EGB211 – Assessment Item No 2: Problem solving

Solve all the following exercises, specifying and justifying all the hypotheses, assumptions and approximations you are considering (make assumption if needed). Also, for all exercises draw the free body diagrams indicating which forces act on the particles in the different situations. A total of 25 points are split among the three exercises as specified in each exercise title. Please highlight the final answers to each part of each question.

Exercise 1. The crane (10 points)

You work in a construction company which has just built a new crane. The duty cycle of the crane is constant: every upward journey needs to lift a load of 1 tonne (metric) of bricks from the ground to a height h?????? = 100 m using its motorised pulley. The motorised pulley consists of an electric motor, a transmission and a negligible-mass pulley, with the following characteristics:

Motor Power 650 kW

Transmission efficiency 0.9

The acceleration/speed profiles chosen to perform the upward journey are qualitatively sketched on the right with ???????? = 10 m/s2. They should consider that:

• the cable cannot transmit any compression force to slow down the mass in the final stage of the upward journey

• therefore, some time (from ???????? to ??????????) needs to be allowed for the mass to decelerate and reach the top with zero final speed.

A. Neglecting the motor power constraint, calculate the values ???????? (total time for lifting) and the time ???????? (when the motor should stop pulling).

B. In the choice of the cable, which is the minimum rated load of the cable ???????????? you will look for?

C. Will the motor have sufficient power to perform the task? How would you modify the shape of the acceleration/speed profile to reduce the power requirements without increasing the total time ?????????? (only qualitative answer required)?

D. During a night, the load is oscillating in the top position, swinging at a constant cable length of ???????? = 6 m. The cable snaps when perfectly vertical, and with an angular speed

. How far will the body impact

with the ground and with which speed?

Exercise 2. The car (7 points)

A ??1 = 2000 kg car is exiting a 20 m radius turn at the maximum speed allowed by its tyres, which can withstand a maximum lateral acceleration of 1 g. After the end of the turn, the narrow single lane mountain road continues straight for 1 kilometre with a downward slope of 10%, but a barrier of mass 2 = 1800 kg is blocking the road 20 metres ahead.

A. Find the speed of the car at the end of the turn (maximum speed in the turn)?

B. Considering that the coefficient of friction between tyres and road in case of sliding is ?????? = 0.95, is there any chance of the driver being able to stop the car before impact (no ABS available) with a reaction time of 1 second?

C. In the worst case scenario of the driver not providing any breaking after the turn (neglect any effect of the engine on speed), calculate the loss of energy in the impact (under the assumption of coefficient of restitution ?? = 0.3).

D. Determine where the vehicle and barrier are going to stop after the impact considering a coefficient of friction ?????? = 0.5 between the barrier and the road. Consider the car sliding on its tyres (full breaking after the impact).

Exercise 3. The platform (8 points)

An object of mass = 10 kg is dropped on a negligible-mass platform from an altitude h1 = 1 m. The platform is mounted on two purely elastic springs with stiffness ??1 = ??2 = 1000 N/m.

Under the hypothesis of the mass never separating from the platform after the first contact, determine:

A. The maximum displacement of the platform downwards and upwards (from first contact position)

B. The natural frequency and damping ratio of the system composed by the springs, and the mass. Also draw the position of the platform ??(??) in time (after the maximum downward displacement has been reached).

Then:

C. Check if the mass is actually adhering to the platform during the whole vibration, considering that the mass is simply laying on the platform. Qualitatively discuss the validity of the results of parts A and B (no re-calculation required). Sketch for parts A-C (not to scale)

Finally:

D. Considering the case of a system with the same mass , the same stiffness elements and two additional dampers with damping coefficient ??1 = ??2 = 10 Ns/m, determine natural frequency, damping ratio and nature of the vibrations (undamped, underdamped, overdamped, etc…). Assume the mass to be fastened to the platform.

Sketch for part D (not to scale)

Solve all the following exercises, specifying and justifying all the hypotheses, assumptions and approximations you are considering (make assumption if needed). Also, for all exercises draw the free body diagrams indicating which forces act on the particles in the different situations. A total of 25 points are split among the three exercises as specified in each exercise title. Please highlight the final answers to each part of each question.

Exercise 1. The crane (10 points)

You work in a construction company which has just built a new crane. The duty cycle of the crane is constant: every upward journey needs to lift a load of 1 tonne (metric) of bricks from the ground to a height h?????? = 100 m using its motorised pulley. The motorised pulley consists of an electric motor, a transmission and a negligible-mass pulley, with the following characteristics:

Motor Power 650 kW

Transmission efficiency 0.9

The acceleration/speed profiles chosen to perform the upward journey are qualitatively sketched on the right with ???????? = 10 m/s2. They should consider that:

• the cable cannot transmit any compression force to slow down the mass in the final stage of the upward journey

• therefore, some time (from ???????? to ??????????) needs to be allowed for the mass to decelerate and reach the top with zero final speed.

A. Neglecting the motor power constraint, calculate the values ???????? (total time for lifting) and the time ???????? (when the motor should stop pulling).

B. In the choice of the cable, which is the minimum rated load of the cable ???????????? you will look for?

C. Will the motor have sufficient power to perform the task? How would you modify the shape of the acceleration/speed profile to reduce the power requirements without increasing the total time ?????????? (only qualitative answer required)?

D. During a night, the load is oscillating in the top position, swinging at a constant cable length of ???????? = 6 m. The cable snaps when perfectly vertical, and with an angular speed

. How far will the body impact

with the ground and with which speed?

Exercise 2. The car (7 points)

A ??1 = 2000 kg car is exiting a 20 m radius turn at the maximum speed allowed by its tyres, which can withstand a maximum lateral acceleration of 1 g. After the end of the turn, the narrow single lane mountain road continues straight for 1 kilometre with a downward slope of 10%, but a barrier of mass 2 = 1800 kg is blocking the road 20 metres ahead.

A. Find the speed of the car at the end of the turn (maximum speed in the turn)?

B. Considering that the coefficient of friction between tyres and road in case of sliding is ?????? = 0.95, is there any chance of the driver being able to stop the car before impact (no ABS available) with a reaction time of 1 second?

C. In the worst case scenario of the driver not providing any breaking after the turn (neglect any effect of the engine on speed), calculate the loss of energy in the impact (under the assumption of coefficient of restitution ?? = 0.3).

D. Determine where the vehicle and barrier are going to stop after the impact considering a coefficient of friction ?????? = 0.5 between the barrier and the road. Consider the car sliding on its tyres (full breaking after the impact).

Exercise 3. The platform (8 points)

An object of mass = 10 kg is dropped on a negligible-mass platform from an altitude h1 = 1 m. The platform is mounted on two purely elastic springs with stiffness ??1 = ??2 = 1000 N/m.

Under the hypothesis of the mass never separating from the platform after the first contact, determine:

A. The maximum displacement of the platform downwards and upwards (from first contact position)

B. The natural frequency and damping ratio of the system composed by the springs, and the mass. Also draw the position of the platform ??(??) in time (after the maximum downward displacement has been reached).

Then:

C. Check if the mass is actually adhering to the platform during the whole vibration, considering that the mass is simply laying on the platform. Qualitatively discuss the validity of the results of parts A and B (no re-calculation required). Sketch for parts A-C (not to scale)

Finally:

D. Considering the case of a system with the same mass , the same stiffness elements and two additional dampers with damping coefficient ??1 = ??2 = 10 Ns/m, determine natural frequency, damping ratio and nature of the vibrations (undamped, underdamped, overdamped, etc…). Assume the mass to be fastened to the platform.

Sketch for part D (not to scale)

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