Week2-VibrationsDynamics_S11

Week2-VibrationsDynamics_S11 - 1 CE 316 Geotechnical...

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Unformatted text preview: 1 CE 316 Geotechnical Earthquake Engineering Week 2 Vibrations & Dynamics Vibratory Motions Appendix A (Kramer, 1996) Periodic repeat at regular intervals (e.g., simple harmonic motions, - engines ) Non-periodic do not repeat at regular intervals (e.g., impulses - blasting ; and long duration transient loads - traffic and earthquakes ) General Types of Vibratory Motion Periodic Motion Non-Periodic Motion Wave Animation Animation courtesy of Dr. Dan Russell, Kettering University Vibration Time History-0.8-0.4 0.4 0.8 5 10 15 20 25 30 35 40 45 50 Time (seconds) Blasting time history Earthquake time history (Kobe) Vibration Time History-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 10 10.5 11 11.5 12 12.5 Time (seconds) Acceleration (g) Kobe Earthquake Portion of Blast Record-30-20-10 10 20 30 10 10.5 11 11.5 12 12.5 Time (seconds) Velocity (in/sec) Vibration Time History Kobe Earthquake Portion of Blast Record Basics of vibrations Trigonometric Notation Vector representation of a simple harmonic motion. Simple Harmonic Motion Amplitude, A Frequency:- circular ( )- cycles/sec u(t) = A sin( t + ) Period, T Phase angle, Definitions 2 T = A = amplitude or magnitude of max displacement Displacement time history = u(t) = A sin( t + ) = period or angular distance divided by speed or time for one cycle to take place. 1 2 f T = = = frequency or cycles per time Superposition of Harmonic Motion u(t) = a cos( t) + b sin( t) a u(t) = A sin( t + ) where, = tan-1 (a/b) or Complex Notations 1 i =- cos sin 2 2 i i i i e e e e i -- +- = = - sin cos : Law s Euler' i e i + = Imaginary real Substituting above into: u(t) = a cos( t) + b sin( t) ( ) ... 2 2 ( ) 2 2 i t i t i t i t i t i t e e e e u t a bi rearranging a ib a ib u t e e - -- + - = -- + = + Complex Notation Argand Diagram a vector representation of a simple harmonic motion complex notation. 2 i t a ib e - 2 i t a ib e - + 2 2 A a b = + If you do the math, the imaginary parts cancel and the amplitude results in: Complex Notation Animation Animation courtesy of Dr. Dan Russell, Kettering University Displacement: Velocity: Acceleration: Other Measures of Motion ) sin( ) ( + = t A t u ) cos( ) ( ) ( + = = = t A t t u t u v u a t A t t u t u a 2 2 2 2 ) sin( ) ( ) ( - = +- = = = Summary ( ) i t u t e = ( ) i t t i Ae u = ( ) sin( ) u t A = ( ) cos( ) v u t A t = = 2 ( ) sin( ) a u t A t = = - 2 2 2 ( ) i t i t u t i Ae Ae = = - Trigonometric Complex: Summary ( ) sin( ) u t A = 2 ( ) sin( ) a u t A t = = + ( 29 ( ) sin 2 v u t A t = = + Tripartite...
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Week2-VibrationsDynamics_S11 - 1 CE 316 Geotechnical...

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