alt=""/>(4)
Otherwise, by Lagrange is obtained equation 5, in which Fi is the coefficient that helps to get losses approximation from the shaft. However, both equations that were described above are given for a component of its matrix form. Therefore, in this work it is proposed a high modulation from sensors to actuators, it means in Figure 3 is represented the effect to improve a mechanic system by nanostructures. It means in Figure 3a is depicted a simple energy transmission among the mechanical systems, for which also must to be given lost energy, such as by the heat. How to avoid this? It could be reduced through many mechanisms of evacuating energy. However, it makes systems to be expensive without total reduction of the heat, because of the natural thermodynamic behavior of systems.
Figure 3
General rotor (shaft) scheme under the energy transmission and control
Source: Own elaboration.
Notwithstanding, while it is controlled by intelligent algorithms through sensors and actuators, the energy transmission enhances, but that transmission gets costing of the produced heat even AMB used over the rotor. It is because the energy transmitted through wires that produce electromagnetic field. Therefore, by faster sensors/actuators as it was proposed from sensors that were based in nanostructures (as in this research), the heat is reduced and movement transmission between rotating machines can get better efficiency. For this reason, it was necessary to quantify the heat that was produced over wires by Joule effect and the evacuated heat as the dependence of geometrical parameters and temperatures, as it was depicted in following equations. For this, Q is the heat evacuation due to friction, T is the temperature, K is the conductivity coefficient, A is cross sectional area to the heat flow, U is the excitation signal and Kp is the proportional gain of the thermal model.
For which, the proposal solution is given by equation 7. It means that for steady state:
And from which temperatures matrix:
By other side, it can be improved by control systems. Nevertheless, this is not totally good, due to in sophisticated systems that need fast and robust response, the controller (even so sophisticated) could be that it cannot reduce in total the heat transmission (it is depicted by Figure 3b). Therefore, what to do? It could be enhanced through faster sensors and actuators, as it is depicted in Figure 3c, it because sensors/actuators that were based in nanostructures.
The physical parameters identification and error analysis to get position control are given by a general system identification, as it is shown in equation 11, for a general expression of the rotor dynamic, in which I is the matrix of electrical current values for every component i. Furthermore, the stiffness coefficient Ky and electrical coefficient KIL.
Such as it was described above, every component of last equation has matrix form. Also the polynomial solution gets the error as the dependence of desired signal in order to identify M2, Ky and KIL that are matrices as the equations 2 and 3, which were described in paragraphs above.
Equation 12 helps to evaluate the error matrix as the dependence of desired signal and the measured signal S in every instance n.
For every S is composed by adaptive weight coefficients that was correlated with the input signal (as measured or expected/simulated) X as it was described in equation 13. For this reason, the matrix of error is defined as:
By other side, the general response y(t) correlated with x(t) and u(t) through a nonlinear function , it is because to look for the optimal trajectory and to get the best position control, as it is represented in equation 15.
It means to solve the costing equation 16 by J, the expected trajectory R, in which this expected position is given by equation 17. And the optimal excitation signal in order to find the optimal response is given by equation 18.
Anodic Aluminum Oxide (AAO) membranes have quite good mechanical, optical and chemical properties, hence, it helps to study designed sensors that were based in them. By other side, AAO membranes have fast response and robustness that means sensors that were based in this kind of membranes, which can achieve these characteristics. In Figure 4 is shown an AAO membrane and, as indication of yellow raw, it is amplified in nanoscale view (from SEM PUCP) its porous [7].
Figure 4
The position sensor that was based in AAO membranes
Source: Own elaboration.
In Figure 5 is depicted the setup to measure positions of the Active Magnetic Bearing (AMB), due to get its control, which is the machine used to evaluate the algorithms that were proposed in this research.
Figure 5
Experimental setup
Source: Own elaboration.
1. Results
The position of the rotor (shaft) around its own axis was selected as physical variable in order to get a desired control, for which there were fixed four position sensors in opposite side to every electromagnet actuator, owing to capture its position as the reference from the shaft axis Y. However, through the position sensors that were based in nanostructure, it
