Perspective Chapter: Optimal Analysis for the Correlation between Vibration and Temperature through an Intelligent Sensor/Transducer Based in Amorphous Nanostructures to Measure Vibrating Surfaces Temperature

Abstract

The vibration is an oscillatory movement caused by a propagation of waves through fluids or solids, and this consequence is achieved in many mechanic systems by the energy transmission between the movement source with the machine that needs the transmission movement, such as the vibration produced by a combustion engine, by a compressor system and by a result of movement transmission over rotor systems. However, if it is not a controlled mechanism to moderate the produced decibels, the main system that is affected by the vibration can reduce its performance; moreover, it can increase the surface temperature of the vibrating source and systems around. In spite of this, when it uses contact sensors to measure the vibration and temperature over the surface vibrating system, the measured data are under disturbance caused by the vibration source. Therefore, in this research is proposed an intelligent sensor/transducer based in amorphous nanostructures owing to measure the vibration of the surface through infrared (IR) emitter/receiver and the absorbance of the receiver sample has a quite range of work and robustness under disturbance of vibrating signals. This proposed sensor also has the possibility to charge energy by itself because of sun/warmth energy conversion.

1. Introduction

The measurement of vibration and temperature is quite important according to get information regarding the oscillations [1, 2] and the molecular kinetic energy of the movement source and heat sources of engines internal components, or combustion motor’s external surfaces. However, there are tasks, in which is not possible to get contact between the sensor with the movement source and thermal source, hence the temperature measurement is given through IR sensors. In other side, it was possible to find the wave parameters from the IR signal of the thermal source owing to get an estimation of the vibration frequency of the movement source. Notwithstanding, there is a trouble concerning the transduction stage in the measurement while there is not a transducer algorithm designed as a consequence of mathematical model which correlates the calibration data with theoretical model of the heat transfer and the surface vibration of the movement source and thermal source. For this reason, it was proposed in this research to analyze a mathematical procedure of the measurement instrumentation according adaptive coefficients in MF strategies [3]. In this research is explained and analyzed the temperature measurement process and the transduction process as the strict correlation with the IR signal from the thermal source.

Therefore, the proposed sensor was evaluated for the measurement of the temperature and vibration of a combustion motor because of getting the understanding of its combustion and the motor user could achieve its diagnostic. There are many temperature sensors based in passive measurement such as thermocouples and thermistors, which proportionate the correlation of temperature in electrical equivalence of voltage and electrical resistance respectively, by other side, there are different piezoelectric to measure the vibration (frequency and amplitude) of surfaces. Nevertheless, there are tasks to measure vibration and temperature of systems that are located in intricate places and it is not suggested to use contact sensors [4, 5, 6, 7, 8, 9, 10, 11], hence IR sensors are the appropriated solution. For example combustion motors have many components inside and the combustion process can produce vibrations in them and in all the motor. Moreover, a not fuel good quality can cause damage in the combustion motors which are plenty used in industry and transport such as in Peru, where the transport (public and private) depends of this kind of motor and will continue using them during many years yet, in spite of the new technology in motors are enhanced by electric motors (or hybrid), hence, it is necessary to understand the combustion process in combustion motors. Therefore, temperature and vibration sensors can give information of the operation of the combustion motor according to repair them when its components are not working appropriated and as a consequence to look for their reparation avoiding pollution. To visualize the measured information, the sensor data (electrical equivalence of temperature and vibration) needs electronic devices due to compensate, amplify and linearize the electrical equivalent of temperature transduced to thermal units. In this research is proposed a mathematical model strategy based in MF in order to get the transduction result. The complexity of the transduction is replaced by the mathematical model designed.

The proposed sensor is part of an integrated system, which is depicted by the Figure 1. The proposed sensor is represented as IS (Intelligent Sensor) in order to measure temperature and vibration (T&V) and the measured data can be sent through IR to an external computer that can be at many meters of distance and 2 antennas A1 and A2 have the tasks of the data transmission. Moreover, the IS has independence of its own energy to be operating because of this proposed sensor has integrated a sample of sun panel to obtain electrical energy through sun/heat energy conversion.

Figure 1.

Communication system of the proposed IS.

Hence, in this research is proposed an intelligent sensor/transducer (as part of an integrated system) based in nanostructures due to measure the temperature and vibration of the combustion motor surface depicted by Figure 2, in which “A” is the IR emitter in controlled frequency that could not be confused with the IR signal produced by temperature of the combustion motor operation. “B” is the sample transducer to receive the vibration signal, for which “D” is the vibration sensor based in nanotubes amorphous. “C” is the Anodic Aluminum Oxide (AAO) sample transducer to receive the temperature signal for which “E” is the temperature sensor based in nanoholes amorphous. “F” is the battery to proportionate energy of the proposed sensor, and “G” is data transmitter according to send the temperature signal and vibration signal from the surface of the combustion motor to a receptor which can be used by the user due to get the diagnostic of the combustion motor. Moreover, H is the sample based in nanostructures to receive the sun energy and I is the converter to electrical equivalent signal due to store the energy in the batteries of the sensor. Many systems can improve their monitoring variable by advanced sensors such as sensors based in nanostructures [1] thereby the understanding of the geometry and material of the sensor is quite important due to get the optimal transduction as consequence of the measurements.

Figure 2.

Temperature and vibration transducer design.

2. Analysis of the optimal transduction design

Optimization analysis through MF is briefly studied in this chapter, in order to find the best solution for the data interpretation from the designed intelligent sensor, whereby the theoretical models of the physical process and analytic interpretation of the experimental results give the support to achieve the costing function by multivariable systems, because of the correlation among the internal variables of the system with the costing function.

Therefore, it was analyzed by polynomial equations as it is described by the general model in Eq. (1), for which “$D^n$” is the derivative $\frac{{\partial }^n}{\partial t^n}$, $y\left(t\right)$ and it is the variable output matrix, “$u\left(t\right)$” is the variable input matrix, “$e\left(t\right)$” is the variable error matrix, “a” and “b” are the adaptive coefficients of the system [3, 12].

$$D^{n}y\left(t\right)+\sum _{j=1}^n{a}_j{D}^{n-j}y\left(t\right)=\sum _{j=1}^n{b}_j{D}^{n-j}u\left(t\right)+e\left(t\right)$$E1

Where solution error analysis, “$e\left(t\right)$”, is discrete error, and “V” keeps the Fourier series coefficients, which is given by the Eq. (2) [3, 12].

$$e_n\left(m\right)=\sum _{k=m}^{n+m}\alpha \left(k,m,{\theta }_a\right)V\left(k\right)$$E2

Furthermore, $\alpha$ is the frequency parameter function given by Eq. (3) [3, 12].

$$\alpha \left(k,m,{\theta }_a\right)=C_{k-m}\sum _{j=0}^n{a}_j{\left(\mathit{jk}{w}_0\right)}^{n-j}$$E3

For which, the nonlinear model for the error analysis is given by the Eq. (4) [3, 12].

$$\sum _{j=0}^{n_1}\sum _{k=1}^{n_2}{g}_j\left(\theta \right)F_{\mathit{jk}}\left(u,y\right)P_{\mathit{jk}}\left(p\right)E_k\left(u,y\right)=0$$E4

Therefore, the costing function is given by Eq. (5) [3, 12].

$$J\left(\theta \right)=\sum _{j=0}^{n_1}\sum _{k=0}^{n_1}{r}_{\mathit{jk}}{g}_j\left(\theta \right)g_k\left(\theta \right)$$E5

Also, according to get parameters of the main model, it was achieved the derivation showed by Eq. (6) [3, 12].

$$\frac{\partial J}{\partial \theta }={\left(\mathrm{{\rm Y}}-\mathrm{\Gamma }\theta \right)}^T{W}^{-1}\left(\mathrm{{\rm Y}}-\mathrm{\Gamma }\theta \right)$$E6

Where parameters are showed in Eq. (7), as the dependence on the adaptive coefficients, in which $\mathrm{{\rm Y}}$ is the response matrix, $\mathrm{\Gamma }$ is the internal variables matrix, $W$ is the weight matrix and $\theta$ is the sensor parameters matrix [3, 12].

$$\theta ={\left({\mathrm{\Gamma }}^T{W}^{-1}\mathrm{\Gamma }\right)}^{-1}{\mathrm{\Gamma }}^T{W}^{-1}\mathrm{{\rm Y}}$$E7

The interpretation and scheme for optimization is depicted by Figure 3, in which are presented 3 paths C1 (green color curve), C2 (violet color curve) and C3 (red color curve) according to achieve the position B from the position A. C2 represents the theoretical path such as the theoretical variable of a process, C3 represents the experimental path due to an experimental data, therefore C1 is the optimal path owing to achieve the position B.

Figure 3.

Scheme of the optimal path.

3. Modeling

It is necessary to design the mathematical model of the proposed sensor that is possible to do through the interpretation of the problematic described in chapters above, moreover the mathematical analysis summarized previously helped to get the understanding of the static behavior and dynamic response of the sensor/transducer system.

In Figure 4 is depicted a first order system based in a thermal system owing to the heating transfer has the characteristic to not have overshoots and not faster response time. Hence, Th represents the temperature of the thermal focus HT, which is bigger than Tl (temperature of the body s) whereby heat is transferred from HT to LT. Furthermore, K is the thermal resistivity, A is the section area crossed by the heating transfer and db its thickness.

Figure 4.

Heat transfer in temperature sensor scheme.

Therefore, a temperatures sensor can be modeled by a first order system due to the heat transfer behavior and that model can be explained by Eq. (8) [12].

$$\mathrm{\Theta }\left(t\right)={\mathrm{\Theta }}_f\left(1-e^{-\frac{t}{\mathit{RC}}}\right)$$E8

Eq. (9) is obtained after the Laplace transformation in Eq. (8), in which $K_p$ is the proportional gain, $L$ is the sensor delay in temperature measurement, $\tau$ is the sensor response time for the temperature measurement [12].

$$\frac{T^{°}}{U\left(S\right)}=\frac{K_p{e}^{\mathit{LS}}}{\mathit{\tau S}+1}$$E9

In second side, for a second order model, such as described in Figure 5, in which, X0 means the displacement registered by the sensor and X1 means the real displacement because of the mass M and the overshoots are depending of K and β (deformation coefficient and damping coefficient), it means that a sensor with second order response can be depicted by Figure 5.

Figure 5.

Displacement in vibration sensor scheme.

Therefore, by Eq. (10) is possible to model a second order system according to understand its dynamic in time domain [12].

$$M\left(\frac{d^2{X}_1}{d{t}^2}-\frac{d^2{X}_0}{d{t}^2}\right)=K{X}_0+\beta \frac{d{X}_0}{\mathit{dt}}$$E10

By Laplace domain, it is obtained the following model due to interpret the experimental data as second order response, which is given by Eq. (11) [12].

$$M{S}^2{X}_1\left(S\right)=X_0\left(S\right)\left[K+\mathit{\beta S}+M{S}^2\right]$$E11

Thereby, Eq. (12) summarizes the parameters for a vibration sensor in Laplace domain [12].

$$\frac{X_0\left(S\right)}{S^2{X}_1\left(S\right)}=\frac{M}{K}\frac{\frac{K}{M}}{S^2+\frac{\beta }{M}S+\frac{K}{M}}$$E12

The mathematical models for sensors described in paragraphs above are enhanced by adaptive models achieved from experimental analysis and the improvement of their physical parameters also can be achieved from dynamic and geometry properties by dependence of the material of the designed sensor (nanostructures).

Such as for example, the theoretical model of the temperature sensor is given by Eq. (9) that was compared by the polynomial analysis of Eq. (1) and the coefficients of the theoretical model can be compared with the MF parameters from Eq. (9), which were obtained by the measured temperature data. Moreover, for the context of the vibration sensor, its theoretical model given by Eq. (12) also was compared with the experimental information described by the parameters Eq. (9) from Eq. (1). In this context, Eq. (13) is the model of the temperature sensor, in which, its parameter “k” is the temperature sensor gain and “τ” is its response time, this expression is achieved as a consequence of the parameters correlation from Eqs. (1) and (9).

$$\mathit{Gs}=\frac{k}{\mathit{\tau S}+1}$$E13

It is known the Zero Order Hold (ZOH) by Eq. (14).

$$G\left(\mathit{zoh}\left(S\right)\right)=\frac{1-e^{-\mathit{jw}{T}_s}}{\mathit{jw}}$$E14

That is equivalent to Eq. (15) in Laplace domain.

$$G\left(\mathit{zoh}\left(S\right)\right)=\frac{1-e^{-S{T}_s}}{S}$$E15

Looking for the digital model by Z transform in Eq. (16), which is achieved from Eq. (15).

$$\mathrm{H}\left(\mathrm{Z}\left\{\frac{\mathrm{k}}{\mathrm{\tau S}+1}\right\}\right)=G\left(\mathit{Zoh}\right)Z\left\{\frac{k}{\mathit{\tau S}+1}\right\}$$E16

Replacing the ZOH in last Eq. (16) is obtained Eq. (17).

$$H\left(Z\left\{\frac{k}{\mathit{\tau S}+1}\right\}\right)=Z\left\{\frac{1-e^{-S{T}_s}}{S}\right\}Z\left\{\frac{k}{\mathit{\tau S}+1}\right\}$$E17

It is known $\mathit{z}={\mathit{e}}^{\mathit{S}{\mathit{T}}_{\mathit{s}}}$ to replace in last Eq. (17) according to obtain Eq. (18).

$$H\left(Z\left\{\frac{k}{\mathit{\tau S}+1}\right\}\right)=\left(1-Z^{-1}\right)Z\left\{\frac{1}{S}\left(\frac{k}{\mathit{\tau S}+1}\right)\right\}$$E18

Eq. (19) is obtained reducing Eq. (18).

$$H\left(Z\left\{\frac{k}{\mathit{\tau S}+1}\right\}\right)=k\left(1-Z^{-1}\right)Z\left\{\frac{1}{S}\left(\frac{\frac{1}{\tau }}{S+\frac{1}{\tau }}\right)\right\}$$E19

By Z transform in Eq. (19) is obtained Eq. (20).

$$H\left(Z\left\{\frac{k}{\mathit{\tau S}+1}\right\}\right)=k\left(1-Z^{-1}\right)Z\left\{\frac{1}{S}\left(\frac{\frac{1}{\tau }}{S+\frac{1}{\tau }}\right)\right\}$$E20

Therefore, by Z transform is achieved the digital model of the proposed temperature sensor that is given by Eq. (21), because of Z transform in Eq. (20).

$$H\left(Z\left\{\frac{k}{\mathit{\tau S}+1}\right\}\right)=\frac{k{Z}^{-1}\left(1-e^{-\frac{T_s}{\tau }}\right)}{1-e^{-\frac{T_s}{\tau }}{Z}^{-1}}$$E21

By other side, the Tustin model is given by Eq. (22), in which “$T_s$” is the sampling time.

$$S=\frac{2}{T_s}\frac{Z-1}{Z+1}$$E22

Hence, the digital model of first order transfer function of the proposed temperature sensor (because of Tustin reduction) is given by Eq. (23) that was obtained replacing in Eq. (13) the Eq. (22).

$$H\left(Z\right)=\frac{T_{s}Z+T_s}{\left(2\tau +T_s\right)Z+\left(T_s-2\tau \right)}$$E23

Eqs. (21) and (23) are the digital model for the transfer function of the proposed sensor, for which Figure 6 shows the comparisons among the theoretical model of the temperature measurement for the operating combustion motor with the experimental temperature measurement, which were obtained and processed by the proposed temperature sensor. The blue color curve is the theoretical result based in heat transfer (Eq. (9)) from the surface motor to the sensor surface, the red color curve is the measurement data obtained by the execution of Eq. (21) in the processor of the temperature sensor, while the green color curve is the measurement data obtained by the execution of Eq. (23) in the processor of the temperature sensor. The error achieved by the measurement data represented by the red color curve was 0.5 percent approximately, and the error obtained by the measurement data represented by the green color curve was 0.9 percent approximately. For both contexts, the error analysis was made by the comparison of the measurement data with the theoretical curve (blue color).

Figure 6.

Theoretical curve (blue) versus experimental curves (red and green) for the measurement temperature analysis.

Therefore, it can be possible to choose Eq. (23) in order to be the base of the temperature monitoring algorithm for the proposed sensor, also because it has more simple expression for the programming in comparison with Eq. (21), which has not simple elements for the programming and it can cause consequences in the computing time. However, the result is much better by the programming of Eq. (21) because it produced less error than the measurement by the processing of Eq. (23), moreover the consequence in the computing time is solved by the short response time of the sensor owing to the nanostructures characteristics of the sample that received the IR signal of the measured temperature.

In order to compare the theoretical measurement with the experimental data of the combustion motor vibration surface, there were achieved the parameters of the second order system for the proposed vibration sensor by comparison of the Eqs. (1), (7) and (12), in similar context to the temperature sensor (described in paragraphs above), it was analyzed the digital equation by Z transform and Tustin reduction according to compare with the theoretical result that is given by the blue color curve and showed in Figure 7. The experiment was made by measuring the vibration of the combustion motor surface while it was pushed the accelerator due to keep stability of the RPM (Revolution Per Minute) and the red color curve is the data from the experimental measurement evaluated by the processor of the proposed sensor and the algorithm executed was supported by the Z transform, nevertheless, the green color curve was achieved by the experimental data that was evaluated by the Tustin model (reduction). Hence the less error value was obtained by the model based in Z transform even though the complication in its programming (in comparison of the model based by Tustin) was not a problem, because of the short response time of the sensor surface based in nanostructures.

Figure 7.

Theoretical curve (blue) versus experimental curves (red and green) for the measurement vibration analysis.

Thus, the error for both models were less than 1 percent (Error of 0.5 percent for Z transform and 1.8 percent for Tustin reduction) and for the operating work of the combustion motor there was not necessity to use digital model expressions for the algorithms analysis of the transduction, even though the sampling time was around 200 uS (less value than the minimal response time: 2mS), hence it was continued the analysis by Laplace domain. However, it can be used for high values of operating work, maybe for future applications.

The mathematical model of sensors are evaluated also as part of a control system for vibration and temperature analysis of a combustion motor, in which was necessary to identify the system and to keep a good performance of the vibration control for a combustion motor. Therefore, an interesting evaluation is given by PID (Proportional Integral Derivative) control as part of the identification system of the combustion motor, and the physical variables (vibration and pressure) are measured by the designed sensor.

Figure 8 shows the adaptive cascade algorithm that is represented by a block diagram to suit the measured signal received from the vibration motor. The input signal is given by the IR measured signal U1 that is adjusted by the matrix weights W1 and M1, controlled by C1 over the sensor/transducer P1.

Figure 8.

Block diagram scheme for the main control algorithm.

The response signal U2 (electrical value of the IR signal measured from the surface motor) is controlled by C2 and adjusted by M2, W2 due to obtain the temperature transduced as a consequence to know the function P2, moreover U3 (which also is U2) in concurrent time is compensated by C3 and adapted by M3, W3 in order to achieve the vibration response of the designed sensor as a consequence to know the transfer function P3.

The internal controller PID in block diagram scheme is depicted by Figure 9, which as necessary for the identification system of the combustion motor parameters that are part of the motor adaptive control. The input signal In(S) gives information of the desired value in temperature and vibration of the motor surface, Con(S) is the PID control (parameters), Pt(S) is the transfer function of the plant (motor surface), S(S) is the transfer function of the designed sensor, and Y(S) is the response signal.

Figure 9.

PID controller used for the identification system.

Eq. (24) is obtained as a result of the algebra analysis from the block diagram above.

$$\left(\mathit{\text{In}}\left(S\right)-S\left(S\right)Y\left(S\right)\right)\mathit{Con}\left(S\right)\mathit{Pt}\left(S\right)=Y\left(S\right)$$E24

Thus, the transfer function obtained from the reduction of Eq. (24) is given by Eq. (25).

$$\frac{Y\left(S\right)}{\mathit{\text{In}}\left(S\right)}=\frac{\mathit{Con}\left(S\right)\mathit{Pt}\left(S\right)}{1+S\left(S\right)\mathit{Con}\left(S\right)\mathit{Pt}\left(S\right)}$$E25

In Eq. (26), it is generalized the transfer function for temperature/vibration combustion internal control, in which $K_S$ is the gain parameter of the designed sensor, ${\tau }_s$ is the response time of the sensor. $K_P$ is the controller proportional gain, $K_D$ is the Controller derivative gain, $K_I$ is the Controller integral gain. $K_{\mathit{pt}}$ is the gain parameter of the plant (combustion motor), ${\tau }_{\mathit{pt}}$ is the plant response time.

$$\left(\frac{K_S}{{\tau }_s\mathrm{S}+1}\right)\left(K_P+K_{D}S+\frac{K_I}{S}\right)\left(\frac{K_{\mathit{pt}}}{{\tau }_{\mathit{pt}}\mathrm{S}+1}\right)+1=0$$E26

The reduction from the equation is given by Eq. (27), for which was decided a Proportional Derivative (PD) controller due to achieve a fast control response under the vibration motor.

$${\tau }_s{\tau }_{\mathit{pt}}{S}^2+\left({\tau }_s+{\tau }_{\mathit{pt}}+K_s{K}_D{K}_{\mathit{pt}}\right)\mathrm{S}+\left(K_s{K}_p{K}_{\mathit{pt}}+1\right)=0$$E27

In Eq. (28) is organized Eq. (27) as a polynomial in second descending order.

$$S^2+\frac{\left({\tau }_s+{\tau }_{\mathit{pt}}+K_s{K}_D{K}_{\mathit{pt}}\right)}{{\tau }_s{\tau }_{\mathit{pt}}}S+\frac{\left(K_s{K}_p{K}_{\mathit{pt}}+1\right)}{{\tau }_s{\tau }_{\mathit{pt}}}=0$$E28

The control parameters can be obtained by different methodologies such as the stability analysis, furthermore the comparison with the theoretical model of the system dynamic given by Eq. (29) [3, 12], in which ${\omega }_0$ is the natural frequency for the system and є is the damping effect.

$$S^2+\left(2\mathrm{є}{\omega }_0\right)\mathrm{S}+{{\omega }_0}^2=0$$E29

Hence, the control parameters $K_p$ and $K_I$, can be obtained by the comparison of the coefficients from Eqs. (28) and (29), from which are proposed the following Eqs. (30) and (31) that are functions of ${\tau }_s,{\tau }_{\mathit{pt}},K_s,K_{\mathit{pt}}$,${\omega }_0$ and є.

$$\frac{\left({\tau }_s+{\tau }_{\mathit{pt}}+K_s{K}_D{K}_{\mathit{pt}}\right)}{{\tau }_s{\tau }_{\mathit{pt}}}=2\mathrm{є}{\omega }_0$$E30

$$\frac{\left(K_s{K}_p{K}_{\mathit{pt}}+1\right)}{{\tau }_s{\tau }_{\mathit{pt}}}={{\omega }_0}^2$$E31

Also, Eq. (32) is the proportional parameter of the PD controller obtained from Eq. (31).

$$K_P=\frac{{{\omega }_0}^2{\tau }_s{\tau }_{\mathit{pt}}-1}{K_s{K}_{\mathit{pt}}}$$E32

Finally, the derivative gain is obtained from Eq. (30) and showed by Eq. (33).

$$K_D=\frac{2\mathrm{є}{\omega }_0{\tau }_s{\tau }_{\mathit{pt}}-{\tau }_s-{\tau }_{\mathit{pt}}}{K_s{K}_{\mathit{pt}}}$$E33

After to obtain the control parameters, it is possible to warrant the influence of the designed sensor in the stability of the system, thus, analyzing Lyapunov stability from equation previous, for which Eq. (34) is the complement of Eq. (17) in which U(S) is the input excitation signal and R(S) is the small displacement in Laplace domain.

$$S^2+\frac{\left({\tau }_s+{\tau }_{\mathit{pt}}+K_s{K}_D{K}_{\mathit{pt}}\right)}{{\tau }_s{\tau }_{\mathit{pt}}}S+\frac{\left(K_s{K}_p{K}_{\mathit{pt}}+1\right)}{{\tau }_s{\tau }_{\mathit{pt}}}=\frac{U\left(S\right)}{R\left(S\right)}$$E34

Eq. (35) is a reduction from Eq. (34) but in time domain.

$$\frac{d^{2}r\left(t\right)}{d{t}^2}+\frac{\left({\tau }_s+{\tau }_{\mathit{pt}}+K_s{K}_D{K}_{\mathit{pt}}\right)}{{\tau }_s{\tau }_{\mathit{pt}}}\frac{\mathrm{d}r\left(t\right)}{\mathit{dt}}+\frac{\left(K_s{K}_p{K}_{\mathit{pt}}+1\right)}{{\tau }_s{\tau }_{\mathit{pt}}}r\left(t\right)=u\left(t\right)$$E35

In addition, preparing variable changes and showed by Eq. (36).

$$\mathrm{y}\left(\mathrm{t}\right)=\frac{\mathrm{d}\mathit{r}\left(\mathit{t}\right)}{\mathit{dt}}$$E36

Eq. (37) is achieved replacing the Eqs. (36) and (34) in Eq. (35), for u(t) null:

$$\frac{\mathrm{d}\mathit{y}\left(\mathit{t}\right)}{\mathit{dt}}=-\frac{\left({\mathit{\tau }}_{\mathit{s}}+{\mathit{\tau }}_{\mathit{pt}}+{\mathit{K}}_{\mathit{s}}{\mathit{K}}_{\mathit{D}}{\mathit{K}}_{\mathit{pt}}\right)}{{\mathit{\tau }}_{\mathit{s}}{\mathit{\tau }}_{\mathit{pt}}}\mathit{y}\left(\mathit{t}\right)-\frac{\left({\mathit{K}}_{\mathit{s}}{\mathit{K}}_{\mathit{p}}{\mathit{K}}_{\mathit{pt}}+\mathrm{1}\right)}{{\mathit{\tau }}_{\mathit{s}}{\mathit{\tau }}_{\mathit{pt}}}\mathit{r}\left(\mathit{t}\right)=\mathrm{0}$$E37

Organizing the last equation by energy analysis, $\mathit{E}\left(\mathit{r},\mathit{y}\right)$, in order to find the Lyapunov equation, which is positive and can achieve the first Lyapunov condition given by Eq. (38).

$$\mathit{E}\left(\mathit{r},\mathit{y}\right)=\mathrm{0.5}{\left(\mathit{y}\left(\mathit{t}\right)\right)}^{\mathrm{2}}+\mathrm{0.5}\frac{\left({\mathit{K}}_{\mathit{s}}{\mathit{K}}_{\mathit{p}}{\mathit{K}}_{\mathit{pt}}+\mathrm{1}\right)}{{\mathit{\tau }}_{\mathit{s}}{\mathit{\tau }}_{\mathit{pt}}}{\left(\mathit{r}\left(\mathit{t}\right)\right)}^{\mathrm{2}}=\mathrm{0}$$E38

Looking for the second Lyapunov condition by the inequality (39).

$$\frac{\mathrm{d}\mathit{E}\left(\mathit{r}\left(\mathit{t}\right),\mathit{y}\left(\mathit{t}\right)\right)}{\mathit{dt}}\le \mathrm{0}$$E39

Therefore, the inequality (40) is obtained replacing Eq. (38) in the inequality (39).

$$\mathit{y}\left(\mathit{t}\right)\frac{\mathrm{d}\mathit{y}\left(\mathit{t}\right)}{\mathit{dt}}+\frac{\left({\mathit{K}}_{\mathit{s}}{\mathit{K}}_{\mathit{p}}{\mathit{K}}_{\mathit{pt}}+\mathrm{1}\right)}{{\mathit{\tau }}_{\mathit{s}}{\mathit{\tau }}_{\mathit{pt}}}\mathit{r}\left(\mathit{t}\right)\frac{\mathrm{d}\mathit{r}\left(\mathit{t}\right)}{\mathit{dt}}\le \mathrm{0}$$E40

Also, replacing Eqs. (36) and (37) in the inequality (40) is obtained the inequality (41).

$$\mathit{y}\left(\mathit{t}\right)\left(-\frac{\left({\mathit{\tau }}_{\mathit{s}}+{\mathit{\tau }}_{\mathit{pt}}+{\mathit{K}}_{\mathit{s}}{\mathit{K}}_{\mathit{D}}{\mathit{K}}_{\mathit{pt}}\right)}{{\mathit{\tau }}_{\mathit{s}}{\mathit{\tau }}_{\mathit{pt}}}\mathit{y}\left(\mathit{t}\right)-\frac{\left({\mathit{K}}_{\mathit{s}}{\mathit{K}}_{\mathit{p}}{\mathit{K}}_{\mathit{pt}}+\mathrm{1}\right)}{{\mathit{\tau }}_{\mathit{s}}{\mathit{\tau }}_{\mathit{pt}}}\mathit{r}\left(\mathit{t}\right)\right)+\frac{\left({\mathit{K}}_{\mathit{s}}{\mathit{K}}_{\mathit{p}}{\mathit{K}}_{\mathit{pt}}+\mathrm{1}\right)}{{\mathit{\tau }}_{\mathit{s}}{\mathit{\tau }}_{\mathit{pt}}}\mathit{r}\left(\mathit{t}\right)\mathit{y}\left(\mathit{t}\right)\le \mathrm{0}$$E41

Finally, it is obtained the inequality (42) due to achieve the second Lyapunov condition, moreover while ${\mathit{\tau }}_{\mathit{s}}$ is small the control system get better stability, it can be possible by sensors with short response time such as the sensors based in nanostructure (as it is designed the proposed sensor of this research).

$$-\frac{\left({\mathit{\tau }}_{\mathit{s}}+{\mathit{\tau }}_{\mathit{pt}}+{\mathit{K}}_{\mathit{s}}{\mathit{K}}_{\mathit{D}}{\mathit{K}}_{\mathit{pt}}\right)}{{\mathit{\tau }}_{\mathit{s}}{\mathit{\tau }}_{\mathit{pt}}}\mathit{y}{\left(\mathit{t}\right)}^{\mathrm{2}}\le \mathrm{0}$$E42

All the analysis was made in Laplace domain, because the response time is enough bigger than the sampling time of both systems “temperature and vibration” by the processor of the advanced sensor. Furthermore, the robustness and short response time of the sensors based in nanostructures give possibility to execute complicated algorithms, however this computing task can be prioritized for future analysis.

4. Sensor design

A good sensitivity of the temperature and vibration of the combustion motor surface depends of the internal membranes of the designed smart sensor, which are prepared by nanotructures of AAO, hence the elaboration of the nanostructures (nanoholes and nanotubes) are prepared by a sequence of steps over aluminum: ultrasound cleaning of aluminum, electropolishing, anodization, and atomic load deposition. Nevertheless, if there is not a good quality of aluminum, the base of the nanostructure could not have robustness over their geometry, in Figure 10 is showed some aluminum samples at 99.9 percent for the designed vibration sensor [13]. Sensors based in samples of nanostructures give the possibility to achieve robustness and short response time [14].

Figure 10.

Aluminum samples.

Electropolishing is an electrochemical process according to achieve more sophisticated cleaning over every sample by electrolysis and from anode to cathode [13], that is showed by Figure 11.

Figure 11.

Electropolishing.

After the electropolishing, the samples achieve very much shining like a mirror whereby Figure 12 shows 4 samples, in which one of them (up left side) is showed the best electropolished sample.

Figure 12.

Electropolished samples.

The anodization produces chemical effects on the cleaned aluminum such as the holes in nanoscales, this process was made in controlled electro-chemical perchloric acid reaction, and adjusting the electrical source between 20 V to 30 V in environment around 0 Celsius degrees, which is showed by Figure 13. Moreover, by electrochemical deposition was possible to prepare nanostructures amorphous over the AAO samples.

Figure 13.

Anodization and electrochemical deposition process for the sensor elaboration.

As a result were obtained nanostructures samples based in AAO, which are showed by Figure 14. The subfigures “A” and “C” are the samples prepared in the Applied Nanophysics, Institute for Physics of TU Ilmenau by the cooperation research between PUCP and TU Ilmenau, “B” is the sample prepared in the researching laboratories 1 and 2 of the Mechanical Department of PUCP by the optimal procedure discovered by Prof. Lei group.

Figure 14.

AAO samples for the sensor design.

The samples prepared are adapted through its own geometry in nanoscale due to obtain amorphous nanotubes for the vibration sensor and amorphous nanoholes for the temperature sensor. There is an IR emitter as part of the sensor/transducer design, which send IR signal to the combustion motor surface in controlled frequency due to recognize the differences with the IR signal caused by the temperature changes of the combustion motor surface. Therefore, the temperature measurement is correlated between the IR signal with its own IR signal caused because of the temperature changes in the motor surface (temperature measured by IR [6]).

The receptors are given by the nanostructures samples that send the measured data (temperature and vibration) to the microcontroller owing to execute the mathematical model of the correlation among the theoretical model with the experimental mathematical analysis, finally that data is sent to the user by wireless port. In order to obtain the electrical response of the measured variable, it was fixed some cables in every corner of the sample. The electrical resistance equivalent is the physical variable to correlate the measured variable. In Figure 15 are showed 3 prototypes of designed sensors/transducers (A, B and C).

Figure 15.

Prototypes of proposed sensors/transducers.

The sample B of Figure 12 is showed under a microscope Litz in the scale 25 micrometers, thereby the Figure 16 shows some amorphous structures with maximal scales are around 1000 nanometers.

Figure 16.

Amorphous nanostructures over the surface of the designed sensor/transducer.

In Figure 17 is depicted the algorithm scheme for the sensor transduction by the flowchart of the sensor/transducer operation described in paragraphs above, in which physical variables temperature and vibration are measured and processed by concurrence and finally both signals are correlated according to obtain the final transduction result.

Figure 17.

Algorithm scheme for the sensor transduction.

Algorithm scheme in operation during the measurement of the physical variables vibration and temperature of motor surfaces is depicted in the following Figure 18, thereby the predictions of the adaptive algorithm can be obtained by the interpretation of the IR reception [6].

Figure 18.

Algorithm scheme in operation during the measurement.

In the following Figure 19, by the curve A is depicted the measurement data from the designed sensor, the curve B represents the measurement data received by the personal computer at 50 meters of distance with a delay L1 because of the medium used was internet. Nevertheless, if it is used radiofrequency medium communication the delay is reduced in L2 as it is depicted by the curve C.

Figure 19.

Representation of the measured data (a), its transmission by internet (B) and by radiofrequency (C).

It is showed by the Figure 20 the motor used for the experiments, which is a Nissan frontier 2005. The sensors (4 of them) were positioned around the motor by non-contact in every Cartesian axis (X, Y, Z).

Figure 20.

Combustion motor of a Nissan frontier 2005, in which were made the experiments.

Fixing a small electro-pneumatic actuator over the accelerator pedal, which receive the control signal according the main control algorithm that receive the vibration and temperature signal from the designed sensor (through the radiofrequency antennas).

In Figure 21 is showed the vibration of the combustion motor measured by the designed sensor. The vibration signal was transduced from IR to electrical signal (Voltage) that is showed by the blue color curve, and its amplification in equivalent of Decibels by the red color curve in the Figure 18. The combustion motor was evaluated in operation around 4000 RPM and the maximal spectral density was obtained approximately in 68 Hz that can be seen by the green color curve, which in addition can justify the operation frequency of the internal combustion motor.

Figure 21.

Vibration curves achieved by the designed sensor.

The vibration of the motor surface was captured by the designed sensor/transducer and it was sent by IR to the emitter antenna that sent the data by radiofrequency to the receptor antenna, which is at 50 meters outside, moreover the receptor antenna sent the measured vibration to a personal computer by IR, also, according the control signal to activate the electro-pneumatic actuator to change the position of the accelerator pedal. That curves information can be interpreted by the user according to get understanding of the behavior of the motor as a consequence of the combustion. It is necessary to remind that the intelligent algorithm of the adaptive correlation to achieve the physical variables transductions had 87 SNR (Signal to Noise Ratio) in average.

The operating frequency is quite dependent of the vibration frequency of the combustion motor, because of this is the main source of changes in the described system. The vibration measurement can be simpler due to its correlation with the vibrating motor, however, the temperature measurement depends not only from the operating vibration motor, it also depends from its delay caused due to the thermal inertia. Furthermore, the sample frequency is part of the designed sensor analysis.

It was evaluated the performance of the sensor/transducer by different temperature changes, which were caused by accelerating the motor during 45 minutes approximately. Figure 22 shows the optimal estimated temperature of the surface motor that is given by the green color curve. The optimal estimation is achieved as a consequence of the correlation between the experimental measurement (blue color curve) with the temperature measurement by a thermocouple type k (red color curve). Hence, the designed sensor/transducer can measure the temperature of the motor surface by optimal estimations according to answer in front of disturbances, moreover the measured data was sent through IR to the emitter antenna that sent the data to the receptor antenna by radiofrequency at 50 meters outside, from which the information is received by a personal computer through IR with the receptor antenna. The user can interpret the data received, such as for example the diagnostic of the motor by the combustion effects because of the temperature changes. The delay obtained by the radiofrequency data monitoring (vibration and temperature of the surface motor) was between 700mS to 800mS, and the delay obtained by internet data monitoring was between 1.2S to 1.4S.

Figure 22.

Temperature curve from the designed sensor in Celsius degrees.

5. Conclusions

It was designed an intelligent temperature and vibration sensor/transducer based in nanostructures of AAO owing to measure the temperature and vibration over the surface of combustion motors, the achieved data is sent to the user according to get a diagnostic of the combustion motor performance and the user can understand whether the motor could need reparation or not.

The novelty of this proposed article is given in the mathematical analysis to design sophisticated transducers, the support of the analysis is the polynomial structure of the mathematical modeling due to the correlation between the theoretical equation of heating and vibration transfer with the experimental data achieved during the calibration. The MF in adaptive coefficients of the final model gives the advantage to optimize the data filtering while it is supported by the calibration information.

The algorithm designed as a consequence of the mathematical model can be programmed by different language programming because of the simple instructions and the weights achieved from the mathematical model and calibration data help to adapt the measured data according to estimate the right measured temperature and vibration by non-contact transduction. Hence, the transducer designed optimize electronic components of instrumentation, and as a consequence can enhance the effect over pollution caused by combustion motors in Peru, which are used in big quantities by the public and private transport without a continuous and practical monitoring of their operation, moreover it was possible to evaluate the performance of the designed sensor through wireless communication by IR and radiofrequency, which was possible to achieve because of the short response time and robustness of the designed sensor give enough time for the data communication according to get telemetric monitoring of the measured variables.

Finally, the designed sensor can use the energy stored from its own sun energy converter, which gives more independence and autonomy to the designed sensor.

The sample time was also part of the analysis because of it was obtained the digital models for the equations that can be programmed by the processor of the intelligent sensor (to measure the temperature and vibration of the motor surface). Nevertheless, for the RPM operating work the achieved error was less than 1 percent (for the Z transform reduction), hence it was decided to prioritize the equation analysis by Laplace domain, but the model can be used in their equivalent digital expressions whether the total computing response time could be near the system response time (temperature and vibration of the computer motor) even though the fast response in the transductions because of the sensor is based in nanostructures is a good advantage for the communication time when it was sent the measurement by wireless.

6. Future work

It is proposed for a future research that the designed transducer can improve its performance for more amplitude of range of work due to the mathematical analysis can adapt the characteristics parameters of the sensor during the calibration, furthermore, the adaptive transduction can give faster and good response in nonlinear range of work. In addition, with the mathematical model designed for the vibration/temperature transducer can be adapted for complex correlations or control tasks that could be made on wireless.

Moreover, it is proposed for a future research to enhance the applications of the designed sensor in telemetry control and increase the distance between the antennas which support for the measured data transmission.

The heat produced due to the combustion motor operation (the maximal temperature value of the motor surface) can be used to be transformed in electrical energy to be used by the control system, thus this is a target for a future work of this research.

Acknowledgments

It is expressed deep warm gratefulness to Mrs. Aleksandra Ulianova de Calderón due to her total support for the development of this research due to find the understanding of the compromise among new technologies with the environment cares.

There is expressed special thankful to the Mechatronic Engineering Master Degree Program at PUCP, to the Engineering Department PUCP, and DGI (“Dirección de Gestión de la Investigación”) researching office from PUCP because of its financial support in this research through the financing FONCAI.