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Mode Superposition Methods for Dynamic System Analysis

Explore efficient mode superposition methods for structural dynamics and vibration control. Learn about advantages, drawbacks, and improved techniques for classical and non-classically damped systems through numerical examples.

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Mode Superposition Methods for Dynamic System Analysis

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  1. 1998년도 대한 토목학회 학술발표회1998. 10. 23 비비례 감쇠시스템에 대한 효율적인 모드 중첩법 *조상원석사과정, 한국과학기술원 토목공학과 김만철박사, 한국과학기술원 토목공학과 박선규조교수, 성균관대학교 토목공학과 이인원교수, 한국과학기술원 토목공학과 Structural Dynamics & Vibration Control Lab., KAIST

  2. CONTENTS • Introduction • Mode Superposition Method for Classically Damped Systems • Mode Superposition Method for Non-Classically Damped Systems • Numerical Examples • Conclusions Structural Dynamics & Vibration Control Lab., KAIST

  3. INTRODUCTION Background • Dynamic Equations of Motion M : Mass matrix of order n C : Damping matrix of order n K : Stiffness matrix of order n u(t): Displacement vector f(t) : Load vector (1) where • Methods of Dynamic Analysis • Direct integration method • Mode superposition method Structural Dynamics & Vibration Control Lab., KAIST

  4. Advantages of Mode Superposition Method • Effective because of using a few modes • Gives the dynamic characteristics of each mode • Effective for long duration loading • Drawbacks of Mode Superposition Method • Fail to give an accurate solution • Need to consider the effects oftruncated high modes • Improved Mode Superposition Methods • Mode acceleration method (MA method) • Modal truncation augmentation method (MT method) Structural Dynamics & Vibration Control Lab., KAIST

  5. Non-classically Damped System • Decoupling the System (1) (2) where • If Cg has off-diagonal elements, C is called as non-classical damping approximate to classically damped system so, off-diagonal terms are ignored Structural Dynamics & Vibration Control Lab., KAIST

  6. Objective In this study, improved mode superposition methods are applied to non-classically damped system Structural Dynamics & Vibration Control Lab., KAIST

  7. MODE SUPERPOSISTION METHOD FOR CLASSICALLY DAMPED SYSTEM Mode Superposition Method • The Dynamic Equations of Motion (1) • Decoupled Equations by Eigenvectors (2) • Displacements us(t) (3) where Structural Dynamics & Vibration Control Lab., KAIST

  8. Mode Acceleration Method (MA Method) • The Solution by MA Method (4) (5) (6) where us(t) : displacements modallyrepresented ut(t) : displacements not represented by the modes r(t) : time varying portion of f(t) R0 : invariant spatial portion of f(t) Structural Dynamics & Vibration Control Lab., KAIST

  9. The Portion of MA Solution • us(t) : displacements modally represented • ut(t) : displacements not represented by the modes (7) (8) where :force truncation vector : modally represented spatial load vector MA method approximates ut(t) by a static solution Structural Dynamics & Vibration Control Lab., KAIST

  10. Modal Truncation Augmentation Method (MT Method) • The Solution by MT Method (4) • The Portion of Solution • us(t) : displacements modally represented • ut(t) : displacements not represented by the modes (7) (9) MT method approximates ut(t) by a dynamic solution Structural Dynamics & Vibration Control Lab., KAIST

  11. Derivation of Pseudo Eigenvector P (10) where Structural Dynamics & Vibration Control Lab., KAIST

  12. MODE SUPERPOSISTION METHOD FOR NON-CLASSICALLY DAMPED SYSTEM • State Space Equations State Space Equations (11) where Structural Dynamics & Vibration Control Lab., KAIST

  13. Eigenvalue and Eignevector • Associated Eigenvalue Problem (12) (13) where Structural Dynamics & Vibration Control Lab., KAIST

  14. Mode Superposition Method in Non-Classically Damped System • State space equation (11) • Decouple the eqn.(3) by complex eigenvector (14) • Displacements ys(t) (15) Eigenvectors are conjugate pairs (16) Structural Dynamics & Vibration Control Lab., KAIST

  15. MA Method • The Solution by MA Method (17) (18) where ys(t) : displacements modallyrepresented yt(t) : displacements not represented by the modes r(t) : time varying portion of F(t) R0 : invariant spatial portion of F(t) Structural Dynamics & Vibration Control Lab., KAIST

  16. The Portion of Solution • ys(t) : displacements modally represented • yt(t) : displacements not represented by the modes (19) (20) where :force truncation vector : modally represented spatial load vector y(t) is calculated from conjugate pair MA method approximates yt(t) by a static solution Structural Dynamics & Vibration Control Lab., KAIST

  17. MT Method • The Solution by MT Method (17) • The Portion of Solution • ys(t) : displacements modally represented • yt(t) : displacements not represented by the modes (19) (21) MT method approximates yt(t) by a dynamic solution Structural Dynamics & Vibration Control Lab., KAIST

  18. Derivation of Pseudo Eigenvector P (22) where Structural Dynamics & Vibration Control Lab., KAIST

  19. Structures • Four-story shear building • Cantilever beam with multi-lumped dampers • Comparisons • Displacement responses • Displacement error of each method NUMERICAL EXAMPLES Structural Dynamics & Vibration Control Lab., KAIST

  20. R0cost M1 U1 K1 M2 U2 K2 M3 U3 K3 M4 U4 K4 Four-Story Shear Building • Input load ( R0cosWt ) • R0=1 • W = 7rad/sec (≒0.5 w1 ) • w1 = - 0.0317 ±13.2935 i • w2 = - 0.0007 ±29.6597 i Structural Dynamics & Vibration Control Lab., KAIST

  21. Displacement Responses (using 1 mode) Displacement Time(sec) Structural Dynamics & Vibration Control Lab., KAIST

  22. Displacement Responses(using 2 modes) Displacement Time(sec) Structural Dynamics & Vibration Control Lab., KAIST

  23. Displacement Error Using 1 Mode Using 2 Modes Difference Time(sec) Time(sec) Structural Dynamics & Vibration Control Lab., KAIST

  24. R0sin(W t) 1 2 3 45 49 50 14 Cantilever beam with multi-lumped dampers • Material property • Tangential damper, c : 0.3 • Young’s modulus : 100 • Mass density : 1 • Moment of inertia : 1 • Cross-section area : 1 • Input load ( R0cosWt ) • R0=1 • W = 5rad/sec (≒0.6w1 ) • w1 = -0.2696877077±7.7304416035 i • w2 = -0.4278691331±9.7835115896 i Structural Dynamics & Vibration Control Lab., KAIST

  25. Displacement Responses (using 1 mode) Displacement Time(sec) Structural Dynamics & Vibration Control Lab., KAIST

  26. Displacement Responses (using 2 modes) Displacement Time(sec) Structural Dynamics & Vibration Control Lab., KAIST

  27. Displacement Error Using 1 Mode Using 2 Modes Time(sec) Time(sec) Structural Dynamics & Vibration Control Lab., KAIST

  28. Improved mode superposition methods are applied to non-classically damped systems. • MA method and MT method aremore efficientthan simple mode superposition method. • MA method and MT method havesame convergence ratein non-classically damped system. CONCLUSIONS Structural Dynamics & Vibration Control Lab., KAIST

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