Sensitivity Analysis of Fiber-Reinforced Lamina Micro-Electro-Mechanical Switches with Nonlinear Vibration Using a Higher Order Hamiltonian Approach

Sensitivity Analysis of Fiber-Reinforced Lamina Micro-Electro-Mechanical Switches with Nonlinear Vibration Using a Higher Order Hamiltonian Approach

A. Kabiri ^a, S. Sadeghzadeh ^b*

^a School of Railway Engineering, IUST, 16846-13114, 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), 16846-13114, Tehran, Iran

1. Introduction   

Micro-electro-mechanical systems (MEMSs) are used in fields such as aerospace, optical, and biomedical engineering, particularly in applications such as micro-switches, transistors, accelerometers, pressure sensors, micro-mirrors, micro-pumps, micro-grippers, and bio-MEMSs [1-4]. 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 [5-8]. Ghalambaz et al. [9] studied the effects of Van der Waals attraction, Casimir force, small size stretching, fringing field, mid-plane 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 fourth-order Runge-Kutta 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, single-degree-of-freedom 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 mass-spring system. Khan and Akbarzade [21] used a VA, a Hamiltonian approach, and an AFF to analyze a nonlinear oscillator equation in a double-sided clamped micro-beam-based electromechanical resonator. Moreover, Fu et al. [22] applied the EBM to study a nonlinear oscillation problem in a micro-beam model. They used an equation for the free vibration of a micro-beam, based on the Euler–Bernoulli hypothesis, and they compared the results with a fourth-order Runge-Kutta method.

Bayat et al. [23] investigated He’s VA to solve the nonlinear vibration of an electrostatically actuated doubly–clamped micro-beam 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 pull-in 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 pull-in voltage of a stepped cantilever-type 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].

Micro-composites are a new class of materials used for the mechanical components of MEMSs. Fiber-reinforced 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 bulk-form properties. Ashrafi et al. [28] presented a detailed theoretical investigation of the utility of carbon nanotube-reinforced 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 nanotube-based 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 nanotube-based 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 fiber-reinforced 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 micro-beam system.

Table 1. Comparison of natural frequency formulas for micro-beams from recent related works

1. Mathematical Model

Figures 1 and 2 depict the fiber-reinforced lamina clamped–clamped micro-beam, with length , width , constant thickness h, initial gap , and applied voltage . The micro-beam 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 micro-beam [33]. For more design options and capabilities, computational studies are essential in addition to experiments. However, there are no exact (analytical) closed-form 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 mid-plane deformation, the nonlinear partial differential equation of transverse motion could be expressed as follows [34]:

where  is the transverse deflection,  is the Young’s modulus,  is the Poisson's ratio, and  is the effective modulus of the micro-beam. The quantity of  changes with the different thicknesses of the micro-beam, as follows [25]:

Then, by using a theoretical approach to determine the engineering constants of a continuous fiber-reinforced composite material based on whether the applied loads are parallel or perpendicular to the fiber direction, the effect of fiber-reinforced 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]:

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 micro-beam can be implemented by the following equation for the direction on the laminar plane to be parallel to the fiber:

For the direction on the laminar plane to be transverse to the fiber, we would have

 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 micro-beam.  is the normalized motivating force that is derived from electrostatic excitation, as follows [36]:

where  is the dielectric constant of the interface. The boundary conditions are as follows:

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

Then, dimensionless boundary conditions can be written as follows:

Based on the presented formulas, a dimensionless equation of motion can be implemented for MEMS resonators using the following equation:

By using the assumed modes method, the dimensionless deflection solution of Eq. (12) can be introduced as follows:

where  is the th Eigen function of a micro-beam that fulfills the appropriate boundary conditions,  is the th time-dependent deflection coordinate, and  is the supposed degree of freedom of the micro-beam.

To solve Eq. (12), we consider a single-degree-of-freedom model , and deflection function  is assumed to be as follows:

The trial function is

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]:

where

1. Solution Procedure

For the following general oscillator,

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 semi-inverse [37, 38] and He [13, 39] methods, the variation parameter can be written as

where  is the oscillator vibration period and . Thus, the Hamiltonian approach in the presented problem can be expressed as follows:

Then, by defining a new function, we have

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:

For greater convenience, a new function, , is defined as follows:

Then, for the natural frequencies of the system, the following relation is used:

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:

3.1. First-order Hamiltonian approach

After satisfying the initial conditions, utilizing as the trial function in Eq. (25), we obtain

This leads to the following:

Then, the frequency–amplitude relationship can be obtained from the following:

Therefore, after some approximations and simplifications, Eq. (28) can be solved, and the natural frequency can be obtained as follows:

That is approximately equal to

Both the VA [23] and the analytical approximate solution [25] have the same result for this problem.

3.2. Second-order Hamiltonian approach

To improve the accuracy of this approach, a higher order periodic solution was assumed as the time response function:

where the initial condition is

By Substituting Eq. (32) into Eq. (25), we can obtain

Subsequently, the frequency–amplitude relationship can be obtained from the following equation:

To obtain the natural frequency, substituting Eq. (32) into (34) as , a second-order 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. Third-order Hamiltonian approach

A third-order time response can be used for the micro-beam, as follows

where the initial condition is

Similar to the second-order Hamiltonian approach, with some mathematical simplification, values of can be obtained for various values of A and V (see Table 3 for examples).

1. Validation, Results, and Discussion

4.1. Computational efficiency

The nonlinear algebraic equations presented were solved using Wolfram Mathematica software on an Intel(R) Core™ i5-3230M CPU 2.6-GHz processor, including 6 GB of installed memory on a 64-bit 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 third-order Hamiltonian approaches, respectively. In terms of accuracy and computational efficiency, the second-order solution was the best.

Table 2. parameters for different  and values                   (

Table 3. parameters for different  and values                    (

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 micro-beam under electric excitation (V = 24 Volts), with parameters  obtained with the first-, second-, and third-order 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 micro-beam

Simplifying the convolution of a nonlinear system to a pseudo-linear model can provide a useful view on the stability and controllability of a system. For example, let us assume that a nonlinear MEMS micro-beam has a linear-wise model, as below, that includes all nonlinearities in the second part of its dynamics:

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 second-order 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 micro-beam 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 micro-beam.

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 second-order 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 second-order 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 micro-beam. It is apparent that the frequency is decreased with an increase in voltage. The results of the second-order 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 micro-beam 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 micro-beam.

Figures 8 and 9 show the effect of the thickness of an electrostatically actuated micro-beam on the natural frequency at various initial amplitudes when geometrical and structural lamina properties at a fiber-volume fraction of 0.67 are chosen (see Table 5). As demonstrated clearly, natural frequency decreases with an increase in the thickness of a micro-beam.

Table 5. Nominal inputs and standard deviations for the micro-switch parameters

1. Sensitivity analysis

When a sensor or actuator is designed based on micro-beams, 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 micro-beams. 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 micro-beam 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 :

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 micro-beam 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 micro-beam reacts less easily to external excitations. Increasing the thickness increases the system rigidity, which makes the micro-beamless sensitive. In contrast, decreasing the gap between the beam and the electrodes of an electrostatically actuated micro-beam results in a lower excitation output.

Figure 11 depicts micro-beam 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 micro-beam with respect to the applied voltage. By applying more voltage, a less-sensitive 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 micro-beam 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 micro-beam improves its efficiency. Carbon-fiber reinforced nanostructures improve the electro-mechanical properties [44].

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 one-at-a-time 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, variance-based 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:

where n is the number of independent parameters.

The input independent parameter region should be determined as follows to explain the Sobol’s method:

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:

where for any input ,  is called its first-order 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:

where is constant and is determined as follows:

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:

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

Partial variances are computed as follows:

According to Eq. (41), the sensitivity measures  are given by

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 micro-beam 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 sample-based methods (e.g., a variance-based 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 micro-beam.

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 total-order 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 second-greatest 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.

1. Conclusion

The sensitivity of a micro-switch containing a doubly clamped micro-beam 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 mid-plane 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 semi-inverse method and exploiting He’s method, an approximate frequency–amplitude relationship has been obtained. Differentiating the Hamiltonian approach reveals the natural frequency of micro-switches. 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 third-order Hamiltonian approaches, respectively. The second-order 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 second-order 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 second-order 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 second-order Hamiltonian approach are extremely close to the EBM solution aside from considerable discrepancies in high amplitudes and voltages.

Figure 14. Pie chart diagrams of Sobol’s sensitivity analysis (SA) of an electrostatically actuated micro-beam 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 micro-beam; 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 micro-beam reacts less to external excitations. By increasing the thickness, rigidity also increases, which leads to less sensitivity of the micro-beam. However, decreasing the gap between the beam and the electrodes an of electrostatically actuated micro-beam 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 less-sensitive 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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