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