Development of Piezoelectric Gyroscopes of Monolithic Configuration for Detection of Three-dimensional Rotation

利用有限元方法，研制了压电式陀螺仪的数值模拟器。给出了三维模型的有限元公式，其中包括绕任意轴旋转的科里奥利力的影响。各向异性弹性也包括在公式中。对于振动模式和回转轴的方向没有限制。通过数值算例，分别给出了由x切割石英晶体制成的音叉陀螺仪和l形陀螺仪。与激振器相关的模式，面内和面外模式，被利用，激振器被激发，并由提供在激振器表面上的电极对检测。在这些陀螺仪中，激励模式（面内振动）和检测模式（面外振动）的固有频率本质上是不同的，因此通过优化振子的形状来人为地使两种模式简并。结果表明，陀螺仪可以使用由两种不同模态组成的简并模态来工作。结果还表明，陀螺仪对任意轴的旋转具有灵敏度。使用这些陀螺仪可以分别检测绕z轴和x或y轴的旋转角频率。上述结果表明，所开发的三维仿真器对压电陀螺仪的分析和设计是有用的。

A numerical simulator for piezoelectric gyroscopes is developed using the finite element method. The finite element formulation is given for three-dimensional models including the effect of Coriolis force due to rotation around the arbitrary axis. Anisotropic elasticity is also included in the formulation. There is no limitation for the vibrational modes and the direction of the gyratory axis.For numerical examples, a tuning fork gyroscope and a L-shaped gyroscope made of X-cut quartz crystal are respectively demonstrated. The modes associated with the vibrator, in-plane and out-of-plane modes, are utilized for which the vibrator is excited and detected by pairs of the electrodes provided over the vibrator surfaces. In these gyroscopes the natural frequencies of the excitation mode (in-plane vibration) and that of the detection mode (out-of-plane vibration) are inhelently different, so it is made to artificially degenerate both modes by optimizing the shape of the vibrator. It is shown that the gyroscopes can be operated by using the degenerate mode consisting of two different modes. It is also shown that the gyroscopes has a sensitivity to the rotation around the arbitrary axis. It is possible to separately detect the rotational angular frequency around the z axis and x or y axes using these gyroscopes. It is shown in the above results that the developed three-dimensional simulator is useful to analyze or design of the piezoelectric gyroscopes.