Lecture04-Prof.Ju

Lecture04-Prof.Ju - CEE M237A / MAE M269A Lecture 4:...

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CEE M237A / MAE M269A Lecture 4: Dynamic SDOF Response to General Dynamic Loading Professor J. Woody Ju
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Duhamel Integral (undamped) 2 τ Δ d Δ 0 p () Recall pure impulse (short duration) At time short duration Duhamel Integral for undamped system impulse For sin td p d t pd dv t t m ττ ξ ωτ ω =, >, ⎡⎤ = •⇒ 0 =−
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Duhamel Integral (undamped) (Cont’d) 3 t p(t) 0 d τ ( ) p Infinitesimal rectangular impulse () ( ) ( ) 0 0 For 0, at rest I.C.'s Linear SDOF; valid only for linear systems! 1 sin I f 00 , 0 Superposition exa sin 0 cos 1 + sin ct : t t t vt p t d m vv v t v t pt d m τω ω ωω = =− ≠≠ =+ ± ±
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Numerical Integration of Duhamel Integral 4 () 00 recall sin sin cos cos sin , For at rest I.C.'s, Need to compute and numerical 11 sin cos cos sin ly tt Bt At t vt t p d t p d mm ωω τ ω τω −= =− ∫∫ ±²²²³²²²´ Now cos 1 equal spacing A A tpdY d t m ς ωτ τ Δτ ως = ⎡⎤ ⎢⎥ µ symbolic representation of the numerical summation process order of integr. approx. =
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Numerical Integration of Duhamel Integral (Cont’d) y Rectangular Summation 5 () 012 1 1 constant within the increment The first order approx. =1 ... A N tY Y Y Y ς ++ + + ± 4 Y 3 Y 0 Y 1 Y 2 Y 5 Y
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Numerical Integration of Duhamel Integral (Cont’d) y Trapezoidal Rule 6 () 012 1 2 =2 linear within the increments 2 2 ... 2 A NN tY Y Y Y Y ς ++ + + + ± 4 Y 3 Y 0 Y 1 Y 2 Y 5 Y straight line
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Lecture04-Prof.Ju - CEE M237A / MAE M269A Lecture 4:...

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