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A rocking mass gyroscope (RMG) is a kind of vibrating mass gyroscope with high sensitivity, whose driving mode and sensing mode are completely uniform. MEMS RMG devices are a research hotspot now because they have the potential to be used in space applications. Support loss is the dominant energy loss mechanism influencing their high sensitivity. An accurate analytical model of support loss for RMGs is presented to enhance their

A rocking mass gyroscope is a kind of dual-axial symmetric vibrating mass gyroscope, consisting of four slender beams attached to a rocking mass post in the middle [

Another important factor, influencing the high sensitivity of RMG, is its _{air}_{support}_{ted}_{surface}_{other}_{air}

_{ted}_{surface}_{other}_{support}_{support}

This paper thus aims to provide an accurate analytical model of support loss for an RMG to enhance its

Support loss depends mainly on the anchor types and materials. The energy lost from micro resonators into the support structure is summarized by three cases: the first case that acts as semi-infinite elastic medium with effectively infinite thickness, the second case that is treated as a plate with in-plane flexural vibrations, and the last case that can be treated as a plate with out-of-plane flexural vibrations. The first case, which is applicable to NEMS resonators, can be solved by modeling as a semi-infinite elastic medium with loads applied at a single point on surface of the half space [

The operational modes of RMG are simulated as shown in

The dimensions of the attachment point are small compared with the vibration wavelength in the substrate at the natural frequency of RMG. So the support case of an RMG can be treated as a 3-D problem with a finite thickness plate. For the case of a support with finite thickness, we consider a semi-infinite plate with a thickness that need not be the same as that of the vibrating structure itself. Using the plate-edge admittance results [

The power lost from a RMG flows mainly into the support structure. Support loss for the RMG can be simplified as a model with a beam attached to the rigid support structure at its end. The model and its main structural parameters are shown in _{1}, _{p}

In such cases, the effects of the microgyroscope on its substrate can be modeled as harmonic point forces and moments acting at the attachment point. An assumption is given: the energy propagated into the support structure would not be reflected, viz., all energy that reaches the support is considered lost.

The estimated

The operational modes of a RMG can be considered equivalent to a superposition of the rocking mode and torsional mode of an equivalent single degree of freedom (SDOF) system, which has the rocking mass post and two beams in line. The loads of the attachment points are described in _{z}_{b}_{t}_{0}.

Using micro scanners’ vibration mode frequency results [_{0}_{1} is the length of the beams, _{2} is the length of the center support. _{p}_{y}

The energy loss can be found by considering the plate support responding to the loads applied to its edge because the support structure is modeled as a plate. Consider the shear force _{z}_{b}_{t}

The point mobility matrix Y relates the normal angular velocity _{b}_{t}_{z}_{p}^{3}/12(1 − ^{2}), _{p}_{p}^{1/2}]^{1/2}. The corresponding coefficients have been calculated, for

The resulting expressions for the power radiated into the plate support are:

When each load is considered individually, the

Using Raleigh’s method, the kinetic energy stored and the effective inertia of the bending beams can be expressed, respectively as [

Then, the kinetic energy and effective inertia of the torsion beams and rocking mass post are also solved:
_{0} sin(_{0} cos(_{m1} is the mass of the center support, and _{m2} is the mass of the rocking mass post. The formulae of the vibratory energy are also scaled by _{0}, the arbitrary amplitude of the rocking vibration mode shape.

The vibration energy and the resonant frequency for the operational modes of RMG are expressed, respectively as:
_{xx} = J_{yy} = J_{m} + 2J_{b} + 2J_{t}

Given _{p}^{3}, _{2}/_{1})^{2}/2 _{2}/_{1} + 2/3. It can be seen from _{p}_{z}_{b}_{t}

For _{1} = 2,200, _{2} = 2,200, _{1} = 5,000, _{p}_{p}_{p}

In this section, the sensitivities of support loss to the main structural parameters involved, such as sizes of the rocking mass post, center support, and beams are studied.

A plot of support loss for _{p}_{1}, _{2}, _{1} and the radius _{1} and _{2} increasing; _{1} dominates the increasing of _{1} must be large enough to achieve high _{1} and the radius

Furthermore, the thickness _{p}_{1} = 2,200, _{2} = 2,220, _{1} = 3,000, a plot of support loss relating

The analytical model derived provides the design guidelines for achieving high

Choice of materials: in order to increase _{support}

Geometrical dimensions:

_{support}_{1}_{2}_{1}_{support}

_{support}_{1}_{support}

_{support}_{p}^{2} and (_{p}^{3}) of RMG. The thicker support structure and the slenderer beams will increase _{support}

The predicted results of the analytical model are compared with the experimental results to demonstrate validity of the model for RMG. Experiments are conducted on several rocking mass microgyroscope prototypes. The prototypes are fabricated by using same silicon wafers and the same structural parameters, except for the different thicknesses of the vibratory structure changed by a thickness reducing technology. The electrostatic actuation and capacitance detection method is used to measure the natural frequencies and

All the measured results, which are measured at a lower pressure of less than 10 Pa in the vacuum chamber, are compared with the theoretical predictions in _{total}

Besides, the silicon structure and the Pyrex base plate are bonded together by coating epoxy resin. The poor coating uniformity will induce serious energy losses. Some Q values are thus much smaller than other Q values of the prototypes with the same dimensions and in the same batch, and 2# prototype is one of the cases. Compared with the experimental results in [

Bae presented some measurements of

By comparing all the experimental results, and to comment on their utility in various thicknesses of the microgyroscopes, we conclude that all the cases indicated that the dominant loss mechanism for RMG may be the radiation into the support structure. The expressions for the power flow, presented in Section 3.2, are applicable for any other resonator geometries for which the attachment to the support structure acts essentially as a point source for vibration in the support.

An accurate analytical model of support loss for a RMG is presented. The anchor types and the support loss mechanism of the RMG are firstly analyzed, and the support loss is simplified as a model with a beam attached to the support plate at its end. The support loads of the RMG are analyzed, and then the powers flowing into the support and the vibration energy of the RMG are also derived. The analytical model of the support loss for RMG is developed, and its sensitivities of the support loss to the main structural parameters are analyzed. Finally, the high-

The authors would like to thank the Laboratory of Microsystems, National University of Defense Technology, China, for access to equipment and technical support. This work was supported by National Natural Science Foundation of China (Grant No. 51005239 and 51175506).

The two operational modes and the main structural parameters of RMG.

The support model and its main structural parameters.

The support loads of RMG and the equivalent SDOF system.

Semi-infinite plate and the applied loads to attachment point.

Support loss for rocking mass gyroscope. _{p}_{p}

Support loss for rocking mass gyroscope.

The prototype and support, mode measuring system and frequency response curve.

The numerical values of coefficients for Y.

Re(y_{11}) = Re(y_{22}) |
0.21645 | 0.22172 |

Re(y_{23}) = Re(y_{32}) |
−0.29149 | −0.28546 |

Re(y_{33}) |
0.46198 | 0.45735 |

The predicted

_{p} |
|||||||
---|---|---|---|---|---|---|---|

_{1} |
_{2} |
_{1} |
_{2} | ||||

2# | 240/2,500 | 5,760.0 | 5,755.4 | 76.8 | 78.8 | ||

4# | 240/2,500 | 5,853.4 | 5,683.2 | 5,678.6 | 226.8 | 123.3 | 125.6 |

5# | 240/2,500 | 5,858.8 | 5,842.8 | 128.4 | 135.9 | ||

3# | 120/2,500 | 2,697.9 | 2,706.2 | 355.0 | 337.4 | ||

4# | 120/2,500 | 2,996.4 | 2,768.9 | 2,778.1 | 668.6 | 325.4 | 321.9 |

7# | 120/2,500 | 2,760.5 | 2,756.2 | 270.4 | 256.9 | ||

01# | 60/2,500 | 1,726.2 | 1,799.8 | 722.8 | 789.4 | ||

06# | 60/2,500 | 1,828.1 | 1,707.0 | 1,793.2 | 1,992.0 | 742.2 | 674.1 |

11# | 60/2,500 | 1,622.1 | 1,689.2 | 811.1 | 796.8 |