On the dynamic modelling of a planetary chain speed increaser for R.E.S.

Saulescu, Radu ; Jaliu, Codruta ; Neagoe, Mircea 等

1. INTRODUCTION

Most small hydropower plants include a gearbox to increase the speed of the turbine shaft to the generator. The range in which the input angular speed must be increased is 3 - 5 (Harvey, 2005; Jaliu et al., 2008). There are two basic types of gearboxes used in hydro-plants: parallel-shaft gearboxes and planetary gearboxes. In the first case, the higher the transmission ratio is, the bigger the overall dimensions are. In the second case, the input and output shafts are coaxial, reducing the overall dimension; the gearboxes are relatively light and compact but the cost of the transmission increases. New innovative solutions of planetary speed increasers were proposed by the authors in the previous papers (Diaconescu, 2005; Jaliu et al., 2008; Neagoe et al., 2008), with the aim to compensate the existent gearboxes disadvantages. This paper presents the dynamic modelling of a proposed planetary chain increaser for renewable energy systems (RES); the dynamics of the increaser represents the starting point for the control system design. The authors will accomplish the design, manufacturing and testing of the speed increaser for stand-alone hydropower stations, in the frame of a research project.

2. STRUCTURAL AND KINEMATICAL ASPECTS

A first step in dynamical modelling is done by defining the structural and kinematical aspects of the transmission.

[FIGURE 1 OMITTED]

Thus, the planetary transmission consists of: a fixed sun gear (3,3'), a satellite gear (2), a semi-coupling with pins (1) and a carrier (H); it has two exterior links (L=2): 1-input, H output.As a consequence of this aspect the obtained degree-of freedom equals 1, thus this planetary unit is being defined by: M=1, a single external independent motion: [[phi].sub.13], [[omega].sub.13], [[epsilon].sub.13];

a force transmission function [T.sub.1] = [T.sub.1] ([[phi].sub.13], [T.sub.H]); L-M=1 a movement transmission function:

[[phi].sub.H3] = [[phi].sub.H3] ([[phi].sub.13]), [[omega].sub.H3] = [[omega].sub.H3] ([[omega].sub.13]), [[epsilon].sub.H3] = [[epsilon].sub.H3] ([[epsilon].sub.13]);

an external independent force [T.sub.H].

The force transmission function is being determined by means of the internal kinematical ratio:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where: [i.sup.z.sub.xy] is the transmission ratio from element x to element y, while element z is considered blocked; [[omega].sub.xy] represents the relative speed between elements x and y;

The transmission ratio of the planetary transmission is given by relation (2), while the multiplication ratio, as the reverse of the previous one, by relation (3):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

i = 1/[i.sup.3.sub.1H] = 1 [1 - [i.sub.0] (3)

The kinematical modelling is solved using relation (2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

3. DIYNAMIC ASPECTS

The interior efficiency ( [[eta].sub.0] ) of the planetary transmission must be determined, in order to obtain the dynamic modelling:

[[eta].sub.0] = [[eta].sup.H.sub.13] = [[eta].sup.H.sub.12] x [[eta].sup.H.sub.23] (5)

The efficiency of the transmission is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

in which the exponent w is being calculated with:

[FIGURE 2 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (6) the force transmission function results as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The kinematical and static functions for the planetary chain transmissions are given in relations (4) and (7). An aggregate composed from a turbine, the analyzed planetary speed increaser and a generator is considered for the dynamic simulations.

4. PREMISES FOR DYNAMIC MODELLING

The dynamic modelling relies on the following premises:

* In the dynamic modelling, the inertial effects due to the satellite gears rotation are neglected (their masses being considered in the axial inertial moment of the afferent carrier shaft), while the inertial effects of the mobile central elements are considered integrated into the shafts that materialize the external links of the planetary gears; under this premise, the static correlations between the external torques of each planetary gear are valid, while the dynamic correlations interfere only for the shafts that materialize the planetary gears external links. The mechanical inertia momentums of the two shafts are (see Fig. 1 - Saulescu et al, 2009):

[J.sub.1] = 0,03; [J.sub.H] = 0,02 [[Kgm.sup.2]] (8)

* the rubbing effect is considered by means of the efficiency [eta];

* The real machine (water turbine--speed increaser generator) is replaced on the experimental stand by a machine of type: DC Motor--increaser--brake, in which the motor and the brake have the following mechanical characteristics:

[T.sub.m] =-0,1237[[omega].sub.m] +10; [T.sub.b] = -[[omega].sub.b] [Nm] (9)

* In the numerical simulations, the following values for the kinematical and dynamic parameters are considered:

--the satellite and sun gears teeth numbers are [z.sub.2] = 30, [z.sub.3] = 24,

- the efficiencies of the pin coupling and chain transmission are [[eta].sub.12] = 0.995, [[eta].sub.23] = 0,92.

5. CONCLUSIONS

The theoretical aspects from which was deducted the analytical kinematical and static calculus were presented in the paper; the mechanical momentums of inertia and the motor's mechanical characteristics (belonging to the turbine) and brake's mechanical characteristics (belonging to the generator) were considered in the dynamic modelling. As the dynamic modelling is being accomplished in order to compare the theoretical results with the experimental ones, obtained on the stand, the turbine is being replaced with a DC engine, and the generator is being replaced with a brake. The considered inertia mechanical momentums (see Fig. 1,a and 2) have as an input element 1, the engines rotor, the entering shaft with it's appropriate rolls, while for the output element H, the exit shaft, brake's rotor and the inertial effect of the satellite considered as the concentrated mass.

The presented parameters are being used in order to design the dynamic model, (Saulescu et al., 2009).

The dynamical model is conceived on a DC motor type machine--planetary chain speed increaser--brake (see Fig.2), on a representative case, in which the multiplication ratio is the one recommended in the specialized literature (Harvey, 2005; Von Schon, 2007; Jaliu et al., 2008; Amer, 2005) for small hydro.

The dynamical modelling is done to be used for pushing forward the researches on two directions:

* The theoretic direction, meaning: attaining a controlling program for the system;

* The experimental direction, meaning: obtaining the main product (the planetary chain increaser) and implementing it on a system of type: turbine - transmission - generator, for a particular case.

6. ACKNOWLEDGEMENT

The authors will accomplish the design, manufacturing and testing of the speed increaser for stand-alone hydropower stations in the framework of the research project "Innovative mechatronic systems for micro hydros, meant to the efficient exploitation of hydrological potential from off-grid sites", ID_140. The preparation and publishing of this paper were possible with the financial support of this research project.

7. REFERENCES

American Chain Association, (2005). Standard Handbook of Chains: Chains for Power Transmission and Material Handling, 2nd Edition, Dekker Mechanical Engineering.

Diaconescu, D. (2005), Products Conceptual Design (Romanian), Transilvania University Publishing House, Brasov

Harvey, A. (2005). Micro-hydro design manual, TDG Publishing House.

Jaliu, C., Diaconescu, D.V., Neagoe, M. and Saulescu R. (2008). Dynamic features of speed increasers from mechatronic wind and hydro systems, Proc. of EUCOMES 08, 2nd European Conf. on Mechanism Science, pp. 355-373, September 2008, Springer Publishing House, Cassino, Italy

Neagoe, M. et al. (2008). A Conceptual Design Application Based On A Generalized Algorithm. The 19th International DAAAM Symposium. pp. 0953-0956, Trnava, Slovakia

Neagoe, M., Diaconescu, D.V., Jaliu, C., Pascale, L., Saulescu, R. and Sisca, S. (2008). On a New Cycloid Planetary Gear used to Fit Mechatronic Systems of RES, OPTIM 2008. Proc. of the 11th Int. Conf. on Optimization pf Electrical and Electronic Equipment. Vol. II-B. Renewable Energy Conversion and Control, pp. 439-449, Brasov, Romania, May 2008, IEEE Catalogue 08EX1996

Saulescu, R. et al. (2009). Dynamic simulations of a planetary chain speed increaser for RES, Proceedings of 20th DAAAM International Symposium, Katalinic, B. (Ed.), pp. -, ISBN 978-3-901509-70-4, Vienna, November 2009, DAAAM International Vienna

Von Schon, H.A.E.C. (2007). Hydro-Electric Practice--A Practical Manual of The Development of Water, its Conversion to Electric Energy, and its Distant Transmission, France Press Publishing House