Laplace+TransferFunctions

M.D. Bryant ME 344 notes 03/25/08 7
• Apply inverse Laplace: x(t) = L -1 {X(s); s→ t }
x(t) = H o
ω 2 n
L -1 {
ω 2 n
s (s2 + 2 ζ ωn s + ω 2 n ) }
+ x o
ω 2 n
s ω 2 n
L -1 {
s2 + 2 ζ ωn s + ω 2 n
}
+ x 1 + 2 ζ ωn xo
ω 2 n
ω 2 n
L -1 {
s2 + 2 ζ ωn s + ω 2 n
}
o Use Laplace transform tables for L -1 :
x(t) = H o
ω 2 n
{1 - e-ζ ω n t sin(ωn 1 - ζ2 t + arccos ζ )
1 - ζ2
}
+ x o
ω 2 n
{ - ω2 n e-ζ ω n t sin(ωn 1 - ζ2 t - arccos ζ )
1 - ζ2
}
+ x 1 + 2 ζ ωn xo
ω 2 n
{ ω n e -ζ ω n t sin(ωn 1 - ζ2 t )
1 - ζ2
}