- Email: [email protected]
Dynamic Analysis of Piezoelectric Fan with Random System Parameters
Available online at www.sciencedirect.com Available online at www.sciencedirect.com
ScienceDirect ScienceDirect
Procediaonline Computer 00 (2018) 000–000 Available at Science www.sciencedirect.com Procedia Computer Science 00 (2018) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Computer Science 133 (2018) 45–54
International Conference on Robotics and Smart Manufacturing (RoSMa2018) International Conference on Robotics and Smart Manufacturing (RoSMa2018)
Dynamic Analysis of Piezoelectric Fan with Random System Dynamic Analysis of Piezoelectric Fan with Random System Parameters Parameters Ankita Kotaaa, Sujoy Mukherjeebb* Ankita Kota , Sujoy Mukherjee * a Mechatronics, IIITDM Jabalpur, India. a Mechanical Department, IIITDM Jabalpur, Mechatronics, IIITDM Jabalpur, India.India. b Mechanical Department, IIITDM Jabalpur, India. b
Abstract Abstract Now a day’s electronics is becoming more and more compact. Electronic circuits and devices need cooling means in order to Now a day’s morethese and days. more Piezoelectric compact. Electronic cooling means in order to dissipate heat.electronics Heat sinksisarebecoming mostly used fans cancircuits also beand useddevices for thisneed purpose. Piezoelectric fans are dissipate heat. Heat sinks mostly containing used these substrate days. Piezoelectric fans can also used of forpiezoelectric this purpose.material. Piezoelectric fans are cantilevered uni-morph or are bi-morph, sandwiched between twobelayers As sinusoidal voltage is applied to piezoelectric material, cantilever starts to vibrate and generates air flow. The advantage of usingAspiezofans is cantilevered uni-morph or bi-morph, containing substrate sandwiched between two layers of piezoelectric material. sinusoidal voltage applied to piezoelectric cantilever starts vibrate andand generates air play flow.aThe advantage of using piezofans is that theyisconsume less power and material, are low noise devices. Thetodeflection frequency major role in flow generation. This that they consume less power are low noise The frequency play a major role in flow Carlo generation. This paper predicts the behavior ofand displacement and devices. frequency ondeflection variation and of material properties. Classical Monte simulation is usedpredicts for prediction of uncertainty in performance parameter. It is found that uncertainty material Monte properties large paper the behavior of displacement and frequency on variation of material properties.inClassical Carlocauses simulation is used forinprediction of uncertainty performance parameter. deviations performance parameters,infrom the predicted ones. It is found that uncertainty in material properties causes large deviations in performance parameters, from the predicted ones. © 2018 The Authors. Published by Elsevier Ltd. © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) © 2018 The Authors. Published by Elsevier Ltd.committee This is an open access article under thescientific CC BY-NC-ND license Peer-review under responsibility of the of the(https://creativecommons.org/licenses/by-nc-nd/4.0/). International Conference on Robotics and Smart Manufacturing. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords:Smart material, Piezo, Dynamics,Uncertainty,Mathematical Model. Keywords:Smart material, Piezo, Dynamics,Uncertainty,Mathematical Model.
* Sujoy Mukherjee. Tel.+91-761-2794421. [email protected] *E-mail Sujoyaddress: Mukherjee. Tel.+91-761-2794421. E-mail address: [email protected] 1877-0509© 2018 The Authors. Published by Elsevier Ltd. This is an open access under the CC by BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). 1877-0509© 2018 Thearticle Authors. Published Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). 1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the International Conference on Robotics and Smart Manufacturing. 10.1016/j.procs.2018.07.007
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Author name / Procedia Computer Science 00 (2018) 000–000
46 2
1.
INTRODUCTION
Piezoelectric material has the property of generating electric charge on applying pressure on it. The converse of this is also possible [1].The conversion rate depends on coupling factor. As devices are getting compact day by day, piezoelectric fans are gaining more and more importance. Direct piezoelectric effect has proved advantageous for energy harvesting. Piezofan can be unimorph or bimorph, with substrate and piezo patch. As varying voltage is applied to beam, piezo patch generates force and generates deflection in structure; this in turn induces air flow. These piezoelectric fans have their own natural frequency, which depends on the material used. As the external frequency matches to natural frequency, resonance occurs. Piezo fans have been the area of research since many years. Toda and Osaka[2] found out that as amplitude and strain were increased, resonance was shifted to lower frequency, and hysteresis appeared at low frequency which resulted in decrease of Young’s modulus. Yao and Uchino [3] analyzed the composite piezoelectric structure using lumped mass analysis. The frequency of vibration also depends on length of fan. Maximum deflection can be obtained by optimizing blade length ratio, thickness ratio [4-5]. Dynamic response of symmetric and asymmetric resonators was investigated by Basak et al. [6]. An analytical model was developed for both configurations using beam theory. The performance of piezoelectric fans at higher modes was studied in ref [7]. It was seen that at higher resonance modes, power consumed was more and fluid flow was decreased. It was concluded that these fans can only be implemented in their first resonance mode. Yoo et al. [8] established the relation between tip deflection and wind velocity when fan is operated at fundamental resonance of the metal plate. Uncertainties of the structural and geometric parameters affecting the performance of piezofans are determined by Classical Monte Carlo method [910]. In present study, piezoelectric cantilevered beam was considered as a continuous system. Uncertainties in the structure have a great influence on the dynamic performance of piezofan.The effect on the natural frequency and deflection was determined using Monte Carlo Method and the maximum and minimum values were determined. 2. MATHEMATICAL MODELING 2.1. Structure
Fig.1.Structure of piezofan
The structure of piezoelectric unimorph used is as shown in fig.1. A thin plate of Mylar sheet sandwiched between two patches of piezoelectric materials. The total length is divided into three regions. Starting from the clamped end
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Ankita Kota et al/ Procedia Computer Science 00 (2018) 000–000
47 3
to start of piezo patch is region 1, region 2 is from starting of piezo patch to end of piezo patch and region 3 extends from the end of piezo patch to the free end of the elastic beam. 2.2. Modeling Euler beam theory is used to derive equations of motion. There are two equations for each region along with boundary conditions [6]. ‘u’ and ‘w’ represent longitudinal and transverse displacements respectively. Fig. 2 and Fig.3.show the free body diagram for transverse and longitudinal loads. For simplification a small area dx is taken into consideration. Equations of motion for a particular section are obtained by balancing the moments and forces along the section dx. Here, M is bending moment, V is shear force and P represents the longitudinal force acting on the section. Similarly equations of motion are derived for each region.
Fig.1 Free body diagram for transverse loads.
Region1 − + = 0,
(1)
+ = 0,
(2)
Region 2 − + + + = 0
− + − 2 + 2 + = 0
(3) (4)
Region 3 − + = 0,
(5)
+ = 0,
(6)
Boundary Conditions 0, = 0,
0, = 0 ,
0, = 0,
, = , ,
48 4
, = , ,
, = , ,
, = , .
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Author name / Procedia Computer Science 00 (2018) 000–000
, = ,
, = ,
Natural boundary conditions + , + = ,
2 + 2 + , + = , ,
,
2 + 2 + , = , ,
Fig.2 Free body diagram for longitudinal loads.
+ , + = , ,
2 + 2 + , − = , , 2 + 2 + , = , ,
, = 0,
, = 0,
, = 0.
(7)
Terms with dot and dash on top represent differentiation with respect to time and x axis respectively. The terms with double dot represent inertia force acting on the section. Terms with dash in equation (2),(4) and (6) represent moments acting on the section. Terms with dash in equation (1),(3) and (5) represent net force acting on that particular section. Solution for these equations using variable separable method is given by = sin + + ℎ + ℎ + sin + , = , + , , Where i=1,2,3.
(8) (9)
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Ankita Kota et al/ Procedia Computer Science 00 (2018) 000–000
, = =
/
,
/
, = ,
=
/
49 5
, /
(10)
On applying first three boundary conditions, we get , =0, A1= -C1 and B1= -D1. On applying remaining 15 boundary conditions 15 equations are obtained, using which 15 x 15 matrix containing coefficients of the constants is constructed. By solving the determinant of this matrix natural frequency is found out. Natural frequency obtained is 42 Hz.Using this frequency magnitude of displacement obtained is 3.30 mm. Table1. Material Properties and dimensions [6]. Physical Quantity
Value
Eb
Young’s Modulus of the substrate(Mylar)
Ep
Young’s Modulus of the actuator
ρb
4.6 ∗ 10 N/m
Density of substrate
ρp
Density of the actuator
1240 ⁄
6.2 ∗ 10 N/m 7800 ⁄
tb
Thickness of the beam
0.1016 mm
tp
Thickness of the actuator
0.127 mm
B
Width of the beam and actuator
8.89 mm
L1
Length of region 1
0.5 mm
L2
Length of region2
0.03 mm
L3
Overall length of the structure L3
27.424 mm
K33
Relative dielectric constant
3800
ε0
Permittivity of the free space
d31
Piezoelectric strain coefficient
8.8542 ∗ 10 ⁄
Table1. Shows the values of parameters used in analysis.
−320 ∗ 10 ⁄
3. UNCERTAINTY ANALYSIS Monte Carlo method is used for solving probabilistic problems. Randomness in the parameters is considered in terms of coefficient of variation (COV). COV is measure of relative variability. It is ratio of standard deviation () to average (). Specific value of coefficient of variation is selected for each parameter [9-10]. Random values of the inputs are generated using the equation below (11) = + Where is the random number generated between 0 and 1.The number of samples is selected based on the convergence of standard deviation of performance parameters. Fig.4 shows the convergence diagram of standard deviation for frequency. From Fig.4 it can be clearly seen that the curve starts to converge at 10,000 samples. Therefore number of samples considered for further analysis is 10,000.
Table 2.shows the values of mean and coefficient of variance for system parameters. COV for geometric parameters is always taken as one while for other parameters is determined experimentally The values for the parameters used for numerical simulation are taken from the literature [9]. Fig. 5. (a), (b) and (c) show the frequency pattern, standard deviation in frequency and probability distribution after consideration of variations in parameters. Standard deviation curve shows the deviation of a parameter from the true value and the probability curve shows the distribution of samples in the particular range. It can be seen from fig.5(c) that
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Author name / Procedia Computer Science 00 (2018) 000–000
50 6
probability distribution is of normal type and is maximum in range of 40-44 Hz.
Fig.3 Divergence of standard deviation curve for frequency. Table 2. Coefficient of variance for different parameters [9]. Random input variable
Mean Value
Distribution Type
COV (%)
L1 L2 L3 W tp tb Ep
0.5 mm 0.03 mm 27.423 mm 8.8894mm 0.127mm 0.1016mm
Normal Normal Normal Normal Normal Normal Normal
1 1 1 1 1 1 3
Eb
4.6 ∗ 10 N/m
Normal
3
Normal
5
6.2 ∗ 10 N/m
d31
−3.1998 ∗ 10
Fig.6. (a), (b) and (c) shows the displacement pattern, standard deviation of displacement and probability distribution. It can be seen that probability distribution is maximum in interval of 3.2 mm to 3.4 mm. Table 3.Randomness of output parameters Parameters
Std Deviation
Mean
COV (%)
Max Value
Min Value
Frequency(Hz)
1.49
42.01
3.5
47.57
35.76
Displacement(mm)
0.10
3.37
3.0
3.82
2.99
Table.3 shows the results of simulation. It is found that COV for frequency and displacement is 3.5% and 3.0% respectively. Maximum and minimum displacement values for piezo fan are found to be 3.8 mm and 2.9 mm respectively with mean of 3.37 mm. This shows the influence of randomness of input parameters in output of
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Ankita Kota et al/ Procedia Computer Science 00 (2018) 000–000
piezofans.
(a)
(b)
51 7
et al. /Computer Procedia Computer (2018) 45–54 AuthorAnkita nameKota / Procedia Science 00Science (2018) 133 000–000
852
(c) Fig.5. (a).Frequency Pattern, (b).Standard deviation in frequency, (c).Probability distribution for frequency.
4. CONCLUSIONS MCS is carried out for 10,000 considering the piezo fan as a continuous system. Randomness for geometric parameters is considered to be 1. It is found that COV for frequency and displacement is 3.5% and 3.0% respectively. Similarly minimum and maximum values of frequency are 35.76 Hz and 47.57 Hz respectively with mean of 42.01 Hz. Maximum and minimum displacement values for piezo fan are found to be 3.82 mm and 2.9 mm respectively with mean of 3.37 mm. This shows the influence of randomness of input parameters in output of piezofans. Thus it is needed to incorporate randomness of input parameters in analysis of piezo fan and Monte Carlo method has proven to be an efficient method. Uncertainty in any system introduces errors in engineering data and leads to failure of system. The numerical results provide useful bounds on parameters that effect performance of piezoelectric cooling fan.
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Ankita Kota et al/ Procedia Computer Science 00 (2018) 000–000
(a)
(b)
53
9
Author name / Procedia Computer Science 00 (2018) 000–000
10
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54
54
(c) Fig.6. (a).Displacement Pattern, (b).Standard deviation in displacement, (c).Probability distribution for displacement.
References [1] Leo, D.J., 2007. Engineering analysis of smart material systems. John Wiley & Sons. [2] Toda, M., 1980. Elastic properties of piezoelectric PVF2.Journal of Applied Physics, 51(9), pp.4673-4677. [3] Yao, K. and Uchino, K., 2001. Analysis on a composite cantilever beam coupling a piezoelectric bimorph to an elastic blade. Sensors and Actuators A: Physical, 89(3), pp.215-221. [4] Burmann, P., Raman, A. and Garimella, S.V., 2002. Dynamics and topology optimization of piezoelectric fans.IEEE Transactions on Components and Packaging Technologies, 25(4), pp.592-600. [5] Wolf, K.D., 2000. Electromechanical energy conversion in asymmetric piezoelectric bending actuators. PhD Thesis, TU Darmstadt, pp.1 -65. [6] Basak, S., Raman, A. and Garimella, S.V., 2005. Dynamic response optimization of piezoelectric ally excited thin resonant beams. Journal of Vibrations and Acoustics, 127(1), pp. 18-27. [7] Wait, S.M., Basak, S., Garimella, S.V. and Raman, A., 2007. Piezoelectric fans using higher flexural modes for electronics cooling applications. IEEE Transactions on Components and Packaging Technologies, 30(1), pp.119-128. [8] Yoo, J.H., Hong, J.I. and Cao, W., 2000. Piezoelectric ceramic bimorph coupled to thin metal plate as cooling fan for electronic devices. Sensors and Actuators A: Physical, 79(1), pp.8-12. [9] Srivastava, S., Yadav, S.K. and Mukherjee, S., 2015. Effect of material uncertainties on dynamic analysis of piezoelectric fans. PIE Smart Structures/NDE, San Diego, USA. [10] Siva, C., Murugan, M.S. and Ganguli, R., 2010. Effect of uncertainty on helicopter performance predictions. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 224(5), pp.549-562.
ScienceDirect ScienceDirect
Procediaonline Computer 00 (2018) 000–000 Available at Science www.sciencedirect.com Procedia Computer Science 00 (2018) 000–000
ScienceDirect
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
Procedia Computer Science 133 (2018) 45–54
International Conference on Robotics and Smart Manufacturing (RoSMa2018) International Conference on Robotics and Smart Manufacturing (RoSMa2018)
Dynamic Analysis of Piezoelectric Fan with Random System Dynamic Analysis of Piezoelectric Fan with Random System Parameters Parameters Ankita Kotaaa, Sujoy Mukherjeebb* Ankita Kota , Sujoy Mukherjee * a Mechatronics, IIITDM Jabalpur, India. a Mechanical Department, IIITDM Jabalpur, Mechatronics, IIITDM Jabalpur, India.India. b Mechanical Department, IIITDM Jabalpur, India. b
Abstract Abstract Now a day’s electronics is becoming more and more compact. Electronic circuits and devices need cooling means in order to Now a day’s morethese and days. more Piezoelectric compact. Electronic cooling means in order to dissipate heat.electronics Heat sinksisarebecoming mostly used fans cancircuits also beand useddevices for thisneed purpose. Piezoelectric fans are dissipate heat. Heat sinks mostly containing used these substrate days. Piezoelectric fans can also used of forpiezoelectric this purpose.material. Piezoelectric fans are cantilevered uni-morph or are bi-morph, sandwiched between twobelayers As sinusoidal voltage is applied to piezoelectric material, cantilever starts to vibrate and generates air flow. The advantage of usingAspiezofans is cantilevered uni-morph or bi-morph, containing substrate sandwiched between two layers of piezoelectric material. sinusoidal voltage applied to piezoelectric cantilever starts vibrate andand generates air play flow.aThe advantage of using piezofans is that theyisconsume less power and material, are low noise devices. Thetodeflection frequency major role in flow generation. This that they consume less power are low noise The frequency play a major role in flow Carlo generation. This paper predicts the behavior ofand displacement and devices. frequency ondeflection variation and of material properties. Classical Monte simulation is usedpredicts for prediction of uncertainty in performance parameter. It is found that uncertainty material Monte properties large paper the behavior of displacement and frequency on variation of material properties.inClassical Carlocauses simulation is used forinprediction of uncertainty performance parameter. deviations performance parameters,infrom the predicted ones. It is found that uncertainty in material properties causes large deviations in performance parameters, from the predicted ones. © 2018 The Authors. Published by Elsevier Ltd. © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) © 2018 The Authors. Published by Elsevier Ltd.committee This is an open access article under thescientific CC BY-NC-ND license Peer-review under responsibility of the of the(https://creativecommons.org/licenses/by-nc-nd/4.0/). International Conference on Robotics and Smart Manufacturing. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords:Smart material, Piezo, Dynamics,Uncertainty,Mathematical Model. Keywords:Smart material, Piezo, Dynamics,Uncertainty,Mathematical Model.
* Sujoy Mukherjee. Tel.+91-761-2794421. [email protected] *E-mail Sujoyaddress: Mukherjee. Tel.+91-761-2794421. E-mail address: [email protected] 1877-0509© 2018 The Authors. Published by Elsevier Ltd. This is an open access under the CC by BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). 1877-0509© 2018 Thearticle Authors. Published Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/). 1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the International Conference on Robotics and Smart Manufacturing. 10.1016/j.procs.2018.07.007
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Author name / Procedia Computer Science 00 (2018) 000–000
46 2
1.
INTRODUCTION
Piezoelectric material has the property of generating electric charge on applying pressure on it. The converse of this is also possible [1].The conversion rate depends on coupling factor. As devices are getting compact day by day, piezoelectric fans are gaining more and more importance. Direct piezoelectric effect has proved advantageous for energy harvesting. Piezofan can be unimorph or bimorph, with substrate and piezo patch. As varying voltage is applied to beam, piezo patch generates force and generates deflection in structure; this in turn induces air flow. These piezoelectric fans have their own natural frequency, which depends on the material used. As the external frequency matches to natural frequency, resonance occurs. Piezo fans have been the area of research since many years. Toda and Osaka[2] found out that as amplitude and strain were increased, resonance was shifted to lower frequency, and hysteresis appeared at low frequency which resulted in decrease of Young’s modulus. Yao and Uchino [3] analyzed the composite piezoelectric structure using lumped mass analysis. The frequency of vibration also depends on length of fan. Maximum deflection can be obtained by optimizing blade length ratio, thickness ratio [4-5]. Dynamic response of symmetric and asymmetric resonators was investigated by Basak et al. [6]. An analytical model was developed for both configurations using beam theory. The performance of piezoelectric fans at higher modes was studied in ref [7]. It was seen that at higher resonance modes, power consumed was more and fluid flow was decreased. It was concluded that these fans can only be implemented in their first resonance mode. Yoo et al. [8] established the relation between tip deflection and wind velocity when fan is operated at fundamental resonance of the metal plate. Uncertainties of the structural and geometric parameters affecting the performance of piezofans are determined by Classical Monte Carlo method [910]. In present study, piezoelectric cantilevered beam was considered as a continuous system. Uncertainties in the structure have a great influence on the dynamic performance of piezofan.The effect on the natural frequency and deflection was determined using Monte Carlo Method and the maximum and minimum values were determined. 2. MATHEMATICAL MODELING 2.1. Structure
Fig.1.Structure of piezofan
The structure of piezoelectric unimorph used is as shown in fig.1. A thin plate of Mylar sheet sandwiched between two patches of piezoelectric materials. The total length is divided into three regions. Starting from the clamped end
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Ankita Kota et al/ Procedia Computer Science 00 (2018) 000–000
47 3
to start of piezo patch is region 1, region 2 is from starting of piezo patch to end of piezo patch and region 3 extends from the end of piezo patch to the free end of the elastic beam. 2.2. Modeling Euler beam theory is used to derive equations of motion. There are two equations for each region along with boundary conditions [6]. ‘u’ and ‘w’ represent longitudinal and transverse displacements respectively. Fig. 2 and Fig.3.show the free body diagram for transverse and longitudinal loads. For simplification a small area dx is taken into consideration. Equations of motion for a particular section are obtained by balancing the moments and forces along the section dx. Here, M is bending moment, V is shear force and P represents the longitudinal force acting on the section. Similarly equations of motion are derived for each region.
Fig.1 Free body diagram for transverse loads.
Region1 − + = 0,
(1)
+ = 0,
(2)
Region 2 − + + + = 0
− + − 2 + 2 + = 0
(3) (4)
Region 3 − + = 0,
(5)
+ = 0,
(6)
Boundary Conditions 0, = 0,
0, = 0 ,
0, = 0,
, = , ,
48 4
, = , ,
, = , ,
, = , .
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Author name / Procedia Computer Science 00 (2018) 000–000
, = ,
, = ,
Natural boundary conditions + , + = ,
2 + 2 + , + = , ,
,
2 + 2 + , = , ,
Fig.2 Free body diagram for longitudinal loads.
+ , + = , ,
2 + 2 + , − = , , 2 + 2 + , = , ,
, = 0,
, = 0,
, = 0.
(7)
Terms with dot and dash on top represent differentiation with respect to time and x axis respectively. The terms with double dot represent inertia force acting on the section. Terms with dash in equation (2),(4) and (6) represent moments acting on the section. Terms with dash in equation (1),(3) and (5) represent net force acting on that particular section. Solution for these equations using variable separable method is given by = sin + + ℎ + ℎ + sin + , = , + , , Where i=1,2,3.
(8) (9)
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Ankita Kota et al/ Procedia Computer Science 00 (2018) 000–000
, = =
/
,
/
, = ,
=
/
49 5
, /
(10)
On applying first three boundary conditions, we get , =0, A1= -C1 and B1= -D1. On applying remaining 15 boundary conditions 15 equations are obtained, using which 15 x 15 matrix containing coefficients of the constants is constructed. By solving the determinant of this matrix natural frequency is found out. Natural frequency obtained is 42 Hz.Using this frequency magnitude of displacement obtained is 3.30 mm. Table1. Material Properties and dimensions [6]. Physical Quantity
Value
Eb
Young’s Modulus of the substrate(Mylar)
Ep
Young’s Modulus of the actuator
ρb
4.6 ∗ 10 N/m
Density of substrate
ρp
Density of the actuator
1240 ⁄
6.2 ∗ 10 N/m 7800 ⁄
tb
Thickness of the beam
0.1016 mm
tp
Thickness of the actuator
0.127 mm
B
Width of the beam and actuator
8.89 mm
L1
Length of region 1
0.5 mm
L2
Length of region2
0.03 mm
L3
Overall length of the structure L3
27.424 mm
K33
Relative dielectric constant
3800
ε0
Permittivity of the free space
d31
Piezoelectric strain coefficient
8.8542 ∗ 10 ⁄
Table1. Shows the values of parameters used in analysis.
−320 ∗ 10 ⁄
3. UNCERTAINTY ANALYSIS Monte Carlo method is used for solving probabilistic problems. Randomness in the parameters is considered in terms of coefficient of variation (COV). COV is measure of relative variability. It is ratio of standard deviation () to average (). Specific value of coefficient of variation is selected for each parameter [9-10]. Random values of the inputs are generated using the equation below (11) = + Where is the random number generated between 0 and 1.The number of samples is selected based on the convergence of standard deviation of performance parameters. Fig.4 shows the convergence diagram of standard deviation for frequency. From Fig.4 it can be clearly seen that the curve starts to converge at 10,000 samples. Therefore number of samples considered for further analysis is 10,000.
Table 2.shows the values of mean and coefficient of variance for system parameters. COV for geometric parameters is always taken as one while for other parameters is determined experimentally The values for the parameters used for numerical simulation are taken from the literature [9]. Fig. 5. (a), (b) and (c) show the frequency pattern, standard deviation in frequency and probability distribution after consideration of variations in parameters. Standard deviation curve shows the deviation of a parameter from the true value and the probability curve shows the distribution of samples in the particular range. It can be seen from fig.5(c) that
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Author name / Procedia Computer Science 00 (2018) 000–000
50 6
probability distribution is of normal type and is maximum in range of 40-44 Hz.
Fig.3 Divergence of standard deviation curve for frequency. Table 2. Coefficient of variance for different parameters [9]. Random input variable
Mean Value
Distribution Type
COV (%)
L1 L2 L3 W tp tb Ep
0.5 mm 0.03 mm 27.423 mm 8.8894mm 0.127mm 0.1016mm
Normal Normal Normal Normal Normal Normal Normal
1 1 1 1 1 1 3
Eb
4.6 ∗ 10 N/m
Normal
3
Normal
5
6.2 ∗ 10 N/m
d31
−3.1998 ∗ 10
Fig.6. (a), (b) and (c) shows the displacement pattern, standard deviation of displacement and probability distribution. It can be seen that probability distribution is maximum in interval of 3.2 mm to 3.4 mm. Table 3.Randomness of output parameters Parameters
Std Deviation
Mean
COV (%)
Max Value
Min Value
Frequency(Hz)
1.49
42.01
3.5
47.57
35.76
Displacement(mm)
0.10
3.37
3.0
3.82
2.99
Table.3 shows the results of simulation. It is found that COV for frequency and displacement is 3.5% and 3.0% respectively. Maximum and minimum displacement values for piezo fan are found to be 3.8 mm and 2.9 mm respectively with mean of 3.37 mm. This shows the influence of randomness of input parameters in output of
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Ankita Kota et al/ Procedia Computer Science 00 (2018) 000–000
piezofans.
(a)
(b)
51 7
et al. /Computer Procedia Computer (2018) 45–54 AuthorAnkita nameKota / Procedia Science 00Science (2018) 133 000–000
852
(c) Fig.5. (a).Frequency Pattern, (b).Standard deviation in frequency, (c).Probability distribution for frequency.
4. CONCLUSIONS MCS is carried out for 10,000 considering the piezo fan as a continuous system. Randomness for geometric parameters is considered to be 1. It is found that COV for frequency and displacement is 3.5% and 3.0% respectively. Similarly minimum and maximum values of frequency are 35.76 Hz and 47.57 Hz respectively with mean of 42.01 Hz. Maximum and minimum displacement values for piezo fan are found to be 3.82 mm and 2.9 mm respectively with mean of 3.37 mm. This shows the influence of randomness of input parameters in output of piezofans. Thus it is needed to incorporate randomness of input parameters in analysis of piezo fan and Monte Carlo method has proven to be an efficient method. Uncertainty in any system introduces errors in engineering data and leads to failure of system. The numerical results provide useful bounds on parameters that effect performance of piezoelectric cooling fan.
Ankita Kota et al. / Procedia Computer Science 133 (2018) 45–54 Ankita Kota et al/ Procedia Computer Science 00 (2018) 000–000
(a)
(b)
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(c) Fig.6. (a).Displacement Pattern, (b).Standard deviation in displacement, (c).Probability distribution for displacement.
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