Nonlinear vibration and performance analysis of a hybrid-driving T-beam micro-gyroscope with 2:1 internal resonance

Abstract

This paper is based on a T-beam resonator with the addition of a piezoelectric structure. T-beam resonator is redesigned as the T-beam micro-gyroscope with a hybrid electrostatic-piezoelectric driving. The sensing and driving modes of the gyroscope are designed to have a 1:2 frequency ratio with the lower-order sensing modes excited by two-to-one internal resonance. Specifically, the dynamics of the gyroscope with hybrid-driving is modeled by using Lagrange’s equations. The two flexural modes of in-plane modes of T-beam are obtained by using Hamilton's principle and the beam is dimensioned to obtain an accurately tuned two-to-one internal resonance structure. Lagrange’s equation is used to develop a dynamical model and the Galerkin method is used to get a reduced-order model with two degrees of freedom. The model preserves the quadratic nonlinearities of the coupled modes while retaining the high-order terms of the electrostatic force to study the static pull-in voltage. The response of the gyroscope for three sensing mode bandwidths in vacuum and non-vacuum environments with two quality-factors is analyzed. The results show that the internal resonance can significantly improve the response bandwidth of the gyroscope, while the input energy of gyroscope increases significantly under the effect of hybrid-driving, which makes the performance of the gyroscope improve in both operating environments. In particular, the advantage of the hybrid-driving is more obvious in the lower quality-factor operating environment. It proves the great potential of T-beam gyroscope with hybrid-driving.

Appendix

Coefficient (Г₁₁, Г₁₂, Г₂₂) of kinetic energy term and coefficients (W₁₁, W₁₂, W₂₂) of potential energy term are as follows:

$$ \Gamma_{ij} = \tfrac{1}{2}\sum\limits_{k = 1}^{3} {r_{k} } \vartheta_{k} \int_{0}^{1} {\phi_{ik} \phi_{jk} {\text{d}}s_{k} } + \tfrac{1}{2}r_{k} \vartheta_{k} $$

(52)

$$ W_{ij} = \tfrac{1}{2}\sum\limits_{k = 1}^{3} {\tfrac{{\alpha_{k} }}{{\vartheta_{k}^{3} }}\int_{0}^{1} {\phi^{\prime\prime}_{ik} } \phi^{\prime\prime}_{jk} {\text{d}}s_{k} } $$

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Bi-mode approximation for axial displacement:

$$ u_{k} = \Lambda_{11k} A_{1}^{2} + 2\Lambda_{12k} A_{1} A_{2} + \Lambda_{22k} A_{2}^{2} ,\quad k = 1,2,3 $$

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Λ_ijk is a spatial function of the dimensionless arc length $s_{k}$:

$$ \begin{gathered} \Lambda_{ijk} = \tfrac{1}{2}\tfrac{{\vartheta_{k} }}{{\vartheta_{1} + \vartheta_{2} }}s_{k} \left(\sum\limits_{n = 1}^{2} {\tfrac{1}{{\vartheta_{n} }}} \int_{0}^{1} {\phi^{\prime}_{in} } \phi^{\prime}_{jn} {\text{d}}s_{n} \right) - \tfrac{1}{2}\tfrac{1}{{\vartheta_{k} }}\int_{0}^{{s_{k} }} {\phi^{\prime}_{ik} } \phi^{\prime}_{jk} {\text{d}}s_{k} ,\quad k = 1,2 \hfill \\ \Lambda_{ij3} = - \tfrac{1}{2}\tfrac{1}{{\vartheta_{3} }}\int_{0}^{{s_{3} }} {\phi^{\prime}_{i3} } \phi^{\prime}_{j3} {\text{d}}s_{3} \hfill \\ \end{gathered} $$

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Define the coefficient $Q_{ij}$ as follows:

$$ Q_{ij} = r_{3} \vartheta_{3} \int_{0}^{1} {\tilde{Q}_{ij} } {\text{d}}s_{3} $$

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where,

$$ \begin{aligned} \tilde{Q}_{00} & = 2(\phi_{11} \left| {_{{s_{1} = 1}} } \right.\Lambda_{113} - \Lambda_{111} \left| {_{{s_{1} = 1}} } \right.\phi_{13} )\quad \quad \, \tilde{Q}_{10} = 2(\phi_{11} \left| {_{{s_{1} = 1}} } \right.\Lambda_{123} - \Lambda_{121} \left| {_{{s_{1} = 1}} } \right.\phi_{13} ) \\ \tilde{Q}_{01} & = 2(\phi_{11} \left| {_{{s_{1} = 1}} } \right.\Lambda_{123} + \phi_{21} \left| {_{{s_{1} = 1}} } \right.\Lambda_{113} \quad \quad \, \tilde{Q}_{11} = 2(\phi_{11} \left| {_{{s_{1} = 1}} } \right.\Lambda_{223} + \phi_{21} \left| {_{{s_{1} = 1}} } \right.\Lambda_{123} \\ & \quad - \Lambda_{121} \left| {_{{s_{1} = 1}} } \right.\phi_{13} - \Lambda_{111} \left| {_{{s_{1} = 1}} } \right.\phi_{23} ) \, \quad \quad \quad \quad \quad - \Lambda_{221} \left| {_{{s_{1} = 1}} } \right.\phi_{13} - \Lambda_{121} \left| {_{{s_{1} = 1}} } \right.\phi_{23} ) \\ \tilde{Q}_{02} & = 2(\phi_{21} \left| {_{{s_{1} = 1}} } \right.\Lambda_{123} - \Lambda_{121} \left| {_{{s_{1} = 1}} } \right.\phi_{23} )\quad \quad \tilde{Q}_{12} = 2(\phi_{21} \left| {_{{s_{1} = 1}} } \right.\Lambda_{223} - \Lambda_{221} \left| {_{{s_{1} = 1}} } \right.\phi_{23} ) \\ \end{aligned} $$

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Coefficients in Eq. (33):

$$ \begin{aligned} n_{11} & = \tfrac{{Z_{01} }}{{2\Gamma_{11} }};n_{12} = \tfrac{1}{{\Gamma_{11} }}( - \omega_{1}^{2} (Q_{00} A_{10} + Q_{10} A_{20} )); \\ n_{13} & = \tfrac{1}{{2\Gamma_{11} }}(E_{12} - \omega_{2}^{2} (2\Gamma_{12} + Q_{01} A_{10} \\ & \quad + Q_{11} A_{20} ) + (2W_{12} - f_{22} F_{01} - h_{22} F_{02} )); \\ n_{14} & = - \tfrac{1}{{\Gamma_{11} }}\omega_{1}^{2} Q_{00} ;n_{15} = - \tfrac{1}{{2\Gamma_{11} }}\omega_{2}^{2} Q_{11} ; \\ n_{16} & = - \tfrac{1}{{2\Gamma_{11} }}(\omega_{2}^{2} Q_{01} + 2\omega_{1}^{2} Q_{10} );n_{17} = \tfrac{1}{{2\Gamma_{11} }}Q_{00} ; \\ n_{18} & = \tfrac{1}{{2\Gamma_{11} }}(Q_{02} + Q_{11} );n_{19} = \tfrac{1}{{2\Gamma_{11} }}(Q_{01} + Q_{10} ); \\ n_{21} & = \tfrac{{Z_{10} }}{{2\Gamma_{22} }};n_{22} = \tfrac{1}{{2\Gamma_{22} }}( - \omega_{1}^{2} (2\Gamma_{12} + Q_{01} A_{10} \\ & \quad + Q_{11} A_{20} ) + (2W_{12} - f_{22} F_{01} - h_{22} F_{02} )); \\ n_{23} & = \tfrac{1}{{\Gamma_{22} }}( - \omega_{2}^{2} (Q_{02} A_{10} + Q_{12} A_{20} ));n_{24} = - \tfrac{1}{{2\Gamma_{22} }}\omega_{2}^{2} Q_{01} ; \\ n_{25} & = - \tfrac{1}{{\Gamma_{22} }}\omega_{1}^{2} Q_{12} ;n_{26} = - \tfrac{1}{{2\Gamma_{22} }}(\omega_{2}^{2} Q_{11} + 2\omega_{1}^{2} Q_{02} ); \\ n_{27} & = \tfrac{1}{{2\Gamma_{22} }}(Q_{01} + Q_{10} );n_{28} = \tfrac{1}{{2\Gamma_{22} }}Q_{12} ; \\ n_{29} & = \tfrac{1}{{2\Gamma_{22} }}(Q_{02} + Q_{11} ) \\ \end{aligned} $$

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