THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

600 20. Vibration and Vibration ControlFigure 20.7. A spring-damper system that is subjected to motion excitation.exciting force, we now have√T = F 1 +(2ξω / ) 2ωTn= √F 0 [ ( ) ](20.38)22 ( ) 2ω1 −ω n+ 2ξ ω ω nMotion ExcitationAs the corollary to the force excitation model of Figure 20.6, the system for motionexcitation is given in Figure 20.7. Variable x represents the motion of the dynamicsystem and variable y represents the harmonic displacement of the supportingbase. The dynamics of this system is characterized by the following equation:mẍ + C(ẋ − ẏ) + k(x − y) = 0 (20.39)The ratio of the magnitudes of the two displacements as a function of frequencydenotes the transmissibility given by√)T = x √ 2y = k2 + (Cω)√ 21 +(2ξ ω(k − mω2 ) 2 + (Cω 2 ) =ω n√ 2 [ ( ) ](20.40)22 ( ) 21 − ωω n+ 2ξ ω ω nThe right-hand side of Equation (20.40), which was expressed in terms of ω nand ξ, is identical to the transmissibility of Equation (20.38). This equality indicatesthat the methodologies employed to protect the supporting structure underforce excitation are also applicable to insulating the dynamic system from motionexcitation.Equations (20.38) and (20.40) are used to plot Figure 20.8 in order to illustratethe interrelation between the damping ratio ξ, the ratio of disturbing frequency to