18
2 Single Degree of Freedom (SDOF) Systems
arbitrary forcing functions. In contrast to standard methods which are abstract (see
Appendix B), our approach is intuitive and based on mechanics.
Example 2.4 Consider an undamped SDOF which is subjected to a constant forcing
function f (t) = q, t ≥ 0, and has the ICs (x 0 , ˙
x 0 ). The general solution has the form
x(t) = A cos(ω t) + B sin(ω t) + x p (t), t ≥ 0,
where the particular solution is x p (t) = q/
m ω 2 = q/k = x st . That this is a
particular solution results by checking that it satisfies the equation of motion ¨
x +
ω 2 x = q/m. We have ¨
x p + ω 2 x p = 0 + ω 2 (q/k) = q/m so that x p (t) = q/k
is a particular solution. The initial conditions x(0) = x 0 and ˙
x(0) = ˙
x 0 imply
A + x st = x 0 and B ω = ˙
x 0 so that
x(t) =
x 0 − x st
cos(ω t) +
˙
x 0
ω
sin(ω t) + x st .
For zero initial conditions, the displacement x(t) = x st
1 − cos(ω t)
is shown
in Fig. 2.5 for ω = π . Note that the dynamic amplification factor DAF =
max t {x(t)}/x st = 2. This means that the response of system to a suddenly applied
force q is twice that of the system if the force is applied statically.
Example 2.5 Consider a damped SDOF which is at rest at the initial time t = 0,
i.e., x 0 = 0 and ˙
x 0 = 0, and is subjected to a forcing function which is constant and
equal to q in a time interval (t 1 , t 2 ) and zero outside this interval (see Fig. 2.6, left
panel).
0
2
4
6
8
10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t
x(t)/x
st
Fig. 2.5 Displacement of an undamped SDOF system under a suddenly applied force