Theoretical and experimental investigation of the nonlinear response of an electrically actuated imperfect microbeam

In this study a theoretical and experimental investigation of the nonlinear response of an electrically actuated microbeam is performed. A mechanical model is proposed, which accounts for two common imperfections of microbeams, due to microfabrications, which are the compliant support conditions and the initially deformed profile. A computationally efficient single-mode reduced-order model is derived by combining the Ritz technique and the Padé approximation. Numerical simulations of the harmonic response of the device near primary resonance are shown illustrating nonlinear phenomena arising in the device response. Experimental investigation is conducted on a polysilicon imperfect microbeam confirming the simulation results. The concurrence between the theoretical results and the experimental data reveals that this model, while simple, is capable of properly capturing the response both at low and, especially, at higher electrodynamic voltages.


Introduction
Microelectromechanical systems (MEMS) are largely used in a wide spectrum of applications and in a variety of different fields, where they have many different functions, including sensing, actuation, and signal processing [1,2].
Simulating their complex dynamics accurately and properly requires careful modelling.Numerous studies emphasise the importance of generating reduced-order models, which are able to produce reliable results without excessive computational effort.Many approaches based on the Galerkin technique have been proposed in the literature.They mainly focus on the electric term, which is difficult to be analyzed directly and has been addressed using many different procedures.Younis et al. [3] developed a Galerkin reduced-order model based on multiplying the equation of motion with the denominator of the electrostatic force term.This approach requires only few modes to converge.It was adopted in several case-studies, such as electrically actuated microbeams and arches, with good agreement reported between theoretical results and experimental data for both the static and dynamic problems [3,4].Krylov and Dick [5] adopted a suitable shape function, which was able to considerably simplify the expression for the electric contribution.Ouakad and Younis [6], investigating carbon nanotube resonators, developed a reduced-order model, which relies on numerical integration of the electrostatic force term at every time step, which was proven to yield fairly accurate results, although timeconsuming.Rhoads et al. [7], before applying the Galerkin method, approximated the electric term through a truncated Taylor expansion.This leads to a rather simple reduced-order model, which is accurate for a "small" displacement.Ruzziconi et al. [8] proposed a simple and efficient reduced-order model based on combining the Galerkin technique and the Padé approximation.It is able to analyze the device response not only at low electrodynamic voltages, but also at large values, where the inevitable escape (dynamic pull-in) becomes impending.
This paper deals with the MEMS device of figure 1, which is an electrically actuated imperfect microbeam.The actual dimensions are experimentally measured and an experimental parametric study of its response in a neighbourhood of the natural frequency is performed.A mechanical model is presented, which takes into account both the initial deformed configuration of microbeam and the flexible supports.A reduced-order model is introduced, according to the procedure proposed in [8].We show a satisfactory concurrence between theoretical and experimental results, which is achieved both at low and, remarkably, at higher electrodynamic excitation.
The paper is organized as follows.The MEMS device is experimentally tested (Sec.2), the mechanical model is introduced (Sec.3), a single d.o.f.reduced-order model is derived (Sec.4), nonlinear dynamic simulations are performed, which validate the model (Sec.5), and the main conclusions are summarized (Sec.6).

Experimental data
Figure 1 shows a top-view of the experimentally tested MEMS device, which consists of a polysilicon microbeam actuated by an electrode placed directly underneath it on a substrate.The figure also illustrates the actuation pads on both sides of the microbeam, which are used to actuate the device electrically.Typically, these pads have some flexibility, due to under-etching from fabrications, which needs to be accounted for in the model.Since the model results are sensitive to the geometry, the actual dimensions of the device are measured directly.For this, a Wyko profilometer is used to map the topography of the device, as shown in figure 2, which visualizes the measured profile at rest and without any electric load.The measured length is = 392.82µm, the width is = 43.7 µm, and the maximum initial rise is 5.3 µm.We present data of dynamic testing conducted using a high-frequency laser Doppler Vibrometer (MSA-MEMS motion analyzer) under reduced pressure (low damping conditions).Using a random signal excitation, the linear fundamental resonance frequency is found to be Ω  145.86 kHz (figure 3).
The other parameters values are derived in order to have a good agreement between the theoretical results and the experimental frequency response.In particular, we consider the microbeam thickness equal to ℎ = 1.1 µm and the initial rise as constituted by 3.6 µm of arched configuration and 1.7 µm of deflection due to residual stresses.For the gap, we use = 1 µm.The microbeam is described in the framework of the Euler-Bernoulli theory and a linearly elastic isotropic and homogeneous material is supposed.We assume the shallow arched approximation.The nondimensional potential energy is which consists of the stain energy of the microbeam, the potential of the external electrostatic force and the potential of the springs.The axial displacement ( , ) is "condensed" by a classical procedure [9].Applying Lagrange's method and adding damping and force, the resulting nondimensional equation of motion becomes where = − (∫ ( ( ′( , )) + ′( , ) ′ ( )) ) (3) and the boundary conditions are The superimposed dot and prime in the formulas (1)-( 4) denote the derivative with respect to the nondimensional time and space , respectively.The nondimensional variables used in (1)-( 4) are (denoted by hats, which are dropped in ( 1)-( 4) for convenience) and the nondimensional parameters are where is the mass density per unit length, is the axial stiffness, is the bending stiffness, and are the area and the moment of inertia of the cross section, is the effective Young's modulus, is the material density, is the gap width between the stationary electrode and the undeformed straight position of the microbeam, = is the overlapped area between the microbeam and the stationary electrode, is the dielectric constant in the free space, is the relative permittivity of the gap space medium with respect to the free space, is the viscous damping coefficient, and are respectively the translational and rotational stiffness.Referring to the microbeam experimentally tested in Sec. 2, the parameters of Eq. ( 2) are calculated as: = 9.73947, = 0.0000592566, = 0.08, = 6.00525.The nondimensional stiffness coefficients are estimated as = 12600 and = 9.45, which are based on the obtained natural frequency of the microbeam.

Reduced-order model
After analyzing the static and the linearized unforced undamped dynamics, we use these results to generate a reduced-order model [10].
The microbeam deflection ( , ) is approximated by ( , ) ≅ ( ) + ∑ ( ) ( ) .We consider the single (first symmetric) mode reduced-order model (M = 1), and we select the shape functions ( ) and ( ), respectively, equal to the stable static configuration and to the aforementioned mode shape.We apply the Ritz method to the potential form (1). The total potential energy function becomes where ( ) is the amplitude of the displacement of the mode shape.In (7) we can directly compute the coefficients in the part due to the geometrical nonlinearity, while this is not possible for the part due to the electric force term, because the integral cannot be solved analytically.Since this expression for the electric potential is too complicated and time-consuming for nonlinear dynamic simulations, we attain an approximation of this term by the Padé expansion.To determine the Padé coefficients we require conditions, which retain the significant quantitative and qualitative aspects of the dynamics, such as the presence of a maximum in ( ), after which there is the transition to the escape.This yields to the total potential energy ( ) = 1497.

Numerical simulations
We calculate the frequency response curves for the same voltage values previously investigated in figure 4.These numerical results have been obtained by using selfdeveloped codes, based on direct integration and Runge-Kutta method.The results are reported in dimensions to facilitate the comparison with the experimental data.
These results show a good matching between the theoretical and the experimental curves.In a neighborhood of the resonance, the device exhibits a softening behavior.The non-resonant and resonant branches have the characteristic bending toward lower frequencies.The experimental branches exist for a smaller range than the theoretical one, but along these interval, the curves nearly coincide.Both the gap between the two branches, the range where the two attractors coexist and the bending are adequately represented.This occurs not only at low electrodynamic voltages (figure 7a) but also at higher ones (figure 7b).These results may be considered as an effective validation of the proposed reduced-order model.In fact, this reveals that, despite the apparent simplicity, the model is able to catch the most relevant nonlinear phenomena arising in the MEMS response and to represent them properly and accurately.
For sake of completeness, we note that the presence of a longer range of existence of both the branches in the theoretical simulations is due to dynamical integrity reasons, as observed in the case-study of a MEMS capacitive accelerometer [11].

Conclusions
A MEMS device based on an electrically actuated microbeam has been experimentally investigated.Frequency response curves have been obtained by a sweeping process in a neighborhood of the first symmetric natural frequency.
After introducing a mechanical model which takes into account imperfections both in the profile and in the boundary conditions, a single d.o.f.reduced-order model has been proposed.It is based on combining the Ritz technique and the Padé approximation.Despite the apparent simplicity of the model, the theoretical results show a satisfactory agreement with the experimental frequency response curves.The model is able to catch, properly and accurately, the most relevant aspects of the nonlinear MEMS response, both at low and, remarkably, at higher electrodynamic voltage excitations.This validates the proposed approach.

Fig. 1 .
Fig. 1.A picture of the tested MEMS device showing the laser dot of the laser Doppler vibrometer.

Fig. 3 .
Fig. 3. Measured first resonance frequency.Experimental frequency sweep tests are conducted at a DC voltage load of = 0.7 , which yield frequency response curves of the maximum amplitude of the oscillations.In the frequency-sweeping process, the dynamic voltage (AC load amplitude V AC ) is kept constant and the frequency is increased (forward sweep) or decreased (backward sweep) slowly, i.e. quasistatically.The frequency step is 5 Hz and the pressure is 60-70 mtorr.Examples are reported in figure 4.

Fig. 4 .
Experimental frequency response curves at (a) = 1.5 ; (b) = 2.6 .The forward and backward sweep are respectively in red and blue.3 Mechanical model A schematic of the tested MEMS device is illustrated in figure 5. To simulate non-ideally fixed-fixed boundary conditions, translational and rotational springs have been added at each edge.The shallow arched initial shape is delineated by ( ) = (1 − cos(2)), where is the maximum initial rise.The electrode generates both an electrostatic and an electrodynamic excitation.Only the viscous damping contribution is considered, since is generally the most relevant source of energy loss in typical MEMS[2].