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

20.3 General Solution for the One-Degree Model of Simple System 59120.3 General Solution for the One-Degree Modelof Simple SystemAn initial displacement is introduced to the model system of Figure 20.1, and itis desired to determine the response of this system. The differential equation instandard mathematical shorthand notation isẍ(t) + 2ξω n ẋ(t) + ωn 2 x(t) = 0 (20.11)which undergoes a Laplacean transformation intos 2 X(s) − sx(0) − ẋ(0) + 2ξω n [sX(s) − x(0)] + ωn 2 X(s) = 0Let the initial displacement be represented by x(0) = x 0 and the initial velocityẋ(0) = 0. The preceding equation reduces tos + 2ξω nX(s) = x 0(20.12)s 2 + 2ξω n s + ωn2The graphical residue technique, described by Example Problem 4 in AppendixC, can be used to find the inverse transform for Equation (20.12) and hence thesolution of Equation (20.11):x(0√x(t) = √1 − ξ2 e−ξω nt cos ω n 1 − ξ 2t − π )2 + cos−1 ξwhich reduces tox(0√ )x(t) = √1 − ξ2 e−ξω nt sin ω n 1 − ξ 2t + θ(20.13)from the trigonometric relationship cos (ϕ − π/2) = sin ϕ and by settingθ = cos −1 ξ (20.14)Example Problem 3Consider the system of Figure 20.1 which is initially displaced and then suddenlyreleased. Its resultant motion is described byx(t) = 3.0e −3.43t sin (11.4t + 60 ◦ )Find the system’s damping ratio, natural frequency, and the initial displacement.SolutionApplying Equations (20.13) and (20.14),θ = 60 ◦ = cos −1 ξ, ξ = 0.500, and ξω n = 3.43Hence ω n = 6.86. The initial condition is established fromx 03.0 = √ = x 0√ = 1.155x 01 − ξ2 1 − (0.500)2x 0 = 2.597