A review on vibrating beam-based micro/nano-gyroscopes

Abstract

A comprehensive review of the modeling approaches used to simulate the behaviors of micro/nano-gyroscopes is presented. The performance and sensitivity of these inertial sensors can be significantly improved through understanding their governing dynamics and exploiting specific phenomena and distinctive features. Such understanding can be developed by solving and analyzing their governing equations and boundary conditions that may comprise a set of highly nonlinear partial differential equations. The operating principle of vibrating beam gyroscopes is described and their main actuation and sensing mechanisms are reviewed and discussed. The multi-fidelity modeling approaches that have been used for the design, performance analysis, and control of vibratory micro/nano-gyroscopes are consolidated and reviewed. The use of these mathematical models has opened doors for the development of new sensing designs with unprecedented sensitivity and extended operating range. To date, extensive research has been conducted on modeling and simulations of micro/nano-gyroscopes. However, several open research topics have not been thoroughly explored yet. These include nanoscale experimentation for model validation, damage/fatigue modeling, and self-powered energy harvesting gyroscope systems. This review presents the current state of the art and highlights promising research directions for continued technological advancement of micro/nano-gyroscopes.

Introduction

Gyroscopes are inertial sensors present in several moving engineering systems, such as air vehicles, cars, and satellites to detect their orientation and adjust their path. Recent advances in micro and nanotechnology have boosted the development of miniaturized gyroscopes at low fabrication cost with a wide operating range, high precision, possible implementation of multiple sensors on a single chip, integration with electronic circuits, and low power requirements. These small size, reliable, cost-effective micromachines sensors are conquering the markets held by the conventional gyroscopes (Kuehnel 1995). Microelectromechanical systems (MEMS) enable the achievement of a small form factor, at the same time as a technology for mass manufacture to reduce the price of a single unit provided that there are enough quantities for production. For this reason, the design of the micro-gyroscope has been always an intense research field in the industry (Yuan 2017) as well as in academia (Saukoski et al. (2008)). This has also extended the scope of new features and applications including mobile phones, micro-satellites, and portable navigation devices in which large-scale and conventional gyroscopes cannot be deployed.

In the last few decades, several research studies have investigated the performance of micro/nano-gyroscopes in terms of the scale factor, dynamic range, bandwidth, temperature sensitivity, and deploying vibrating structures, such as beams, plates, shells, and tuning forks (Jain and Gopal 2017). The following reported studies have concentrated on improving a few performance criteria in micro/nano-gyroscopes (Ouakad (2019); Ghommem and Abdelkefi 2017a, b, c; Yoon et al. 2015; Tatar et al. 2014; Ozer et al. 2013; Tatar et al. 2012; Prikhodko et al. 2012; Prikhodko et al. 2013; Trusov et al. 2011; Sharma et al. 2008; Alper et al. 2007; Acar and Shkel 2008; Acar et al. 2009; Kawai et al. 2001; Adams et al. 1999; Geiger et al. 1999; Zhou et al. 2006). These designs rely on the energy transfer between two vibration modes via an induced motion due to the Coriolis force of the sensing element vibrating in a rotating frame of reference (Menon et al. (2018)). The operation of these inertial sensors requires exciting a vibrating element along one direction by the application of different possible actuation mechanisms including electrostatic, electrothermal, and piezoelectric (Satz and Hammerschmidt 2007). This is usually referred to as drive mode which maintains a constant momentum. The induced motion (sense mode) due to the Coriolis force caused by a combination of the drive and angular rate input is detected via a transduction mechanism to measure the angular speed of the rotating base (Acar et al. 2009). Two main sensing modes have been proposed to determine the rotation rate. This can be captured by tracking either the microbeam displacement via capacitive, resistance change, or piezo-resistive transduction techniques or the shift in the resonance frequency between the two vibrations modes (driving and induced motions). The shift in the resonant frequency can be obtained from the Fast Fourier Transform (FFT) analysis of the dynamic response of the vibrating structure. A third mode combining the two aforementioned detection mechanisms can be also realized (Cao et al. 2020a).

The design of micro/nano-gyroscopes can be investigated through sophisticated mathematical models. Such models may include a set of nonlinear partial differential equations (PDEs). The sources of the nonlinearities in these equations may be material or constitutive, geometric, inertial, driving forces, fluid–structure interactions, or dissipation mechanisms. The significant progress in computer hardware and software have enabled the numerical implementation of the complex equations governing the behaviors of these inertial sensors. These computational models allowed researchers to discern their complex mechanisms, perform parametric and optimization studies to improve their performance and robustness, and explore the potential of new sensing techniques and designs. Modeling approaches with different fidelity levels have been proposed and used to analyze the dynamic behaviors of micro/nano-gyroscopes, ranging from lumped-parameter models to coupled multi-physics models. Similarly, modeling techniques, such as finite element analysis (FEA) are common use for analyzing complex microelectromechanical systems based devices. FEA is constituting a reliable technique to predict the functionality and performance of the device based on material and structural designs (Mian et al. 2015). For micromechanical vibrating member gyroscopes, the prediction of the dynamical characteristics of the structure is crucial for the sensor application (Mian et al. 2015).

This review study is organized as follows. First, the operating principle of vibratory N/MEMS gyroscopes is presented and discussed in Sect. 2. In Sect. 3, the various sensing and actuation mechanisms used in gyroscopes are consolidated. Then, in Sect. 4, the various modeling approaches for simulating the static and dynamic operations of micro/nano-gyroscopes are reviewed and discussed. These include lumped-parameter, distributed-parameter (continuous), and finite element models. Next, possible topics for future research are proposed and discussed in Sect. 5. Finally, some conclusions are drawn in Sect. 6.

Operating principle of vibratory micro/nano-gyroscopes

The operating principle of vibrating beam gyroscopes is based on the transfer of the mechanical energy among two vibrations modes via the Coriolis effect which takes place in the presence of a combination of rotational and flexural motions. The flexural motion (drive mode) can be generated using different actuation mechanisms. Several sensing techniques, such as capacitive, piezoresistive, optical can be deployed to detect the lateral motion (sense mode) and then obtain the applied rotation rate. An example of a vibratory gyroscope design that has been widely studied in the literature over the last two decades is composed of a cantilever beam attached to a rotating base and connected to a rigid tip mass at the free end (Ouakad 2019; Ghommem and Abdelkefi 2017a,b,c; Yoon et al. 2015; Tatar et al. 2012, 2014; Ozer et al. 2013; Prikhodko et al. 2012,2013; Trusov et al. 2011; Sharma et al. 2008; Alper et al. 2007; Acar and Shkel 2008; Acar et al. 2009; Kawai et al. 2001; Adams et al. 1999; Geiger et al. 1999), as shown in Fig. 1. The tip mass is coupled to two electrodes powered by voltage sources. One of these voltage sources is deployed to generate an electrostatic force and excite the microbeam in the transverse direction and the other voltage source is utilized to detect the induced vibrations in the lateral direction, commonly converted to a change in capacitance, and hence extract the base rotation rate.

Fig. 1
figure1

Schematic of a vibratory micro-gyroscope made of a cantilever beam with a tip mass subject to electric actuation (Ghommem and Abdelkefi 2017a)

The gyroscope can operate either in amplitude- or frequency-based mode. In the first operating mode, the magnitude of the vibrations along the sense direction, which can be measured as capacitance change between the side electrode and the tip mass, is used to determine the angular speed. The latter operating mode is based on extracting the base rotation rate from the differential frequency between the drive and the sense modes. This sensing approach was originally proposed in Nayfeh et al. (2015) and further studied for microbeams made of nanocrystalline silicon in Ghommem and Abdelkefi 2017b.

Sensing and actuation mechanisms for vibrating micro/nano gyroscopes

Achieving large displacements at low power consumption constitutes the major concern for the implementation of the vibrating beam based MEMS gyroscopes. Electrostatic, piezoelectric, and electrothermal are commonly used actuation mechanisms. Electrostatic actuation constitutes the prevailing approach as it requires a low voltage and can excite high-frequency resonant modes. However, the resulting deflection amplitude is relatively small. To detect the Coriolis-induced vibrations in the secondary sense mode, capacitive, piezoresistive, or piezoelectric pick-off mechanisms are commonly used (Acar and Shkel 2008).

Piezoelectric sensing and actuation

When mechanical stress is applied to a piezoelectric material, an electrical charge is produced. This is known as the piezoelectric effect. The piezoelectric norm, by the IEEE standard, has provided various forms of piezoelectric constituent equations. Piezoelectricity has a complex relationship between the electrical, mechanical, thermal, and elastic properties. The relationships are schematically shown in Fig. 2. The corners of each of the triangles are functions representing the “E” electric field, “D” electric displacement, “σ” stress, “ε” strain, “T” temperature, and “S” entropy (Bevan 1998). The linear piezoelectric theory assumes constant entropy and then resulting in an adiabatic operation. In this case, the mechanical strains do not contribute to the piezoelectric substrate thermodynamic state.

Fig. 2
figure2

Elastic, thermal, and electrical properties of piezoelectric (Bevan 1998)

There are two approaches used in the design of a piezoelectric: transversal and longitudinal approaches. The first approach, as illustrated in Fig. 3a, makes use of the d31 piezoelectric coefficient (transverse strain constant). The in-plane strain \({X}_{1}\) in the piezoelectric film is induced by an external electric field \({E}_{3}\) normal to the plane. When a voltage is applied to the electrodes, the piezoelectric film contracts laterally which produces the bending of the beam in the upward direction. As shown in Fig. 3b, the in-plane electrical field \({E}_{3}\) induces a transverse piezoelectric strain \({X}_{3}\). This mode is known as the longitudinal strain, d33 mode. The d33 coefficient is approximate twice the d31 coefficient. Therefore, the expected deflection actuated by the d33 coefficient is larger (Smith et al. 2012).

Fig. 3
figure3

A schematic diagram of a cantilever actuated in a d31 mode and b d33 mode (Smith et al. 2012)

Many architectures for these systems have been developed and a single model that represents all systems does not exist. A general representation of the piezoelectric constitutive equations for thin beams is presented as (Kagawa et al. 1996):

$$\sigma = E_{p} \varepsilon_{ij} - e_{ij} E_{i}$$
(1)
$$D_{i} = e_{ij} \varepsilon_{ij} + \in_{ij} E_{i}$$
(2)

where εij is the mechanical strain, \({E}_{p}\) is the Young’s modulus at the constant electric field, \(\sigma\) represents the mechanical stress, \({E}_{i}\) denotes the electric field, \({D}_{i}\) is the electrical displacement, εij is dielectric permittivity at constant strain, and \({e}_{ij}\) represents the piezoelectric stress coefficient, where i is the polarization direction and j is the strain direction.

Many research studies have been conducted to investigate the performance of novel gyroscope systems (Giannini et al. 2020; Bestetti et al. 2020; Menéndez 2019; Yi-Hsuan and Peng 2020). Piezoelectric sensing and actuation methods have been deeply investigated for use in MEMS gyroscope system applications (Mian et al. 2015; Mojahedi et al. 2013a; Ghommem and Abdelkefi 2017; Rasekh and Khadem 2013). Cao et al. (Cao and Sepúlveda 2019) designed a piezoelectric bending sensor consisting of four identical flexible piezoelectric gyroscopes for structural health monitoring using both theoretical analysis and experimental demonstration. Each of the four sensors is made of an 80 μm thick, 50 mm long, and 15 mm wide PPFE film, which has a d31 value of approximately 2 pC/N, sandwiched between two 500 nm thick sputtered silver metal electrodes. Based on the transverse piezoelectric effect of PPFE and the inherent polarity characteristic of piezoelectric materials, their work presented a designed non-destructive sensing network able to provide a 2D full description of an arbitrary deformation. In another recent study, Khodaei et al. (Khodaei et al. 2018) investigated a cantilever beam gyroscope under piezoelectric actuation both theoretically and experimentally under steady-state base rotation and reported good agreement between the two sets of data, as shown in Fig. 4.

Fig. 4
figure4

Stationary system response (bending acceleration) to different piezoelectric excitation frequencies: a f = 5 Hz and b 10 Hz (Bestetti et al. 2020)

As shown in Fig. 5, the simulation results no longer match the experimental data as the rotational speed is increased. However, experimentally they demonstrated that the beam vibrates even in the absence of forced excitations or significant Coriolis acceleration. This points to the presence of simplifications, such as the straight beam assumption and unmodeled dynamics. These results indicated the deficiency in the current models and motivate further investigations to improve the theory by capturing the underlying dynamics.

Fig. 5
figure5

System response (bending acceleration) piezoelectric excitation frequency of 5 Hz and different rotational speeds: a Ω = 0 rad/s, b Ω = 4 rad/s, c Ω = 8 rad/s, and d Ω = 12 (Bestetti et al. 2020)

Edamana et al. (Edamana et al. 2015) presented a strategy for controlling a piezoelectric actuator to generate periodic reference trajectories for calibrating a MEMS gyroscope, as a method of improving performance. Their work revealed that incorporating a capacitive threshold sensor with conventional analog position sensing in the piezoelectric stage, substantially improve the angular velocity estimation accuracy by 80%.

Electrothermal actuation

During the last few years, there has been significant research on electrothermally-actuated beams that rely on the thermal expansion effect triggered by Joule heating (Alcheikh et al. 2018; Hajjaj et al. 2017; Saqib et al. 2018; Shakoor et al. 2011a, b, 2018; Bazaei and Moheimani 2014). This is achieved by passing a current through the vibrating structure. Such an actuation mechanism can produce a large driving force acting along with both directions parallel and perpendicular to the substrate and resulting in large deflection. However, in general, the Joule heating process is relatively slow in comparison with other actuation mechanisms (depending on the thermal time constant of the used material). It is highly sensitive to the environmental temperature, consumes high power, and may disturb the operation of the surrounding electronics (Alcheikh et al. 2018; Hajjaj et al. 2017). The power consumption of an electrothermal actuator is given by (Alcheikh et al. 2018; Hajjaj et al. 2017):

$$P = \frac{{V^{2} }}{R} = \frac{{V^{2} A}}{\rho L}$$
(3)

where \(V\) denotes the excitation voltage, \(A\) is the cross-sectional area, \(L\) is the length of the actuator (anchor to anchor), and \(\rho\) represents the resistivity.

Electrothermal actuators have been recently deployed for MEMS gyroscopes (Saqib et al. 2018; Shakoor et al. 2011a, b, 2018). Shakoor et al. (Shakoor et al. 2011a) designed and tested a vibratory gyroscope that uses a Chevron-shaped thermal actuator made of electroplated nickel (Ni), as shown in Fig. 6. Their experimental study demonstrated the capability of the actuator to generate large displacements (7.36 \(\mathrm{\mu m}\)) at low excitation voltage (0.7 V) thanks to the thermo-physical properties of the electroplated Ni. They also showed the potential of the electrothermal actuator to produce a high-quality factor gyroscope with higher sensitivity in comparison with electrostatic actuators. In the dynamic operation mode, they were able to achieve a driving direction displacement of 4.2 \(\mathrm{\mu m}\) with an actuation power of 360 mW. In a recent study, Saqib et al. (Saqib et al. 2018) performed a numerical analysis of an electro-thermally actuated MEMS gyroscope comprising three-mass coupled resonators for the drive mode and 1-DOF sense mode. They used silicon-based Chevron-shaped thermal actuators which consumes 60% less power than those made of nickel. They developed and used an analytical model to optimize the performance of the gyroscope. The analytical results were verified using FEA simulations. The power consumption was 22 mW while securing a high gain: 0.15 displacement of the sense mass resulting in a capacitance change of 28.11 fF when applying an angular velocity of 50 rad/s.

Fig. 6
figure6

SEM image of the electrothermally actuated micro-gyroscope developed by Shakoor et al. (Shakoor et al. 2011a)

Electrostatic sensing and actuation

Electrostatic transduction is the most used actuation and sensing method in micro/nano-gyroscopes given its low power requirement, fast response, simplicity, and high energy density. It deploys capacitors of parallel plate electrodes and requires only a voltage source for actuation. Several forms of electrostatic force have been developed to account for the fringing field. This force is characterized by a nonlinearity that may lead to undesirable effects, such as pull-in and stiction resulting in the failure of the sensor. As for electrostatic sensing of rotational motion, the induced vibrations (due to the Coriolis effect) alter the capacitance of the capacitor made of a fixed electrode (substrate) and a flexible structure. In Table 1, the most common forms of electrostatic forces used for the actuation of vibrating beam MEMS devices are depicted. In this table, \({{\upepsilon }_{r}\mathrm{ and \epsilon }}_{0}\) are the dielectric coefficient and permittivity of the dielectric vacuum, respectively. \(B\) and h are the width and thickness of the beam, respectively. \(V\) represents the applied voltage and \(g\) denotes the gap distance between the fixed electrode and the beam.

Table 1 Summary of the most common electrostatic force representations used in MEMS modeling

The performance analysis of vibrating beam operating in both amplitude-based and frequency-based modes while employing capacitive-based sensing technique has been the subject of several research studies (Ouakad 2019; Ghommem and Abdelkefi 2017b, c; Nayfeh et al. 2015; Mojahedi et al. 2013a, b, c, 2014a, b; Ghommem and Abdelkefi 2017; Rasekh and Khadem 2013; Mokhtari et al. 2017; Ghommem et al. 2010,2013; Lajimi et al. 2015,2017a,b). Most of the aforementioned studies used Palmer’s model to account for the fringing effect. However, as shown in Fig. 7, there are significant discrepancies between the expressions of the electrostatic force depending on the considered assumptions. Kimbali et al. (Kambali and Pandey 2015) found that the Palmer model may underestimate the fringing field effects when compared to numerical solutions if the beam undergoes large deflections.

Fig. 7
figure7

Adapted from (Kambali and Pandey 2015)

Comparison between the a capacitance and b electrostatic force approximated using Kimbali (Ceff and Feff) model, Palmer model, a numerical solution, and the solution without fringing field effects

Modeling approaches for design enhancement of micro/nano-gyroscopes

The design of vibratory beam-based micro/nano-gyroscopes can be efficiently assisted by mathematical models and simulation tools. Several numerical studies have investigated the dynamic responses of rotating beams for micro-gyroscope applications (Ouakad 2019; Ghommem and Abdelkefi 2017a,c; Nayfeh et al. 2015; Mojahedi et al. 2013a, b, c, 2014a, b; Ghommem and Abdelkefi 2017; Rasekh and Khadem 2013; Ghommem et al. 2010, 2013; Lajimi et al. 2015,2017a, b, c; Li et al. 2012; Esmaeili et al. 2007; Bhadbhade et al. 2008; Ghayesh et al. 2016; Zhao and Wu 2017). These numerical analyses are useful and helpful in addressing the following points: (1) to gain a solid understanding of the underlying physics and dynamics that govern these micro/nano-systems’ responses; (2) to reduce the prototyping cost and decrease the design time and the development cycle; (3) to predict the performance of existing designs and come up with recommendations for design enhancement; (4) to explore a large design space within a reasonable time; and (5) to develop and assess the potential feasibility of new sensing concepts.

In this section, modeling approaches of different fidelity levels used for the design and performance analysis of vibratory gyroscopes are consolidated and discussed. Their use depends on the level of physics and dynamics to be captured. Figure 8 summarizes the modeling approaches of gyroscopes as reported in the literature. The need to properly capture the nano- and micro-scale physical features of these inertial sensors may require the application of non-classical mechanics’ theories and this would significantly increase the complexity of the associated models that may include a set of coupled multi-physics partial differential equations. The numerical integration of these models usually requires the deployment of powerful hardware and software capabilities.

Fig. 8
figure8

Modeling approaches of vibrating beam micro/nano gyroscopes

Lumped-parameter models

Lumped-parameter models constitute a simplified mathematical representation of the physical system that is described by a finite number of state variables (or degrees of freedom). They have several advantages including:

  • Can assist in the design phase and the identification of high-impact system parameters

  • Can be easily combined with control strategies

  • Lesser demands on the computational resources and time

  • Suitable for optimization studies that require the simulation of many configurations

On the other hand, this type of models has several shortcomings such as (Hong et al. 2003):

  • Cannot reflect any geometry characteristics and does not include parasitic mechanical modes

  • May fail to embody the inherent physical phenomena that are interrelated in a complex manner

  • Limited capability to test new design and sensing mechanisms

  • Limited capability to gain insight and understanding of complex systems

For illustration, a basic model of a vibratory gyroscope is presented: that is, a proof mass supported by springs and dashpots along with two directions and undergoing rotational motion, as shown in Fig. 9. X–Y and x–y denote the stationary and rotating frames, respectively. The proof mass is subject to an external force F(t) to excite it along the drive direction. The rotational motion induces vibrations along the lateral direction due to the Coriolis effect.

Fig. 9
figure9

(adopted from Younis 2011)

A typical vibratory gyroscope

The equations of motion can be obtained by applying Newton’s second law. Assuming harmonic excitation, these equations are expressed as (Younis 2011):

$$m\ddot{x} + c_{x} \dot{x} + k_{x} x = F_{d} \sin \left( {\omega_{d} t} \right) + m\dot{\Omega }y + 2m\dot{y}\Omega + {\text{m}}\Omega^{2} x$$
(4)
$$m\ddot{y} + c_{y} \dot{y} + k_{y} y = - m\dot{\Omega }x - 2m\dot{x}\Omega + m\Omega^{2} y$$
(5)

where \(m\) denotes the mass, \({c}_{x}\) and \({c}_{y}\) are the damping coefficients along with the drive and sense directions, respectively. \({k}_{x}\) and \({k}_{y}\) are the spring constants along with the drive and sense directions, respectively. \({F}_{d}\) and \({\omega }_{d}\) are the driving force amplitude and frequency, respectively.

The two differential equations are coupled through the angular speed and acceleration of the rotating base. In practical situations, the angular velocity \(\Omega\) is usually much smaller than the natural frequency of the micro/nano-system and the induced response \(y\) is much smaller than the primary response \(x\) (excited motion). As such, the terms including \({\Omega }^{2}\) and \(2\dot{y}\Omega\) can be neglected. Furthermore, assuming constant angular velocity \(\Omega\), the equations of motion are given by (Younis 2011):

$$m\ddot{x} + c_{x} \dot{x} + k_{x} x = F_{d} \sin \left( {\omega_{d} t} \right)$$
(6)
$$m\ddot{y} + c_{y} \dot{y} + k_{y} y = - m\Omega \dot{x}$$
(7)

Introducing the natural frequency \({\omega }_{x}=\sqrt{\frac{{k}_{x}}{m}}\), the damping ratio \({\zeta }_{x}=\frac{{c}_{x}}{2m{\omega }_{x}}\), and the mass-normalized forcing amplitude \({f}_{d}=\frac{{F}_{d}}{m}\), Eq. (6) can be rewritten as:

$$\ddot{x} + 2\zeta_{x} \omega_{x} \dot{x} + \omega_{x}^{2} x = f_{d} \sin \left( {\omega_{d} t} \right)$$
(8)

The analytical solution of Eq. (8) is:

$$x\left( t \right) = X\sin \left( {\omega_{d} t - \phi } \right)$$
(9)

where

$$X = \frac{{f_{d} }}{{m \omega_{x}^{2} }}\frac{1}{{\sqrt {\left( {1 - r_{x}^{2} } \right)^{2} + \left( {2\zeta_{x} r_{x} } \right)^{2} } }}\quad {\text{and}}\;\phi = \tan^{ - 1} \left( {\frac{{2\zeta_{x} r_{x} }}{{1 - r_{x}^{2} }}} \right)$$
(10)

The frequency ratio is defined as \({r}_{x}=\frac{{\omega }_{d}}{{\omega }_{x}}\). The sense-mode equation can be then expressed as follows:

$$m\ddot{y} + c_{y} \dot{y} + k_{y} y = f_{s} \cos \left( {\omega_{d} t - \phi } \right)$$
(11)

where \({\omega }_{y}=\sqrt{\frac{{k}_{y}}{m}}\), \({\zeta }_{y}=\frac{{c}_{y}}{2m{\omega }_{y}}\), and \({f}_{s}=-2\Omega {\omega }_{d}X\). The amplitude of the response of the sense mode is given by (Younis 2011):

$$Y = \frac{{f_{s} }}{{m \omega_{y}^{2} }}\frac{1}{{\sqrt {\left( {1 - r_{y}^{2} } \right)^{2} + \left( {2\zeta_{y} r_{y} } \right)^{2} } }} = \left[ {\frac{{\frac{{2m\omega_{d} X}}{{k_{y} }}}}{{\sqrt {\left( {1 - r_{y}^{2} } \right)^{2} + \left( {2\zeta_{y} r_{y} } \right)^{2} } }}} \right] \Omega$$
(12)

where \({r}_{y}=\frac{{\omega }_{d}}{{\omega }_{y}}\).

The sensitivity of the induced amplitude to the base rotation is given by (Younis 2011):

$$S = \frac{\partial Y}{{\partial \Omega }} = \frac{{\frac{{2m\omega_{d} X}}{{k_{y} }}}}{{\sqrt {\left( {1 - r_{y}^{2} } \right)^{2} + \left( {2\zeta_{y} r_{y} } \right)^{2} } }}$$
(13)

Vibratory gyroscopes are commonly driven at resonance to maximize the sensitivity. At resonance, the sensitivity can be expressed in terms of the quality factors of the drive and sense modes \({Q}_{x}\) and \({Q}_{y}\) as (Younis 2011):

$$S = \frac{{2m\omega_{d} F_{d} Q_{x} Q_{y} }}{{k_{x} k_{y} }}$$
(14)

As such, the sensitivity of the gyroscope can be enhanced by considering high-quality factors (low damping). Several research studies (Yeh et al. 2001; Verma et al. 2015; Riaz et al. 2011) have relied on similar simple representations of the gyroscopic effect to assist in the design phase of gyroscopes. Table 2 summarizes some research works conducted on micro-/nano-gyroscopes using lumped-parameter models. Verma et al. (Verma et al. 2015) developed a set of coupled differential equations to simulate the response of a 3 degree-of-freedom (DOF) gyro-accelerometer system comprising a one DOF for the driving mode and 2 DOFs for the sensing modes. They used the model to demonstrate the feasibility of a novel detection mechanism of angular motion and linear acceleration. Li et al. (Li et al. 2019a) designed a tuning fork gyroscope with an anchored leverage mechanism. A lumped-parameter model was solved analytically and used to analyze the dynamics and mechanical sensitivity of the gyroscope. The analytical results were found in good agreement with the finite element method (FEM) simulations. Their numerical study revealed that the proposed anchored leverage mechanism enables significant improvement in the mechanical sensitivity of the gyroscope. Kwon et al. (Kwon et al. 2017) developed a dynamic model of a MEMS vibratory gyroscope in the form of an input–output equation to relate the rotation rate to the current output. The model was obtained by integrating a lumped mass-spring-damper system with an electrical circuit. The numerical simulations results qualitatively compared well with their experimental counterparts, as shown in Fig. 10 Furthermore, the dynamic model enabled the optimization of the electrical circuit parameters to achieve higher sensitivity.

Table 2 Summary of micro-/nano-gyroscope research using lumped parameter models
Fig. 10
figure10

a Frequency response curves and b output voltage of a MEMS gyroscope comparing the FEM simulation and experimental results performed by Kwon et al. (Kwon et al. 2017). Discrepancies in the frequency predictions were thought to be caused by manufacturing defects (Kwon et al. 2017)

Modeling continuous gyroscope systems using classical continuum mechanics theories

The use of classical continuum mechanics theories to simulate the operation of micro/nano gyroscopes has several advantages including:

  • Increased accuracy compared to lumped-parameter models

  • Improved representation of the physical system

  • Easily analyzed using common numerical methods

However, this modeling approach has also drawbacks such as:

  • Can become mathematically cumbersome when accounting for multi-dimensional and nonlinear effects

  • Increased computational complexity and expense when compared to the lumped-parameter model

The derivation of the governing equations of vibrating beam micro-gyroscopes is commonly performed by expressing the potential and kinetic energies followed by the application of the extended Hamilton’s principle. For instance, considering Euler–Bernoulli beam assumption, the equations of motion and associated boundary conditions of the electrically-actuated micro-gyroscope, shown in Fig. 1, are expressed as follows (Ghommem and Abdelkefi 2017a, b; Nayfeh et al. 2015; Mojahedi et al. 2013b, c):

$$m\ddot{w} + \mu_{w} \dot{w} + EIw^{IV} + 2m\Omega \dot{v} - m\Omega^{2} w + m\dot{\Omega }v - J\Omega^{2} w^{\prime\prime} - J\dot{\Omega }v^{\prime\prime} - J\ddot{w}^{^{\prime\prime}} = 0$$
(15)
$$m\ddot{v} + \mu_{v} \dot{v} + EIv^{IV} - 2m\Omega \dot{w} - m\Omega^{2} v - m\dot{\Omega }w - J\Omega^{2} v^{\prime\prime} - J\dot{\Omega }w^{\prime\prime} - J\ddot{v}^{\prime \prime} = 0$$
(16)

At \(x=0\)

$$w = v = 0\quad {\text{and}}\quad w^{\prime} = v^{\prime} = 0$$
(17)

At \(x=L\)

$$w^{\prime \prime} = v^{\prime \prime} = 0$$
(18)
$$EIw^{\prime\prime\prime} + M\Omega^{2} w - M\dot{\Omega }v - 2M\Omega \dot{v} - M\ddot{w} - J\dot{\Omega }v^{\prime} - J\Omega^{2} w^{\prime} - J\ddot{w}^{\prime \prime} = - \frac{{\smallint_{0} A_{W} }}{{\left( {g_{w} - w} \right)^{2} }}V_{w}^{2} \left( {1 + 0.65\frac{{g_{w} - w}}{{b_{w} }}} \right)$$
(19)
$$EIv^{\prime\prime\prime} + M\Omega^{2} v + M\dot{\Omega }w + 2M\Omega \dot{w} - M\ddot{v} - J\dot{\Omega }w^{\prime} - J\Omega^{2} v^{\prime} - J\ddot{v}^{\prime \prime} = - \frac{{\varepsilon_{0} A_{v} }}{{\left( {g_{v} - v} \right)^{2} }}V_{v}^{2} \left( {1 + 0.65\frac{{g_{v} - v}}{{b_{v} }}} \right)$$
(20)

where \(w\) and \(v\) denote the flexural displacements along with the transverse (drive) and lateral (sense) directions, respectively. \(\Omega\) is the base rotation rate, \(\mu\) represents the damping coefficient, \(J\) is the mass moment of inertia, \(E\) is the Young’s modulus of the material, \(I\) is the moment of inertia, and \(m\) is the beam mass per unit length. (\({A}_{W}\),\({g}_{w}\)) and (\({A}_{v}\),\({g}_{v}\)) denote the areas and initial gap distances of the drive and sense capacitors, respectively. \(M\) represents the mass of the tip. The parameter \({\epsilon }_{0}\) = 8.85 × 10−12 C2N−1 m−2 is the permittivity of the dielectric vacuum between the tip mass and the electrode. \({V}_{W}\) and \({V}_{v}\) are the voltages applied along with the drive and sense directions, respectively.

Equations (1520) describe the coupled bending modes of the vibrating beam gyroscope (shown in Fig. 1) subject to electrostatic and Coriolis forces. This design has been widely investigated in the literature. Ghommem et al. (Ghommem et al. 2010) solved analytically the linearized problem to produce the calibration curves of the gyroscope under different electric actuation. Operating the microsystem at large DC voltage (away from pull-in) near resonance was found to increase the sensitivity of the sense-mode response to base rotations. The method of multiple scales was used in (Ghommem et al. 2013) to develop a nonlinear reduced-order model. This model was utilized to simulate the nonlinear response of the electrically actuated micro-gyroscope under the varying angular speed of the rotating base. Nayfeh et al. (Nayfeh et al. 2015) applied perturbation techniques to derive an analytical expression of the frequency differential between the drive and sense modes. The rotational motion was found to break the symmetry of the eigenvalue problem and transform the identical natural frequencies of the coupled modes into a pair of closely spaced frequencies. As such, they proposed to use the frequency difference of this pair to detect the angular speed. This is usually referred to as the frequency-based sensing approach.

Later, this previous work (Nayfeh et al. 2015) was extended in (Ghommem and Abdelkefi 2017b) by incorporating the fringing field of the electrostatic forcing and considering the microstructure of the beam made of nanocrystalline silicon. This material comprises two main phases, grain, and interface as shown in Fig. 11. Their effective properties are dependent on the size of the grain. This effect was incorporated from a size-dependent two-phase micromechanical beam model developed in (Shaat and Abdelkefi 2015; Shaat et al. 2016) that gives an estimation of the effective material properties of nanocrystalline silicon when varying its grain size. When applying the same DC voltage for the drive and sense modes, a linear variation of the differential frequency with the base rotation was observed. However, applying different DC voltages along with the drive and sense directions was found to decrease noticeably the sensitivity of the differential frequency to the angular speed. Larger the DC voltage bias led to further degradation of the gyroscope sensitivity and resulted in nonlinear calibration curves. Operating the sensor at higher-order modes was found to mitigate the effect of the DC voltage bias on the sensitivity of the micro-gyroscope.

Fig. 11
figure11

a Schematic of a multiphase nanocrystalline material containing grains, voids, and interface phases (Larkin et al. 2018) and b low-resolution SEM image of nanocrystalline silicon film (Liao et al. 2003)

Ghommem and Abdelkefi (Ghommem and Abdelkefi 2017a) considered the operation of the micro-gyroscope in the amplitude-based mode, i.e., the angular speed is obtained from the induced vibrations. They used the differential quadrature method for space discretization and finite difference method (FDM) for time marching to analyze the nonlinear dynamical response of the electrically actuated micro-gyroscope made of nanocrystalline silicon. In particular, they investigated the impact of the grain size of the nanocrystalline silicon on the micro-gyroscope frequency response and calibration curves. Considering a microbeam with a bigger grain size of the constituent nanocrystalline silicon was observed to reduce the motion of the sense mode, increase the natural frequency, and reduce the pull-in bandwidth. Furthermore, when operating with microbeams having small nanocrystalline grain size, high angular speeds were observed to switch the dynamic behavior of the sense mode from the nearly linear to the nonlinear softening type. A similar microbeam design shown in Fig. 1 was adopted in (Ghommem and Abdelkefi 2017c) and proposed a resistance-based sensing technique to measure the induced vibrations. This technique is based on transmitting the Coriolis force applied on the tip mass to a tiny probe that alters the resistance of a voltage divider electrical circuit. They conducted a numerical analysis to examine the nonlinear dynamics of the rotating electrically actuated microbeam and demonstrate the capability of the proposed sensing approach to obtain reliable measurements of the base rotation rates.

Lajimi et al. (Lajimi et al. 2015,2017a, b, c; Li et al. 2012; Esmaeili et al. 2007; Bhadbhade et al. 2008; Ghayesh et al. 2016) developed a mathematical model of coupled bending-bending vibration of a rotating cantilever beam with a proof mass at its end treated as a rigid body, as shown in Fig. 12. Their model accounted for the eccentricity and the rotary inertia of the end mass. They applied the method of multiple scales and shooting methods to analyze the nonlinear dynamic behavior and investigate the operation of the gyroscope under amplitude-based and frequency-based modes. Frequency responses were generated for different quality factors, applied AC voltages, and rotation rates. Fold-bifurcation points and dynamic solution multivalued were identified and analyzed.

Fig. 12
figure12

Schematic of the micro-gyroscope: a rotating cantilever beam carrying a rigid body at its free end (Lajimi et al. 2015)

Bhadbhade et al. (Bhadbhade et al. 2008) considered a vibratory gyroscope comprising a cantilever microbeam undergoing coupled flexural–torsional vibrations (see Fig. 13). The primary motion (flexural vibrations) was triggered using a piezoelectric patch actuator. The base rotation induces torsional motion which is used as a detector of the angular speed. They reported a detailed derivation of the governing equations using the extended Hamilton’s principle. The Galerkin approach was used then to transform the coupled partial differential equations into a set of ordinary differential equations. They defined the gyroscope sensitivity as the ratio of the variations of the torsional deflection at the free end with respect to the variations of the base rotation. The sensitivity was computed for a range of 0–70 rad/s.

Fig. 13
figure13

Schematic of a piezoelectrically-actuated gyroscope: flexural–torsional beam (Bhadbhade et al. 2008)

Mojahedi et al. (Mojahedi et al. 2013b) formulated the mathematical model of beam micro-gyroscope (similar design as shown in Fig. 1) while accounting for the nonlinearities arising from the electrostatic forcing, geometry, and inertia. The model showed good predictive capability when compared to previously published experiments and finite element simulations. Rasekh and Khadem (Rasekh and Khadem 2013) conducted a numerical investigation of the performance of a nano-gyroscope made of a rotating cantilever beam subject to electric actuation. They developed an electromechanical model coupling the vibrating microstructure to an electric circuit made of a capacitor and transimpedance amplifier to obtain the output voltage from the induced vibrations. Based on the simulation results, the gyroscope sensitivity was found equal to 0.28 mV/degree/s in a linear operational range up to 400 degree/s. Their device showed superior performance in terms of resolution, sensitivity, and operational range when compared to commercial MEMS gyroscopes.

Table 3 provides a detailed outline of several reduced-order models of MEMS gyroscopes that have been studied over the last few decades including the proposed structural design sensing and actuation mechanisms and solution methods for analyzing their performance.

Table 3 Summary of micro-/nano-gyroscope research using classical continuum models

Size-dependent phenomena for improved modeling of continuous micro-/nano-gyroscopes

As the size of the gyroscope shirks to micro/nano- or nanoscale, material microstructure and intermolecular forces (usually ignored in the macro-scale) become significant. Two types of size-dependent phenomena have been added to reduced-order models of continuous gyroscope systems. Non-classical continuum mechanics approaches allow researchers to include microstructural effects, such as residual stresses and grain rotations, deformations, and interactions. Intermolecular forces have been also added to the micro/nano-gyroscope model to account for the small but significant forces generated between surfaces that are separated by nano-scale gaps. The major advantages of adding size-dependent effects include:

  • Enhanced prediction of stiffening/softening effects from the material structure

  • Increased accuracy of predicted static deflections and natural frequencies

  • Improved prediction of dynamic operations

  • Increased reliability of nano-scale gyroscope designs

  • Improved nano-gyroscope design capabilities

The major disadvantages of including size-dependent effects are:

  • Increase in the model complexity

  • Lack of experimental validation/verification

Non-classical continuum approaches are used to describe size-dependent material properties and intermolecular interactions in a reduced-order continuum mechanics framework. As the size of a system is decreased from macro-scale down to the nano-scale, the assumptions made by classical continuum approaches are no longer valid. For example, the commonly used Euler–Bernoulli beam theory represents individual material particles within the elastic domain as particles that can only translate. However, as the size of the gyroscopes is decreased from micro- to the nano-scale, the scale of the system approaches that of the individual material particles or grains. Thus, the particles are more accurately described as volumes that can undergo micro-deformations, micro-translations, and non-local interactions, as well as translations. Research has shown that classical continuum theories alone are inadequate to describe small-scale systems (Li et al. 2020). Several researchers have applied non-classical continuum approaches to model micro-and nano-gyroscopes. The most extensively used size-dependent theory used to model micro/nano-gyroscopes is the couple stress theory. Ghayesh et al. (Ghayesh et al. 2016) were the pioneer to use the couple stress theory to model a cantilever beam-type micro-gyroscope. The coupled stress can be expressed as follows:

$$\mu_{ij} = \mu l^{2} \left( {\gamma_{ji} + \eta \gamma_{ij} } \right)$$
(21)

where \(\mu\) represents the shear modulus of the material, \(l\) is the couple stress length scale parameter, and \({\gamma }_{ji}\) are the rotation vectors.

Three common variations of the couple stress equation are determined by the selection of the parameter \(\eta\). The classical couple stress theory (Mindlin and Tiersten 1962; Toupin 1962) assumes that \(\eta\) can vary between − 1 and 1. If \(\eta\) is set equal to 1, the modified couple stress theory (Yang et al. 2002) is recovered. Similarly, consistent couple stress theory (Hadjesfandiari and Dargush 2011) is formulated if \(\eta\) is chosen to equal − 1. Following the derivation for the modified couple stress theory presented by Ghayesh et al. (Ghayesh et al. 2016), Eq. (21) becomes:

$$\mu_{ij} = 2\mu l^{2} \gamma_{ji}$$
(22)
$$\gamma_{ji} = \frac{1}{2}\left( {\theta_{i,j} + \theta_{j,i} } \right) ; \theta_{x} = 0 ; \theta_{y} = \frac{1}{2}\left( {u_{x,z} - u_{z,y} } \right) ; \theta_{z} = \frac{1}{2}\left( {u_{y,x} - u_{x,y} } \right)$$
(23)
$$\mu_{xy} = - \mu l^{2} w^{\prime\prime} ; \mu_{xz} = \mu l^{2} v^{\prime\prime}$$
(24)

As the scale of a system is diminished from macro- to micro/nanoscale, the ratio of bulk material to surface area is increased. Therefore, the effect of residual stresses on the material surface becomes important. Residual stresses on the free surfaces of a gyroscope beam can be determined using the surface elasticity theory of Gurtin–Murdoch (Gurtin and Murdoch 1975) as follows:

$$\tau_{\alpha \beta }^{ \pm } = \tau_{0} \delta_{\alpha \beta } + \left( {\lambda_{0} + \tau_{0} } \right)\varepsilon_{ll}^{s \pm } \delta_{\alpha \beta } + 2\left( {\mu_{0} - \tau_{0} } \right)\varepsilon_{\alpha \beta }^{s \pm } + \tau_{0} u_{\alpha ,\beta }$$
(25)

Following the derivation for the surface stresses present on each surface of a gyroscope with a square cross-section presented in (Larkin et al. 2018), the non-zero stress components become:

$$\tau_{xx}^{ + } = \tau_{0} + \frac{b}{2}\left( {2\mu_{0} + \lambda_{0} } \right)v^{\prime\prime} + \frac{h}{2}\left( {2\mu_{0} + \lambda_{0} } \right)w^{\prime\prime}$$
(26)
$$\tau_{xx}^{ - } = \tau_{0} - \frac{b}{2}\left( {2\mu_{0} + \lambda_{0} } \right)v^{\prime\prime} - \frac{h}{2}\left( {2\mu_{0} + \lambda_{0} } \right)w^{\prime\prime}$$
(27)
$$\tau_{xy}^{ \pm } = \tau_{0} v^{\prime} ; \tau_{xz}^{ \pm } = \tau_{0} w^{\prime}$$
(28)

where \({\tau }_{0}\) denotes the residual surface stress, and \({\lambda }_{0}\) and \({\mu }_{0}\) are the surface Lame’ constants. The + and—signs indicate the top and bottom surfaces of the beam, respectively. The new expressions for the axial stress including couple stress and surface elasticity can then be written as:

$$\sigma_{xx} = - \frac{{2\mu \left( {1 - \nu } \right)}}{1 - 2\nu } (yv^{\prime \prime} + zw^{\prime \prime} ) + \left( {\frac{\nu }{1 - \nu }} \right)\left( {\frac{{2z\tau_{0} }}{h}} \right)w^{\prime \prime} + \left( {\frac{\nu }{1 - \nu }} \right)\left( {\frac{{2y\tau_{0} }}{b}} \right)v^{\prime \prime}$$
(29)

This leads to the following set of governing equations and boundary conditions for a cantilever beam gyroscope:

$$D_{v} v^{iv} - mv\Omega^{2} - j\Omega^{2} v^{\prime\prime} - S_{v} v^{\prime\prime} - 2m\Omega \dot{w} + m\ddot{v} - j\ddot{v}^{\prime \prime} = 0$$
(30)
$$D_{w} w^{iv} - mw\Omega^{2} - j\Omega^{2} w^{\prime\prime} - S_{w} w^{\prime\prime} + 2m\Omega \dot{v} + m\ddot{w} - j\ddot{w}^{\prime \prime} = 0$$
(31)

For x = 0,

$$v = w = 0\, {\text{and}}\, v^{\prime} = w^{\prime} = 0$$
(32)

For x = L,

$$v^{\prime\prime} = w^{\prime\prime} = 0$$
(33)
$$D_{v} v^{\prime\prime\prime} + M\Omega^{2} v + 2M\Omega \dot{w} - M\ddot{v} - j\Omega^{2} v^{\prime} - S_{v} v^{\prime} - j\ddot{v}^{\prime} = - \frac{{\varepsilon A_{v} V_{DCv}^{2} }}{2}\left( {\frac{1}{{\left( {g_{v} - v\left( L \right)} \right)^{2} }} + \frac{0.65}{{h_{m} \left( {g_{v} - v\left( L \right)} \right)}}} \right)$$
(34)
$$D_{w} w^{\prime\prime\prime} + M\Omega^{2} w - 2M\Omega \dot{v} - M\ddot{w} - j\Omega^{2} w^{\prime} - S_{w} w^{\prime} - j\ddot{w}^{\prime} = - \frac{{\varepsilon A_{w} \left( {V_{DCw} + v_{AC} \left( t \right)} \right)^{2} }}{2}\left( {\frac{1}{{\left( {g_{w} - w\left( L \right)} \right)^{2} }} + \frac{0.65}{{b_{m} \left( {g_{w} - w\left( L \right)} \right)}}} \right)$$
(35)

where

$$D_{v} = D_{c,v} + D_{cs} + D_{se,v} ;D_{w} = D_{c,w} + D_{cs} + D_{se,w} ;D_{cs} = bh\mu l^{2}$$
(36)
$$D_{se,v} = - \left( {\frac{{hb^{2} \tau_{0} }}{6}} \right)\left( {\frac{\nu }{1 - \nu }} \right) + \left( {\frac{{hb^{2} }}{2} + \frac{{b^{3} }}{6}} \right)\left( {2\mu_{0} + \lambda_{0} } \right) ; D_{c,v} = \frac{{2\mu I_{v} \left( {1 - \nu } \right)}}{1 - 2\nu }$$
(37)
$$D_{se,w} = - \left( {\frac{{bh^{2} \tau_{0} }}{6}} \right)\left( {\frac{\nu }{1 - \nu }} \right) + \left( {\frac{{bh^{2} }}{2} + \frac{{h^{3} }}{6}} \right)\left( {2\mu_{0} + \lambda_{0} } \right) ;\; D_{c,w} = \frac{{2\mu I_{w} \left( {1 - \nu } \right)}}{1 - 2\nu }$$
(38)
$$S_{v} = 2h\tau_{0} ; S_{w} = 2b\tau_{0}$$
(39)

It is clear from Eqs. (3638) that the couple stress and surface elasticity contribute to the structural stiffness of the system and create a stiffer response to the electrostatic loading. Furthermore, the surface elasticity introduces an additional force into the governing equations of motion (Eqs. 30 and 31) and shear boundary condition (Eqs. 34 and 35) which serve to further increase the system’s rigidity in small length scales.

Ghayesh et al. (Ghayesh et al. 2016) used the modified couple stress theory to study the size-dependent on the static pull-in, natural frequencies, and nonlinear frequency response of micro-gyroscopes. They found that the characteristic structural stiffening behavior caused by the couple stress leads to increased static pull-in voltages and increased natural frequencies. They used the Galerkin method for eight modes to numerically study the frequency response and showed that increasing couple stress caused a rightward shift in the resonance region and a significant reduction in the sensitivity of the gyroscope. Later, Keivani et al. (Keivani et al. 2017) used the consistent couple stress theory and van der Waals forces to model the size-dependent behaviors. They employed the Duan-Adomian method (DAM), analytical model, and the differential quadrature method (DQM), numerical model, to calculate the pull-in voltage for paddle-shaped and double-sided NEMS sensors under centrifugal forces. They concluded that the couple stress effects increased pull-in voltage. Additionally, positive centrifugal force increased the pull-in voltage while a negative rotation caused a decrease in the pull-in voltage.

Larkin et al. (Larkin et al. 2018) used the modified couple stress and surface elasticity theories to account for grain rotation effects and residual surface stresses within the nanocrystalline silicon. They used experimentally determined values for the size-dependent parameters and found a pronounced structural stiffening effect that resulted in the addition of these size-dependent phenomena. This led to a substantial increase in the static pull-in voltage and natural frequencies as the scale of the gyroscope was decreased from micro- to nano-scale. Later, Larkin et al. (Larkin et al. 2020a) used DQM to study the effects of couple stress and surface elasticity on the dynamic operation of micro/nano-gyroscopes. They showed that the stiffening effect from the size-dependent properties caused large shifts in the resonance regions of the electrostatically driven beam. It was also demonstrated that the amplitude of the driving and sensing displacements were highly diminished by the small-scale effects. These findings led the authors to conclude that size-dependent effects should be considered in the modeling of micro/nano-gyroscopes in order to increase the effectiveness of future nano-scale gyroscope designs.

In a recent paper, Ouakad (Ouakad 2019) developed a size-dependent model of a beam micro-gyroscope based on non-classical mechanics. They investigated the effect of couple stress and the impact of a nontrivial shaped tip mass on the nonlinear structural behavior. Previously, most researchers have represented the gyroscopes proof mass as a point mass thus, reducing the electrostatic force to a point force acting on the beam tip. However, Ouakad (Ouakad 2019) used a distributed electrostatic force model that considers tip mass geometry. They found that the tip mass geometry plays an important role in the static and nonlinear dynamic operations of cantilever micro/nano-gyroscopes.

The proximity of the gyroscope tip mass, driving, and sensing electrodes in micro/nano-scale systems also led to the development of intermolecular forces, such as Casimir interactions and van der Waals forces. The interaction of quantum fields between two adjacent uncharged plates causes a change in the zero-point energy between the proximal surfaces. Electromagnetic field fluctuations in the vacuum cavity with frequency \(n\pi /\lambda\), where \(\lambda\) denotes the distance between the two surfaces, can pass between the two plates while a greater number of waves can pass through the larger vacuum cavity, as shown in Fig. 14a. This leads to the formation of an attractive force between the two surfaces called the Casimir force. While this phenomenon takes place at the atomic level, significant Casimir forces have been measured when objects are separated by a few nanometers. The Casimir force between the tip mass and the sensing, and driving electrodes can be represented as follows (Mojahedi et al. 2014a):

$$F_{Cv} = \frac{{\pi^{2} A_{v} \hbar c}}{{240\left( {g_{v} - v} \right)^{4} }}; F_{Cw} = \frac{{\pi^{2} A_{w} \hbar c}}{{240\left( {g_{w} - w} \right)^{4} }}$$
(40)

where \(\mathrm{\hslash }\) is Planck’s constant and \(c\) represents the speed of light.

Fig. 14
figure14

Diagrams of intermolecular forces, a the Casimir force, and b the van der Waals force

The tip mass surface and electrode are assumed to be parallel plates. When uncharged molecules come into proximity a transient dipole is formed between them, as shown in Fig. 14b. This causes a weak electrostatic force to form between the adjacent molecules known as the van der Waals force. The van der Waals intermolecular force between a nano-gyroscope tip mass and its sensing and driving electrodes was formulated by Mojahedi et al. (Mojahedi et al. 2014a) as:

$$F_{vdWv} = \frac{{AA_{v} }}{{6\pi \left( {g_{v} - v} \right)^{3} }}; F_{vdWw} = \frac{{AA_{w} }}{{6\pi \left( {g_{w} - w} \right)^{3} }}$$
(41)

where A is Hamaker’s constant.

The effects of intermolecular forces for nano-gyroscope applications have been extensively studied by Mojahedi and his collaborators. Starting in 2013, Mojahedi et al. (Mojahedi et al. 2013d) performed an analysis of the static pull-in and natural frequency of a nano cantilever gyroscope subjected to Casimir and van der Waals forces. Then Mojahedi et al. (Mojahedi et al. 2013c) examined the static pull-in instability of a cantilever nano-gyroscope including nonlinear geometric effects and intermolecular forces. Later, Mojahedi et al. (Mojahedi et al. 2014b) studied the pull-in instability of micro/nano-bridge gyroscopes with nonlinear mid-plane stretching effects, fringing field effects, and intermolecular forces. They determined that the Casimir and van der Waals forces significantly decreased the pull-in voltage of nano-gyroscopes. Mokhtari et al. (Mokhtari et al. 2017) analyzed the effects of the Casimir forces on the dynamic pull-in instability of a micro-bridge gyroscope and found that the intermolecular force played a significant role in decreasing the onset voltage of dynamic pull-in for small electrode gaps.

In 2013, Mojahedi et al. (Mojahedi et al. 2013a) made another contribution to the modeling enhancement of micro/nano-gyroscopes by introducing the squeeze film damping effect into their cantilever beam gyroscope model. The squeeze film damping effect occurs in micro-/nano-scale air gaps. When air is unable to escape from in between the electrode and proof mass, while the system is vibrating, a film of compressed air is formed, resulting in a damping force. Mojahedi et al. (Mojahedi et al. 2013a) showed that the squeeze film damping created a stiffening effect that significantly altered the nonlinear dynamic behavior of the cantilever micro-gyroscope system. Many MEMS are encapsulated in high vacuum environments; therefore, most researchers assume their system to be in a vacuum and neglect the squeeze film damping effect. Table 4 provides a summary of recent size-dependent models developed or the analysis of micro/nano-gyroscopes.

Table 4 Summary of micro-/nano-gyroscope research using size-dependent models

Finite element and computational modeling approaches for gyroscopes

The finite element method (FEM) is a numerical mathematical method involving the transformation of a continuum system to an equivalent discretized system represented by elements or cells (Kagawa et al. 1996). Such a technique can be implemented to develop computational models of micro/nano-gyroscopes using well-established software tools, such as ANSYS, COMSOL, NASTRAN, and SOLIDWORKS, etc… Nowadays, there exist several possible simulation options and selecting a method falls upon the application of the system itself and user preference. Since there exist many differences between software packages, one might excel in one type of stimulation but lack in another, it is up to the user to identify the advantages and disadvantages before integrating their models into that specific software. There are several advantages associated with the use of finite element modeling/analysis in comparison with its lumped-parameter modeling or reduced-order modeling counterparts. Some of these advantages include:

  • Possible representation of complex system geometry

  • More external parameters can be included (thermal flux, flow boundary layer, mechanical-electro coupling)

  • Capability to capture all coupled vibration modes

  • Visual representation of system displacements, mode shapes, etc.

Some possible drawbacks to computational models include:

  • High computational cost compared to lumped parameter and reduced order models

  • Long simulation run times

Advantageously commercially available computers are becoming more powerful every year and the availability of supercomputing clusters is also increasing. FEM constitutes one of the most flexible and accurate numerical approach to solve a multi-physics problem despite the high computational cost (Shakoor et al. 2008). Dynamic characteristics of the MEMS gyroscopes and impacts of the operating environment, such as thermal fluctuations can be analyzed using this method (Chang et al. 2010). With the use of finite element software, such as mentioned above the geometric complexity has little to no bounds, however, when it comes to modeling gyroscopes conceptually, the cantilever shaped beam is one of the most employed designs, due to its simplicity and flexibility for further performance enhancement (Erturk and Inman 2008). A cantilever beam can be defined as a structure with one end fixed and the other end free to vibrate made up of elastic metal (Kumari and Khanna 2016). Similarly, thin plate system configurations paved the way for the development of the gyroscopes of monolithic configuration. For representation of the characteristic prediction using finite element analysis, numerically, the work of Kagawa et al. (Kagawa et al. 1996) for piezoelectric vibratory gyroscopes is highlighted.

First, the case of the simplest monolithic gyroscope made of a thin piezoelectric square plate is considered, as shown in Fig. 15. The plate is driven by the pairs of four electrodes distributed around its center and the electrodes for detection are placed in its four comers. Pairs of partial electrodes are placed on both surfaces of the piezoelectric thin plate. Electrode arrangements and the finite element division are shown in Fig. 16.

Fig. 15
figure15

a In-plane motion (cross-sectional view) and b coordinates and Coriolis force in rotation for thin a plate piezoelectric gyroscope (Kagawa et al. 1996)

Fig. 16
figure16

Thin plate piezoelectric gyroscope-electrode arrangement and finite element division (Kagawa et al. 1996)

As the thin plate vibrates in its z-y plane, it is assumed that the strain is independent of the thickness or z-direction, and the electric field is parallel to the x-direction and constant. The work considered that the plate is divided into triangular elements. Within an element e, displacement vector \({u}_{e}=[{u}_{x}\left(x,y\right),{u}_{y}\left(x,y\right)]\) and the potential \({\varphi }_{e}=\varphi (x,y)\) are, respectively, defined and expressed in terms of the linear combination of the nodal displacement vector \({\xi }_{e}\) and nodal potential vector \({\phi }_{e}\) defined at the nodal coordinates \(({x}_{i},{y}_{i})\) where \(i=\mathrm{1,2},\dots ,6\) as (Kagawa et al. 1996):

$$u_{e} = \left[ {\begin{array}{*{20}c} {N^{*} } & 0 \\ 0 & {N^{*} } \\ \end{array} } \right]\xi_{e}$$
(42)
$$\varphi_{e} = N^{*}$$
(43)

where \({\xi }_{e}\) represents the nodal displacement, \({\phi }_{e}\) is the nodal potential vector, N represents the interpolation function vector, and (*) refers to the transpose of the matrix or vector. The strain vector \({\epsilon }_{e}\), which is the derivative of the displacement with respect to the coordinates, is expressed as (Kagawa et al. 1996):

$$\varepsilon_{e} = \frac{1}{{2\Delta_{e} }}\left[ {\begin{array}{*{20}c} {n^{*} } & 0 & 0 \\ 0 & {n^{*} } & 0 \\ 0 & 0 & {n^{*} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {B_{e} } & 0 \\ 0 & {C_{e} } \\ {B_{e} } & {C_{e} } \\ \end{array} } \right]\xi_{e}$$
(44)

where \({\Delta }_{e}\) denotes the area of the triangular element, n is the vector associated with the derivative of the interpolation function, and \({B}_{e}\), \({C}_{e}\), are the coefficient matrices associated with the derivative of the interpolation function with respect to the x- or y-direction. For a piezoelectric plate of thickness \(2{t}_{3}\). the z directional electric field \({E}_{3}\), in the element is expressed with the nodal potential vector \({\phi }_{e}\), as (Kagawa et al. 1996):

$$E_{3} = - \frac{{2\varphi_{e} }}{{2t_{3} }} = - \frac{1}{{t_{3} }}N^{*} \phi_{e}$$
(45)

The energy functional for a piezoelectric vibrator consists of strain energy, kinetic energy, electrostatic energy, and work done by external forces in the finite element method, the energy functional is minimized applying the variational principle. However, when the system dissipates or rotates, the energy function becomes complex. Here, an adjoint system is introduced to establish that the function is real (Gladwell 1966). Strain energy \({U}_{e}\), electrostatic energy \({H}_{e}\) and kinetic energy \({T}_{e}\), are thus defined for the element e as follows (Kagawa et al. 1996):

$$\begin{aligned} U_{e} & = \frac{1}{8}\iiint\limits_{e} {\left( {\varepsilon_{e}^{*}{\overline{\tilde{\sigma}}} + \tilde{\varepsilon }_{e}^{*} \overline{\sigma }_{e} + \overline{\varepsilon }_{e}^{*} \tilde{\sigma }_{e} + \overline{\tilde{\varepsilon }}_{e}^{*} \sigma_{e} } \right)dxdydz} \\ & = \frac{{t_{3} }}{2}\left( {\xi_{e}^{*} K_{e} \overline{{\tilde{\xi }}}_{e} + \tilde{\xi }_{e}^{*} K_{e} \overline{\xi }_{e} } \right) \\ & + \frac{{t_{3} }}{4}\left( {\xi_{e}^{*} P_{e} \overline{{\tilde{\phi }}}_{e} + \tilde{\xi }_{e}^{*} P_{e} \overline{\phi }_{e} + \overline{\xi }_{e}^{*} P_{e} \tilde{\phi }_{e} + \overline{{\tilde{\xi }}}_{e}^{*} P_{e} \phi_{e} } \right) \\ \end{aligned}$$
(46)
$$\begin{aligned} H_{e} & = \frac{1}{8}\iiint\limits_{e} {\left( {E_{e}^{*} {\overline{\tilde{D}}}_{e} + \tilde{E}_{e}^{*} \overline{D}_{e} + \overline{E}_{e}^{*} \tilde{D}_{e} + \overline{{\tilde{E}}}_{e}^{*} D_{e} } \right)}dxdydz \\ & = \frac{{t_{3} }}{2}\left( {\phi_{e}^{*} G_{e} \overline{{\tilde{\phi }}}_{e} + \tilde{\phi }_{e}^{*} G_{e} \overline{\phi }_{e} } \right) \\ & - \frac{{t_{3} }}{4}\left( {\phi_{e}^{*} P_{e} \overline{{\tilde{\xi }}}_{e} + \tilde{\phi }_{e}^{*} P_{e} \overline{\xi }_{e} + \overline{\phi }_{e}^{*} P_{e} \tilde{\xi }_{e} + \overline{{\tilde{\phi }}}_{e}^{*} P_{e} \xi_{e} } \right) \\ \end{aligned}$$
(47)
$$\begin{aligned} T_{e} & = \frac{1}{8}\iiint\limits_{e} {\left( {u_{e}^{*} \tilde{f} + u_{e}^{*} \overline{f}_{e} + \overline{u}_{e}^{*} \tilde{f}_{e} + \overline{{\tilde{u}}}_{e}^{*} f_{e} } \right)dxdydz} \\ & = \frac{{t_{3} }}{2}\omega^{2} \left[ {\xi_{e}^{*} \left( { I_{12} + \frac{{j^{2} \Omega }}{\omega }\Theta } \right) M_{e} \xi_{e} + \tilde{\xi }_{e}^{*} \left( {I_{12} + \frac{{j^{2} \Omega }}{\omega }\Theta } \right) M_{e} \overline{{\xi_{e} }} } \right] \\ \end{aligned}$$
(48)

where \({K}_{e}\) denotes the stiffness matrix, \({P}_{e}\) is the electromechanical coupling matrix, \({G}_{e}\) represents the electrostatic matrix, \(\Theta\) represents the matrix of rotation, \({M}_{e}\) is the mass matrix, and \({I}_{n}\) denotes the \(nxn\) unit matrix. The accent character, \((\stackrel{-}{})\) refers to the adjoint system and \(\left(\stackrel{\sim }{}\right)\) to its complex conjugate. No mechanical work externally applied is considered. As the system is mechanically and electrically damped due to structural and dielectric loss, both mechanical \({R}_{me}\) and electrical \({R}_{ge}\) energy losses must be included, which are given as follows (Kagawa et al. 1996):

$$\begin{aligned} R_{me} & = \frac{j}{{8Q_{me} }}\iiint\limits_{e} {\left( { - \varepsilon_{e}^{*} \tilde{\overline{\sigma }}^{E} + \tilde{\varepsilon }_{e}^{*} \overline{\sigma }_{e}^{E} + \overline{\varepsilon }_{e}^{*} \tilde{\sigma }_{e}^{E} - \overline{\tilde{\varepsilon }}_{e}^{*} \sigma_{e}^{E} } \right)dxdydz} \\ & = \frac{{jt_{3} }}{{2Q_{me} }}\left( { - \xi_{e}^{*} K_{e} \overline{{\tilde{\xi }}}_{e} + \tilde{\xi }_{e}^{*} K_{e} \overline{\xi }_{e} } \right) \\ \end{aligned}$$
(49)
$$\begin{aligned} R_{ge} & = \frac{j}{8}\tan \delta_{e} \iiint\limits_{e} {( - E_{e}^{*} \overline{\tilde{D}}_{e}^{T} + \tilde{E}_{e}^{*} \overline{D}_{e}^{T} + \overline{E}_{e}^{*} \tilde{D}_{e}^{T} - \overline{{\tilde{E}}}_{e}^{*} D_{e}^{T} )dxdydz} \\ & = j\frac{{t_{3} }}{2}\tan \delta_{e} \left( { - \phi_{e}^{*} G_{e} \overline{\tilde{\phi }}_{e} + \tilde{\phi }_{e}^{*} G_{e} \overline{\phi }_{e} } \right) \\ \end{aligned}$$
(50)

where \({Q}_{me}\) represents the mechanical quality factor of the material, \({\sigma }^{E}\) is the stress vector \(({E}_{3}=0)\), \(tan{\delta }_{e}\), is the tangent of dielectric loss factor, and \({D}^{T}\) is the dielectric flux density \((T=0)\). The dielectric loss is considered only over the region where the electrodes are provided. External work \(W\) is only electrically made through the electrodes \({A}_{P}\), due to the charges applied, which is given as follows (Kagawa et al. 1996):

$$\begin{aligned} W & = \frac{{t_{3} }}{2}\iint\limits_{{A_{p} }} {\left( {\varphi \hat{\overline{\tilde{D}}}_{n} + \tilde{\varphi }\hat{\overline{{D}}}_{n} + \overline{\varphi }\hat{{\tilde{D}}}_{n} + \overline{{\tilde{\varphi }}}\hat{D}_{n} } \right)dA} \\ & = \frac{{t_{3} }}{2}\left( {\phi_{e}^{*} \hat{{\tilde{q}}} + \tilde{\phi }^{*} \widehat{{\overline{q}}} + \overline{\phi }^{*} \hat{{\tilde{q}}} + \overline{\tilde{\phi }}^{*} \hat{q}} \right) \\ \end{aligned}$$
(51)

where \(D\) is the electric displacement flux density normal to the boundary and \(q\) is the nodal charge vector corresponding to the electrodes. (^) refers to the known value. Lagrangian \(\mathcal{L}\) for the whole system is thus given with the compatibility implied on the connecting nodes as follows (Kagawa et al. 1996):

$$\mathop \sum \limits_{e} \left( {U_{e} - H_{e} - T_{e} - R_{me} - R_{ge} } \right) - W$$
(52)

The application of the variational principle \((\delta \mathcal{L}=0)\) to the above equation yields four sets of the discretized equations of motion for the whole system; the one corresponding to the real system is given in as:

$$\left( {\left[ {\begin{array}{*{20}c} {K - \omega^{2} M} & {P_{1} } & {P_{2} } \\ {P_{1}^{*} } & { - G_{11} } & { - G_{12} } \\ {P_{2}^{*} } & { - G_{12}^{*} } & { - G_{22} } \\ \end{array} } \right] + j\left[ {\begin{array}{*{20}c} {\frac{1}{{Q_{m} }}\left( {K - 2\omega^{2} \frac{\Omega }{\omega } } \right)Q_{m} \Phi M} & 0 & 0 \\ 0 & {\tan \delta G_{11} } & {\tan \delta G_{12} } \\ 0 & {\tan \delta G_{12}^{*} } & {\tan \delta G_{22} } \\ \end{array} } \right]} \right)x\left[ {\begin{array}{*{20}c} \xi \\ {\phi_{1} } \\ {\phi_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ {\hat{q}} \\ 0 \\ \end{array} } \right]$$
(53)

Several works reported in the literature, for the past two decades (2000–2020), have been conducted to provide insights on the development in finite element modeling and analysis for gyroscope systems. An in-depth review of the types of equations and software used in the finite element analysis (FEA) of electrostatic and piezoelectric gyroscopes and their results are presented. Kagawa et al. (Kagawa et al. 1996) considered a two-dimensional in-plane vibration piezoelectric thin plate (Fig. 15), crossbar (Fig. 17), and ring-type gyroscope configuration.

Fig. 17
figure17

Cross bar gyroscope (Kagawa et al. 1996)

A finite element approach to the simulation of piezoelectric vibrator gyroscopes was presented for characteristic prediction. Equation 53 is implemented on a computer code to simulate the behavior of the piezoelectric vibrator gyroscopes. The gyroscopes considered here are made of a thin plate and have pairs of electrodes on both their faces for driving and detecting. They examined the effects of the rotation on the mode shapes, the resonant frequencies, and the transmission characteristics demonstrating the sensing capability against the rotation. Equation 53 suggests that the natural frequency must be chosen as low as possible for realizing the high sensitivity. The configuration made of crossed bars shown in Fig. 17 is a candidate in which two bars are coupled at their center, supported at some places. It was found that the sensitivity is improved as much as about ten times compared with that of the square plate gyroscope.

Kagawa et al. (Kagawa et al. 2001) developed a three-dimensional finite element model for a thin plate gyroscope. The solution characteristics were compared to those of a two-dimensional model, as shown in Fig. 18. The examination shows that, with this configuration, both the rotation and the gyratory axis can be detected in the three-dimensional space.

Fig. 18
figure18

Characteristics of the voltage in the vicinity of second mode: 3D and 2D comparison (Kagawa et al. 2001)

The work (Kagawa et al. 2001) was then extended to include a bimorph-type resonator vibrating out of the plane, which is only numerically considered for another example of demonstration. Kagawa et al. (Kagawa et al. 2006) proposed a tubular piezoelectric gyroscope which was examined experimentally and numerically using FEM simulations. Leakage output observed in the experimental system was identified through the simulation, which proves why it is a powerful tool for the design and analysis of the piezoelectric devices. Shakoor et al. (Shakoor et al. 2008) reported a comparative study of simulated three-degree of freedom full transient start-up analysis of a non-resonant micromachined MEMS gyroscope using FEA and System Model Extraction (SME) techniques. Initially, the MEMS system was analyzed at the device level, using FEM, to determine the natural frequencies of the micro-gyroscope using both static and dynamic responses of the system. After FEA simulations were analyzed, a generated system model of the gyroscope using SME was developed and used to run full transient response analysis and compared the two techniques (Shakoor et al. 2008). They found that by using SME, one can quickly and easily optimize the device design through parametric analysis. The SME technique took only a few hours for device optimization as compared to FEM with comparative and reliable results. It was also found that for transient analysis, FEA optimization took 45 h. For the same time interval, SME took 2–3 h, with is almost 15 times faster. Shakoor et al. (Shakoor et al. 2008) recommended using SME for optimization and FEA for comprehensive of the overall behavior of the system.

Wu et al. (Wu et al. 2009) proposed the first working piezoelectric micromachined modal gyroscope (PMMG) and modeled the system based on two different configurations. One of which is the piezoelectric rectangular parallelepiped with driving electrodes or sensing electrodes on the top and bottom surface of the body (Fig. 19a) and the other is the piezoelectric rectangular parallelepiped with concentrated masses at four corners of the top and bottom surface of the body (Fig. 19b). The modal and harmonic analyses were conducted, and the working model of the PMMG was obtained based on the finite element method analysis results for the two models. A kind of PMMG model is introduced as shown in Fig. 19a, which was first proposed by Maenaka et al. (Maenaka et al. 2006).

Fig. 19
figure19

Model 1 (a) and Model 2 (b) of PMMG (Wu et al. 2009)

Wu et al. (Wu et al. 2009) found that in order to increase the kinetic energy of the working resonant mode and to improve the vibration directivity of the piezoelectric block, concentrated masses in the corners of the top and bottom surfaces of the piezoelectric block are added, as shown in Fig. 19b, the sensing electrodes of S1, S2, S3, and S4 in model 1 are replaced by m1, m2, m3 and m4, respectively. It was found that the frequency of the working resonance of the system without concentrated masses is 326.549 kHz, while the system with concentrated masses resonated at 259.947 kHz. The decreased working frequency is favorable when designing a driving circuit with large output voltage and diminishing the noise influence. Adding concentrated masses also increased the modal displacement, stored energy, and sensitivity of the PMMG.

A piezoelectric driven mechanical actuator commonly known as mechanical dither in ring laser gyroscope (RLG) is commonly engaged to avoid the dead band. Yu and Long (Yu and Long 2015) presented an analytical model and a finite element model to investigate the resonant frequency as a function of geometry and material for mechanical dithers, referred to as A-C type, in a ring laser gyroscope. Yu and Long (Yu and Long 2015) verified their numerical results through comparison with experimental data. The maximum errors in the resonant frequencies were 3.14% and 1.79%, for analytical and FEM simulations respectively. The study was conducted using ANSYS 12.0 finite element program for modal analysis of the mechanical dither with bimorph piezoelectric actuators. The comparison of resonant frequencies between the analytical and FEM models is shown in Table 5. Table 6 contains a summary of previous computational works related to MEMS gyroscope design.

Table 5 Comparison of resonant frequencies for mechanical dither with piezoelectric actuators (Yu and Long 2015)
Table 6 Outline of computational models for multi-degree-of-freedom MEMS gyroscopes

Limitations and recommendations for future research

While there have been extensive modeling and simulation efforts for beam type vibratory gyroscopes, experimental studies are largely missing from the literature, especially for nano-scale systems. Experimental validation is a vital component for modeling any dynamical system. The main purpose of modeling micro/nano-gyroscopes is to match the real-world physics as close as possible so these models can be used to effectively design functional N/MEMS devices. Experimental studies have been mainly conducted on gyroscopes made of coupled and suspended masses driven by actuators (Hong et al. 2000; Painter and Shkel 2003; Cao et al. 2016; Giner et al. 2018; Liu et al. 2019a), tuning fork gyroscopes (Lavrik and Datskos 2019; Xu et al. 2019; Cao et al. 2020b), or vibrating ring gyroscopes (Cao et al. 2019; Efimovskaya et al. 2018, 2020; Zhang et al. 2012). Currently, there are many cantilever beam micro/nano-gyroscope models that have been developed without direct comparison to experiments. This can be mainly attributed to the difficulty of manufacturing high aspect ratio micro/nano-beams with tip masses with high degrees of precision (Gao and Huang 2017; Gentili et al. 2005).

Manufacturing defects and damage are both common in fabricated N/MEMS devices (Skogström et al. 2020). However, there are few studies in the literature concerning the operation of imperfect gyroscope systems. The micromachining processes used to build N/MEMS devices often create surface defects, such as grooves or pits (Cook et al. 2019). Micromachining may also produce imperfect structures with tampering sides (Liu et al. 2019b). Fatigue cracking and damage from shock loads have been demonstrated in metallic and silicon MEMS devices (Choa 2005). More research into the fatigue behavior and the operation of imperfect gyroscopes is needed to more accurately model the performance of real-world N/MEMS systems. Recently, Liu et al. (Liu et al. 2020) developed models for studying the degradation in the resonant frequency and sensitivity of butterfly resonator gyroscopes subjected to long-term static loading. They projected that the sensitivity of gyroscopes under these test conditions could degrade as much as 20.27% in 10 years. Recently, Larkin et al. (Larkin et al. 2020b) studied the static and nonlinear dynamic behaviors of beam micro-gyroscopes with multiple surface cracks. It was determined that crack location and severity greatly affect the operability of micro-gyroscopes. It should be mentioned that conventional vibrating micro-gyroscopes require tuning of both driving and sensing mode resonant frequencies to attain high sensitivity. Given the outstanding dynamic features associated with coupled vibrating beams, the development of a full mathematical model for a hybrid vibratory N/MEMS inertial sensor for simultaneous measurements of acceleration constitutes a promising research line to significantly advance the field of MEMS resonators. Further model enhancement can be achieved by verifying its robustness to noise, which may arise from microfabrication issues, measurement limitations, or perturbation factors.

Another open area for research is multifunctional energy harvesting operations. It has been extensively demonstrated that electrostatic (Mohanty et al. 2019), piezoelectric (Liang et al. 2017), and flexoelectric (Deng et al. 2014) energy harvesting methods are viable for small-scale power generations. However, limited efforts have been made towards the development of multifunctional harvesters where the output voltage could be generated using more than one input source, such as mechanical vibrations and magnetic energy (Gupta et al. 2018). For more examples regarding such systems, the reader is referred to the following references (Gupta et al. 2018; Karan et al. 2019; Khandelwal et al. 2019; Bai et al. 2017; Li et al. 2019b; Kim et al. 2017). Another category of a multifunctional energy harvester is for systems that can perform two separate tasks simultaneously. Here, it is referred to as the primary operating principle of the system, whatever that may be, and the act of harvesting energy at the same time. For example, gyroscopes have been commonly used as inertial sensors. However, gyroscopes constitute perfect candidates for multifunctional energy harvesting or self-powered systems because they are deployed in dynamic environments with abundant wasted mechanical vibrations that in such a way they can harvest energy while in operation. The idea of such multifunctional energy harvesters has become more and more appealing as there have been many different applications of NEMS/MEMS devices. For example, Cho et al. (Cho et al. 2019) demonstrated a road-compatible piezoelectric energy harvester (RPEH) that harness unused energy to power self-powered sensors and vehicle indicators. Similarly, Chew et al. (Chew et al. 2016) presented, in the context of wireless sensors autonomous in energy, a single macro-fiber composite (MFC) piezoelectric transducer which was used for the first time as a multifunctional device as both sensor and energy harvester in a time-multiplexing manner. Furthermore, Khandelwal et al. (Khandelwal et al. 2019) demonstrated a multifunctional device that can harvest biomechanical as well as wind energy. In addition to various applications as a continuous energy source, energy harvesters have demonstrated to be effective at different scales and applications. This concept of designing a single device based on a multi-functional energy harvester may open a new route to harvest energy together through multifunctional activities to the scientific community.

Conclusions

The current review consolidated and compared the existing modeling approaches used for the design and performance analysis of vibratory N/MEMS gyroscopes. Lumped-parameter models constituted useful tools to obtain simplified representations of complex nonlinear multi-degree of freedom systems. However, the broad assumptions taken to derive such models may lead to inaccurate predictions of the inertial sensor response. A more accurate approach is to use distributed-parameter models of continuous N/MEMS gyroscopes. These models can capture the inherent physical phenomena and size-dependent effects such as non-classical continuum mechanics theories and small-scale effects. However, the mathematical complexity of these models limits their applicability to simple geometries, such as micro-beams or -plates. Conversely, computational models, such as finite element simulations enable researchers to accurately study the performance of MEMS gyroscopes with complex geometries involving a large number of degrees of freedom. These types of simulations have the potential to best represent real-world N/MEMS devices. However, their associated high computation time limits their usage for optimization and design purposes.

The need for small, low power, accurate inertial sensors for modern electronics has driven the advancement of micro/nano-gyroscope sensing, and modeling techniques. Several sensing and actuating mechanisms such as electrostatic, electrical resistance, and piezoelectric transduction have been widely incorporated into N/MEMS gyroscope design. The quest for ever smaller and lower power gyroscopes has prompted the development of nanogyroscopes. To accurately model nano-scale gyroscopes, special considerations such as material microstructure, non-classical continuum models, and proximity effects have been incorporated into gyroscope models.

While many avenues of N/MEMS gyroscopes have been explored, there is still a room for improvement for this kind of inertial sensor. In this review, several open research topics have been identified including nanoscale experimentation for model validation, damage/fatigue modeling, and multifunctional gyroscope systems. This review stands as a point of reference for future researchers to highlight the current state of the art and expose possible directions for continued technological advancement of vibratory micro/nanogyroscopes.

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Acknowledgements

K. Larkin acknowledges the financial support from the Advanced Simulation and Computing (ASC)—Integrated Codes (IC) Program, through the Lagrangian Applications Project at Los Alamos National Laboratory.

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Larkin, K., Ghommem, M., Serrano, M. et al. A review on vibrating beam-based micro/nano-gyroscopes. Microsyst Technol (2021). https://doi.org/10.1007/s00542-020-05191-z

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