Representation of Dynamic Stiffness Matrix with Orthogonal Polynomials

직교다항식을 이용한 구조계의 축약된 동강성행렬 표현

  • 양경택 (대림산업(주) 기술연구소) ;
  • 최계식 (대림산업(주) 기술연구소)
  • Published : 1993.06.01

Abstract

A modeling method is described to provide a smaller structural dynamic model which can be used to compare finite element model of a structure with its experimental counterpart. A structural dynamic model is assumed to be represented by dynamic stiffness matrix. To validate a finite element model, it is often necessary to condense a large degrees of freedom (dofs) to a relatively small number of dofs. For these purpose, static reduction techniques are widely used. However, errors in these techniques are caused by neglecting frequency dependent terms in the functions relating slave dofs and master dofs. An alternative method is proposed in this paper in which the frequency dependent terms are considered by expressing the reduced dynamic stiffness matrix with orthogonal polynomials. The reduced model has finally a minimum set of dofs, such as sensors and excitation points and it is under the same condition as the physical system. It is proposed that the reduced model can be derived from finite element model. The procedure is applied to example structure and the results are discussed.

노후 구조물의 안전진단을 위하여 동적재하시험을 수행하고 그 결과를 유한요소모델과 같은 해석적 모델과 결합하여 기존구조물의 강성평가 및 파손부위 색출에 적용하고 있는데 측정점의 제한성과 유한요소모델의 많은 자유도가 측정데이타를 유한요소모델과 연계하는데 커다란 문제점으로 대두된다. 본 연구에서는 유한요소모델과 같이 많은 자유도를 갖는 구조계의 해석적 모델을 측정데이타와 결합하기 위하여 축약된 좌표계에서 구조계의 동강성행렬(dynamic stiffness matrix)표현방법을 제시하였다. 유한요소모델로부터 좌표계를 축약시 필연적으로 발생되는 주파수의존성(frequency dependency)을 고려하기 위하여 주파수영역에서 Chebyshev다항식으로 축약된 동강성행렬을 표시하였고 특이점에서 발생되는 악조건(ill-condition)을 극복하기 위하여 특이해분리(singular value decomposition)기법을 사용하였다. 제시된 방법의 검증을 위하여 간단한 구조계에 대하여 시뮬레이션을 수행하였으며 본 방법으로 수립된 구조계의 동적모델은 축약이전의 전체계에 대한 동적특성을 비교적 정확히 유지하고 있고 일반적으로 사용되는 정적축약 형태의 수학적 모델보다 우수함을 알 수 있었다.

Keywords

References

  1. AIAA Journal v.12 Statistical Identification of Structures Jon D. Collins;Gary C. Hart
  2. Transactions of The SAE v.81 The Treatment of Randomness in Finite Element Modelling Hart,G.C.;Collins,J.D.
  3. AIAA Journal v.8 no.7 Comment on the Eigenvalue Problem for Structural Systems with Statistical Properties Kiefling
  4. ASCE Journal Modal Analysis and Random Structural Systems T.K.Hasselman;G.C.Hart
  5. The Spring Meeting/STAR Symposium Structural Damage Detection by Measurement of Dynamic Response J.K.Vandiver
  6. AIAA Journal v.21 Improvement of a Large Analytical Model Using Test Data A.Berman;E.J.Nagy
  7. Int'J for Numerical Methods in Engineering v.18 Analytical Selection of Masters for Reduced Eigenvalue Problem Shah,V.N.;Raymund,M.
  8. AIAA Journal v.3 no.2 Reduction of Stiffness and Mass Matrices Robert J. Guyan
  9. AIAA Journal v.3 no.5 Structural Eigenvalue Problems: Elimination of Unwanted Variables Bruce;Irons
  10. The 10th Internal Modal Analysis Conferences Dynamic Substructuring by Guyan Condensation Selection of The Master DOF N.Bouhaddi;S.Cogan;R.fillod
  11. AIAA Journal v.11 Reduction of Structural Frequency Equations R.L.Kidder
  12. Numerical Recipes : The Art of Scientific Computing W.H.Press;B.P.Flannery;S.A.Teukolsky;W.T.Vetterling
  13. 전산구조공학회지 v.2 no.2 구조물의 동특성 추정방법에 관한 연구 윤정방;Masanobu Shinozuka
  14. Int'J for numerical Methods in Engineering v.12 An Accurate Method of Dynamic Condensation in Structural Analysis Andrew Yee-Tuk Leung
  15. Int'J for Numerical Methods in Engineering v.19 A New Approach to the Dynamic Analysis of Structures using Fixed Frequency Dynamic Stiffness Matrices A.J.Fricker