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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The bell-shaped vibratory angular rate gyro (abbreviated as BVG) is a novel shell vibratory gyroscope, which is inspired by the Chinese traditional bell. It sensitizes angular velocity through the standing wave precession effect. The bell-shaped resonator is a core component of the BVG and looks like the millimeter-grade Chinese traditional bell, such as QianLong Bell and Yongle Bell. It is made of Ni43CrTi, which is a constant modulus alloy. The exciting element, control element and detection element are uniformly distributed and attached to the resonator, respectively. This work presents the design, analysis and experimentation on the BVG. It is most important to analyze the vibratory character of the bell-shaped resonator. The strain equation, internal force and the resonator's equilibrium differential equation are derived in the orthogonal curvilinear coordinate system. When the input angular velocity is existent on the sensitive axis, an analysis of the vibratory character is performed using the theory of thin shells. On this basis, the mode shape function and the simplified second order normal vibration mode dynamical equation are obtained. The coriolis coupling relationship about the primary mode and secondary mode is established. The methods of the signal processing and control loop are presented. Analyzing the impact resistance property of the bell-shaped resonator, which is compared with other shell resonators using the Finite Element Method, demonstrates that BVG has the advantage of a better impact resistance property. A reasonable means of installation and a prototypal gyro are designed. The gyroscopic effect of the BVG is characterized through experiments. Experimental results show that the BVG has not only the advantages of low cost, low power, long work life, high sensitivity, and so on, but, also, of a simple structure and a better impact resistance property for low and medium angular velocity measurements.

The bell-shaped vibratory angular rate gyro (abbreviated as BVG) is a novel shell vibratory gyroscope, which is inspired by the Chinese traditional bell. It senses angular velocity through the standing wave precession effect. The Bell-shaped resonator is a core component of the BVG and looks like the millimeter-level Chinese traditional bell, such as QianLong Bell and Yongle Bell. BVG has not only the advantages of low cost, low power, long work life, high sensitivity, and so on, but also a simple structure and good anti-impact performance for low and medium velocity angular measurements [

The classic vibratory shell solid wave gyro includes: a hemispherical resonator gyro, a cylinder gyro, a ring vibratory gyro, a disc plant vibratory gyro, and so on [

The proposed bell-shaped vibratory angular rate gyro is inspired by the traditional Chinese bell. Changing the resonator structure of the traditional vibratory gyro improves accuracy and impact resistance. There are many characteristics of the traditional Chinese bell, which includes beautiful sound, vibration stability, a steady mode shape, good impact resistance,

There are many problems using a bell structure to make a gyro. It is difficult to analyze the bell-shaped resonator, because the vibratory character is very complex and the middle surface is not described using the single linear function. To summarize, the revolution vibratory shell finds a method to solve the vibratory problem of a bell-shaped resonator. For the HRG, VA Matveev and other researchers used the theory of a thin shell to analyze the character of a hemispherical resonator. He presents the equilibrium differential equation and derives the governing equation [

In this article, we explore bell-shaped resonator structure and present the design, analysis and experimentation of the BVG. The important work is as follows: to design the structure of a bell-shaped resonator and present the principle of the work, shown in Section 2. In Section 3, a theoretical analysis of the proposed resonator is performed using the theory of thin shells. The strain equation, internal force and resonator's equilibrium differential equation are derived in the orthogonal curvilinear coordinate system. On this basis, the mode shape function and the simplified second order normal vibration mode dynamical equation are obtained. In Section 4, the coriolis coupling relationship about the primary mode and secondary mode is established. The method of the signal processing of the BVG and the control loop of the prototype are presented. In Section 5, to analyze the impact resistance property of the bell-shaped resonator, which is compared with other shell resonators using the Finite Element Method, demonstrates that the BVG has the advantage of better impact resistance. In Section 6, a reasonable method of installation and a prototypal gyro are designed. The gyroscopic effect of the BVG is characterized through experimentation. Experiment results show that the BVG has not only the advantages of low cost, low power, long work life, high sensitivity, and so on, but also a simple structure and better impact resistance for low and medium angular velocity measurements. Finally, some important conclusions are drawn.

The core component of the BVG is a bell-shaped resonator, which influences the performance of the gyro directly. The key of the BVG is the structure of the bell-shaped resonator. The bell-shaped resonator is inspired by the traditional Chinese bell, as shown in

The bell-shaped resonator looks like the traditional bell, which is designed by a parabolic structure intuitively. In order to reduce the cost and design procedure, the structure of the top, middle and bottom merge into a parabola, and this is achieved to integrate the design. This merge does not change the vibratory character of the structure. The position of the excitation electrode and the detection electrode is very important for the BVG. For the actual bell, to excite bell vibration, one uses a stick to hit the wall of the bell. The vibration effect is very significant. However, the bell-shaped resonator selects the position on the wall of bell. In order to ensure the power consumption and cost of the gyro, it controls the mode shape and detects the precession of the standing wave based on piezoelectricity. The piezoelectric elements are attached to the wall of the bell-shaped resonator to excite the primary mode and detect the second mode. Using the isolated hole reduces the disturbance between neighboring electrodes, as shown in

The BVG sensitizes angular velocity through the standing wave precession effect. The eight piezoelectric elements are attached to the wall of the bell-shaped resonator uniformly, as shown in

During practical application, the stress wave propagates on the resonator, causing the standing wave, and the piezoelectric elements sensitize the stress wave and solve the angular velocity. In order to reduce the influence of piezoelectric elements when the resonator is rotated and to improve the excitation efficiency, the elements should be set near the constrained boundary. Such an excitation type is like the principle of leverage, using a small amount of force to achieve a large deformation to improve the excitation efficiency.

The piezoelectric elements chose the PZT5A, which was polarized in the thickness direction. The first and fifth elements contract and expand while applying an alternating current signal (in

For the shell vibratory gyro, the analysis of the resonator vibratory character is very important and focuses on the mode shape function, precession factor and natural frequency. The BVG is a novel structure in the vibratory gyro field and has a unique vibratory character. The analysis of internal force distribution and the equilibrium differential equation are presented using the theory of thin shells. It derives the simplified second normal vibration mode dynamic equation based on the parameters of the gyro.

The bell-shaped resonator is a parabolical structure and described in the orthogonal curvilinear coordinate system (^{2}/(2

In

In the orthogonal curvilinear coordinate system, it is not only convenient to describe the point motion, but also to express the thickness of the resonator. The vibratory character of the bell-shaped resonator is presented in this coordinate.

Based on the theory of thin shells and the Kirchhoff-Lyav hypothesis, the bell-shaped resonator's character as presented includes the relationship of internal force, the equilibrium differential equation and the dynamic equation [

For the isotropic three-dimensional elastic shell element, the middle surface geometric equation of the bell-shaped resonator is obtained in the orthogonal curvilinear coordinate system, (_{φ}_{θ}_{φθ}_{φ}_{θ}_{φ}_{θ}_{φθ}

Substituting

Based on the transformation relation in the orthogonal curvilinear coordinate system and the expression of internal force in the theory of thin shells, the internal force of the bell-shaped resonator is presented. Using the relations between the mechanical force and stress, one can derive the mechanical membrane forces and bending moments for the bell-shaped resonator [^{2}^{−2};

According the the D'Alembert principle, if the sum of force and inertial force and the torque are all equal to zero, which is applied to the isolated unit, then the unit is in a state of equilibrium. The bell-shaped resonator is fit for this principle. As far as we know, the middle surface of the bell-shaped resonator cannot stretch. The three components of tangential displacement in the shell bending equation are equal to zero. It can be expressed as [

The normal force and shearing force are equal to zero:

Therefore, we can derive the equilibrium differential equation (for specific derivation, see _{φ}_{θ}_{v} is the middle surface load of the bell-shaped resonator, which is in relation to the input angular velocity. For the point,

The angular velocity,

According to the Coriolis Theorem, the absolute velocity of point

The absolute acceleration is as follows:

The load of rotation is the load of the middle surface on the bell-shaped resonator and derived in:

Substituting

To solve the

According to

_{n}

Substituting

For _{0}, _{0},

The precession factor is:

The typical analytical method does not solve this complex problem, but it clearly describes the vibratory procedure, the force and equilibrium relation. In practice, the Finite Element Method is widely used for the bell-shaped resonator to analyze the natural frequency, mode shape and work mode.

Analysis of the vibratory character is based on the theory of thin shells. It is very necessary to know the overall mode shape of the bell-shaped resonator. The modal simulation is presented using FEM. The main structure parameters of the resonator are as follows:

Using the mode shape function of Section 3.3, the simulation result is shown in

The mode shape control about the standing wave on the circumference is very important for the BVG.

The transversal surface of the standing wave is the vibratory ring. On the ring, there are two rigid axes, _{n}_{p}_{n}_{p}_{p}_{q}_{p}_{q}_{ω}_{τ}_{p}_{q}

The diagram of the BVG signal process method is shown in

The bell-shaped vibratory angular rate gyro's greatest strength is its remarkable impact resistance property. Using FEM to compare the hemispherical resonator gyro [

The maximum stress of HRG is 181 MPa; the cylinder vibratory gyro is 1,600 MPa; the novel ring vibratory gyro is 3,000 MPa; the BVG is 151 MPa. The result shows that the bell-shaped resonator has the best impact resistance property. Based on the mechanics of materials, the tensile strength of Ni43CrTi is 500 MPa. Therefore, the bell-shaped resonator does not get damaged in 20,000 × g. During the impact process, the resonator produces a small deformation. It is quickly restored after the impact. The control loop also can restrain the influence of the small deformation, too.

The main structure of the BVG includes the bell-shaped resonator, a fixed axis and a foundation. These connect together through mechanical technology, as shown in

Finally, the 9 wires connected to the signal process system include 8 wires of piezoelectric elements and GND. The wires connect to the circuit board through the insulated joint, which is installed on the foundation. The prototypal bell-shaped resonator is shown in

The experiment of the bell-shaped resonator includes: natural frequency test, mode shape test, coriolis test and gyroscopic effect test. The natural frequency and mode shape have been tested in [

The sine alternating current signal (_{pp}_{p}_{p}

During the test, the bell-shaped resonator is in an uncontrolled situation. The frequency split of the tested bell-shaped resonator is 0.5 Hz. In the static state, the two axes exit the vibratory coupling. This described Lissajous pattern is not the line. When the angular velocity is applied to the sensitive axis, the couple effect is very obvious and the standing wave produces precession. The bell-shaped resonator is already sensitive to the coriolis effect.

The prototypal BVG includes power, a signal sample board and a signal processing board, as shown in

The BVG is fixed on the high-precision angular velocity turntable to test the gyroscopic effect. The results are shown in

The manufacturing technology has imperfections. Specifically, the piezoelectric element is difficult to stick on the resonator with a curved surface of variable curvature.

There is no temperature compensation for the bell-shaped resonator. The long work time leads to a change of the frequency and influences the drift. The bell-shaped resonators vibratory characteristics change as the temperature changes.

Structure processing error and control loop error exist. Improving the method of restraining the frequency split and designing the control loop is needed.

The parameters of the governing equation of the bell-shaped resonator are not exact.

In the future, important work is needed: improving the manufacturing technology and the control loop design, designing a temperature compensation method,

Based on the orthogonal curvilinear coordinate system, the equilibrium differential equation using the theory of thin shells is presented, and the motion of the point on the bell-shaped resonator is described. The vibratory character of the resonator is analyzed with the input angular velocity. On this basis, the mode shape function and the simplified second order normal vibration mode dynamical equation are obtained. The Coriolis coupling relationship about the primary mode and the secondary mode is established. The method of signal processing of the BVG and the control loop of the prototype is presented. To analyze the impact resistance property of the bell-shaped resonator, which is compared with the other shell resonator using the Finite Element Method, demonstrates that the BVG has the advantage of better impact resistance property. A reasonable means of installation and the prototypal gyro are designed. The gyroscopic effect of the BVG is characterized through experimentation. The experimental results shows that the zero drift instability is 22.5°/h and linearity is 1.24%. The index is poorer than other shell vibratory gyros. The main reason is: the manufacturing technology has imperfections; there is not any temperature compensation; structure processing error and control loop error exist; the parameters of the governing equation of the bell-shaped resonator are not exact. In the future, important work includes improvement of the performance of the BVG. In conclusion, the BVG has not only the advantages of low cost, low power, long work life, high sensitivity, and so on, but also a simple structure and better impact resistance for low and medium angular velocity measurements.

This work is supported by the National Nature Science Foundation of China (Grant No.61031001, 61261160497).

The variable basic relationship is as follows:

Differential of arc length:

Differential of volume:

Curvature:

Radius of curvature:

Lame constants:

Domain of definition:
_{t}_{b}

According to the theory of thin shells, the basic equilibrium is as follows in

For the orthogonal curvilinear coordinate system:

Using the hypothesis condition in the paper, the equilibrium differential equation is derived.

The solution of _{1} (_{2} (_{3} (_{4} (

The _{φ}_{θ}_{v}

Substituting the variables into

For the orthogonal curvilinear coordinate system:

Simplifying,

The dynamic equation of the bell-shaped resonator is derived through the integral operation. However, the equation is too complex to solve. Using the difference method solves this equation, maybe. This work is ongoing.

The authors declare no conflict of interest.

Schematic of a traditional Chinese bell.

Schematic of a parabolic structure.

Schematic of the resonator.

Schematic of mount electrode.

Schematic of the working principle. (

Schematic of the standing wave precession.

Schematic of piezoelectric work principle. (

Schematic of the working principle. (

Structure parameters of the bell-shaped resonator.

Schematic of the work mode shape. (

Numerical simulation of the mode shape.

The transversal surface of the standing wave.

Schematic of the signal process.

The stress cloud charts under 20,000 × g. (

Schematic of fabrication. (

Prototypal bell-shaped resonator.

Picture of coriolis test. (

The photo of circuit. (

Experiment result. (

The parameters with simulation.

Material | Ni43CrTi (3J53) |

Density (^{3}) |
8,170 |

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 |