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기전 연성 효과를 고려한 전기모터의 진동 및 디자인 민감도 해석

Title
기전 연성 효과를 고려한 전기모터의 진동 및 디자인 민감도 해석
Other Titles
Vibration and Design Sensitivity Analysis of an Electric Motor Considering Mechanical and Electromagnetic Interaction
Author
임형빈
Alternative Author(s)
Im, Hyung-Bin
Advisor(s)
정진태
Issue Date
2011-02
Publisher
한양대학교
Degree
Doctor
Abstract
본 연구에서는 모터의 회전자와 고정자 사이의 공극변화에 인한 기전연성효과를 고려하여 BLDC모터의 동적 특성을 분석하였다. 공극변화가 고정자에 대한 회전자의 상대적 병진 변위를 일으켰을 때, 운동에너지 및 위치에너지, 레이리 감쇠함수, 그리고 자기에너지를 회전자 변위와 고정자의 전류로 표현하였다. 구하여진 에너지와 감쇠함수를 가지고 라그란지 방정식을 이용하여 새로운 운동방정식을 유도하였다. 제안된 모델의 방정식은 변위와 전류가 연성된 비선형 방정식이다. 그리고 나서, 1960년대 이후에 적용된 퓨리에 급수이론에 기초한 해석적 방법을 이용하여 기존의 운동방정식을 소개한다. 기존모델에서는 자기장에 대한 방정식을 먼저 풀고 불평형 자기력을 결정한다. 결정된 그 힘은 모터의 병진방향 운동방정식에 적용한다. 이는 회전자의 변위가 고정자의 전류에 영향을 줄 수 없다는 것을 의미한다. 기존 모델과 제안된 모델에 대한 변위 및 전류 등의 시간응답을 계산하였고 회전자 편심에 대한 효과를 조사하였다. 공극이 시간에 변함에 따라, 전류, 전자기 토크, 그리고 회전속도에 대한 기존 모델과 제안된 모델의 시간응답의 차이는 적었다. 하지만, 회전자 변위에 대한 차이는 상당히 큼을 알 수 있다. 그러므로, 제안된 모델이 기존의 모델보다 더 정확한 동적 특성을 나타낼 수 있음을 알 수 있다. 회전자의 편심은 고정자 전류의 주기와 전자기 토크를 증가시키는 반면에 회전자의 회전속도를 감소시킨다는 것을 확인하였다. 새로운 운동 방정식을 유도한 후에, 유도된 운동 방정식을 이용하여 민감도 해석을 수행하였다. 민감도 해석은 어떤 모델의 구조의 변화나 혹은 변수값의 변화에 따라 그 모델의 변화가 어느 정도 민감한지를 결정하는 해석 방법이다. 그 모델의 특성 응답의 변수값 변화에 따라 얼마나 변하는지를 보여줌으로써, 모델의 개선뿐만 아니라 최적화에 있어서 민감도 해석은 유용한 도구이다. 본 연구에서는 BLDC모터에 대한 변수 민감도에 초점을 맞추었다. 변수 민감도는 변수에 대한 응답의 미분값이다. 본 논문에서는 직접미분법을 이용하여 모터 변수에 대한 편미분 민감도 방정식을 유도하였다. 민감도 방정식을 얻기 위해서 이전 장에서 유도된 지배 미분방정식이 사용되었다. BLDC모터는 동적 시스템이다. 그래서 모터에 대한 민감도 역시 시간에 따라 변하게 된다. 모터에 대한 민감도의 시간응답은 Newmark 시간적분법을 적용하여 얻어졌다. 민감도 해석에서 일반적으로 변수와 목적함수를 고려해야만 한다. 본 연구에서는 전자기적 변수로 저항, 코일 턴수, 그리고 영구자석의 잔류자장을 고려하였고, 기계적 변수로 회전자의 질량과 편심, 축과 베어링의 강성과 감쇠를 고려하였다. 실제 모터는 기계적인 특징과 전자기적인 특징이 서로 연성된다. 그러므로, 정확한 민감도 해석을 위해서 기계적 변수와 전자기적 변수를 동시에 고려해야만 한다. 이러한 변수들이 고려된 뒤에 목적함수들이 고려된다. 목적함수들은 시스템에 영향을 주는 중요한 출력성능이어야만 한다. 모터 시스템에서 중요한 출력은 전자기 토크, 회전속도 그리고 회전자의 진동이다. 본 연구에서는 토크, 회전속도, 그리고 회전자 진동에 대한 민감도를 확인하고 분석한다. 본 연구의 결과로 토크와 회전자 진동은 변수들에 대한 민감도가 속도보다는 상대적으로 민감하다는 것을 알 수 있다. 민감도 해석을 통해서 본 연구에서는 몇 가지 모터 디자인에 대한 개선안을 제시한다. 모터의 높은 토크를 얻기 위해서 우리는 다음과 같은 사항들을 제안할 수 있다. 큰 질량을 가진 회전자, 편심이 큰 회전자, 작은 저항의 모터 코일, 적은 모터 코일 턴수, 그리고 작은 잔류자장을 가진 영구자석이 그것이다. 높은 회전속도를 얻기 위해서 우리는 작은 회전자 질량, 작은 회전자 편심, 작은 모터 코일 저항, 적은 코일 턴수, 그리고 작은 잔류자장을 가진 영구자석을 제안할 수 있다. 마지막으로 모터의 진동을 저감하기 위해서 우리는 작은 회전자 질량, 높은 댐핑 계수 및 강성, 회전자의 작은 편심, 높은 모터 코일 저항, 많은 코일 턴수, 그리고 큰 잔류자장을 가진 영구자석을 제안할 수 있다. 본 연구의 또 다른 목적은 BLDC모터의 개선된 새로운 디자인의 제시이다. 본 논문에서는 다중 목적함수법을 사용하였다. 그 목적함수는 최적화된 변수를 찾기 위한 최적함수이다. 민감도 해석을 결과를 이용해서 우리는 설계 변수 중 둔감한 변수들을 제거하고 BLDC모터의 설계범위를 결정하였다. 다중 목적함수법을 통해서, 목적함수들의 시간응답들을 설계범위 내에서 계산하였다. 그 함수값은 얻어진 시간응답의 rms값에 의해서 계산되고 그 값들 중 최소값을 확인할 수 있다. 확인된 최소값에서 최적 디자인 변수들을 얻을 수 있다.| In this study, the dynamic behaviors of a BLDC motor are analyzed, when the motor undergoes mechanical and electromagnetic interaction due to an air gap variation between the stator and rotor. When considering the air gap variation caused by the translational motion of the rotor relative to the stator, the kinetic and potential energies, Rayleigh dissipation function, and the magnetic coenergy are expressed in terms of the rotor displacements and stator currents. With these energies and function, new equations of motion are derived using Lagrange’s equation. The equations for the proposed model are nonlinear equations in which the displacements and currents are coupled. And then, we introduce the previous equations of motion obtained by the analytical method based on the Fourier series which have been used frequently since 1960’s. In previous model, the equations for the magnetic flux density are solved first for the stator currents from which the unbalanced magnetic forces is determined. The determined forces apply to the equations of translation motion for a motor. This means that the rotor displacements cannot affect the stator currents. The time responses for the displacements and currents are computed for the proposed and previous models. Furthermore, the effects of rotor eccentricity are also investigated. It is found that, when the air gap varies with time, the time responses for the proposed and previous models have small differences in the stator currents, electromagnetic torques, and rotating speeds. However, the time responses have large differences in the rotor displacements. Therefore, this paper claims that the proposed model describes the dynamic behaviors of the motor more accurately than the previous model. It is also shown that rotor eccentricity increases the stator current period and the electromagnetic torque, while it decreases the rotating speed of the rotor. Sensitivity analysis is used to determine how sensitive a model is to changes in the value of the parameters of the model and to changes in the structure of the model. By showing how the model behaviour responds to changes in parameter values, sensitivity analysis is a useful tool in model building as well as in model evaluation. In this study, we focus on parameter sensitivity for the BLDC motor. Parameter sensitivity is the derivative of the response with respect to the parameter. We derived the partial differential sensitivity equations with respect to the motor parameters using the direct differential method. The governing equations of motion for the proposed model derived by the Lagrange’s equation is used when the sensitivity equations is derived. The BLDC motor is the dynamic system so the sensitivity for the motor varies with time. Time histories of the sensitivity for it are obtained by applying the Newmark time integration method. In sensitivity analysis, we have to consider the parameters and object functions. In this research, we considered the resistance, number of coil turns and the residual magnetic flux density of the permanent magnet as the electromagnetic parameters and the mass and eccentricity of the rotor and the stiffness and damping coefficient of the shaft and bearing system as the mechanical parameters. The real motor has the mutual interaction between the mechanical and electromagnetic behaviours. Therefore, to perform the accurate sensitivity analysis, we have to consider the mechanical and electromagnetic parameters simultaneously. After the parameters considering, the object functions are considered. The object functions must be important outputs affecting a system. In the motor system, the important outputs are the torque, speed and vibration of the rotor. In this study, the sensitivities for the torque, speed, and vibration of the motor is checked and analyzed. As a result, the torque and vibration are sensitive considerably for the electromagnetic parameters and the speed is relatively insensitive for the parameters. Through the sensitivity analysis, this study presents some guidelines to design a motor. To design a motor with a high torque, we have to consider as follows: the large mass of the rotor, the large eccentricity of the rotor, the small resistance of the motor coil, the small number of coil turns, and the permanent magnet with the large residual magnetic flux density. To design a motor with a high speed, we can suggest as follows: the small mass of the rotor, the rotor with the small eccentricity, the small resistance of the motor coil, the small number of coil turns, and the permanent magnet with the small residual magnetic flux density. To reduce the motor vibration, we can present as follows: the small mass of the rotor, the rotor system with the high damping, the shaft and bearing system with the high stiffness, the rotor with the small eccentricity, the high resistance of the motor coil, the large number of the motor coil turns, and the permanent magnet with the large residual magnetic flux density. Another purpose of this study is to propose a new design for an improved BLDC motor. The multi objective function is presented in this paper. The function is the optimal function which can find the optimal design parameters. Using the results of the sensitivity analysis, we eliminate the most insensitive design parameter and determine the design range of the BLDC motor. Through the multi objective function method, time responses of the objective functions are computed within the determined ranges. The function value is calculated by using the root mean square values of the obtained time responses and we check the minimum of it. In the checked minimum value, the optimum design parameters are obtained.; In this study, the dynamic behaviors of a BLDC motor are analyzed, when the motor undergoes mechanical and electromagnetic interaction due to an air gap variation between the stator and rotor. When considering the air gap variation caused by the translational motion of the rotor relative to the stator, the kinetic and potential energies, Rayleigh dissipation function, and the magnetic coenergy are expressed in terms of the rotor displacements and stator currents. With these energies and function, new equations of motion are derived using Lagrange’s equation. The equations for the proposed model are nonlinear equations in which the displacements and currents are coupled. And then, we introduce the previous equations of motion obtained by the analytical method based on the Fourier series which have been used frequently since 1960’s. In previous model, the equations for the magnetic flux density are solved first for the stator currents from which the unbalanced magnetic forces is determined. The determined forces apply to the equations of translation motion for a motor. This means that the rotor displacements cannot affect the stator currents. The time responses for the displacements and currents are computed for the proposed and previous models. Furthermore, the effects of rotor eccentricity are also investigated. It is found that, when the air gap varies with time, the time responses for the proposed and previous models have small differences in the stator currents, electromagnetic torques, and rotating speeds. However, the time responses have large differences in the rotor displacements. Therefore, this paper claims that the proposed model describes the dynamic behaviors of the motor more accurately than the previous model. It is also shown that rotor eccentricity increases the stator current period and the electromagnetic torque, while it decreases the rotating speed of the rotor. Sensitivity analysis is used to determine how sensitive a model is to changes in the value of the parameters of the model and to changes in the structure of the model. By showing how the model behaviour responds to changes in parameter values, sensitivity analysis is a useful tool in model building as well as in model evaluation. In this study, we focus on parameter sensitivity for the BLDC motor. Parameter sensitivity is the derivative of the response with respect to the parameter. We derived the partial differential sensitivity equations with respect to the motor parameters using the direct differential method. The governing equations of motion for the proposed model derived by the Lagrange’s equation is used when the sensitivity equations is derived. The BLDC motor is the dynamic system so the sensitivity for the motor varies with time. Time histories of the sensitivity for it are obtained by applying the Newmark time integration method. In sensitivity analysis, we have to consider the parameters and object functions. In this research, we considered the resistance, number of coil turns and the residual magnetic flux density of the permanent magnet as the electromagnetic parameters and the mass and eccentricity of the rotor and the stiffness and damping coefficient of the shaft and bearing system as the mechanical parameters. The real motor has the mutual interaction between the mechanical and electromagnetic behaviours. Therefore, to perform the accurate sensitivity analysis, we have to consider the mechanical and electromagnetic parameters simultaneously. After the parameters considering, the object functions are considered. The object functions must be important outputs affecting a system. In the motor system, the important outputs are the torque, speed and vibration of the rotor. In this study, the sensitivities for the torque, speed, and vibration of the motor is checked and analyzed. As a result, the torque and vibration are sensitive considerably for the electromagnetic parameters and the speed is relatively insensitive for the parameters. Through the sensitivity analysis, this study presents some guidelines to design a motor. To design a motor with a high torque, we have to consider as follows: the large mass of the rotor, the large eccentricity of the rotor, the small resistance of the motor coil, the small number of coil turns, and the permanent magnet with the large residual magnetic flux density. To design a motor with a high speed, we can suggest as follows: the small mass of the rotor, the rotor with the small eccentricity, the small resistance of the motor coil, the small number of coil turns, and the permanent magnet with the small residual magnetic flux density. To reduce the motor vibration, we can present as follows: the small mass of the rotor, the rotor system with the high damping, the shaft and bearing system with the high stiffness, the rotor with the small eccentricity, the high resistance of the motor coil, the large number of the motor coil turns, and the permanent magnet with the large residual magnetic flux density. Another purpose of this study is to propose a new design for an improved BLDC motor. The multi objective function is presented in this paper. The function is the optimal function which can find the optimal design parameters. Using the results of the sensitivity analysis, we eliminate the most insensitive design parameter and determine the design range of the BLDC motor. Through the multi objective function method, time responses of the objective functions are computed within the determined ranges. The function value is calculated by using the root mean square values of the obtained time responses and we check the minimum of it. In the checked minimum value, the optimum design parameters are obtained.
URI
https://repository.hanyang.ac.kr/handle/20.500.11754/140630http://hanyang.dcollection.net/common/orgView/200000416135
Appears in Collections:
GRADUATE SCHOOL OF ENGINEERING[S](공학대학원) > MECHANICAL & INDUSTRIAL ENGINEERING(기계 및 산업공학과) > Theses(Ph.D.)
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