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Parametric Resonance Amplification in a MEMGyroscope Laura A. Oropeza-Ramos
Kimberly L. Turner
Mechanical and Environmental Engineering University of California, Santa Barbara [email protected]
Mechanical and Environmental Engineering University of California, Santa Barbara [email protected]
Abstract— A feasibility analysis of using parametric amplification in order to improve the performance of a micromachined rate gyroscope is presented. The actuation is based on variable force actuators that depend on the force-time variation relation, thus the nonlinear Mathieu equation governs the driving motion. We demonstrated by numerical simulations that in a parametric amplified system with linear mechanical stiffness there exist a flattened region that can be used as the operation region of a 2DOF microgyroscope. In this way the system is able to achieve a wider bandwidth of operation without sacrificing the sense mode gain: a robust micro sensor that is less sensitive to parameter variations.
parameter fluctuations. This work motivates the interest of analyzing other types of amplification schemes in the mechanical domain. Parametrically excited systems are described by differential equations in which the input appears as a time dependent coefficient. One of the most notable peculiarity of this systems is that large responses can be generated even if the excitation frequency is far away from the natural frequency. Parametric resonance studies in micro-oscillators have been previously presented in [5], [6], [7] In this paper we present a mathe-
I. I NTRODUCTION Leon Foucault (1819-1868), a 19th-century French scientist and physicist, is responsible for giving the name gyroscope to a wheel or rotor mounted in gimbal rings. Leon Foucault named his gyroscope in 1852 [1], instrument that was used by him as a tool to study the earth’s rotation. All vibratory gyroscopes are based on the transfer of energy between two vibration modes of a structure caused by Coriolis [2]. There is a wide variety in the design and operating principles of micromachined gyroscopes [3]. However, almost all reported use vibrating mechanical elements to sense rotation. Because they are not based on rotational parts are suitable for batch micro-fabrication. Significant effort has been put into development of conventional micro angular rate sensors operated in the linear regime (Simple Harmonic Oscillators) [2], [3]. With this design type there is always a tradeoff between the signal amplitude (quality factor Q) and the bandwidth of operation. In order to achieve high sensitivity, the drive and the sense resonant frequencies are typically designed and tuned to match, and the device is controlled to operate at or near the peak of the response curve. A system that requires this mode matching is very sensitive to parameter variations due to fabrication imperfections and operating conditions. Acar [4] proposed a 4-DOF nonresonant micromachined gyroscope design based on forming mechanically decoupled 2DOF drive-mode oscillator and 2-DOF sense-mode oscillator using three interconnected proof masses. The system utilizes dynamical amplification in the drive and sense directions to achieve large oscillation amplitudes without resonance resulting in increased bandwidth and reduced sensitivity to
Fig. 1.
Model of a 2-DOF vibratory gyroscope.
matical analysis and simulations of the response of a 2-DOF micro gyroscope amplified by parametric resonance. We show that the use of an actuator which driving force is proportional to displacement allows for pumping the system far from its natural frequency. At the same time the implementation of a linear mechanical restoring force will generate a flattened sensing response. This behavior implies a system with large amplitude and wide bandwidth, less sensitive to parameter variations, that can be used in an augmented range of operation. This analysis on parametric amplification provides insight into possible design rules to achieve robust micro-gyroscopes, using parametric pumping. II. M ODEL D ESCRIPTION A. Dynamics of a Vibratory Micro Gyroscope A conventional vibratory gyroscope can be modeled as two independent spring-mass-damper oscillators coupled by
TABLE I N ONDIMENSIONAL PARAMETERS
βx = δx =
2cx mω
2 2r1 VA mω 2
2 4(kx1 +r1 VA ) mω 2
δy = γ=
4ky mω 2 4Ωz ω
2cy mω
αy = βx3 =
c0
2 2r3 VA mω 2
c0.1
Ε
αx =
2 4(kx3 +r3 VA ) mω 2
c0.2
δy3 =
4ky3 mω 2
c0.3
(·) =
d(·) dτ
δx3 =
∆1
Coriolis force (see Fig.1). The governing equations of this 2DOF system in the primary mode (x-direction) and secondary mode (y-direction) can be expressed as: m¨ x + cx x˙ + Fr (x) = Fa + 2mΩz y˙ m¨ y + cy y˙ + Fr (y) = −2mΩz x˙
(1)
where m is the mass, cx and cy are the damping coefficients, Fr (x) and Fr (y) represent the elastic restoring forces provided by the springs in the x and y direction respectively and Fa correspond to actuation force. Note that the terms 2mΩz x˙ and 2mΩz y˙ represent the rotation-induced Coriolis forces and the angular rotation is considered to be constant. The restoring force from the mechanical springs is generally modeled as: (2) Fr (·) = k1 (·) + k3 (·)3 In this particular case, in order to generate a parametric resonance response, the actuation force is generated by a set of non-interdigitated comb-fingers described by [8]: Fa (x, t) = −(r1 x + r3 x3 )(V (t))2
(3)
where r1 and r3 are electrostatic coefficients that depend on the physical dimensions and spacings of the electrostatic comb drives, and V (t) is the voltage applied across the drives. Note that Fa is a function of displacement x and time t. Since the electrostatic force has a square dependence on the applied voltage we use a square rooted sinusoidal signal 1 V = VA (1 + cos(ωt)) 2 in order to isolate the parametric effects from the harmonic response. Substituting (3) and (2) in (1) and rescaling time as 2τ = ωt: x + αx x + (δx + 2βx cos 2τ )x + +(δx3 + 2βx3 cos 2τ )x3 − γ y˙ = 0 y + αy y + δy y + δy3 y 3 + γx = 0
(4)
where the derivative operator and the nondimensional parameters are defined as stated in Table I. The expression for the primary mode (x-direction) in (4) is in the form of a nonlinear Mathieu equation which is a second order nonlinear differential equation with periodic coefficients. In contrast with the harmonic response, in which the excitation appears as an inhomogeneity in the governing differential equations, in this case the excitation appears as a coefficient in the governing differential equations. In a harmonic oscillator a
Fig. 2. Theoretical regions of parametric resonance and amplification in the (−δ) parameter space for the principal parametric resonance, as the damping coefficient is varied.
small excitation can produce a large response only when the driving frequency is close to the linear natural frequency and the amplitude depends on the damping coefficient (Q-factor). In contrast, a small parametric excitation can produce a large response when the driving frequency is away from the natural frequency of the system and the amplitude is not dependent on the damping term. B. Nonlinear Mathieu Equation coupled to Harmonic response It is important to notice that the sensing oscillator that resonates along the y-direction is still expressed as a harmonic differential equation as can be seen in (4); however, the motion is induced by the transfer of energy through the coupler Coriolis normalized term γ that relates the angular rotation ˙ Since it is not possible to Ωz and the parametric velocity x. obtain an exact closed form solution of the nonlinear Mathieu equation, the stability of solutions of this class of equations has been examined and characterized in terms of transition curves that define the boundaries of instability regions [9], [10]. The general procedure of perturbation theory is to identify a small parameter, usually denoted by , such that when is zero the problem becomes solvable. Considering that x
Kimberly L. Turner
Mechanical and Environmental Engineering University of California, Santa Barbara [email protected]
Mechanical and Environmental Engineering University of California, Santa Barbara [email protected]
Abstract— A feasibility analysis of using parametric amplification in order to improve the performance of a micromachined rate gyroscope is presented. The actuation is based on variable force actuators that depend on the force-time variation relation, thus the nonlinear Mathieu equation governs the driving motion. We demonstrated by numerical simulations that in a parametric amplified system with linear mechanical stiffness there exist a flattened region that can be used as the operation region of a 2DOF microgyroscope. In this way the system is able to achieve a wider bandwidth of operation without sacrificing the sense mode gain: a robust micro sensor that is less sensitive to parameter variations.
parameter fluctuations. This work motivates the interest of analyzing other types of amplification schemes in the mechanical domain. Parametrically excited systems are described by differential equations in which the input appears as a time dependent coefficient. One of the most notable peculiarity of this systems is that large responses can be generated even if the excitation frequency is far away from the natural frequency. Parametric resonance studies in micro-oscillators have been previously presented in [5], [6], [7] In this paper we present a mathe-
I. I NTRODUCTION Leon Foucault (1819-1868), a 19th-century French scientist and physicist, is responsible for giving the name gyroscope to a wheel or rotor mounted in gimbal rings. Leon Foucault named his gyroscope in 1852 [1], instrument that was used by him as a tool to study the earth’s rotation. All vibratory gyroscopes are based on the transfer of energy between two vibration modes of a structure caused by Coriolis [2]. There is a wide variety in the design and operating principles of micromachined gyroscopes [3]. However, almost all reported use vibrating mechanical elements to sense rotation. Because they are not based on rotational parts are suitable for batch micro-fabrication. Significant effort has been put into development of conventional micro angular rate sensors operated in the linear regime (Simple Harmonic Oscillators) [2], [3]. With this design type there is always a tradeoff between the signal amplitude (quality factor Q) and the bandwidth of operation. In order to achieve high sensitivity, the drive and the sense resonant frequencies are typically designed and tuned to match, and the device is controlled to operate at or near the peak of the response curve. A system that requires this mode matching is very sensitive to parameter variations due to fabrication imperfections and operating conditions. Acar [4] proposed a 4-DOF nonresonant micromachined gyroscope design based on forming mechanically decoupled 2DOF drive-mode oscillator and 2-DOF sense-mode oscillator using three interconnected proof masses. The system utilizes dynamical amplification in the drive and sense directions to achieve large oscillation amplitudes without resonance resulting in increased bandwidth and reduced sensitivity to
Fig. 1.
Model of a 2-DOF vibratory gyroscope.
matical analysis and simulations of the response of a 2-DOF micro gyroscope amplified by parametric resonance. We show that the use of an actuator which driving force is proportional to displacement allows for pumping the system far from its natural frequency. At the same time the implementation of a linear mechanical restoring force will generate a flattened sensing response. This behavior implies a system with large amplitude and wide bandwidth, less sensitive to parameter variations, that can be used in an augmented range of operation. This analysis on parametric amplification provides insight into possible design rules to achieve robust micro-gyroscopes, using parametric pumping. II. M ODEL D ESCRIPTION A. Dynamics of a Vibratory Micro Gyroscope A conventional vibratory gyroscope can be modeled as two independent spring-mass-damper oscillators coupled by
TABLE I N ONDIMENSIONAL PARAMETERS
βx = δx =
2cx mω
2 2r1 VA mω 2
2 4(kx1 +r1 VA ) mω 2
δy = γ=
4ky mω 2 4Ωz ω
2cy mω
αy = βx3 =
c0
2 2r3 VA mω 2
c0.1
Ε
αx =
2 4(kx3 +r3 VA ) mω 2
c0.2
δy3 =
4ky3 mω 2
c0.3
(·) =
d(·) dτ
δx3 =
∆1
Coriolis force (see Fig.1). The governing equations of this 2DOF system in the primary mode (x-direction) and secondary mode (y-direction) can be expressed as: m¨ x + cx x˙ + Fr (x) = Fa + 2mΩz y˙ m¨ y + cy y˙ + Fr (y) = −2mΩz x˙
(1)
where m is the mass, cx and cy are the damping coefficients, Fr (x) and Fr (y) represent the elastic restoring forces provided by the springs in the x and y direction respectively and Fa correspond to actuation force. Note that the terms 2mΩz x˙ and 2mΩz y˙ represent the rotation-induced Coriolis forces and the angular rotation is considered to be constant. The restoring force from the mechanical springs is generally modeled as: (2) Fr (·) = k1 (·) + k3 (·)3 In this particular case, in order to generate a parametric resonance response, the actuation force is generated by a set of non-interdigitated comb-fingers described by [8]: Fa (x, t) = −(r1 x + r3 x3 )(V (t))2
(3)
where r1 and r3 are electrostatic coefficients that depend on the physical dimensions and spacings of the electrostatic comb drives, and V (t) is the voltage applied across the drives. Note that Fa is a function of displacement x and time t. Since the electrostatic force has a square dependence on the applied voltage we use a square rooted sinusoidal signal 1 V = VA (1 + cos(ωt)) 2 in order to isolate the parametric effects from the harmonic response. Substituting (3) and (2) in (1) and rescaling time as 2τ = ωt: x + αx x + (δx + 2βx cos 2τ )x + +(δx3 + 2βx3 cos 2τ )x3 − γ y˙ = 0 y + αy y + δy y + δy3 y 3 + γx = 0
(4)
where the derivative operator and the nondimensional parameters are defined as stated in Table I. The expression for the primary mode (x-direction) in (4) is in the form of a nonlinear Mathieu equation which is a second order nonlinear differential equation with periodic coefficients. In contrast with the harmonic response, in which the excitation appears as an inhomogeneity in the governing differential equations, in this case the excitation appears as a coefficient in the governing differential equations. In a harmonic oscillator a
Fig. 2. Theoretical regions of parametric resonance and amplification in the (−δ) parameter space for the principal parametric resonance, as the damping coefficient is varied.
small excitation can produce a large response only when the driving frequency is close to the linear natural frequency and the amplitude depends on the damping coefficient (Q-factor). In contrast, a small parametric excitation can produce a large response when the driving frequency is away from the natural frequency of the system and the amplitude is not dependent on the damping term. B. Nonlinear Mathieu Equation coupled to Harmonic response It is important to notice that the sensing oscillator that resonates along the y-direction is still expressed as a harmonic differential equation as can be seen in (4); however, the motion is induced by the transfer of energy through the coupler Coriolis normalized term γ that relates the angular rotation ˙ Since it is not possible to Ωz and the parametric velocity x. obtain an exact closed form solution of the nonlinear Mathieu equation, the stability of solutions of this class of equations has been examined and characterized in terms of transition curves that define the boundaries of instability regions [9], [10]. The general procedure of perturbation theory is to identify a small parameter, usually denoted by , such that when is zero the problem becomes solvable. Considering that x
