of the impulse response vector operator and the external drive vector. A sequential procedure is introduced in which the complex structure and the external drive are further decomposed in the hope of achieving further simplifications. In this procedure one recognizes that a dynamic system may be associated with each type of response in an elemental structural component. With this recognition the complex structure is modeled by a set of dynamic systems. The formalism is then stated in terms of matrices and vectors, e.g., the response is a vector (each element represents the response of a specific dynamic system), the impulse response operator is a matrix (the off-diagonal elements describe the couplings between the dynamic systems), etc. If the dynamic systems are chosen so that each, in isolation, can be described in terms of an eigen-impedance operator, then, in addition, a modal analysis can be applied to the multiple dynamic systems that compose the model of the structure. In the modal analysis, however, the ranks of the impulse response matrix, the response vector, and the drive vector are swollen by the modal count, usually rendering the matrix equation unwieldy. In the parallel wave analysis, the propagations in the dynamic systems are described by impulse response operators that are commensurate with those pertaining to boundlessly extrapolated dynamic systems. The finiteness of the dynamic systems is accounted for by junction matrices; a junction defines the boundaries through which dynamic systems interact either with each other (transmissions) or with self (reflections). As in the modal approach, in this wave approach, the resulting formalism is, again, rather unwieldy. It is shown that considerable reductions and simplifications are attained if the structure can be modeled by spatially one-dimensional dynamic systems.