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Analysis of pull-in instability of geometrically nonlinear microbeam using radial basis artificial neural network based on couple stress theory.

Heidari M, Heidari A, Homaei H - Comput Intell Neurosci (2014)

Bottom Line: The output obtained from neural network model is compared with numerical results, and the amount of relative error has been calculated.Based on this verification error, it is shown that the radial basis function of neural network has the average error of 4.55% in predicting pull-in voltage of cantilever microbeam.The results reveal significant influences of size effect and geometric parameters on the static pull-in instability voltage of MEMS.

View Article: PubMed Central - PubMed

Affiliation: Mechanical Engineering Group, Islamic Azad University, Aligudarz Branch, P.O. Box 159, Aligudarz, Iran.

ABSTRACT
The static pull-in instability of beam-type microelectromechanical systems (MEMS) is theoretically investigated. Two engineering cases including cantilever and double cantilever microbeam are considered. Considering the midplane stretching as the source of the nonlinearity in the beam behavior, a nonlinear size-dependent Euler-Bernoulli beam model is used based on a modified couple stress theory, capable of capturing the size effect. By selecting a range of geometric parameters such as beam lengths, width, thickness, gaps, and size effect, we identify the static pull-in instability voltage. A MAPLE package is employed to solve the nonlinear differential governing equations to obtain the static pull-in instability voltage of microbeams. Radial basis function artificial neural network with two functions has been used for modeling the static pull-in instability of microcantilever beam. The network has four inputs of length, width, gap, and the ratio of height to scale parameter of beam as the independent process variables, and the output is static pull-in voltage of microbeam. Numerical data, employed for training the network, and capabilities of the model have been verified in predicting the pull-in instability behavior. The output obtained from neural network model is compared with numerical results, and the amount of relative error has been calculated. Based on this verification error, it is shown that the radial basis function of neural network has the average error of 4.55% in predicting pull-in voltage of cantilever microbeam. Further analysis of pull-in instability of beam under different input conditions has been investigated and comparison results of modeling with numerical considerations shows a good agreement, which also proves the feasibility and effectiveness of the adopted approach. The results reveal significant influences of size effect and geometric parameters on the static pull-in instability voltage of MEMS.

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Pull-in voltage versus size effect for cantilever beam with gap of 2.5 μm, a thickness of 1 μm, length of 300 μm, and width of 0.5 μm, for linear and nonlinear geometry model.
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fig15: Pull-in voltage versus size effect for cantilever beam with gap of 2.5 μm, a thickness of 1 μm, length of 300 μm, and width of 0.5 μm, for linear and nonlinear geometry model.

Mentions: When the applied voltage between the two electrodes increases beyond a critical value, the electric field force cannot be balanced by the elastic restoring force of the movable electrode and the system collapses onto the ground electrode. The voltage and deflection at this state are known as the pull-in voltage and pull-in deflection, which are of utmost importance in the design of MEMS devices. The pull-in voltage of cantilever and fixed-fixed beams is an important variable for analysis and design of microswitches and other microdevices. Typically, the pull-in voltage is a function of geometry variable such as length, width, and thickness of the beam and the gap between the beam and ground plane. To study the instability of the nanoactuator, (22) is solved numerically and simulated. To highlight the differences between linear and nonlinear geometry model results of Euler-Bernoulli microbeam, we first compare the pull-in voltage for fixed-fixed and cantilever beams with a length of 100 μm, a width of 50 μm, a thickness of 1 μm, and two gap lengths. For a small gap length of 0.5 μm (shown in Figure 4), we observe that linear and nonlinear geometry model give identical results. However, for a large gap length of 2 μm (shown in Figure 5), we observe that pull-in voltage for fixed-fixed beam is significantly different. As shown in Figure 6, the difference in the pull-in voltage is even larger when a gap length of 4.5 μm is considered. In Figures 7, 8, and 9, pull-in voltages of fixed-free beams are shown. It is evident that pull-in voltage of fixed-fixed beam is larger than fixed-free beam. More extensive studies for the cantilever beam with lengths varying from 100 to 500 μm and thicknesses varying from 1 to 4 μm are shown in Figures 10 and 11. The gap lengths used vary from 5 to 30 μm. For gaps smaller than 15 μm and lengths larger than 350 μm, we observe that the pull-in voltages obtained with linear and nonlinear geometry model are very close. However, for large gaps (such as the 15 μm case) and for short beams (such as the 100 μm case), we observe that the difference in the pull-in voltage obtained with linear and nonlinear geometry model is not negligible. In Figures 12 and 13, we investigate the fixed-fixed beam example with lengths varying from 100 to 500 μm and thicknesses varying from 0.5 to 2 μm. We observe that, for all cases, the pull-in voltages obtained with linear model are with significant error (larger than 5.5%) compared to the pull-in voltages obtained with nonlinear geometry model. When the gap increases, the error in pull-in voltage with linear model increases significantly. Furthermore, contrary to the case of cantilever beams, the thickness has a significant effect on the error in pull-in voltages. The thinner the beam, the larger the error. Another observation is that the length of the beam has little effect on the error in pull-in voltage. This observation is also different from the case of cantilever beams. From the results, it is clear the linear model is generally not valid for the fixed-fixed beams case, except when the gap is very small, such as the 0.5 μm case as shown in Figure 4. Effect of the size effect on the pull-in voltage of fixed-fixed and fixed-free beam is illustrated in Figures 12 and 13, respectively. These figures represent that the size effect increases the pull-in voltage of the nanoactuators. Figures 14 and 15 shows the pull-in voltage versus size effect for fixed-fixed beam and cantilever beam respectively.


Analysis of pull-in instability of geometrically nonlinear microbeam using radial basis artificial neural network based on couple stress theory.

Heidari M, Heidari A, Homaei H - Comput Intell Neurosci (2014)

Pull-in voltage versus size effect for cantilever beam with gap of 2.5 μm, a thickness of 1 μm, length of 300 μm, and width of 0.5 μm, for linear and nonlinear geometry model.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4016879&req=5

fig15: Pull-in voltage versus size effect for cantilever beam with gap of 2.5 μm, a thickness of 1 μm, length of 300 μm, and width of 0.5 μm, for linear and nonlinear geometry model.
Mentions: When the applied voltage between the two electrodes increases beyond a critical value, the electric field force cannot be balanced by the elastic restoring force of the movable electrode and the system collapses onto the ground electrode. The voltage and deflection at this state are known as the pull-in voltage and pull-in deflection, which are of utmost importance in the design of MEMS devices. The pull-in voltage of cantilever and fixed-fixed beams is an important variable for analysis and design of microswitches and other microdevices. Typically, the pull-in voltage is a function of geometry variable such as length, width, and thickness of the beam and the gap between the beam and ground plane. To study the instability of the nanoactuator, (22) is solved numerically and simulated. To highlight the differences between linear and nonlinear geometry model results of Euler-Bernoulli microbeam, we first compare the pull-in voltage for fixed-fixed and cantilever beams with a length of 100 μm, a width of 50 μm, a thickness of 1 μm, and two gap lengths. For a small gap length of 0.5 μm (shown in Figure 4), we observe that linear and nonlinear geometry model give identical results. However, for a large gap length of 2 μm (shown in Figure 5), we observe that pull-in voltage for fixed-fixed beam is significantly different. As shown in Figure 6, the difference in the pull-in voltage is even larger when a gap length of 4.5 μm is considered. In Figures 7, 8, and 9, pull-in voltages of fixed-free beams are shown. It is evident that pull-in voltage of fixed-fixed beam is larger than fixed-free beam. More extensive studies for the cantilever beam with lengths varying from 100 to 500 μm and thicknesses varying from 1 to 4 μm are shown in Figures 10 and 11. The gap lengths used vary from 5 to 30 μm. For gaps smaller than 15 μm and lengths larger than 350 μm, we observe that the pull-in voltages obtained with linear and nonlinear geometry model are very close. However, for large gaps (such as the 15 μm case) and for short beams (such as the 100 μm case), we observe that the difference in the pull-in voltage obtained with linear and nonlinear geometry model is not negligible. In Figures 12 and 13, we investigate the fixed-fixed beam example with lengths varying from 100 to 500 μm and thicknesses varying from 0.5 to 2 μm. We observe that, for all cases, the pull-in voltages obtained with linear model are with significant error (larger than 5.5%) compared to the pull-in voltages obtained with nonlinear geometry model. When the gap increases, the error in pull-in voltage with linear model increases significantly. Furthermore, contrary to the case of cantilever beams, the thickness has a significant effect on the error in pull-in voltages. The thinner the beam, the larger the error. Another observation is that the length of the beam has little effect on the error in pull-in voltage. This observation is also different from the case of cantilever beams. From the results, it is clear the linear model is generally not valid for the fixed-fixed beams case, except when the gap is very small, such as the 0.5 μm case as shown in Figure 4. Effect of the size effect on the pull-in voltage of fixed-fixed and fixed-free beam is illustrated in Figures 12 and 13, respectively. These figures represent that the size effect increases the pull-in voltage of the nanoactuators. Figures 14 and 15 shows the pull-in voltage versus size effect for fixed-fixed beam and cantilever beam respectively.

Bottom Line: The output obtained from neural network model is compared with numerical results, and the amount of relative error has been calculated.Based on this verification error, it is shown that the radial basis function of neural network has the average error of 4.55% in predicting pull-in voltage of cantilever microbeam.The results reveal significant influences of size effect and geometric parameters on the static pull-in instability voltage of MEMS.

View Article: PubMed Central - PubMed

Affiliation: Mechanical Engineering Group, Islamic Azad University, Aligudarz Branch, P.O. Box 159, Aligudarz, Iran.

ABSTRACT
The static pull-in instability of beam-type microelectromechanical systems (MEMS) is theoretically investigated. Two engineering cases including cantilever and double cantilever microbeam are considered. Considering the midplane stretching as the source of the nonlinearity in the beam behavior, a nonlinear size-dependent Euler-Bernoulli beam model is used based on a modified couple stress theory, capable of capturing the size effect. By selecting a range of geometric parameters such as beam lengths, width, thickness, gaps, and size effect, we identify the static pull-in instability voltage. A MAPLE package is employed to solve the nonlinear differential governing equations to obtain the static pull-in instability voltage of microbeams. Radial basis function artificial neural network with two functions has been used for modeling the static pull-in instability of microcantilever beam. The network has four inputs of length, width, gap, and the ratio of height to scale parameter of beam as the independent process variables, and the output is static pull-in voltage of microbeam. Numerical data, employed for training the network, and capabilities of the model have been verified in predicting the pull-in instability behavior. The output obtained from neural network model is compared with numerical results, and the amount of relative error has been calculated. Based on this verification error, it is shown that the radial basis function of neural network has the average error of 4.55% in predicting pull-in voltage of cantilever microbeam. Further analysis of pull-in instability of beam under different input conditions has been investigated and comparison results of modeling with numerical considerations shows a good agreement, which also proves the feasibility and effectiveness of the adopted approach. The results reveal significant influences of size effect and geometric parameters on the static pull-in instability voltage of MEMS.

Show MeSH
Related in: MedlinePlus