2.4 Time Domain Analysis
11
Note that undamped systems are not realistic since all mechanical/structural
systems dissipate energy during deformation so that damping is always present,
although it can be very small.
If ζ ∈ (0, 1), the displacement has the form
x(t) = x h (t) + x p (t) = e
−ζ ω t
A cos(ω d t) + B sin(ω d t)
+ x p (t), t ≥ 0,
(2.13)
where ω d = ω
1 − ζ 2 . The bracket in the expression of the homogeneous
solution is periodic with period T d = 2 π/ω d . However, the homogeneous
solution is not periodic. It decays to zero as time increases indefinitely so that
x(t) x p (t) for large times. This limit case is referred to as the steady-state
solution. Similar considerations hold for damping ratios ζ ≥ 1 (see Sect. 2.4.1).
3. The ICs have to be imposed on the system solution x(t) = x h (t) + x p (t), i.e.,
x(0) = x h (0) + x p (0) = x 0 and ˙
x(0) = ˙
x h (0) + ˙
x p (0) = ˙
x 0 . These conditions
deliver the undetermined constants in the expression of the general solution x(t).
For example, consider a SDOF system subjected to a constant action, i.e., f (t)
is a constant q so that Eq. 2.8 becomes
¨
x(t) + 2 ζ ω ˙
x(t) + ω
2 x(t) = q/m, t ≥ 0.
(2.14)
This equation suggests that we can assume that x p (t) = c is a constant.
Indeed, this trial particular solution and Eq. 2.14 give ω 2 c = q/m so that
c = q/
m ω 2 = q/k = x st is the deformation of the system under the static
load q. The general solution is
x(t) = e
−ζ ω t
A cos(ω d t) + B sin(ω d t)
+ x st , t ≥ 0,
(2.15)
so that
x(t) = e
−ζ ω t
(x 0 − x st ) cos(ω d t) +
˙
x 0 + ζ ω (x 0 − x st )
ω d
sin(ω d t)
+ x st , t ≥ 0.
(2.16)
by imposing the initial conditions x(0) = x 0 and ˙
x(0) = ˙
x 0 which require A +
x st = x 0 and −ζ ω A + B ω d = ˙
x 0 or A = x 0 − x st and B =
˙
x 0 + ζ ω (x 0 −
x st )
/ω d . The steady-state solution x ss (t) = x st since x(t) → x st as t → ∞.
The rate of the convergence to x st is controlled by the product ζ ω.
Example 2.2 The free vibration solution of the oscillator in Eq. 2.8 and the initial
conditions (x 0 , ˙
x 0 ) result from Eq. 2.16 by setting x st = 0 and have the expression
x h (t) = e
−ζ ω t
x 0 cos(ω d t) +
˙
x 0 + ζ ω x 0
ω d
sin(ω d t)
.
11
Note that undamped systems are not realistic since all mechanical/structural
systems dissipate energy during deformation so that damping is always present,
although it can be very small.
If ζ ∈ (0, 1), the displacement has the form
x(t) = x h (t) + x p (t) = e
−ζ ω t
A cos(ω d t) + B sin(ω d t)
+ x p (t), t ≥ 0,
(2.13)
where ω d = ω
1 − ζ 2 . The bracket in the expression of the homogeneous
solution is periodic with period T d = 2 π/ω d . However, the homogeneous
solution is not periodic. It decays to zero as time increases indefinitely so that
x(t) x p (t) for large times. This limit case is referred to as the steady-state
solution. Similar considerations hold for damping ratios ζ ≥ 1 (see Sect. 2.4.1).
3. The ICs have to be imposed on the system solution x(t) = x h (t) + x p (t), i.e.,
x(0) = x h (0) + x p (0) = x 0 and ˙
x(0) = ˙
x h (0) + ˙
x p (0) = ˙
x 0 . These conditions
deliver the undetermined constants in the expression of the general solution x(t).
For example, consider a SDOF system subjected to a constant action, i.e., f (t)
is a constant q so that Eq. 2.8 becomes
¨
x(t) + 2 ζ ω ˙
x(t) + ω
2 x(t) = q/m, t ≥ 0.
(2.14)
This equation suggests that we can assume that x p (t) = c is a constant.
Indeed, this trial particular solution and Eq. 2.14 give ω 2 c = q/m so that
c = q/
m ω 2 = q/k = x st is the deformation of the system under the static
load q. The general solution is
x(t) = e
−ζ ω t
A cos(ω d t) + B sin(ω d t)
+ x st , t ≥ 0,
(2.15)
so that
x(t) = e
−ζ ω t
(x 0 − x st ) cos(ω d t) +
˙
x 0 + ζ ω (x 0 − x st )
ω d
sin(ω d t)
+ x st , t ≥ 0.
(2.16)
by imposing the initial conditions x(0) = x 0 and ˙
x(0) = ˙
x 0 which require A +
x st = x 0 and −ζ ω A + B ω d = ˙
x 0 or A = x 0 − x st and B =
˙
x 0 + ζ ω (x 0 −
x st )
/ω d . The steady-state solution x ss (t) = x st since x(t) → x st as t → ∞.
The rate of the convergence to x st is controlled by the product ζ ω.
Example 2.2 The free vibration solution of the oscillator in Eq. 2.8 and the initial
conditions (x 0 , ˙
x 0 ) result from Eq. 2.16 by setting x st = 0 and have the expression
x h (t) = e
−ζ ω t
x 0 cos(ω d t) +
˙
x 0 + ζ ω x 0
ω d
sin(ω d t)
.