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Spring Semester Examinations 2011/ 2012
Exam Code(s) 3BM, 3BES
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 8 Department(s) Mechanical & Biomedical Engineering Course Co-ordinator(s) Dr. Conchúr Ó Brádaigh
Requirements: Statistical/Log tables Mathematical tables Laplace Transform Tables Handout (2 pages) Supplementary Equations (1 page) Graph Paper
Release to Library: Yes
Question 1
A sledge-hammer strikes an anvil with a velocity of 30 m/sec (Figure Q1). The hammer and the anvil weigh 50 kg and 500 kg, respectively. The anvil is supported on four springs as shown, each of stiffness k = 200 kN/m.
Find (i) the equation of motion of the system; (ii) the natural frequency of vibration of the system; and (iii) the expression for vertical motion of the system in time; for each of the following cases:
(a) the hammer remains in contact with the anvil after impact [15]
(b) the hammer does not remain in contact with the anvil after the initial impact [18]
You may assume that the anvil is restrained to move in the vertical direction only.
Figure Q
Question 3
An experiment is conducted to find the dynamic response characteristics of an automobile wheel assembly system. For this, the wheel is connected to a shaft through a tie rod and is subjected to a harmonic force f(t), as shown in Figure Q3.
The shaft offers a torsional stiffness of k (^) t , while the wheel undergoes torsional vibration about the axis of the shaft. Given the following data:
Torsional stiffness of shaft, k (^) t = 5000 N.m/rad Polar mass moment of inertia of wheel assembly, J 0 = 1.5 kg.m^2 Harmonic force applied to wheel, f(t) = 1500 sin 50t N
(a) Write the equation of motion of the system in terms of the rotation angle, θ , the shaft torsional stiffness k (^) t and the applied moment M(t). [8]
(b) Find the Laplace transform function of the system. You may assume the initial conditions θ(0) = 0; dθ(0)/dt = 0. [10]
(c) Find the rotational response of the system, θ(t). [15]
Figure Q
Question 4
An airplane standing on a runway is shown in Figure Q4.
The airplane has a total mass, m = 25,000 kg and a mass moment of inertia J 0 = 75 x 10^6 kg.m^2.
The main landing gear is located a distance l 1 = 20 m from the Centre of Gravity (C.G.) of the airplane and has a spring stiffness coefficient of k 1 = 20 kN/m.
The nose landing gear is located a distance l 2 = 30 m from the Centre of Gravity (C.G.) of the airplane and has a spring stiffness coefficients of k 2 = 10 kN/m.
(a) Derive the equations of motion of the airplane in matrix format. [15]
(b) Find the undamped natural frequencies of the system [10]
(c) Calculate the modal vectors of the airplane [8]
NOTE: Damping may be neglected.
Figure Q
ME 347 MECHANICAL VIBRATIONS
SUPPLEMENTARY EQUATIONS
Free Vibration of Single DOF Undamped Systems
; or
; or
Free Vibration of Single DOF Damped Systems
Critical Damping:
Under-Damping:
or
Logarithmic Decrement:
Forced Vibration of Single DOF Damped Systems
Response of system to force input of F 0 cos wt:
t n
xt x x x te ^ n
0 cos^0
( ) sin
X e t
xt Xe t
d t
d t
n
n
