結構動力學大連理工土木水利學院

院DalianUniversityof第I單自由第2－3工程抗震InstituteofEarthquakeEngineering院§2-1基本動力體系的kcmkcmkcmDaliankcmDalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院DalianUniversityofmcc工程抗震InstituteofEarthquakeEngineering

2014-12-01 院體系質量 彈性特性（剛度或柔度）k能量耗散機制或阻 DalianUniversityof每個特性都假設集結于DalianUniversityof研究單自由度體系的自由振動重要性在塔、單層廠房1、它代表了許多實際塔、單層廠房InstituteofEarthquakeEngineering2014-12- 2、它是分析多自由度InstituteofEarthquakeEngineering2014-12- 工程抗院v(t院v(tkmp(tcv(t受力Fsmp(tv(t受力Fsmp(tF(tdDalianUniversityof

2014-12-01 院

FsFdDalianUniversityof粘滯（或粘性）阻尼（ViscousDalianUniversityofRcdv(t)外力（External

InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院FFsFdp(t)kv(t)cv(t)p(t)DalianDalianUniversityof單自由度系統(tǒng)的運動方程（Equationof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院DalianUniversityDalianUniversityof由于δv的任意性，且不等工程抗震InstituteofEarthquakeEngineering

2014-12-01 院DalianUniversityofFFs(t)Fs(t)vv(t)v(t)工程抗震InstituteofEarthquakeEngineering

2014-12-01 院

Fs(t)kv(t)kst動力平衡DalianUniversityofcv(t)kv(t)kstp(tDalianUniversityof去掉靜力平衡Δst與時間無工程抗震InstituteofEarthquakeEngineering

2014-12-01 院院DalianUniversityDalianUniversityofInstituteofEarthquakeEngineering2014-12- 性、彈性、InstituteofEarthquakeEngineering2014-12- 性、彈性、小變形的情工程抗院k2ck2f(t)v(t)v(t)vtgItfs(t)/

fD

fs(t)/DalianUniversityoffI(tDalianUniversityof

(t)

(t)阻尼力和彈性力只與相對

構破壞的原因質越大，慣性力越InstituteofEarthquakeEngineering2014-12- InstituteofEarthquakeEngineering2014-12- 工程抗院gvt(t)v(t)vg

DalianUniversityof DalianUniversityof地面運動一般測量的是加速度，此時需要 加速度積分得地面運動的速度和位InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院§院DalianUniversityDalianUniversityof

自由Particular自由振

Free!＝0強迫復復

v(t)齊線性方程，通解取工程抗震InstituteofEarthquakeEngineering

2014-12-01 院

RealGGRG

θ為幅DalianUniversityofGGcosiGsin(ei)DalianUniversityof(ei)

eicosisineicosisincos[eiei]/sini[eiei]/工程抗震InstituteofEarthquakeEngineering

2014-12-01 院院kcsms2k/m2

DalianUniversityof方DalianUniversityof相對于k和m的

2

css2m

運動方程的

s1v(t)G1

c2G2工程抗震InstituteofEarthquakeEngineering

2014-12-01 v(t)GeitG12v(t)(G1v(t)GeitG12院(G2RiG2I)(cos院v(t)(G1RG2R)cost(G1IG2I)sinti[(G1IG2I)cost(G1RG2R)sint]自由振動必須是實的，虛部項必DalianUniversityofG1RG2RGRG1IG2DalianUniversityofA2GR;B

v(t)2GRcost2GIsint2Gcos(tAcostBsinv(t) iG)eit iG 常微分方程中有如下定G

G2;

如果方程有復值解，則它的共數(shù)、復值解的實部和虛部也都InstituteofInstituteofEarthquakeEngineering2014-12-

相位

程的解 院Dalian院DalianUniversityof

v(t)2GRcost2GIsint2Gcos(tAcostB iGI

iGI v(0)A2G;v(0)B v(t)v(0)costv(0)sin

v(t)cos(tv(0)2v(0)2[tan1BA

工程抗震InstituteofEarthquakeEngineering

2014-12-01 院

v(t)cos(t[

tan1DalianUniversityoff;T2 自振周期或固有周期（NaturalPeriodof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院院2css2m

sc(c(c)22臨界c2m

低阻

阻尼cc c cDalianUniversityofs阻1DalianUniversityofs阻

c/cc2222

超阻

11D工程抗震InstituteofEarthquakeEngineering

2014-12-01 院c2mccs1s2c2mccs1s2初始條件v(0)

個相同的實

G1和G2均為實G2v(0)DalianUniversityofDalianUniversityof

G v(t)[v(0)(v(0)

2mω是系統(tǒng)不出現(xiàn)振蕩的最工程抗震InstituteofEarthquakeEngineering

2014-12-01 超臨界院DalianUniversityof

c/cc2s22

工程抗震InstituteofEarthquakeEngineering

2014-12-01 院v(t)c c

v(t)(G GeiwDtc cs

為使反應是實的，G1和G2必須為共軛D1阻尼體系的自D1阻尼體系的自振頻采用三角函數(shù) v(t)(AcoswtBsinw DalianUniversityofv(t)(w AsinwtwBcoswt)et(AcoswtBsinwt)et DalianUniversityofv(0)

Bv(0)Dv(t)(v(0)coswtv(0)v(0)sinw D工程抗震InstituteofEarthquakeEngineering

2014-12-01 院v(t)(v(0)coswtv(0)v(0)sinw Dv(t)cos(t D

[[]2D ;

具有不變的振動周期具有不變的振動周期工程抗InstituteofEarthquakeEngineering2014-12- DalianUniversityof院通常的工程結構，阻尼比ξ的值一般在1%~10%之間。故近似地認為ωD＝ξξDalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 院t,tt,t2/v(t)cos(Dt)e相鄰兩個正波峰的比：

/

e2DalianUniversityofDalianUniversityof11

ln

/

2小阻尼情況近 Taylor

vn/

12

2!取前兩

由此可見，對于粘滯阻尼自

動，任意相鄰兩振幅之間的是一個常工程抗震InstituteofEarthquakeEngineering

2014-12-01 院阻尼院1lnvn/ 1近似

DalianUniversityof這種測定阻尼系數(shù)的方法稱為自振衰減法(FreeDalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院

例p=20kips;v=0.20in

無阻尼振動頻f1/T1/1.402f1 v=0.16in;1

阻尼特DalianUniversityofDalianUniversityofT2 1.40

阻尼系

ln(0.20/0.16)/2cccW 0.0496386

阻尼頻11

0.9991/2

0.0355219201.548kips

6周后振

(v/v)6 工程抗震InstituteofEarthquakeEngineering

2014-12-01 院m求自振頻m

DalianDalianUniversityof

m k

KK3EI工程抗震InstituteofEarthquakeEngineering

2014-12-01 院kkalDalianUniversityofml2kaDalianUniversityofm工程抗震InstituteofEarthquakeEngineering

2014-12-01 院1DalianUniversityofU DalianUniversityof2

T2

T1(mM)v2工程抗震InstituteofEarthquakeEngineering

2014-12-01 院2-DalianUniversityDalianUniversityof

第2工程抗震InstituteofEarthquakeEngineering

2014-12-01 院無阻尼自由振動v(t2GRcost2GIsint2Gcos(tAcostBsinDalianUniversityofDalianUniversityof臨界阻

v(t)(GRiGI k/m2c2m

iGI

c c 超阻

11

低阻 v(t)(AcoswtBsinw 工程抗震InstituteofEarthquakeEngineering

2014-12-01 院

v(t 0 0

m

p0sin(tDalianDalianUniversityof

vc(t)AcostBsintvp(t)Csint0m2CsintkCsint0

sin確定常數(shù)

Cp0

;頻率k12 頻率 v(t)AcostBsintCsin工程抗震InstituteofEarthquakeEngineering

2014-12-01 院 v(t)AcostBsintCsin系數(shù)A和B通過初始條件確定。對于由靜止開始的運動

Cp0 112;Cp0 112; 12 DalianDalianUniversityof

v(t)p0

(sintsin

R(t)v(t)

12R(t) (sin 12

p0/

12 工程抗震InstituteofEarthquakeEngineering

2014-12-01 院v(t) 1院v(t) 112(sintDalianUniversityofDalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院p(t)院Cp0 1 2Cp0 1 2v(t)v0sintvcost

costcost

22/2/DalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院院22/2/DalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 院v(院v(t)v0sintvcost0costcost 非常小，但不等于

ε為一非常小的v(t)

m(22

costcostDalianUniversityof 20 sin()tsin()t DalianUniversityofm(22

sintsin因為最后一個等式中的ε非常小，所以函數(shù)sinεt變化緩慢，它的周期等于2π/ε，該值很大。因此，上式可以視為周期為2π/?，可變振幅等于p0/(2mε?)sinεt的振動，這種振動按下圖所工程抗震InstituteofEarthquakeEngineering

2014-12-01 院v院v(t)v0sintvcost0costcost率sintsinDalianUniversityof因為最后一個等式中的ε非常小，所以函數(shù)sinεt變化緩慢，它的周期等于2π/ε，該值很大。因此，上式可以視為周期為2π/?，可變振幅等于p0/(2mε?)sinεt的振動，這種振動按下圖所DalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院v(院v(t)sintvcostP 0 m(22costcost 20

t tv(t)0sintvcostlim

mInstituteofEarthquakeEngineering2014-12- 0L’Hospital上下求導隨著時間線性增長，這種InstituteofEarthquakeEngineering2014-12- 0L’Hospital上下求導隨著時間線性增長，這種稱為 DalianUniversityof工程抗院InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- DalianUniversityof動力放大系數(shù)DynamicMagnification院v(t)院v(t)p0 112(sintsinD

D11β1DalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 院DalianUniversityof工程抗震InstituteofEarthquakeEngineeringDalianUniversityof

2014-12-01 院DalianUniversityof工程抗院DalianUniversityof

2014-12-01 11Dalian11DalianUniversityof

152.32/院工程抗震院

2014-12-01 D D院DalianUniversityof院DalianUniversityof

2014-12-01 院0 0m

用待定系數(shù)法求該微分方程DalianUniversityofvp(t)A1costA2DalianUniversityof (t)A2costA2sin A2costA2sint2AsintAcost

AcostAsint

工程抗震InstituteofEarthquakeEngineering

2014-12-01 ((21A2A)21t(A221A)sin22t0msin院DalianUniversityofDalianUniversityof

22A2A A22A2

m2224222

P0 k(12)2(2AP0(22A

1 m2224222

k(12)2(2工程抗震InstituteofEarthquakeEngineering

2014-12-01 院vp(t)sintA212A212 DalianUniversityof相位tg相位

2 1工程抗震InstituteofEarthquakeEngineering

2014-12-01 院方院方程的通

sin

t

tsint v(t)

BsintB

tAcostAsin 代入初始

v(0) 可DalianUniversityofDalianUniversityofv(t) 0 0sintvcos

初始條件的自由振瞬態(tài)反應很快 eesinsinDtsin sin tD伴生的自由振瞬態(tài)反應很快純強迫振穩(wěn)態(tài)諧振反工程抗震InstituteofEarthquakeEngineering

2014-12-01 院v(t)sint(12)2(2

[(12)sint2DalianUniversityofDalianUniversityofk

[(1

2

]1/

tg

1 k(12

sin

p0/工程抗震InstituteofEarthquakeEngineering

2014-12-01 院D [(12)2(2)2]1/2p0/DalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 院tg

1 DalianUniversityof工程抗震InstituteofEarthquakeEngineeringDalianUniversityof

2014-12- 院P0[(1院P0[(12)2(2)2ktg1DalianUniversityof

2014-12-01 院DalianUniversityof工程抗院DalianUniversityof

2014-12-01 Determine院

2DalianUniversityofThuswiththedataofDalianUniversityoftg(22)tg15(27.92162)

227.9Thesameresult(withinengineeringaccuracy)isgivenbythedataofthesecondtestInstituteofEarthquakeEngineering2014-12- ThesameresultsInstituteofEarthquakeEngineering2014-12- 工程抗院

D [(12)2(2)2]1/2R(t)

1

(sintsin

p0/ 無阻尼時，外載的頻率等于DalianUniversityofDalianUniversityof對阻尼

D11/動力放大系dDd

12峰βDmax峰

21

DDInstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院DalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12- 院DalianUniversityof工程抗震院DalianUniversityof

2014-12-01 院DalianUniversityof院DalianUniversityof

2014-12-01 院§院?InstituteofEarthquakeEngineering2014-12-InstituteofEarthquakeEngineering2014-12- DalianUniversityof工程抗院

輸(t)輸(t)ggAP0 1k(12)2(2P1

k(12)2(2DD [(12)2(2)2]1/2p0/DDalianUniversityDalianUniversityofInstituteofEarthquakeInstituteofEarthquakeEngineering2014-12-

在β<0.6，ξ＝0.7時，放倍數(shù)接近常量。儀器反應入幅值成正比。這種儀器作為低頻加速度院

vg(t)vg02 t)2 g0D2v g0DalianUniversityofDalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院這里只介紹簡單的DalianUniversityofDalianUniversityof基礎產(chǎn)生的InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院m p(t)p0DalianUniversityDalianUniversityof

彈性

穩(wěn)態(tài)相對位v(t)p0DsintkfS(t)kv(t)p0DsintfD(t)cv(t)

p0Dcostk

2p0Dcost總反力幅

(t)[f2

p1(2)21(2)21(2)2支撐系統(tǒng)的傳導比 TRfmax(t)/1(2)2工程抗震InstituteofEarthquakeEngineering

2014-12-01 k2ck2代入基底的運動vg0k 2p0單自由度隔振體系(支座擾動vg(t)vg0

運動方院院DalianUniversityofpDalianUniversityof

kvg

穩(wěn)態(tài)響g)2gg)2g1(22

vt(t)

m2

傳導比TR

g(t)/ g1(21(2 工程抗震InstituteofEarthquakeEngineering

2014-12-01 院Dalian院DalianUniversityof當頻率較高時，傳導比要較低頻時低很多，因此，使系高頻運動是有利的工程抗震InstituteofEarthquakeEngineering

2014-12-01 院隔振率 IE1 If

TRTRD1(2D[(12)2(2)2

Ifβ=

隔振體系只有在β>21/2時才有DalianUniversityof對于小阻尼情 DalianUniversityofIE22/2 22IE/1IE2 2/22m/k2W/kg / 21 21工程抗震InstituteofEarthquakeEngineering

2014-12-01 院院 21

隔振體系越柔越 DalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 例題3-例題3-院DalianUniversityofDeflectionssometimesdevelopinconcretebridgegirdersduetocreep,andifthebridgeconsistsofalongseriesofidenticalspans,thesedeformationswillcauseaharmonicexcitationinavehicletravelingoverthebridgeatconstantspeed.院DalianUniversityofInstituteofEarthquakeEngineering2014-12- is40percentofFigureE31showsahighlyidealizedmodelofthistypeofsystem,inwhichthevehicleweightis4000lb[1814kg]anditsspringstiffnessisdefinedbyatestwhichshowedthatadding100lb[45.36kg]causedadeflectionof0.08in[0.203cm].Thebridgeprofileisrepresentedbyasinecurvehavingawavelength(girderspan)of40ft[12.2m]anda(single)amplitudeof1.2in[3.05cm].FromthesedataitisdesiredtopredictthesteadystateverticalmotionsinthecarInstituteofEarthquakeEngineering2014-12- is40percentof工程抗院穩(wěn)態(tài)響vt(t)院穩(wěn)態(tài)響vt(t)vg12Dsint2DalianUniversityof

2014-12-01 院傳導隔振率TR80/500IE10.16f彈簧剛kW/4stDalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 院第2章曾用自由振動衰減法計算結構的阻取自然對

lnvn/

m對數(shù)衰減 DalianUniversityDalianUniversityof放半功率每 能量損失InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院D [(12)2(2)2]1/2p0/dD 12d DalianDalianUniversityofDmax

21DmaxD11/max/0D0/需 使用諧振荷載的幅值InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院v(t)sintkp0[(12)2(2)2k反應幅值由阻尼控阻尼比由反應幅值降到峰2DalianUniversityof1/ 水平時的2DalianUniversityof

2

阻尼耗

2/20

)dtm2與22 成正工