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One degree freedom<br />
linear system unforced<br />
response<br />
y t c m y t k m yt 0<br />
y t 2 n y t n2 yt 0<br />
Some<br />
defintions<br />
k<br />
n m rad/sec<br />
damping ratio<br />
c c c<br />
r<br />
c<br />
2 km 2 n m<br />
Solution roots<br />
s 1,2 n n 2 1<br />
3 cases<br />
d n 1 2 30<br />
Defined only<br />
for under<br />
damped<br />
case(else zero)<br />
1 1 1<br />
Underdamped<br />
2 roots,<br />
complex<br />
conjugate<br />
Critical damped<br />
(one real root,<br />
double<br />
multiplicity)<br />
Over damped,<br />
2 real roots,<br />
distinct<br />
yt e nt Acos d t Bsin d t<br />
A y0, B y 0A n<br />
d<br />
yt A Bte nt<br />
A y0, B y 0 A n<br />
T d 2<br />
d<br />
1<br />
n<br />
Nasser M. Abbasi<br />
May 25, 2011<br />
One_DOF_system.vsd<br />
Damped period of oscillation<br />
Time constant<br />
yt Ae 2 1 n t Be 2 1 n t<br />
A y 0y0 n<br />
2 n<br />
B y 0y0 n<br />
2 n<br />
2 1<br />
2 1<br />
2 1<br />
2 1