Complexity of distributed parameter bond graph models

Abstract

Modeling metrics and algorithms that assist the development of dynamic system models are essential for efficient use of modeling and simulation in everyday engineering tasks. Various modeling procedures have been proposed by researchers in order to make the modeling procedure more systematic and easier to use by inexperienced modelers. The author, in particular, has previously developed an energy-based modeling metric called "element activity" that was implemented in the Model Order Reduction Algorithm (MORA). The motivation underlying the development of the element activity and MORA was to address model reduction of nonlinear systems. However, it was also developed for linear systems where analytical expressions for the activity are derived. This property makes it particularly appealing for linear systems where activity is easily calculated by avoiding numerical simulations. The dynamic behavior of distributed parameter systems is approximated by either finite-segment or finite-mode model representations. These models are able to accurately predict the dynamic behavior of the system given that "adequate" segments or modes are included in the model. Frequency-based methodologies can be used to address the complexity of these models. In this case, natural frequencies within the frequency range of interest are retained in the model and modes outside this range are not included. The purpose of the current work is to address the complexity of distributed parameter bond graph models of simple continuous structures such as the longitudinal vibration of a bar. Bond graph models of this system already exist in the literature and the objective of this paper is to identify the necessary complexity (number of segments) using the activity metric. A new modeling procedure is proposed for this type of system where the model starts from simple and the number of segments is increased until the activity based criterion is satisfied. An illustrative example is provided to demonstrate the effectiveness of this new methodology.