School of Mechanical and Electrical Engineering,

MEC2401 Dynamics I, S2, 2015, Assignment 2

Answer all questions (200/1000) Due Date 26th October 2015

Assignment must be submitted electronically and student should follow the instructions given in the file “instructions for assignment preparation” which is posted on the course study desk Take gravitational acceleration g = 9.81 m/s2

Q1. (Marks 40/200)

A carriage of mass M is at rest on a smooth rigid floor as shown in figure Q1. A point mass m is attached to a slender rigid of length L with negligible mass is pivoted to the carriage at point A trough a frictionless pin. The point mass & rod release from the rest in horizontal position at time t =0. At time t = T seconds the rod is at vertical position as shown in dotted lines in figure Q1. M = 5 mB and mB=10 kg and L=1.5 m.

i. Determine the absolute velocity of the carriage and the point mass B when the rod is at vertical position.

(15 Marks) ii. Determine the distance travel by carriage during time T seconds. (15 Marks)

iii. What is the angular velocity of the mass B at time T seconds (10 Marks)

Figure Q1

Q2. (Marks 40/200)

At the instant shown in the Figure Q2, link AB has an angular velocity ?AB = 3 rad/s. angular acceleration aAB = 2 rad/s2. Each link is considered as a uniform slender bar each with a mass of 2.0 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) (10 Marks)

III. Determine the horizontal and vertical components of acceleration at point C (10 Marks)

Figure Q2

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 in a inclined surface (170 to horizontal) at a constant acceleration of 0.5 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. What is the maximum acceleration can develop by the front end loader to move forward on the inclined plane without tipping over. (10 Marks)

III. Investigate the motion of the front end loader case (i) if the inclination of the surface become 260 to the horizontal and the surface condition with loose soil i.e static friction between the wheels and the ground is 0.43. (10 Marks)

Figure Q3

Q4. (Marks 40/200)

A slender uniform bar AB shown in Figure Q4 has a mass of 5 kg and length L = 2.5m and at rest in vertical plane. A thin 1.5 kg circular disk of radius 100 mm is fixed to the bar AB through a smooth pin at it centre B. Two springs support the rod at A, and C, AC = 0.8L. The rod is pivoted to ground using a frictionless pin at point E and it oscillates about its equilibrium position. K1=K2= 1600 N/m. A dashpots which have a damping coefficient 600 N s/m is attached to the rod AB at D, AD=0.35L as shown in Figure Q4.

(i) Write the equation of motion for small oscillations of the rod& disk attachment about its equilibrium

position (20 Marks)

(ii) Determine the damp-free natural frequency of vibration of the bar & attached disk arrangement. (10 Marks)

(iii) Calculate damping coefficient ? (damping factor) and comment on the damping condition. (10 Marks)

Elevation

Figure Q4

Q5. (Marks 40/200)

Figure Q5 shows a collision of a high speed locomotive engine and an empty oil tanker at an unprotected railway crossing. The locomotive A has mass MA and was travelling in constant velocity of 100 km/hr and the Tanker B has mass MB and was travelling in constant velocity of 80 km/hr. MA = 4.5 Mg and MB = 1.3 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 (no-derailment happened). (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. (10 Marks)

III. Calculate the possible energy loss for Case (i) and Case (II) for values of e = 0.8. Comment on the severity of collision depending on your calculated values. (10 Marks)

IV. In the Case ii with e=0.8 , estimate the angular velocity of the tanker if the tanker has started rotating (approx) about point O just after it moving away from the locomotive. Take the radius of gyration k of the empty tanker (about point O) = 2.48 m (10 Marks)

List all the assumptions clearly for each case.

Figure Q5

MEC2401 Dynamics I, S2, 2015, Assignment 2

Answer all questions (200/1000) Due Date 26th October 2015

Assignment must be submitted electronically and student should follow the instructions given in the file “instructions for assignment preparation” which is posted on the course study desk Take gravitational acceleration g = 9.81 m/s2

Q1. (Marks 40/200)

A carriage of mass M is at rest on a smooth rigid floor as shown in figure Q1. A point mass m is attached to a slender rigid of length L with negligible mass is pivoted to the carriage at point A trough a frictionless pin. The point mass & rod release from the rest in horizontal position at time t =0. At time t = T seconds the rod is at vertical position as shown in dotted lines in figure Q1. M = 5 mB and mB=10 kg and L=1.5 m.

i. Determine the absolute velocity of the carriage and the point mass B when the rod is at vertical position.

(15 Marks) ii. Determine the distance travel by carriage during time T seconds. (15 Marks)

iii. What is the angular velocity of the mass B at time T seconds (10 Marks)

Figure Q1

Q2. (Marks 40/200)

At the instant shown in the Figure Q2, link AB has an angular velocity ?AB = 3 rad/s. angular acceleration aAB = 2 rad/s2. Each link is considered as a uniform slender bar each with a mass of 2.0 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) (10 Marks)

III. Determine the horizontal and vertical components of acceleration at point C (10 Marks)

Figure Q2

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 in a inclined surface (170 to horizontal) at a constant acceleration of 0.5 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. What is the maximum acceleration can develop by the front end loader to move forward on the inclined plane without tipping over. (10 Marks)

III. Investigate the motion of the front end loader case (i) if the inclination of the surface become 260 to the horizontal and the surface condition with loose soil i.e static friction between the wheels and the ground is 0.43. (10 Marks)

Figure Q3

Q4. (Marks 40/200)

A slender uniform bar AB shown in Figure Q4 has a mass of 5 kg and length L = 2.5m and at rest in vertical plane. A thin 1.5 kg circular disk of radius 100 mm is fixed to the bar AB through a smooth pin at it centre B. Two springs support the rod at A, and C, AC = 0.8L. The rod is pivoted to ground using a frictionless pin at point E and it oscillates about its equilibrium position. K1=K2= 1600 N/m. A dashpots which have a damping coefficient 600 N s/m is attached to the rod AB at D, AD=0.35L as shown in Figure Q4.

(i) Write the equation of motion for small oscillations of the rod& disk attachment about its equilibrium

position (20 Marks)

(ii) Determine the damp-free natural frequency of vibration of the bar & attached disk arrangement. (10 Marks)

(iii) Calculate damping coefficient ? (damping factor) and comment on the damping condition. (10 Marks)

Elevation

Figure Q4

Q5. (Marks 40/200)

Figure Q5 shows a collision of a high speed locomotive engine and an empty oil tanker at an unprotected railway crossing. The locomotive A has mass MA and was travelling in constant velocity of 100 km/hr and the Tanker B has mass MB and was travelling in constant velocity of 80 km/hr. MA = 4.5 Mg and MB = 1.3 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 (no-derailment happened). (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. (10 Marks)

III. Calculate the possible energy loss for Case (i) and Case (II) for values of e = 0.8. Comment on the severity of collision depending on your calculated values. (10 Marks)

IV. In the Case ii with e=0.8 , estimate the angular velocity of the tanker if the tanker has started rotating (approx) about point O just after it moving away from the locomotive. Take the radius of gyration k of the empty tanker (about point O) = 2.48 m (10 Marks)

List all the assumptions clearly for each case.

Figure Q5

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