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# From Bond Graph to Equations

With a causal bond graph model, equivalent dynamic equations can easily be derived. To create the dynamic equations, the following steps have to be performed.

 • Denote for every bond its effort and flow pair. You can use names that are obvious or methodically use numbers. Some examples are shown at the right. • Replace elements by corresponding dynamic equations. Use the correct effort and flow description which can be found in the element tables. As an example descriptions for the R-element are given. • Replace the junctions by the correct equations. For 0 junctions the efforts are equal. All flows of the bonds pointing towards the 0 junction should be added and all flows of the bonds pointing from the 0 junction should be subtracted. For 1 junctions the flows are equal. All efforts of the bonds pointing towards the 0 junction should be added and all efforts of the bonds pointing from the 0 junction should be subtracted. Some examples are shown at the right. • Rearrange the equations by removing redundant variables.

## Example

As an example in the figure below a causal bond graph model is shown. We will derive the set of dynamic equations out of this model. 1 Denote all efforts and flows. This is shown below: 2 Write the dynamic equations of the elements and junctions:

## equations

Se

uin = constant , iin = free

R

ur = Re ir

I

il = (1/L) int(ul)

1 junction

iin = ir = il = im

uin - ur - ul - um = 0

GY

um = km ωm

Tm = km im

I

Tj = (1/J) int(ωj)

R

Tr = d ωr

1 junction

ωm = ωj = ωr

Tm - Tl - Tr = 0

 3 Reduce the amount of equations. We replace the equalities of the 1 junctions: iin = ir = il = im= i and ωm = ωj = ωr= ω .

## equations

Se

uin = constant

R

ur = Re i

I

i = (1/L) int(ul)

1 junction

uin - ur - ul - um = 0

GY

um = km ω

Tm = km i

I

Tj = (1/J) int(ω)

R

Tr = d ω

1 junction

Tm - Tl - Tr = 0

The resulting dynamic equations can be used for simulation. 20-sim can automatically extract the dynamic equations out of a bond graph model. The equations can be shown using the Show Equations command or used in the Simulator for simulation.