Analogy in modelling mechatronic systems.

Dolga, Valer ; Dolga, Lia ; Filipescu, Hannelore 等

1. INTRODUCTION

The mechatronic systems represent an integration of the electronic and information technologies in constructing advanced mechanical systems. In accordance to the authors' evaluation, based on experts' opinions, the essential characteristics of the mechatronic systems are high speed and accuracy, enhanced efficiency, robustness and miniaturization (Amerongen, 2007); (Dolga & Dolga, 2007).

A judicious design is essential, by balancing modelling, analysis, experiment validation and construction. Modelling, analysis and prediction ensure efficiency and quality, just from the first stages, after the launching of the design theme. A common working language plays an important role (Bishop, 2002). This language origins in the analogies between the physical systems defining the objective of mechatronics; it will be also useful in implementing efficient working methods and tools in mechatronics. The systemic study of the dynamic behaviour expects the modelling of various electromechanical systems (Krus, 2003); (Hostert, 2007). Extending the utility of several principles from within the analytical mechanics over the electromechanical systems is vital (Preumont, 2006).

2. THE MECHATRONIC SYSTEMS STRUCTURE

Mechatronics expresses the integration of mechanical, electromagnetic and computer elements to produce devices and systems that monitor and control machine and structural systems. Figure 1 shows a concise representation of a mechatronic system, omitting the interdependencies between components and distinguishing four sub-systems: mechanical-, informational-, electrical- and computing system. Within the mechanical process, which is a basic component of the mechatronic system, mainly three flows are present: substance-, power (energy)--and informational flow. Modelling and simulation of the mechatronic systems must consider analogies between the subsystems, in order to apply a unitary approach.

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3. ANALOGY AND MODELLING

The systems theory provides a fundamental "instrument" in modelling and analysis: the transfer function. The methodology based on the transfer function involves the advantage of an acknowledged theory for the systems analysis and synthesis as well as for a software package in which Matlab/ Simulink is the preferred working environment. Figure 2 shows the starting window of a Matlab/ Simulink library, called "MODEL_RI" created by the authors during this study.

However, the approach of representing a system by the transfer function, referring to an input quantity and an output quantity leaves aside the energetic aspects that are specific and important in physical systems and may not be neglected.

The theory of physical systems is based on the concept of energy (E), defined as power accumulated along the time. Starting from this viewpoint, one can define the notion of "generalized" power, [PI], as being the product of two physical quantities, observable and complementary: quantities across two points ([alpha]) and through a point respectively ([tau]). These approaches are not singular. The theory of physical systems adds to the previous topics, named by some authors "pressurelike" and "flowlike", the equivalency between the effort e and the flow f. There is a very close similarity (almost an identity) between these two modes of definition.

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In accordance to the physical principles referring to systems operation, the topic of "energy ports" can be introduced, through which systems interact one with each other, by an energy exchange (multiport) (see Figure 3) (Amerongen, 2007).

Figure 4 proposes the stages in modelling physical systems starting from the physical system towards the corresponding equations. The design stage comes in a chronological order between the analysis stage and the implementation stage. The object-oriented development of the mechatronic product incorporates all these three stages.

The object-oriented approach is a perfect tactic for the teamwork, which is a fundamental characteristic of the mechatronic philosophy on design. The paradigm involves an efficient communication between the members of different groups that work together at the same project.

The authors worked in a dedicated modelling environment, Dymola (Dynamics modeling laboratory), to develop a mechatronic project, because it provides modelling- and simulation tools based on object-oriented paradigm (Figure 5). Using a principle similar to the bond-graph method, Dymola considers the energetic flow, a common characteristic for the majority of the physical systems (Elmqvist et al. 2003).

The principles of Newtonian mechanics are overall simple in their form, but sometimes difficult to apply. These aspects brought Lagrange to create the principles of the analytical mechanics "a mechanics of the systems--with a finite number of parameters which should not use any concrete figure or picture". The mechanics of Lagrange, the mechanics of the non-holonome systems, the Hamiltonian mechanics, and the variational principles use the analytical mechanics.

Variational methods and the associated extremum principles are fundamental scientific techniques of a notorious physical significance. Yet, this treatment will be restricted to variational methods of analysis for lumped physical systems, characterized by the energy variables "effort" and "flow".

Figure 6 shows a case of integrating a mechanical and an electrical system, a proper mathematical model being defined.

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A condenser microphone consists of a movable plate mounted on a circular spring parallel to a fixed back plate (Preumont, 2006). The plates form a variable capacitor charged by a constant voltage source through an RL circuit. The dynamics of the moving plate is modelled by a spring-mass system of constants m, k, c. The equations (1), (2) govern the mathematical model developed from the analytical mechanics principles, where the symbols correspond to those in Figure 6:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

4. CONCLUSION

Mechatronics is not only the sum of the implied engineering domains; consequently, the concepts, the language, the methods and tools in mechatronics are more advanced. The product design approach requires consistent concepts and methodologies based on an appropriate language. Analogies between physical systems and quantities of different nature can lead to a common language and various possibilities are available for a homogeneous approach, but the advantage of applying analytical mechanics principles is certain.

The authors outlined that a successful modelling and simulation is conditioned by an integrated approach and an appropriate experiment that should confirm/ infirm the developed model and the performed simulation.

The result of the study is useful in team working on a mechatronic product. The study will continue in view of improving the mechatronic design philosophy by specific concepts.

5. REFERENCES

Amerongen, J. (2007). Mechatronic design--a port-based approach, Proc. Of the 4th Int. Symp. On Mechatronics and Applications, Sharjah, March 26-29, Available from: http://www.ce.utwente.nl/rtweb/publications/2007/pdf-files/ Sharjah.pdf. Accessed: 2008-05-12

Bishop, H. (2002). The mechatronics handbook, CRC Press, ISBN 0-849-300-665, London-New York-Whashington

Dolga, V.; Dolga, L. (2007). The principles of analytical mechanics applied to the dynamical analysis of the mechatronic systems, Proc. of 2nd Intern. Conf. "Computational mechanics and virtual engineering", pp.221-226, ISBN 978-973-598-117-4, Brasov, 11-13 X 07

Elmqvist, H. et at. (2003). Real-time simulation od detailed automotive models, Proc. of 3rd Intern. Modelica Conference, pp.29-38, Linkoping, 3-4 November 2003

Hostert, C.; Maas, S. (2007). Dynamic simulation of mechatronic systems, Available from: http://www.aliai.lu / rt/rt20071/ rt20071d.pdf. Accessed 2008-05-10

Krus, P. (2003). Modelling and Simulation of heterogenous engineering systems, Available from: http://www.machine. ikp.liu.se/edu/post/ multidomain/ modelling_simulation.pdf Accessed 2008 -04-20

Preumont, A. (2006). Mechatronics. Dynamics of Electromechanical and piezoelectric systems, Springer, ISBN 1-4020-4695-2, Dordrecht (Netherlands)