다중 격자 기법이 적용된 부분 내재적 조화 균형법을 이용한 비정상 유동 해석

임동균(D. K. Im),박수형(S. H. Park),권장혁(J. H. Kwon) 한국전산유체공학회 2010 한국전산유체공학회 학술대회논문집 Vol.2010 No.11

In order to analyze the periodic unsteady flow problem efficiently the partially implicit harmonic balance(PIHB) method was developed. Contrary to the existing harmonic balance method, this method handles the harmonic source term explicitly and deals with flux terms implicitly. This method has a good convergence in comparison with the full explicit harmonic method and it is easy to apply this method because there is no need to calculate the complicated flux Jacobian term by comparing with the full implicit harmonic method. With the multigrid method about the each harmonic it turns out that this method has a good convergence regardless method. With the multigrid method about the each harmonic it turns out that this method has a good convergence regardless of the number of harmonics. The oscillating flows over NACA0012 airfoil is considered to verivy this method then the result correponsed to both the result of dual time stepping and explicit Runge-Kutta method.

부분 내재적 조화 균형법을 이용한 주기적인 2차원 비정상 유동 해석

임동균(Dong-Kyun Im),박수형(Soo Hyung Park),권장혁(Jang Hyuk Kwon) 한국항공우주학회 2010 韓國航空宇宙學會誌 Vol.38 No.12

본 연구에서는 주기적 비정상 유동 해석을 위해 푸리에 변환을 이용하는 조화 균형법의 효율적인 해법을 제안한다. 내재적으로 유속항을 처리하고 외재적으로 조화 원천항을 처리하였다. 외재적 조화 균형법 보다 더 빠르게 수렴 시킬 수 있으며 내재적 조화 균형법을 적용할 때 추가되는 자코비안 행렬을 처리할 필요가 없다. 또한 완전 내재적 기법에 상응하는 수준의 수렴안정성을 확인할 수 있었다. 2차원 비정상 유동 문제로 피칭하는 NACA0012 익형에 적용하였으며 이중 시간 적분법 및 외재적 Runge-Kutta기법의 해와 매우 일치하는 결과를 얻었다. An efficient solution method for harmonic balance techniques with Fourier transform is presented for periodic unsteady flow problems. The present partially-implicit harmonic balance treats the flux terms implicitly and the harmonic source term is solved explicitly. The convergence of the partially Implicit method is much faster than the explicit Runge-Kutta harmonic balance method. The method does not need to compute the additional flux Jacobian matrix from the implicit harmonic source term. Compared with fully implicit harmonic balance method, this partial approach turns out to have good convergence property. Oscillating flows over NACA0012 airfoil are considered to verify the method and to compare with results of explicit R-K(Runge-Kutta) and dual time stepping methods.

다중격자 기법이 적용된 효율적인 조화 균형법 개발

임동균(D.K. Im),박수형(S.H. Park),권장혁(J.H. Kwon) 한국전산유체공학회 2010 한국전산유체공학회지 Vol.15 No.4

In order to analyze the periodic unsteady flow problem efficiently the partially implicit harmonic balance (PIHS) method was developed. Contrary to the existing harmonic balance method, this method handles the harmonic source term explicitly and deals with flux terms implicitly. This method has a good convergence in comparison with the full explicit harmonic method and it is easy to apply this method because there is no need to calculate the complicated flux Jacobian term by comparing with the full implicit harmonic method. With the multigrid method about the each harmonic it turns out that this method has a good convergence regardless of the number of harmonics. The oscillating flows over NACA0012 airfoil is considered to verify this method then the result correponsed to both the result of dual time stepping and explicit Runge-Kutfa method.

투영 기반 모델 차수 축소를 적용한 회전 구성품의 매개 변수 조화 균형 해석

강승훈,이상민,황민호,조해성,김용세,신상준 한국항공우주학회 2024 韓國航空宇宙學會誌 Vol.52 No.6

본 논문은 회전 구성품의 조화 균형 해석을 위한 매개 변수 모델 차수 축소(model-order reduction, MOR) 절차 및 하이퍼 축소(hyper-reduction, HR) 기법과의 결합을 제안하였다. 적합 직교 분해(proper orthogonal decomposition, POD) 모드를 축소 기저로 활용하였다. 비선형 하중 벡터 및 기울기 행렬의 계산 비용 절감을 위해 에너지 보존 요소 선별 및 가중치 부여(energy conserving sampling and weighting, ECSW) 기법을 도입하였다. 약 90만 자유도 예제에 대해 수치 해석이 수행되었다. 본 논문의 HR-MOR 절차에선 기해석 결과 수집, POD 모드 추출, ECSW 기법에 요구되는 전처리 계산 비용이 발생하지만, 상당히 신속한 매개 변수 조화 균형 해석이 수행되었다. 또한, 비선형 고조화 조화 성분을 포함한 진동 응답이 기존 전체 차수 조화 균형 해석 결과 대비 1% 이내의 상대 오차 내에서 정확하게 예측되었다. 전처리 계산 비용을 포함한 누적 계산 비용을 감안하였을 때, 본 논문의 HR-MOR 절차는 2회 이상의 다중 매개 변수 조화 균형 해석 시 효율적임을 확인하였다. In this paper, parametric model-order reduction (MOR) framework is utilized for the harmonic balance analysis of a rotating component, in conjunction with the hyper-reduction (HR) method. Proper orthogonal decomposition (POD) mode is used as a reducing basis. Energy conserving weighting and sampling (ECSW) method is also employed to reduce the computational cost for the nonlinear force vector and its gradient matrix. Numerical simulation is attempted for an example with approximately 0.9 million degrees of freedom. The additional off-line cost is required for the collection of the pre-computed results, POD mode extraction, and ECSW method. Nevertheless, significant efficiency is demonstrated. In addition, vibratory response, including the nonlinear super-harmonic component, is accurately obtained within the relative discrepancy of 1% when compared against those by the full-order analysis. Considering the required computational cost including that for the off-line stage, it is found that the present HR-MOR framework will be efficient for the parametric harmonic balance simulation.

Current harmonic selection for torque ripple suppression based on analytical torque model of PMSMs

Zhong, Zaimin,Zhou, Shuihua,Hu, Chengyu,You, Junming The Korean Institute of Power Electronics 2020 JOURNAL OF POWER ELECTRONICS Vol.20 No.4

For permanent magnet synchronous motors (PMSMs) without sufficient design optimization, it is necessary to inject current harmonics to suppress torque ripple. Due to the influence of magnetic saturation, the current harmonics for torque ripple suppression changes with the electrical loads. Based on an existing analytical torque model of a PMSM considering spatial harmonics and magnetic saturation, this paper established a torque harmonic coupling model through a Taylor expansion and a harmonic balance method. Then, based on the torque harmonic coupling model, current harmonic selection for torque ripple minimization was proposed. The validity of the torque harmonic coupling model and current harmonic selection was verified by a finite element analysis (FEA) and experiments on a laboratory PMSM drive system.

Approximate Response of a Non-linear Vibration Isolation System Using the Harmonic Balance Method

Gun-Myung Lee(이건명) 한국기계가공학회 2018 한국기계가공학회지 Vol.17 No.6

A non-linear vibration isolation system which is composed of a non-linear spring and a linear damper was proposed in past research. When the support of the isolation system is excited harmonically, the response component of the isolation system mass at the excitation frequency has been calculated approximately using the harmonic balance method. The response was approximated by a single mode, and the result was compared with a numerical result which is assumed as an accurate one. Next, the response was approximated by two modes, and the result was compared with the former one.

Prediction method for steady-state response of local rubbing blade-rotor systems

Qian Zhao,Hongliang Yao,Qi Xu,Bang-chun Wen 대한기계학회 2015 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.29 No.4

Blade-rotor systems frequently encounter the problem of blade-to-case rubbing, which affects their safety and stability. Numericalsimulation can be used to predict the steady-state response of these systems. However, such simulation is frequently computationallyexpensive because of the high dimensions of the dynamic model of a blade-rotor system. To overcome this problem, a new method thatcombines the receptance-based dimension-reduction approach with the incremental harmonic balance (IHB) method is presented in thisstudy. First, a dynamic model of a blade-rotor system is developed using the finite element method, and the number of dimensions of themodel is reduced by the receptance method. Subsequently, the steady-state response is obtained by the improved IHB method to convenientlymanage the large number of super-harmonic components of the local rubbing system. Finally, the precision and efficiency of theproposed method is verified by comparing its results with those obtained by the Newmark-b method. The proposed method is found to beefficient in analyzing local rubbing blade-rotor systems with high dimensions, local nonlinearities, and rich super-harmonics.

Dynamic Analysis of Harmonically Excited Non-Linear Structure System Using Harmonic Balance Method

문병영,강범수,김병수,Mun, Byeong-Yeong,Gang, Beom-Su,Kim, Byeong-Su 대한기계학회 2001 KSME International Journal Vol.15 No.11

An analytical method is presented for evaluation of the steady state periodic behavior of nonlinear structural systems. This method is based on the substructure synthesis formulation and a harmonic balance procedure, which is applied to the analysis of nonlinear responses. A complex nonlinear system is divided into substructures, of which equations are approximately transformed to modal coordinates including nonlinear term under the reasonable procedure. Then, the equations are synthesized into the overall system and the nonlinear solution for the system is obtained. Based on the harmonic balance method, the proposed procedure reduces the size of large degrees-of-freedom problem in the solving nonlinear equations. Feasibility and advantages of the proposed method are illustrated using the study of the nonlinear rotating machine system as a large mechanical structure system. Results obtained are reported to be an efficient approach with respect to nonlinear response prediction when compared with other conventional methods.

Nonlinear vibration analysis of a type of tapered cantilever beams by using an analytical approximate method

Weipeng Sun,Youhong Sun,Yongping Yu,Shaopeng Zheng 국제구조공학회 2016 Structural Engineering and Mechanics, An Int'l Jou Vol.59 No.1

In this paper, an alternative analytical method is presented to evaluate the nonlinear vibration behavior of single and double tapered cantilever beams. The admissible lateral displacement function satisfying the geometric boundary conditions of a single or double tapered cantilever beam is derived by using Rayleigh-Ritz method. Based on the Lagrange method and the Newton Harmonic Balance (NHB) method, analytical approximate solutions in closed and explicit form are obtained. These approximate solutions show excellent agreement with those of numeric method for small as well as large amplitude. Moreover, due to brevity of expressions, the present analytical approximate solutions are convenient to investigate effects of various parameters on the large amplitude vibration response of tapered beams.

Period doubling of the nonlinear dynamical system of an electrostatically actuated micro-cantilever

Y.M. Chen,J.K. Liu 국제구조공학회 2014 Smart Structures and Systems, An International Jou Vol.14 No.5

The paper presents an investigation of the nonlinear dynamical system of an electrostatically actuated micro-cantilever by the incremental harmonic balance (IHB) method. An efficient approach is proposed to tackle the difficulty in expanding the nonlinear terms into truncated Fourier series. With the help of this approach, periodic and multi-periodic solutions are obtained by the IHB method. Numerical examples show that the IHB solutions, provided as many as harmonics are taken into account, are in excellent agreement with numerical results. In addition, an iterative algorithm is suggested to accurately determine period doubling bifurcation points. The route to chaos via period doublings starting from the period-1 or period-3 solution are analyzed according to the Floquet and the Feigenbaum theories.