University of Exeter
ENG3015 Structural Dynamics
Year: 2023/24 and onwards
Coursework 1
Q1: Figure Q1 shows a structural frame with rigid cap beam AB vibrating horizontally. The
beam is supported vertically by three columns (AE, GF and BH) and restrained horizontally
by a system of horizontal steel tubes (BC and CD) supported by another vertical column (CI).
Apart from the steel tubes, all other structural members may be assumed not to deform
axially.
Figure Q1
a) Sketch the development of the equivalent SDOF system for horizontal vibration of
beam AB
b) Using the notation provided, derive expression for the undamped natural frequency
corresponding to the horizontal vibration of the cap beam.
c) Using the developed expression and the following design data, calculate the natural
frequency of this SDOF system in the horizontal direction
m’=12,000 kg/m
E=210 GPa
I=104170 cm4
D=0.5 m; d=0.4 m
h1=4 m; h2=6 m
l1=2 m; l2=4 m
2
Q2: The system shown in Figure Q2 may be assumed to have rigid levers, no mass and
frictionless pivot. For small vertical vibrations of the point mass:
Figure Q2
a) Derive expression for the damping ratio.
b) Derive expression for damped natural frequency.
c) Derive expression for the critical damping coefficient.
Note: all expressions should be given in terms of variables m, c and k, describing the physical
properties of the system, and p, q and r relating to the geometry of the system. There are no
specific numerical values associated with these six variables.
3
Q3: In a type approval test, a helicopter (Figure Q3) is dropped under gravity from a height,
ℎ, with its rotors stationary. The pilot's seat is mounted on springs of combined stiffness