MAE147-lecture+8

MAE147-lecture+8 - MAE 147 "Vibrations" Lecture 8...

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MAE 147 “Vibrations” Lecture 8 Instructor: Professor Andrei Shkel Email: ashkel@uci.edu Office: EG4208
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MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 L8-2 Before we get started ± For this week ² Finishing Chapter 4 and starting Chapter 5 ² Quiz today, Monday, Feb 4 ² Reading: Sections 4.2, 4.8(due Feb. 6 th ) ² HW4 due on Feb. 6 th ± Last Time ² Solving non-homogeneous ODE ² Step response ² Harmonic forced excitations
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MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 L8-3 Lecture Outline ± Analysis of Harmonic Response ² Resonance ² Phase-amplitude relations ² Response to Base excitation ² Transmissibility
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MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 L8-4 Demo of Free Vibrations with Damping ± The abination shows two 1-DOF mass-spring systems initially at rest, but displaced from equilibrium by x=x max . The black mass is undamped and the blue mass is damped (underdamped). After being released from rest the undamped (black) mass exhibits simple harmonic motion while the damped (blue) mass exhibits an oscillatory motion which decays with time. ± Observation: frequency and phase of oscillations are nearly the same
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MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 L8-5 10 -1 10 0 10 1 10 -1 10 0 10 1 ζ =0 () ( ) 2 1 2 2 2 2 1 1 2 tan ; sin 4 1 n n n n ss t K y ωω ω ς φφ ςω = + + = Frequency Response (Cont) 2 2 2 2 1 4 1 1 n n K Y + = ( ) 2 1 1 2 tan n n φ = =0.1 =0.2 =0.4 =0.8 =1 =1.6 =0 10 -1 10 0 10 1 -180 -150 -120 -90 -60 -30 0 =0.1 =0.2 =0.4 =0.8 =1 =1.6 n n n ω= n
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MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 L8-6 Characteristics of Forced Vibration 2/26/2001 Prof. Y. Shkel, ME 340, Spring 2001 7 10 -1 10 0 10 1 10 10 0 10 1 ζ =0 () 2 1 2 2 2 2 1 1 2 tan ; sin 4 1 n n n n ss t K y ωω ωως φφω = + + = Frequency Response 2 2 2 2 1 4 1 1 n n K Y + = 2 1 1 2 tan n n φ = =0.1 =0.2 =0.4 =0.8 =1 =1.6 10 0 1 -180 -150 -120 -90 -60 -30 0 n ω n n = n = With =0 1 K Y n ω→ 0 = n ω< 180 = n ω> With ζ≠ 0 2 2 1 ς = n 2 1 = n d Compare with the damped natural frequency d : n Straight-Line Approximation ω<< 0 . 1 1 K Y : n ω>> 2 1 n K Y log log 2 log 1 n K Y 0 log 1 K Y =0.6
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MAE 147 “Vibrations” Professor Andrei M. Shkel Winter 2008 L8-7 Response to an applied harmonic force ± Three identical damped 1-DOF mass- spring oscillators, all with natural frequency f0=1, are initially at rest. A
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This note was uploaded on 04/07/2008 for the course MAE 147 taught by Professor Shkel during the Winter '08 term at UC Irvine.

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MAE147-lecture+8 - MAE 147 &quot;Vibrations&quot; Lecture 8...

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