| In the following, assume that all systems have only
one degree of freedom
Question 1
A car plus driver travelling
over a rough road may be modelled as a 250 kg mass on a spring of stiffness 12 kNm-1.
The road has a "waviness" of amplitude 5 cm and wavelength 5 m. At what speeds
would the driver experience an amplitude greater than 10 cm, and what speed should be
exceeded to reduce the amplitude to less than 5 mm?
Answer: Between
3.9 and 6.7 ms-1; above 18.3 ms-1
Question 2
In order to determine the damping coefficient
produced by a viscous damper, and experiment is set up whereby the damper is mounted with
a spring between a fixed support and a mass of 500 grams. The amplitude of free vibration
decayed too quickly to be easily measured, so a sinusoidal force was applied to the mass
of variable frequency. The displacement of the mass was monitored with a suitable
transducer, and also the phase angle between the exciting force and the displacement. The
frequency was observed to be 1.3kHz when the phase angle was 90°, and the maximum
amplitude occurred when the frequency was 700Hz. What was the damping coefficient of the
damper?
Answer: 4.9
kNs m-1
Question 3
A machine weighs 1600 kg and
generates a disturbing force of 1200 N at a frequency of 1400 cycles min-1. It
is supported on four mountings each carrying an equal proportion of the total load. Each
mounting has a spring constant of 1.2 MNm-1, and negligible damping.
Calculate (a) the amplitude of
vertical motion of the machine, and (b) the amplitude of the force transmitted to the
foundations.
Answer: (a) 0.04 mm (b) 195 N
Question 4
A delicate instrument of mass 3.5 kg is to be
installed in an office. The office however is situated on the ground floor above the
basement in which the central heating pump is installed. It is found that when the pump is
operating at a speed of 1450 rev min-1 the table on which the instrument is
placed experiences unwanted vibration, the amplitude of which is observed to be 5 mm. To
prevent this the instrument is to be mounted on a spring-damper system, of stiffness 75 Nm-1
and damping coefficient, c = 2.3 Nsm-1. (a) What will be the amplitude
of vibration of the instrument when the pump is running? (b) What pump speeds should be
avoided if the amplitude is to be kept below 0.5 mm?
Answer:
(a) 0.02 mm (b) lower than 154 rpm
Question
5
A variable speed motor of mass 600
kg is supported on rubber mountings which have a stiffness of 250 kNm-1. The
mountings also provide a viscous damping force. The motor produces a sinusoidal force of
1.3W2sinWt, where W is the
rotational speed of the motor in rad s-1. The amplitude of vibration of the
system at resonance is observed to be 14 mm.
(a) Estimate the damping ratio of
the mountings
(b) Calculate the force transmitted
to the foundations when the motor speed is 450 rev min-1.
(c) Find the amplitude and phase lag
of the displacement of the system at this speed.
Answer: (a)
0.0774 (b) 706 N (c) 2.7 mm, 175.3°
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