基于线性位移模式的环形谐振子理论研究

Theoretical Research on Ring-Shaped Harmonic Oscillator Based on Linear Displacement Mode

环形谐振子理论是谐振陀螺研究领域的重要内容,然而在目前经典环形谐振子理论中,仍存在很多问题尚未解决,例如挠度控制方程不能精确求解环形结构的弯曲问题,获得的二阶弯曲角频率及进动系数理论解不能充分反映出其随尺寸参数(如高度h、曲率半径r等)变化的影响规律等。为了克服上述不足,从环形结构理论的基本假设和基本条件出发,基于环形结构的线性位移模式假设、虚功原理和哈密顿原理,建立新的环形谐振子理论,包括广义本构关系、平衡方程和边界条件等,获得环形结构静态弯曲问题的解析求解方法和动态问题的理论解。最后,针对典型的环形结构静、动态问题,通过与其他理论结果进行对比,验证了理论和求解方法的正确性。工作表明,进动系数不是恒定的,其值会随着结构尺寸的不同在0.4附近很小范围内发生变化。

The theory of ring-shaped harmonic oscillators is an important part for the research on resonant gyroscopes.However,as for the current classical theory,there are still many defects,for example,the deflection governing equation cannot accurately handle bending problems,and solutions of the bending angular frequency and precession coefficient cannot reflect the influence of structure dimension parameters(e.g.height h and curvature radius r).In order to overcome the above shortcomings,this paper establishes a new theory for ring-shaped resonators,including generalized constitutive relations,equilibrium equations and boundary conditions,based on the basic assumptions and conditions,the linear displacement mode,the virtual work principle and the Hamilton's principle for ring-shaped structures.And then the analytical solution method is built for static bending problems while the theoretical solution is proposed for dynamic problems.Finally,as for typical static and dynamic problems,the new theory and the solutions are verified by comparing the results with those from other theories.This paper indicates that the precession coefficient is not constant,which will vary around 0.4 within a small range for different structural dimensions.