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Bell-shaped vibratory angular rate gyro (abbreviated as BVG) is a new type Coriolis vibratory gyro that was inspired by Chinese traditional clocks. The resonator fuses based on a variable thickness axisymmetric multicurved surface shell. Its characteristics can directly influence the performance of BVG. The BVG structure not only has capabilities of bearing high overload, high impact and, compared with the tuning fork, vibrating beam, shell and a comb structure, but also a higher frequency to overcome the influence of the disturbance of the exterior environment than the same sized hemispherical resonator gyroscope (HRG) and the traditional cylinder vibratory gyroscope. It can be widely applied in high dynamic low precision angular rate measurement occasions. The main work is as follows: the issue mainly analyzes the structure and basic principle, and investigates the bell-shaped resonator's mathematical model. The reasonable structural parameters are obtained from finite element analysis and an intelligent platform. Using the current solid vibration gyro theory analyzes the structural characteristics and principles of BVG. The bell-shaped resonator is simplified as a paraboloid of the revolution mechanical model, which has a fixed closed end and a free opened end. It obtains the natural frequency and vibration modes based on the theory of elasticity. The structural parameters are obtained from the orthogonal method by the research on the structural parameters of the resonator analysis. It obtains the modal analysis, stress analysis and impact analysis with the chosen parameters. Finally, using the turntable experiment verifies the gyro effect of the BVG.

A solid-state wave gyroscope can be used to measure the angular velocity of a rotating body based on the inertia effect of the standing wave in two vibration modes of the axisymmetric resonator, which have advantages, such as small size, high operation accuracy, low cost, low power consumption, good shock resistance, and long life [

The hemispherical resonator gyroscope (HRG) was developed rapidly in Delco, Litton, and Northrop Grumman Co. [

The gyroscope is a simple millimeter-scale metallic gyro, which is comprised of a resonator, piezoelectric elements and a capacitor. It is easier to manufacture and fabricate than that with micro-machining technology;

The bell-shaped resonator can bear a higher impact than others;

The accuracy is improved using capacitor-detecting technology;

It can be widely applied in the high dynamic low precision angular rate measurement occasions.

The schematic sketch of the traditional bell is shown in

The schematic sketch of the presented bell-shaped resonator is shown in the figure (see

The working principle of the BVG is based on the inertia effect of the standing wave in two vibration modes of the axisymmetric resonator caused by the Coriolis force. The schematic diagram of the working principle is shown in

In this case, if the gyroscope is rotating about the symmetry axis with an angular velocity, Ω (to measured), the Coriolis force, _{c}_{s}_{s}

The paraboloidal resonator and coordinate system are shown in

The resonator's middle surface is generated by rotating the meridian line about the ^{2}/4_{t}_{b}_{t}_{b}_{t}_{b}

The two principal radii of curvature for the midsurface of the paraboloidal shell are:

Where _{1} describes the curvature of the midsurface in the meridional (_{2} is in a plane normal to the meridian. The latter is the distance along the normal plane from the middle surface to the axis of rotation (

Generating equations of the motion of the paraboloidal body (Ω = 0) are [_{ϕ}_{z}_{θ}_{z}_{1} + _{z}_{2} +

Assuming a linearly elastic, isotropic material, the stress-strain equations are:
_{ϕϕ}_{zz}_{θθ}

The strain-displacement equations are found to be:

In the present work, paraboloidal shells of revolution are analyzed by a 3-D approach. Instead of attempting to solve equations of motion, an energy approach is followed, which, as sufficient freedom is given to the three displacement components, yields frequency values as close to the exact ones as desired.

An elastic body undergoing free, undamped vibration has potential energy and kinetic energy. The potential energy (_{ij}_{ij}

The kinetic energy of the thick shell is:

For the free, undamped vibration, the time (

In this paper, our work is simulated by FEM (finite element method) software, which could solve the modal frequency, the modal displacements distribution, the electric displacement distribution due to response displacements and the couple field analysis.

The benefit of BVG is the ability to load a higher impact, which can adapt well to the high dynamic environment. The impact is the system's sudden change of force, displacement, speed and acceleration in transient excitation. The FEM could help us in analyzing how the resonator changes in the impact process. The impact value, corresponding to the time in transient excitation, is shown in

The maximum stress disappears at 10 ms during the impact process, corresponding to the Von Mises stress distribution and the displacements distribution, as shown in

In order to get the appropriate modal frequency and modal shape, the dimensions of the entire structure are optimized in the modal analysis module. The modal analysis results are shown in

According to the modal contour of the FEM model, we can find the active mode and sense mode of the resonator. In fact, there are some calculation errors and grid errors that influence the frequency.

In fact, there are other physical factors and geometrical factors that influence the sensitivity of the BVG, such as the Young's modulus, piezoelectric constants, natural frequency and other dimension parameters. The influence of the main structure parameters on static and dynamic characteristics is investigated comprehensively by using an orthogonal test design. The steps of the orthogonal test are as follows.

Confirm the experiment factor on the target index.

The active mode and sense mode frequency of the resonator (target: 6, 000

the difference between the previous mode frequency with the active mode (target: > 1, 500

the difference between the follow mode frequency with the sense mode (target: > 1, 500

Confirm the experiment factor (see

The top cylinder length (L1)

The radius of the resonator (R1)

The thickness of the resonator (H1)

The length between the top and the isolation hole (L2)

The length between the isolation hole and the bottom edge (L3)

Confirm the factor level of the test design.

Confirm the orthogonal table. (see

Confirm the headers of the

Program the test plan and experiment based on

Results analysis.

Next test.

This method got the optimal parameter after the experiment as follows: H1 = 0.7 mm; R1 = 11 mm; L3 = 9 mm; L2 = 6 mm; L1 = 6 mm.

The metallic resonator of the prototypal gyroscope is made out of a Cu-based alloy doped with Ni and Cr. In order to obtain high elasticity, a high Q-factor, and homogeneous density of the metallic resonator, some thermal treatment procedures, such as solid solution and age strengthening, are carefully programmed before precision machining.

A modal test is used to validate the vibration of the sensor's work modes. The harmonic response is tested using a TD1250-C frequency response analyzer. In active mode test, the inducing voltage is applied on piezoelectric elements A and E, and the response voltage is picked up from piezoelectric elements C and G; the inducing voltage is applied on piezoelectric elements B and F, and the response voltage is picked from piezoelectric elements D and H. The resonating response is shown in

From the response curves of the active mode and sense mode, the gyroscope resonates at the frequency of 6,064.5 Hz and 6,077.1 Hz, approximately, which is close to the FEM simulation result. The frequency split of two work modes is 12.6 Hz. Using the method to restrain the frequency split, we use the method in [

To verify the resonator mode shape, we use the Polytec PSV-400 scanning vibrometer to test the radial displacement amplitude (see

The circumference of the resonator was divided into 72 parts. The active frequency is 6,064.5 Hz; the amplitude is 5 V. We get the curve as shown in

As shown, the circumference of the resonator produces the four-wave mode shape. However, it is not the standard mode shape. The reasons for the bias are as follows:

The influence of piezoelectricity;

The errors of the signal generator;

The noise of the experiment circuit;

The position of the piezoelectric element on the surface of the resonator;

The manufacture errors of the resonator.

The control circuit is very important in the overall performance of the BVG. It is designed on the classical vibratory gyroscope, and the diagram is shown in

The variable angular rate was applied to a single axis of a turntable to verify that BVG produced the gyroscopic effect. The test results are given in

Although a lot of effort is being spent on improving these weaknesses, an efficient and effective method has yet to be developed. Next, it is important that we research the “Real” bell-shaped resonator, as shown in

This article presents the modeling, simulation and fabrication of BVG. BVG based on the piezoelectric effect of active and capacitor detection has a solid metallic structure on the millimeter scale, which can be manufactured by conventional machining technology. Compared with the conventional vibratory gyroscope, the novel structure could load a higher impact. It can be widely applied in high dynamic low precision angular rate measurement occasions. It obtained reasonable structural parameters with the finite element analysis and an intelligent platform. The main work is as follows: the issue mainly analyzes the structure and basic principle and investigates the bell-shaped resonator's mathematical model. Using the current solid vibration gyro theory analyzes the structural characteristics and principles of BVG. The bell-shaped resonator is simplified as a paraboloid of the revolution mechanical model, which has a fixed closed end and a free opened end. It obtains the natural frequency and vibration modes based on the theory of elasticity. The structural parameters are obtained from the orthogonal method by research on the structural parameters of the resonator analysis. It obtains the modal analysis, stress analysis and impact analysis with the chosen parameters. Finally, using the turntable experiment verifies the gyro effect of the BVG. However, the quality factor and performances of this gyro are not so attractive now compared to other similar gyros. The main reason are as follows: the influence of material temperature character, the frequency split of the resonator, the algorithm of the control loop and so on. Next, the important work is as follow: restrain the frequency split, compensate for the temperature, research the advanced algorithm about the signal of the gyro and so on.

This work was supported by the National Natural Science Foundation of China (Grant No. 61031001 and 61261160297).

Traditional bell.

Cross-section of bell-shaped resonator with a variable thickness axisymmetric multi-curve.

Cross-section of a bell-shaped resonator with a variable thickness axisymmetric multi-curve.

Schematic of a bell-shaped resonator.

Schematic of the working principle.

Cross-section of an open paraboloidal resonator with variable thickness.

The diagram of impact.

Schematic of the working principle.

The result of mode analysis.

The schematic diagram of the resonator structure.

BVG sensor assembly unit: (

Resonating response of active mode: (

The resonator test.

Mode shape of the resonator.

The block diagram of the control method and the photo of the control circuit.

The turntable test.

The parameters with simulation.

Material | Ni43CrTi (3J53) |

Density (^{3}) |
8170 |

Poisson's ratio | 0.3 |

Young modulus ( |
196.76 |

Yield strength ( |
500 |

Simulation method | Transient dynamic |

Mesh generation method | Free |

Release ratio | 1 |

Piezoelectric element | PZT-5A |

The parameters with simulation.

First | 5909.3 |

Second | 5914.6 |

Third | 8924.8 |

Fourth | 8929.2 |

Fifth | 10,478.6 |

Sixth | 10,479.2 |

Seventh | 15,603.7 |

Eighth | 15,626.8 |

^{16}(2^{15}) orthogonal table.

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

1 | (1) | 3 | 2 | 5 | 4 | 7 | 6 | 9 | 8 | 11 | 10 | 13 | 12 | 15 | 14 |

2 | (2) | 1 | 6 | 7 | 4 | 5 | 10 | 11 | 8 | 9 | 14 | 15 | 12 | 13 | |

3 | (3) | 7 | 6 | 5 | 4 | 11 | 10 | 9 | 8 | 15 | 14 | 13 | 12 | ||

4 | (4) | 1 | 2 | 3 | 12 | 13 | 14 | 15 | 8 | 9 | 10 | 11 | |||

5 | (5) | 3 | 2 | 13 | 12 | 15 | 14 | 9 | 8 | 11 | 10 | ||||

6 | (6) | 1 | 14 | 15 | 12 | 13 | 10 | 11 | 8 | 9 | |||||

7 | (7) | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | ||||||

8 | (8) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||||||

9 | (9) | 3 | 2 | 5 | 4 | 7 | 6 | ||||||||

10 | (10) | 1 | 6 | 7 | 4 | 5 | |||||||||

11 | (11) | 7 | 6 | 5 | 4 | ||||||||||

12 | (12) | 1 | 2 | 3 | |||||||||||

13 | (13) | 3 | 2 | ||||||||||||

14 | (14) | 1 | |||||||||||||

15 | (15) |

Headers of the table.

Factors | L1 | H1 | R1 | L2 | L3 | ||||||||||

Column | A | B | A×B | C | A×C | B×C | D×E | D | A×E | E |