10
2 Single Degree of Freedom (SDOF) Systems
2.4 Time Domain Analysis
Consider the equation of motion in the x-coordinate, i.e., Eq. 2.8. The general
solution x(t) of this equation is the sum
x(t) = x h (t) + x p (t)
(2.9)
of the general solution of the homogeneous equation x h (t), i.e., Eq. 2.8 with
f (t) = 0, and a particular solution x p (t) of the inhomogeneous equation, i.e.,
Eq. 2.8 with the actual input f (t). The functions x h (t) and x p (t) are referred to as
general homogeneous and particular inhomogeneous solutions. The homogenous
solution is known up to some constants which need to be determined. The particular
solution is any function x p (t) which satisfies the equation of motion, i.e., ¨
x p (t) +
2 ζ ω ˙
x p (t) + ω 2 x p (t) = f (t)/m is satisfied at all times. This solution can be found
simply in some cases but has to be constructed in most realistic situations.
The formulation of the equations of motion for SDOF systems and the solutions
of these equations require to select systems of coordinates, account for solution
properties, and satisfy initial conditions. The following items provide details on
these topics.
1. The equations of motions are meaningless in the absence of system of coordinates. Moreover, the functional forms of these equations and their solutions
depend on the system of coordinates. For example, in the y-coordinate defined
by y(t) = x(t) + a, the equations of motion Eqs. 2.7 and 2.8 of the SDOF system
in Fig. 2.2 take the form
m ¨
y(t) = f (t) − k
y(t) − a
− c ˙
y(t)
(2.10)
or, equivalently,
¨
y(t) + 2 ζ ω ˙
y(t) + ω
2 x(t) =
f (t) + k a
/m
(2.11)
with the initial conditions y(0) = a + x 0 and ˙
y(0) = ˙
x 0 . Note that, if f (t) = 0,
the SDOF system is in free vibration in the x-coordinate and forced vibration in
the y-coordinate.
2. The functional forms of the general solutions of damped (ζ > 0) and undamped
(ζ = 0) SDOF systems have similarities and notable differences.
If ζ = 0, the general homogeneous solution has the expression x h (t) =
A cos(ω t) + B sin(ω t). It is periodic with period T = 2 π/ω and oscillates
indefinitely. The displacement function of these systems is
x(t) = A cos(ω t) + B sin cos(ω t) + x p (t), t ≥ 0.
(2.12)
2 Single Degree of Freedom (SDOF) Systems
2.4 Time Domain Analysis
Consider the equation of motion in the x-coordinate, i.e., Eq. 2.8. The general
solution x(t) of this equation is the sum
x(t) = x h (t) + x p (t)
(2.9)
of the general solution of the homogeneous equation x h (t), i.e., Eq. 2.8 with
f (t) = 0, and a particular solution x p (t) of the inhomogeneous equation, i.e.,
Eq. 2.8 with the actual input f (t). The functions x h (t) and x p (t) are referred to as
general homogeneous and particular inhomogeneous solutions. The homogenous
solution is known up to some constants which need to be determined. The particular
solution is any function x p (t) which satisfies the equation of motion, i.e., ¨
x p (t) +
2 ζ ω ˙
x p (t) + ω 2 x p (t) = f (t)/m is satisfied at all times. This solution can be found
simply in some cases but has to be constructed in most realistic situations.
The formulation of the equations of motion for SDOF systems and the solutions
of these equations require to select systems of coordinates, account for solution
properties, and satisfy initial conditions. The following items provide details on
these topics.
1. The equations of motions are meaningless in the absence of system of coordinates. Moreover, the functional forms of these equations and their solutions
depend on the system of coordinates. For example, in the y-coordinate defined
by y(t) = x(t) + a, the equations of motion Eqs. 2.7 and 2.8 of the SDOF system
in Fig. 2.2 take the form
m ¨
y(t) = f (t) − k
y(t) − a
− c ˙
y(t)
(2.10)
or, equivalently,
¨
y(t) + 2 ζ ω ˙
y(t) + ω
2 x(t) =
f (t) + k a
/m
(2.11)
with the initial conditions y(0) = a + x 0 and ˙
y(0) = ˙
x 0 . Note that, if f (t) = 0,
the SDOF system is in free vibration in the x-coordinate and forced vibration in
the y-coordinate.
2. The functional forms of the general solutions of damped (ζ > 0) and undamped
(ζ = 0) SDOF systems have similarities and notable differences.
If ζ = 0, the general homogeneous solution has the expression x h (t) =
A cos(ω t) + B sin(ω t). It is periodic with period T = 2 π/ω and oscillates
indefinitely. The displacement function of these systems is
x(t) = A cos(ω t) + B sin cos(ω t) + x p (t), t ≥ 0.
(2.12)