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Inertial navigation devices include star sensor, GPS, and gyroscope. Optical fiber and laser gyroscopes provide high accuracy, and their manufacturing costs are also high. Magnetic suspension rotor gyroscope improves the accuracy and reduces the production cost of the device because of the influence of thermodynamic coupling. Therefore, the precision of the gyroscope is reduced and drift rate is increased. In this study, the rotor of liquid floated gyroscope, particularly the dished rotor gyroscope, was placed under a thermal field, which improved the measurement accuracy of the gyroscope. A dynamic theory of the rotor of liquid floated gyroscope was proposed, and the thermal field of the rotor was simulated. The maximum stress was in

MEMS (microelectromechanical systems) gyroscopes, such as micromachined vibrating gyroscope [

The angular vibration of the MEMS gyroscope is mainly used to measure angle and velocity. The University of Minnesota presented a highly accurate angular vibratory gyroscope [

As shown in Figure ^{−1}, scaling factor of 39.8 mV^{−1}Hz^{−1/2}. And the bias stability is 50.95°/h [

Maglev gyroscope.

Tsinghua rotor gyroscopes.

In recent years, the researchers from the University of Electronic Science and Technology in China investigated the LC tuning of magnetic suspension rotor gyroscope. This gyroscope mainly consists of a stator and a rotor. Its suspended rotor is driven, as shown in Figure

Rotor gyroscope.

In 2012, Tsinghua University and Harbin Industrial University investigated the rotor of liquid floated gyroscope. This gyroscope has high accuracy and mainly consists of a stator, rotor, coil, and test version. The edge of its structure is shown in Figure

Maglev rotor gyroscopes.

The heat coupling of gyroscope’s rotor is produced by a magnetic field heat, heat flow field, and the composition of heat flow field and magnetic field coupling. The principle of these fields is shown in Figure

Coupling moment of rotor gyroscope.

This is the moment of gyroscope equation:

The dynamic torque equation of the dishing rotor gyroscope in the stator coordinate system

The dynamic torque equation of the dishing rotor gyroscope in the rotor coordinate system

The dynamic torque equation of the dishing rotor gyroscope in the stator coordinate system

The dynamic torque equation of the dishing rotor gyroscope in the rotor coordinate system

In order to establish an equation about gyroscope torque, firstly, the hypothesis is as follows. The quality of the dishing rotor is represented by

The

The front dished rotor about the equation of mechanical characteristics is calculated as follows:

The dished rotor moves at a high-speed in a closed chamber filled with #3 industrial white oil. The rotor is in the

The dynamic equation is simplified. Consider

Solving (

The dynamic equation (

Substituting (

At the same time, (

These three equations are angular velocity calculated by coupling torque about this kind of liquid floated gyroscope. At the same time, (

Equation (

For the theory coupled thermal field of dished rotor, these are as follows. First is the coupling thermodynamics calculation; second is the establishment of boundary conditions.

The volume of the dishing rotor is assumed to consist of two parts: spherical and cylindrical. The volume of the sphere is calculated as follows:

The whole dishing rotor is in the thermal field, and the temperature is simulated. Consider

The equation above can be reduced. The following was established and the dished rotor was in coupled thermal field about the calculation of volume and temperature. Consider

The thermodynamic analysis of the dishing rotor gyroscope shows a continuous connection between the well-posed equation of differential thermodynamics and the continuum mechanics of the problems, because the dishing rotor gyroscope is filled with #3 industrial white oil; thus, a Godunov-type equation can be applied [

This gyroscope operates under high-speed rotation: a form of nonequilibrium thermodynamics and state variables can be established as follows:

The Boltzmann equation of thermodynamics of the dishing rotor heat entropy in the growth equation is as follows [

The medium oil film of the dishing rotor gyroscope is produced by a model of the turbulent flow of inhomogeneous superfluid thermodynamics. The specific liquid properties of #3 industrial white oil are given. The average vortex line length is measured in unit volume and heat flux restructuring expression. The second principle of thermodynamics is combined with the Lagrange equation using the Legendre transformation through the constitutive theory. The complete expression of nonequilibrium and its entropy flux has been established [

This is a part of the calculated heat about dished rotor, where

The turbulence theory of nonlinear superfluid involves entropy flux with different temperatures and heat flux of the dishing rotor [

In thermodynamic analysis, the lattice Boltzmann method is successfully applied in various problems related to isothermal fluid dynamics [

Balance the function of density distribution for these two models. Consider

The boundary of the dished rotor gyroscope was assumed to be in the #3 industrial white oil. The function of the particle density distribution is expressed as follows:

A pressure oil body exists but is applicable only along the edge of the density distribution function. As proposed by Fermi [

The common thermodynamic framework extensive key depends on the assumption of entropy [

Dynamic torque of liquid floated gyroscope.

The thermodynamic theory of simulation for dished rotor, we demonstrate respectively from two aspects of 2D and 3D simulation.

When the dishing rotor gyroscope is in a confined space, the pressure of the fluid in the COMSOL is simulated, particularly the temperature field of the gyroscope’s rotor. It can be considered by the rotor flow field; these are for the gyroscope that can produce the Eddy current. The temperature of this gyroscope’s rotor is high. In the COMSOL simulation, the inner cavity joins the laminar flow in the four corners of the stator. The maximum temperature is shown in Figure

Thermal field of temperature under different rotor structures ((a), (b)).

The coupled thermal simulation of the dishing rotor and heat coupling were performed in the 3D simulation of COMSOL. They were conducted using a grid, as shown in Figure

Grid of the dished rotor gyroscope.

The rotor’s coupling heat was simulated with the gyroscope’s rotor speeds of 5000 rad/min and 5549 rad/min, and the results are shown in Figures

Thermal simulation of the dishing rotor gyroscope. (a) 5000 rad/min and (b) 5549 rad/min.

The coupling force field of the dishing rotor gyroscope and the rotor’s coupling heat were simulated with the gyroscope’s rotor speeds of 5000 and 5549 rad/min, as shown in Figures

Bonding force simulation of the liquid floated gyroscope. (a) 5000 rad/min and (b) 5549 rad/min.

The rotor is driven by a magnetic field, and the gyroscope’s rotor speeds are 5000 and 5549 rad/min. The analysis of the thermal coupling characteristic curve is shown in Figures

Bonding force distribution of the liquid floated gyroscope. (a) 5000 rad/min and (b) 5549 rad/m.

ADAMS has established a model of the liquid floated gyroscope. A grid should be constructed for the gyroscope during the simulation of the dishing rotor. The gyroscope’s rotor is placed in the grid, as shown in Figure

Grid of the liquid floated gyroscope.

The simulation model is established by measuring the rotation of the dishing rotor, particularly its angular velocity and the force and speed of the gyroscope’s rotor. The results are shown in Figure

Simulation of the liquid floated gyroscope in ADAMS.

This experiment adopts the phase field equation that can be derived from the extremum principle of thermodynamics, which describes the state of the system. The quantity can be obtained by applying the extremum principle model for thermodynamics and the evolution of the constitutive equation for the thermodynamic system. The phase field method is a potential simulation tool. The microstructure evolution of a system is complex, and several parameters are introduced as the standards in thermodynamics. The relationship between thermodynamics and phase field parameters is analyzed, simulated, and verified [

The total Gibbs energy of the GS mechanochemical system is given by

The value of the flow of

The rotor’s coupling heat is given in the theoretical calculation and the simulation above. The coupled heat is related to an experiment using PIV measurements, and the results are shown in Figures

Thermal fluid measurement of the dished rotor. (a) 5000 rad/min and (b) 5549 rad/min.

Consider

For the rotor operates at speeds of 5000 rad/min and 5549 rad/min, and the rotor’s flow field velocity distribution of heated. Now, it is marked for analysis, and the thermal maximum edge away from the speed of rotor is included

Thermodynamic analysis of the measuring fluid for the dishing rotor. (a) 5000 rad/min and (b) 5549 rad/min.

Consider

Thermal fluid experimental results of the dishing rotor measurements (a) 5549 rad/min and (b) thermal fluid experimental of rotor in 5549 rad/min.

Stress analysis of the rotor in the experiment with the bonding thermal force of rotor in 5549 rad/min.

The experimental results of the coupled thermal simulation and theoretical results are as follows. The maximum rotor coupled border coupling heat for the coupled thermal simulation results is

The dishing rotor gyroscope was coupled and was established in different coordinates of the system for the thermodynamic analysis. Dynamic equations were solved, and the dynamic equation of the disc rotor angular rate was obtained. The model theory was solved in the COMSOL through 2D and 3D simulations coupled with thermal simulation, and it was in the mechanics of the dynamic system simulation in the ADAMS components. Maximum stress was found in

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Ministry of Science and Technology of China (Grant nos. 2012CB934103 and 2011ZX02403).