结构动力学大连理工土木水利学院

院DalianUniversityof第I单自由第2－3工程抗震InstituteofEarthquakeEngineering院§2-1基本动力体系的kcmkcmkcmDaliankcmDalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院DalianUniversityofmcc工程抗震InstituteofEarthquakeEngineering

2014-12-01 院体系质量 弹性特性（刚度或柔度）k能量耗散机制或阻 DalianUniversityof每个特性都假设集结于DalianUniversityof研究单自由度体系的自由振动重要性在塔、单层厂房1、它代表了许多实际塔、单层厂房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粘滞（或粘性）阻尼（ViscousDalianUniversityofRcdv(t)外力（External

InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院FFsFdp(t)kv(t)cv(t)p(t)DalianDalianUniversityof单自由度系统的运动方程（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动力平衡DalianUniversityofcv(t)kv(t)kstp(tDalianUniversityof去掉静力平衡Δst与时间无工程抗震InstituteofEarthquakeEngineering

2014-12-01 院院DalianUniversityDalianUniversityofInstituteofEarthquakeEngineering2014-12- 性、弹性、InstituteofEarthquakeEngineering2014-12- 性、弹性、小变形的情工程抗院k2ck2f(t)v(t)v(t)vtgItfs(t)/

fD

fs(t)/DalianUniversityoffI(tDalianUniversityof

(t)

(t)阻尼力和弹性力只与相对

构破坏的原因质越大，惯性力越InstituteofEarthquakeEngineering2014-12- InstituteofEarthquakeEngineering2014-12- 工程抗院gvt(t)v(t)vg

DalianUniversityof DalianUniversityof地面运动一般测量的是加速度，此时需要 加速度积分得地面运动的速度和位InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院§院DalianUniversityDalianUniversityof

自由Particular自由振

Free!＝0强迫复复

v(t)齐线性方程，通解取工程抗震InstituteofEarthquakeEngineering

2014-12-01 院

RealGGRG

θ为幅DalianUniversityofGGcosiGsin(ei)DalianUniversityof(ei)

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

2014-12-01 院院kcsms2k/m2

DalianUniversityof方DalianUniversityof相对于k和m的

2

css2m

运动方程的

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]自由振动必须是实的，虚部项必DalianUniversityofG1RG2RGRG1IG2DalianUniversityofA2GR;B

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

G2;

如果方程有复值解，则它的共数、复值解的实部和虚部也都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临界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初始条件v(0)

个相同的实

G1和G2均为实G2v(0)DalianUniversityofDalianUniversityof

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

2mω是系统不出现振荡的最工程抗震InstituteofEarthquakeEngineering

2014-12-01 超临界院DalianUniversityof

c/cc2s22

工程抗震InstituteofEarthquakeEngineering

2014-12-01 院v(t)c c

v(t)(G GeiwDtc cs

为使反应是实的，G1和G2必须为共轭D1阻尼体系的自D1阻尼体系的自振频采用三角函数 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 ;

具有不变的振动周期具有不变的振动周期工程抗InstituteofEarthquakeEngineering2014-12- DalianUniversityof院通常的工程结构，阻尼比ξ的值一般在1%~10%之间。故近似地认为ωD＝ξξDalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 院t,tt,t2/v(t)cos(Dt)e相邻两个正波峰的比：

/

e2DalianUniversityofDalianUniversityof11

ln

/

2小阻尼情况近 Taylor

vn/

12

2!取前两

由此可见，对于粘滞阻尼自

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2014-12-01 院阻尼院1lnvn/ 1近似

DalianUniversityof这种测定阻尼系数的方法称为自振衰减法(FreeDalianUniversityof工程抗震InstituteofEarthquakeEngineering

2014-12-01 院

例p=20kips;v=0.20in

无阻尼振动频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

阻尼频11

0.9991/2

0.0355219201.548kips

6周后振

(v/v)6 工程抗震InstituteofEarthquakeEngineering

2014-12-01 院m求自振频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 院无阻尼自由振动v(t2GRcost2GIsint2Gcos(tAcostBsinDalianUniversityofDalianUniversityof临界阻

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确定常数

Cp0

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2014-12-01 院 v(t)AcostBsintCsin系数A和B通过初始条件确定。对于由静止开始的运动

Cp0 112;Cp0 112; 12 DalianDalianUniversityof

v(t)p0

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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为一非常小的v(t)

m(22

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

sintsin因为最后一个等式中的ε非常小，所以函数sinεt变化缓慢，它的周期等于2π/ε，该值很大。因此，上式可以视为周期为2π/ϖ，可变振幅等于p0/(2mεϖ)sinεt的振动，这种振动按下图所工程抗震InstituteofEarthquakeEngineering

2014-12-01 院v院v(t)v0sintvcost0costcost率sintsinDalianUniversityof因为最后一个等式中的ε非常小，所以函数sinεt变化缓慢，它的周期等于2π/ε，该值很大。因此，上式可以视为周期为2π/ϖ，可变振幅等于p0/(2mεϖ)sinεt的振动，这种振动按下图所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上下求导随着时间线性增长，这种InstituteofEarthquakeEngineering2014-12- 0L’Hospital上下求导随着时间线性增长，这种称为 DalianUniversityof工程抗院InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- DalianUniversityof动力放大系数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

用待定系数法求该微分方程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

初始条件的自由振瞬态反应很快 eesinsinDtsin sin tD伴生的自由振瞬态反应很快纯强迫振稳态谐振反工程抗震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/ 无阻尼时，外载的频率等于DalianUniversityofDalianUniversityof对阻尼

D11/动力放大系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

输(t)输(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时，放倍数接近常量。仪器反应入幅值成正比。这种仪器作为低频加速度院

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

2014-12-01 院这里只介绍简单的DalianUniversityofDalianUniversityof基础产生的InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院m p(t)p0DalianUniversityDalianUniversityof

弹性

稳态相对位v(t)p0DsintkfS(t)kv(t)p0DsintfD(t)cv(t)

p0Dcostk

2p0Dcost总反力幅

(t)[f2

p1(2)21(2)21(2)2支撑系统的传导比 TRfmax(t)/1(2)2工程抗震InstituteofEarthquakeEngineering

2014-12-01 k2ck2代入基底的运动vg0k 2p0单自由度隔振体系(支座扰动vg(t)vg0

运动方院院DalianUniversityofpDalianUniversityof

kvg

稳态响g)2gg)2g1(22

vt(t)

m2

传导比TR

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

2014-12-01 院Dalian院DalianUniversityof当频率较高时，传导比要较低频时低很多，因此，使系高频运动是有利的工程抗震InstituteofEarthquakeEngineering

2014-12-01 院隔振率 IE1 If

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

Ifβ=

隔振体系只有在β>21/2时才有DalianUniversityof对于小阻尼情 DalianUniversityofIE22/2 22IE/1IE2 2/22m/k2W/kg / 21 21工程抗震InstituteofEarthquakeEngineering

2014-12-01 院院 21

隔振体系越柔越 DalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 例题3-例题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工程抗院稳态响vt(t)院稳态响vt(t)vg12Dsint2DalianUniversityof

2014-12-01 院传导隔振率TR80/500IE10.16f弹簧刚kW/4stDalianUniversityof工程抗震InstituteofEarthquakeDalianUniversityof

2014-12-01 院第2章曾用自由振动衰减法计算结构的阻取自然对

lnvn/

m对数衰减 DalianUniversityDalianUniversityof放半功率每 能量损失InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院D [(12)2(2)2]1/2p0/dD 12d DalianDalianUniversityofDmax

21DmaxD11/max/0D0/需 使用谐振荷载的幅值InstituteofEarthquakeInstituteofEarthquakeEngineering2014-12- 院v(t)sintkp0[(12)2(2)2k反应幅值由阻尼控阻尼比由反应幅值降到峰2DalianUniversityof1/ 水平时的2DalianUniversityof

2

阻尼耗

2/20

)dtm2与22 成正工