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
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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.
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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 𝑘,
and of dampers of combined rate 𝑐. The combined mass of the (dummy) pilot and seat is 𝑚.
Figure Q3
a) Using clear sketches and maximum 100 words, show that the equation of motion of the
pilot and seat after impact with the ground, which occurs at t=0 s, is:
𝑚w**9;̈(w**5;) + 𝑐w**9;̇(w**5;) + 𝑘w**9;(w**5;) = 0
with the following initial conditions: w**9;(0) = −
𝑚𝑔
𝑘
and w**9;̇(0) = √2𝑔ℎ, the positive
coordinate direction being as shown in Figure Q3. In the given equation of motion w**9;(w**5;) is
the dynamic displacement from the static equilibrium position of mass 𝑚 after the
helicopter strikes the ground.
b) Consider the case where m=100 kg, k=2.5x105 N/m, c=3000 Ns/m and h=5 m. Using NDOF
software, calculate the maximum displacement, velocity and acceleration experienced by
the pilot after the helicopter strikes the ground. Present relevant annotated NDOF plots
indicating maximum values calculated during the first 1 s after the helicopter strikes the
ground.
c) Compare the acceleration calculated under b) with that which would be obtained without
dampers. Use a pair annotated plots NDOF for this comparison. Explain in no more than 50
words the effect of damping and lack of it for this particular structural dynamics design
problem.
d) What would be the maximum acceleration of the pilot if the spring stiffness were reduced
to 6x104 N/m with no dampers fitted? Use NDOF plots to compare the undamped responses
under c) and d). In no more than 50 words describe advantages and disadvantages of using a
softer spring?
1
University of Exeter
ENG3015 Structural Dynamics
Year: 2022/23 and onwards
Coursework 2
Q1: A two-storey frame is shown in Figure Q1. The horizontal beams are infinitely sMff
relaMve to the verMcal columns.
Figure Q1
Design data:
• Bending sMffness of columns: EI=81.35 MNm2
.
• Mass per unit length of the horizontal beams: m’=10,000 kg/m.
• SMffness of the horizontal spring restraining the top floor: k=50 MN/m.
• It can be assumed that damping raMo is 2% for all relevant modes of horizontal
vibraMon.
For the structural frame shown in Figure Q1:
a) Calculate the mass and sMffness matrices of this system.
b) Calculate the natural frequencies of the system.
c) Determine and sketch the unity-scaled 2nd mode shape of this system.
d) Calculate the modal mass corresponding to the unity-scaled 2nd mode shape calculated
in c) above.
e) Which modelling parameter needs changing and how to enable modelling of this system
in the NDOF so_ware, considering the limitaMons of that so_ware?
f) Introduce the assumpMon made in e), sketch the new system and using the same
degrees of freedom as shown Figure Q1 re-calculate:
i) mass and sMffness matrices of the system; check the calculaMons using NDOF so_ware.
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ii) natural frequencies of the system; check the calculaMons using NDOF so_ware.
iii) unity scaled 2nd mode shape of this system; check the calculaMons using NDOF
so_ware.
iv) modal mass corresponding to the unity-scaled 2nd mode shape calculated; check the
calculaMons using NDOF so_ware.
g) Compare modal masses calculated under d) and under f-iv) and explain the difference, if
any. Which physical feature of the structure is causing the difference and why, if any?
3
Q2: A three-storey frame shown in Figure Q2 is subjected to the harmonic force . The
horizontal beams are infinitely sMff relaMve to the verMcal columns.
Figure Q2
.
Modal damping raMos for the system are: .
Using NDOF and EI=4.5x106 Nm2 answer following quesMons.
a) Calculate natural frequencies and unit normalised mode shapes.
b) Determine modal masses. Check the obtained modal mass for the 3rd mode by hand.
c) Plot the forced displacement response of the system in the first 15 s.
d) The forced displacement response plot produced under c) has some specific clearly
observable features which determine the type of the response. How is response
plofed called?
a. Transient response,
b. BeaMng response,
c. Resonant response?
Select one answer. Explain your answer referring to the relaMonship between the excitaMon
frequency and natural frequencies of the system.
University of Exeter
ENG3015 Structural Dynamics
Year: 2022/23 and onwards
Coursework 3
Figure Q3(a) shows a finite element model of a floor structure featuring six floor panels,
each panel being 8 m long and 4.5 m wide. Therefore, the total length of the floor plate is
27 m (6x4.5 m).
Figure Q3(a).
Figure Q3(b) shows modal properSes for the first four modes of vibraSon.
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Figure Q3(b)
Using Appendix G - Concrete Society Technical Report 43 (2nd EdiSon) answer the following
quesSons related to the vibraSon serviceability of this floor structure.
a) Is this a low- or a high-frequency floor? Explain the reasoning behind your answer and
what kind of vibraSon response does this floor need to be design for under human
walking: transient, random, free or resonant vibraSon?
b) Calculate maximum response factor for verScal vibraSon at point 2 due to single person
walking at point 1 of this floor. Show all calculaSons and clearly state the assumpSons
made. Data needed for calculaSons:
• Pedestrian weight is 700 N.
• The minimum pacing rate is 1.4 Hz.
• The maximum pacing rate is 2.0 Hz.
• Modal damping raSo is 1.5% in all relevant modes of vibraSon.
• Assume that the walking path is 8m long across the floor through point 1.
Mode 1: f1=3.8Hz, m1=37,000kg Mode 2: f2=5.1Hz, m2=25,000kg
Mode 3: f3=7.1Hz, m3=20,000kg Mode 4: f4=9.0Hz, m4=18,000kg
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• Mode shape amplitudes at point 1 are:
o For mode 1: 0.7
o For mode 2: -0.8
o For mode 3: 0.3
o For mode 4: -0.6
• Mode shape amplitudes at point 2 are:
o For mode 1: 1.0
o For mode 2: 0
o For mode 3: 0
o For mode 4: -0.5
c) Check your floor vibraSon response calculaSons using independently the ‘App G’ opSon
in the NDOF sodware. Submit screenshot of the acceleraSon response envelope as a
funcSon of the pacing frequency with annotated interpretaSon of the relevant maximum
response in terms of R-factor and compare with the calculaSons under b) above.
d) Can this floor be used as a workshop?
Note:
Some design data provided my not be needed in calculaSons. If any design data is missing,
make reasonable assumpSons and clearly state such assumpSons.
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