Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
8 Derivation of the solutions of the equations of motion<br />
8.1 Free vibration<br />
8.1.1 Undamped free vibration<br />
u<br />
K<br />
M<br />
mu ′′ + ku = 0<br />
u ′′ + ω 2 u = 0<br />
The solution is<br />
u (t) = u (0) cos ωt + u′ (0)<br />
ω<br />
sin ωt<br />
8.1.2 under-damped free vibration c < c r , ξ < 1<br />
u<br />
K<br />
M<br />
C<br />
The solution is<br />
mu ′′ + cu ′ + ku = 0<br />
u ′′ + 2ξωu ′ + ω 2 u = 0<br />
u = e −ξωt (A cos ω d t + B sin ω d t)<br />
Applying initial conditions gives A = u (0) and B = u′ (0)+u(0)ξω<br />
ω d<br />
. Therefore the solution becomes<br />
(<br />
)<br />
u (t) = e −ξωt u (0) cos ω d t + u′ (0) + u (0) ξω<br />
sin ω d t<br />
ω d<br />
33
Change language