CONTENTS l Introduction l Mode Superposition Method for

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CONTENTS l Introduction l Mode Superposition Method for Classically Damped Systems l Mode Superposition

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

INTRODUCTION Background l Dynamic Equations of Motion (1) where M C K u(t) f(t)

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

l Advantages of Mode Superposition Method w Effective because of using a few modes

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

Non-classically Damped System l Decoupling the System (1) (2) where l If Cg has

Non-classically Damped System l Decoupling the System (1) (2) where l 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 5

Objective In this study, improved mode superposition methods are applied to non-classically damped system

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

MODE SUPERPOSISTION METHOD FOR CLASSICALLY DAMPED SYSTEM Mode Superposition Method l The Dynamic Equations

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

Mode Acceleration Method (MA Method) l The Solution by MA Method (4) (5) (6)

Mode Acceleration Method (MA Method) l The Solution by MA Method (4) (5) (6) where us(t) : displacements modally represented ut(t) : displacements not represented by the modes r(t) : time varying portion of f(t) R 0 : invariant spatial portion of f(t) Structural Dynamics & Vibration Control Lab. , KAIST 8

l The Portion of MA Solution w us(t) : displacements modally represented (7) w

l The Portion of MA Solution w us(t) : displacements modally represented (7) w ut(t) : displacements not represented by the modes (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 9

Modal Truncation Augmentation Method (MT Method) l The Solution by MT Method (4) l

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

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

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

MODE SUPERPOSISTION METHOD FOR NON-CLASSICALLY DAMPED SYSTEM State Space Equations l State Space Equations

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

l Associated Eigenvalue Problem (12) l Eigenvalue and Eignevector (13) where Structural Dynamics &

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

l Mode Superposition Method in Non-Classically Damped System w State space equation (11) w

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

MA Method l The Solution by MA Method (17) (18) where ys(t) : displacements

MA Method l The Solution by MA Method (17) (18) where ys(t) : displacements modally represented yt(t) : displacements not represented by the modes r(t) : time varying portion of F(t) R 0 : invariant spatial portion of F(t) Structural Dynamics & Vibration Control Lab. , KAIST 15

l The Portion of Solution w ys(t) : displacements modally represented (19) w yt(t)

l The Portion of Solution w ys(t) : displacements modally represented (19) w yt(t) : displacements not represented by the modes (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 16

MT Method l The Solution by MT Method (17) l The Portion of Solution

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

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

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

NUMERICAL EXAMPLES l Structures w Four-story shear building w Cantilever beam with multi-lumped dampers

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

Four-Story Shear Building R 0 cos t M 1 K 2 K 3 K

Four-Story Shear Building R 0 cos t M 1 K 2 K 3 K 4 M 2 M 3 M 4 U 1 U 2 U 3 • Input load ( R 0 cos. Wt ) R 0=1 W = 7 rad/sec (≒ 0. 5 w 1 ) w 1 = - 0. 0317 ± 13. 2935 i w 2 = - 0. 0007 ± 29. 6597 i U 4 Structural Dynamics & Vibration Control Lab. , KAIST 20

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

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

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

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

Displacement Error Using 2 Modes Difference Using 1 Mode Time(sec) Structural Dynamics & Vibration

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

Cantilever beam with multi-lumped dampers R 0 sin(W t) 1 2 3 45 14

Cantilever beam with multi-lumped dampers R 0 sin(W t) 1 2 3 45 14 • Material property Tangential damper, c : 0. 3 Young’s modulus : 100 Mass density : 1 Moment of inertia : 1 Cross-section area : 1 • Input 49 50 load ( R 0 cos. Wt ) R 0=1 W = 5 rad/sec (≒ 0. 6 w 1 ) w 1 = -0. 2696877077± 7. 7304416035 i w 2 = -0. 4278691331± 9. 7835115896 i Structural Dynamics & Vibration Control Lab. , KAIST 24

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

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

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

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

Displacement Error Using 1 Mode Time(sec) Using 2 Modes Time(sec) Structural Dynamics & Vibration

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

CONCLUSIONS u Improved mode superposition methods are applied to non-classically damped systems. u MA

CONCLUSIONS u Improved mode superposition methods are applied to non-classically damped systems. u MA method and MT method are more efficient than simple mode superposition method. u MA method and MT method have same convergence rate in non-classically damped system. Structural Dynamics & Vibration Control Lab. , KAIST 28

Superposition of waves Superposition principle the superposition principleSuperposition of waves Superposition principle the superposition principle
Traveling Waves Superposition Wave Superposition Add the twoTraveling Waves Superposition Wave Superposition Add the two
16 Superposition and Standing Waves Superposition of Waves16 Superposition and Standing Waves Superposition of Waves
Lecture 9 Superposition Superposition Theorem 12 In aLecture 9 Superposition Superposition Theorem 12 In a
Superposition Free SHM EOM Solutions Superposition the sumSuperposition Free SHM EOM Solutions Superposition the sum
Superposition Theorem Objective of Lecture Introduce the superpositionSuperposition Theorem Objective of Lecture Introduce the superposition
Superposition Theorem Superposition theorem is one of theSuperposition Theorem Superposition theorem is one of the
Superposition of Waves Principle of Superposition What happensSuperposition of Waves Principle of Superposition What happens
Superposition Free SHM EOM Solutions Superposition the sumSuperposition Free SHM EOM Solutions Superposition the sum
Les relations gomtriques superposition recoupement inclusion Superposition recoupementLes relations gomtriques superposition recoupement inclusion Superposition recoupement
SUPERPOSITION RELATIVE DATING IS NOT EXACT SUPERPOSITION RELATIVESUPERPOSITION RELATIVE DATING IS NOT EXACT SUPERPOSITION RELATIVE
Geologic Time LAW OF SUPERPOSITION Super what SuperpositionGeologic Time LAW OF SUPERPOSITION Super what Superposition
PORTC MODE DIR DATA Bit MODE 0 MODEPORTC MODE DIR DATA Bit MODE 0 MODE
Mode Questce que cest la mode La modeMode Questce que cest la mode La mode
Contents 2 Contents 3 Contents 4 Contents 5Contents 2 Contents 3 Contents 4 Contents 5
TITLE CONTENTS 1 CONTENTS 2 CONTENTS 3 CONTENTSTITLE CONTENTS 1 CONTENTS 2 CONTENTS 3 CONTENTS
CONTENTS 1 CONTENTS 2 CONTENTS 3 CONTENTS 4CONTENTS 1 CONTENTS 2 CONTENTS 3 CONTENTS 4
contents 1 contents 2 contents 3 contents 4contents 1 contents 2 contents 3 contents 4
The Bisection Method Introduction Bisection Method Bisection MethodThe Bisection Method Introduction Bisection Method Bisection Method
LPP Simplex Method Simplex method Introduction Simplex methodLPP Simplex Method Simplex method Introduction Simplex method
Problem statements The superposition method is employed TheProblem statements The superposition method is employed The
TransmissionPropagation Operator Theory TipWave Superposition Method compared toTransmissionPropagation Operator Theory TipWave Superposition Method compared to
TransmissionPropagation Operator Theory and Tip Wave Superposition MethodTransmissionPropagation Operator Theory and Tip Wave Superposition Method
Reflections and Superposition Contents Reflections Why they occurReflections and Superposition Contents Reflections Why they occur