Electrostatic nonlinear dispersive parametric mode interaction

Understanding and controlling the nonlinear coupling in micro/nanomechanical resonators are of great importance to the exploitation of advanced devices. The recently observed electrostatic nonlinear parametric coupling is a very interesting topic. However, the theoretical model of the electrostatic parametric coupling remains unclear. This paper explicitly derives the model and the electrostatically induced dispersive parametric coupling which reveals the ability to tune the bifurcation topology of capacitive resonators is analyzed based on the multiple-time-scale method. A novel displacement-to-frequency transduction scheme based on this electrostatic dispersive parametric coupling effect is proposed. The transduction sensitivity is theoretically given, which indicates that this electrostatic dispersive transduction scheme can provide even more design freedoms than the existing displacement-to-frequency transduction scheme based on tension modulation. In addition, a bifurcation reversal effect is predicted in the strong actuated states of the dispersive parametric coupled system, which reveals the ability to tune the bifurcation topology of capacitive resonators.


Introduction
Micro-and nanomechanical resonators are widely used as the cores of timing [1,2], sensing [3][4][5], information processing [6,7], and quantum experiments [8,9], because of their abundant mechanisms of coupling with various physical fields. The mutual interactions between different degrees of freedom in micro-and nanomechanical resonators have attracted great attention in recent years. Understanding and efficiently manipulating modal coupling effects are very important for developing novel devices and improving the performance of existing devices [7,[9][10][11][12]. Modal coupling effects exist in many different mechanisms, such as mechanical linkages, dielectric coupling, tension-induced parametric coupling, internal resonance, electrostatic coupling and so on [13][14][15][16][17][18][19]. Modal interactions induced by nonlinear effects are of special interest, which takes the tensioninduced parametric coupling as the typical representative [20][21][22]. Among the vibrating mechanical resonators with tension generated, the displacement of one mode can affect the others' dynamics resulting from the stiffness hardening characteristic. The couplings usually consist of dispersive parametric coupling and dissipative parametric coupling, which Correspond to the affected resonant frequency and damping ratio, respectively.
The dispersive parametric coupling caused by displacement-induced tension in resonators with clamped-clamped topology has been extensively studied. Ref. [13] demonstrated that the displacement-induced tension coupling in a doubly clamped beam resonator could detect the displacement of any other mode by measuring the response of one mode. Ref. [15] engineered a strain-coupled nanomechanical beam with a high degree of linearity between the frequency shift of the coupled modes. Ref. [18] revealed the emergence of both dispersive parametric coupling and dissipative parametric coupling in a micromechanical resonator embedded with a nanomechanical resonator. Ref. [23] realized mechanically induced transparency and mode cooling in a phonon cavity on account of strain-induced parametric coupling with a sufficient rate, simulated by solving the coupled-Van der Pol-Duffing equations numerically based on rotating frame approximation. Ref. [24] analyzed the bifurcation near the fixed points and derived the threshold of vibration coupling between the second and third modes of an electrically actuated clamped-clamped microbeam caused by geometric nonlinearity. Though the aforementioned tensioninduced parametric coupling is usually limited to clamped-clamped or thin film resonators and always negligible for most centrally anchored or bulk-fabrication-process resonators, they can provide great enlightenment for the research of other kinds of modal coupling.
With the well-known stiffness-softening characteristic, electrostatic nonlinearity often occurs in most capacitive micromechanical resonators. Due to the existence of shared capacitance in these devices, the response of any mode (oscillator) can affect the electric potential energy of the whole system, so different modes (oscillators) might interact with each other. Recently, an electrostatic-nonlinearity induced parametric modal coupling effect in a capacitive microelectromechanical ring resonator was observed [19], which also showed that the mode coupling strength can be dynamically tuned with great flexibility in the control of the coupling stiffness. This kind of nonlinear parametric coupling is regarded as a very efficient coupling and does not have topology restriction of the resonator, which is even more widespread than the tension-induced one. The following-up researches indicate that the same coupling can also be found on other kinds of capacitive resonators [25,26]. They analyzed the direction and range of the frequency shift as well as the location of the frequency hopscotch in the employed tuning fork resonator. Though the experimental results are substantial, the detailed theoretical model, more precise physical pictures, and potential intriguing applications for this electrostatic-induced nonlinear parametric modal coupling are yet to be given.
In this paper, we provide a thorough theoretical model for the electrostatic nonlinearity induced parametric modal coupling effect. The explicit expressions of the frequency responses of the coupled modes are given. Numerical analyses based on those expressions are provided as well. The electrostatically induced dispersive parametric coupling effect is perfectly simulated numerically. A novel displacement-to-frequency transduction scheme based on this electrostatic dispersive parametric coupling effect is proposed, and the transduction sensitivity is given theoretically. The ability of electrostatic dispersive parametric coupling to tune the bifurcation topology of capacitive resonators is shown by demonstrating a bifurcation reversal effect. This paper is organized as follows. In Sect. 2, the theoretical model for the electrostatic nonlinear coupling is developed, which is then solved based on the multiple time scale analysis. Section 3 analyses the electrostatic nonlinear coupling effect based on the theoretical model and a novel electrostatic dispersive parametric transduction scheme is proposed. A bifurcation reversal effect is predicted as well. Finally, this work is ended with a conclusion and outlook in Sect. 4.
2 Theoretical model for electrostatic nonlinear parametric coupling

Coupling model
To introduce the electrostatic nonlinear parametric coupling that occurs in capacitive micro-electromechanical resonators, we consider models depicted in Fig. 1. Figure 1a shows two mechanical oscillators sharing a common capacitor biased with a constant voltage of V 0 . The two oscillators can represent two distinct mechanical resonators [27], or two individual normal modes in a common resonator [19]. For the latter case, the modulation of the capacitive distance is equal to the superposition of the displacements of the two normal modes. In Fig. 1b, a flexible electrically charged micro/nano-scale rod, beam, or wire resonator [28][29][30] is placed near an oppositely charged rigid base electrode. The charge on the resonator is approximately constant q. The superposed displacements of different normal modes in the resonator modulate the capacitive distance simultaneously.
Here, we consider a two-mode system. The effective stiffness and mass of each mode are represented by k j and m j , where j = 1, 2 indicates the mode label. d 0 is the capacitive distance at equilibrium when the capacitor is not charged. X j indicates the effective displacement of mode j. For the systems that the biased voltage is kept constant in Fig. 1a, the additional electrostatic potential energy is given by [19] where A is the effective area of the common capacitor; e 0 is the permittivity of the vacuum. For the systems that the charge is kept constant in Fig. 1b, the electrostatic potential energy is given by [31] À where j is an effective parameter of order unity that is related to the dimensions and shapes of the resonator and electrode. No matter whether the common capacitor is charged with constant voltage or constant charge, its electrostatic potential energy may always contain the nonlinear terms expressed by where factor C is given by Ae 0 V2 0/2 for the constantly biased case or jq 2 /4pe 0 for the constantly charged case. For shared capacitors with more complicated geometries, we need to introduce effective parameters in (3). For example, the capacitive area A in Eq. (1) should be replaced by effective area A eff to consider the irregular capacitance, and the effective parameter j in (2) changes for different capacitance geometry. The Lagrangian of the coupled-two-modes system is given by Substituting (4) into the Lagrange's equation, we obtain the dynamical equations of motion, When the electric voltage or charge is applied, a steady electrostatic force will cause an offset for the Fig. 1 Schematics of the micro/nano-mechanical systems that exhibit electrostatic nonlinear parametric coupling. a The constant-voltage case. Two oscillators share a common capacitor biased with a constant voltage V 0 . The displacements of both oscillators will affect the capacitive distance and the direction of the arrow indicates the positive direction we artificially specify.
b The constant-charge case. A flexible electrically charged micro/nano-scale rod, beam, or wire resonator is placed in the vicinity of an oppositely charged rigid base electrode. The superposed displacements of different normal modes in the resonator modulate the capacitive distance simultaneously equilibrium position of the resonator. The new equilibrium position X Ã j of the two modes is given by We define the new displacement from the new equilibrium position x j = X j -X Ã j and denote the capacitive distance at new equilibrium The nonlinear electrostatic restoring forces in (5) and (6) can be expanded into the Taylor series to X 1 and X 2 at the new-equilibrium positions. By introducing damping and actuation terms, the equations of motion are further given by where x j = (k j -2C/m j d 3 1 indicates the angular resonant frequency considering the electrostatic-negative-stiffness effect, c j represents the damping rate of mode j. F j and x dj are the amplitude and angular frequency of the external force independently applied upon mode j. x dj is very close to x j . The above equations of motion with nonlinear coupling terms can fully describe the parametric coupling.

Multiple time scale analysis
We solve those equations using the multiple time scale method [32,33]. First, we define dimensionless variables T = x 1 t * , u = x 1 /d 0 , v = x 2 /d 0 . A dimensionless small parameter e is introduced additionally. Most applicable micro or nano-resonators usually possess high-quality factors (Q = x 1 /c 1 ) to obtain better performance. Q » 1 holds for most of the cases. Thus, we can define e = 1/Q = c 1 /x 1 . The equations of motion (8) and (9) are nondimensionalized as where D = d/dT and D 2 = d 2 /dT 2 denote the differentiation operators, The solutions of Eqs. (10) and (11) can be expressed in the forms Based on the Chain Rule, we have D = D 0 ? eD 1 , . The higherorder terms about e have been neglected. D n m (m = 0, 1, 2, n = 1, 2) denotes the n-th order differentiation operators to T m . Substituting (12) and (13) into (10) and (11), and then equating the coefficients of the like powers of e on both sides, we obtain. Order e 0 Order e 1 Order e 2 The general solutions of (14) and (15) can be written in the forms where M, N[C. The bar indicates the complex conjugation. Substituting (20) and (21) into (16) and (17) leads to where c.c. denotes the complex conjugation of the terms in front. Suppose 1 ? X 2 and 1 -X 2 are far from both 1 and X 2 , so they will not contribute to the secular terms. Eliminating the secular terms of (22) and (23) yields which means that M and N are functions of only T 2 . We can further obtain the particular solutions of u 1 and v 1 from (22) and (23) after the secular terms have been eliminated. Substituting the particular solutions of u 1 and v 1 and the general solutions (20) and (21) into (18) and (19), and eliminating the secular terms give where the detuning parameters r 1 and r 2 are introduced according to X d1 = 1 ? er 1 and X d2 = X 2-? er 2 . Based on (24) and (25), we have D 2 1 M = 0 and D 2 1 N = 0, which can be further used to simply (26) and (27).
To obtain the approximative solution, the secular conditions of the first order (24) and (25) and those of the second order (26) and (27) should hold simultaneously. We can expand those equations into the time scale of T, and combine (24) with (26) and (25) with (27), which gives We introduce polar notation to the complex amplitude of the first-order approximation where |u|, |v|, u 1 , u 2 [ R. |u| and |v| are the first-order approximations of the nondimensionalized amplitudes of the working modes. u 1 and u 2 are the first-order approximations of the nondimensionalized phases of the working modes. Substituting (30) and (31) into (28) and (29), and separating the result into real and imaginary parts, we obtain where U = er 1 T -u 1 , W = er 2 T -u 2 , and The approximations in the above expressions are made based on the assumptions that d 1 & d 0 and the electrostatic frequency tuning is much smaller than the resonant frequencies, x 2 j ) 2C/m j d 3 1 , which are true in practice. We can conclude that the third-order nonlinearity coefficients are dominant in this system. The steady-state response corresponds to d|u|/dT = d|v|/dT = dU/dT = dW/dT = 0, which further corresponds to the solutions of All the frequency responses of the coupled modes are described in (40)-(43). In this paper, we are only interested in amplitude-frequency responses. Eliminating the phase variables U and W in (40)-(43), we obtain the amplitudes |u| and |v| as implicit functions of r 1 and r 2 . Then, transforming the dimensionless parameters to the practical ones, we obtain the displacement amplitudes of the modes |x 1 | and |x 2 |, which are implicitly given by where d 1 = x d1 -x 1 and d 2 = x d2 -x 2 are the detuning parameters of the driving forces. The amplitude-frequency responses of the coupled nonlinear modes can be simulated by calculating |x 1 | and |x 2 | with different values of x d1 and x d2 . It should be noted that the results of frequency responses may contain unstable branches, which can be handled with various common methods [18,34,35]. Based on the above results, the resonant frequency of one mode can be modulated by the vibration of the other mode. The frequency shift of one mode caused by another mode can be explicitly given. We rewrite Eqs. (44) and (45) into the forms of In the square roots, the amplitudes of the coupled modes should satisfy |x 1 |B F 1 /m 1 x 1 c 1 and |x 2 |B F 2 / m 2 x 2 c 2 . When both modes are at resonance, which indicates that the amplitudes of both modes reach the maximum, the frequency shifts can be obtained by calculating the detunings d 1 and d 2 in Eqs. (46) and (47) and applying the resonance conditions |x 1 |= F 1 / m 1 x 1 c 1 and |x 2 |= F 2 /m 2 x 2 c 2 . The maximum frequency shift b d 1 of mode 1 caused by the resonance of mode 2 and that b d 2 of mode 2 caused by mode 1 are given bŷ respectively. The interaction condition of this electrostatic dispersive parametric coupling is that the coupled modes share a common biased or charged capacitor and both modes can modulate the capacitive gap of the shared capacitor. It is not affected by the frequency or phase relations between the coupled modes. However, this electrostatic dispersive parametric coupling can only reside in capacitive resonators, and it is very highly susceptible to the characteristics of the shared capacitor and the nonlinear electrostatic potential, which accordingly provide great freedom for engineering.
3 Dynamical analyses of the electrostatic dispersive parametric coupling

Frequency responses analysis
In this Section, we analyze the electrostatic nonlinear parametric coupling in typical capacitive electromechanical systems shown in Fig. 2 based on the theoretical model of (44) and (45). Figure 2a shows the schematic transient pattern of the simultaneously actuated normal modes 1 and 2 of a resonator. Their displacements affect the capacitive gap in superposing form. Dispersive parametric modal interaction occurs when a nonlinear electrostatic potential is applied [19]. Figure 2b shows a more apparent example. Two oscillators actuated independently share a common capacitor that is biased with a constant voltage V 0 can produce dispersive parametric coupling as well.
During the above-mentioned cases, the vibrational displacement of one mode (oscillator) can influence the resonant frequency of the other mode (oscillator).
Here, we consider a pair of modes (oscillators) labelled by 1 and 2 with typical parameters as follows: the resonant frequencies x 1 = 2p 9 135,000 Hz and x 2 = 2p 9 165,000 Hz, the corresponding damping rates c 1 = 2p 9 1 Hz and c 2 = 2p 9 2.5 Hz, and modal masses m 1 = 5.8 9 10 -9 kg and Fig. 2 Typical systems that can produce electrostatic dispersive parametric coupling. a Schematic transient pattern of the simultaneously actuated modes 1 and 2 of a ring resonator [19]. b Schematic pattern of two distinct mechanical oscillators actuated independently sharing a common capacitor biased with a constant voltage. Displacements of both modes (oscillators) contribute to the variation of the capacitive gap. Dispersive parametric interaction occurs when a nonlinear electrostatic potential is applied m 2 = 4 9 10 -9 kg. The electrostatic factor is given by C = Ae 0 V 2 0 =2, where V 0 = 30 V and the effective capacitive area is given by A = 5.65 9 10 -9 m 2 [19].
The two modes (oscillators) are actuated simultaneously. For the case of considering different modes in a single resonator (Fig. 2a), the co-excitation of two modes can be realized by applying two driving signals.
The frequency of each signal should be close to the resonant frequency of the corresponding mode. Each signal should be applied to electrodes located at the anti-nodal deforming parts of the corresponding mode to guarantee actuation efficiency. For the case considering different oscillators sharing a common capacitor (Fig. 2b), the co-excitation can be realized by independently actuating the two oscillators. Take Mode 1 for example, it is actuated with the efficient push-pull driving method shown in Fig. 3. Electrodes at a pair of antinodes marked Drive? are applied with driving voltage V 0 ? V d1 cos(x 1 t), providing driving force F? . And the Drive-electrodes at another pair of antiphase antinodes are applied with driving voltage V 0 -V d1 cos(x 1 t) to provide F-. The difference between F? and F-leads to a resultant driving force.
Thus, we can express the electrostatic driving force of the two modes as where j = 1, 2. This push-pull driving scheme not only enhances the driving efficiency but also eliminates the common-mode offset and higher-harmonic force terms compared to the single-driving case. A dj is the effective area of the driving capacitive electrodes of mode j. In the following analyses, we assume A dj = A for simplicity. If mode j is actuated while the other mode k is in free, which indicates that F j = 0 and F k = 0, there is no frequency response in mode k because |x k |B F k / m k x k c k = 0. Then the frequency response of the mode j is given by which is the ordinary Duffing solution [32]. The amplitude-frequency responses of the solely driven modes 1 or 2 are depicted in Fig. 4a or b, respectively. Under the influence of the electrostatic potential, both

Interaction analysis and electrostatic dispersive parametric transduction scheme
Here, we study the simultaneously actuated cases. First, we consider the condition that both modes are weakly actuated so that both of them are not driven into the bifurcation condition [36]. Assuming V d1-= 1 mV and V d2 = 3 mV, the amplitude-frequency responses |x 1,2 | of modes 1 and 2 are functions of detunings d 1 and d 2 , as shown in Fig. 5a and b, which are simulated based on (44) and (45), respectively. It is observed that the displacement of one mode will cause a frequency dispersion to another mode. The resonant frequencies of two coupled modes in simultaneously actuated case (red solid curves) are both lower than that when they are actuated alone (red dotted line).
According to the analysis results of (48) and (49), the frequency shifts reach the maximum value when both modes are at resonance, corresponding to the red point in Fig. 5. Though the modes are weakly driven, this electrostatic parametric interaction causes maximum frequency shifts of -0.21 Hz and -0.20 Hz for modes 1 and 2, respectively, which all exceed their linewidths. This electrostatic-potential-induced parametric interaction is very similar to the tensioninduced parametric interaction previously found in the clamped-clamped resonator [13]. The difference is that the frequency only shifts to lower values for this electrostatic nonlinear interaction, while it may shift to both higher [13] or lower [18] values for the tensioninduced nonlinear interaction. Next, we increase the driving voltage of one mode to excite it into nonlinear vibration, while keeping the other mode vibrating linearly. For instance, mode 2 is driven into a nonlinear region by V d2 = 18 mV, while mode 1 is still weakly driven by V d1 = 1 mV. The simulated amplitude-frequency responses are depicted in Fig. 6. If the driving frequency (detuning) of mode 2 is changed from high to low, which indicates that mode 2 lands on the higher bifurcation branch in Fig. 3b, the dispersions of modes 1 and 2 are shown as Fig. 6a and b, respectively. The intensity of mode 2 is reflected by the resonant frequency of mode 1 illustrated in Fig. 6a, which shows a maximum frequency shift of -6.1 Hz. Since mode 1 is weakly actuated, its back action to mode 2 is relatively weaker, as shown in Fig. 6b, which illustrates the higher bifurcation branches of mode 2, and a negligible shift of the resonant frequency is observed. If the driving frequency (detuning) of mode 2 is changed from low to high and mode 2 is in the lower bifurcation branch in Fig. 4b, the dispersions of the coupled modes are shown in Fig. 6c and d, showing a resonant frequency shift of -2.1 Hz in mode 1. Due to the weak back action from mode 1 to mode 2, Fig. 6d illustrates the lower bifurcation branches of mode 2, and a very small shift of the resonant frequency is observed. These simulation results are consistent with the experiments reported in Ref. [19], in which two low-frequency modes are actuated in the linear region while a high-frequency mode is simultaneously actuated in the stiffness-softening Duffing nonlinear condition. The intensity of the high-frequency mode is reflected by the decreasing of the resonant frequencies of the two low-frequency modes. The bistability of the nonlinear bifurcation of the high-frequency mode can be fully revealed by the dispersion of the low-frequency modes, similar to simulations in Fig. 6a and c. Likewise, if mode 1 is actuated into the nonlinear condition while mode 2 is driven linearly, we can obtain similar results. We consider such a case by setting V d1 = 3 mV and V d2 = 3 mV. The amplitudefrequency responses |x 1,2 | are shown in Fig. 7. The intensity and bistability of the nonlinear bifurcation of mode 1 can be revealed by the frequency dispersion of mode 2, as shown in Fig. 7b and d. While the weak back action of mode 2 to mode 1 still causes a very small dispersion of resonant frequency, as shown in Fig. 7a and c.
The aforementioned analytical results concerning electrostatic modal coupling establish the foundation for displacement-to-frequency transduction, which utilizes the frequency dispersion of one linear mode to detect the displacement intensity of the other mode. In theory, this coupling is reciprocal and each one of the coupled modes can be selected to be the monitor or objective mode. But the mode with a larger quality factor (Q = x/c) can possess lower line width and therefore higher resolution for frequency detection, so it would be more appropriate to be the monitor mode. In this case, mode 1 (Q 1 = 135 k) is chosen to be the monitor mode to detect the actuation intensity of the objective mode 2 (Q 2 = 66 k) by monitoring the frequency shift.
The theoretical model for this detection scheme is given based on the coupling model (44) and (45). Applying the resonance condition |x 1 |= F 1 /m 1 x 1 c 1 to the monitor mode 1, we have Substituting (52) into (45), we obtain frequency modulation of the monitor mode 1 (d 1 ) as a function of the frequency detuning of the objective mode 2 (d 2 ) given by By using the weak actuation assumption of monitor mode 1, |x 1 | (|x 2 |, Eqs. (52) and (53) can be further simplified to Equations (54) are equivalent to the Duffing solution (51) for j = 2. The above analysis indicates that the frequency modulation d 1 of the weakly driven monitor mode 1 is a very good estimation of the power intensity |x 2 | 2 of the objective mode 2. The detection sensitivity of this dispersive parametric transduction is given by which is calculated to be 46 Hz lm -2 for the current design. The variation range of the frequency modulation d 1 of the weakly driven monitor mode 1 is given by The frequency modulation d 1 of the monitor mode 1 as functions of the frequency detuning d 2 of mode 2 with different actuation voltages are depicted in  Fig. 8, which is calculated based on Eq. (53). Mode 1 is weakly actuated with V d1 = 1 mV. The curves in Fig. 8 represent the red solid curves in Fig. 5a and Fig. 6a, and c. The power intensity |x 2 | 2 of the objective mode 2 is reflected by the frequency modulation d 1 of the monitor mode 1. As V d2 increases to drive mode 2 into a nonlinear region, the frequency modulation d 1 shows typical bistable bifurcations similar to those of the stiffness-softening Duffing response to detuning d 2 . If the driving frequency of mode 2 is changed from high to low (low to high), the frequency modulation d 1 locates at the high (low) branch.
The electrostatic dispersive parametric coupling can be used to engineer a displacement transducer.
The key part of this transducer is a particularly designed monitor oscillator capacitively coupled to the objective oscillator. The power intensity of the objective oscillator can be measured by detecting the frequency modulation of the monitor oscillator. The dispersive parametric transduction sensitivity S given by (56) is only related to the parameters of the monitor oscillator and the coupling capacitance, which can be optimally designed to obtain higher sensitivity. The dispersive parametric transduction sensitivity (56) can be further expanded for the constant-voltage biased or Fig. 6 Simulated amplitude-frequency responses if mode 1 is weakly actuated while mode 2 is strongly actuated. a Amplitude-frequency responses |x 1 | and b |x 2 | if the driving frequency of mode 2 is changed from high to low. c and d are the results if the driving frequency of mode 2 is changed from low to high constant-charged conditions to S V and S q , respectively, which are given by where Q 1 = x 1 /c 1 and k 1 = x 2 1 m 1 are the quality factor and effective stiffness of the monitor resonator, respectively.
To have a better sensitivity, the monitor resonator with higher quality factor, lower stiffness, and lower mass should be designed. Here, we propose a physical design with optimized sensitivity of the electrostatic dispersive transduction, shown in Fig. 9. A silicon MEMS double-end-tuning-fork (DETF) resonator is Fig. 7 Simulated amplitude-frequency responses if mode 1 is strongly actuated while mode 2 is weakly actuated. a Amplitude-frequency responses |x 1 | and b |x 2 | if the driving frequency of mode 1 is changed from high to low. c and d are the results if the driving frequency of mode 1 is changed from low to high used as the monitor oscillator, and the objective oscillator is represented with a mass suspended by a spring. The longitudinal displacement of the mass can be transduced by the frequency modulation of the DEFT resonator. By employing interdigital-electrode design, one can obtain a capacitive area of the order of A & 10 -6 m 2 . The capacitive gap is designed to be 4 lm. Owing to its balanced mode shape, the DEFT resonator can have a high quality factor. If we further employ the slotted design [37,38], and stiffness-mass design to mitigate the thermoelastic dissipation [39], a quality factor of Q & 600 k could be obtained. Suppose the resonant frequency and the effective mass of this DETF resonator are x 1 /2p & 100 kHz and 100 lg, respectively. Between the monitor resonator and the mass, we apply a constant voltage of V 0 = 50 V which is less than the pull-in voltage (91 V) of the monitor resonator. Substituting the above parameters into Eq. (57), we can obtain a transduction sensitivity of 784 Hz/lm 2 . The inputoutput curve of this optimized electrostatic dispersive transductor is shown in Fig. 9b.

Bifurcation reversal effect
If the two modes are strongly excited to bifurcation states simultaneously, the characteristic of the coupling changes significantly, as shown in Fig. 10. Here, we drive mode 1 alone with a strong driving voltage of V d1 = 4 mV to obtain a stiffness-softening bifurcation state, as shown by the blue curve corresponding to V d2 = 0 in Fig. 10b. If we maintain the actuation condition of mode 1, while applying a strong drive voltage (V d2 = 20 mV) to mode 2 near its resonant frequency simultaneously, the amplitude-frequency response of mode 1 changes to the detuning d 2 .
Especially, there is a transition of bifurcation at the point d 2 = 0 from the stiffness-softening type to the stiffness-hardening type, shown as the curve i. Curve ii in Fig. 10a corresponding to the d 2 [ 0 case is depicted by the red curve in Fig. 10b, whose bifurcation is reversed compared with the original one (blue curve). The sign transition of Duffing constant from negative to positive in Fig. 10 demonstrates that by driving mode 2 with certain driving voltage and detuning, the nonlinearity of strongly driven mode 1 can be restrained and even reversed. This indicates that we can extend the input range of one mode by driving another mode on resonance at high amplitudes, revealing the possibility of linear transduction of very large amplitudes for capacitive electromechanical resonators. A similar phenomenon was observed by applying a nonlinear electrostatic field to a stiffnesshardening clamped-clamped resonator [40]. This early reported bifurcation reversal effect depends on the mode index and the amount of initial tension in a nanomechanical resonator. Here, we demonstrate that the bifurcation reversal can be realized by the pure coupling method, which is independent of the mode index.

Discussion and conclusions
Here, we give a comprehensive comparison of this electrostatic dispersive parametric coupling with the other typical nonlinear modal coupling effects, as shown in Table 1. Unlike the internal resonance related modal interaction, the electrostatic nonlinear dispersive coupling is not related to the frequency relation between the coupled modes. Besides, it is not related to the phase relation between the coupled modes either. The influence factors of this electrostatic dispersive parametric coupling include the electrostatic potential induced cubic nonlinearity and the superposed displacements affecting the capacitive gap. If the shared capacitor is located at the node of any mode, the electrostatic dispersive parametric coupling can no longer exist. We cannot tell from the downward-shifted frequency that it is an electrostatic dispersive parametric coupling, because the negative cubic nonlinearity may also make the mechanical dispersive parametric coupling act like that [18]. However, the electrostatic nonlinear dispersive coupling is closely related to the electrostatic potential, which can be in-situ regulated by tailoring the biased voltage or charge on the shared capacitor. Moreover, the electrostatic dispersive parametric To sum up, this study gives the theoretical model for the electrostatic dispersive parametric coupling in detail. The solutions of the coupled nonlinear equations are obtained based on the multiple-time-scale analysis. By analyzing the amplitude-frequency responses of the coupled model, the dispersive parametric coupling effect is simulated. Based on this dispersive parametric coupling, a novel transduction scheme is explored. The sensitivity of this dispersive parametric transduction is given explicitly, which is only related to the design of the monitor resonator and the coupling capacitor and can be specially engineered according to the requirement of the sensitivity. This novel displacement-to-frequency transduction scheme based on the electrostatic dispersive parametric coupling is different from the existing counterpart that is based on tension modulation [41,42], which can provide even more design freedoms. This electrostatic dispersive parametric transduction scheme has the potential to be used for resonant accelerometers, gyroscopes and other MEMS devices. Moreover, the strongly actuated dispersive parametric coupling reveals the ability to tune the bifurcation topology of capacitive resonators, and a bifurcation reversal effect is predicted. Future studies can investigate the more detailed mechanism and experimental observations of the bifurcation reversal caused by electrostatic-fieldinduced modal interaction.  [13,18] Mechanical dissipative parametric coupling [18] Internal resonance [12] Manifestation The intensity of one mode will make the resonant frequency of the other mode shift to a lower value The intensity of one mode will make the resonant frequency of the other mode shift to higher [13] or lower [18] values The intensity of one mode will make the quality factor of the other mode shift to a lower value