2.6 Energy Method and Free Torsional Vibration
43
Further
a + b = L
(b)
Combining Eqs. (a) and (b) gives
a =
I P 2 L
I P 1 + I P 2
and b =
I P 1 L
I P 1 + I P 2
Therefore, frequency of torsional vibration of the shaft is
f =
1
2π
k 1
I P 1
=
1
2π
π d 4 G (I P 1 + I P 2 )
32L I P 1 I P 2
It may be noted that
k 1 =
π d
4 G
32 a
=
π d
4 G
32L I P 2
(I P 1 + I P 2 )
2.6.2 Rayleigh’s Method
A real system is rarely of single degree of freedom. It is treated either as multiple
degrees of freedom system or a system having a distributed mass. The calculation
of natural frequencies can be simplified by using Rayleigh’s method. The vibration amplitudes are assumed a priori. On the basis of this assumed distribution, it
is possible to calculate natural frequencies of the system with the help of energy
principles. The results thus obtained are not exact.
Another interpretation of energy principle also exists. The natural frequency is a
function of the rate of change of kinetic and potential energies of a system. As such,
when the mass passes through the mean position, its potential energy is zero. The
kinetic energy at that instant is maximum and is equal to the total mechanical energy.
When the mass is at its maximum displacement, its kinetic energy is zero and the
total mechanical energy consists only of the potential energy. As the total energy of
the system is constant [Eq. (2.41)], we can write
T E max = U max
(2.55)
Example 2.15 Determine the natural frequency of an undamped spring–mass system
by incorporating the effect of mass of the spring (Fig. 2.24).
43
Further
a + b = L
(b)
Combining Eqs. (a) and (b) gives
a =
I P 2 L
I P 1 + I P 2
and b =
I P 1 L
I P 1 + I P 2
Therefore, frequency of torsional vibration of the shaft is
f =
1
2π
k 1
I P 1
=
1
2π
π d 4 G (I P 1 + I P 2 )
32L I P 1 I P 2
It may be noted that
k 1 =
π d
4 G
32 a
=
π d
4 G
32L I P 2
(I P 1 + I P 2 )
2.6.2 Rayleigh’s Method
A real system is rarely of single degree of freedom. It is treated either as multiple
degrees of freedom system or a system having a distributed mass. The calculation
of natural frequencies can be simplified by using Rayleigh’s method. The vibration amplitudes are assumed a priori. On the basis of this assumed distribution, it
is possible to calculate natural frequencies of the system with the help of energy
principles. The results thus obtained are not exact.
Another interpretation of energy principle also exists. The natural frequency is a
function of the rate of change of kinetic and potential energies of a system. As such,
when the mass passes through the mean position, its potential energy is zero. The
kinetic energy at that instant is maximum and is equal to the total mechanical energy.
When the mass is at its maximum displacement, its kinetic energy is zero and the
total mechanical energy consists only of the potential energy. As the total energy of
the system is constant [Eq. (2.41)], we can write
T E max = U max
(2.55)
Example 2.15 Determine the natural frequency of an undamped spring–mass system
by incorporating the effect of mass of the spring (Fig. 2.24).