Document Type: Research Paper
Authors
^{1} School of Railway Engineering, Iran University of Science and Technology (IUST), 1684613114, Tehran, Iran
^{2} Assistant Professor, Head of Smart Micro/Nano Electro Mechanical Systems Lab (SMNEMS Lab), School of New Technologies, Iran University of Science and Technology (IUST), 1684613114, Tehran, Iran
Abstract
Keywords

Mechanics of Advanced Composite Structures 5 (2018) 25–39


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Sensitivity Analysis of FiberReinforced Lamina MicroElectroMechanical Switches with Nonlinear Vibration Using a Higher Order Hamiltonian Approach
A. Kabiri^{ a}, S. Sadeghzadeh ^{b}^{*}
^{a }School of Railway Engineering, IUST, 1684613114, Tehran, Iran
^{b }Assistant Professor, Head of Smart Micro/Nano Electro Mechanical Systems Lab (SMNEMS Lab), School of New Technologies, Iran University of Science and Technology (IUST), 1684613114, Tehran, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 2016‐09‐28 Revised 2017‐08‐21 Accepted 2017‐09‐08 
In this paper, the nonlinear free vibration of fiberreinforced lamina microswitches is investigated, and a sensitivity analysis (SA) is given. The switches are modeled as solid rectangular beams consisting of an isotropic matrix with transversely and longitudinally isotropic reinforcements, incorporating a higher order Hamiltonian approach. An SA of the proposed microswitch is presented by calculating the numerical derivatives of the presented nonlinear model with respect to the design parameters. The SA of geometric parameters, such as dimensionless length, thickness, initial gap, applied voltage, axial load, and effective modules of the system, was conducted using the Sobol method. It was found that the natural frequency varied when changes were made to the proposed parameters; this finding can be used to optimize future designs.




Keywords: nonlinear vibration, MEMS switches, higher order Hamiltonian, sensitivity analysis 


© 2018 Published by Semnan University Press. All rights reserved. 
Microelectromechanical systems (MEMSs) are used in fields such as aerospace, optical, and biomedical engineering, particularly in applications such as microswitches, transistors, accelerometers, pressure sensors, micromirrors, micropumps, microgrippers, and bioMEMSs [14]. MEMSs are merged devices that connect electrical and mechanical components. Studying the dynamic and static behaviors of atomic force microscope (AFM) cantilevers and controlling the vibration of these cantilevers are examples of challenges that involve both electrical and mechanical components [58]. Ghalambaz et al. [9] studied the effects of Van der Waals attraction, Casimir force, small size stretching, fringing field, midplane stretching, and axial load on the oscillation frequency of resonators.
The inherent intricacy of the nonlinear vibration of MEMSs makes numerical solutions a better choice than analytical ones. The shooting method [10], the differential quadrature method [11], the homotopy analysis method (HAM) [12], the variational approach (VA) [13], the max–min approach (MMA) [14, 15], and the energy balance method (EBM) [16] are some of the numerical and approximate analytical approaches than can be addressed. Ganji et al. [17] applied the EBM and an amplitude frequency formulation (AFF) to govern the approximate analytical solution for the motion of two mechanical oscillators. They showed that in comparison with a fourthorder RungeKutta method, their solution is intuitive and useful for solving strongly nonlinear oscillators. When Ganji and Azimi [18] used both the MMA and an AFF to derive an approximate analytical solution for the free vibrating motion of nonlinear, conservative, singledegreeoffreedom systems, they concluded that both methods had the same results. Both of these methods are convenient for solving nonlinear equations and can also be utilized for a wide range of time and boundary conditions for nonlinear oscillators.
Yildirim, Saadatnia et al. [19] applied the Hamiltonian approach to obtain the natural frequency of a Duffing oscillator; the obtained results were in complete agreement with the approximate frequencies and the exact solution. Askari [20] utilized a higher order Hamiltonian approach to elicit approximate solutions for the model of buckling in a column and massspring system. Khan and Akbarzade [21] used a VA, a Hamiltonian approach, and an AFF to analyze a nonlinear oscillator equation in a doublesided clamped microbeambased electromechanical resonator. Moreover, Fu et al. [22] applied the EBM to study a nonlinear oscillation problem in a microbeam model. They used an equation for the free vibration of a microbeam, based on the Euler–Bernoulli hypothesis, and they compared the results with a fourthorder RungeKutta method.
Bayat et al. [23] investigated He’s VA to solve the nonlinear vibration of an electrostatically actuated doubly–clamped microbeam that was equivalent to the first order of a higher Hamiltonian method [24]. They demonstrated that the VA is a good candidate for the precise periodic solution of nonlinear systems. Final results of mentioned works are listed in Table 1.
The pullin instability of a cantilever nanoactuator model incorporating the effects of surface, fringing field, and Casimir attraction force was investigated in [26]. Furthermore, an approximate analytical model for calculating the pullin voltage of a stepped cantilevertype radiofrequency (RF) MEMS switch was developed based on the Euler–Bernoulli beam and a modified couple stress theory, and it was validated by a comparison with the finite element results [27].
Microcomposites are a new class of materials used for the mechanical components of MEMSs. Fiberreinforced composite materials for structural applications are often made in the form of a thin layer, called lamina. It is known that fibers are stiffer and stronger than the same material in bulk form, whereas matrix materials share their common bulkform properties. Ashrafi et al. [28] presented a detailed theoretical investigation of the utility of carbon nanotubereinforced composites for designing actuators with low stiffness and high natural frequencies of vibration. The authors investigated the effects of the nanotube aspect ratio, dispersion, alignment, and volume fraction of the elastic modulus and longitudinal wave velocity, and they calculated the bounds on Young's modulus and wave velocity, capturing the trends of other experimental results reported in the literature. Thostenson and Chou [29] simulated the mechanical and physical properties of nanotubebased composite materials. The focus of this research was to develop a fundamental understanding of the structure/size influence of aligned multiwalled carbon nanotubes on the elastic properties of nanotubebased composites. The experimental characterization results were compared with numerical predictions. Hautamaki et al. [30] conducted an experimental evaluation of MEMS strain sensors embedded in composites, examining the effects of wafer geometry and composite plate stiffness on MEMS strain sensors. In another study, Spearing [31] discussed the effects of length scale and material characterization on MEMS design. He presented the MEMS materials set that is derived from three fabrication routes.
In this study, in order to design actuators made from fiberreinforced composites, a higher order Hamiltonian method [32] was used to obtain an approximate numerical solution. Contrarily to some recent research, such as [23], we showed that the second order result is extremely close to the EBM and exact solutions. The methodology of using a higher order Hamiltonian approach for solving an ordinary differential equation with high nonlinearity is also presented. Numerical comparisons and results were carried out to confirm the correctness and accuracy of the applied method. The ability of the solution for estimating the effect of various parameters on natural frequency is shown and discussed. Sensitivity analysis (SA) of the proposed MEMS device was studied by considering various parameters in the operation of the microbeam system.
Table 1. Comparison of natural frequency formulas for microbeams from recent related works
Figures 1 and 2 depict the fiberreinforced lamina clamped–clamped microbeam, with length , width , constant thickness h, initial gap , and applied voltage . The microbeam is doubly clamped and placed between two completely fixed electrodes. The applied voltage results from the electric field, which can be divided into a DC polarization and an AC electric field.
Applying an AC electric field or a periodic mechanical load causes the dynamic deflection and vibration of the microbeam [33]. For more design options and capabilities, computational studies are essential in addition to experiments. However, there are no exact (analytical) closedform solutions for all boundary conditions of mechanical systems. As a good alternative, the free vibration of MEMSs can be simulated by applying a Galerkin method (GM) and utilizing classical beam theory.
Regarding the effect of midplane deformation, the nonlinear partial differential equation of transverse motion could be expressed as follows [34]:
, 
(1) 
where is the transverse deflection, is the Young’s modulus, is the Poisson's ratio, and is the effective modulus of the microbeam. The quantity of changes with the different thicknesses of the microbeam, as follows [25]:


Figure 1. Schematics of doublyclamped microelectromechanical resonator 



Figure 2. Schematics of deformed microelectromechanical resonator with applied voltage 

(2) 
Then, by using a theoretical approach to determine the engineering constants of a continuous fiberreinforced composite material based on whether the applied loads are parallel or perpendicular to the fiber direction, the effect of fiberreinforced composite material on the quantity of and can be expressed in terms of a modulus, Poisson's ratios, and volume fractions of the constituents [35]:
(3) 
is is the longitudinal modulus, is the transverse modulus to the fiber direction in the plane of the lamina, is the major Poisson's ratio, is the modulus of the fiber, and is the modulus of the matrix; is the Poisson's ratio of the fiber, is the Poisson's ratio of the matrix, is the fiber volume fraction, and is the matrix volume fraction. Hence, an effective modulus for the microbeam can be implemented by the following equation for the direction on the laminar plane to be parallel to the fiber:
(4) 
For the direction on the laminar plane to be transverse to the fiber, we would have
(5) 
Symbolizes the tensile or compressive axial load and is related to the discrepancy of both the thermal expansion coefficient and the crystal lattice period between the substrate and the microbeam. is the normalized motivating force that is derived from electrostatic excitation, as follows [36]:
, 
(6) 
where is the dielectric constant of the interface. The boundary conditions are as follows:

(7) 

(8) 
The following dimensionless parameters are used to normalize Eq. (1):

(9) 
Then, dimensionless boundary conditions can be written as follows:

(10) 
(11) 
Based on the presented formulas, a dimensionless equation of motion can be implemented for MEMS resonators using the following equation:
(12) 
By using the assumed modes method, the dimensionless deflection solution of Eq. (12) can be introduced as follows:
, 
(13) 
where is the th Eigen function of a microbeam that fulfills the appropriate boundary conditions, is the th timedependent deflection coordinate, and is the supposed degree of freedom of the microbeam.
To solve Eq. (12), we consider a singledegreeoffreedom model , and deflection function is assumed to be as follows:

(14) 
The trial function is
. 
(15) 
This function satisfies the boundary conditions.
Then, by substituting the presented functions into the dimensionless equation of motion and integrating from 0 to 1, the dimensionless equation of motion changes [22]:
, 
(16) 
where

(17) 
For the following general oscillator,
, 
(18) 
where u and t are the generalized dimensionless displacement and dimensionless time, respectively, and is the oscillator amplitude. Based on the variational principle, by implementing the semiinverse [37, 38] and He [13, 39] methods, the variation parameter can be written as
, 
(19) 
where is the oscillator vibration period and . Thus, the Hamiltonian approach in the presented problem can be expressed as follows:
(20) 
Then, by defining a new function, we have
(21) 
By choosing any arbitrary point, such as , and setting , an approximate frequency–amplitude relationship can be obtained. This approach is much simpler than other traditional methods and has become widely used [40]. The accuracy of this location method, however, strongly depends upon the chosen location point. To overcome the shortcomings of the EBM, a new approach, based on the Hamiltonian one, has been suggested [32]. Differentiating the Hamiltonian approach leads us to natural frequency of the system:
(22) 
For greater convenience, a new function, , is defined as follows:
(23) 
Then, for the natural frequencies of the system, the following relation is used:
or 
(24) 
From Eq. (24), we can obtain the approximate frequency–amplitude relationship of a nonlinear oscillator [32, 41]. For the current special problem, we have the following Hamiltonian equation:
(25) 
3.1. Firstorder Hamiltonian approach
After satisfying the initial conditions, utilizing as the trial function in Eq. (25), we obtain
(26) 
This leads to the following:
(27) 
Then, the frequency–amplitude relationship can be obtained from the following:
(28) 
Therefore, after some approximations and simplifications, Eq. (28) can be solved, and the natural frequency can be obtained as follows:

(29) 
That is approximately equal to
(30) 
Both the VA [23] and the analytical approximate solution [25] have the same result for this problem.
3.2. Secondorder Hamiltonian approach
To improve the accuracy of this approach, a higher order periodic solution was assumed as the time response function:
, 
(31) 
where the initial condition is

(32) 
By Substituting Eq. (32) into Eq. (25), we can obtain
(33) 
Subsequently, the frequency–amplitude relationship can be obtained from the following equation:
(34) 
To obtain the natural frequency, substituting Eq. (32) into (34) as , a secondorder algebraic equation set becomes solvable, allowing the natural frequency and a and b values to be obtained for various values of and . Some of the results are listed in Table 2.
3.3. Thirdorder Hamiltonian approach
A thirdorder time response can be used for the microbeam, as follows
, 
(35) 
where the initial condition is
(36) 
Similar to the secondorder Hamiltonian approach, with some mathematical simplification, values of can be obtained for various values of A and V (see Table 3 for examples).
4.1. Computational efficiency
The nonlinear algebraic equations presented were solved using Wolfram Mathematica software on an Intel(R) Core™ i53230M CPU 2.6GHz processor, including 6 GB of installed memory on a 64bit operating system. The required time for calculating natural frequencies was 5–10 seconds, 30–40 seconds, and 3.5–4 minutes for first, second, and thirdorder Hamiltonian approaches, respectively. In terms of accuracy and computational efficiency, the secondorder solution was the best.
Table 2. parameters for different and values (
(0. 3, 10) 

(0. 4, 10) 

(0. 5, 10) 

(0. 6, 10) 

(0. 7, 10) 
Table 3. parameters for different and values (
(0. 3, 10) 

(0. 4, 10) 

(0. 5, 10) 

(0. 6, 10) 

(0. 7, 10) 
4.2. Validation
In comparison with previous works, where higher order approximations were not used, more accurate dynamic responses and natural frequencies were observed. The EBM is the best criterion for such comparisons. Figure 3 depicts a comparison of the dynamic response of a microbeam under electric excitation (V = 24 Volts), with parameters obtained with the first, second, and thirdorder Hamiltonian approaches. These comparisons were repeated (see Fig. 4) after changing the A value to 0.5.
Table 4 compares the frequencies commensurate for different parameters of the system, obtained from the Hamiltonian method and the EBM [22]. Exact values are also reported for some cases. Accuracy increases with an increase in the order of approximations. When the order increases, more accurate results are achieved. Increasing the applied voltage or initial amplitude leads to a greater number of errors. Thus, in the case of larger initial amplitude and applied voltage, higher order approximations would be more useful than lower order ones.
Figure 3. Comparison of dynamic responses obtained using higher order Hamiltonian approaches and an EBM solution (
Figure 4. Comparison of dynamic responses obtained using higher order Hamiltonian approaches and an EBM solution ( )
4.3. Phase diagram of microbeam
Simplifying the convolution of a nonlinear system to a pseudolinear model can provide a useful view on the stability and controllability of a system. For example, let us assume that a nonlinear MEMS microbeam has a linearwise model, as below, that includes all nonlinearities in the second part of its dynamics:

(37) 
A nonlinear term ( ) may contain a high level of nonlinearity because it includes space, time, and input variables as operands. Figure 5 shows the effect of parameters and V on the phase plane of a system for simulated using a secondorder Hamiltonian method. As can be seen in Figure 5, by increasing the order of Hamiltonian approach, the amplitude parameters (a, b, c) decrease; thus, the overall amplitude also decreases. When a microbeam resonates near the zero point as the basal condition, a notable reduction in the velocity of the resonator is observable. This phenomenon disappears immediately after it passes from the basal condition. This means that the dynamics of this nonlinear system also depend on the position of the point that is being measured on the MEMS microbeam.

Table 4. Comparison of natural frequencies (rad/s) for various parameters of the system ( 


(0. 3, 0) 
26. 3644 
26. 3644 
26. 3672 
26. 3669 
26. 3867 
26.8372 


(0. 3, 10) 
24. 2526 
24. 2526 
24. 2547 
24. 2543 
 


(0. 3, 20) 
16. 3556 
16. 3556 
16. 3552 
16. 3547 
16. 3829 
16.6486 


(0. 4, 0) 
27. 2053 
27. 2053 
27. 2214 
27. 2195 
27. 2759 
 


(0. 4, 10) 
25. 0500 
25. 0500 
25. 0639 
25. 0621 
25. 1217 
 


(0. 4, 20) 
17. 0187 
17. 0187 
17. 0238 
17. 0219 
17. 1023 
 


(0. 5, 0) 
28. 0019 
28. 0019 
28. 0657 
27. 0605 
28. 1758 
 


(0. 5, 10) 
25. 7611 
25. 7611 
25. 8203 
25. 8155 
25. 9365 
 


(0. 5, 20) 
17. 3839 
17. 3839 
17. 4270 
17. 4241 
17. 5835 
 


(0. 6, 0) 
28. 5579 
28. 5579 
28. 7564 
28. 7499 
28. 9227 
28.5382 


(0. 6, 10) 
26. 1671 
26. 1671 
26. 3600 
26. 3562 
26. 5324 
 


(0. 6, 20) 
17. 0940 
17. 0940 
17. 2901 
17. 3013 
17. 5017 
18.5902 


Figure 5. Effect of and V parameters on the phase plan of the system for simulated using a secondorder Hamiltonian approach 
Figure 6. Effect of and parameters on the frequency of an electrostatically actuated microbeam with and various values of A 
4.4. Free vibration
Figure 6 shows the effect of and parameters on natural frequency. It can be observed that the frequency is proportional to N. However, it decreases when initial amplitude (A) increases. The secondorder Hamiltonian approach has nearly same response as that of the EBM solution, even for higher values of N and A. Furthermore, it can be observed that the frequency increases with increasing . Results obtained by the secondorder Hamiltonian approach are close to the EBM solution, especially for low amplitudes and values.
Figure 7 shows the effect of applied voltage on the natural frequency of an electrostatically actuated microbeam. It is apparent that the frequency is decreased with an increase in voltage. The results of the secondorder Hamiltonian approach are extremely close to the EBM solution, but they have considerable discrepancies in terms of high amplitudes and applied voltages.
The nonlinear behavior of the system leads to an abrupt fall in high applied voltage. The natural frequency decreases dramatically at high voltages. However, natural frequency also increases with an increase in amplitude. This is due to the following effect: when the initial amplitude increases, the equivalent linear system of the electrostatically actuated microbeam is hardened. It was also observed that for higher amplitudes, the discrepancy is less than that of lower values.
Several simulations and plots can be introduced to facilitate fundamental design requirements before any manufacturing process. Based on the presented examples, the proposed nonlinear model based on a Hamiltonian approach is efficient and sufficiently acceptable to determine the effects of parameters on the natural frequency and the phase plane diagram of an electrostatically actuated microbeam.
Figures 8 and 9 show the effect of the thickness of an electrostatically actuated microbeam on the natural frequency at various initial amplitudes when geometrical and structural lamina properties at a fibervolume fraction of 0.67 are chosen (see Table 5). As demonstrated clearly, natural frequency decreases with an increase in the thickness of a microbeam.
Figure 7. Effect of parameter on the natural frequency of an electrostatically actuated microbeam with , and various values of A 
Table 5. Nominal inputs and standard deviations for the microswitch parameters
Parameter 
Value 
Figure 8. Variation of natural frequency due to the thickness of the microbeam, thickness (h), where the applied load is parallel to the fiber direction 
Figure 9. Variation of natural frequency due to the thickness of the microbeam, (h), where applied load is perpendicular to the fiber direction 
When a sensor or actuator is designed based on microbeams, recognizing the parameters can help the designer to obtain the best resolution and accuracy. This issue can be addressed by conducting SA on electrostatically actuated microbeams. Thus, SA of the model, with respect to the model parameters, is a key step. SA can classify the values of various significant parameters according to the proposed research preferences [43].
Various parameters contribute to the final responses in any nonlinear system under consideration. Parameters include the length , width 𝐛, thickness , Young’s modulus, and Poisson's ratio of a microbeam on the one hand and the initial gap , electrostatic load , on the other. This model requires the identification of many parameters and inputs at varying levels of sensitivity, and various SA methods can be used to achieve this requirement. The Sobol method is a famous SA approach based on variance that is widely used in various studies and scientific fields, such as hydrogeology, geotechnics, ocean engineering, biomedical engineering, hybrid dynamic simulation, and electromagnetism.
5.1. Simple form of sensitivity
A simple form of determining sensitivity for natural frequency respecting these parameters could be achieved by differentiating the natural frequency with respect to the parameters and plotting the results (e.g., Figure 10). Below, represents the sensitivity of with respect to the variation of parameter :

(38) 
Using the parameters introduced in Table 5, Figs. 10 to 13 are presented to depict variations in sensitivity with respect to various parameters. Figure 10 shows the sensitivity of an electrostatically actuated microbeam with respect to parameter . It can be observed that sensitivity decreases with increases in parameter . Since in the presented sensitivity plot, this means that if increases or decreases, the microbeam reacts less easily to external excitations. Increasing the thickness increases the system rigidity, which makes the microbeamless sensitive. In contrast, decreasing the gap between the beam and the electrodes of an electrostatically actuated microbeam results in a lower excitation output.
Figure 10. Sensitivity plot of an electrostatically actuated microbeam with respect to the variation of parameter with and various amplitudes 
Figure 11 depicts microbeam sensitivity versus the external load. With increased external loads, sensitivity decreases because of the rigidity. The rigidity increases due to system hardening, which is caused by higher external loads.
Figure 12 shows a sensitivity plot of a microbeam with respect to the applied voltage. By applying more voltage, a lesssensitive sensor and actuator can be achieved. Although more sensitivity is a good option for low applied voltages, supplementary equipment for detecting and exciting the system would be extremely challenging. Therefore, it is not suggested for practical purposes.
In Figure 13, a sensitivity plot of a microbeam with respect to the modulus of elasticity is shown. The modulus of elasticity can be considered to be about 700 GPa. Greater hardness leads to greater sensitivity, so any approach that increases the stiffness of a microbeam improves its efficiency. Carbonfiber reinforced nanostructures improve the electromechanical properties [44].
Figure 11. Sensitivity plot of an electrostatically actuated microbeam with respect to the variation of parameter with and various amplitudes 
Figure 12. Sensitivity plot of an electrostatically actuated microbeam with respect to the variation of parameter V with , and various amplitudes 
Figure 13. Sensitivity plot of an electrostatically actuated microbeam with respect to the variation of parameter with various amplitudes 
5.2. The Sobol method of global sensitivity
The Sobol method for nonlinear models is used here to achieve an optimal design. This method was used to investigate the effects of a geometrical parameter on the natural frequency when all other parameters are changing at the same time. The inputs are changed, and the effect of each one on the model output is analyzed.
For nonlinear mathematical models with numerous inputs, it is often difficult to anticipate the response of the model output to changes in the inputs. Complicated interactions of physical processes mean that varying one input parameter at a time does not adequately characterize the range of possible model outputs, although a oneatatime analysis may be valuable as a preliminary screening exercise. A comprehensive SA must examine the response of the model to changes in all parameters across their range of values; this is known as global SA.
Between different general sensitivity analyses, variancebased methods are gaining the most attention. In these methods, the sensitivity index is computed as the share of each parameter in the overall output variance of the model. The Sobol method [45] is an independent general SA method based on variance analysis.
We consider a computer model , and thus, this relation can be written as follows:
, 
(39) 
where n is the number of independent parameters.
The input independent parameter region should be determined as follows to explain the Sobol’s method:
, 
(40) 
where and are the minimum and maximum values for , respectively.
The Sobol sensitivity indices are ratios of partial variances to the total variance. We can subdivide them into partial variances of increasing dimensionality:
, 
(41) 
where for any input , is called its firstorder or main effect sensitivity index, includes all the partial variances of interaction of two input parameters, and so on.
The representation of function is derived from the sum of the following functions:
, 
(42) 
where is constant and is determined as follows:
(43) 
Sobol showed that the decomposition of Eq. (25) is unique. Also, all terms of the mentioned equation can be evaluated via the following multidimensional integrals:
(44) 

, 
(45) 
where show the integration over all the variables, excluding and , respectively. Hence, for higher order terms, a continuous formula can be obtained. In the sensitivity indices based on variance, the total variance of , is expressed as
(46) 
Partial variances are computed as follows:
(47) 
According to Eq. (41), the sensitivity measures are given by
(48) 
The measure of the first order evaluates the contribution of the variation of to the total variance of Y. The measure of the second order of evaluates the contribution of the interaction of and on the output, and so on.
By applying Sobol’s sensitivity, the effect of the electrostatically actuated microbeam parameters on the frequency of the system can be obtained. The first step in SA is to determine the ranges of the model inputs. Defining the probability distribution of inputs requires the use of samplebased methods (e.g., a variancebased method or the Monte Carlo SA method). How the input parameters are chosen and how their ranges are determined basically depends on the objectives of the SA. In this study, we have analyzed four dimensionless parameters ( ) to predict the influence of various parameters of an electrostatically actuated microbeam.
5.2.1 Steps for implementation
Sobol’s algorithm can be summarized by the following steps:
1. Select the total number of simulations to be performed.
2. Select the parameters for SA.
3. Determine ranges for test variables.
4. Choose a distribution for each of the parameters. In this case, a uniform distribution is chosen for all four parameters.
5. Calculate the variance of the parameters using Eq. (47).
6. Compute the partial variance or first order effects for each parameter by fixing the values of that parameter and varying the remaining parameters.
7. Calculate the totalorder index of the parameters using Eq. (48).
8. Sort the parameters according to their sensitivities.
Figure 14 shows the effects of dimensionless parameters on the natural frequency, including Young’s modulus, an axial force toward the beam, applied voltage, and the ratio of the gap between the beam and the electrodes to the thickness of the beam, via the Sobol method. As is observable, by increasing the applied initial amplitude, the effects of different parameters change, and at higher frequencies, the term ( ) has the greatest influence on the natural frequency.
In clamped–clamped beams, the abovementioned ratio (beam–electrode gap to beam thickness) exerts the greatest influence on the natural frequency, and the length of the beams has the secondgreatest influence on this frequency. The applied voltage and Young’s modulus are effective on frequencies at low amplitudes, but these effects can be ignored at high amplitudes. The axial force also has an ignorable influence on the frequency.
The sensitivity of a microswitch containing a doubly clamped microbeam with length , width , and constant thickness h ( ); effective modulus ; initial gap and electrostatic applied voltage is studied by using a higher order Hamiltonian approach. A nonlinear partial differential equation of the transverse motion resulting from midplane deformation has been expressed, and the normalized motivating force has been calculated based on electrostatic excitation. A dimensionless equation of motion has been derived based on the variational principle. By implementing a semiinverse method and exploiting He’s method, an approximate frequency–amplitude relationship has been obtained. Differentiating the Hamiltonian approach reveals the natural frequency of microswitches. The overall results of this study are listed below.
6.1. Comparisons
1. The VA and the analytical approximate solution had the same results for this problem.
2. The time required for calculating natural frequencies using the proposed specific computational platform was 5–10 seconds, 30–40 seconds, and 3.5–4 minutes for first, second, and thirdorder Hamiltonian approaches, respectively. The secondorder solution was the most efficient computationally and in terms of accuracy.
3. The obtained results have been validated in comparison with the EBM, in which higher order approximations are neglected.
4. By increasing the order of approximation, the accuracy of the proposed method increases.
5. Increasing the applied voltage or initial amplitude leads to more errors. Thus, in the case of a higher initial amplitude and applied voltage, higher order approximations are essential.
6.2. Natural frequency calculations
1. The natural frequency increases as N increases. However, it decreases in accordance with an increase of the initial amplitude (A). Nevertheless, the secondorder Hamiltonian approach produces an extremely close response to that of the EBM solution, even for higher values of N and higher amplitude. Natural frequency also decreases with amplitude increasing.
2. The natural frequency increases with increasing values of . Results of the secondorder Hamiltonian approach are close to the EBM solution except for high amplitude and values. The natural frequency also decreases as the voltage increases. The results of the secondorder Hamiltonian approach are extremely close to the EBM solution aside from considerable discrepancies in high amplitudes and voltages.
A = 0.3 
A = 0.4 
A = 0.5 
A = 0.6 
A = 0.7 
Figure 14. Pie chart diagrams of Sobol’s sensitivity analysis (SA) of an electrostatically actuated microbeam with respect to the variation of parameters with various amplitudes
1. The nonlinear behavior of the system leads to an abrupt fall in high applied voltages. The natural frequency decreases dramatically at high voltages and also decreases at low voltages.
2. The natural frequency increases with an increase in the modulus of elasticity and decreases with increasing thickness of the microbeam; however, these variations are not significant
6.3. Sensitivity analysis
1. Sensitivity decreases when increases. As , the presented sensitivity plot (i.e. Fig. 10) means that if increases or decreases, the microbeam reacts less to external excitations. By increasing the thickness, rigidity also increases, which leads to less sensitivity of the microbeam. However, decreasing the gap between the beam and the electrodes an of electrostatically actuated microbeam results in a lower excitation output.
2. With increased external loads, the sensitivity decreases because of increased rigidity; the rigidity increases due to system hardening caused by increased external loads.
3. By applying more voltage, a lesssensitive sensor and actuator can be achieved. Although greater sensitivity seems to be a good option for low levels of applied voltage, providing supplementary equipment for detecting and exciting the system would be extremely challenging; therefore, it is not suggested for practical purposes.
4. By the Sobol method, it was shown that at higher frequencies, the variable ( ) has the greatest influence on the natural frequency. In clamped–clamped beams, the ratio of the gap between the beam and the electrodes to the thickness of the beam has the most influence on the natural frequency, followed by the length of the beams (ranked second most influential). The axial force also has an insignificant effect on frequency.
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[9] Ghalambaz M, Ghalambaz M, Edalatifar M. Nonlinear oscillation of nanoelectromechanical resonators using energy balance method: considering the size effect and the van der Waals force. Applied Nanoscience 2016;6(3): 30917.
[10] AbdelRahman EM, Younis MI, Nayfeh AH. Characterization of the mechanical behavior of an electrically actuated microbeam. Journal of Micromechanics and Microengineering 2002;12(6): 759.
[11] Kuang JH, Chen CJ. Dynamic characteristics of shaped microactuators solved using the differential quadrature method. Journal of Micromechanics and Microengineering 2004;14(4): 647.
[12] Beléndez A, Beléndez T, Márquez A, Neipp C. Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators. Chaos, Solitons & Fractals 2008;37(3): 77080.
[13] He JH. Variational approach for nonlinear oscillators. Chaos, Solitons & Fractals 2007;34(5): 14309.
[14] He JH. Maxmin approach to nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation 2008;9(2): 20710.
[15] Zeng D, Lee Y. Analysis of strongly nonlinear oscillator using the maxmin approach. International Journal of Nonlinear Sciences and Numerical Simulation 2009;10(10): 13618.
[16] He JH. Preliminary report on the energy balance for nonlinear oscillations. Mechanics Research Communications 2002;29(23): 10711.
[17] Ganji DD, Azimi M, Mostofi M. Energy balance method and amplitude frequency formulation based simulation of strongly nonlinear oscillators. 2012.
[18] Ganji DD, Azimi M. Application of max min approach and amplitude frequency formulation to nonlinear oscillation systems. UPB Scientific Bulletin 2012;74(3): 13140.
[19] Yildirim A, Askari H, Yazdi MK, Khan Y. A relationship between three analytical approaches to nonlinear problems. Applied Mathematics Letters 2012;25(11): 172933.
[20] Askari H, Nia ZS, Yildirim A, Yazdi MK, Khan Y. Application of higher order Hamiltonian approach to nonlinear vibrating systems. Journal of Theoretical and Applied Mechanics 2013;51(2): 28796.
[21] Khan Y, Akbarzade M. Dynamic analysis of nonlinear oscillator equation arising in doublesided driven clamped microbeambased electromechanical resonator. Zeitschrift für Naturforschung A 2012;67(89): 43540.
[22] Fu Y, Zhang J, Wan L. Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Current applied physics 2011;11(3): 4825.
[23] Bayat M, Bayat M, Pakar I. Nonlinear vibration of an electrostatically actuated microbeam. Latin American Journal of Solids and Structures 2014;11(3): 53444.
[24] Yildirim A, Saadatnia Z, Askari H, Khan Y, KalamiYazdi M. Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach. Applied Mathematics Letters 2011;24(12): 204251.
[25] Rafieipour H, Lotfavar A, Masroori A. Analytical approximate solution for nonlinear vibration of microelectromechanical system using he's frequency amplitude formulation. Iranian Journal of Science and Technology. Transactions of Mechanical Engineering 2013;37(M2): 83.
[26] Duan J, Li Z, Liu J. Pullin instability analyses for NEMS actuators with quartic shape approximation. Applied Mathematics and Mechanics 2016;37(3): 30314.
[27] Zhu J, Liu R. Sensitivity analysis of pullin voltage for RF MEMS switch based on modified couple stress theory. Applied Mathematics and Mechanics 2015;36(12): 155568.
[28] Ashrafi B, Hubert P, Vengallatore S. Carbon nanotubereinforced composites as structural materials for microactuators in microelectromechanical systems. Nanotechnology 2006;17(19): 4895.
[29] Thostenson ET, Chou TW. On the elastic properties of carbon nanotubebased composites: modelling and characterization. J. Phys. D: Appl. Phys. 2003;36(5): 573.
[30] Hautamaki C, Zurn S, Mantell SC, Polla DL. Experimental evaluation of MEMS strain sensors embedded in composites. Journal of microelectromechanical systems 1999;8(3): 2729.
[31] Spearing S. Materials issues in microelectromechanical systems (MEMS). Acta Mater. 2000;48(1): 17996.
[32] He JH. Hamiltonian approach to nonlinear oscillators. Phys. Lett. A 2010;374(23): 23124.
[33] Younis MI, Nayfeh A. A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dynamics 2003;31(1): 91117.
[34] Rao SS. Vibration of continuous systems. John Wiley & Sons; 2007.
[35] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. CRC press; 2004.
[36] Pelesko JA, Bernstein DH. Modeling Mems and Nems. CRC press; 2002.
[37] He JH. Semiinverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics. International Journal of Turbo and Jet Engines 1997;14(1): 238.
[38] He JH. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons & Fractals 2004;19(4): 84751.
[39] Bayat M, Pakar I. Application of Hes energy balance method for nonlinear vibration of thin circular sector cylinder. International Journal of Physical Sciences 2011;6(23): 556470.
[40] Jamshidi N, Ganji D. Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire. Current Applied Physics 2010;10(2): 4846.
[41] Shou DH. Variational approach for nonlinear oscillators with discontinuities. Computers & Mathematics with Applications 2009;58(1112): 24169.
[42] Qian Y, Ren D, Lai S, Chen S. Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam. Communications in Nonlinear Science and Numerical Simulation 2012;17(4): 194755.
[43] Saltelli A, Homma T. Sensitivity analysis for model output: performance of black box techniques on three international benchmark exercises. Computational statistics & data analysis 1992;13(1): 7394.
[44] Hong Wl, Guo Xm. Deformation of metallic singlewalled carbon nanotubes in electric field based on elastic theory. Applied Mathematics and Mechanics 2010;31(3): 2718.
[45] Sobol' IyM. On sensitivity estimation for nonlinear mathematical models. Matematicheskoe modelirovanie 1990;2(1): 1128.
[4] Jain A, Goodson KE. Thermal microdevices for biological and biomedical applications. Journal of Thermal Biology 2011;36(4): 20918.
[5] Korayem M, Sadeghzadeh S, Homayooni A. Semianalytical motion analysis of nanosteering devices, segmented piezotube scanners. International Journal of Mechanical Sciences 2011;53(7): 53648.
[6] Korayem M, Homayooni A, Sadeghzadeh S. Semianalytic actuating and sensing in regular and irregular MEMs, single and assembled micro cantilevers. Applied Mathematical Modelling 2013;37(7): 471732.
[7] Korayem M, Sadeghzadeh S, Homayooni A. Coupled dynamics of piezotube and microcantilever in scanning probe devices and sensitive samples imaging. Micro & Nano Letters 2012;7(9): 98690.
[8] Korayem M, Karimi A, Sadeghzadeh S. GDQEM analysis for free vibration of Vshaped atomic force microscope cantilevers. International Journal of Nanoscience and Nanotechnology 2014;10(4): 20514.
[9] Ghalambaz M, Ghalambaz M, Edalatifar M. Nonlinear oscillation of nanoelectromechanical resonators using energy balance method: considering the size effect and the van der Waals force. Applied Nanoscience 2016;6(3): 30917.
[10] AbdelRahman EM, Younis MI, Nayfeh AH. Characterization of the mechanical behavior of an electrically actuated microbeam. Journal of Micromechanics and Microengineering 2002;12(6): 759.
[11] Kuang JH, Chen CJ. Dynamic characteristics of shaped microactuators solved using the differential quadrature method. Journal of Micromechanics and Microengineering 2004;14(4): 647.
[12] Beléndez A, Beléndez T, Márquez A, Neipp C. Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators. Chaos, Solitons & Fractals 2008;37(3): 77080.
[13] He JH. Variational approach for nonlinear oscillators. Chaos, Solitons & Fractals 2007;34(5): 14309.
[14] He JH. Maxmin approach to nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation 2008;9(2): 20710.
[15] Zeng D, Lee Y. Analysis of strongly nonlinear oscillator using the maxmin approach. International Journal of Nonlinear Sciences and Numerical Simulation 2009;10(10): 13618.
[16] He JH. Preliminary report on the energy balance for nonlinear oscillations. Mechanics Research Communications 2002;29(23): 10711.
[17] Ganji DD, Azimi M, Mostofi M. Energy balance method and amplitude frequency formulation based simulation of strongly nonlinear oscillators. 2012.
[18] Ganji DD, Azimi M. Application of max min approach and amplitude frequency formulation to nonlinear oscillation systems. UPB Scientific Bulletin 2012;74(3): 13140.
[19] Yildirim A, Askari H, Yazdi MK, Khan Y. A relationship between three analytical approaches to nonlinear problems. Applied Mathematics Letters 2012;25(11): 172933.
[20] Askari H, Nia ZS, Yildirim A, Yazdi MK, Khan Y. Application of higher order Hamiltonian approach to nonlinear vibrating systems. Journal of Theoretical and Applied Mechanics 2013;51(2): 28796.
[21] Khan Y, Akbarzade M. Dynamic analysis of nonlinear oscillator equation arising in doublesided driven clamped microbeambased electromechanical resonator. Zeitschrift für Naturforschung A 2012;67(89): 43540.
[22] Fu Y, Zhang J, Wan L. Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Current applied physics 2011;11(3): 4825.
[23] Bayat M, Bayat M, Pakar I. Nonlinear vibration of an electrostatically actuated microbeam. Latin American Journal of Solids and Structures 2014;11(3): 53444.
[24] Yildirim A, Saadatnia Z, Askari H, Khan Y, KalamiYazdi M. Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach. Applied Mathematics Letters 2011;24(12): 204251.
[25] Rafieipour H, Lotfavar A, Masroori A. Analytical approximate solution for nonlinear vibration of microelectromechanical system using he's frequency amplitude formulation. Iranian Journal of Science and Technology. Transactions of Mechanical Engineering 2013;37(M2): 83.
[26] Duan J, Li Z, Liu J. Pullin instability analyses for NEMS actuators with quartic shape approximation. Applied Mathematics and Mechanics 2016;37(3): 30314.
[27] Zhu J, Liu R. Sensitivity analysis of pullin voltage for RF MEMS switch based on modified couple stress theory. Applied Mathematics and Mechanics 2015;36(12): 155568.
[28] Ashrafi B, Hubert P, Vengallatore S. Carbon nanotubereinforced composites as structural materials for microactuators in microelectromechanical systems. Nanotechnology 2006;17(19): 4895.
[29] Thostenson ET, Chou TW. On the elastic properties of carbon nanotubebased composites: modelling and characterization. J. Phys. D: Appl. Phys. 2003;36(5): 573.
[30] Hautamaki C, Zurn S, Mantell SC, Polla DL. Experimental evaluation of MEMS strain sensors embedded in composites. Journal of microelectromechanical systems 1999;8(3): 2729.
[31] Spearing S. Materials issues in microelectromechanical systems (MEMS). Acta Mater. 2000;48(1): 17996.
[32] He JH. Hamiltonian approach to nonlinear oscillators. Phys. Lett. A 2010;374(23): 23124.
[33] Younis MI, Nayfeh A. A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dynamics 2003;31(1): 91117.
[34] Rao SS. Vibration of continuous systems. John Wiley & Sons; 2007.
[35] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. CRC press; 2004.
[36] Pelesko JA, Bernstein DH. Modeling Mems and Nems. CRC press; 2002.
[37] He JH. Semiinverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics. International Journal of Turbo and Jet Engines 1997;14(1): 238.
[38] He JH. Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos, Solitons & Fractals 2004;19(4): 84751.
[39] Bayat M, Pakar I. Application of Hes energy balance method for nonlinear vibration of thin circular sector cylinder. International Journal of Physical Sciences 2011;6(23): 556470.
[40] Jamshidi N, Ganji D. Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire. Current Applied Physics 2010;10(2): 4846.
[41] Shou DH. Variational approach for nonlinear oscillators with discontinuities. Computers & Mathematics with Applications 2009;58(1112): 24169.
[42] Qian Y, Ren D, Lai S, Chen S. Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam. Communications in Nonlinear Science and Numerical Simulation 2012;17(4): 194755.
[43] Saltelli A, Homma T. Sensitivity analysis for model output: performance of black box techniques on three international benchmark exercises. Computational statistics & data analysis 1992;13(1): 7394.
[44] Hong Wl, Guo Xm. Deformation of metallic singlewalled carbon nanotubes in electric field based on elastic theory. Applied Mathematics and Mechanics 2010;31(3): 2718.
[45] Sobol' IyM. On sensitivity estimation for nonlinear mathematical models. Matematicheskoe modelirovanie 1990;2(1): 1128.