# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 2 Modeling in a Bond Graph  
The connection of two components always determines the power interactions. Furthermore, power can flow in any direction. In the bond graph methodology, effort \(e\) and flow \(f\) are denoted as the power variables. The product of these variables represents the power \(P\) flowing into or out of a port. The representation of a bond is shown in Figure 3[40; 41]. Furthermore, the energy variables are necessary, denoted by momentum \(p(t)\) and displacement \(q(t)\) where \(p(t)=\int e(t)dt\) and \(q(t)=\int f(t)dt\). At each port, both an effort and a flow exist. If one of the effort or flow variables is an input, the other will be the output. The relationship is called causality. Hence, effort and flow are in opposite directions. The causal stroke is represented by a short and perpendicular line made at one end of a bond. The direction of the effort variable is indicated by the causal stroke, as shown in Figure 4[40; 41]. Moreover, the sources, dissipation, and storage elements can be modeled in the bond graph, and Table 2 gives these elements with their causal relations. The basic elements of the multibond graphs are described in the next section. \begin{table}
\begin{tabular}{c c c} \hline System & Effort \((e)\) & Flow \((f)\) \\ \hline Mechanical & Force \((F)\) & Ang.velocity \((\omega)\) \\  & Torque \((\tau)\) & Velocity \((\
u)\) \\ \hline Electrical & Voltage \((v)\) & Current \((i)\) \\ \hline Hydraulic & Pressure \((P)\) & Volume flow rate \((Q)\) \\ \hline \end{tabular}
\end{table} 
Table 1: Power variables. Figure 4: Causal bond. Figure 3: Power bond. Then, the composition of bonds results in a multibond according to Figure 3. The multibond corresponds to three axes \((x,y,z)\), and the power variables are given by:  
\[\underline{\varepsilon}(t)=\left[ \begin{array}{c}e^{x}(t)\\ e^{y}(t)\\ e^{z}(t)\end{array} \right];\ \underline{f}(t)=\left[ \begin{array}{c}f^{x}(t)\\ f^{y}(t)\\ f^{z}(t)\end{array} \right]\]  
The power in any multibond is defined by \(P(t)=\underline{\varepsilon}^{T}(t)\underline{f}(t)\), where \(\underline{\varepsilon}^{T}(t)\) is the transpose of \(\underline{\varepsilon}(t)\). To express that the symbols, \"\(1\)\" and \"\(0\)\" represent arrays of \(1\)-junctions and \(0\)-junctions, respectively; they are given by underscores. Then, the junctions in a multibond graph are \((\underline{1},\underline{0})\). A basic multibond graph showing power variables, general multiport elements represented by \(MP\), and junctions is illustrated in Figure 6.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element** & **Causal Form** & **Causal Relation** \\ \hline Effort Source & \(\mathsf{M}\,\mathsf{S}\,\mathsf{e}\,\mathsf{e}\,\mathsf{e}\,\mathsf{e}\, \mathsf{e  
Furthermore, a multibond graph model is constructed of multiport elements, which can be multiport storage elements \((\mathbb{C},\ \mathbb{I})\), multiport dissipation elements \(\mathbb{R}\), and/or multiport sources \(\left(\underline{MS_{e}},\ \underline{MS_{f}}\right)\). These elements are shown in Figure 7[15]. Multiport gyrators are essential elements in some multibond graph models that determine the Eulerian junction structure, as shown in Figure 8[15]. This multiport gyrator is the triangle structure of three one-junctions and three gyrators, internally modulated by the opposite junctions, which represent the gyroscopic forces described by the exterior product in [PERSON]'s equations for rotating coordinate frames. The corresponding constitutive relationship of the multiport gyrator is defined by:  
\[\underline{M}(t) = \left[ \begin{array}{c}M_{1}(t)\\ M_{2}(t)\\ M_{3}(t)\end{array} \right]=X(Jw_{b})\underline{w}(t)\] \[= \left[ \begin{array}{ccc}0&J_{3}w_{b3}&-J_{2}w_{b2}\\ -J_{3}w_{b3}&0&J_{1}w_{b1}\\ J_{2}w_{b2}&-J_{1}w_{b1}&0\end{array} \right]\left[ \begin{array}{c}w_{1}(t)\\ w_{2}(t)\\ w_{3}(t)\end{array} \right]\]  
where \(w_{b}=\left[ \begin{array}{cc}w_{b1}&w_{b2}&w_{b3}\end{array} \right]^{T}\) is constant for this paper.