Considerations about the modelling and simulation processes for mechatronic systems.

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

Abstract: Mechatronics is at the crossroad of four domains: mechanics, electronics, and control & information technology. Therefore, mechatronic systems require a specific approach that bypasses the physical nature of the components and handles them in a generalized manner. A common working language plays an essential role. This language uses the existing analogies between the physical systems defining the objective of mechatronics and is helpful in implementing efficient working methods and tools in mechatronics. This paper pleads for an interdisciplinary study in mechatronics.

Key words: Mechatronics, modelling, generalized impedance, quadripole, bond-graph.

1. INTRODUCTION

The IFToMM technical committee for mechatronics defined "mechatronics" as a synergic combination of the precision mechanics, electronic control and intelligent systems destined to the product design and process planning (Nobuhiro, 1996). The definition is major for the philosophical approach of mechatronics knowledge, since it involves both creation and manufacturing.

Integrating electronics, computers and control elements in the mechanical system expects a completely innovative design approach. In accordance to the authors' evaluation, which is based on experts' opinions, the following characteristics of the mechatronic systems are essential: high speed and accuracy, enhanced efficiency, robustness and miniaturization.

A careful, judicious design is strongly required, by balancing modelling, analysis, experiment validation and construction. Modelling, analysis and prediction ensure to achieve efficiency and quality, starting still from the design theme. An appropriate approach is required and a common working language plays an essential role. This language uses the existing analogies between the physical systems defining the objective of mechatronics and is helpful in implementing efficient working methods and tools in mechatronics. Based on these arguments, the paper pleads for an interdisciplinary study specific to mechatronics.

2. THE GENERALIZED IMPEDANCE & THE DIPOLE

The systems theory provides a basic instrument in modelling and analysis: the transfer function. However, the approach of representing a system by the transfer function, referring to an input function and an output function leaves aside the energetic aspects that are specific and important in physical systems and may not be neglected.

A passive linear dipole (Figure 1) is assimilated in the electrical domain by a positive quantity that depends upon the working frequency and the circuit parameters. This quantity is called circuit impedance. The dependence Z(s) is obtained by using the equation of the circuit and the Laplace transform (Figure 1).

[FIGURE 1 OMITTED]

One can extend the term impedance and apply it for non-electrical domains too.

A mechanical system has two electrical circuits analogue to it (dual circuits), if considering the analogies:

* velocity [left and right arrow] current intensity and force [left and right arrow] voltage;

* velocity [left and right arrow] voltage and force [left and right arrow] current intensity.

The mechanical impedance can be defined as follows:

[Z.sub.1] = F/v and [Z.sub.2] = v/F (1)

If considering a translation system and the analogy displacement X--electrical charge in the mechanical domain, the corresponding impedances are shown in Figure 2. The only criterion of choosing one or the other form is the comfort in working and studying the phenomenon.

One can write in a similar manner appropriate relationships for the equivalent impedances of the rotational mechanical systems.

In an analogous way, the acoustic impedance can be defined (Z is the acoustic pressure; v is the particle velocity):

Z = p/v (2)

One can define similar analogies to the electric circuit for other systems too: magnetic systems (see Table 1), thermal systems (see Table 2).

[FIGURE 2 OMITTED]

Approaching the analysis and the modelling process based on the impedance concept allows to beneficiate in non-electrical systems of the advantages offered by the systemic equivalences that are specific for electrical circuits.

The energetic aspects can be included in the study by introducing a new generalized power, [PI] = [alpha] x [tau] defined as a product of two physical quantities across two points ([alpha]) and through a single point ([tau]) respectively.

Table 3 reveals examples concerning the classification of different physical quantities from this point of view.

Corresponding to the energetic aspect on one side and to the analogy with the electric domain on the other side, a new essential idiom in the modelling philosophy is the "quadripole". Representing a system by a quadripole corresponds to a complex interpretation. The system is symbolized by a pair of input quantities and a pair of output quantities (Figure 3): the input gate with the terminals 1 and 1' and the output gate with the terminals 2 and 2'. One associates to each gate an instant power (Timotin, 1970).

[FIGURE 3 OMITTED]

The main role of a quadripole is that of belonging to a power transmission chain. For a quadripole, one can define the impedance Z = [U.bar]/[I.bar], the admittance Y = 1/Z and the instant terminal power p = u x i. The fundamental equation of a quadripole is given by the relationship (3), where A and D are non-dimensional coefficients, B is the impedance and D is the admittance:

[[U.bar].sub.1] = [A.bar] x [[U.bar].sub.2] + [B.bar] [[I.bar].sub.2] (3)

The reciprocity condition of the quadripole is given by a relationship of the following type:

[A.bar] x [D.bar] - B x [C.bar] = 1 (4)

A special case of the quadripole is the gyrator, defined as an anti-reciprocal passive linear quadripole:

[A.bar] x [D.bar] - B x [C.bar] = -1 (5)

The topics of "generalized impedance" and "gyrator" are remarkable in mechatronics, when modelling the mechatronic systems by the bond graph technique and object oriented programming (Verge & Jaume, 2003, DYMOLA, 2006).

[FIGURE 4 OMITTED]

The real performances of the quadripole depend upon the elements before and after the quadripole. To realize the quadripole, it is necessary to use the impedance concept, and to consider more criteria, like the imposed transfer function, the maximum transfer of power. Figure 4 presents an objectification of these ideas; the quadripole is objectified once by the gyrator GY and once by the transformer TF.

3. CONCLUSIONS

The authors outline the high degree of generality involved by the presented approach for any non-electrical system. This is an essential aspect due to the synergic combination accepted for the mechatronic idiom. In this context, it becomes possible to represent a mechanical physical system by a theoretical quadripole. This representation is particularly advantageous:

* In case of rapid systems, when and where the torsion deformations within systems are not neglected,

* When using controlled clutches in the kinematic linkage,

* When employing mechanical transmissions (i.e. gear train) in the kinematic linkage.

The authors will use the results of this study in further design of the mechatronics systems and particularly within the National Excellence Research Grant "CONMEC" destined to a network for control and simulation in mechatronics.

4. REFERENCES

Dolga, V. & Dolga, L. (2004). Modelling and simulation of mechatronic systems. Mecatronica, no. 1, June 2004, pp. 34-39, ISSN 1583-7653

Kyuru Nobuhiro, Oho Hirosuke, Mechatronics An Industrial Perspective, IEEE/ASME Trans. on Mechatronics, vol. I, no.1, pp.10-15, March 1996

Soderman U. (1995). Conceptual Modelling of Mode Switching Physical Systems, Dissertation no.375, Dep. of Comp. and Inf. Science, Linkoping University, Sweden

Timotin, A.; Hortopan, V. ; Ifrim, A. & Preda, M. (1970). Lectii de bazele electrotehnicii, EDP Bucuresti

Verge, M. & Jaume, D. (2003). Modelisation structuree des systemes avec les Bond Graphs, Editions Technip, ISBN 2-1708-0838-2

*, (2006). Getting started with DYMOLA, Available from: http://www.dynasim.com/documents/GettingStarted5.pdf, Accessed: 2007-04-23

Table 1. An analogy magnetic system--electric system

Magnetic circuit Electric circuit

* Magnetic flux--[PHI] * Current--I
* Magnetomotive force--[THETA] * Voltage--U

Table 2. An analogy thermal system--electric system

Thermal circuit Electric circuit

* Thermal flux--[PSI] * Current--I
* Temperature--T * Potential--V

Table 3. Physical quantities for defining the generalized power.

Domain The quantity [alpha] The quantity [tau]

Mechanical Velocity [m/s] Force [N]
translation
Mechanical Angular velocity [rad/s] Torque [Nm]
rotation
Electric Voltage [V] Current [A]
Hydraulic Pressure [N/[m.sup.2]] Volume debit [[m.sup.3]/s]