Lecture09-Prof.Ju

Lecture09-Prof.Ju - CEE M237A / MAE M269A Lecture 9:...

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CEE M237A / MAE M269A Lecture 9: Examples on Strings, and Longitudinal Vibrations of Rods Professor J. Woody Ju
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Example 1–Free Vibrations of a String Supported on Both Ends 2 () ( ) 2 2 1 2 2 1 22 B.C. The homogeneous solution is ; : 0 0 and 0 : 0 sin c sin 's 0 os sin 0 0 0 s cm T Y Yx C Y Y x YL L C C L L x C β ββ ωω == = =⇒ = = + = =
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Example 1–Free Vibrations of a String Supported on Both Ends (Cont’d) 3 () 2 1 2 22 2 2 (fundamental frequency) associated mode s ; 1, ha 2, For each frequency , the is sin ; 1,2, 2 sin ; 1,2, (orthonormalizatio pe i ) sn n s jj j j j j T cj LL m jx xj L Lm L c L π βω β φ ω =⇒ == = = = ⇒= " " " 00 sin 0 , cos cos (Orthogon ality conditions) ; ix dx dx i j L L ππ = =≠ ∫∫
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Example 2–Free Vibrations of a String with a Spring/Mass System on One End 4 () 22 2 12 2 2 The homogeneous solution is: with The bound sin cos 0 ary condition at 0 is: 0 0 s cm Yx C x Y T xC x C βω ω ββ == = + = = =
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Example 2–Free Vibrations of a String with a Spring/Mass System on One End (Cont’d) 5 () ( ) 2 '2 2 The equilibrium of vertical forces lead to: The boundary condit tan 1 tan ions lead to: If 1 and 1 1 s s s s T L L kL Tk L M TY L kY L MY LM m L L L MY L m L L L L β ω −− = ⎡⎤ = ⎢⎥ = ⎣⎦ = = = ±±
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Example 2–Free Vibrations of a String with a Spring/Mass System on One End (Cont’d) 6 ( ) 2 The roots of tan 1 are plotted as LL L L ββ β ⎡⎤ =
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Example 2–Free Vibrations of a String with a Spring/Mass System on One End (Cont’d) y Orthogonality: consider any two eigenpairs (i and j) 7 () () ( ) ( ) 2' ' 2 '2 00 ' 0 ; Multiply the 1st eqn. by and the second by and integrating both over the length of the string y 0 and ield: ; 0 ji LL i ij ii i jj j cd x d x i j x x cx x += + = ∫∫ φφ φ ωφ φω 2 0 '' ' ' ' 0 ' ' ' 0 ; Integrating the first term in both eqns above by parts yields: | | j L L dx i j dx dx dx dx =−
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Example 2–Free Vibrations of a String with a Spring/Mass System on One End (Cont’d) y Orthogonality: 8 () ( ) () () '2 22 0 the first orthogonality rel At 0, 0 0 and 0 0 At , 0 Since the frequencies are dist ation takes the form inct, ij ii i jj j L i j j i x xL L L k M T LL k M T M dx L L m φ φφ ω ωω == = =− ⎡⎤ ⇒∴ = ⎢⎥ ⎣⎦ 0 '' 0 : ;
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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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Lecture09-Prof.Ju - CEE M237A / MAE M269A Lecture 9:...

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