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3 Forced Vibration of Single Degree of Freedom System
Exercise 3
3.1 A machine weighing 600 N is supported by springs of stiffness k = 20 N/mm
and has a damper attached, whose coefficient is c =0 .010 N-s/mm. A harmonic
force of amplitude 20 N is applied by the machine. Determine the maximum
and resonant amplitudes of steady-state vibration.
3.2 With the help of a harmonic-loading machine, the mass, damping and stiffness
of a SDF system can be determined. But for so doing, it has to be operated twice
at two different frequencies. If for a single bay portal, the machine was operated
at frequencies ω 1 = 20 rad/s and ω 2 = 30 rad/s with a force amplitude of
2.5 kN, the response amplitude and phase relationships measured for the two
cases were
x 1 = 3.34 × 10
− 3 mm and φ 1 = 14
◦
x 2 = 6.94 × 10
− 3 mm and φ 2 = 52
◦
Determine the mass, stiffness and damping of the SDF system.
3.3 For the basic SDF system of Fig. 2.4, m = 500 kg, k =5000 N and c = 100 N
s/mm. If the system is started from rest, determine the static displacement after
4 cycles.
3.4 A viscously damped mass of 2 kg undergoes a resonant amplitude of 1.4 m
with a period of 0.25 s when subjected to a harmonically excited force of 25 kg.
Determine the damping coefficient.
3.5 If for problem 1, the harmonic force has a frequency of 5 Hz, what will be
the percentage increase in the amplitude of forced vibration when damping is
removed?
3.6 A SDF system having viscous damping has a spring of stiffness 500 N/m.
When the weight is displaced and released, the period of vibration is 2 s and
ratio of successive amplitudes is 4–1. Determine the amplitude of the motion
and the phase angle when a force F = 4 sin 4t is applied to the system.
3.7 The mass of a SDF system is subjected to a force Cω
2 sin ωt. Allowing
viscous damping, obtain an expression for steady-state forced vibration. Also,
determine the value of ω, at which the amplitude is maximum, in terms of
damping ratio.
3.8 A harmonic motion is applied to the system shown in the figure. Derive the
equations of motion.
3 Forced Vibration of Single Degree of Freedom System
Exercise 3
3.1 A machine weighing 600 N is supported by springs of stiffness k = 20 N/mm
and has a damper attached, whose coefficient is c =0 .010 N-s/mm. A harmonic
force of amplitude 20 N is applied by the machine. Determine the maximum
and resonant amplitudes of steady-state vibration.
3.2 With the help of a harmonic-loading machine, the mass, damping and stiffness
of a SDF system can be determined. But for so doing, it has to be operated twice
at two different frequencies. If for a single bay portal, the machine was operated
at frequencies ω 1 = 20 rad/s and ω 2 = 30 rad/s with a force amplitude of
2.5 kN, the response amplitude and phase relationships measured for the two
cases were
x 1 = 3.34 × 10
− 3 mm and φ 1 = 14
◦
x 2 = 6.94 × 10
− 3 mm and φ 2 = 52
◦
Determine the mass, stiffness and damping of the SDF system.
3.3 For the basic SDF system of Fig. 2.4, m = 500 kg, k =5000 N and c = 100 N
s/mm. If the system is started from rest, determine the static displacement after
4 cycles.
3.4 A viscously damped mass of 2 kg undergoes a resonant amplitude of 1.4 m
with a period of 0.25 s when subjected to a harmonically excited force of 25 kg.
Determine the damping coefficient.
3.5 If for problem 1, the harmonic force has a frequency of 5 Hz, what will be
the percentage increase in the amplitude of forced vibration when damping is
removed?
3.6 A SDF system having viscous damping has a spring of stiffness 500 N/m.
When the weight is displaced and released, the period of vibration is 2 s and
ratio of successive amplitudes is 4–1. Determine the amplitude of the motion
and the phase angle when a force F = 4 sin 4t is applied to the system.
3.7 The mass of a SDF system is subjected to a force Cω
2 sin ωt. Allowing
viscous damping, obtain an expression for steady-state forced vibration. Also,
determine the value of ω, at which the amplitude is maximum, in terms of
damping ratio.
3.8 A harmonic motion is applied to the system shown in the figure. Derive the
equations of motion.