New Bond Graph Primitive Elements for Modeling Systems Modeled by Practical Derivatives

Abstract

This paper describes our work in creating and using new bond graph primitive elements to represent time-varying and/or frequency-dependent effects in engineering systems. These phenomena can be mathematically represented by fractional-order differential and integral operators. Equations with such operators arise from the analysis and application of several classes of partial differential equations [1]. Previous researchers (Bagley, Torvik, et. al.) have used this approach to further the modeling of fluid-structure interactions, heat transfer, and related control systems [3–6]. These new primitive elements represent visco-inertial and visco-elastic phenomena, whose constitutive laws are dictated by half-order derivatives and integrals. After a brief overview of the fractional derivative, we continue with a formalized mathematical development of these new primitive elements using an impedance-based approach, which provides further support in the argument for their necessity. This approach provides the system modeler with new tools to widen the range of systems that he can accurately model using a lumped-parameter bond graph approach. We illustrate the application and utility of the approach with an example problem in fluid-structure interactions by presenting bond graph models and corresponding simulations. The simulations reveal that the use of these new elements accurately captures the frequency-dependent behavior of the physical system.