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Unformatted text preview: CEG CEG5065C Soil Dynamics 5065C Soil Dynamics Lecture 08 Lecture 08 Forced Vibrational Sy stems o (1) Undamped Forced Vibrations o (2) Damping Forced Vibrations Luis A. PrietoPortar 2009 Greek Symbols used in Vibrational Analysis: & (alpha) = Attenuation constant; coefficient of thermal expansion; (beta) = Percentage of critical damping (c/ 2 & km); the tuning ratio; (gamma) = Shear strain; (delta) = Relative displacement = u x ; (epsilon) = Strain or change in length per unit length; (zeta) (eta) (theta) = Phase angle; slope angle; (iota) (kappa) (lambda) = Wavelength; (mu) (nu) = Poisson ratio; (xi) = Damping ratio = c / c c = c/2 km (omicron) (pi) (rho) = Mass density ( m/V ); (sigma) = Stress; standard deviation; (tau) = Shear stress; time variable of integration; (upsilon) (phi) = Blast hole diameter; angle of sliding resistance; (chi) (psi) = Angle of phase difference; (omega) = Dominant ground motion circular frequency (= 2 f ). Case (3). A steadystate forcedvibration system without damping. The previously studied free vibration foundationsoil system has now an additional external alternating force Q o sin( & t + ). This is shown below, where the spring constant is still k . This type of problem is typical of footings supporting internal combustion engines that have reciprocating pistons. The general equation of motion for a forced and undamped system is, The general solution will consist of the sum of the solution and the solution. The complement + = && mu ku Q sin t complementary particular ary (homogeneous) equation of motion will be, + = && mu ku 1 2 o which will have the known solution, Basically, this is a simple harmonic oscillation with the undamped natural frequency of = + c o o u C sin t C cos t the system. A particular solution handles the response to the external loading. In general, this equation of motion could be of the harmonic loading form, where is the amplitude of the harmonic resp = p o o u ( t ) U sin t U 2 2 onse. Substituting both and into the general equation of motion yields, and setting yield the value of the amplitude  + = = c p o o o o u u m U sin t kU sin t Q sin t k / m U 2 2 2 2 1 1 where is call = = o o Q / k Q / k U / 1 2 2 ed the . The general solution is thus, 1 = + = + + c p o o tuning ratio Q / k u( t ) u ( t ) u ( t ) C sin t C cos t sin t 1 2 2 The general solution must satisfy the initial conditions for displacement and velocity The velocity is, 1 Given an initial displacement and velocity = = + & & o o o o o u u....
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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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