Research on Nonlinear Dynamics of Drive Mode in Z-Axis Silicon

A novel drive method is proposed to increase the vibration amplitude of drive mode of silicon vibratory gyroscopes based on the nonlinear dynamics research made in this paper. A nonlinear dynamic equation, considering the nonlinear mechanical-elastic and electrostatic force, was firstly deduced according to the simplified structure schematic of drive mode. Then, experiments were made to characterize the nonlinear amplitude-frequency response characteristic as well as its dependence on drive voltage. Besides, by solving the nondimensional nonlinear dynamics using Matlab and Simulink, the influence of initial status of drive mode, that is, the displacement and the velocity, on the stable vibration amplitude was presented. The amplitude-frequency characteristic was also analyzed by simulation. The result of simulation shows good agreement with that of experiments. When nonlinearity is present in the amplitude-frequency characteristic, there is a wide flat regionwith larger amplitude than the peak amplitude obtained in the linear characteristic. Finally, a novel drive method was presented to excite the nonlinear vibration of large amplitude in the downwards-sweeping characteristic curve. It is numerically proved that this merit can be utilized to enlarge the drive mode’s vibration amplitude so as to enhance silicon gyroscopes’ performance.


Introduction
As part of a new generation of inertial sensors, silicon microgyroscopes, used for measurement of angular rate or angle of a rotating carrier, have a lot of merits, such as low power consumption, small volume and weight, high shock resistivity, and suitability for batch fabrication [1][2][3][4][5].However, the performance of silicon microgyroscopes is lower than conventional mechanical, rotating wheel, fiberoptical, and ring laser gyroscopes [6].Thus, it is necessary to improve the performance so as to broaden their application market.
Large vibration amplitude of drive mode is one of the conditions for silicon gyroscopes to achieve high performance [7,8].Therefore, capacitive actuators of more combs and smaller gaps are designed in microgyroscopes so as to increase the vibration amplitude when the same drive voltage is applied.However, when gaps become small and the vibration amplitude gets large, the nonlinear terms of the force applied on proof mass can become sufficiently high to bend the amplitude-frequency characteristic curve.For the traditional drive methodology, drive mode works at the natural frequency in the linear amplitude-frequency characteristic curve.However, when the amplitude-frequency characteristic curve is nonlinear, the drive methodology for the drive mode of silicon gyroscopes is not yet noticed.
Nonlinear dynamics of MEMS resonators has been studied analytically, numerically, and experimentally by some institutions [9][10][11][12][13][14].In these literatures, dynamic models are constructed to match the theoretical characteristic curves with the experimental phenomena of real resonator samples [11,12].The models can be solved by different methods, such as the theoretical method of multiple scales [10], the semianalytical harmonic method [9], the numerical integration method [9][10][11], and the electrical equivalent-circuit modelling method [14].It is known that the nonlinear phenomena refer to the spring softening and hardening phenomena in the amplitude-frequency curves.In capacitive MEMS resonators, the phenomena mainly result from the nonlinear mechanical-elastic reaction force when the mechanical support system is out of its linear work range, as well as the nonlinear electrostatic actuation force produced when the squeeze gaps of capacitors are small.As the drive mode of gyroscope is considered as a resonator, some more research is made in this paper on the nonlinear characteristics of the gyroscope prototype designed by our research team.
Based on the nonlinear amplitude-frequency characteristic, a novel drive method is provided to make drive mode vibrate at large amplitude.The outline of the paper is as follows.First, in Section 2, the nonlinear dynamics is derived based on the simplified schematic of the drive mode.Then, in Section 3, the experimental amplitude-frequency curves are obtained by using a detect circuit and sweeping the actuation frequency increasingly and decreasingly, separately.Besides, the nonlinear phenomena are also illustrated by solving the dynamics numerically using the Matlab and Simulink model in Section 4. Finally, in Section 5, a novel drive method is proposed for actuating the vibration of large amplitude in the nonlinear amplitude-frequency characteristic curve.

Dynamics of Drive Mode
2.1.Structure Schematic for Drive Mode.In the -axis silicon microgyroscope designed by our research team, the proof mass is supported by straight beams, as well as U-shaped beams, and driven and detected by capacitors of comb fingers [15][16][17].The dimensions of the gyroscope sample, fabricated by SOG process, are around 8 mm × 8 mm × 60 m and the SEM photos of beams and capacitors are displayed in Figure 1. Figure 1 shows the simplified structure schematic for the drive mode of the microgyroscope, consisting of the proof mass, comb actuators, sense combs, and supporting beams.As seen in the graph, Beam 1 is a fixed-guided straight beam, and Beam 2 and Beam 3 are the parallel and the serial combination of Beam 1, respectively.The proof mass is displaced by the electrostatic force produced when differential voltage is applied on drive electrodes.Since system's stiffness in the sense direction (-axis) is much larger than that in the drive direction (-axis), the motion of drive mode is simplified as 1-DOF translation in the -axis, as displayed in (1). is the displacement of the proof mass along the drive direction,   and   are the electrostatic force and the mechanical-elastic reaction force, and   =   ẋ is the damping force with   being the damping coefficient: (1)

Mechanical-Elastic Reaction Force.
In Figure 1, as the displacement of the proof mass in the drive direction is , the deformation of the straight beam in Beams 3 is /2.Since the equivalent stiffness of Beam 2 is twice the stiffness of Beam 1, the assumption is made that the deformation in Beam 1 is twice that in Beam 2. Thus, the deformation of the straight beam in Beams 1 and 2 is 2/3 and /3, respectively.The elastic reaction force applied on proof mass by all Beams 2 is approximated through a cubic polynomial relation, as displayed in (2) [10], where  is the elastic modulus of silicon and , ℎ, and  are the width, depth, and length of a straight beam.Similarly, the reaction force produced by all Beams 3 is shown in (3).Thus, the total mechanical-elastic reaction force applied on the proof mass in the drive direction is deduced in (4), where  linear is defined as the linear mechanical stiffness and  non-linear as the quadratic mechanical stiffness coefficient.Consider  vibrate only in the -axis.Then, the electrostatic force applied on the proof mass along the -axis is derived in (5) [5]. ℎ is generated by the overlapped areas with a slide gap of  and   is generated by overlapped areas with a squeeze gap of  2 .It is easily noticed that  ℎ is independent of  while   has a nonlinear relationship with it.Thus, the nonlinear electrostatic force applied on proof mass is relevant to the squeeze gap  2 .
2.4.Nonlinear Dynamics.Substituting ( 4) and ( 5) into (1), the nonlinear dynamics for drive mode is displayed in (6), which can be simplified as (7) when   is much larger than   .Based on (7), the natural frequency of drive mode is approximated in (8), where  linear = 6ℎ 2  / 2 2 is the linear electroelastic stiffness and  2non-linear = 12ℎ 2  / 5 2 is the quadratic electroelastic stiffness coefficient.It is easily seen that the natural frequency increases with the mechanical-elastic stiffness and decreases with the electroelastic stiffness.Besides, the mechanical-elastic and electroelastic stiffness both cause the nonlinearity between the natural frequency and the displacement.The nondimensional differential dynamics, as seen in ( 9), is obtained from ( 6) by the replacement of  = / 2 ,  = ω  = 2 f , f = (1/2)√ linear /  , and   = 2 f ⋅   /  .The nondimensional dynamics is used in the following numeric analysis:

Experimental Characterization
3.1.Experimental Setup.As presented in Figure 3, the carrier voltage   =   sin 2   is applied on the anchor, that is, the moving combs, and two differential sense electrodes are connected with their own preamplifiers.The current,  1 and i s2 , is integrated and the output voltage,  + and  − , passes through the differential amplifier.The process is shown as below: Since the squeeze gap in the sense combs is much larger than that in the drive combs, the sense capacitance between the squeeze faces is ignored and the approximate sense capacitance change is proportional to the displacement ; that is, Δ  = (  /).Thus, if  =  sin(2  +), the output voltage is Then,   is filtered by a high-pass filter and displayed by a spectrum analyzer on which the vibration information of drive mode is at twice the actuation frequency.The practical experimental setup is illustrated in Figure 4. Through the probe station, a packaged gyroscope sample is connected with the detect circuit.During testing, the actuation frequency of the carrier voltage is consecutively tuned by a step of 0.1 Hz increasingly or decreasingly between 3310 Hz and 3330 Hz, and the corresponding amplitude information is recorded by acquiring the value at twice the actuation frequency on the analyzer screen.

Experimental Results.
When   = 5 V,   = 0.5 V, and   sweeps up slowly from 3310 Hz to 3330 Hz, the frequency characteristic curve of the drive mode is fitted in red, as shown in Figure 5, using the amplitude data acquired at each frequency step.Similarly, the characteristic curve, while frequency sweeps down, is fitted in black.It is noticed that when the actuation frequency sweeps down, the natural frequency is around 3313 Hz, much lower than that obtained during upwards sweeping, that is, around 3319 Hz.Both frequency response curves bend towards lower frequency.
Experiments are also made on the influence of applied voltage on the nonlinearity of frequency response.In Figure 6, when   increases from 5 V to 6 V, the natural frequency of upward-sweeping reduces from 3319 Hz to 3316 Hz and that of downward-sweeping decreases from 3313 Hz to 3308 Hz.Thus, increasing   makes the spring softening phenomena more evident.Similarly, as seen in Figure 7, increasing   also aggravates the nonlinear softening characteristic.Frequency (Hz)

Numerical Characterization
4.1.Simulink Model.Matlab and Simulink are used to determine the motion of the proof mass as well as its frequency response characteristic.Parameters, which are given in Table 1, are firstly defined in the Matlab code and then transferred to the Simulink model for simulation.Figure 8 gives the Simulink model, in which an S-Function module is used to construct (9), differential voltage is applied, and the response is monitored by oscilloscopes.By simulating the model, the stable solution of displacement and velocity is obtained.Besides, the amplitude-frequency characteristic curve is determined by fitting the stable vibration amplitude acquired at each actuation frequency.This model is also used to determine and compare the response under different initial status and voltage loads.Here, the actuation frequency   is swept in two ways.Within the frequency range of interest, one sweeping method is to increase frequency consecutively by a certain step and the other is to decrease it by the same step.It must be pointed out that, at each frequency sweeping step, the initial status of the displacement and velocity as well as the phase of applied voltage is maintained equal to the value at the finish time of the previous frequency step.

Influence of Initial Status. When drive voltage is applied
with   = 5V,   = 0.5 V, and   = 3315 Hz, the phase plane portrait of ( 9) is illustrated in Figure 9, in which the arrow tells the progressing trend of the nondimensional displacement and velocity along with time [18].Besides, three   curves in the graph refer to the equivalent transient response trajectories under different initial status.It is observed that the nondimensional stable amplitude of displacement is around 0.1 when initial status is ( 0 ,   0 ) = (0.20, −0.20), while it is around 0.85 when ( 0 ,   0 ) = (0.00, 0.92).In addition, the trajectory is divergent when ( 0 ,   0 ) = (0.20, 0.99).It is easily concluded that the stable motion of drive mode is dependent on the initial motion status of the proof mass when voltage is applied as here.

Amplitude-Frequency
Response.The Simulink model in Figure 8 and the parameters in Table 1 are reused to obtain the characteristic curve of frequency response while voltage     is applied as   = 5V,   = 0.5 V, and the actuation frequency   is increased and decreased consecutively by a step size of 0.5 Hz between 3305 Hz and 3330 Hz.Besides, the initial status at the start frequency is ( 0 ,   0 ) = (0, 0).The corresponding stable vibration amplitude at each frequency step is collected to fit characteristic curves.
Figure 10 gives the simulated characteristic curves, in which the pink solid line and the blue dashed line refer to the responses while frequency sweeps up and down, respectively.It is noticed that two natural frequencies  up and  down appear in frequency response curves.The curves both bend towards lower frequency, manifesting high nonlinearity.During sweeping up, when the actuation frequency gets near  up , the amplitude rushes to the peak  up and then descends gradually.Similarly, during sweeping down, the amplitude ascends slowly to the peak  down and then drops dramatically at the frequency  down .It is also seen that two natural frequencies are different with  down <  up < f , and their corresponding peak amplitude is not the same with  down >  up .Besides, when   = 3315 Hz, the obtained downwards-sweeping amplitude is over 7 times larger than the upwards-sweeping amplitude  and even larger than  up .In the downwards-sweeping curve, there is a flat region with relatively large amplitude before  down is reached.Thus, the nonlinear downwards-sweeping amplitude-frequency characteristic curve is very attractive for performance enhancement of silicon gyroscopes.By using the same Simulink model, some investigation is made on the dependence of the amplitude-frequency characteristic on applied voltage.Simulations are made by changing parameters   and   separately.The simulated results are shown in Figures 11 and 12.When   and   increase, the two natural frequencies obtained by different sweeping methods both descend but their corresponding peak amplitude ascends.Besides, the decrement of  down is much larger than that of up .It can be concluded that the frequency response nonlinearity can be adjusted by   and   and the flat region of the downwards-sweeping curve grows larger when nonlinearity is intensified.

Excitation of Nonlinear Vibration
According to Figure 9, when the initial status of vibration is ( 0 ,   0 ) = (0.00, 0.92) and differential voltages 0.5 sin 6.2737 + 5 and 0.5 sin 6.2737 − 5 are applied on drive combs, the vibration of the drive mode will be stabilized at relatively large amplitude, which is especially attractive for improving the scale factor of silicon gyroscopes.Therefore, Figure 13 illustrates how to excite the nonlinear vibration with larger amplitude.A triangular impulse is firstly loaded to start the motion of drive mode and the vibration displacement and velocity are recorded.As shown in Figure 14, the impulse intensity is adjusted to make the motion status ( 0 ,   0 ) = (0.00, 0.92) appear at some time.Thus, the differential drive voltage is loaded at this time.The nonlinear response, as illustrated in Figure 15, is obtained by simulating the model in this way.It is noticed that when   changes consecutively from 3315 Hz and 3314 Hz to 3316 Hz, the stable displacement amplitude varies from 0.848 and 0.867 to 0.823 and the phase relative to the normalized sine voltage is maintained around −108.6 degrees.
According to the numerical and experimental characteristics, the nonlinear vibration with larger amplitude can be actuated by decreasing the drive frequency consecutively with a certain frequency step and time interval.Thus, the nonlinear vibration in the downward sweeping curve can be excited by monitoring or gradually accumulating the status of vibration.Finally, the novel drive methodology makes drive mode vibrate at large amplitude and it is robust with no drops of amplitude and phase when the drive frequency fluctuates.

Discussion
When differential voltages   sin 2  +  and   sin 2  −   are applied on drive stators with   = 5 V and   = 0.5 V, the amplitude-frequency characteristic curves obtained by experiments and simulations are in good agreement.It is displayed that the frequency characteristic curves bend towards lower frequency with the natural frequency much smaller than the initial value.Besides, the peak vibration amplitude and the natural frequency, obtained by different sweeping methods, are not the same.
According to (8), it is known that  linear and  2non-linear lead to reduction of the natural frequency   while  non-linear raises it.In addition,  2non-linear makes the frequency response curves bend towards lower frequency while  non-linear makes them bend towards higher frequency.Thus, the spring softening phenomena in Figures 5 and 10 imply the relationship between the nonlinear stiffness coefficients in (8); that is,  2non-linear >  non-linear .
Based on the definition of  linear and  2non-linear , increasing   or decreasing  2 makes electroelastic stiffness larger, leading to more evident spring softening phenomena.Besides, according to (8), the nonlinear stiffness is also increased by displacement enlargement.Thus, increasing   and   , which makes the vibration amplitude of the proof mass larger, aggravates spring softening phenomena.The same influence of applied voltage on the amplitude-frequency characteristic is indicated by the experimental results in Figures 6 and 7 as well as the numerical analysis in Figures 11 and 12.
In the nonlinear amplitude-frequency characteristic curves, though the vibration amplitude and the natural frequency are multivalued, the flat region of downwardssweeping curves with extremely large amplitude is useful for improving gyroscope's performance, such as sensitivity and robustness.
It is numerically proved that the vibration of the drive mode ends with different stable amplitude under different initial motion status, when applied voltage is large enough.Thus, the nonlinear vibration with larger amplitude can be excited by loading drive voltage at the specific status or decreasing the drive frequency gradually by a certain frequency and time step.The obtained amplitude in the downward sweeping curve is much larger than the amplitude in the linear frequency response curve.Besides, the amplitude and the phase of vibration are less influenced by the fluctuation of the actuation frequency.It can be concluded that this novel drive methodology is beneficial for performance improvement of silicon vibratory gyroscopes.

Conclusion
In this paper, the nonlinear frequency softening characteristic of the drive mode in the -axis silicon microgyroscope designed by our research team is determined by analytical, experimental, and numerical methods, results of which show good agreement qualitatively.It is illustrated that the spring softening phenomena of the amplitude-frequency characteristic appear when applied voltage and the vibration amplitude are large enough.Besides, it is also proved that applied voltage plays a significant role in the nonlinear vibration of the gyroscope, so the nonlinearity can be tuned by drive voltage.
Moreover, under nonlinear vibration, the wide flat region with large amplitude in the downwards-sweeping amplitudefrequency characteristic curve is attractive for improving gyroscope's performance.The nonlinear vibration of large amplitude can be excited by applying drive voltage at a certain motion status or gradually decreasing the actuation frequency.Besides, the large amplitude and the phase of vibration are maintained when the actuation frequency fluctuates.Thus, for silicon microgyroscopes, it can be a promising drive method to make the drive mode vibrate at the large amplitude in the flat region of the nonlinear amplitudefrequency characteristic curve.

Figure 2 :
Figure 2: One group of comb fingers.

Figure 3 :
Figure 3: Schematic of signal-detecting interface circuitry for drive mode vibration.

Figure 5 :
Figure 5: Measured amplitude-frequency characteristic of drive mode.

Figure 6 :
Figure 6: Measured dependence of amplitude-frequency characteristic on DC bias   .(a) Frequency sweeps up.(b) Frequency sweeps down.

Figure 7 :
Figure 7: Measured dependence of amplitude-frequency characteristic on amplitude of AC voltage   .(a) Frequency sweeps up.(b) Frequency sweeps down.

Figure 10 :
Figure 10: Amplitude-frequency characteristic curve of drive mode.

Figure 11 :
Figure 11: Dependence of amplitude-frequency characteristic on DC bias   .(a) Frequency sweeps up.(b) Frequency sweeps down.

Figure 12 :
Figure 12: Dependence of amplitude-frequency characteristic on amplitude of AC voltage   .(a) Frequency sweeps up.(b) Frequency sweeps down.

Figure 13 :
Figure 13: Systematic schematic for exciting the nonlinear vibration of drive mode.

Figure 14 :
Figure 14: Response of drive mode when a single triangular pulse strike is loaded.

Figure 15 :
Figure 15: Nonlinear vibration of drive mode when differential drive voltage is loaded.
sin 2   +   and   sin 2   −   relative to the proof mass, in which   is the DC bias,   is the amplitude of AC voltage, and   is the actuation frequency.N groups of comb fingers are attached to one drive stator.Figure2illustrates one group of comb fingers, made up of one moving comb and two fixed combs.As explained in Section 2.1, the moving combs are assumed to

Table 1 :
Structural parameters for drive mode of silicon gyroscope.