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

592 20. Vibration and Vibration ControlFigure 20.4. Response of a second-order system that is underdamped.Because ξ x 0 , i.e., a slope of x existsinitially, an initial velocity is present. The actual damping is ξω n , and the dampedperiod is defined as2πτ d = √ (20.15)ω n 1 − ξ2In Figure 20.4, it is evident that the decay rate e −ξωnt of the free oscillation dependson the system damping. The greater the damping, the faster is the rate of decay.Let us now establish the relationship between the rates of decay and damping.First, the time response at two distinct points, each of which is a quarter periodfrom the crossover points in the first two lobes, must be determined. These pointsare the points where the sine function equals unity and not the peak points of thedamped response in Figure 20.4. From Equations (20.4) and (20.13), it is notedthatx(0√ )x 1 = √1 − ξ2 e−ξω nt 1sin ω n 1 − ξ 2t 1 + θ(20.16)andx 2 =x(0√ )√1 − ξ2 e−ξω n(t 1 +τ d ) sin ω n 1 − ξ 2(t 1 + τ d ) + θ(20.17)Taking the ratio of the two amplitudes represented by Equations (20.16) and (20.17)and noting that in this case the sine functions equal unity, we obtainx 1x 2= e−ξω nt 1e −ξω n(t 1 +τ d ) = eξω nτ d