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Efficient Mode Superposition Methods for Non-Classically Damped System

12th KKNN Seminar Taejon, Korea, Aug. 20-22, 1999. Efficient Mode Superposition Methods for Non-Classically Damped System. Sang-Won Cho, Graduate Student, KAIST, Korea Ju-Won Oh, Professor, Hannam University, Korea In-Won Lee, Professor, KAIST, Korea. CONTENTS. Introduction

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Efficient Mode Superposition Methods for Non-Classically Damped System

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  1. 12th KKNN Seminar Taejon, Korea, Aug. 20-22, 1999 Efficient Mode Superposition Methods for Non-Classically Damped System Sang-Won Cho, Graduate Student, KAIST, Korea Ju-Won Oh, Professor, Hannam University, Korea In-Won Lee, Professor, KAIST, Korea

  2. CONTENTS • Introduction • Mode Superposition Methods for Classically Damped System • Mode Superposition Methods for Non-Classically Damped System • Numerical Examples • Conclusions

  3. INTRODUTION • Dynamic Equations of Motion where M : Mass matrix of order n C : Damping matrix of order n K : Stiffness matrix of order n u(t) : Displacement vector R0: Invariant spatial portion of input load r(t) : Time varying portion of input load (1)

  4. Introduction • Methods of Dynamic Analysis • Direct integration method • Short duration loading as an impulse • Mode superposition method • Long duration loading as an earthquake

  5. Introduction • Improved Mode Superposition Methods • Mode acceleration (MA) method • Modal truncation augmentation (MT) method • Limitation of MA and MT methods • Applicable only to classically damped systems

  6. Introduction • Objective To expand MA and MT methods to analyze non-classically damped systems

  7. Previous Studies: Mode Superposition Methods for Classically Damped System

  8. Classically Damped System (1) • Mode Displacement (MD) Method • Dynamic Equations of Motion • Modal Transformation • Modal Equations (2) where (3)

  9. Classically Damped System • MA Method (Williams, 1945) • Displacement (4) where

  10. Classically Damped System • MT Method (Dickens & Wilson, 1980) • Displacement (5) where P : MT vector : modal displacement

  11. Classically Damped System - For P - For , solve (6) (7) (8) (9)

  12. Classically Damped System • Summary

  13. This Study: Mode Superposition Methods for Non-Classically Damped System

  14. Non-Classically Damped System (1) • Non-Classically Damped System • Dynamic Equations of Motion • State Space Equations (10) where • Eigenvalue Problem (11) where and : complex conjugate pairs

  15. Non-Classically Damped System (10) • MD Method • State Space Equations • Modal Transformation • Modal Equations (12) (13)

  16. Non-Classically Damped System • MA Method • Displacement (14) where

  17. Non-Classically Damped System • MT Method • Displacement (15) where : MT vector : modal displacement

  18. Non-Classically Damped System • For • For , solve (16) (17) (18) (19)

  19. Non-Classically Damped System (19) • Stability of MT method • Modal equation • Solution ( r(t) = sin ( t), z(0)=0 ) • Stability condition (20) where (21)

  20. Non-Classically Damped System (19) • Characteristics of MT Solution • Solution • Property of • Simplification (22) (23)

  21. Non-Classically Damped System • Comparison MT Solution with MA Solution • MT solution • MA solution • Coefficient of MT solution (24) (25) (26) where (27)

  22. Non-Classically Damped System • Summary

  23. NUMERICAL EXAMPLES • Structures • Cantilever Beam with Lumped Dampers • To compare the MA and MT methods with MD method • 10-Story Shear Building • To show the divergent case of MT method

  24. Numerical Examples 11 10 9 100 IN 3 2 1 • Cantilever Beam with Lumped Dampers • El-Centro Earthquake Table 1 Eigenvalues E = 3.0107 L = 100 A = 4 C = 0.1 I = 1.25  = 7.4110-4 10 Beam Elements Fig. 1 Beam Configuration

  25. Numerical Examples 11 11 1 1 5 5 6 6 8 8 2 2 7 7 9 9 10 10 3 3 4 4 • Moment at Each Node MD Method MA & MT Methods Mm / Md Node Number Node Number Mm: Moment by mode superposition methods Md : Moment by direct integration method

  26. Numerical Examples 11 11 1 1 5 5 6 6 8 8 2 2 7 7 9 9 10 10 3 3 4 4 • Shear Force at Each Node MD Method MA & MT Methods Sm / Sd Node Number Node Number Sm: Shear force by mode superposition methods Sd : Shear force by direct integration method

  27. Numerical Examples m1=1Ksec2/IN k1=800 K/IN m2=2 Load Case 2 m3=2 m4=2 m5=3 k2=1600 m6=3 m7=3 m8=4 m9=4 m10=4 Load Case 1 • 10-Story Shear Building • Harmonic Loading ( = 32.0 rad/sec) Table 2 Eigenvalues Fig. 2 10-Story Shear Building

  28. Numerical Examples MA Method MT Method Displacement • Load Case 1 Time (sec ) Time (sec ) MA and MT solutions are same

  29. Numerical Examples • Load Case 2 MA Method MT Method Displacement No solution Time (sec ) Time (sec ) MT method gives no solution

  30. CONCLUSIONS • Expanded MA and MT methods were applied to non-classically damped system. • MA method is stable, whereas MT method is conditionally stable. • MT method gives same results with MA method when MT method is stable.

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