4.3 System Model Structure for Bond Graph Simulations

MADE models that are compatible with Bond Graph Simulations recreate the physical structure of a system and display how items are connected in a system. Figure 17 shows a simplified representation of effort or flow sources that provide power to a system. The system contains a junction which links power sources to the dissipative elements (R-type elements) and to energy storage elements (C-type or I-type elements), eventually exiting the system to an effort or flow sink.

Figure 17: General Structure of Bond Graphs — Sources → System Junction(s) → {Energy Store(s), System Dissipater(s)} → Sinks, bounded by System Boundary (MUX bars) on both sides.

The modeling approach for Bond Graph Simulations is demonstrated using the Driveline Subsystem from the Vehicle System training model. Figure 18 shows the block diagram: SE (IN) → Transmission (1-IR) → Driveshaft (0-C) → Differential (1-IR) → splits into two branches (Front/Rear): Half Shaft (0-CR) → Planetary Gearbox (1-IRC) → Wheel Resistance (0-R) → Wheel (1-I) → SES (OUT).

4.3.1 Analyze Power Flows to Build Model Structure

The initial step prior to modeling is to classify all items in the system in terms of power flow to determine which Bond Graph elements to use (see Appendix B for a list of Bond elements and associated power flow/effort for each domain). Based on physical assumptions and the way each item functions regarding power flows, the user can determine whether it can be considered as a power source, storage element, dissipative element, transformer or gyrator.

Transmission component. The function of a Transmission is to provide controlled application of power. The Transmission component receives mechanical effort (torque) from the coupling component (external source – see vehicle system) to provide an output of angular velocity. To capture the inductance of rotating mass and resistance from moving oil and friction, the Transmission is designated as a 1-IR Bond type.

Driveshaft component. The function of the Driveshaft is to transmit torque and rotation to other components in a drivetrain. The Driveshaft component receives mechanical flow (angular velocity) from the Transmission component to provide an output torque. To capture the capacitance of a Driveshaft in torsion, it is designated as a 0-C Bond type.

Differential component. The function of the Differential is to allow the outer drive Wheel to rotate faster than the inner drive Wheel during a turn. This is done by a gear train with three shafts, where the angular velocity of one shaft is either the average of the angular velocities of the other two shafts or is a fixed multiple of that average. The Differential component receives mechanical effort (torque) from the Driveshaft to provide an output of angular velocity. To capture the inductance of rotating gears and resistance from friction, the Differential is designated as a 1-IR Bond type.

Half Shaft component F & R. The Half Shaft components transmit torque from the Differential to the Planetary Gearbox. Each Half Shaft component receives mechanical flow (angular velocity) from the Differential component to provide an output torque. To capture the Half Shafts in torsion, these components are designated as a 0-C Bond type.

Planetary Gearbox component F & R. The Planetary Gearboxes function as reduction Gearboxes, which prevents the shafts and Differential from being subject to high torques. Each Planetary Gearbox receives mechanical effort (torque) from the Half Shafts to provide an output of angular velocity. To capture the gear box in torsion and resistance from friction these components are designated as a 1-IR Bond type.

Wheel Resistance component F & R. The Wheel Resistance Component captures the torsion of the Wheels. Each Wheel Resistance Component receives mechanical flow (angular velocity) from the Planetary Gearboxes and transmits angular velocity as its output. The Wheel Resistance components are designated as a 0-R Bond type.

Wheel component F & R. The Wheel Component functions as the final item that stores and transmits angular velocity. Two components are used to model both the power transformation effects and the resistive effects associated with the Wheel. Each Wheel Component receives mechanical flow (angular velocity) from the Wheel Resistance components and transmits angular velocity as its output. The Wheel components are designated as a 1-I Bond type.

Sources & Sinks (MUX bars). Sources and sinks are captured in the form of input and output flows from MUX bars. The input and output MUX bars are designated as effort sources (SE) and effort sinks (SESINK) respectively. The input MUX bar provides a mechanical rotational effort (torque) while the output MUX bar provides two mechanical rotational flows (angular velocity) for the two component branches.

4.3.2 Create FBD Blocks (Step 2)

Each component in the Driveline Subsystem is recreated as logical blocks using the 'Add New Component' button (Figure 19 shows the resulting logical block layout: Transmission, Driveshaft, Differential, then Half Shaft Front/Rear, Planetary Gearbox Front/Rear, Wheel Resistance Front/Rear, Wheel Front/Rear, bracketed by IN/OUT MUX bars).

4.3.3 Assign Functions & Flows (Step 3 - 6)

For Bond Types to be assigned to an item, the user must first assign functions and flows. To do this:

• Open the Functions Editor
• Assign Functions from the Functions Taxonomy
• Assign input and output flows
• Select and connect input flow properties to output flow properties
• Save changes

4.3.4 Select Bond Type & Enter Passive Variables (Step 8)

To assign a Bond Type to an item:

• Select an item from the System Model editor or Project Explorer Viewer
• Select the Bond tab from the Properties Viewer
• Select the Bond Type drop-down menu and select the appropriate Bond Type from the list
• Enter values and limits for Passive Variables in their respective fields

Passive variables per Bond Type:

Bond Type	C	I	R	Value	Initial Value	Upper	Lower
0-C	✓	-	-	Yes	Yes	Yes	Yes
0-CR	✓	-	✓	Yes	Yes (C only)	Yes (C only)	Yes (C only)
0-J	-	-	-	N/A	N/A	N/A	N/A
0-R	-	-	✓	Yes	N/A	N/A	N/A
1-C	✓	-	-	Yes	Yes	Yes	Yes
1-CR	✓	-	✓	Yes	Yes (C only)	Yes (C only)	Yes (C only)
1-I	-	✓	-	Yes	Yes	Yes	Yes
1-IR	-	✓	✓	Yes	Yes (I only)	Yes (I only)	Yes (I only)
1-J	-	-	-	N/A	N/A	N/A	N/A
1-R	-	-	✓	Yes	N/A	Yes (Limiter only)	N/A
1-IRC	✓	✓	✓	Yes	Yes (C & I only)	Yes (C & I only)	Yes (C & I only)
GY	-	-	-	Yes (Ratio)	N/A	N/A	N/A
SE	-	-	-	Yes (Amplitude)	N/A	N/A	N/A
SES	-	-	-	Yes (Amplitude)	N/A	N/A	N/A
SF	-	-	-	Yes (Amplitude)	N/A	N/A	N/A
SFS	-	-	-	Yes (Amplitude)	N/A	N/A	N/A
TF	-	-	-	Yes (Ratio)	N/A	N/A	N/A

Passive variable term definitions:

• Capacitance (Value): A parameter defining the item's ability to store and release energy
• Inductance (Value): A parameter defining the item's ability to store and release energy
• Resistance (Value): A parameter defining the item's ability to dissipate power
• Initial Value (Capacitance): Initial value of flow on the capacitor
• Upper (Value): Upper limit value of effort on the capacitor/inductor
• Lower (Value): Lower limit value of effort on the capacitor/inductor
• Ratio (Gyrator): Constant value defining the gyrator ratio of effort to output flow or input flow to output effort
• Amplitude (SE/SF/SES/SFS): A value assigned to the effort/flow source/sink
• Ratio (TF): Constant value defining the transformer ratio of input effort to output effort or input flow to output flow

4.3.5 Open Bond Graph Model Editor (Step 10)

This step requires opening the Bond Graph editor to view the Bond Graph and verifying that the model is controllable. To do this:

• Right click on the system model or item in the Project Explorer
• Select Bond Graph from the right-click menu

These actions open the Bond Graph Editor. The editor shows a notification on the top-left side indicating whether the system is controllable based on the Bond types configured. Controllability describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any other final state in a finite time interval.

Notifications shown: "Bond Model is Controllable" or "Bond Model is not Controllable".

The Bond Graph consists of power Bonds connected by Bond elements. The item name is displayed under each Bond element, while each power Bond is automatically labelled with a unique number (used in Bond equations).

The Power Bonds and Bond elements have three distinct colors which indicate different states of a model:

• Black: These power Bonds and Bond elements are valid and are on the same level of indenture (LOI)
• Red: Indicates an invalid Bond type assigned to a Bond junction which causes a mismatch of causal strokes. Power Bonds affected by this error are also highlighted in red.
• Blue: Indicates a Bond type assigned to an item originating from a different level of indenture (LOI)

Figure 24 shows an uncontrollable Vehicle System model with a Problems panel listing errors, e.g.:

• "Causal strokes do not match the bond group assigned to 1 - I (Driveshaft)"
• "Causal strokes do not match the bond group assigned to 1 - IR (Differential)"
• "Convert Fuel to Mechanical Motion (Engine) is missing an enabled flow property"

4.3.6 Generate Response Simulation Graphs (Step 11)

The user can determine response of an item or flow based on Bond Graph configuration of the model. This response is visualized through a graph plotted in the Response Simulation viewer. To do this:

• Right-click on a Bond junction or Bond element in the Bond Graph viewer
• Select Response Simulation from the right-click menu – this will expand an additional sub-menu
• Select the desired flow property to plot in the Response Simulation viewer

(Figure 25 shows a right-click context menu on a Bond junction offering: Response Simulation submenu with items like "6E.Torque", "6F.Angular velocity", "7E.Torque", "7F.Angular velocity", "16E.Torque", "16F.Angular velocity", "23E.Torque", "23F.Angular velocity", "24E.Torque", "24F.Angular velocity"; also Zoom In/Out and Verify Controllability options.)

Figure 26 shows an example Flow Response graph in the Response Simulation viewer, plotting a value rising from 0 toward ~0.7 with a "FAILURE ACTIVATION" marker at x=50, oscillating before settling near 0.7.