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Unformatted text preview: Soil Dynamics Lecture 02 Simple Vibrations Luis A. PrietoPortar, August 2006. Some simple initial definitions: A free vibration is any system that vibrates under the action of forces that are part of, or are inherent to the system itself. A forced vibration is any system that vibrates under the action of an external force to the system itself. The degree of freedom : The figure at left below can be described by a single coordinate z so it is a single degree of freedom system; the figure in the middle needs z 1 and z 2 to describe the motion of the system, so it is a twodegrees of freedom system; the figure at right is also a twodegree of freedom system, requiring z and to describe the motion. The basic design criterion for foundations subjected to vibrations (whether seismic, machinery or impact loadings) is to control their displacements . These displacements are of two kinds, 1) temporary cyclic elastic displacements (that is, they return to their original position after the loading stops, and 2) permanent plastic displacements (the foundation remains displaced from its original position after the loading ceases). Foundations can vibrate in all six possible modes, as shown below. These six modes of vibration may contribute to unbalanced forces in a simple foundation. These unbalanced forces in turn generate the vibrations. Each mode is analyzed separately. The most common simplification is to represent the foundationsoil system subjected to a dynamic loading Q with a spring and a dashpot analog model system (also known as a lumped parameter vibration system ). For example, a foundation subjected to a vertical axis dynamic loading, show below at left is represented by the figure on the right: this is equivalent to a lumped parameter vibration system Q In this lecture, we will consider the following four cases, in progressively increasing complexity: 1) A free vibration system without damping; 2) A steadystate forced vibration system without damping; 3) A free vibration system with viscous damping; and 4) A steadystate forced vibration system with viscous damping. (1) A freevibration system (with only a springmass). 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 = 3 s s s q W lb k z Az in = = expressed as an equality by using the spring constant k , The coefficient of subgrade reaction k s is, When the foundation is disturbed from its static equilibrium, the foundationsoil system will vibrate. The resulting equation of motion can be written from Newtons second law of motion, + = + = k mz kz or z z m 1 2 2 1 2 1 + = + = = + + + n n n n n n W k z kz or z z g m with a solution z A cos t A sin t where is the undamped natural circular frequency....
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This note was uploaded on 08/29/2011 for the course CEG 5905 taught by Professor Staff during the Summer '10 term at FIU.
 Summer '10
 STAFF

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