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

594 20. Vibration and Vibration Controlsolution is in the form of Equation (20.13) and it is determined by the residues ofthe complex poles, wherein the poles constitute the solution ofms 2 + Cs + k = 0The steady-state solution of Equation (20.19) is a sinusoidal oscillation expressedasx(t) = x(ω) sin (ωt − φ) (20.22)and this solution is determined by the residues at the complex poles s = s =±iω.While the exact solution can be derived by using the residue method, the solutionof Equation (20.19) can be readily obtained through the standard differentialmethods. The advantage of the Laplace transform method that it facilitates findingthe frequency and stability information.Through Equations (20.21) and (20.22), the magnitude of the steady-state oscillationcan be determined fromF 0X(s) =(20.23)ms 2 + Cs + kFrom Equation (20.23), the magnitude of the oscillation can be determined as afunction of frequency. Because s = σ + iω and ω = 2πf, substituting s = iω intoEquation (20.23) yieldsF 0x(iω) =−mω 2 + iωC + kThe denominator is a complex number, so the magnitude of this number is equalto√(k − mω2 ) 2 + (ωC) 2The magnitude of the oscillation as a function of frequency isx(ω) =F 0√(k − mω2 ) 2 + (Cω) = F 0 /k√ (20.24)2 [1 − (m/k)ω2 ] 2 + (Cω/k) 2From the definitions of Equations (20.7) through (20.10), Equation (20.24) can beexpressed asX(ω)F 0 /k = 1√ (20.25)[1 − (ω/ωn ) 2 ] 2 + [2ξ(ω/ω n )] 2The magnitude is now a function of only two quantities, the ratio (ω/ω n ) and thedamping ratio ξ. The phase angle φ is also a function of these two parameters andit is given bytan φ = 2ξ(ω/ω n)(20.26)1 − (ω/ω n ) 2The system undergoes a resonance when the excitation frequency f = ω/2πequals the natural frequency of the dynamic system f n = ω n /2π. The damping