This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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, outofphase 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 Steadystate response • Particular solution • Constant with time • Applied load motion frequency, but, outofphase 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...
View
Full
Document
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.
 Spring '11
 Louis

Click to edit the document details