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SD-Lecture07-Free-Vibrational-Systems

# SD-Lecture07-Free-Vibrational-Systems - CEGCEG-5065C Soil...

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CEG CEG-5065C Soil Dynamics 5065C Soil Dynamics Lecture #07 Free Vibrational Systems o (1) Undamped Free Vibrations o (2) Damped Free Vibrations Luis A. Prieto-Portar 2009

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Case (1). A free-vibration 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 = q W lb expressed as an equality by using the spring constant k , The coefficient of sub-grade reaction k s is, 3 s s s k z Az in = = When the foundation is disturbed from its static equilibrium, the foundation-soil system will vibrate. When damping is not present c=0, and the equation of motion is simply 0 0 ° ± + = + = ² ³ ´ µ °° °° k mu ku or u u m
1 2 The solution to this first-order 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 ω = + = frequency are, 2 1 m k ω π 1 2 2 and 2 2 The solution in terms of the circular frequency is, o o o o o o T f k m u C sin t C cos t π ω π π ω ω ω = = = = = +

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( ) ( ) ( ) ( ) 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 0 0 0 0 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 ω ω ω ω = = + = = = - = = ° ° ° Therefore, the equaton of motion becomes, o o o o o o u u sin t u cos t ω ω ω ω = + ° The displacement of an undamped free vibrating system is,
The maximum displacement Z is called the single amplitude . The peak-to-peak 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 = = = ° ± ° ± ° ± = = = = ² ³ ² ³ ² ³ ´ µ ´ µ ´ µ n n n n s s T and the frequency of oscillation f is, f T k Wg g The undamped natural frequency f m z W z ω π ω π ω π ω π π π π π The plots for the velocity and the acceleration of the foundation can be found from the

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SD-Lecture07-Free-Vibrational-Systems - CEGCEG-5065C Soil...

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