You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Since t 1 is very small, it can be assumed that u changes is negligible as well as the change in velocity, hence the<br />
above reduces to the same result as in the case of undamped. Therefore, the system is solved as free system, but<br />
with initial velocity u ′ (0) = F 0 /m and zero initial position.<br />
Initial conditions are u (0) = 0 and u ′ (0) = 0 then the solution is<br />
applying initial conditions gives A = 0 and B =<br />
u impulse = e −ξωt (A cos ω d t + B sin ω d t)<br />
( F0<br />
)<br />
m<br />
ω d<br />
, hence<br />
u impulse (t) = e −ξωt (<br />
F0<br />
mω d<br />
sin ω d t<br />
If the initial conditions were not zero, then ( the solution for these are added ) to the above. From earlier, it<br />
was found that the solution is u (t) = e −ξωt u(0) cos ω d t + u′ (0)+u(0)ξω<br />
ω d<br />
sin ω d t , therefore, the full solution is<br />
due to IC only<br />
{ (<br />
}} ){<br />
{ ( }} ){<br />
u (t) = e −ξωt u(0) cos ω d t + u′ (0) + u(0)ξω<br />
sin ω d t + e −ξωt F0<br />
sin ω d t<br />
ω d<br />
mω d<br />
)<br />
due to impulse<br />
critically damped with impulse input ξ = c<br />
c r<br />
= 1 with initial conditions u (0) = 0 and u ′ (0) = 0 then the<br />
solution is<br />
u (t) = (A + Bt) e −( cr<br />
2m<br />
)<br />
t<br />
= (A + Bt) e −ωt<br />
where A, B are found from initial conditions A = u (0) = 0 and B = u ′ (0) + u (0) ω = F 0<br />
m<br />
, hence the solution<br />
is<br />
u impulse (t) = F 0t<br />
m e−ωt<br />
If the initial conditions were not zero, then the solution for these are added to the above. From earlier, it<br />
was found that the solution is u (t) = (u 0 (1 + ωt) + u ′ (0) t) e −ωt , therefore, the full solution is<br />
u (t) =<br />
due to IC only<br />
{ }} {<br />
(<br />
u (0) (1 + ωt) + u ′ (0) t ) e −ωt +<br />
due to impulse<br />
{ }} {<br />
F 0 t<br />
m e−ωt<br />
over-damped with impulse input ξ = c<br />
c r<br />
> 1 With initial conditions are u (0) = 0 and u ′ (0) = 0 the<br />
solution is<br />
u impulse (t) = Ae λ 1ωt + Be λ 2ωt<br />
where A, B are found from initial conditions and<br />
λ 1 = −ωξ + ω √ ξ 2 − 1<br />
λ 2 = −ωξ − ω √ ξ 2 − 1<br />
A = u′ (0) − u (0) λ 2<br />
2ω √ ξ 2 − 1<br />
B = −u′ (0) + u (0) λ 1<br />
2ω √ ξ 2 − 1<br />
Hence the solution is<br />
(<br />
−ξ+ √ (<br />
ξ<br />
u impulse (t) = Ae<br />
2 −1<br />
)ωt −ξ− √ )<br />
ξ + Be 2 −1 ωt<br />
42