《结构动力学》课程教学课件（讲稿）03 Response to harmonic loading

Wuhan University of TechnologyPartISingle-degree-of-freedom systems(SDOF)Chapter3Response to harmonicloading4-1

4-1 Wuhan University of Technology Chapter 3 Response to harmonic loading Part I Single-degree-of-freedom systems (SDOF)

Wuhan University of Technology?Contents3.1 Undamped system3.2 System with viscous damping3.3 Resonant response system3.4 Accelerometers and displacement meters3.5 Vibration isolation3.6 Evaluation of viscous-damping ratio3.7Review and problems3.8Appendix4-2

4-2 Wuhan University of Technology 3.1 Undamped system 3.2 System with viscous damping 3.3 Resonant response system 3.4 Accelerometers and displacement meters 3.5 Vibration isolation 3.6 Evaluation of viscous-damping ratio 3.7 Review and problems 3.8 Appendix Contents

Wuhan University of Technology3.1 Undamped systemy(t)Y-p(t)m00000kBasiccomponentsofaidealizedSDoFsystemmi(t)+cv(t)+kv(t)= po sin @t4-3

4-3 Wuhan University of TechnologyBasic components of a idealized SDOF system y ( t ) c m p ( t ) k y ( t ) c m p ( t ) k 0 mv t cv t kv t p t   ( ) ( ) ( ) sin     3.1 Undamped system

Wuhan Universityof Technology3.1 Undamped systemmi(t)+ kv(t) = po sin @tComplementarysolutionv(t) = Acos ot + Bsin otParticular solutionThegeneralsolutionmustalsoincludetheparticularsolutionwhichdependsupontheformof dynamicloading.·In this case of harmonic loading, it is reasonable to assume that thecorresponding motion is harmonic and in phase with the loading; thus,theparticularsolutionis4-4

4-4 Wuhan University of Technology 3.1 Undamped system 0 mv t kv t p t ( ) ( ) sin    ( ) cos sin c vt A t B t     Complementary solution Particular solution • The general solution must also include the particular solution which depends upon the form of dynamic loading. • In this case of harmonic loading, it is reasonable to assume that the corresponding motion is harmonic and in phase with the loading; thus, the particular solution is

Wuhan Universityof Technology3.1 Undamped systemy,(t)=Csinatma'Csinのt+kCsinat=PsinatkConsidering0minwhichβ isdefined astheratio of theappliedloadingfrequencytothenatural freevibrationfrequencyβ=の/の4-5

4-5 Wuhan University of Technology ( ) sin p vt C t   2 0 -m C t kC t P t  sin sin sin     k 2 m  0 2 1 1 P C k        -  in which is defined as the ratio of the applied loading frequency to the natural freevibration frequency   / Considering  3.1 Undamped system

Wuhan University of Technology3.1 Undamped systemGeneral solutionPsintv(t)=v(t)+y,(t)=Acosot+Bsinot+Bkv(0) = 0 i(0)= 01PoβA=0B1-B2k4-6

4-6 Wuhan University of Technology 0 2 1 ( ) ( ) ( ) cos sin sin 1 c p P vt v t v t A t B t t k               - A  0 0 2 1 1 P B k         -  v(0) 0  v(0) 0  General solution 3.1 Undamped system

Wuhan Universityof Technology3.1 Undamped systemPsinat-βsinot)kwhere p. / k = ys, is the displacement which would be produced bythe load po applied staticallyi1is the magnificationfactor (MF)representing theamplification effect of the harmonically applied loading.1-Brepresents the response component at the frequency ofthesinのtappliedloading;itiscalledthesteadystateresponseandisdirectly related to the loading.represents is the response component at the natural vibrationβ sin ot frequency and is the freevibration effect controlled by theinitial conditions.4-7

4-7 Wuhan University of Technology     0 2 1 sin sin 1 P vt t t k             - where is the displacement which would be produced by the load applied statically; 0 / st p k v  0 p 2 1 1- is the magnification factor (MF) representing the amplification effect of the harmonically applied loading. sint  sint represents the response component at the frequency of the applied loading; it is called the steadystate response and is directly related to the loading. represents is the response component at the natural vibration frequency and is the freevibration effect controlled by the initial conditions. 3.1 Undamped system

Wuhan Universityof Technology3.1 Undamped systemPsin@t-βsinottSince inapractical case, damping will cause the last term tovanisheventually,it istermedthetransientresponse.Forthishypotheticalundampedsystem（假想的无阻尼体系），however,thistermwill notdampoutbutwill continueindefinitely4-8

4-8 Wuhan University of Technology     0 2 1 sin sin 1 P vt t t k             - Since in a practical case, damping will cause the last term to vanish eventually, it is termed the transient response. For this hypothetical undamped system（假想的无阻尼体系）, however, this term will not damp out but will continue indefinitely. 3.1 Undamped system

Wuhan University of Technology3.1 Undamped systemResponseRatio.Aconvenientmeasureoftheinfluenceofdynamicloadingisprovidedbytheratioofthedynamicdisplacementresponsetothedisplacementproducedbystaticapplicationofload Po ,i.e.,v(t)v(t)R(t)Po/kVstR(t):sin@t-βsin ot1-B24-9

4-9 Wuhan University of Technology Response Ratio. A convenient measure of the influence of dynamic loading is provided by the ratio of the dynamic displacement response to the displacement produced by static application of load , i.e., 0 p   0 () () / st vt vt R t v pk     2 1 ( ) sin sin 1 Rt t t             - 3.1 Undamped system

Wuhan University of Technology3.1 Undamped systemR,(t)MF(a)2元0R,(t)MF(b)2元0R(0)(c)Frequency ratioβ=4-10

4-10 Wuhan University of Technology 3.1 Undamped system