tw12 university of bolton school of … · questions from part a and 2 questions from part b. ......

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)

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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

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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


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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

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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

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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


(iii) 2𝑥𝑑𝑦

𝑑𝑥= 𝑦


(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

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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

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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

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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

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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

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∫1

𝑎2 + 𝑥2𝑑𝑥 =

1

𝑎𝑡𝑎𝑛−1 (

𝑥

𝑎) + 𝑐

𝒚 = 𝒖 ⋅ 𝒗 𝒕𝒉𝒆𝒏 𝒅𝒚

𝒅𝒙= 𝒗

𝒅𝒖

𝒅𝒙+ 𝒖

𝒅𝒗

𝒅𝒙

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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)

tw12 university of bolton school of … · questions from part a and 2 questions from part b. ......

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