Dynamic simulations of a planetary chain speed increaser for R.E.S.

Saulescu, Radu ; Diaconescu, Dorin ; Jaliu, Codruta 等

1. INTRODUCTION

This paper presents the dynamic modelling of a planetary chain speed increaser proposed by the authors, transmission which will be manufactured in order to be included in a small hydropower plant. The first step in the development of the dynamic model (the establishment of the increaser structural and the theoretical kinematical features) was presented in (Saulescu et al., 2009). This paper presents the dynamic response of the planetary chain speed increaser by means of Matlab-Simulink software. The dynamic model will be used in the design of the control system for the small hydro plant. 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. THE DYNAMIC MODEL

In the dynamic modelling, the main objective is to determine the angular speed transmission functions and the moments, both relative to time (Jaliu et al., 2008; Meriam et al., 2006), considering the planetary chain increaser from Fig. 1,a, the functions are:

[[omega].sub.13] = [[omega].sub.13] (t), [[omega].sub.H3] = [[omega].sub.H3] (t )> [T.sub.m] = [T.sub.m] (t) [T.sub.b] = [T.sub.b] (t) (1)

The dynamic modelling is made in the following cases:

* The motion equation is modelled by neglecting rubbing, and

* The motion equation is modelled by considering rubbing effects.

2.1 Case I (friction is neglected)

In this case the Lagrange method is used (Meriam et al., 2006):

d/dt ([partial derivative][E.sub.c]/[partial derivative][omega] - [partial derivative][E.sub.c] [partial derivative][phi]) = Q (2)

According to Fig. 1,b the kinetic energy Ec and the generalized force Q, have the following expressions:

[E.sub.c] = 1/2 ([J.sub.1] x [[omega].sup.2.sub.1] + [J.sub.H] x [[omega].sup.2.sub.H]); Q = [T.sub.m] x [i.sup.3.sub.1H] + [T.sub.b] (3)

in which [J.sub.l], represents the mechanical inertia momentum of the input element 1, respectively [J.sub.H], for the output element H:

Deriving Ec, relative to time, it is obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

From relations (2) and (3), it results:

[[epsilon].sub.1][J.sub.1] + [[epsilon].sub.H][J.sub.H] = [T.sub.m] x [i.sup.3.sub.1H] + [T.sub.b] (5)

In the premise of the neglected friction, it is finally obtained:

[[epsilon].sub.1] [J.sub.1] + [[epsilon].sub.H] [J.sub.H] = [T.sub.m] x [i.sup.3.sub.1H] + [T.sub.b] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

For the considered case, by means of numerically replacement of the known parameters, the following motion equation is obtained:

[[epsilon].sub.H] x 0.0212 + [[omega].sub.H] x 1.0049 - 2 = 0 (7)

2.2 Case II (friction is considered

In this case, according to Fig. 1,c and d and 2, the following system of equations can be written using the Newton-Euler method (Meriam et al., 2006):

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

In the final stage, by solving system (8) and by taking into account friction, the equation used for modelling the dynamical system of the machine is obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

By replacing the known parameters, it results the motion equation in the case of considering friction:

[[epsilon].sub.H] x 0.0207 + [[omega].sub.H] x 1.0031 -1.2606 = 0 (10)

In order to establish if the results obtained in the previous case are correct, friction is neglected in relation (9) and equation (7) is obtained.

3. NUMERICAL SIMULATIONS

Considering [[epsilon].sub.H] = 0 during the steady-state regime, the angular speeds of the input and output shafts from the analyzed transmission are being calculated in the two premises:

* When friction is neglected

[[omega].sub.H] = 1.9901; [[omega].sub.1] = 0.3980 [[s.sup.-1]]

* When friction is considered

[[omega].sub.H] = 1.2567; [[omega].sub.1] = 0.2513 [[s.sup.-1]]

The machine dynamic response, considering the speeds, accelerations and moments as functions of time, in both premises are drawn in the Fig. 3:

[FIGURE 3 OMITTED]

4. CONCLUSIONS

1) The gearboxes for small hydropower plants increase the speed of the turbine shaft to the generator between 3 and 5 times (Harvey, 2005).

2) The analyzed planetary chain transmission increases the input speed 5 times and decreases the input moment 3.1516 times (see Fig. 3).

3) The dynamic modelling is made in the premise that the speed increaser is used in a system of type: motor (turbine) + increaser + brake (generator).

4) The system consisting of motor, speed increaser, brake starts practically, in both cases, in about 0.16 s, after which enters in the steady-state regime.

5) The dynamic model is useful in the design of the control system for the wind turbines and hydro stations. The system control program can be established by considering certain environmental conditions/seasons and by replacing the motor and the brake from the dynamic modelling with a turbine and a generator.

6) Based on the dynamic modelling, the authors will accomplish the design, manufacturing and testing of a speed increaser for off-grid hydropower stations in the framework of a research project.

5. ACKNOWLEDGEMENT

The preparation and publishing of this paper were possible with the financial support of the research project ID_140 'Innovative mechatronic systems for micro hydros, meant to the efficient exploitation of hydrological potential from off-grid sites".

6. REFERENCES

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

Jaliu, C. et al (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, Sept. 2008, Springer Publishing House, Cassino, Italy

Meriam, J. L. & Kraige, L. G. (2006). Engineering mechanics: Dynamics. 6th edition. John Wiley & Sons

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