Week3-VibDyn_cont_Responses_S11

Week3-VibDyn_cont_Responses_S11 - 1 CE 316 Geotechnical...

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Unformatted text preview: 1 CE 316 Geotechnical Earthquake Engineering Week 3 Vibrations & Dynamics (continued) & Response Spectra Undamped Forced Vibrations Harmonic Loading sin mu ku Q t ϖ + = && t k Q t u t k Q u u ϖ β ϖ ϖ β β ϖ sin 1 cos sin ) 1 ( 2 2- + + -- = & ϖ ϖ β = Solution… Where , β is the tuning ratio … Circ. Frequency of forcing function Undamped Forced Vibrations Harmonic Loading t k Q t u t k Q u u ϖ β ϖ ϖ β β ϖ sin 1 cos sin ) 1 ( 2 2- + + -- = & Solution… Complementary solution • Due to initial conditions Particular solution • Constant with time • Applied load motion frequency, but, out-of-phase Undamped Forced Vibrations Harmonic Loading t k Q t u t k Q u u ϖ β ϖ ϖ β β ϖ sin 1 cos sin ) 1 ( 2 2- + + -- = & = = u u & ( 29 t t k Q u 2 sin sin 1 1 ϖ β ϖ β-- = If… MF = magnification factor 2 β = 2 ϖ ϖ < For the amplitude is greater that static condition. Undamped Forced Vibrations Example: From an initial stationary state, the undamped SDOF system of Example B.l is subjected to a harmonic base acceleration of 0.20g at a frequency of 2 Hz. Compute the response of the system. Solution… harmonic base acceleration of 0.20g at a frequency of 2 Hz ( ) sin b u t A t ϖ = && Damped Forced Vibrations Harmonic Loading t m Q u u u ϖ ϖ ξϖ sin 2 2 = + + & & & 2 1 2 2 2 2 1 ( ) ( sin cos ) (1 )sin 2 cos (1 ) (2 ) t d d Q u t e C t C t t t k ξϖ ϖ ϖ β ϖ ξβ ϖ β ξβ- = + + - - - + Solution… Transient response • Complementary solution • Decays with time • Due to initial conditions Steady-state response • Particular solution • Constant with time • Applied load motion frequency, but, out-of-phase sin Q t m ϖ Damped Forced Vibrations Harmonic Loading [ ] t t k Q t C t C e t u d d t ϖ ξβ ϖ β ξβ β ϖ ϖ ξϖ cos 2 sin ) 1 ( ) 2 ( ) 1 ( 1 ) cos sin ( ) ( 2 2 2 2 2 1-- +- + + =- Total Damped Forced Vibrations Harmonic Loading, Steady State ) sin( φ ϖ + = t A u 2 2 2 ) 2 ( ) 1 ( 1 ξβ β +- = k Q A -- =- 2 1 1 2 tan β ξβ φ Damped Forced Vibrations...
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This note was uploaded on 02/27/2011 for the course CIV ENG 316 taught by Professor Louis during the Spring '11 term at Missouri S&T.

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Week3-VibDyn_cont_Responses_S11 - 1 CE 316 Geotechnical...

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