A minimal set of dynamic equations in systems modelled with bond graphs

Abstract

The current paper presents a procedure for converting to a minimal set the dynamic equations of a system modelled with bond graphs starting from a large set of differential-algebraic equations including any type of topological loops and zero-order causal paths (ZCPs). This minimal set is composed only of differential equations expressed in terms of the degrees of freedom of the system. These equations are suitable for use in real-time simulations. Initially, flows corresponding to inertances and displacements associated with compliances are used to establish the dynamic equations and to find the ZCPs of the system. Two different methods are used to solve the ZCPs. With the first method, Lagrange multipliers are introduced by means of new flows and efforts as break variables of causal paths, adding constraint equations. With the second method, break variables are used directly to open the ZCPs. The procedure developed in the present paper solves the problem originated by the use of Lagrange multipliers with the introduction of break variables that imply the presence of new variables and constraint equations. This procedure, before numerical integration of the dynamic equations, converts the equations to a minimal set in order to gain computational efficiency. Several examples are also provided to illustrate the conversion steps.