tw12 university of bolton school of … · questions from part a and 2 questions from part b. ......
TRANSCRIPT
TW12
UNIVERSITY OF BOLTON
SCHOOL OF ENGINEERING
BENG (HONS) IN MECHANICAL ENGINEERING
SEMESTER 2 EXAMINATION 2016/2017
ENGINEERING PRINCIPLES 2
MODULE NO: AME4053
Date: Monday 15 May 2017 Time: 10.00 – 12.00
INSTRUCTIONS TO CANDIDATES: This paper is split into two parts; Part
A and Part B. There are three
questions in each Part.
Answer 4 questions in total; 2 questions from Part A and 2 questions from Part B.
All questions carry equal marks.
Marks for parts of questions are
shown in brackets.
This examination paper carries a total
of 100 marks.
CANDIDATES REQUIRE : Formula sheet (attached)
Page 2 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
Part A Q1 A flywheel 0.9m diameter has its initial angular velocity of 6 rad/s increased to its final angular velocity with an angular acceleration of 12 rad/s2 whilst making 100 revolutions.
Calculate:-
a) the final angular velocity of the flywheel (8 marks)
b) the time taken for the 100 revolutions (8 marks)
c) the linear acceleration and final linear velocity of a point on the rim of the flywheel (9 marks)
Total 25 marks
Q2 a) Using the parallel axis theorem prove that the second moment
of area for a rectangular section (figure Q2a) IGG = 𝑏𝑑3
12
(15 marks)
Figure Q2a
b) A disc of mass 15kg and radius of gyration 300min is accelerated uniformly from 300 rev/min to 1000 rev/min in 8 seconds. Neglecting any drive friction, determine the torque required to accelerate the disc.
(10 marks)
Total 25 marks
Page 3 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
Please turn the page
Q3 a) A spherical shell 80mm internal diameter has a wall thickness of 10mm.
Calculate the internal pressure if the stress in the material is not to exceed 30
MPa
(10 marks) b) A thin cylinder has an internal diameter of 100mm and a wall thickness
of 4.5mm. Calculate the circumferential and longitudinal stresses when
the cylinder is subjected to an internal pressure of 20 bar
(15 marks)
Total 25 marks
Please turn the page
Page 4 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
PART B
Q4) a) Find the first and second derivatives of each of the following;
i) 3𝑥4 − 2𝑥 + 1 (2 marks)
ii) 6 sin(3𝑡) − 2cos (7𝑡) (2 marks)
b) Find the derivatives of each of the following;
i) √4𝑥2 − 2𝑥 + 5 (4 marks)
ii) (2𝑥 − 1)cos (𝑥) (4 marks)
iii) 𝑥2+5
sin (𝑥) (4marks)
c) Find and classify the stationary points of the following functions;
i) 𝑥3 − 9𝑥2 + 27𝑥 − 27 (4 marks)
ii) 𝑥3 − 12𝑥 − 2 (5 marks)
Total 25 marks
Page 5 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
Please turn the page
Q5) a) Use the Newton Raphson Method to find the solutions of
𝑥3 − 8.2𝑥 − 4 = 0
using starting values of 𝑥 = −4, 𝑥 = −1 and 𝑥 = 2.
(9 marks)
b) Evaluate the following definite integrals;
(i) ∫ 4𝑥3𝑑𝑥2
1 (3 marks)
(ii) ∫ (3 sin(2𝑥) − 4cos (3𝑥))𝑑𝑥𝜋
0 (4 marks)
c) The table below gives corresponding values of x and y. Plot the graph
and by using Simpson’s rule, find the area under the graph.
(9 marks)
Total 25 marks
x 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1
y 800 730 622 528 438 366 306 262 241
Page 6 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
Please turn the page
Q6) (a) Find the particular solution of the following differential equations:
(i) 𝑑𝑦
𝑑𝑥− 4𝑥3 −
2
𝑥= 0
given that 𝑦 = 1 where 𝑥 = 1. (5 marks)
(ii) 𝑑𝑦
𝑑𝑥= 𝑦3
given that 𝑦 = 1 where 𝑥 = 0. (5 marks)
(iii) 2𝑥𝑑𝑦
𝑑𝑥= 𝑦
given that 𝑦 = 1 where 𝑥 = 1. (5 marks)
(b) Find the particular solution of the differential equation:
𝑑2𝑦
𝑑𝑥2− 2
𝑑𝑦
𝑑𝑥+ 5𝑦 = 0
given that 𝑦 = 1 and 𝑑𝑦
𝑑𝑥= 3 when 𝑥 = 0
(10 marks)
Total 25 marks
END OF QUESTIONS
Page 7 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
Formula Sheet. Stress and Strain:
E
L
u
A
F
G
strainshear
A
F
lat
long
lat
long
volumetric
E
V
V
KV
V
etcE
etcEE
EEE
zz
xy
y
zyxx
.................
.......
12
213
EG
EK
Static Equilibrium:
∑ 𝐹𝑥 = 0 ; ∑ 𝐹𝑦 = 0 ; ∑ 𝐹𝑧 = 0 ; ∑ 𝑀𝑥 = 0 ; ∑ 𝑀𝑦 = 0 ; ∑ 𝑀𝑧 = 0
Page 8 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
Thin Pressure Vessels:
hoop
pd
t
2
longitudinal
pd
t
4
longitudinal
pd
tEl
41 2
diametral
p
tEd
41 2
For Cylindrical Shells: Vpd
tEV
45 4
For Spherical Shells: Vpd
tEV
3
41
2nd Moments of Area
Rectangle I = 12
bd3
Circle I = 64
4πd Polar J =
32
4d
Parallel Axis Theorem
Ixx = IGG + Ah2
Bending
R
E
yI
M
Torsion
G
rJ
T
Motion
v = u + at 2 = 1 + t
v2 = u2 + 2as 221
22
s = tvu
2 t
2
21
s = ut + ½ at2 21
2
1tt
Page 9 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
Speed = Distance Acceleration = Velocity
Time Time
s = r
V = r
a = r
Torque and Angular
TP
mkI
IT
2
Energy and Momentum
Potential Energy = mgh
Kinetic Energy
Linear = ½ mv2
Angular =½ I2
Momentum
Linear = mv
Angular = I
Vibrations
Linear Stiffness
Fk
Circular frequency m
kn
Frequency n
nn
Tf
1
2
maF
T
Tf
xa
trxrv
trx
2
1
sin
cos
2
22
Page 10 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
Integration
∫ 𝑥𝑛 𝑑𝑥 = 𝑥𝑛+1
𝑛 + 1+ 𝐶 (𝑛 ≠ −1)
∫ (𝑎𝑥 + 𝑏)𝑛 𝑑𝑥 =(𝑎𝑥 + 𝑏)𝑛+1
𝑎(𝑛 + 1)+ 𝑐 (𝑛 ≠ −1)
∫1
𝑥𝑑𝑥 = 𝑙𝑛|𝑥| + 𝐶
∫1
𝑎𝑥 + 𝑏𝑑𝑥 =
1
𝑎ln|𝑎𝑥 + 𝑏| + 𝑐
∫ ⅇ𝑥 𝑑𝑥 = ⅇ𝑥 + 𝑐
∫ ⅇ𝑚𝑥 𝑑𝑥 =1
𝑚ⅇ𝑚𝑥 + 𝑐
∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝑐
∫ cos 𝑛𝑥 𝑑𝑥 =1
𝑛sin 𝑛𝑥 + 𝑐
∫ sin 𝑥 𝑑𝑥 = −cos 𝑥 + 𝑐
∫ sin 𝑛𝑥 𝑑𝑥 = −1
𝑛cos 𝑛𝑥 + 𝑐
∫ sec2 𝑥 𝑑𝑥 = tan 𝑥 + 𝑐
∫ sec2 𝑛𝑥 𝑑𝑥 =1
𝑛tan 𝑛𝑥 + 𝑐
∫1
√1 − 𝑥2𝑑𝑥 = sin−1 𝑥 + 𝑐
∫1
√𝑎2 − 𝑥2𝑑𝑥 = sin−1 (
𝑥
𝑎) + 𝑐
∫1
1 + 𝑥2𝑑𝑥 = tan−1 𝑥 + 𝑐
Page 11 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
∫1
𝑎2 + 𝑥2𝑑𝑥 =
1
𝑎𝑡𝑎𝑛−1 (
𝑥
𝑎) + 𝑐
𝒚 = 𝒖 ⋅ 𝒗 𝒕𝒉𝒆𝒏 𝒅𝒚
𝒅𝒙= 𝒗
𝒅𝒖
𝒅𝒙+ 𝒖
𝒅𝒗
𝒅𝒙
Page 12 of 12
School Of Engineering BEng (Hons) in Mechanical Engineering Semester 2 Examination 2016/2017 Engineering Principles 2 Module No. AME4053
𝒚 =𝒖
𝒗 𝒕𝒉𝒆𝒏
𝒅𝒚
𝒅𝒙=
𝒗𝒅𝒖
𝒅𝒙−𝒖
𝒅𝒗
𝒅𝒙
𝒗𝟐
Differential Equations
Auxiliary equations for differential equations of the form
𝑎𝑑2𝑦
𝑑𝑥2 + 𝑏𝑑𝑦
𝑑𝑥+ 𝑐𝑦 = 0
1) Real and Different routes 𝛼 and 𝛽
𝑦 = 𝐴ⅇ𝛼𝑥 + 𝐵ⅇ𝛽𝑥
2) Repeated (Real and Equal) root 𝛼
𝑦 = ⅇ𝛼𝑥(𝐴 + 𝐵𝑥)
3) Complex roots (𝑝 ± 𝑖𝑞)
𝑦 = ⅇ𝑝𝑥(𝐴 cos 𝑞𝑥 + 𝐵 sin 𝑞𝑥)
Numerical Methods
Simpson Rule
𝐴𝑟ⅇ𝑎 ≈𝛥𝑥
3(𝑦0 + 4𝑦1 + 2𝑦2 + 4𝑦3 + 2𝑦4 … + 4𝑦𝑛−1 + 𝑦𝑛)
Newton-Raphson Method
𝑥𝑖+1 = 𝑥𝑖 −𝑓(𝑥𝑖)
𝑓/(𝑥𝑖)