Break variables

Chapter 9 gives a review on how multibody systems can be modelled by means of multibond graphs. A major contribution of the chapter is a procedure that provides a minimum number of break variables in multibond graphs with ZCPs. For the state variables and these break variables (also called semi-state variables) a DAE system can be formulated that can be solved by means of the backward differentiation formula (BDF) method implemented in the widely used DASSL code. The approach is illustrated by means of a multibond graph with ZCPs of the planar physical pendulum example. 

Keywords Break variables Differential-algebraic systems Lagrange multipliers Multi-body system Multi-bond graphs Zero-order causal paths... 

Initially, flows corresponding to inertances and displacements associated with compliances are used to establish the dynamic equations and to find the zero-order causal paths of the system. Two different methods are used to solve the ZCPs [1, 2]. With the first one, Lagrange multipliers are introduced by means of new flows and efforts as break variables of causal paths, adding constraint equations. With the second one, break variables are used directly to open the ZCPs. 

Algebraic equations that relate the break variables with one another by means of algebraic assignments along the existing topological loops in systems with ZCP classes 2, 3, and 4... 

In (9.26), the lower the number of break variables, the easier it is to solve this system. The purpose of the following algorithms is to open all the existing causal loops in the bond graph by means of the minimum number of break variables. In a later step, the mathematical model will be automatically obtained based on these break variables. These algorithms have been conceived to deal with one-dimensional and multi-bond graph systems. 

The purpose of this first algorithm is to obtain the smallest number of break variables that open all the ZCPs of the system in a systematic way. Achieving this is made pos-... 

Class 2 ZCPs, i.e., those causal paths that begin and end in resistor ports, are sought. Once all the resistors associated with classes 2 or 3 have been analyzed, the program will select as break variable the flow or effort associated with the resistor whose origin port belongs to the greatest number of topological loops of classes 2, 3, and 4 (sum of them). This sum is required to obtain a minimum number of break variables. 

Therefore, the effort in Ri, is chosen as break variable since it opens the only topological loop that appears. So, the first constraint equation of the example is... 

While class 4 ZCPs exist with resistors and storage elements as path initiators If no class 2 or 3 ZCPs exist or they have already been opened, the mechanism to open the class 4 ZCPs will be the following from all the GJS ports belonging to this class of ZCP, those flows or efforts in the ports with most associated class 4 topological loops will be chosen as break variables. 

The same example of Fig. 9.23 will be considered. As the effort in the origin port of (cii) has been chosen as break variable of the class 4 topological loop in efforts, there still remains one class 4 topological loop in flows having Ri as initiator. The possible port candidates to open this algebraic loop are the flows gathering in Oi, O2, and at the input port of the transformer. 

The purpose of this second algorithm is to follow in an iterative way all the causal paths to compute the values of the auxiliary variables that Algorithm 1 will use as a reference to obtain the break variables. These variables are able to open all the existing topological loop classes in the model. 

Basically, the procedure used for one-dimensional bond graph can be used to automatically obtain the minimum number of break variables to generate the system equations in MBG systems. The peculiarities of the procedure in MBG will be the following ... 

Break variable of class 5.4 ZCP effort in P-axis of l2-Therefore, the mathematical model is composed of... 

Two different methods are used to solve the ZCPs. With the first one, Lagrange multipliers are introduced by means of new flows and efforts as break variables of... 

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