2.6 Energy Method and Free Torsional Vibration
45
The maximum potential energy of the system is
U max =
1
2
kx
2
max =
1
2
k A
2
Equating the maximum kinetic energy to the maximum potential energy, we get
1
2
k A
2
=
1
2
m + ρ
L
3
(Ap)
2
or
p =
k
m + ρ
L
3
The natural frequency is given by
f =
1
2 π
k
m + ρ
L
3
2.7 Logarithmic Decrement
Logarithmic decrement is a concept used to measure the damping factor of a
SDF system. It is based on the measure of the decay of two successive amplitudes. Logarithmic decrement is defined as the natural logarithm of two successive
amplitudes.
At any instant t = t 1 , the amplitude from Eq. (2.31) is given by (Fig. 2.25)
x 1 = e
− n t 1 (C 1 cos
p 2 − n 2 t 1 + C 2 sin
p 2 − n 2 t 1 )
(2.56)
Next positive amplitude will occur at t = t 1 + T = t 1 +
2π
√
p 2 −n 2
The value of this amplitude at t = t 1 + T is given by
x 2 = exp
− n
t 1 +
2π
p 2 − n 2
C 1 cos
p 2 − n 2
t 1 +
2π
p 2 − n 2
+ C 2 sin
p 2 − n 2
t 1 +
2π
p 2 − n 2
(2.57)
or
45
The maximum potential energy of the system is
U max =
1
2
kx
2
max =
1
2
k A
2
Equating the maximum kinetic energy to the maximum potential energy, we get
1
2
k A
2
=
1
2
m + ρ
L
3
(Ap)
2
or
p =
k
m + ρ
L
3
The natural frequency is given by
f =
1
2 π
k
m + ρ
L
3
2.7 Logarithmic Decrement
Logarithmic decrement is a concept used to measure the damping factor of a
SDF system. It is based on the measure of the decay of two successive amplitudes. Logarithmic decrement is defined as the natural logarithm of two successive
amplitudes.
At any instant t = t 1 , the amplitude from Eq. (2.31) is given by (Fig. 2.25)
x 1 = e
− n t 1 (C 1 cos
p 2 − n 2 t 1 + C 2 sin
p 2 − n 2 t 1 )
(2.56)
Next positive amplitude will occur at t = t 1 + T = t 1 +
2π
√
p 2 −n 2
The value of this amplitude at t = t 1 + T is given by
x 2 = exp
− n
t 1 +
2π
p 2 − n 2
C 1 cos
p 2 − n 2
t 1 +
2π
p 2 − n 2
+ C 2 sin
p 2 − n 2
t 1 +
2π
p 2 − n 2
(2.57)
or