ENG3015 TURN OVER 1
ENG3015
UNIVERSITY OF EXETER
FACULTY OF ENVIRONMENT, SCIENCE AND
ECONOMY
May 2023
Structural Dynamics
Module Convenor: Professor A Pavic
Duration: TWO HOURS + 30 upload time
No Word Count specified
Answer ALL questions.
This is an OPEN BOOK examination.
ENG3015 CONTINUE 2
Question 1 (30 marks)
Answer the following questions.
a) When spinning at 600 rotations per minute two washing machines produce two
sinusoidal forces with same maximum amplitude of 400N, but their maximum
amplitudes are separated in time by 0.0125s. The two machines are placed side by
side and to check their effect on a suspended laundry floor it is necessary to find
their resultant vertical force. Find time-varying function describing the resultant of the
two vertical forces including the amplitude and phase shift relative to one of the two
excitation forces.
(8 marks)
b) A rigid structural beam is supported by linear elastic springs as shown in Figure
Q1b. The masses of the beam and springs are negligible compared to the lumped
mass m shown. The beam is restrained from moving horizontally at point A and can
rotate. The movement of the beam in the vertical direction can be represented as an
undamped spring-mass system.
Design data:
k=20 MN/m
m=2,000 kg
Figure Q1b
b-i) Develop expression for calculating the undamped natural frequency of this
SDOF system in terms of m and k.
(6 marks)
b-ii) Calculate the undamped natural frequency and express it in Hz.
(3 marks)
c) For the structure in Q1b) and its mass and stiffness properties, calculate the value
of the viscous damping coefficient required to give damping ratio of 2%.
(3 marks)
ENG3015 TURN OVER 3
d) Figure Q1d shows modulus and phase plots of a receptance FRF corresponding
to a damped mass-spring SDOF system. Negative phase indicates lagging of the
steady-state sinusoidal response behind the sinusoidal excitation of the same
frequency, as appropriate.
Figure Q1d
d-i) Take the relevant values from the FRF plot using two significant digits and find
mass, stiffness and damping coefficients of the SDOF system.
(5 marks)
d-ii) Using the calculated mass stiffness and damping properties check the values of
the receptance FRF modulus (0.000067m/N) and phase (-12.5o
) values
corresponding to the unit sinusoidal excitation at 0.8Hz.
(5 marks)
ENG3015 CONTINUE 4
Question 2 (40 marks)
In the 3-storey shear-type building below, each column has the same lateral stiffness
and each bay of floor is rigid and has the same mass, so that the total storey
stiffnesses and masses are as shown. Assume that m = 10000 kg, k = 2000 kN/m.
Figure Q2
a) Neglecting damping, write the matrix equations of free vibration motion for the
structure in Figure Q2.
(10 marks)
b) It is found that the natural frequencies for the structure in Figure Q2 are: 1.339;
3.062; and 4.665 Hz. Determine the mass-normalised mode shape vector
corresponding to the fundamental frequency and sketch the mode of vibration.
(12 marks)
c) For the structure in Figure Q2, determine the damping matrix required to give 5%
of critical damping at the first and third modes of vibration.
(10 marks)
d) Determine the percentage of critical damping achieved at the second modal
frequency.
(3 marks)
e) Use the modal superposition method to determine the maximum displacement at
the top floor of the building in Figure Q2 if it is known that the steady-state
solution for the modal equations is X(t) = {5 cos( 1t); 0; 0}T in m, and where 1 is
the natural frequency of the first mode.
(5 marks)
ENG3015 5
Question 3 (30 marks)
An external fuel tank on a prototype helicopter resonates at 700 rad/s when it is full
due to the driving force of the rotors operating at a constant speed. Assume that the
total mass of the fuel is 1,200 kg when the tank is full. It can be assumed that the mass
of the tank is negligible relative to the mass of the fuel in it. The tank is mounted on
isolation mountings that provide an inherent damping ratio of 3%. It has been decided
to attach a vibration absorber to the fuel tank, in form of a tuned mass damper to
reduce the level of vibration.
(a) Determine the appropriate damping ratio d, mass ratio d and thereby the required
mass md of the tuned mass damper to be attached to the fuel tank such that it
increases the system’s effective damping ratio to 10%. Determine the optimum
tuning frequency d of the damper and its required spring stiffness kd and viscous
damping coefficient cd.
(18 marks)
(b) For the tuned mass damper designed in 3(a), if the tank is partially filled with fuel
such that its total mass is of its mass at full capacity, estimate:
i. The frequency at which the rotors must operate such that the tuned mass
damper is most effective.
(4 marks)
ii. Determine the resonance frequencies corresponding to the case where the
fuel tank is filled to of full capacity.
(8 marks)
END OF QUESTION PAPER
