
6 Single Degree-of-Freedom Vibration
Section 6.2: Deterministic SDOF Vibration
6.1. A single degree-of-freedom system with ω
n
= 1 rad/s and ζ = 0.24 is excited by an external force
per unit mass, F (t) /m = 2 sin ω
f
t. Derive the response if the forcing frequency coincides with the
damped natural frequency.
Solution: The damped natural frequency is ω
n
1 − ζ
2
= 0.97 rad/s. The particular solution is given by
x
p
(t) = D sin (ω
f
t − φ) ,
where
D =
2
ω
2
n
1
*
(1 − ω
2
f
/ω
2
n
)
2
+ (2ζω
f
/ω
n
)
2
,
θ = tan
−1
2ζω
f
/ω
n
1 − ω
2
f
/ω
2
n
.
When ω
f
= ω
d
, we have
ω
f
ω
n
=
*
1 − ζ
2
.
Then, the amplitude and phase of the particular solution become
D =
2
ω
2
n
1
ζ
*
ζ
2
+ 4
1 − ζ
2
= 4.26
θ = 1.45 rad/s.
The particu