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Week3-VibDyn_cont_Responses_S11

Week3-VibDyn_cont_Responses_S11 - CE 316 Geotechnical...

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1 CE 316 Geotechnical Earthquake Engineering Week 3 Vibrations & Dynamics (continued) & Response Spectra
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Undamped Forced Vibrations Harmonic Loading 0 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 0 0 0 0 2 0 0 0 - + + - - = & 0 ϖ ϖ β = Solution… Where , β is the tuning ratio Circ. Frequency of forcing function
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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 0 0 0 0 2 0 0 0 - + + - - = & Solution… Complementary solution Due to initial conditions Particular solution Constant with time Applied load motion frequency, but, out-of-phase
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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 0 0 0 0 2 0 0 0 - + + - - = & 0 0 0 = = u u & ( 29 t t k Q u 0 2 0 sin sin 1 1 ϖ β ϖ β - - = If… MF = magnification factor 2 β = 0 2 ϖ ϖ < For the amplitude is greater that static condition. 0
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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 0 ( ) sin b u t A t ϖ = &&
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Damped Forced Vibrations Harmonic Loading t m Q u u u ϖ ϖ ξϖ sin 2 0 2 0 0 = + + & & & 0 2 0 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 0 sin Q t m ϖ
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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 0 2 1 0 - - + - + + = - Total
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Damped Forced Vibrations Harmonic Loading, Steady State ) sin( φ ϖ + = t A u 2 2 2 0 ) 2 ( ) 1 ( 1 ξβ β + - = k Q A - - = - 2 1 1 2 tan β ξβ φ
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Damped Forced Vibrations Example: The SDOF system shown in Figure EB.4a is at rest when the sinusoidal load is applied. Determine the transient, steady state, and total motion of the system.
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