14
2 Single Degree of Freedom (SDOF) Systems
We conclude with the observations that (1) the general homogeneous solution
x h (t) has the physical meaning of free vibration and (2) the features of the free
vibration solution are determined by damping, i.e., x h (t) → 0 as time t → ∞ if
ζ > 0, x h (t) has no oscillation if ζ ≥ 1, x h (t) exhibits oscillation if ζ < 1, and
x h (t) is periodic with period 2 π/ω if ζ = 0 (see Eq. 2.20). The following subsection
examines in details under-damped SDOF systems.
2.4.2 General Homogeneous Solution (ζ < 1)
The general solution of Eq. 2.8 with f (t) = 0 and ζ < 1 constitutes the free
vibration solution of an under-damped SDOF system and has the expression x h (t)
in Eq. 2.20. These types of systems are commonly encountered in applications. The
section presents the standard form of the general homogeneous solution and its
amplitude-phase version. It also illustrates a method for estimating the damping
ratio ζ from measurements.
2.4.2.1 Standard Form
The constants A and B in the expression of x h (t) can be found by imposing
the initial conditions x h (0) = x 0 and ˙
x h (0) = ˙
x 0 . These conditions give
A = x 0 from the expression of x h (t) and −ζ ω A + B ω d = ˙
x 0 from
˙
x h (t) = −ζ ω e −ζ ω t
A cos(ω d t) + B sin(ω d t)
+ e −ζ ω t − A ω d sin(ω d t) +
B ω d cos(ω d t)
. This system of linear equations gives A = x 0 and
B =
˙
x 0 + ζ ω A
ω d
=
˙
x 0 + ζ ω x 0
ω d
so that
x h (t) = x free vibration (t) = e
−ζ ω t
x 0 cos(ω d t) +
˙
x 0 + ζ ω x 0
ω d
sin(ω d t)
.
(2.21)
This is the most general expression of the free vibration solution. It can be used
to find the free vibration solutions in special cases. For example, the free vibration
solution of undamped systems (c = 0 which implies ζ = 0) is
x h (t) = x free vibration (t) = x 0 cos(ω t) +
˙
x 0
ω
sin(ω t).
(2.22)
There is a significant difference between the free vibration solutions of Eqs. 2.21
and 2.22. The free vibration in Eq. 2.22 is periodic with period T = 2 π/ω while
2 Single Degree of Freedom (SDOF) Systems
We conclude with the observations that (1) the general homogeneous solution
x h (t) has the physical meaning of free vibration and (2) the features of the free
vibration solution are determined by damping, i.e., x h (t) → 0 as time t → ∞ if
ζ > 0, x h (t) has no oscillation if ζ ≥ 1, x h (t) exhibits oscillation if ζ < 1, and
x h (t) is periodic with period 2 π/ω if ζ = 0 (see Eq. 2.20). The following subsection
examines in details under-damped SDOF systems.
2.4.2 General Homogeneous Solution (ζ < 1)
The general solution of Eq. 2.8 with f (t) = 0 and ζ < 1 constitutes the free
vibration solution of an under-damped SDOF system and has the expression x h (t)
in Eq. 2.20. These types of systems are commonly encountered in applications. The
section presents the standard form of the general homogeneous solution and its
amplitude-phase version. It also illustrates a method for estimating the damping
ratio ζ from measurements.
2.4.2.1 Standard Form
The constants A and B in the expression of x h (t) can be found by imposing
the initial conditions x h (0) = x 0 and ˙
x h (0) = ˙
x 0 . These conditions give
A = x 0 from the expression of x h (t) and −ζ ω A + B ω d = ˙
x 0 from
˙
x h (t) = −ζ ω e −ζ ω t
A cos(ω d t) + B sin(ω d t)
+ e −ζ ω t − A ω d sin(ω d t) +
B ω d cos(ω d t)
. This system of linear equations gives A = x 0 and
B =
˙
x 0 + ζ ω A
ω d
=
˙
x 0 + ζ ω x 0
ω d
so that
x h (t) = x free vibration (t) = e
−ζ ω t
x 0 cos(ω d t) +
˙
x 0 + ζ ω x 0
ω d
sin(ω d t)
.
(2.21)
This is the most general expression of the free vibration solution. It can be used
to find the free vibration solutions in special cases. For example, the free vibration
solution of undamped systems (c = 0 which implies ζ = 0) is
x h (t) = x free vibration (t) = x 0 cos(ω t) +
˙
x 0
ω
sin(ω t).
(2.22)
There is a significant difference between the free vibration solutions of Eqs. 2.21
and 2.22. The free vibration in Eq. 2.22 is periodic with period T = 2 π/ω while