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Hence at t = 0 the phase of the response will be<br />
( ) ( ) 2ζr<br />
arg (y) = arg ˆF − tan −1 (1 − r 2 )<br />
So when ˆF<br />
( )<br />
is real, the phase of the response is simply − tan −1 2ζr<br />
(1−r 2 )<br />
4.3 Undamped case<br />
When ζ = 0 the above becomes<br />
For real force this becomes<br />
∣<br />
The magnitude ∣Ŷ ∣ = F k<br />
4.4 damped cases<br />
ζ > 0<br />
1<br />
(1−r 2 )<br />
( ˆF<br />
y = Re<br />
k<br />
∣ ˆF<br />
∣<br />
=<br />
k<br />
)<br />
1<br />
(1 − r 2 ) eiϖt<br />
1<br />
(<br />
(1 − r 2 ) cos<br />
y = F k<br />
and phase zero.<br />
∣<br />
∣Ŷ<br />
y = Re<br />
( ))<br />
ϖt + arg ˆF<br />
1<br />
(1 − r 2 cos (ϖt)<br />
)<br />
( )<br />
ˆF 1<br />
k (1 − r 2 ) + i2ζr eiϖt<br />
∣ ˆF<br />
∣ 1<br />
∣ = √<br />
k<br />
(1 − r 2 ) 2 + (2ζr) 2<br />
) (<br />
arg<br />
(Ŷ = φ = arg ˆF<br />
Hence for real force and at t = 0 the phase of displacement is<br />
( ) 2ζr<br />
− tan −1 1 − r 2<br />
)<br />
− tan −1 ( 2ζr<br />
1 − r 2 )<br />
+ ϖt<br />
lag behind the load.<br />
When r < 1 then φ goes from 0 to −90 0 Therefore phase of displacement is 0 to −90 0 behind force. The<br />
minus sign at the front was added since the complex number is in the denominator. Hence the response will<br />
always be lagging in phase relative for load.<br />
For r > 1<br />
Now 1 − r 2 is negative, hence the phase will be from −90 ◦ to −180 ◦<br />
When r = 1<br />
( ) ˆF 1<br />
y = Re<br />
k i2ζ eiϖt<br />
∣ ∣<br />
∣Ŷ<br />
)<br />
arg<br />
(Ŷ<br />
∣ ˆF<br />
∣<br />
∣ =<br />
k<br />
= −90 ◦<br />
1<br />
2ζ<br />
Now phase is −90 ◦ 21