5
FOUR BAR LINKAGES WITH LINEAR ACTUATORS USED FOR LARGE DIURNAL SOLAR TRACKING STROKES
Moldovan Macedon Dumitru1, Vișa Ion1, Săulescu Radu1, Comşiţ Mihai1
Abstract: This paper deals with four-bar linkages, driven by linear actuators, used in solar tracking systems with large diurnal strokes. Based on the analysis of the existing solutions in the literature, a dimensional synthesis algorithm of a four-bar linkage actuated by a linear actuator connected to the connecting rod and to the base is proposed. The results are useful for designers and specialists working in the field of solar tracking systems implementation.
Keywords: solar tracking systems, photovoltaic platform, four-bar linkage, large angular stroke, synthesis algorithm, pressure angle.
1. Introduction
Solar tracking systems maximize the normal component of the solar beam radiation captured by the photovoltaic modules. Depending on the number of tracking axis (single- or dual-axis), an energetic gain up to 45% can be obtained [1, 2].
Mainly, solar tracking systems are classified in azimuthal, pseudo-azimuthal, equatorial and pseudo-equatorial type [4, 5]. The tracking system must adjust, continuously or sequentially, the orientation of PV modules in order to track the apparent path of the sun in the sky. The current position of the sun is characterized by a pair of solar angles: diurnal and respectively elevation angle. The stroke range of before mentioned solar angles depends on the tracking system type. In this paper, the case of an azimuthal solar tracking system, defined by the azimuthal angle ψ* corresponding to the azimuthal (diurnal) solar angle ψ and by the altitude angle α* corresponding to the solar altitude (elevation) angle α, is analysed. An example of such a tracking system is presented in Fig. 1a [6] and the variations of solar angles for the summer solstice (N=172) in Brasov / Romania location, are plotted in Fig. 1b.
a) b)
Fig. 1. a) Azimuthal dual-axis tracker [6]; b) Sun-ray azimuth and altitude angles and their variations during the Summer Solstice (day no. 172) at Brasov / Romania.
If the angular stroke of the solar altitude, α, varies between 0° and 90° depending on geographic location (between 0° and 67.8° in Brasov, Romania), for the solar azimuthal angle, ψ, the maximum stroke is much higher than 90º, varying upon geographic location between +180° and -180° (between + 125° and - 125° in Brasov, Romania). Usually, the elevation of a photovoltaic platform is assured by a triangle linkage with linear actuator (Fig. 2a) and the diurnal tracking by a gear reducer with rotary actuator. Linear actuators have usually lower costs comparing to the rotary ones. In the same time, gear reducers have some disadvantages: difficulty to control the clearances and lower efficiency. Therefore, research focuses on designing relatively simple linkages, with linear actuators, able to reach large angular strokes with favourable values of the pressure angles. Due to the reduced angular stroke, Δψ*, of a triangle linkage, presented in Fig. 2a, limited at 120º…130º by the minimum accepted values for pressure angles β = 60°...65°, this linkage cannot be usually used to achieve large angular stroke for the diurnal tracking angle. In Fig. 2b, the linear actuator is connected with a four-bar (planar or spatial) linkage; in this case, the linear actuator drives one of the rockers and the four-bar linkage works as angular amplifier if a large angular stroke (ϕ ≥ 130°) is required [5]. An azimuthal stroke of 190º was obtained with the linkage presented in Fig. 2b, for a minimum transmission angle γmin = 38º and an amplification ratio of ϕ / φ = 190º / 86º = 2.20. The same azimuthal angular stroke was reached by a spatial linkage with γmin = 20º and an amplification ratio of ϕ / φ = 190º / 72º = 2.64 [7].
The paper presents the kinematical synthesis of an azimuthal tracking linkage using a linear actuator positioned between the basis and the connecting rod of the linkage in the general case. An optimisation algorithm is presented and also the numerical simulations done for a linkage that can achieve large angular stroke (up to 235º) maintaining pressure angles under the maximum admitted value (60°... 65°).
2. Problem formulation
The aim of this study is to design a four-bar planar linkage, driven by a linear actuator for the diurnal movement of an azimuthal tracking system for a photovoltaic platform attached in joint A to the rocker AB. For an azimuthal solar tracking system should be taken into account the following requirements:
- the diurnal angular stroke Δψ* should be larger than 180°;
- the pressure angles are limited to 65° ... 65° to avoid excessive forces in the linear actuator and in linkage’s elements.
From the above condition results:
(7)
where:
(8)
Once obtained the main parameters of the linkage (l0, l2, l3 and Δψ*), the analytical expression of the pressure angles βB, βC can be thus determined as functions of the angular stroke φ1 of the crank AB
3. Results and discussions
Based on the presented methods, it was design a four-bar planar linkage (Fig. 4) for the diurnal movement of an azimuthal solar tracking system for a photovoltaic platform intended to be installed in Brasov, Romania (latitude 45.65° N).
The analytical model was used to evaluate the maximum angular stroke Δψ* of the crank AB for an eccentricity e = 0.1·l1, where l1 is the length of the rocker AB, and for pressure angles of 60° both in initial and final position of the linkage. A maximum diurnal angular stroke Δψ* = 235° was obtained, along with the main linkage parameters: the angular stroke of the crank CD Δφ3 ≈ 118º, resulting an amplification factor of 1.99, the reduced length of the connecting rod BC / AB = l2 / l1 ≈ 1.93, the reduced length of the crank CD, CD / AB = l3 / l1 ≈ 2.17, and the reduced length of the base AD / AB = l0 / l1 ≈ 1 (Fig. 4).
The variations of the pressure angles βB, in joint B, and βC, in joint C, (Fig. 4) are lower than imposed value of 60° for the entire angular stroke of the crank AB. These angles are not dependent on position of the joint M but only on angular displacement φ1 of the crank AB. For all three solutions (M1, M2 and M3) the pressure angles βM1, βM2, and βM3 reach only once, at different angular displacement φ1 of the crank AB, the admitted value of 60°, in rest, all the values are below. Therefore, these solutions are valid and for their differentiation the average pressure angles for the entire angular stroke of the crank AB were calculated and results presented in Table 1. Thus, M3 (1.19, -0.9) results as the optimal position of the linear actuator’s joint with the connecting rod, for which the average value of the pressure angle βM is 25.25°. For this solution, the average load of the linear actuator will be the smallest and also the energy needs to drive the linkage.
Table 1. Average pressure angles for the entire angular stroke of the crank AB.
Solution / M1 / M2 / M3Average pressure angle / 28.26° / 33.65° / 25.25°
4. Conclusions
The geometrical and analytical synthesis of a four-bar linkage, used in a tracking system for a photovoltaic platform with large azimuthal angular stroke, driven by a linear actuator, can be concluded with:
a) the use of linear actuator, instead of rotary actuator, is a viable solution when a four-bar linkage is used for the azimuthal movement of the platform;
b) the four-bar linkage, due to the eccentricity imposed to reduce the complexity of the linkage (the crank CD should not interfere with the joint A), and to the maximum admitted pressure angle of 60°... 65°, allows a maximum angular stroke Δψ* = 235°, wherever the linear actuator is connected (on the crank or on the connecting rod);
c) if the four-bar linkage is driven by a linear actuator positioned between the basis and the connecting rod, the optimal positions of its revolute joints were identified in the half-plane of the connecting rod containing the basis; thus the overall sizes of the mechanism decrease compared with the case of the linkage actuated on the rocker;
d) the best positions of the linear actuator’s joints depend by the average pressure angle for the entire stroke of the crank AB, resulting a minimal average load of the linear actuator and minimal energy needs;
e) thus, the four-bar linkage driven by a linear actuator positioned between the basis and the connecting rod can be used as efficient amplifier mechanism, in the specific application of solar tracking systems, being able to output an amplifying factor up to 2, while maintaining the pressure angles lower than maximum admitted pressure angle of 60°... 65°.
Acknowledgments
This work was done in the frame of the Program: Cooperation in Priority Fields - PNII, developed with the support of ANCS, CNDI-UEFISCDI, Romania in the project EST IN URBA, PN-II-PT-PCCA-2011-3.2-051.
References
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