MAE147-lecture 6

MAE147-lecture 6 - MAE 147 "Vibrations" Lecture 6...

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MAE 147 “Vibrations” Lecture 6 Instructor: Professor Andrei Shkel Email: ashkel@uci.edu Office: EG4208
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L6-2 MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 Before we get started ± For this week ² Make-up lecture on Feb. 1 st , 9:00-10:20am, MDEA ² Detailed notes for make-up lecture will be available on-line ² Reading: Sections 4.1-4.2 (due Feb. 1 st ) ² HW3 is due today ² HW4 assigned, due on Feb. 6 th ± Last Time ² Free Undamped Response: 1 st and 2 nd order systems ² Relationship between x,v,a in pure harmonic motion ² Equivalent mass/inertia ² Finding natural frequency (E.O.M and Rayleigh’s method)
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L6-3 MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 Linear or Nonlinear ODE 2 5 yy yt ++= && sin 5 += 2 5 y t ++ = 40 &
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L6-4 MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 0 1 2 3 0 1 2 3 3e -2t t 1 st and 2 nd Order Linear ODE Example 1: 0 2 = + y dt dy 3 0 = = t y 0 2 = + p 2 = p t e y 2 3 = Solution: pt Ce y = 2/05/2001 Prof. Y. Shkel, ME 340, Spring 2001 7 Solution of Homogeneous Linear ODE 0 ' 1 ) 1 ( 1 ) ( 0 = + + + + y a y a y a y a n n n n L 1. Characteristic equation: 0 1 1 1 0 = + + + + n n n n a p a p a p a L 2. Find all N roots: N p p , , 1 L 3. Find N independent solutions: N k e C y t p k k k K 1 , = = ( ) t p r k r k k k k e C t tC C y + + + + + + = 1 2 1 L If p k is r -repeated root 5. General solution as sum of N solutions = = N k k y y 1 4. Real coefficients For each complex conjugates i p m m β α ± = + + 2 , 1 t e C t e C y y t m t m m m sin cos 2 1 2 1 + + + + + = + C k
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MAE147-lecture 6 - MAE 147 "Vibrations" Lecture 6...

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