Lecture03-Prof.Ju

Lecture03-Prof.Ju - CEE M237A / MAE M269A Lecture 3:...

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CEE M237A / MAE M269A Lecture 3: Dynamic SDOF Response to Periodic and Impulsive Loading Professor J. Woody Ju
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Response to Periodic Loading 2 () [] P 0 periodic 2 0 11 non-harmonic loading period of excitation (loading) = , satisfy: Trigonome Use the Four tric ier 22 expansion for : si m cos n p p p p T nn p m pt nT pt nt nt vc vk v p t T aa b TT π ω ππ ∞∞ == += =+ + ++= ± ²² ² ³´´´µ ´ ´´¶
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Response to Periodic Loading (Cont’d) 3 () 0 0 0 0 1 22 cos ; 1,2,3,. .. sin ; 1, where 2,3,. .. p p p T p T n pp T n ap t d t T nt t d t n TT nt bp t d t n π = ==
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Example 1 4 t p(t) 2 p T 2 p T 2 p T 2 p T 0 p 0 2 sin , 0 2 p p T pt t T π < < periodic loading
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Example 1 (Cont’d) 5 0 2 00 0 2 0 0 0 2 0 0 Nothing between 2 and 12 sin 0, odd 22 2 sin cos 2 , even 1 , 2 sin sin 2 p p pp T T n p n p TT p t ap d t n tn t d t p n T n p t bp d t T π ππ ⎛⎞ == ⎜⎟ ⎝⎠ = = 2 0 1 0, 2,3,. .. p T n n = =
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Example 1 (Cont’d) 6 () 0 00 0 1s i n c o s 2 2 cos4 cos6 ... 2 where 2 3 2 Note that the coefficients decrease very fa 2 15 st. 35 p p pt t t tt T π ωω ω =+ −+ ±
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Example 1 (Cont’d) y Case I (undamped) 7 0 00 0 12 3 2 Assume: , 42 2 2 3 If 44 2 34 33 23 9 ; ; . .. 4 p p k T m T TT T π ξω ω ππ ωω ββ β =0, =⇒ = = = == = = = = ±± ±
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Example 1 (Cont’d) y S.S. response only: 8 () () () () 00 22 0 01 2 0 0 11 sin cos ... 88 1 1 sin cos2 cos4 ... 71 5 6 0 nn n ba vt nt kk Pa v v t v t P tt t k ωω ββ π ω ⎡⎤ =+ ⎢⎥ −− ⎣⎦ == + + + + +
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Example 1 (Cont’d) y Case II (underdamped) 9 () () () 1 2 2 2 2 0 1 2 2 2 2 0 1 0 0 2 0 1 12 s i n + 1 2 cos 2 where , tan 1 n nn n n n n n n n b vt n t k a nt k a v k v v t βξ β ω θ
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This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

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Lecture03-Prof.Ju - CEE M237A / MAE M269A Lecture 3:...

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