UNIVERSITY OF GLASGOW

Degrees of MEng, BEng, MSc and BSc in Engineering

Vibration M4 (ENG4137)

??? May, 2014
?? – ??

Attempt THREE questions

The numbers in square brackets in the right-hand margin indicate the marks allotted to the
part of the question against which the mark is shown. These marks are for guidance only.

An electronic calculator may be used provided that it does not have a facility for either
textual storage or display, or for graphical display.

Continued overleaf
Page 1 of 6

Q1 A body of mass m, moment of inertia I (about an axis through S) and centre of mass
at C is supported by a spring system at S, which is distance r from C as shown in
Figure Q1. At S the spring system consists of a linear spring of stiffness and a
rotational spring of rotational stiffness . A force of amplitude 2.5 N and frequency
2.5 Hz is applied to the mass. The following data apply:

= 0.5kg = 0.2kgm2
= 150mm ℎ = 40mm

(a) Draw a free-body diagram for the vibrating system.
[3]

(b) Hence write the equations of motion in matrix form.
[3]

(c) Given that the natural frequencies of the system are 3.8 Hz and 19.4 Hz,
derive the equations of motion by normal mode analysis in the form

̈ + 2 =
[15]

(d) Determine the principal coordinates in the form

= 1(2 − 2)

And hence, for small values of angle , calculate the linear and angular
displacement amplitudes of oscillation of the centre of mass C.
[4]

Figure Q1

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Q2 A machine supported on a factory floor is exciting a resonance condition. An
undamped vibration absorber is to be mounted below the factory floor in order to
remove the resonance condition.

For vibration analysis, the machine and support floor may be regarded as a lumped
mass of 9000 kg, with the support floor sitting on foundations that can be modelled
as springs of total stiffness 12x106 N/m.

If the operating range of the machine is 320 – 400 rpm, determine the minimum
value of absorber mass that will ensure satisfactory operation.
[25]

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Page 3 of 6

Q3 A balance board is used for fitness training. The balance board, shown in Figure Q3,
consists of a uniform square plate attached to ground by four identical load cells,
which act as linear springs in the z direction. The locations of the load cells are at
each corner of the square plate such that their top centres make a square of side 0.5
m. The balance board has mass, m = 2 kg, and the load cells have equal linear
stiffness, k = 500 N/m. The moments of inertia of the plate, about an axis through
the plate centre in the x direction and about an axis through the plate centre in the y
direction, are both given by I = 0.1 kgm2.

(a) Define a set of generalised coordinates for the plate.
[2]

(b) Derive the system’s total kinetic energy, T.
[2]

(c) Derive the systems total potential function, U.
[8]

(d) Hence, write down the equations of motion in matrix form using Lagrange’s
equation as given by:
0=

+

−



iii q
U
q
T
q
T
dt
d

[9]

(e) Comment on the form of the equations of motion. Calculate and comment on
the natural frequencies and describe and sketch the mode shapes. Briefly state
why practical measurement values of the natural frequencies would differ
from these calculated values?
[4]

Figure Q3

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Page 4 of 6

Q4 Noise measurements are carried out in a factory to assess the risk of noise exposure
by the factory workers. The result of the octave band analysis is given as follows:

Centre Frequency (Hz) 63 125 250 500 1k 2k 4k 8k
Band SPL (dB(LIN)) 110 110 108 112 106 115 106 97

(a) Calculate the overall linear sound pressure level in both deciBels and Pascals.
[5]

(b) Show that the bandwidth (∆f) of an octave band filter can be written in terms
of the filter centre frequency (Fc) as:

∆ =
√2

Calculate the spectrum level in the 1 kHz octave band.
[8]

(c) Why is an A-weighting network used in acoustic noise measurements?

The workers in the factory, while exposed to the above noise, wear hearing
protectors. The protectors have an attenuation characteristic which rises from
15 dB at 63 Hz by 2 dB/octave up to 8 kHz. Approximating the dB(A)
network to a fall of 4 dB/octave below 4kHz, calculate the overall A-weighted
sound pressure level experience by a worker in this factory.

Comment on any risk to the worker’s hearing.
[12]

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Page 5 of 6

Q5
(a) When supporting a structure for experimental modal analysis a suspension
technique is sometimes used. How should the relationship between the rigid-
body and structural modes be managed to minimise errors in this case?
[2]

(b) When a structure is suspended the suspension points are often placed at the
nodes, and when the structure is forced by a vibrational exciter flexible
connecting rods are often used. Why are these techniques adopted?
[2]

(c) Transducers on the surface of the structure must be attached to prevent them
falling off, but an insufficiently stiff attachment can act as a low-pass filter.
What does this mean?
[2]

(d) As the frequency of the forcing function is increased, the phase angle between
the forcing function and the response is seen to rise from near-zero to almost
180°. Assuming that the vibration is lightly damped, at approximately what
phase angle would you expect the output to reach (i) half power and (ii)
maximum power?
[2]

(e) If a transduction system does not sample the behaviour of the structure at a
sufficiently high frequency, a discretisation error called ‘aliasing’ may occur.
Briefly discuss and illustrate, using a diagram, how aliasing takes place.
[3]

(f) Given that the equation of motion of a forced spring-mass-damper system
may be written as ̈ + ̇ + = (), and further expressing () and
() as and , show that the transfer function may be expressed
as:

() = 1( − 2) + ()

[5]

(g) Further separate the transfer function in part (f) into real and imaginary parts.
[4]

(f) Using the real and imaginary parts determined in part (g), derive expressions
for the amplitude and phase angle of the system.
[5]

End of question paper
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