Lecture06-Prof.Ju

Lecture06-Prof.Ju - CEE M237A / MAE M269A Lecture 6: MDOF...

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CEE M237A / MAE M269A Lecture 6: MDOF Undamped Free Vibrations Professor J. Woody Ju
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Undamped Free Vibrations 2 () ( ) () 2 2 22 MDOF solve eigenvalues and eigenvectors ˆ Assume sin ˆ sin eigenvalue problem 1. Vibration Frequencies ˆ 0 freque = 0 km vt v t vv t v v ωθ ω =+ ⇒= ⎡⎤ −= ⎣⎦ += ⇒← ⇒∴ = ±± ² ± ² ± ± ± ncy equation (E.O.M. 0 ) mv kv ²² ±± ±± ±
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Undamped Free Vibrations (Cont’d) 3 1 2 3 2 12 123 Frequency vector: Define: Solve for N roots, , , 0 ... N N Ni i ω λω λλ λ ⎪⎪ = ⎨⎬ ⎩⎭ = ..., << < < < = ± #
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Undamped Free Vibrations (Cont’d) 4 2 3 1 ˆ ˆ ˆ For convenienc 2. Vibra e, de tion Mode Shapes: corresp fine (normali onding to zed) ˆ i i ii Ni v v v v φ ω ⎧⎫ ⎪⎪ == ⎨⎬ ⎩⎭ ± ± # ±
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Undamped Free Vibrations (Cont’d) 5 ( ) () () () 11 1 1 10 11 0 01 00 1 1 11 10 0 01 0 2 00 0 If we define: ˆ then: 0 1 0 ˆ 0 ˆ 0 ˆ 0 ˆ n n nn N n NN N n n n n n n n Ev eE v EE v v Ek m v ω × ×− −× −× − = ⎡⎤ ⎧⎫ ⎧ ⎫ ⎢⎥ ⇒= ⎨⎬ ⎩⎭ ⎣⎦ += ⇒∴ =− = ± ± ±± ± ± ± ± ± ± ± ± ± ± [] 1 00 0 234 1 ?? ,,, T nnn N n v E vv v E " 3
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Undamped Free Vibrations (Cont’d) 6 1 22 .. 33 1 1. Divided by the maximum component 2. 1 3. Divided Three ways to normalize mo by any value 1 ˆ 1 ˆ ˆ ˆ de shape vectors: n nn eg n n Nn n Nn T n v v v m v φ φφ = ⎧⎫ ⎪⎪ == ⎨⎬ ⎩⎭ ± ## ± ± 123 =||| | N Φ ⎡⎤ ⎣⎦ " ± ±± ± ±
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Multi Degree of System of Lumped Masses and Springs 7
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Multi Degree of System of Lumped Masses and Springs (Cont’d) 8 () 1 11 1 1 , 1,2,. .., 1 .., 1 jj j j j j j j j j NN N N pF Fm u j N pk u u k uu m u j N m u + ++ +− = = ⇒+ = = −− = ±± 12 2 22 3 3 33 4 .. . . . .... . . . ..... ...... N N kk k k k k k k k ⎡⎤ ⎢⎥ −+ ⎢− + ⇒= ⎣⎦ ### % #### ²
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Multi Degree of System of Lumped Masses and Springs (Cont’d) 9 1 11 22 2 33 3 1 .. .... . . , ...... . ....... NN N N mu p p p p p p −− ⎡⎤ ⎢⎥ == = ⎬⎨ ⎪⎪ ⎣⎦ ### % #### # # ±± ± mu ku p + = ²² ±± ±± ±
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Example 1 10 y Three Story Building
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Example 1 (Cont’d) 11 ( ) ( ) 3 ; 12 ; i ii j i j j j ij i col col Fk u k uu F k u
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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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Lecture06-Prof.Ju - CEE M237A / MAE M269A Lecture 6: MDOF...

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