Modern rotating machines have a tendency to increase capacity, reduce weight and obtain high operating speed. So the machines become more and more sophisticated and intelligent ever. But still there exists many factors not recognized by modeling techniques of the entire rotor system. Even for a carefully designed rotating machinery having high performance, broad stability with highly qualified materials, unexpected excessive vibrations may be produced and eventually result in fatigue failure or damage during their operating life. Therefore, early stage preventive detection for the system faults becomes increasingly important. Many monitoring methods for early detection of symptoms for various failure modes are developed until now. But they do not clearly match the needs of the industry, because the time-varying parameter in the model of the rotors are not modeled adequately.
In this research work, a new modeling and modal analysis method for rotor systems usable in industry is proposed for the purpose of innovation of the theoretical technology for the early monitoring and detecting the critical dynamic failure modes of the rotating machineries.
The essence of the method is to introduce the modulated coordinates such that the periodically time-varying linear differential equations can be effectively transformed to the equivalent time-invariant linear differential equations. The modal analysis procedure for the asymmetric rotor system, of which rotating and stationary parts possess asymmetric and isotropic properties is investigated in the first stage. With a simple asymmetric rotor model, the analytical modal analysis procedure is illustrated and compared with the conventional method based on formulation in the rotating coordinates. A numerical example with a flexible asymmetric rotor model is also provided to demonstrate the effectiveness of the proposed modeling method.
In the second stage, the method is extended for asymmetric rotors with anisotropy, so called, general rotor systems. The core idea of the method is to introduce the modulated coordinates to derive the equivalent infinite order time-invariant matrix equation from the finite order time-varying matrix equation. Time-varying parameter equation is inevitable for general rotors, of which rotating and stationary parts both possess asymmetric properties. An extensive modal analysis method is rigorously developed for the infinite order time-invariant system and an approximate, yet effective modal analysis technique is introduced with the reduced order system. Four types of directional frequency response functions are defined for the proposed method and illustratively obtained for comparison of detection ability of each frequency response functions. It is shown theoretically that they can be effectively used for detection of the presence of system anisotropy and asymmetry. A numerical example with a simple rotor model and a flexible asymmetric rotor are also provided to demonstrate the theoretical findings and practicality of the proposed method. And the sensitivity of the degree of asymmetry and degree of anisotropy in investigated with tedious and steady work based on perturbation methods.
Directional frequency response functions (dFRFs) are re-defined in the basis of proposed modeling method, which is based on modulating coordinates. The conventional results regarding the high sensitivity of rdFRFs of index -l(r-dFRFs of index 0) with respect to degree of asymmetry (anisotropy) is re-visited in terms of modulation coordinates. Furthermore a new normal dFRF is found in the investigation of the method, which represent the coupled effect of anisotropy and asymmetry of the rotor system. The convergence of the infinite sized equivalent time-invariant system is guaranteed by careful analysis of the structure of eigenfrequencies and residues of dFRFs. The results were illustrated using numerical models and simple analytical models to enhance the understanding of the proposed method.
주기적인 시변특성을 갖는 회전체의 새로운 모드해석기법이 이 연구에서 개발되었다. 새로 개발된 기법의 핵심 아이디어는 변조좌표계를 활용하여 주기적인 시변특성의 운동방정식을 효과적으로 등가의 시불변계로 변환하는 방법이다.
회전축부에만 비대칭성이 존재하는 경우로 국한하면 본 방법론은 회전좌표계 변환법과 유사하며 변조좌표계의 변조주파수가 회전속도의 두 배속인 변조좌표를 사용하는 변환법이다. 회전축부 및 정지부 모두 비대칭성이 존재하는 경우에는 무한한 크기의 형태를 갖는 등가의 계로 운동방정식이 귀착된다. 본 연구에서는 위 두 가지 경우에 집중하여 등가의 시불변계의 이론적인 모드해석을 수행하고 이 때 나타나는 고유치의 공액 관계 및 모드형상의 특성을 분석하였다. 특히 비대칭회전체의 경우 고유치의 부호가 바뀌고 회전속도 두 배 만큼 이동된 값이 대응되는 고유치로 나타나고 고유벡터는 상하가 역전된 공액 형태임을 밝혀냈다. 또한 비대칭/비등방 회전체의 경우, 무한형태의 운동방정식을 유한차수로 근사화하여 해석하면서 나타나는 수렴성 문제 및 비대각 항들의 산포와 잔차항의 크기에 관련한 차수분석을 엄밀히 수행하여 무한형태의 행렬에서도 방향성 스펙트럼의 수렴성 및 그 활용도를 보였다.
본 연구결과를 효율적으로 보여주기 위해 단순 회전체 모형을 기반으로 해석을 수행하고 고유치, 잔차항 들의 형태 및 방향성 스펙트럼의 특성을 구하였다. 더 나아가 실용적인 적용방법을 강구하기 위해 검증된 열린균열 모형을 사용하여 유연회전체의 방향성 주파수 응답함수의 특징이 균열의 깊이에 따라서 어떻게 변하는 지를 수치적으로 강구하였다.
이 연구의 주요 성과를 다음과 같이 요약할 수 있다.
가) 주기적인 시변계의 등가적인 무한형태로 확장된 운동방정식 및 관련 이론적인 모드 해석 기법
나) 비대칭/비등방 회전체 방향성 주파수 응답함수의 정립, 이의 잔차 특성 및 활용법
