You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
4 Steady state solutions to single degree freedom system<br />
4.1 Summary:<br />
( )<br />
my ′′ + cy ′ + ky = Re ˆF e<br />
iϖt<br />
{ }<br />
x = Re ˆXe<br />
iϖt<br />
ˆX = ˆF D (r, ζ)<br />
k<br />
1<br />
D (r, ζ) =<br />
(1 − r 2 ) + 2iζr<br />
{ }<br />
ˆF<br />
x = Re<br />
k |D (r, ζ)| ei(ϖt−θ)<br />
θ = tan −1<br />
2ζr<br />
1 − r 2<br />
4.2 Details<br />
( )<br />
Let load be harmonic and represented in general as Re ˆF e<br />
iϖt<br />
where ˆF is the complex amplitude of the force.<br />
Hence system is represented by<br />
( )<br />
my ′′ + cy ′ + ky = Re ˆF e<br />
iϖt<br />
( ) ˆF<br />
y ′′ + 2ζω n y ′ + ωny 2 = Re<br />
m eiϖt<br />
)<br />
)<br />
)<br />
Let y = Re<br />
(Ŷ e<br />
iϖt<br />
Hence y ′ = Re<br />
(iϖŶ eiϖt , y ′′ = Re<br />
(−ϖ 2 Ŷ e iϖt , therefore the differential equation<br />
becomes<br />
(<br />
Re −ϖ 2 Ŷ e iϖt) (<br />
+ 2ζω n Re iϖŶ eiϖt) + ωn 2 Re<br />
(Ŷ ) ( )<br />
e<br />
iϖt ˆF<br />
= Re<br />
m eiϖt<br />
Dividing numerator and denominator ω 2 n gives<br />
Where r = ϖ ω n<br />
, hence the response is<br />
Therefore, the phase of the response is<br />
y = Re<br />
( )<br />
arg (y) = arg ˆF<br />
( ˆF<br />
−ϖ 2 + 2ζω n iϖ + ωn) 2 Ŷ =<br />
m<br />
ˆF<br />
m<br />
Ŷ =<br />
(−ϖ 2 + 2ζω n iϖ + ωn)<br />
2<br />
Ŷ = ˆF 1<br />
k (1 − r 2 ) + i2ζr<br />
( )<br />
ˆF 1<br />
k (1 − r 2 ) + i2ζr eiϖt<br />
( 2ζr<br />
− tan −1 (1 − r 2 )<br />
20<br />
)<br />
+ ϖt