64
3 Forced Vibration of Single Degree of Freedom System
Fig. 3.4 Variation of phase
angle with frequency ratio
η =
1 − 2ζ 2
ζ being a small quantity, the maximum value of μ is obtained when η ∼ = 1. It can
be seen in Fig. 3.3 that when the frequency of the external force is nearly equal to
the frequency of the system in free vibration, magnification factor increases rapidly.
The maximum value of the magnification factor is highly sensitive to the damping
of the system. The condition of maximum amplitude is known as the condition of
resonance.
The phase angle φ given by Eq. (3.11b) is rewritten as follows
ϕ = tan
− 1
2ζ η
1 − η 2
(3.20)
Equation (3.20) is plotted with ζ as a parameter in Fig. 3.4. For small values of
η, the phase angle is 90°. At larger values of η, φ tends to approach 180°.
The following conclusions can be drawn from the study of forced vibrations:
(1) The free vibration part is transient and vanishes, while the forced part persists.
(2) With the increase of ζ, the magnification factor μ decreases.
(3) The magnitude of the maximum value of the magnification factor is very
sensitive to the value of ζ.
(4) The magnification factor assumes significant values for 0.5 < η < 1.5, and the
maximum value is obtained when η ∼ = 1.
(5) Steady-state vibration is independent of the initial conditions in the system.
Example 3.1 A steel rigid frame of Fig. 3.5 supports a rotating machine, which
exerts a horizontal force at the girder level of 50,000 sin 11t N. Assuming 4%
critical damping, what is the steady-state amplitude of vibration? I for columns
= 1500 × 10
− 7 m
4 , E = 21 × 10
10 N/m
2
3 Forced Vibration of Single Degree of Freedom System
Fig. 3.4 Variation of phase
angle with frequency ratio
η =
1 − 2ζ 2
ζ being a small quantity, the maximum value of μ is obtained when η ∼ = 1. It can
be seen in Fig. 3.3 that when the frequency of the external force is nearly equal to
the frequency of the system in free vibration, magnification factor increases rapidly.
The maximum value of the magnification factor is highly sensitive to the damping
of the system. The condition of maximum amplitude is known as the condition of
resonance.
The phase angle φ given by Eq. (3.11b) is rewritten as follows
ϕ = tan
− 1
2ζ η
1 − η 2
(3.20)
Equation (3.20) is plotted with ζ as a parameter in Fig. 3.4. For small values of
η, the phase angle is 90°. At larger values of η, φ tends to approach 180°.
The following conclusions can be drawn from the study of forced vibrations:
(1) The free vibration part is transient and vanishes, while the forced part persists.
(2) With the increase of ζ, the magnification factor μ decreases.
(3) The magnitude of the maximum value of the magnification factor is very
sensitive to the value of ζ.
(4) The magnification factor assumes significant values for 0.5 < η < 1.5, and the
maximum value is obtained when η ∼ = 1.
(5) Steady-state vibration is independent of the initial conditions in the system.
Example 3.1 A steel rigid frame of Fig. 3.5 supports a rotating machine, which
exerts a horizontal force at the girder level of 50,000 sin 11t N. Assuming 4%
critical damping, what is the steady-state amplitude of vibration? I for columns
= 1500 × 10
− 7 m
4 , E = 21 × 10
10 N/m
2