Project D24: Network-Based Remodeling of Mechanical and Mechatronic Devices

In industrial applications, the automatic modeling of systems consisting of mechanical or electrical devices or couplings thereof leads to differential-algebraic equations - DAEs. Besides the large scale, these systems usually are of high index, which is, roughly speaking, a measure of the order of the derivative of a perturbation that enters the solution of a perturbed DAE. A high index causes severe problems in the numerical solution of DAEs like for example the reduction of convergence order; therefore it is necessary to remodel the system as an equivalent but index-reduced DAE. In general, this remodeling leads to an unstructured model that, at first glance, has no reinterpretation as an electrical or mechanical system. Since common simulation packages like SIMPACK, Modelica and SPICE do not require a DAE model but a so-called network list, which is a special data format containing information about the interconnection structure of the sytem components, this is a great disadvantage.

For electrical systems though, there is an index reduction technique developed in the Matheon project D5 that obeys the structure of the specific network. In this approach, the index of the corresponding DAE is reduced by a pure topological analysis of the network, that detects the index increasing sets, i.e., loops of voltage sources and capacitances as well as cutsets consisting of current sources and inductances and replaces them with equivalent structures of lower index.

Correspondence of mechanical and electrical quantities Mechanical ~ Electrical Force ~ Current Velocitiy ~ Node Potential Impulse ~ Charge Position ~ Magnetic Flux Kinetic Energy ~ Electrical Energy Potential Energy ~ Magnetic Energy

The aim of this project is to extend this graph-theoretical approach to mechanical networks and thus to establish a structure-preserving index-reduction of mechanical systems. Furthermore, exploiting the equivalences between electrical and mechanical systems, we aim to derive a  non-index-increasing coupling procedure of electrical and mechanical structures to develope a remodeling strategy for mechatronic systems.   

Correspondence of mechanical and electrical devices Spring ~ Inductor Inerter ~ Capacitor Damper ~ Resistor Mechanical Ground ~ Electrical Ground