Motivation

With the decreasing structural size going along with expanding complexity of technical systems there is an emerging demand for new design methods and modeling support. This becomes even more important because of the increasing heterogeneity of technical systems. In particular, for the design of mechatronical systems this leads to the following problem: There are established tools for the design of the mechanical or electronical parts like FEM, multi-body, or circuit simulators, but those are usually specialized on their physical domain. Therefore, the consideration of interactions between the mechanical and electronical components is extraordinary laborious. These interactions often have to be taken into account because the assembly of independently optimized sub-components usually does not lead to an optimal system. On the other side, considering coupling effects results in new challenges in many aspects, which lead from the need of designers competence in multiple disciplines (electronic and mechanical engineering, physics and mathematics) up to the high complexity of the mathematical models which demand the employment of adapted simulation tools.

An alternative approach is the employment of a single simulator for which the model descriptions are available in a corresponding formulation. Such behavioural modeling languages are e.g. VHDL-AMS or Modelica. First appropriate simulators are already available. They allow a modularized modeling of the complete system or parts of it with arbitrary accuracy. But the modeling process of heterogeneous systems is very time consuming and moreover the resulting mathematical models become very complex even for comparatively small systems, posing numerical problems with respect to robustness, efficiency, and stability. Today the simulation of industrial-sized systems lies beyond the limits of this approach. In order to reduce the numerical effort, model reduction techniques become of interest.

Modeling support for complex mechatronical systems

Within this Fraunhofer MEF project, a modeling approach has been developed which is based on symbolic methods and can be adapted to multi-physical systems due to its general mathematical principle. This includes an automatic generation of behavioural models as well as model reduction for mechatronical components. Such models allow due to their reduced complexity an interactive processing and a more efficient simulation of the overall system.

The application of the automatic setup of analytic model equations with the software Analog Insydes® could be extended for generalized Kirchhoff equations by employing analogies between analog circuits and mechanical systems. For this an implementation of the mapping of the through and across quantities was mandatory. That way the equation setup and the symbolic approximation methods could be extended for the treatment of mechatronical systems. Mechanical components are modeled by finite elements, where the network nodes are strongly connected with each other. Displacement and rotation as a consequence of force and torque are considered to be small. Instead of current and voltage, the mechanical devices are described by force, torque, displacement, and rotation. In cooperation with Fraunhofer IIS/EAS a corresponding symbolic model library for micro-mechanical devices has been developed.

Acceleration sensor

As a simple example for a multi-physical system we consider an acceleration sensor including of mechanical and electronical parts. The sensor consists of three parallel conducting plates which form two serial capacities. The central plate can be moved from its balanced position resulting in a Hook's force. In case of an acceleration, the central plate moves away from its central position resulting in changes of the capacities between the electronical connectors. This yields a potential drop Vout for the central plate with respect to the potential in the idle state.

 

The system is accelerated by a time-dependent force acting towards (0,-1, 0) starting at time t = 10 ms. The dynamic behaviour of the sensor can be mathematically described by a DAE system, consisting of 39 equations. Using symbolic approximation techniques the system can be reduced automatically to only five remaining equations while checking the corresponding output quantity Vout against the generated approximation error. The diagram illustrates a comparison for the quantity Vout for the original and the simplifed DAE system.