# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 1 Introduction
The finite element method is employed to embed distributed dynamics of rotating shafts with bending into constitutive laws of bond graphs resulting in a finite element bond graph model [14]. Some papers using bond graphs to obtain the steady state behavior of a system are cited below. In [15], a bond graph procedure was introduced to obtain whether or not the equilibrium state or steady state of a system exists. Causality gives a certain propagation throughout the bond graph, and the bicausal bond graph represents the generalization of the causality selection [16]. Bicausality decouples effort and flow at a bond so that they can be assigned independently at each bond [17]. Hence, bicausality was applied to get the equilibrium state of a mechatronic system in [18]. The steady state of the system modeled by bond graphs whose storage elements have a derivative causality assignment was presented in [19]. A new approach to compute the equilibria and the steady states of a biomolecular system using a bond graph was proposed in [20]. Currently, the analysis of multibody systems has been a challenge in scientific research. It is common that in rigid bodies with joints that determine their displacements angles in three dimensions, the kinematic and dynamic relationships can be very complex [21]. The multibond graph representation contains multidimensional bonds that determine arrays of ordinary bond vectors. Then, the generalized power variables of effort and flow are represented by vectors. Therefore, multibond graphs result in compact modeling with great potential in three-dimensional mechanical multibody systems [22; 23]. Figure 1 shows how multibond graphs allow the compaction of graphical and mathematical modeling. The following papers related to multibond graphs can be cited: The multibond graph notation becomes a direct way to represent the behavior of energy, power, and other physical properties of multiport systems, which was introduced in [21]. The causality assignment of vector bond graphs was proposed in [24]. The bond graph description of rigid body rotation was described in [25]. Some equivalent procedures for junction structures with gyrators were shown in [26]. Multiport resistors, storage elements, transformers, and gyrators can be decomposed into 1- and 2-port elements, which was proposed in [27]. In [28], it was proven that a bond graph with the junction structure and the mathematical representation corresponds to a port-Hamiltonian system. The representation of the real and imaginary part of the phasors for electrical circuits using multibond graphs was presented in [29]. The linearization of a class of non-linear systems represented by multibond graphs was proposed in [30]. In [31], a pseudo bond graph of a greenhouse was elaborated to simulate temperature and relative humidity inside using multiport (three ports) elements to describe the state of a two element (dry air and water vapor) fluid. Some recent papers with multibond graphs are as follows: A bond graph model of a helicopter'ssemi-active suspension and the associated simulations were proposed in [32]. The application of a three-dimensional multi-body bond graph modeling approach for simulating vibration in a horizontal oil well was presented in [33]. In [34], a method for an explicit port-Hamiltonian formulation of multibond graphs was introduced. In this paper, a direct methodology to obtain the steady state of linear time invariant (LTI) systems modeled by multibond graphs is presented. First, Lemma 1 establishes the multiport junction structure of a multibond graph in an integral causality assignment (MBGI) whose multiport storage elements can have integral and derivative causality assignments. From this MBGI, the state space of this system is obtained. With the advantages of the causality of a system in the physical domain, a multibond graph in a derivative causality assignment (MBGD) is proposed. Based on this MBGD, a theorem to obtain the steady state of the state variables of the system is proposed.