Lecture11-Prof.Ju

Lecture11-Prof.Ju - CEE M237A / MAE M269A Lecture 11:...

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Unformatted text preview: CEE M237A / MAE M269A Lecture 11: Examples on Transverse Vibrations of Beams Professor J. Woody Ju Example 1 – Free Vibrations of a Simply Supported Uniform Beam on Both Ends 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 2 3 4 The homogeneous solution for ( ) is: The BC ; s at sin cos sinh co (0, ) a s r 0 ; e: h d Z d Z Z E Z x C x C x C x C x I dx dx d Z L d Z L Z L E I dx d Z x L x x β β β β = + + + = ≡ = = ≡ = • = • Example 1 – Free Vibrations of a Simply Supported Beam on Both Ends (cont’d) 3 ( ) ( ) 2 4 1 2 3 4 2 2 4 2 1 2 3 4 2 4 1 3 sin cos sinh cosh sin cos sinh cosh 2 s a in 2 sinh nd C C C L C L C L C L C C C L C L C L C L C C C L C L β β β β β β β β β β β β + = + + + = − + = − − + + = = ∴ = = = ∴ Example 1 – Free Vibrations of a Simply Supported Beam on Both Ends (cont’d) 4 ( ) ( ) 4 2 1 1 3 Since for 0,then ; 1,2, , The sinh is never zero mode shapes of vi for simply-supported conditions at both ends are: s brati i o n sin n ; i i i i i i i L i E I L i L A Z x L i C C x C L x β β π β π ω ρ φ β ≠ ⇒∴ ⇒∴ = ⎛ ⎞ = → = ⎜ ⎟ ⎝ ⎠ = = = • = • ∞ " 1 1,2, , where are undetermined amplitudes i i C = ∞ " Example 2 – Free Vibrations of a Uniform Beam Fixed on Both Ends 5 ( ) ( ) ( ) ( ) ( ) 1 2 3 4 sin cos sinh cosh 0 ; 0 ; The homogeneous solution is: The BCs at 0, are : Z x C x C x C x C x dZ Z dx dZ L Z L x x L d β β β β • = + + + = = • = = = Example 2 – Free Vibrations of a Beam Fixed on Both Ends (Cont’d) 6 1 2 3 4 1 1 1 1 sin cos sinh cosh cos sin cosh sinh C C C L L L L C L L L L β β β β β β β β ⎧ ⎫ ⎡ ⎤ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⇒∴ = ⎨ ⎬ ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ − ⎣ ⎦ ⎩ ⎭ frequency determinant Expansion of this cos yi cosh 1 elds L L β β = • Example 2 – Free Vibrations of a Beam Fixed...
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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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Lecture11-Prof.Ju - CEE M237A / MAE M269A Lecture 11:...

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