4 Basic Bond Graph Elements

For every energy domain, effort and flow are linked together by three basic Bond Graph elements:

• Resistance (R)
• Capacitive (C)
• Inductance (I)

These Bond Graph elements form the dynamic equations of a system. These dynamic equations are represented using a tetrahedral figure representing variables and their relationships for the linear mechanical domain. Dynamic equations for other domains can be similarly represented (see Appendix A). Various elements used in Bond Graphs are based on these equations.

Figure 5: Mechanical (Linear) variables and associated equation sets — tetrahedron with corners Relative Velocity v, Momentum m, Force F, Relative Displacement q, and center elements R and I, C:

• v = (1/I) × m
• F = R × v
• v = (1/R) × F
• q = ∫ v dt
• m = ∫ F dt
• F = (1/C) × q

4.1 Bond Ports & Multi-ports

Basic Bond Graph elements are represented as ports in Bond Graph modeling. Ports may contain multiple power Bonds to form multi-ports e.g. a filter component has two ports for two power Bonds leading to capacitance and resistance Bond elements.

Table 1: Basic Bond Graph Elements Summary

Category	No. of Elements	Basic Element Type	Relation
Power Discontinuous	1-Port Elements	C-type	Integral relation between Flow & Effort
Power Discontinuous	1-Port Elements	I-type	Integral relation between Effort & Flow
Power Discontinuous	1-Port Elements	R-type	Algebraic relation between Flow & Effort
Power Discontinuous	1-Port Elements	Effort Source (SE)	Fixes Effort independently of Flow
Power Discontinuous	1-Port Elements	Flow Source (SF)	Fixes Flow independently of Effort
Power Discontinuous	1-Port Elements	Effort Sink (SESINK)	-
Power Discontinuous	1-Port Elements	Flow Sink (SFSINK)	-
Power Continuous	2-Port Elements	Transformer (TF)	-
Power Continuous	2-Port Elements	Gyrator (GY)	-
Power Continuous	n-Port Elements	0-Junction	-
Power Continuous	n-Port Elements	1-Junction	-

Effort and Flow sources are considered active sources and sinks of energy respectively, while resistance, capacitance and inductance items are considered passive elements. Sources and sinks represent the boundary conditions of a system.

4.2 Bond Graph Junctions

Power Bonds are connected at Bond Graph junctions. Two types of junctions exist in Bond Graph Modeling: zero-junctions (0-junction) and one-junctions (1-junction). Each junction is set up such that the amount of power coming into the junction equals the amount of power leaving the junction – power creation or storage is not allowed at a Bond Graph junction.

0-junction (Figure 6): Each of the power Bonds connected to a zero junction has equal Effort (E) terms. The flow (F) terms of the power Bonds connected to the zero-junction sum to zero i.e. F1 + F2 + F3 = 0. There can be only one effort variable at a 0-junction – this is indicated by one causal bar at the junction.

1-junction (Figure 7): Each of the power Bonds connected to a one junction have equal flow terms. The effort (E) terms of the power-Bonds connected to a one-junction sum to zero i.e. E1 + E2 + E3 = 0. There can only be one flow variable at the 0-junction – this is indicated by the power Bond without a causal stroke at the junction.

4.2.1 Bond Graph 1-Port Elements

The possible combination of resistive, capacitive and inductive elements for 0-junctions and 1-junctions, including their respective equations:

Table 2: I-, R- & C-elements and their relationships with effort and flow

Element	Junction	Equation
Resistance	1-R	Flow = R × effort
Resistance	0-R	Effort = flow / R
Capacitance	0-C, 0-CR, 1-CR, 1-IRC	Effort = (1/C) ∫ flow
Inertia	1-I, 1-IR, 1-IRC	Flow = (1/I) ∫ effort

Sources & Sinks. Power variables in a dynamic system are assumed to be constant for simulation analysis purposes e.g. voltage applied to an electric motor, or pressure supplying a control valve or pump drive speed. In Bond Graph, these constants are called sources: effort sources (SE) or flow sources (SF). Power state variables, which are independent of the system (not necessarily constant), are regarded as sources – these idealized sources are active 1-port elements.

The environment surrounding the system is considered a source or reservoir of infinite capacity. For example, ambient temperature remains constant irrespective of the amount of heat flow received from the system.

A boundary condition is imposed on the system by the environment. The environment also receives energy flow from the system, which does not affect the system itself. Both aspects are captured by two types of sink elements: effort sinks (SESINK) or flow sinks (SFSINK). A sink is an element in which positive reference direction of energy flow is directed towards itself. While energy sources deliver energy into a system, sinks consume energy flowing out of the system.

Resistance (R-type) elements. R-type elements are used to model resistive power dissipation. Any effect governed by an algebraic relationship between net effort and flow can be classified as a resistive effect. For example, resistive effects in hydraulic systems include friction and pressure drop. In MADE Bond Graphs, effort elements are designated as causes, while flow elements are designated as effects – resistance elements are designated as effects, hence the causal stroke is located at the 'Resistance' end of the power Bond.

Capacitance (C-type) elements. C-type elements are used to model capacitive power storage. A device provides a capacitive effect if net flow into it causes increased effort within itself. Capacitance implies energy storage and thus power storage characteristics. Capacitive power storage devices include mechanical springs, drive shaft in torsion, mechanical compliance, hydraulic accumulators, electric capacitors, and oil-containing volumes in hydraulic control systems. A C-type element can store and give up energy in manners affecting dynamic performance of the system of which they are part.

To comply with simulation procedures that prefer equations that use integration rather than differential forms, the causal bar is placed away from the 'Capacitance' end of the power Bond in MADE, leading to the following integral relation between flow and effort:

E = Cause(Flow) = (1/C) ∫ Flow dt + Effort(0)

Inductance (I-type) elements. I-type elements are used to model inductance or inductive power storage. Inductive power storage occurs when inertias are accelerated, storing up kinetic energy. For example, a fly wheel acting as a rotary mass is considered an inductance element.

To comply with computer solution procedures, the causal bar is placed against the 'Inductance' end of the power Bond in MADE, leading to the following integral relation between effort and flow:

F = I(E) = (1/I) ∫ Effort dt + F(0)

4.2.2 Bond Graph 2-Port Elements

Two types of 2-port elements exist in Bond Graph modeling: transformers (TF) and gyrators (GY). These elements are used at the boundaries of different engineering domains. These are passive elements which do not store or dissipate power – for these elements power input is equivalent to power output.

Transformers. Many major components in control systems are power transforming devices. For example, a hydraulic pump converts mechanical power to hydraulic power, or an ideal mechanical gear train which does not store or dissipate power but rather multiplies angular velocity of the input gear with the output gear. The input torque is the same constant multiplied by the output torque.

Power transformation is represented in Bond Graph as the symbol TF in the center of two power Bonds (Figure 15). Only one causal stroke can be placed against the TF in the middle: either before the TF when the effort variable is being transformed or after the TF when a flow variable is being transformed.

Gyrators. A variation sometimes arises wherein the input effort variable transforms to a flow variable, and the input flow variable transforms to the output effort variable. An example of a gyrator is an ideal electric motor. Angular velocity of the motor shaft is a multiple of input voltage, while motor current is the same constant multiplied by shaft torque.

This power transduction is called a gyrator and its power Bond Graph symbol is GY (Figure 16). For gyrator power transformation, there must either be two causal bars at the GY or two causal bars at the ends of both power Bonds. Two causal bars at the GY describe the input effort variable transforming into an output flow variable. Two causal bars at the end of the power Bonds describe an input flow variable transforming to an output effort variable.