Fig. 2. Derived bond graph.

　

The diesel engine is modelled as an inertial element. I₂ and a torque source S_e1. The flexible coupling is shown to consist of flexibility, C₅, and damping, R₆, with a common velocity difference, f₄. The O-junction between bond 3 and 7 is used to "compute the velocity difference" which is distributed to both C₅ and R₆. The two gears are shown as transformers (TF) with modulus equal to the gear ratios. The propeller is modelled as an inertial element, 133, with and resistance according to the propeller law, R₃₂. The clutch is shown as a block, which we will treat in detail in the next section.

From the derived bond graph, we see that the causal input to the inertia elements I₁₀ and I₁₉ is different. When effort, e, is input to an I-element, the flow. f, is given by a function of the time integral of the effort, e. When f is input, e is a function of the time derivative of f.

This is called integral and derivative causality, respectively. Integral causality is also called preferred causality since nature only integrates, only mathematicians differentiate. When derivative causality must be assigned to an element, this means that the element is not dynamically independent. Thus, the element gives no state variable. As can be seen in Figure 2, the two inertia elements I₁₀ and I₂₆ have derivative causality, and the momentum belonging to these elements are not state variables. Using the relation for the step up gear and assigned causality the dependency is given by

We will now show how the bond graph deals with the presence of derivative causality when writing the state equation for the inertia element 19 following the standard systematic approach.

The state equation for element 19 is now described by other system state variables and known functions, except for the effort e₁₀, being a dynamically dependent element (e₂₀ is the torque transmitted trough the clutch). We will now eliminate e₁₀ from the state equation using a standard approach.

The effort e₁₀ can be obtained by differentiating p₁₀ in Eq.

Substituting e₁₀ from Eq. (5) into Eq. (4) yields the derivative of p₁₉ on each side of the equation sign, and thus and implicit equation which always results from derivative causality. Rearranging the equation, the state equation for p₁₉ becomes

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