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Bond Graph Modelling, A Quick Learning: Part 4

In this article, I would focus on extracting the differential equations from the bond graph presented earlier (Part 3). Although, a complete bond graph generates set of first order differential equations, we are interested here to illustrate the equations extraction from bond graph model, which means, we would like to go back, and check, whether the bond graph model in figure 1 represents the same equation as in (1). In return, this effort will also enable us to have more in-depth knowledge of system modeling [1].

mass-spring-damper second order equation


Complete bond graph model of mass-spring-damper system
Figure 1: Complete bond graph model of mass-spring-damper system

In bond graph model, system variables are associated with the integrally causal elements which provide set of first order differential equations. For example, momentum associated with the mass and position of the spring are generalized variables for the model shown in figure 1. Since bond graph considers momentum of integrally causal I-element and displacement of integrally causal C-element as state variables, the two differential equations can be written from figure 1 as,

momentum of integrally causal I-element


displacement of integrally causal C-element


Where, dP2 and dQ5 represents time derivative of momentum and displacement respectively. To extract the differential equation, we need to know effort through bond number 2, e2, and flow through bond number 5, f5. From figure 1, since 1-junction is the effort summing junction, we can write,

effort summing junction


Next step is to find out the expression for e1 and e3. From figure 1, we see that e1 is basically input force, SE1 and e3 and e4 are equal since they are connected to the 0-junction. But from figure 1, e4 is,

effort summing junction


Where, e5 is the spring force, and according to the nature of capacitive element, it can be expressed as, e5=Q5/C5. Q5 is the displacement and C5 is the spring compliance which is inverse of spring stiffness. On the other hand, e6 in equation (5) is the damping force which can be expressed according to the nature of resistive element as, e6=f6*R6. Here, R6 represents the damping coefficient and we need to find out f6 in next steps. From figure 1, f6 is equal to f4 which can be express as,

flow summing junction


From figure 1, f7 is equal to f8 (SF8) and is zero since we assumed the wall is fixed. The next term in equation (6) is f3 which is again equal to f2 from figure 1. Now, according to the nature of inductive element, we can write, f2=P2/I2. Now, all the parameters are identified and after substituting those in equation (4), it becomes

bond graph equation

Now, substituting e2 in equation (2) gives,

bond graph mass-spring-damper equation


Equation (7) exactly represents the equation (1). Here, the first term represents the external force (f), the second and third term represent spring (kx) and damping (cx') force consecutively. Thus, the bond graph method derives the same differential equations that was derived by traditional approach in equation (1) with more clarity on the dynamics of system. [1, 2].

[1] E. G. Ovy, Machine dynamics modeling by bond graph approach for condition monitoring, Master’s thesis, University of Calgary, 2017.
[2] Karnopp, D. C., Margolis, D. L., and Rosenberg, R. C., 2012, “System Dynamics: Modeling and Simulation of Mechatronic Systems”, Fifth Edition, John Wiley and Sons, Inc.

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