2.3 Equation of Motion
9
The system in Fig. 2.2 has a single degree of freedom since the position of its
mass is completely defined at all times by, e.g., the function x(t) in the x-coordinate
or the function y(t) in the y-coordinate.
2.3 Equation of Motion
Newton’s law (mass times acceleration equals the applied actions) for the mass in
Fig. 2.2 in the x-coordinate gives the following differential equation:
m ¨
x(t) = f (t) − k x(t) − c ˙
x(t)
(2.7)
for the displacement function x(t). It is assumed that the forcing function f (t) is
positive, i.e., it acts from left to right in the positive direction of the x-coordinate.
The displacement function x(t) satisfies an ordinary differential equation of order
2 with constant coefficients since k and c are assumed to be constant. The single
and double dots denote first- and second-order time derivatives. With the notations
ω 2 = k/m and 2 ζ ω = c/m, the above equation takes the form
¨
x(t) + 2 ζ ω ˙
x(t) + ω
2 x(t) = f (t)/m.
(2.8)
The notation ω 2 = k/m makes sense since k and c are positive. We will see that
the parameters ω > 0 and ζ ≥ 0 have precise physical meaning. The solution of
Eqs. 2.7 and 2.8 requires initial conditions (ICs) which consists of the displacement
x(0) = x 0 and the velocity ˙
x(0) = ˙
x 0 at the initial time, where x 0 and ˙
x 0 are
specified numbers.
The differential equations 2.7 and 2.8 describe the motion of most general
SDOF systems, i.e., the motion of damped oscillators in forced vibration if
f (t) = 0. Special cases of these equations of interest in applications are considered
extensively in this chapter. For example, the SDOF system is said to be in free
vibration if there is no applied force, i.e., f (t) = 0, and its vibrations are undamped
if c = 0 or, equivalently, ζ = 0.
The solution of Eqs. 2.7 and 2.8 can be obtained by analysis in the time domain
and the frequency domain. We begin with the time domain analysis and present
several approaches for calculating the displacement function x(t). The latter part of
this chapter deals with the frequency domain analysis. This approach is particularly
useful for SDOF systems which exhibit steady-state motion, i.e., the motions
observed in some systems for large times. A formal definition of the steady-state
response is given shortly.
9
The system in Fig. 2.2 has a single degree of freedom since the position of its
mass is completely defined at all times by, e.g., the function x(t) in the x-coordinate
or the function y(t) in the y-coordinate.
2.3 Equation of Motion
Newton’s law (mass times acceleration equals the applied actions) for the mass in
Fig. 2.2 in the x-coordinate gives the following differential equation:
m ¨
x(t) = f (t) − k x(t) − c ˙
x(t)
(2.7)
for the displacement function x(t). It is assumed that the forcing function f (t) is
positive, i.e., it acts from left to right in the positive direction of the x-coordinate.
The displacement function x(t) satisfies an ordinary differential equation of order
2 with constant coefficients since k and c are assumed to be constant. The single
and double dots denote first- and second-order time derivatives. With the notations
ω 2 = k/m and 2 ζ ω = c/m, the above equation takes the form
¨
x(t) + 2 ζ ω ˙
x(t) + ω
2 x(t) = f (t)/m.
(2.8)
The notation ω 2 = k/m makes sense since k and c are positive. We will see that
the parameters ω > 0 and ζ ≥ 0 have precise physical meaning. The solution of
Eqs. 2.7 and 2.8 requires initial conditions (ICs) which consists of the displacement
x(0) = x 0 and the velocity ˙
x(0) = ˙
x 0 at the initial time, where x 0 and ˙
x 0 are
specified numbers.
The differential equations 2.7 and 2.8 describe the motion of most general
SDOF systems, i.e., the motion of damped oscillators in forced vibration if
f (t) = 0. Special cases of these equations of interest in applications are considered
extensively in this chapter. For example, the SDOF system is said to be in free
vibration if there is no applied force, i.e., f (t) = 0, and its vibrations are undamped
if c = 0 or, equivalently, ζ = 0.
The solution of Eqs. 2.7 and 2.8 can be obtained by analysis in the time domain
and the frequency domain. We begin with the time domain analysis and present
several approaches for calculating the displacement function x(t). The latter part of
this chapter deals with the frequency domain analysis. This approach is particularly
useful for SDOF systems which exhibit steady-state motion, i.e., the motions
observed in some systems for large times. A formal definition of the steady-state
response is given shortly.