2.4 Time Domain Analysis
27
The latter form of x(t) shows that the deformation x(t) increases in time and that
the increase is linear in ω t for ω t 1. It also shows that resonance does not mean
instant unbounded response. The response is finite in finite time intervals. Yet, it
increases in time and will eventually exceed the system strength.
2.4.5.2 Beating Phenomenon
Consider the oscillator and forcing function of the previous section and assume that
ν = ω but the difference between these frequencies is small, i.e., |ν − ω| | 1.
The displacement x(t) of Eq. 2.41 is valid for ν = ω and can be written in the
form
x(t)
=
x st
ω
1 − (ν/ω) 2
ω + ν
2
sin(ν t) − sin(ω t)
+
ω − ν
2
sin(ν t) + sin(ω t)
x st
ω
1 − (ν/ω) 2
ω 2 sin
(ν − ω) t
2
cos
(ν + ω) t
2
,
where the latter expression results by neglecting the second term in the square
bracket since |ν − ω| | 1 and by using the trigonometric identity
sin(ν t) − sin(ω t) = 2 sin
(ν − ω) t)/2
cos
(ν + ω) t)2
.
This approximation gives
x(t)
2 x st
1 − (ν/ω) 2 sin
(ν − ω) t
2
cos(ω t)
(2.43)
and shows that the high frequency component cos(ω t) with period 2 π/ω is
modulated by the slow varying amplitude, the term in the square bracket, with period
4 π/|ν − ω|.
Example 2.11 An undamped oscillator with unit mass and natural frequency ω = 6
is at rest at the initial time and is subjected to an harmonic force f (t) = q sin(ν t).
The left panel of Fig. 2.9 shows the evolution in time of x(t) for ν = ω = 6 so
that system is at resonance. The rate of increase of the amplitude of x is linear in
time in agreement with our previous comments. The right panel of Fig. 2.9 shows
the displacement x(t) for ν = 5.9 which differs slightly from ω = 6. The plot
illustrates the beating phenomenon. Note that the half of the long period of x(t) is
slightly larger than 60 in agreement with the period of the square bracket in Eq. 2.43
which is 4 π/|ν − ω| = 125. The small period is 2 π/6 1.
27
The latter form of x(t) shows that the deformation x(t) increases in time and that
the increase is linear in ω t for ω t 1. It also shows that resonance does not mean
instant unbounded response. The response is finite in finite time intervals. Yet, it
increases in time and will eventually exceed the system strength.
2.4.5.2 Beating Phenomenon
Consider the oscillator and forcing function of the previous section and assume that
ν = ω but the difference between these frequencies is small, i.e., |ν − ω| | 1.
The displacement x(t) of Eq. 2.41 is valid for ν = ω and can be written in the
form
x(t)
=
x st
ω
1 − (ν/ω) 2
ω + ν
2
sin(ν t) − sin(ω t)
+
ω − ν
2
sin(ν t) + sin(ω t)
x st
ω
1 − (ν/ω) 2
ω 2 sin
(ν − ω) t
2
cos
(ν + ω) t
2
,
where the latter expression results by neglecting the second term in the square
bracket since |ν − ω| | 1 and by using the trigonometric identity
sin(ν t) − sin(ω t) = 2 sin
(ν − ω) t)/2
cos
(ν + ω) t)2
.
This approximation gives
x(t)
2 x st
1 − (ν/ω) 2 sin
(ν − ω) t
2
cos(ω t)
(2.43)
and shows that the high frequency component cos(ω t) with period 2 π/ω is
modulated by the slow varying amplitude, the term in the square bracket, with period
4 π/|ν − ω|.
Example 2.11 An undamped oscillator with unit mass and natural frequency ω = 6
is at rest at the initial time and is subjected to an harmonic force f (t) = q sin(ν t).
The left panel of Fig. 2.9 shows the evolution in time of x(t) for ν = ω = 6 so
that system is at resonance. The rate of increase of the amplitude of x is linear in
time in agreement with our previous comments. The right panel of Fig. 2.9 shows
the displacement x(t) for ν = 5.9 which differs slightly from ω = 6. The plot
illustrates the beating phenomenon. Note that the half of the long period of x(t) is
slightly larger than 60 in agreement with the period of the square bracket in Eq. 2.43
which is 4 π/|ν − ω| = 125. The small period is 2 π/6 1.