Lecture10-Prof.Ju

Lecture10-Prof.Ju - CEE M237A / MAE M269A Lecture 10:...

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CEE M237A / MAE M269A Lecture 10: Examples on Rods, and Transverse Vibrations of Beams Professor J. Woody Ju
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Example 1–Free Vibrations of a Bar with One End Fixed and Other End Free 2 () 2 2 12 Consider a uniform homogeneous, isotropic bar of length L. The homogeneous solution for th sin cos ; e sp atial dependence is 0 0 a : The B.C. ar n : e d0 dX L X d xC x E x Xx C ρω ββ β = = + = =
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Example 1–Free Vibrations of a Bar with One End Fixed and Other End Free (Cont’d) 3 () 2: 2 2 : 1 0 0 0 cos 0 The frequency eqn. and its root 21 cos 2 s are: ; 1,2, , ; 1,2, , The corresponding mode shapes are 2 : n n nn n L XC dX L CL L n E L Xx C x C dx n n π ββ ω β ρ φ =→ =→= = = =∞ ⇒∴ = = = = " " where is an undetermin si ed a n 2 mplitude n nx L C
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Example 2–Free Vibrations of a Bar with Both Ends Fixed 4 () ( ) 2 2 12 The homogeneous solution for the spatial dependence is: sin cos ; 00 a n d The BCs a r 0 e: XX Xx C x C x E L ρω ββ β = = += =
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Example 2–Free Vibrations of a Bar with Both Ends Fixed (Cont’d) 5 () () ( ) 1: 1 2 : 1 0 0 0 =0 sin 0 ; 1,2, , The mode shapes of free sin 0 ; 1,2 vibrations are ,, : ; 1,2, si , n n n nn XC X n Ln L n LC E L nx Xx C L n xC n L π ββ ωρ φ β =→= →= ⇒∴ ⇒∴ == =∞ = = = " " "
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Example 3–Free Vibrations of Two Bars with Fixed and Free Ends 6 () ( ) ( ) ( ) 11 2 2 2 2 2 2 1 1 1 1 2 1 1 1 The interface is fully continuous between two bars. Both bars vibrate at the same frequency so tha wher t the displ , and , sin cos . e ; is: it u x tX x e u x x e Xx C x C x ωω ρω ββ β ω == = =+ 1 2 '' 2 2 21 1 2 2 2 2 2 2 2 sin cos ; E C x C x E =
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Example 3–Free Vibrations of Two Bars with Fixed and Free Ends (Cont’d) 7 () ( ) 12 22 1 11 2 2 2 2 The on and are 0 0 and 0 At the interfa boundary conditions displacement and force continuity ce, require 0 0 XX dX L X dx XL X dX L dX EA dx dx == = =
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Example 3–Free Vibrations of Two Bars with Fixed and Free Ends (Cont’d) 8 () 22 1 ' 11 1 ' '' 12 111 2 2 2 2 2 Therefore, we have: 0c o ss i n sin 0 1 0 cos 0 the determinant of the matrix of co For a nontrivial solu effici tion, 0=0 ents , , s 0 u m LL C LC EA L E CCC C A X C ββ β ⎡⎤ ⎢⎥ −= ⎣⎦ →= 222 2 2 t vanish: tan tan 1 =
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Example 3–Free Vibrations of Two Bars with Fixed and Free Ends (Cont’d) 9 The can be seen in the figure roots of transcendental as the intersections of equa cur tion ves:
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Example 3–Free Vibrations of Two Bars with Fixed and Free Ends (Cont’d) 10 ' 111 1: 222 Knowing the freq. ,amplitude ratios can be computed using any two of the three matrix eqns. For example, using as the reference value, the other two coefficien c t os s are: nn i n EA CC C β ω = () 11 ' 2: 1 1 : 1 1 22 2 1 1 2 2 1 1 2 2 Thus the mode shape has the form: cos sin cos sin + ; 0 sin o ; cs 0 L L xC x L x xL L A x E L x φ ββ = = ⎛⎞ = ⎜⎟ ⎝⎠ ≤≤
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Lecture10-Prof.Ju - CEE M237A / MAE M269A Lecture 10:...

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