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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 ) • Nonperiodic – 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 NonPeriodic Motion Wave Animation Animation courtesy of Dr. Dan Russell, Kettering University Vibration Time History0.80.4 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 History0.80.60.40.2 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 Record302010 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 ( ϖ ) cycles/sec u(t) = A sin( ϖ t + φ ) • Period, T • Phase angle, φ Definitions 2 T π ϖ = A = amplitude or magnitude of max displacement Displacement time history = u(t) = A sin( ϖ t + φ ) = period or angular distance divided by speed or time for one cycle to take place. 1 2 f T ϖ π = = = frequency or cycles per time Superposition of Harmonic Motion u(t) = a cos( ϖ t) + b sin( ϖ t) a u(t) = A sin( ϖ t + φ ) where, φ = tan1 (a/b) or Complex Notations 1 i = cos sin 2 2 i i i i e e e e i α α α α α α + = =  α α α sin cos : Law s Euler' i e i + = Imaginary real Substituting above into: u(t) = a cos( ϖ t) + b sin( ϖ 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 ϖ ϖ ϖ ϖ ϖ ϖ  +  =  + = + Complex Notation • Argand Diagram a vector representation of a simple harmonic motion – complex notation. 2 i t a ib e ϖ 2 i t a ib e ϖ + 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( ) ( φ ϖ + = t A t u ) cos( ) ( ) ( φ ϖ ϖ + = ∂ ∂ = = t A t t u t u v u a t A t t u t u a 2 2 2 2 ) sin( ) ( ) ( ϖ φ ϖ ϖ = ⇒ + = ∂ ∂ = = Summary ( ) i t u t e ϖ = ( ) i t t i Ae u ϖ ϖ = ( ) sin( ) u t A ϖ = ( ) cos( ) v u t A t ϖ ϖ = = 2 ( ) sin( ) a u t A t ϖ ϖ = =  2 2 2 ( ) i t i t u t i Ae Ae ϖ ϖ ϖ ϖ = =  Trigonometric Complex: Summary ( ) sin( ) u t A ϖ = 2 ( ) sin( ) a u t A t ϖ ϖ π = = + ( 29 ( ) sin 2 v u t A t π ϖ ϖ = = + Tripartite...
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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.
 Spring '11
 Louis

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