Free vibration 189
1
u (t) (cm)
0.8 0.6 0.4 0.2
2
4
6
8
1 0
t (s)
Figure 15.3 Free vibration of an overcritically-damped system.
Case 3: ξ < 1, i.e. undercritically-damped systems. In this special case, the two roots in equation 15.5b become: s1,2 ξω ± iωD
(15.12)
where
ωD ω 1 – ξ2
(15.13)
ωD is the damped natural frequency of the system in free vibrations. The solution given in equation 15.5 becomes: u(t) [AcosωDt BsinωDt]e ξωt
(15.14)
Using the initial conditions u(0) and u˙(0), the constants can be determined leading to: u˙(0) u(0)ξω u(t) sinωDt u(0)cosωDt e ξωt ωD
(15.15)
u(t) dcos(ωDt θ)e ξωt
(15.16)
or
where d
u˙(0) + u(0)ξω u(0) ω 2
2
(15.17a)
D
u˙(0) + u(0)ξω u(0) sinθ ; cosθ ωDd d
(15.17b, c)