Dynamic behavior of T-beam resonator with repulsive actuation

Electrostatic MEMS transducer driven by repulsive force is an attractive possibility and has advantages of avoiding the pull-in instability, tuning the natural frequency, and achieving high sensitivity by applying high enough voltages. In this work, a “T”-shaped beam, which is formed by attaching a secondary beam perpendicular to a primary cantilever at the tip, is introduced and its nonlinear dynamics is analyzed. A reduced-order model is derived from mode shapes formed from electromechanical coupling effects respectively. Generalized forms of forced Mathieu equation of motion are derived, and then, dynamic behaviors are investigated through the theory of multiple scales. The resonant responses, including both primary and principal parametric resonances, reveal softening behavior originating from quadratic and cubic nonlinearities in the governing equation. The behavior of the T-beam is compared with traditional cantilever structure. The resonance under repulsive force demonstrates that the T-beam has several advantages over a traditional cantilever: Lower natural frequency but higher resonant responses can improve the signal-to-noise ratio; with an attached micropaddle, the T-beam has a larger surface for absorption of targeted analytes for mass sensing. We conclude that an electrostatic MEMS resonator with a “T”-shaped beam is potentially appropriate for the new generations of sensors and actuators.


Introduction
Capacitive sensing and actuation have been the most common methods of transduction in MEMS transducers for the past few decades [1][2][3]. This transduction scheme is popular because of its compatibility with microfabrication technology, low sensitivity to temperature change, and small power consumption [1][2][3]. The basic design of this kind of transducers has not changed significantly since their introduction in 1970s and 1980s [4]. They are based on a two-electrode design such as parallel plate or comb-drive configurations. These designs suffer from pull-in instability [5], small stroke range [6], and low allowed voltages that hamper the sensitivity of these devices [4].
Repulsive electrode configuration is a multi-electrode capacitive scheme that addresses the pull-in issues associated with conventional two-electrode designs by producing significant out-of-plane actuation [7,8]. Since their introduction by Lee et al. [7], the nonlinear dynamic behavior of this scheme has been extensively explored to produce different MEMS sensors and actuators such as micro-mirrors [9,10], microphones [11], accelerometers [12], MEMS switches [13], and resonators [8]. However, current designs of repulsive electrostatic MEMS transducers based on cantilever should be improved for applications needed for mass sensing.
A small surface area limits the performance in mass sensing. Generally, mass sensor measurements depend greatly on absorption/adsorption of the targeted analytes on and into sensing layer [14,15]. When certain molecules adhere to the surface, the change in deflection (static mass sensing) or resonant frequencies (dynamic mass sensing) can be measured [16][17][18] to determine the concentration of the targeted analytes. Tiny microbeams with limited attaching surface can only accommodate small amount of molecules which generates limited deflection and shift in resonant frequencies. To amplify the shift in the resonant frequency, microcantilevers are designed with a large surface area that contacts the targeted analytes. Larger surface area makes it possible for more molecules to adhere to the microstructure and therefore more significant change in the mass and resonance frequency. One of the common methods to increase the surface area is to attach a micropaddle to the microstructure [19][20][21].
To improve mass sensing capability, researchers have examined parametric resonance as an effective mechanism [22,23]. Unlike conventional MEMS mass sensors with high Q, which have to work in very low pressure and temperature to detect the small frequency shifts in resonance, parametrically excited MEMS oscillators measure the frequency shifts of the parametric resonance tongue. The transition from varying mass attachment at stability boundary is very distinct; therefore, tiny frequency changes can be detected with high resolution. The large responses of parametric resonators, unaffected by damping and without Q limitation, are ideal for high-resolution, highly sensitive mass sensors. Although there are many studies on the parametric resonance of gap-closing electrodes [24][25][26][27], there are just a few studies of parametric resonance of repulsive actuators [28,29].
To enhance mass sensing, we present a T-beam that has a larger surface area compared with cantilever system and study its fundamental and parametric resonances. The "T"-shaped beam is composed of a paddle at the free end of a microcantilever, driven by repulsive electrostatic configuration. Figure 1 shows the Fig. 1 Schematic of the T-beam repulsive force actuator assembly schematic of the T-beam MEMS resonator with dimensions and properties in Table 1. The electrostatic configuration ( Fig. 2) consists of three separate electrodes. The first electrode is the movable T-beam. The second electrode is placed underneath the T-beam and is called the bottom electrode. The third electrode is a continuous side electrode that is placed on the substrate within a certain distance around the perimeter of the bottom electrode as shown in Fig. 1. Applying voltages on the side electrode while grounding the movable and bottom electrodes creates an electrostatic force on the movable electrode. This force is upward, meaning that it pushes the microstructure away from the bottom electrode [8,30]. The electrostatic force naturally eliminates the pull-in possibility between the movable and bottom electrodes [11,31]. Because the beam is pushed above its original position when a DC voltage is applied on the side electrode, the effective gap between the moving and bottom electrodes increases [8]. That provides more room for the vibration of the microstructure, enlarging its stroke [8,10,30].
The organization of the paper is as follows. Section 2 presents the derivation of the governing equation for a microcantilever with paddle that is subjected to repulsive electrode configuration. A lumped parameter model is extracted from the continuous governing equation of motion using Galerkin's decomposition method. The discretized equation resembles the nonlinear Mathieu's equation that is solved using the method of multiple scales. In Sect. 3, the static response of the microbeam to DC voltage excitation is obtained numerically with MATLAB. The dynamic response of the microstructure to DC+AC voltage excitation is investigated as well. The primary resonance and principal parametric resonance of the microstructure are analyzed and discussed in this section. Finally, Sect. 4 concludes the paper.

Problem formulation
The T-beam consists of two coupled beams: a primary microbeam fixed at one end with another shorter secondary microbeam attached at the tip. For reasonable simplification, when doing static and dynamic analyses, here we treat the structure as an Euler-Bernoulli beam (primary beam) of width b, thickness h , crosssectional area A P = bh, and second moment of area I P = 1 12 bh 3 . The micropaddle (secondary beam) is a rigid body with massM, mass moment of inertia C around a line parallel to its width b p passing through its center of mass.

Equation of motion
We derive the equation of motion for the system using the extended Hamilton's principle, shown in Eq. (1), where δT is the variation of kinetic energy, δU is the variation of potential energy, and δW NC is the virtual work of nonconservative forces, such as damping. The beam-paddle hypothesis assumes the micropaddle is a rigid body and hardly contributes to the potential energy but significantly increases the kinetic energy [1]. The rotation angle θ c is assumed equal to that of the microbeam (primary beam) tipŴx (L , t). Here, the T-beam resonator system is treated as a conservative electromechanical system. The work done by the electrostatic forces on the T-beam is considered conservative and included in the potential energy. The virtual work done by nonconservative external force comes from viscous dissipation, mostly on the primary beam, and the portion on the secondary beam is neglected. The kinetic energy, potential energy, and virtual work for the microsystem are represented as Eqs. (2)- (8).
T total = T prim + T sec (4)   Parameter Substitution whereŴ (x,t) is the z-direction beam displacement at locationx and timet and s is a local coordinate attached to the microplate, as shown in Fig. 4. The subscript denotes the partial derivatives with respect tox. d is the initial gap, V is the electrode voltage, and f 1 , f 2 are referring to the electrostatic force on microbeam and micropaddle, respectively. Note that the first-order Taylor expansions are used when it comes to the nonlinear force on the paddle: The numerical profile for repulsive force is obtained and fit as two ninth-order polynomials from a finite element simulation for the intended fabrication dimensions in COMSOL (Fig. 3). The beam dimensions are listed in Table 1, and the nondimensional parameters are indicated in Table 2. Expressing and obtaining the variations of the energy terms and plugging them into Eq. (1) yield the nondimensionalized governing equation of motion for the system shown as Boundary conditions are accordingly shown in Equations (10) for x = 0 and Eqs. (11) and (12) for x = 1,

Mechanical mode shapes
Determining the microstructure's natural frequencies as accurately as possible is crucial for characterizing their actual geometry and boundary conditions, revealing their operating range and restrictions, and enabling accurate calibration of resonator-based devices [32]. By setting damping term c = 0 and forcing terms V = 0, linear eigenvalue problem is obtained: Using separation of variables W (x, t) = φ(x)q(t) and settingq(t) = −ω 2 q(t), where ω stands for the nondimensional natural frequency, the mode shape is written as where β is also a nondimensional parameter defined by ω = β 2 . Combining boundary conditions Eqs. (14)- (17), we can find c 3 = −c 1 and c 4 = −c 2 , and the characteristic equation is described as For eigenvalues and mechanical mode shapes, we need to obtain the nontrivial solutions to Eq. (19). Solving A B C D = 0 with MATLAB, values of β and c 2 /c 1 for the first four mechanical modes are shown in Table 3, where the dimensional natural frequencies are calcu- The results of modal frequencies are compared against COMSOL, which shows a close agreement (Figs. 5 and 6). Meanwhile, voltage-related boundary value problem is solved for mode shapes, but because of the significant nonlinearity from the microstructure as well as the electrostatics, several nonlinear terms have to be dropped in order to simplify the boundary value problem. Thus, the result is very close to that of our mechanical modal analysis and voltage-related modes are not pursued. More details are included in Appendix A.

Reduced-order model
Once the mechanical modes shapes φ m s are known, with Galerkin's method, the partial differential equation in Eq. (1) is reduced to a set of ordinary differential equations (more details are included in Appendix B), Eq. (20), that can be solved numerically.
where m = 1, 2, ..., N and N represents the number of mode shapes in consideration.
For a one-mode approximation, Eq. (20) is reduced to, 2.4 Dynamic mathematical model T-beam structure as a sensor or resonator is mostly utilized when resonating. The actuator voltage V that is applied to the structure becomes the input signal, which superimposes a sinusoidal function V AC on the V DC that operates the device. When applying the combination of two voltages, AC is dropped as well since the AC voltage is typically much smaller than DC voltage. Our efforts are then mostly focused on first-order resonance of the driving frequency, which accounts for the total displacement as where q s stands for the static deflection and q(t) represents the dynamic displacement. Our particular concerns are the dynamic response q(t) about the static equilibrium position. Neglecting the squeezefilm damping as well as contact, a generalized form of forced Mathieu equation of motion is formulated as Eq. (23) based on one-mode approximation, where with δ j s and f j s resulting from DC voltage terms as described Before we apply the perturbation method, it is fundamental to keep linear terms while making nonlinear terms reasonably small. Here, the bookkeeping parameter is assumed to be at the order O(10 −2 ). The solved steady-state amplitude q s , assumed of order O(1), from Eq. (21) when V = V DC , helps predict and assume the initial order of all the magnitude order of terms in Eq. (23) in Tables 4 and 5.
Based on theory of multiple scales in perturbation at most to the second order ( 2 or T 2 ), the terms higher than that have really minimal effect on dynamic behavior, and therefore, we exclude the small nonlinearity of higher orders (O(10 −5 ) or smaller) and keep dynamic-related terms up to cubic terms. Equation (23) is reduced to, We then seek an approximate asymptotic solution to Equation (24) by letting where T 0 = τ , T 1 = τ and T 2 = 2 τ represent fast, slow and very slow timescale terms, respectively. For achieving solution to Eq. (24), it is essential to evaluate the conditions for primary and secondary resonances separately.

Primary resonance
Since the excitation F 0 is O( 2 ), Ω − ω n is assumed to be O( 2 ) for consistency. Hence, we introduce σ as detuning parameter examining the closeness of Ω to 1 by putting Based on Table 4 ( f 3 , f 4 , and higher ones are dropped), the governing Eq. (24) should be, To balance the effect of primary-resonance excitation, we need to reorder the equations so that the order of parametric and primary force and ζ all appears in the 2 order. Substituting Eqs. (25) and (27) into (24) and equating the coefficients of 0 , 1 and 2 on both sides, we obtain T-beam 0.00385 1.8667 0.0133 −0.0315 −4.839e-4 −3.080e-4 −4.738e-6 2.266e-4 3.486e-6 0.147 Table 5 The assumed order of magnitude for the terms shown in Eq. (23) for principal parametric resonance with the values when V DC =90V and V AC =30V. Terms at higher order including f 5 , δ 5 are dropped where D i = ∂ ∂ T i . The general solution to Eq. (23) can be written in the form Inserting Eq. (30) and (26) into Eqs. (28) and (29), eliminating the secular or resonance terms implies that Implementing polar notation of A = 1 2 ae iβ in terms of amplitude a and nonlinear frequency β, introducing variable γ = −β + σ T 2 , and separating real and imaginary parts give derivatives with respect to T 2 as The analytical response for primary resonance is therefore represented as q = a cos(β + ω n τ ).
By setting right-hand side of Eqs. (32) and (33) to be zero for the asymptotic approximation, the amplitude a in terms of detuning parameter is determined by the equations: The amplitude a is obtained by solving the simplified equation: in which

Results and discussion
The static and dynamic behaviors of a T-beam resonator under repulsive force are mostly studied for a one-mode approximation model. The damping is estimated by

Static analysis
For the static analysis, the sensor voltage, V s , is a constant DC voltage upwards of 200V. First, the boundary value problem is derived based on Eq. (9) after removing time-dependent components, which results in governing equation as Eq. (46): and the corresponding boundary conditions for x = 0 as Eq. (47) and for x = 1 as Eqs. (48) and (49): 24 Y. Tian et al.
The closed-form W s = W s (x) that represents the static deflection is obtained using bvp4c in MATLAB after rewriting it to be four first-order ordinary differential equations. The solution W s (x) gives the static T-beam profile (0 ≤ x ≤ 1) for varying DC voltages. The tip deflection is then obtained for increasing DC voltages to capture the static solution shown in Fig. 7. Similar to previous research that revealed that electrostatic levitation could actuate the cantilever upward more than an order of magnitude larger than the initial gap [8,28], the T-beam is free from pull-in instability and can have a large stroke. Moreover, removing the time-dependent terms in Eq. (20), the static deflections of the T-beam using mode approximations are also shown in Fig. 7. Comparison of results from boundary value method and reduced-order model implies that two-mode approximation is closer to the boundary value method. However, we compared the weight of each mode in Table 6. The values are normalized with respect to the total tip displacement. It is evident that weights of the first mode are always hundred times larger than the weights of the second mode. Because it is the dominant mode, we use the first mode approximation to obtain the dynamic responses.
The Jacobian matrix of Eq. (21) is computed at the static equilibrium points, and the corresponding eigenvalues of the Jacobian matrix yield the natural frequencies using a one-mode approximation at many voltages V s in Fig. 8. T-beam resonator driven by repulsive force shows relatively small natural frequency, while it produces a high deflection (Fig. 7). Merging T-beam and electrostatic levitation as one single MEMS mass sensor, therefore, enables detecting a small change in the resonant frequency with less susceptibility to noise. The merger can inspire a new generation of mass sensors using the phenomenon of mode localization [32] by connecting two single resonators with a coupling beam. Meanwhile, the T-beam undoubtedly enlarges the surface which helps detect and measure the presence of different kinds of analytes, including biological species like viruses or bacteria, whether in liquid or gaseous media or specific gases, especially volatile organic compounds (VOCs).

Perturbation analysis
The method of multiple scales (MMS) perturbation technique is applied to perturb the microactuator near its fundamental natural frequency and at twice that frequency to numerically solve the governing equations accordingly and study the actuator's primary and principal parametric resonant behaviors. With σ serving as a detuning parameter and the nondimensional amplitude a, we can investigate more than dynamic T-beam responses by comparing them with cantilevers under the same excitation. For determining the steady-state response amplitudes in the vicinity of the principal parametric resonance, the stability of periodic motions is examined by determining the relevant eigenvalues of the Jacobian matrix in the vicinity of the periodic motions. A positive eigenvalue represents an unstable limit circle in phase portrait, and even slight disturbance leads to boundless increasing amplitudes. This effect does not happen if the eigenvalue is negative, which helps the disturbance decay with time. When analyzing primary resonance, a stable displacement of the T-beam can be reflected by the stable amplitude a. Unstable amplitudes are hard to hold. For principal parametric resonance, things get much intriguing. When amplitude a in our solution (Eq. (44)) is associ-  ated with a negative eigenvalue, the response is defined as principal parametric resonance because the response period is the same as the excitation period. However, when the eigenvalue has a positive sign, it indicates subharmonic resonance. Parametric resonance has a much larger amplitude than subharmonic resonance, which is why most researchers on MEMS resonators see opportunities. In order to study the advantages of utilizing Tshaped beam instead of traditionally applied cantilever, which is highly popular and widely integrated into MEMS sensing due to the well-known excellent properties, in designing the resonator under repulsive configuration, we can compare T-beam's dynamic behaviors with cantilever of same dimensions, meaning 500 microns in length and 20 microns in width. Then, a great potential to improve new generations of sensors and actuators by combining T-beam and electrostatic levitation appears.

Primary resonance
Comparing with results of a cantilever of the same length in primary resonance in Fig. 9, we conclude that the dynamic response of T-beam also shows hysteresis and softening behavior, which is because of a quadratic term and a negative cubic term for the electrostatic force from the forcing profile polynomial fit. These two terms are dominant in the forcing function T-beam Cantilever Fig. 9 Comparison of primary resonant response at V DC = 195V and V AC = 1.5V for T-beam and cantilever. Quality factor is set to 130. Markers (black circle for T-beam and blue circle for cantilever) indicate amplitude of stable responses, and dashed red lines represent unstable branch and explain why the softening is significant even at AC voltage near 1.5V. That means, as the frequency approaches the natural frequency, the oscillation amplitude increases significantly, and when the frequency drops below the resonant frequency, the system experiences hysteresis and drops to a significantly smaller oscillation amplitude. To validate the results, we make a comparison of the primary resonance obtained from the perturbation, long-time integration, and shooting methods [1] as depicted in Fig. 10. All of them are based on Eq. (21) with one-mode approximation. As it is deduced, the perturbation method overestimates the results, which has been reported in previous studies of nonlinear resonators [33]. However, it is a computationally efficient and quick method to roughly estimate the response of the system. For more accurate responses, shooting technique is recommended.

Principal parametric resonance
In general, the parametric excitation actuation is more efficient, requires less voltages than the primary resonance excitation, and makes the response faster as the excitation frequency is twice the resonance. Figure 11 demonstrates that the T-beam's parametric resonance Quality factor is set to 500. Solid lines (black for T-beam and blue for cantilever) indicate the amplitude of stable responses, and dashed red lines represent unstable branch achieves a much higher amplitude than the cantilever. The quality factor for parametric resonance is set to 500 to avoid high AC voltages to trigger parametric resonance. At very low quality factors, parametric resonance is not activated or requires very high AC voltages. At large amplitudes, the beam hovers around the bottom electrode, but does not collide with it because of the squeeze film damping effect as our previous study reported [28]. Because of electrostatic levitation, the usable range of dynamics is now about 25 μm compared to 2 μm of the initial gap, which is a significant range of motion for micro-actuators.
The T-beam resonator can achieve very large dynamic amplitudes from parametric resonance as shown in Figs. 12, 13, and 14 . If the AC voltage is high enough, the unstable branch, which is indicated in red, intersects the stable branch, which is indicated in black. The intersection of stable and unstable branches is called transcritical bifurcation where the two branches exchange stability. In fact, the bifurcations in generalized parametric resonance have been thoroughly investigated [24] and the bifurcation diagram shown here, describ-ing physically achievable amplitudes without hitting the substrate, is a type of the quasi-softening nonlinear characteristics. Based on the simulation, an AC voltage of at least 4V is necessary to overcome the threshold voltage for parametric resonance when the DC voltage is 90V and the quality factor is 500. Since the perturbation method is computationally fast, especially in vicinity of targeted frequency range, and can predict both stable and unstable branches, it can also be applied to produce the principal parametric transition curve shown in Fig. 15. The unstable area (grey area) is where the parametric resonance happens, while in the stable area (white area), only quasi-periodic happens. This transition curve shows a minimum threshold voltage of about V AC = 4V when quality factor is 500. The responses at four points (P1, P2, P3, and P4) in Fig. 15 are shown in Fig. 16 to better indicate whether the beam is showing parametric resonance or quasiperiodic motion. Inlets of Fig. 16 show more detailed waveforms of oscillation over 20 excitation periods. It is evident that the oscillation amplitudes of parametric resonance (P3 and P4) are much bigger than that of quasi-periodic motion (P1 and P2); for parametric resonance, the oscillation period is the twice the excitation period, while for quasi-periodic motion, the oscillation period is the same as the excitation period. It can be easily seen that there are 20 peaks in inlets of P1 and P2 and 10 peaks in inlets of P3 and P4.
The T-beam's ability to achieve unprecedented amplitudes is highly advantageous for resonance sensors where a large signal-to-noise ratio is desired. The DC voltage governs the center frequency, which provides tunability for the proposed system without pull-in instability. This is in contrast to parallel-plate devices that the increase in DC voltage makes the system prone to pull-in collapse.
Future work can investigate internal resonance because the T-beam enables having commensurate relationships between modal frequencies. This property can be very useful for filtering and mass sensing.

Conclusion
A MEMS mass sensor prototype of an electrostatic Tbeam under the influence of a repulsive force is presented in this paper. The proposed model for approximation uses an Euler-Bernoulli beam with a rigid paddle attached to the tip. Analytical and numerical responses were obtained. These results show that it is possible to predict the upper bound of the primary and parametric resonances using a one-mode approximation. The comparison made with cantilever under same conditions shows better properties, including higher amplitude in oscillation of primary and principal parametric resonance, which contributes to large signalto-noise ratios and enhanced resolutions. Especially in principal parametric resonance, the significantly greater amplitude than cantilever beam gives much promising applications combining research achievements in this area. We have also presented the transition curve to specify the region where the system shows distinct large parametric resonances. The fact that the presented repulsive resonator does not suffer from pull-in instability and has large amplitudes makes it suitable for optical sensors and actuators to achieve high resolution. The T-beam resonator can improve the next generations of mass sensors to have better sensitivity.
Simplify the above to be: @x=1: Drop nonlinear terms of φ(x)q(t) and q(t) is canceled: @x=1: The algebraic equations above can be solved, and therefore, voltage-related natural frequencies are obtained as given in Table 7. They are quite close to each other, most probably from dropping too many nonlinear terms, and both static and dynamic results based on them hardly have any difference.