MODALDECOMPOSITIONANDBEHAVIOUROFFREEVIBRATIONRESPONSEWITHGROUNDINGANDUPLIFTING

Taisuke MasunoTaiseiCorporationDesignDivision

16thU.S.‐Japan‐N.Z.WorkshopontheImprovementofStructuralEngineeringandResiliency

UpliftedStructure:Limitingearthquakeforce

2

Fixed Foundation FreeJointFoundation

Buildingholdingonagainstearthquake,resultingislargerinputforce

Inputearthquakeforcewillbeleveledoffwhenuplifted

3

“Unknownrandomvibration”amplifiedandactingontheupliftedstructure

Subject:MechanismofHigherModeVibration

SpringPendulum4

Equilibrium Position

sMs

sK

sysM

Natural Position

5

uplifting grounding

grounding uplifting landing

detatchment

ULUL

ULM

ULM

M

• Uplifting• Landing• Grounding• Detaching• UpliftingUltimateMoment• RockingStiffness

RestoringforceofGround

KR

RestoringforcecharacteristicsofgroundspringR

6

• DetachPhenomenon• LandingDetachPhenomenon

Grounding

MUL

The rightside uplifted

-MUL

UL

UL

M

KR

DetachPhenomenon

Landing DetachPhenomenon

GroundingThe rightside uplifted

Equationofrockingmotionmodel

7

,

,

,

N

iii

N

ii

NNN

HmIsym

Hmm

Hmm

1

2

0

111

M

2

2 2 1

0

00

N N

N

k kk k

k k ksym

K

0fKxxCxM ex

MUL

‐MUL

θUL‐θUL

M

θKR

TranspositionofNon‐linearComponent

1

Nu

uθ

x

に、

RM0

0

exf及び・・上部架構自由度

・・ロッキング自由度 DOFofrocking

DOFofsuperstructure

0fKxxCxM ex

exfKxxCxM

8

exex xMMf

ex1

R 10

0

H

H

M

N

canbeexpressedasmultiplicationofMandexf exx

exxMKxxCxM

N

iiIM

0Rexここに、 ・・rocking gravity acceleration

Rockingreactionforce

Where

TranspositionofNon‐linearComponent

Expressesupliftingandgroundingbehaviour ofastructure

：Structuralresponse：gravityaccelerationvectorexxx

9

exxMKxxCxM

s :Generalizedgravityaccelerations s :Generalizedgravity

s s s s s s s sM y C y K y M

EquilibriumPosition

sMs

sK

sM

NaturalPosition

ModalDecomposition

10

1,2,1 Nssss 0φKφM

Arbitraryt :111

1

00

0

φ

011 1 φφKtK

02 1111 KMhC

111111 yMyM

→1stmodeisrigidrotationmode

2

2 2 1

0

00

N N

N

k kk k

k k ksym

K

N

iii

N

ii

NNN

HmIsym

Hmm

Hmm

1

2

0

111

M

→GeneralizedStiffness,Dampingwillbe0

→1st modebehaviour isuniformacceleration

RigidRotation1st Mode

11

KMyMyKs

ULsssULssss

gULgUL

Sth modespringpendulumwilloscillateaboutlimitedequilibriumpositionandwillbestoppedat afterfullyattenuated

RotationalrestoringforceisconstantM=MUL atupliftingphase.Hence,generalizedgravityandequilibriumpositionwillbeconstant

LimitedEquilibriumPosition

LimitedEquilibriumPosition

sMsUL

sK

sM

NaturalPosition

1st 2nd 3rd 4th ・・・

NaturalPosition

EQP

EQPEQP

η1

η2

η3 η4

SpringPendulumofeachmodeatstaticequilibriumposition

12

LimitedEquilibriumPosition

EQP:Equilibrium Position

UpliftingPhase Changeinthedirectionofgravityoneachmode

Springpendulumofeachmode

13

MechanismofHigherModeVibration

A B B

A

time

Natural position

EQP(the right side)

DetachPhenomenon

Landing DetachPhenomenon

2syg

4syg

ULssM

ULssM

EQP(the left side)syg

gravity

ForEachMode

14

LtdEQPat PhaseA

Max.ofPhaseBwhenLandingDetachPhenomenon

Ltd EQPatPhaseB

Naturalposition

Max.ofPhaseAwhenDetachPhenomenon

TrackofEQP

Grounding Firstuplifting Seconduplifting Thirduplifting

PhaseA PhaseB

MechanismofHigherModeVibration

1.0

1.0

2.0

2.0