244
6 Oscillations
where t c = m/r
A(t) = A 0 e
−t/2t c
(6.59)
Quality factor
Q = ωt c = ωm/r
(6.60)
ω
= ω 0
1 −
1
4Q 2
(6.61)
The value of quality factor indicates the sharpness of resonance.
Q =
ω 0
ω 2 − ω 1
(6.62)
where ω 0 is the resonance angular frequency and ω 2 and ω 1 are, respectively, the
two angular frequencies above and below resonance at which the average power has
dropped to one-half its resonance value. (Fig. 6.6).
Fig. 6.6 Resonance
frequency curve, ω 0 is the
resonance angular frequency.
ω 1 and ω 2 are defined in the
text
Forced vibrations are set up by a periodic force F cos ωt.
Equation of motion of a particle of mass m
md
2 x
dt 2 +
r dx
dt
+ kx = F cos ωt
(6.63)
or
d
2 x
dt 2 + 2b
dx
dt
+ ω
2
0 x = p cos ωt
(6.64)
where
k/m = ω
2
0 , r/m = 2b and F/m = p
(6.65)
6 Oscillations
where t c = m/r
A(t) = A 0 e
−t/2t c
(6.59)
Quality factor
Q = ωt c = ωm/r
(6.60)
ω
= ω 0
1 −
1
4Q 2
(6.61)
The value of quality factor indicates the sharpness of resonance.
Q =
ω 0
ω 2 − ω 1
(6.62)
where ω 0 is the resonance angular frequency and ω 2 and ω 1 are, respectively, the
two angular frequencies above and below resonance at which the average power has
dropped to one-half its resonance value. (Fig. 6.6).
Fig. 6.6 Resonance
frequency curve, ω 0 is the
resonance angular frequency.
ω 1 and ω 2 are defined in the
text
Forced vibrations are set up by a periodic force F cos ωt.
Equation of motion of a particle of mass m
md
2 x
dt 2 +
r dx
dt
+ kx = F cos ωt
(6.63)
or
d
2 x
dt 2 + 2b
dx
dt
+ ω
2
0 x = p cos ωt
(6.64)
where
k/m = ω
2
0 , r/m = 2b and F/m = p
(6.65)