MAE147-lecture 7

# MAE147-lecture 7 - MAE 147 “Vibrations” Lecture 7...

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Unformatted text preview: MAE 147 “Vibrations” Lecture 7 Instructor: Professor Andrei Shkel Email: [email protected] Office: EG4208 L7-2 MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 Before we get started ¡ For next week ¢ Finishing Chapter 4 and starting Chapter 5 ¢ Quiz on Monday, Feb 4 ¢ Reading: Sections 4.3, 4.4, 4.5 (due Feb. 4 th ) ¢ HW4 due on Feb. 6 th ¡ Last Time ¢ Solving ODE ¢ Damped Free Vibrations ¢ Stability of Linear Systems L7-3 MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 Lecture Outline ¡ Solution for the Harmonic Response ¢ Free response ¢ Step Response ¢ Forced response to harmonic excitation ¢ Steady-State Response ¢ Frequency Response and Transfer Function ¢ Examples ¡ Resonance L7-4 MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 Solution of Non-homogeneous ODE ) ( 2 2 t f ky dt dy g dt y d = + + Solution: Particular solutions k m k k t s p y b e t y ∑ = = λ 1. Complementary solution of the associated homogeneous equation, y c 2 2 = + + ky dt dy g dt y d 2. A particular solution of the nonhomogeneous equation, y p 2. f ( t ) has a form t B t A t f p c y y y + = 3. General solution of the nonhomogeneous equation is ω ω cos sin ) ( + = t D t C y p ω ω cos sin + = 1. f ( t ) has a form k m k k t t a e t f ∑ = = ) ( λ s=0, …,r repeats of the root p r = λ f ( t ) is called the driving or the forcing function L7-5 MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 ( ) t C t C e t c 3 cos 3 sin 2 1 + = − 2 4 6 1 2 3 4 5 6 t 2 4 6 8 10 1 2 3 4 5 e-t (sin 3t-3cos 3t)+3 t 2 4 6-0.5 0.5 1 1.5 2e-t sin(3t) t Example 4: 10 2 2 2 = + + y dt dy dt y d 6 ; = = = = t t dt dy y Solution: 1 2 3 6 ; C C = = t e y t 3 sin 2 − = Solution of Nonhomogeneous ODE (Continue)...
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MAE147-lecture 7 - MAE 147 “Vibrations” Lecture 7...

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