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

20.2 Modeling Vibration Systems 587Figure 20.2. Root locations and nomenclature for a second-order dynamic system. Thes-plane to the right shows the domain of the roots as a function of the damping ratio ξ.where2ξω n = C mω 2 n = k m(20.5)Equation (20.4) is the characteristic equation of which roots essentially determinethe response of the system [i.e., the position of the mass as a function of time orx(t)]. Solving for the roots of Equation (20.4) yieldss 1 , s 2 = −2ξω n ± √ (2ξ 2 ω n ) 2 − 4ωn2 (20.6)2These roots can assumed complex values and so can be plotted on a s-plane,where s = σ + iω and ω = 2π f . The general plot is shown in the complex planeof Figure 20.2(a). For this plot it is assumed that the system is stable and the valueof ξ falls between zero and unity. The term ξ is the damping ratio and ω n is theundamped natural frequency. In Figure 20.2(a) it is noted that ξ = cos θ. Bothξ and ω n constitute the key factors that determine the roots of the characteristicequations and hence the response of the system. The following four cases ofdamping are of interest:Case 1: ξ1(Overdamped system)The roots are s 1 , s 2 =−ξω n ± iω √ ξ 2 − 1 and the system response isgiven by( √ ) ( √ )x(t) = Ae − ξω n +ω n ξ 2 −1 t − ξω+ Ben −ω n ξ 2 −1 t