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Text Box: Tp (°C)160

15

0 20 40 60

Time (min)

the flow of hot gases is suddenly terminated, and the rod cools by free convection to ambient air at Too = 300 K and by radiation exchange with large surroundings at

Tsur = 300 K. The free convection

coefficient can be expressed as h (W/m2 · K) = C

LlTn, where C = 4.4 W/m2 · K1.188 and n = 0.188. The emissivity of the rod is 0.5. Determine the subsequent thermal response of the rod. Plot the midspan temperature

as a function of cooling time, and determine the time required for the rod to reach a safe-to-touch temperature of 315 K.

5.135

Consider the acceleration-grid foil (k = 40 W/m · K, a = 3 X 10-5 m2/s, e= 0.45) of Problem 4.72. Develop an implicit, finite-difference model of the foil, which

can be used for the following purposes.

(a) Assuming the foil to be at a uniform temperature of 300 K when the ion beam source is activated, obtain a plot of the midspan temperature–time his- tory. At

what elapsed time does this point on the foil reach a temperature within 1 K of the steady- state value?

(a) Using a time increment of Llt = 60 s and the

implicit method, find the temperature history of the midplane of the book and determine whether curing will occur (170°C for 5 min).

(b) Following the reduction of the platen temperatures to 15°C (t = 50 min), how long will it take for the midplane of the book to reach 37°C, a safe temper- ature

at which the operator can begin unloading the press?

(c) Validate your program code by using the heating schedule of a sudden change of platen tempera- ture from 15 to 190°C and compare results with those from an

appropriate analytical solution (see Problem 5.46).

5.137 A thin circular disk is subjected to induction heating from a coil, the effect of which is to provide a uniform heat generation within a ring section as

shown.

Convection occurs at the upper surface, while the lower surface is well insulated.

act as an interfacial layer of negligible thickness and effective contact resistance Rt”,c = 2 X 10-5 m2 · K/W.

Initial Temperatures (K)

n/m

1

2

3

4

5

6

1

700

700

700

1000

900

800

2

700

700

700

1000

900

800

3

700

700

700

1000

900

800

Interface with solder and flux

l

Copper, pure

Steel, AISI 1010

(a) Derive the transient, finite-difference equation for node m, which is within the region subjected to induction heating.

(b) On T r coordinates sketch, in qualitative manner, the steady-state temperature distribution, identify- ing important features.

5.138 An electrical cable, experiencing uniform volumetric

generation q, is half buried in an insulating material while the upper surface is exposed to a convection

y, n

x, m

1, 3

1, 2

1, 1

2, 2

3, 2

2, 1

3, 1

2, 3

3, 3

4, 3

4, 2

4, 1

5, 2

6, 2

5, 1

6, 1

5, 3

6, 3

process (Too, h).

Df

m – 1, n

m, n + 1

Dr m, n

rm

m, n – 1

m + 1, n

D x = Dy = 20 mm

(a) Derive the explicit, finite-difference equation in terms of Fo and Bic=Llx/kRt”,c for T4,2 and deter- mine the corresponding stability criterion.

(b) Using Fo = 0.01, determine T4,2 one time step after contact is made. What is Llt? Is the stability criterion satisfied?

5.140

Consider the system of Problem 4.92. Initially with no flue gases flowing, the walls (a = 5.5 X 10-7 m2/s) are at a uniform temperature of 25°C. Using the implicit,

finite-difference method with a time increment of 1 h,

find the temperature distribution in the wall 5, 10, 50,

(a) Derive the explicit, finite-difference equations for

an interior node (m, n), the center node (m = 0), and the outer surface nodes (M, n) for the convec- tion and insulated boundaries.

(b) Obtain the stability criterion for each of the finite- difference equations. Identify the most restrictive criterion.

5.139 Two very long (in the direction normal to the page) bars having the prescribed initial temperature distribu- tions are to be soldered together. At time t = 0,

the m = 3 face of the copper (pure) bar contacts the m = 4

and 100 h after introduction of the flue gases.

5.141

Consider the system of Problem 4.86. Initially, the ceramic plate (a = 1.5 X 10-6 m2/s) is at a uniform temperature of 30°C, and suddenly the

electrical heating elements are energized. Using the implicit, finite-difference method, estimate the time required for the difference between the surface

and initial tem- peratures to reach 95% of the difference for steady- state conditions. Use a time increment of 2 s.

5.142

Consider the fuel element of Example 5.11, which operates at a uniform volumetric generation rate of

face of the steel (AISI 1010) bar. The solder and flux

q1 = 107 W/m3

until the generation rate suddenly

143.

ʹ Dynamics I S2, 2014

1

School of Mechanical and Electrical Engineering, Faculty of Health, Engineering and Sciences,

MEC2401 Dynamics I, S2, 2014, Assignment 2

Answer all questions (200/1000) Due Date 27th October 2014

Assignment must be submitted electronically and student should follow the instructions given in the

ĨŝůĞ ͞instructions for assignment preparation͟ which is posted on the course study desk

Take gravitational acceleration g = 9.81 m/s2

Q1. (Marks 30/200)

The free rolling ramp shown in Figure Q1, a has mass of 90 kg . A crate whose mass is 60 kg slides from rest at A

6m down to the ramp to B.

I. Determine the ramps speed when the crate reaches B (Assume that the ramp surface and the floor are

smooth) ŝĨ ɽсϯϱ0

(10 Marks)

II. Determine the distance the ramp travels when the crate reaches B ŝĨ ɽсϯϱ0

(Assume that the ramp surface

and the floor are smooth) (10 Marks)

III. Describe the motion of rolling ramp, If the sliding friction coefficient of the ramp surface and the crate is

0.58 and the inclination angle ɽсϮϱ0

(10 Marks)

6m

ș

A

B

Floor

Figure Q1

MEC2401 ʹ Dynamics I S2, 2014

2

Q2. (Marks 50/200)

At the instant shown in the Figure Q2, link AB has an angular velocity AB = 6 rad/s. angular acceleration

ɲAB = 2 rad/s2

. Each link is considered as a uniform slender bar each with a mass of 1.5 kg/m,

I. Determine the angular velocity and angular acceleration of link BC and CD (20 Marks)

II. Determine the kinetic energy of each link (AB, BC and CD) (15 Marks)

III. Determine the horizontal and vertical components of acceleration at point C (15 Marks)

450 mm

500 mm 750

mm

9090 A

D

B

35 0

70 0

ɲ AB =2 rad/s2

ʘ AB = 6 rad/s

C

Figure Q2

MEC2401 ʹ Dynamics I S2, 2014

3

Q3. (Marks 40/200)

Figure Q3 shows a 10Mg front-end-loader used to move Ore in the loader bucket. The centres of mass (CM) for

the front-end-loader and the Ore load at G and GL

respectively.

I. Determine the reactions exerted by the ground on the pairs of wheels at A and B, if the front-end-loader

is moving forward at a constant acceleration of 1.6 m/s2

from the rest. The Ore Load is 2.5Mg and the CM

(GL) is at height h = 3.5 m and x = 2.0 m (from the centre of front wheel A). (20 Marks)

II. The front-end-loader with a 1 Mg Ore Load CM (GL) at h = 2.85 m and x= 2.6 m is moving forward at a

constant velocity of 40 km/hr. Can the front-end-loader completely come to a rest safely without tipping

over, if the operator suddenly applied brakes? Assume that all wheels are locked when the brakes are

applied and the coefficient of static friction between the wheels and the ground is 0.55. (20 Marks)

170

Figure Q3

MEC2401 ʹ Dynamics I S2, 2014

4

Q4. (Marks 40/200)

Figure Q4 shows a uniform disk attached to a shaft at A (in vertical plane). Disk has a mass (M) 15 kg

and G is the centre of mass. A constant torque of T = 50 N m is exerted on the disk by the shaft at A and

ƚŚĞ ĚŝƐŬ ƐƚĂƌƚ ƌŽƚĂƚŝŶŐ ĂďŽƵƚ ƚŚĞ ƐŚĂĨƚ͛Ɛ ŚŽƌŝǌŽŶƚĂů ĂǆŝƐ ;ƉĞƌƉĞŶĚŝĐƵůĂƌ ƚŽ ƚŚĞ ƐŚĞĞƚͿ͘ IŶŝƚŝĂůůǇ time t = 0 s

ĂŶĚ ɽ с Ϭ0

.

I. Derive an expression for the resulting angular acceleration ߠሷof the disk and calculate the angular

acceleration at the instant ɽ с ϰϱ0

. (15 Marks)

II. Calculate the rotational speed (rpm) of the disk around A, kinetic energy of the disk and the work

done by the torque T after ɽ с ϰϱ0

. (15 Marks)

III. Calculate the radial force acting on the shaft by the disk and its direction with respect to Y axis,

when at ɽ с ϰϱ0

. Illustrate it on a free-body-diagram (Marks 10)

(List all your assumptions)

A

M= 15 kg

Ø1m

T = 50 Nm

G

Y

X

ș

Elevation

MEC2401 ʹ Dynamics I S2, 2014

5

Figure Q4

Q5. (Marks 40/200)

Figure Q5 shows a collision of a high speed locomotive engine and an oil tanker at an unprotected railway

crossing. The locomotive A has mass MA andwas travelling in constant velocity of 120 km/hr and the Tanker B has

mass MB and was travelling in constant velocity of 80 km/hr. MA = 5 Mg and MB = 1.8 Mg

I. Calculate velocities of the locomotive and the tanker after the collision if the locomotive and the tanker

become entangled and move off together after the collision. (10 Marks)

II. Calculate velocities of the locomotive and the tanker after the collision in terms of e (0<e <1) which the

coefficient of restitution between the locomotive and the tanker. (20 Marks)

III. Calculate the possible energy loss for Case (i) and Case (II) for two values of e; 0.2, and 0.8. Comment on

the severity of collision depending on your calculated values. (10 Marks)

List all the assumptions clearly for each case.

Train A

Tanker B

V (train) = 120 km/hr

V (Tanker) 80 km/hr

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