Mode Ordering in Tuning Fork Structures with Negative Structural Coupling for Mitigation of Common-mode g-Sensitivity Brenton R. Simon, Sambuddha Khan, Alexander A. Trusov, and Andrei M. Shkel Microsystems Laboratory, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA E-mail: brent.simon@gmail.com, sambuddk@uci.edu, alex.trusov@gmail.com, and ashkel@uci.edu Abstract— This paper reports a method of mode ordering in tuning fork structures, effectively inducing a negative coupling stiffness between the resonant proof masses. The coupling mechanism selectively stiffens the undesirable in-phase resonance mode and softens the desirable out-of-phase resonance, thus widening the frequency separation between the desirable and undesirable modes of vibration in tuning fork structures. In gyroscopes, the approach leads to improved robustness to fabrication imperfections and immunity to environmental vibrations, while at the same time enhancing the scale factor and reducing the noise. Advantages of the method are illustrated on a Quadruple Mass Gyroscope (QMG) architecture, which was previously reported. It is experimentally demonstrated that the common-mode g-sensitivity can be reduced by over 20 times with design modifications resulting in mode re-ordering. Keywords—mode ordering, g-sensitivity, negative coupling stiffness, Coriolis Vibratory Gyroscopes. I. INTRODUCTION The noise and stability of high-performance micromachined inertial sensors such as gyroscopes and accelerometers are limited by the Quality factor (Q) of devices [1]. The mechanical scale factor of Coriolis vibratory gyroscopes, as well as amplitude modulated accelerometers, is inversely proportional to natural frequency of the devices. The Q-factor can be improved by reducing the natural resonance frequency and thereby reducing thermoelastic damping [1]. However, reduction of operational frequency increases the influence of external acceleration, which is a disadvantage for high-performance inertial sensors. The influence of external common-mode acceleration can be minimized by using the anti-phase vibratory mode of the tuning fork resonators, without raising the natural frequency of the device. However, the in-phase vibratory mode is always the lower in frequency than the operational anti-phase mode, when conventional flexures are used. Such mode ordering [2] leads to a higher than desired anti-phase operational mode, which effectively reduces the mechanical scale factor. The design configuration also leaves the frequency of the in-phase vibratory mode lower in order of resonant frequencies, which is most sensitive to external acceleration, and most likely to respond to common-mode external acceleration. In this paper, we demonstrate that mode ordering is necessary to improve the performance of inertial sensors. Our study includes both analytical modeling and experimental results. A dual-axis anti-phase tuning fork resonator is used for this purpose. Mode ordering is achieved by designing a suitable coupling structure that switches the in-phase and anti- phase resonances of the tuning fork resonator in the frequency spectrum. This leads to a low anti-phase resonance with a high mechanical scale factor, while pushing the acceleration- sensitive in-phase resonance to high frequencies, the frequencies of perturbation that the device is less likely to experience. II. A DUAL-AXIS ANTI-PHASE TUNING FORK RESONATOR A dual-axis tuning fork resonator with eight degrees of freedom uses the same design principals of a classic Coriolis vibratory gyroscope with a higher degree of symmetry. An example of a dual-axis tuning fork device is shown in Fig. 1. Fig. 1. An optical photograph of a fabricated and packaged QMG devices with SEM images of the supporting and coupling springs. This is an in-house fabricated and packaged Quad-Mass- Gyroscope (QMG), [3]. As it can be seen in this image, the structure consists of four identical proof masses (individual This work was supported by DARPA under grant N66001-12-C-4035.