Tuning the primary resonance of vibrating beam micro-gyroscopes based on piezoelectric actuation and multiple nonlinearities

In this paper, the nonlinear dynamic characteristics of statically piezoelectric actuated vibrating beam micro-gyroscopes are studied. The comprehensive nonlinear model including curvature, inertia and electrostatic force nonlinearities is considered. In the research of electrostatic micro-gyroscopes, it’s a novel way to tune the primary resonance by piezoelectric actuation and multiple nonlinearities. The multiple scales method and numerical continuation technique are used to characterize the frequency-amplitude and force-amplitude responses of the micro-gyroscopes. The effect of varying the size-dependent, fringing field, statically piezoelectric voltage and nonlinear curvature and inertia on the dynamic response of the micro-gyroscope is investigated in detail. The frequency-response results show that small vibrations produce a symmetrical frequency response curve in sense direction while the system actually has a significant softening characteristic in drive direction. The nonlinear multi-value problem effectively reduces in sense direction under the size-dependent effect, which plays an important role in the design of detection instruments for micro-gyroscopes. Choosing a positive piezoelectric actuation voltage will obtain a higher sensitivity. Increasing the curvature nonlinearity and reducing the inertial nonlinearity of the gyroscopic system will help the micro-gyroscope obtain better sensitivity, and may eliminate multi-valued responses as much as possible. actuation.


Abstract
In this paper, the nonlinear dynamic characteristics of statically piezoelectric actuated vibrating beam micro-gyroscopes are studied. The comprehensive nonlinear model including curvature, inertia and electrostatic force nonlinearities is considered.
In the research of electrostatic micro-gyroscopes, it's a novel way to tune the primary resonance by piezoelectric actuation and multiple nonlinearities. The multiple scales method and numerical continuation technique are used to characterize the frequencyamplitude and force-amplitude responses of the micro-gyroscopes. The effect of varying the size-dependent, fringing field, statically piezoelectric voltage and nonlinear curvature and inertia on the dynamic response of the micro-gyroscope is investigated in detail. The frequency-response results show that small vibrations produce a symmetrical frequency response curve in sense direction while the system actually has a significant softening characteristic in drive direction. The nonlinear multi-value problem effectively reduces in sense direction under the size-dependent effect, which plays an important role in the design of detection instruments for micro-gyroscopes.
Choosing a positive piezoelectric actuation voltage will obtain a higher sensitivity.
Increasing the curvature nonlinearity and reducing the inertial nonlinearity of the gyroscopic system will help the micro-gyroscope obtain better sensitivity, and may eliminate multi-valued responses as much as possible.

Introduction
In recent years, the investigation and development of micro-inertial sensors based on MEMS technology has been received extensive attention of many researchers [1].
Micro-inertial sensors are widely used in various fields of engineering such as automotive, air vehicles, aerospace, robotics, military systems and consumer electronics [2]. They are more attractive and highly applied due to their many notable advantages on low manufacturing cost, high durability, small size, and low-power [3,4]. As a key component of micro-inertial sensor, vibrating beam micro-gyroscope has been proposed in tracking orientation, guiding direction, and controlling path. A great deal of design models and performance investigations of vibrating gyroscopes including different types, such as forks, beams, and shells, have been published in several studies [5][6][7][8][9]. The main objective of these vibrating micro-gyroscopes is to measure angular rate or angle via detecting Coriolis force of the system.
The actuation and transduction principles of vibrating beam gyroscopes commonly used in previous literatures are electrostatic and piezoelectric. Based on a simply supported piezoelectric beam gyroscope, Yang and Fang [10] analyzed the parameter performance of the system sensitivity. By considering the cantilever beam gyroscope, Li et al. [11] found the optimal combination of geometric parameters for the doubleresonant condition with maximum sensitivity. By analyzing the flexural-torsional vibrations of a piezoelectric beam gyroscope, Bhadbhade et al. [12] studied its gyroscopic effect due to the angular speed of the system by using an assumed mode expansion method. Based on a cantilever beam attached to a proof mass at its free end, Esmaeili et al. [13] used Hamilton principle derive a general 6-dof frequency equation of an electrostatic micro-gyroscope without taking the effect of rotary inertia into account. They utilized a linear approximation method to denote electrostatic forcing and studied the effect of mass and acceleration of the vibratory gyroscope. An electrostatic micro-gyroscope with AC voltage to excite the primary vibration and Coriolis force caused by rotational speed to generate the secondary vibration has been developed by Ghommen et al. [14]. They proposed a closed-form solution to investigate static deflection and linearized amplitude-frequency response in both drive and sense directions of the electrostatic micro-gyroscope. Using the same model, Nayfeh et al.
[15] presented a novel differential frequency-domain method to measure rotational speed of the electrostatic micro-gyroscope. Lajimi et al. [16] derived an improved mathematical model of a rigid-body electrostatic micro-gyroscope to correct the existent simplified model of a beam-mass micro-gyroscope which examined by Rasekh and Khadem [17].
A lot of researchers have introduced the electrostatic force nonlinearity on the nonlinear response of the vibrating micro-gyroscope due to its important impact. Based on the multiple scales method, Ghommem et al. [18] analyzed the nonlinear dynamics of the reduced-order model of the vibrating beam micro-gyroscope. By using differential quadrature and finite difference methods, Ghommem and Abdelkefi [19] studied the primary resonance of the nonlinear vibrating beam micro-gyroscope for varying grain sizes of nanocrystalline materials and several types of electric actuation conditions. Based on the assumed mode method, the nonlinear dynamics such as primary resonance and mechanical-thermal noise of eccentric micro-gyroscopes are investigated by Lajimi et al. [20][21][22]. However, few literatures specifically examined the effect of nonlinear curvature and inertia on the dynamic vibrations of the vibrating beam micro-gyroscope. By considering nonlinear curvature and inertia of the micro/nano gyroscopes, Mojahedi et al. [2,3,[23][24][25] mainly studied the static deflection and pull-in instability analysis, but did not investigate the multiple nonlinearities on nonlinear dynamics of the vibrating beam micro-gyroscope.
Previous studies have shown that the incorporation of the size-dependent effect in the micro-structure system has a significant influence on the dynamic response [26].
Ghayesh et al. [27] developed a mathematical model of the vibrating beam gyroscope and presented that the size-dependent has a notable influence about the dynamic response by using the modified couple stress theory. Meanwhile, the fringing field effect on the pull-in instability is investigated by Ghommem and Abdelkefi [4].
However, they did not analyze the fringing field influence on the dynamic characteristic of electrostatic micro-gyroscopes. Therefore, the overall impact of the size-dependent and the fringing field on the nonlinear dynamic response should be studied for the electrostatic micro-gyroscope.
Axial excitation will be introduced by using piezoelectric layers bond to the four surfaces of the cantilever beam. By applying synchronous periodic voltages to the upper and lower piezoelectric layers, the tensile mechanical stress or compressive stress will be generated along the axial of the beam [28,29]. For a lot of beam models, such as axially moving beams, accelerating beams, periodic structures and conveying fluid pipes, Yang et al. [30][31][32][33][34][35][36][37] investigated the stability in parametric resonance and nonlinear dynamics. By using this method to produce axial excitation of a micro-beam, Azizi et al. [38][39][40][41][42][43] investigated many dynamic responses such as stability, bifurcation, and primary resonance of the piezoelectrically actuated beam under different boundary conditions. However, the beams which they studied were all stationary, without rotational speed. By using a thin layer of piezoelectric film (PZF) encircling the circumferential surface of the ring, Liang et al. [44] found that the attached PZF can enhance the sensitivity of the MEMS ring gyroscope.
A vibrating beam micro-gyroscope model with considering multiple nonlinearities, the size-dependent effect and the fringing field effect is developed in this work. The partial differential governing equations with axial excitation of the vibrating microgyroscope are derived via the extended Hamilton's principle in both drive and sense directions. Then the Galerkin technique is used to truncate the partial differential governing equations to ordinary differential equations [45]. The frequency/force amplitude response curves of the vibrating beam micro-gyroscope are studied by utilizing the multi-scale method and numerical continuation technique. The effect of the size-dependent, fringing field, nonlinear curvature and inertia, and piezoelectric DC voltage on nonlinear behaviors of the micro-gyroscope is investigated. It is found that nonlinear multi-value problem is obviously reduced in sense direction due to the effect of size-dependent. A higher sensitivity can be obtained by choosing a positive piezoelectric actuation DC voltage to the four piezoelectric layers which surrounded the cantilever micro-beam. The influence of nonlinear curvature and inertial on the dynamic response of the vibrating beam micro-gyroscope is also investigated in detail. The micro-cantilever beam is attached to a tip mass M at its free end and surrounded piezoelectric layers on its four surfaces under a base rotation Ω along x axis as shown in Fig. 1. The deformation of the piezoelectric micro-beam is described by means of two transverse displacements v(x, t) and w(x, t) respectively along sense and drive directions. The same VDC1 voltages are applied to two fixed electrodes in sense and drive directions. Furthermore, the tip mass is driven by a VAC·cos(ω) voltage only in drive direction, which produces a secondary vibration in sense direction due to the angular speed. Especially, the fixed VDC2 voltages are applied to the four piezoelectric layers, which will produce axial excitation on the micro-beam due to the piezoelectric

Modeling
( ( )) 22 and ρ is the mass density, I is cross sectional second moments of area By considering the effects of the size-dependent [27] and μ is the shear modulus, l is the length-scale parameter, ε0 is the dielectric constant, bM and hM are the width and thickness of the tip mass, e31 is the piezoelectric constant, βF is the fringing field parameter, (Aw, Av) are the areas and (dw, dv) are initial gap distances of the drive and sense capacitors, respectively.
Applying the Hamilton principle to the total kinetic and potential energy, we can obtain the flexural-flexural differential equations of motion in both sense and drive directions as follows: By introducing the following non-dimensional quantities where V0 is the unitary voltage.
Using the non-dimensional quantities, then one may obtain the equations of motion governing the transverse vibrations in its nondimensional form In Eqs. (9) and (10), η represents the length-scale parameter of size-dependent effect and βF denotes the parameter of fringing field effect; the same DC voltages VDC2 are applied to the four piezoelectric layers; k denotes the coefficient of nonlinear curvature and inertia and the nonlinearities in curvature and inertia are not considered as k=0.

Reduced-order model of the micro-gyroscope
The solutions of the gyroscopic system can be decomposed into static and dynamic and wd(x,t) represent the static and dynamics components in sense and drive directions, respectively. By expanding the right hand side of Eqs. (9) and (10) where r(x) is the rth mode shape of a cantilever beam, qr(t) and pr(t) are the generalized coordinates of sense and drive directions, respectively.
Furthermore, by using the single-mode approximation [16,21,22] and setting dv=dw and Av=Aw, the parameter d equals to 1 and αv=αw, so the 2DOF nonlinear ordinary differential equations are expressed as where   (1) ,

Application of multi-scale method for nonlinear micro-gyroscopes
The multi-scale method has been widely applied in the investigation of the nonlinear micro-gyroscope. The nonlinear gyroscopic Eqs. (13) and (14) are studied by using the procedure of the multi-scale method. Then, the solutions of Eqs. (13) and (14)

q q T T T q T T T q T T T p p T T T p T T T p T T T
where ε is a small parameter, and T0 = t, T1 = εt and T2 =ε 2 t are different time scales.
Damping c, angular speed Ω and voltage VAC are scaled with ε 2 c, ε 2 Ω and ε 3 VAC since they are relatively weak. The time derivatives are expressed as ( 2 ) ...
Order (19)     The general solutions of Eq. (18) are supposed as   q  T  T  T  A  T  T  e  A  T  T  e   p  T  T  T  A  T  T  e  A  T  T  e                 (21) where ωp and ωq are natural frequencies corresponding to the drive and sense directions, respectively.  (22) In order to eliminate the secular terms in the preceding equations, the solvability conditions of the terms are set to zero. Inspection of the results one may find Aq (T1,T2) and Ap(T1,T2) and their complex conjugates to be function of only T2:    (29) where aq(T2), ap(T2) and γq(T2), γp(T2) are real amplitudes and phases of the responses, respectively.

D q q i D A T T i D A T T A T T A T T A A
By equating the coefficient of     2  2 ) ) ..., where G1, G2, H1 and H2 are obtained by Eq. (25) and aq, ap, γ1 and γ2 are given by Eq. (30).
In the calculation of this paper, by considering the geometry and material properties of the micro-gyroscope as follows [4,14,27]

Effects of the size-dependent and the fringing field
In this section, the effects of the size-dependent and the fringing field on the frequency-amplitude and force-amplitude responses of the vibrating beam microgyroscope are investigated. The curves of both drive and sense directions are plotted in both with size-dependent (η=0.6913) and fringing field (βF=0.65).
As can be seen from Fig. 2, only one external excitation force is given to drive direction, the system will cause vibrations in sense direction since the gyroscopic effect produced by Coriolis force. The bifurcation points of the stable solution and unstable solution in Fig. 2 indicate that the system has reached the limit points. It can be seen that the frequency of the system is the same value in drive and sense directions as the bifurcation occurs. Fig. 2a shows that the point where the maximum amplitude appears in drive direction is the above bifurcation point. Fig. 2b illustrates that the point where the maximum amplitude appears in sense direction is when the small parameter σ1 is equal to 0, that is, the external excitation frequency is equal to the frequency in drive direction. The nonlinear response in drive direction always exhibits a softening characteristic, while the nonlinear response in sense direction describes a multi-valued effect as the excitation frequency less than drive natural frequency. When only the sizedependent is considered, it can be seen that the amplitude in drive direction is reduced as shown in Fig. 2a. Although the amplitude in sense direction does not change much, the size-dependent effectively reduces the nonlinear multi-value problem which plays an important role in gyro detection system. It was further found that, only considering the size-dependent, the softening characteristics of the system are weakened. The sizedependent has little effect on the sensitivity of the system dynamic response as depicted in Fig. 2b. It can also be seen from Fig. 2 that when only the fringing field effect is considered, the maximum amplitude in drive direction increases. Although the amplitude in sense direction does not change much, it increases the nonlinear multivalue problem as shown in Fig. 2b. It was further found that, only considering the fringing field, the soft characteristics of the system in drive direction are enhanced as shown in Fig. 2a. Therefore, if the effects of the size-dependent and the fringing field are ignored, the micro-gyroscope measurement will be inaccurate. Under the effects of the size-dependent and the fringing field, Fig. 3 describes the force-amplitude response curves in both drive and sense directions. It is known from Fig. 2 that the multi-valued nonlinear response of the system occurs when the small parameter σ1 is less than 0, so σ1 = -0.1 is taken. It can be seen from Fig. 3 that the trend curves of four types parameters of force amplitudes in both drive and sense directions are the similar, except that the values in sense direction are much smaller than the values in drive direction. As the amplitude of the externally excited voltage increases, the maximum amplitude of the gyroscopic system gradually increases. Within a certain interval (such as [0.1-0.2]), the system appears a multi-valued response, including two stable solutions and an unstable solution. Continuing to increase the amplitude of the external excitation voltage, the system reaches a single value again, and the larger amplitude of the external excitation, the higher the sensitivity.
It can be seen from Fig. 3 that when only the size-dependent is considered, the sensitivity of the original system is better as the amplitude of the external excitation voltage is small; increasing the external excitation amplitude can increase the system's multi-value response interval; if continues to increase external excitation voltage, the maximum amplitude is higher than original system. It can also be seen from Fig. 3 that when only the effect of the fringing field is considered, the sensitivity of the system is better than original system when the amplitude of the external excitation voltage is small; increasing the external excitation amplitude, the multi-valued response interval of the system decreases; continue to increase the magnitude, the maximum amplitude is lower than original system. When considering the effects of the size-dependent and the fringing field, the sensitivity of original system is better when the amplitude of the external excitation voltage is small; increasing the external excitation amplitude increases the system's multi-value response interval; maximum amplitudes of two directions are higher than original system as continuing to increase the external excitation voltage. Therefore, the size-dependent effect can significantly affect the force-amplitude curve of the system. Therefore, the conclusion of Fig. 2 can also be proved, if the effects of the size-dependent and the fringing field are ignored, the gyroscope measurement will be inaccurate. As can be seen from Fig. 5, as the DC voltage VDC2 goes from positive to negative, the multi-value response area of the system decreases. The higher branch represents a stable solution, and the vibrating gyroscope can work at a higher external excitation amplitude.

Effect of nonlinear curvature and inertial
In this section, the effect of nonlinear curvature and inertial on frequency-amplitude and force-amplitude responses of the vibrating beam micro-gyroscope is studied. without curvature (κ4=0) and with inertia nonlinearities (κ5≠0).
From Fig. 6a, it can be seen that when the coefficient value k=0, k=0.005 and k=0.05, the nonlinear effect of the system is not change, that is to say, when the value of k is small, the nonlinear curvature and inertia can be ignored. As the nonlinear coefficient k=0.5 is taken, when considering the case of κ4≠0 and κ5=0, the softening characteristics of the responses in drive direction are weakened compared to the original system; As κ4=0 and κ5≠0, the softening characteristics are enhanced; As κ4≠0 and κ5≠0, the softening characteristics of the responses in drive direction are also increased compared to the original system. These results show that curvature nonlinearity makes the system softer and inertial nonlinearity makes the system harder. Under the influence of the same coefficient k, the effect of curvature nonlinearity is stronger than that of inertial nonlinearity. In addition, we find that the maximum amplitude of the system in drive direction is constant under different nonlinearities.
In Fig. 6b, as coefficient values of the nonlinearity are k=0, k=0.005 and k=0.05, it can also be found that the nonlinear effect of the system in sense direction is the same.
When taking the nonlinear coefficient k=0.5, the maximum amplitude in sense direction of the case of κ4≠0 and κ5=0 is larger than that cases of κ4=0, κ5≠0 and κ4≠0, κ5≠0. One may also find the multi-valued response interval is the smallest for the case of κ4≠0 and κ5=0. Therefore, increasing the curvature nonlinearity and reducing the inertial nonlinearity of the system will help the system obtain better sensitivity and eliminate multi-valued responses as much as possible. inertia, Fig. 7 shows the force-amplitude response curves in both drive and sense directions. As can be seen from Fig. 7, the trend curves of force-amplitude in drive and sense directions are different, which is different from the force-amplitude curves and inertia coefficient k=0.5 is taken and in the case of κ4≠0 and κ5=0, the system's multi-value response interval increases with the increase of the external excitation amplitude; continue to increase the external excitation amplitude, the maximum amplitude in drive direction is higher than original system. When in the cases of κ4≠0, κ5≠0 and κ4=0, κ5≠0, the multi-valued response interval of the system reduces with the increase of the external excitation amplitude; continue to increase the external excitation amplitude, the maximum amplitude in drive direction is lower than original system. From Fig. 7b in sense direction, when taking the nonlinear coefficient k=0.5, the multi-value interval of the system is the biggest in the case of κ4≠0 and κ5=0. As the cases of κ4=0, κ5≠0 and κ4≠0, κ5≠0, the multi-value interval of the system is smaller than original system. However, as the amplitude of the externally excited voltage increases, the maximum amplitude of original system in sense direction is bigger than other nonlinear systems.

Conclusions
In this research, by considering the effect of statically parametric excitation on the model of the vibrating beam micro-gyroscope, the nonlinear dynamics with nonlinearities in curvature, inertia and electrostatic force is studied. The multi-scale method and numerical continuation technique are used to investigate the nonlinear responses of the gyroscopic system. This paper studies the effect of the size-dependent, fringing field, and static axial excitation on the frequency-amplitude and forceamplitude responses of the nonlinear system. For such complex nonlinear microgyroscopes, the main resonance response of the system is investigated under the primary and internal resonance conditions. The following conclusions are drawn: (1) Small vibrations produce a symmetrical frequency response curve in sense direction while the system actually has a significant softening characteristic in drive direction. When only the size-dependent effect is considered, the amplitude in drive direction decreases, and the nonlinear multi-value problem is effectively reduced in sense direction, which plays an important role in the gyroscope detection system. If the effect of the size-dependent and the fringing field is ignored, the micro-gyroscope measurement will be inaccurate.
(2) As the piezoelectric DC voltage decreases from positive to negative, the softening characteristic becomes more and more obvious in drive direction, there is always one DC voltage where the amplitude reaches a maximum value. However, the nonlinear responses in sense direction present multi-valued effect. If we want to obtain a higher sensitivity, it is better to choose a positive VDC2 voltage. The higher branch represents a stable solution, and the vibratory gyroscope can work at higher external excitation amplitude.
(3) The nonlinear responses of the system are constant when the value of nonlinear coefficient k is small and its influence can be ignored. When the coefficient k of nonlinearities in curvature and inertia is big (e.g. k=0.5), the curvature nonlinearity makes the system softer and the inertial nonlinearity makes the system harder. Under the influence of the same coefficient, the effect of curvature nonlinearity is stronger than that of inertial nonlinearity. Increasing the curvature nonlinearity and reducing the inertial nonlinearity of the system help the system obtain better sensitivity, and can eliminate multi-valued responses as much as possible.