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Semester 1 Examinations 2012/ 2013
Exam Code(s) 3BM, 3BSE
Exam(s) 3 rd^ Mechanical Engineering 3 rd^ Energy Systems Engineering
Module Code(s) ME Module(s) Mechanical Vibrations
Paper No. 1 Repeat Paper
External Examiner(s) Professor Robin Clarke Internal Examiner(s) Professor Sean Leen Dr. Conchúr Ó Brádaigh
Instructions: (^) Answer 3 questions. All questions carry equal marks.
Duration 2 hours
No. of Pages 11 Department(s) Mechanical & Biomedical Engineering Course Co-ordinator(s) Dr. Conchúr Ó Brádaigh
Requirements: Statistical/Log tables Mathematical tables Tables from SS Rao (2 pages) Laplace Transform Tables Handout (2 pages) Supplementary Equations (1 page) Graph Paper
Release to Library: Yes
Question 1
(a) Find the undamped natural frequency of a steel sign (Figure Q1) in the z-x plane when the following information is known. Sign width, b = 2.0m; Sign height, d = 0.5m ; Sign thickness, t = 0.02m; Post height, l = 3.5m Inner radius of post, r (^) i = 0.050m Outer radius of post, r (^) o = 0.055m Modulus of steel, E, = 207 GPa Density of steel = 7,800 kg/m^3
The post is embedded 0.25m into the sign.
Hint: consider the post as being a cantilever beam, with an extra mass at its free end. [18 marks]
(b) The sign is disturbed from equilibrium by an imposed velocity in the x direction of 2.0 m/sec. Calculate the maximum value (amplitude) of the resulting displacement. [7 marks]
Figure Q
Question 3
A fixed-fixed steel beam ( E = 207 GPa), of length 7.0m, width 0.6m and thickness 0.15m, carries an electric motor of mass 100 kg and speed 1,500 rpm at its mid-span, as shown in Figure Q3.
A rotating force of magnitude FO = 7,000N is developed due to the unbalance in the rotor of the motor.
(a) Find the amplitude of steady vibrations, assuming damping to be negligible, [18 marks] (b) Find the amplitude of steady vibrations if the damping ratio, ζ = 0.9. [7 marks]
You may disregard the mass of the beam in your calculations.
Figure Q
Question 4
A marine engine is connected to a propeller through gears. A schematic diagram of the engine is shown in Figure Q4.
The mass moments of inertia of the flywheel, engine, gear 1, gear 2 and propeller (in kg.m^2 ) are 20,000, 1,500, 350, 160 and 1,750 respectively.
Find: (a) The natural frequencies of the system in torsional vibration; and [18 marks]
(b) The mode shapes of the system in torsional vibration. Sketch these mode shapes. [7 marks]
Assume that:
1. The flywheel can be considered to be stationary (fixed) as its mass moment of inertia is very large compared to the other rotors. 2. The engine and gears can be replaced by an equivalent single rotor. 3. The steel used in the shafts has a shear modulus, G = 80GPa [25 marks]
Figure Q
ME 347 MECHANICAL VIBRATIONS
SUPPLEMENTARY EQUATIONS (P1)
Free Vibration of Single DOF Undamped Systems
or
or
Free Vibration of Single DOF Damped Systems
Critical Damping:
Under‐Damping:
or
t n
x t x x x te ^ n
0 cos^0
( ) sin
X e t
x t Xe t
d
t
d
t
n
n
ME 347 MECHANICAL VIBRATIONS
SUPPLEMENTARY EQUATIONS (P2)
Logarithmic Decrement:
Forced Vibration of Single DOF Damped Systems
Response of system to force input of F 0 cos wt:
+
