Abstract
This work presents the analysis of a ring-type motion sensor which can sense the angular rate about the three axes besides the three components of linear acceleration. It integrates the functions of three-axis gyroscope and three-axis accelerometer. The design of the sensor is based on a vibrating ring shell structure. Translational modes of the ring are used to sense acceleration, and flexural modes are used to detect angular rates. Derivation of the equations of motion of the sensor using a Lagrangian method is detailed, and final expressions for the sensor’s sensitivity are obtained and verified by finite element analysis.
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Abbreviations
- \(\vec{u}\) :
-
Vibration-induced displacement of ring shell due to excitation of different modes
- \(p\) :
-
Independent spatial coordinate(s)
- \(q_{i} \left( t \right)\) :
-
Generalized (modal) coordinates as a function of time
- \(\varphi_{i} (p)\) :
-
Ring mode shapes
- Xo Yo Zo :
-
Inertial frame of reference
- X Y Z:
-
Translating and rotating (non-inertial) frame of reference
- \(\overrightarrow {U}_{arb}\) :
-
Position of a point on the ring structure relave to the inertial frame (Xo Yo Zo)
- \(\vec {r}_{o}\) :
-
Position vector from the origin of the inertial frame to the origin of the non-inertial frame
- \(\vec {r}_{p}\) :
-
Position vector from the center of the ring structure to a point on the undeformed ring
- \(\vec{u}_{T} ,\vec{u}_{F}\) :
-
Displacements of the center and a point on the ring structure due to translation and flexure respectively
- \(\vec {v}_{p}\) :
-
The velocity of a point on the ring
- \(\vec{\varOmega }\) :
-
Applied rate of turn
- \(C_{{r,\theta_{k} }}\) :
-
Capacitance of the electrodes located radially around the ring shell
- \(C_{{z,\theta_{k} }}\) :
-
Capacitance of the electrodes under the ring shell
- \({\text{R}}_{\text{ring}}\) :
-
Radius of the ring’s centre line in the unstressed stat
- \({\text{W}}_{\text{ring}}\) :
-
Thickness of the ring shell
- \({\text{h}}_{\text{ring}}\) :
-
Height of the ring structure
- \({\text{g}}_{\text{r}}\) :
-
Initial gap between a radial electrode and the ring structure
- \({\text{g}}_{\text{z}}\) :
-
Initial vertical gap between a bottom electrode and the ring structure
- \(\Delta \theta_{i}\) :
-
Electrode arc angle
- \(u_{r}(\theta) ,\,u_{z}(\theta)\) :
-
In-plane and out-of-plane displacements of the ring structure
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Abdul-wahed, A.M., Mahmoud, M.A.E. Vibratory ring six-axis motion sensor. Microsyst Technol 21, 2133–2144 (2015). https://doi.org/10.1007/s00542-014-2308-3
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DOI: https://doi.org/10.1007/s00542-014-2308-3