3.1.3 Damped Free Vibration (< Example [Solution] en by weight?) 大 201510y13 Example 3.2 Forced Vibration [Solution] Harmonic loading Two basic→ Impulse loading vibeaion 0=628 .Harmonic vibration -e--6.281-005-627 Natural vibrafion frequenoy: f=a/2x=lH拉 Impulse vi tion Natural vibrafion period: T-l/f=ls 3.2.1 Harmonic Vibration 3.2.1 Harmonic Vibration Equation of motion m+kx=Fosin at Where Response xt)=e(acoso,t+bsin@t)+ccosot+dsino ,the accelerationc be represented as F(() The equtvalent force amplitude as F and the frequency ratio 大杠 5 5 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 25  1 3.1.3 Damped Free Vibration [3.1] A single story concrete factory with total mass concentrated on the roof, is 204 t, and the lateral stiffness of all columns is 8048.6 kN/m. Please determine the building’s natural period. [Solution] single story —— SDOF Concrete factory —— Damping ratio=5% ( if it given by steel frame? , masonry wall?) Roof mass —— M= 204 t，( if it given by weight? ) Example 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 26 39 43 204 0 2 8048 6 . . . m k  ＝ ＝ ＝ Undamped natural vibration circular frequency: ＝6.281 6.28 1 0.05 6.27 ＝  2＝  2＝ f＝ 2＝1Hz T 1 f＝1s Damped natural vibration circular frequency: Natural vibration frequency: Natural vibration period: 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 27 Example [Solution]  Harmonic vibration : a train of sinusoidal waves having a given amplitude．  Impulse vibration : during a short time, the wave amplitude can be simplified as a constant value, or a triangular shape． Two basic excitation Harmonic loading Impulse loading No. 28 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 28 3.2 Forced Vibration  undamped system subjected to simple harmonic loading Where is the amplitude and a is the circular frequency of the harmonic load． For a ground acceleration, the acceleration can be represented as The equivalent force amplitude as and the frequency ratio 0 F ( ) sin F t mg t 0 0    F mg oe 0      a / No. 29 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 29 3.2.1 Harmonic Vibration m  x   kx  F0 sin at  Equation of motion  Response sin 2 0 n n F x 2 x t m       xt e a t b t c t d t d d t n       cos  sin  cos  sin  Homogeneous solution a and b satisfy i.c.’s Decays exponentially in time “Transient” response 瞬态反应 Particular solution c and d satisfy F(t) Does not decay “Steady-state” response 稳态反应 3.2.1 Harmonic Vibration 2015/10/13 熊海贝，同济大学土木工程学院 xionghaibei@tongji.edu.cn 30