2. 1 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 12th KKNN Seminar Taejon, Korea, Aug. 20-22, 1999

3. 2  Introduction  Mode Superposition Methods for Classically Damped System  Mode Superposition Methods for Non-Classically Damped System  Numerical Examples  Conclusions CONTENTS

4. 3  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 R 0 :Invariant spatial portion of input load r(t):Time varying portion of input load (1)INTRODUTION

5. 4  Methods of Dynamic Analysis Direct integration method Direct integration method -Short duration loading as an impulse Mode superposition method Mode superposition method -Long duration loading as an earthquake Introduction Introduction

6. 5  Improved Mode Superposition Methods Mode acceleration (MA) method Mode acceleration (MA) method Modal truncation augmentation (MT) method Modal truncation augmentation (MT) method  Limitation of MA and MT methods Applicable only to classically damped systems Applicable only to classically damped systems Introduction Introduction

7. 6  Objective To expand MA and MT methods to analyze non-classically damped systems Introduction Introduction

8. 7 Previous Studies: Mode Superposition Methods for Classically Damped System

9. 8  Mode Displacement (MD) Method Dynamic Equations of Motion Dynamic Equations of Motion Modal Transformation Modal Transformation Modal Equations Modal Equations (1) where (3) (2) Classically Damped System Classically Damped System

10. 9  MA Method (Williams, 1945) Displacement Displacement Classically Damped System Classically Damped System where (4)

11. 10  MT Method (Dickens & Wilson, 1980) Displacement Displacement (5) Classically Damped System Classically Damped System where P: MT vector : modal displacement

12. 11 - For P - For P - For, solve - For, solve (6) Classically Damped System Classically Damped System (9) (7) (8)

13. 12 Classically Damped System Classically Damped System  Summary

14. 13 This Study: Mode Superposition Methods for Non-Classically Damped System

15. 14  Non-Classically Damped System Dynamic Equations of Motion Dynamic Equations of Motion State Space Equations State Space Equations (1) Non-Classically Damped System Non-Classically Damped System (10) where Eigenvalue Problem Eigenvalue Problem where and : complex conjugate pairs (11)

16. 15  MD Method State Space Equations State Space Equations Modal Transformation Modal Transformation Modal Equations Modal Equations (10) (12) (13) Non-Classically Damped System Non-Classically Damped System

17. 16  MA Method Displacement Displacement (14) Non-Classically Damped System Non-Classically Damped System where

18. 17  MT Method Displacement Displacement (15) Non-Classically Damped System Non-Classically Damped System where : MT vector : modal displacement

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

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

21. 20 Characteristics of MT Solution Characteristics of MT Solution -Solution -Property of -Simplification (19) (22) (23) Non-Classically Damped System Non-Classically Damped System

22. 21 Comparison MT Solution with MA Solution Comparison MT Solution with MA Solution -MT solution -MA solution -Coefficient of MT solution (24) (25) Non-Classically Damped System Non-Classically Damped System (26) where (27)

23. 22 Non-Classically Damped System Non-Classically Damped System  Summary

24. 23  Structures Cantilever Beam with Lumped Dampers Cantilever Beam with Lumped Dampers -To compare the MA and MT methods with MD method 10-Story Shear Building 10-Story Shear Building -To show the divergent case of MT method NUMERICAL EXAMPLES

25. 24  Cantilever Beam with Lumped Dampers El-Centro Earthquake El-Centro Earthquake 1 2 3 9 10 11 100 IN Fig. 1 Beam Configuration E = 3.0  10 7 L = 100 A = 4 C = 0.1 I = 1.25  = 7.41  10 -4 10 Beam Elements Table 1 Eigenvalues Numerical Examples Numerical Examples

26. 25 Moment at Each Node Moment at Each Node MD Method MA & MT Methods M m / M d 1 2 3 910 11 4 56 7 8 Node Number 1 2 3 910 11 4 56 7 8 Numerical Examples Numerical Examples M m : Moment by mode superposition methods M d : Moment by direct integration method

27. 26 Shear Force at Each Node Shear Force at Each Node MD Method MA & MT Methods 1 2 3 910 11 4 56 7 8 Node Number 1 2 3 910 11 4 56 7 8 Numerical Examples Numerical Examples S m / S d S m : Shear force by mode superposition methods S d : Shear force by direct integration method

28. 27  10-Story Shear Building Harmonic Loading ( = 32.0 rad/sec ) Harmonic Loading ( = 32.0 rad/sec ) m 1 =1K  sec 2 /IN m 2 =2 k 1 =800 K/IN k 2 =1600 m 3 =2 m 4 =2 m 5 =3 m 6 =3 m 7 =3 m 8 =4 m 9 =4 m 10 =4 Fig. 2 10-Story Shear Building Table 2 Eigenvalues Load Case 2 Load Case 1 Numerical Examples Numerical Examples

29. 28 Load Case 1 Load Case 1 MA Method MT Method Displacement MA and MT solutions are same Time (sec ) Numerical Examples Numerical Examples

30. 29 Displacement Load Case 2 Load Case 2 MT method gives no solution MA Method MT Method Time (sec ) Numerical Examples Numerical Examples No solution

31. 30 Expanded MA and MT methods were applied to non-classically damped system. Expanded MA and MT methods were applied to non-classically damped system. MA method is stable, MA method is stable, whereas MT method is conditionally stable. MT method gives same results with MA method MT method gives same results with MA method when MT method is stable. CONCLUSIONS