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Unformatted text preview: CEG CEG5065C Soil Dynamics 5065C Soil Dynamics Lecture #07 Lecture #07 Free Vibrational Systems o (1) Undamped Free Vibrations o (2) Damped Free Vibrations Luis A. PrietoPortar 2009 Case (1). A freevibration system without dampening. The soil subgrade reaction q is the foundation load W over an area A . In the lumped parameter system, the displacement z s of the soil is proportional to the load W , or s W lb k z inch = expressed as an equality by using the spring constant k , The coefficient of subgrade reaction k s is, 3 s s s q W lb k z Az in = = When the foundation is disturbed from its static equilibrium, the foundationsoil system will vibrate. When damping is not present c=0, and the equation of motion is simply & + = + = && && k mu ku or u u m 1 2 The solution to this firstorder differential equation is of the form, The parameter is the undamped natural circular frequency of the system, and thus the natural period and o k k u C sin t C cos t m m k m k m = + = frequency are, 1 2 2 1 2 and 2 2 The solution in terms of the circular frequency is, o o o o o o m k T f k m u C sin t C cos t = = = = = + ( ) ( ) ( ) ( ) 1 2 2 2 1 2 1 1 The boundary conditions reveals the value of the constants. At initial conditions, the time 0 with initial displacement and velocity o o o o o o o o o t u u , u C sin C cos C C u u u C cos C sin C C = = + = = = = = & & & Therefore, the equaton of motion becomes, o o o o o u u sin t u cos t = + & The displacement of an undamped free vibrating system is, o The maximum displacement Z is called the single amplitude . The peaktopeak amplitude is 2Z and is also referred to as the double amplitude . The time required for the sinusoidal displacement to repeat itself is called the period T , which is given by, 2 1 2 1 1 1 2 2 2 2 = = = & & & = = = = n n n n s s T and the frequency of oscillation f is,...
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This note was uploaded on 09/11/2011 for the course CEG 5065c taught by Professor Staff during the Fall '09 term at FIU.
 Fall '09
 STAFF

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