MEC3031S: Dynamics II

Mechanical Engineering Take Home Assignment 2014

Submission Date: Monday, 14 September, 8.30 a.m.

Submit to box outside the BISRU office area, Room 3.10 Elec-Mech Building

There are six (6) questions in this assignment. The marks for each question are shown.

Guidelines

1. Use a pen and pencil – not a PC - apart from where specifically instructed to do so. 2. VERY IMPORTANT: There is no need to derive results unless specifically asked to do

to so in the question. 3. Where there are options, select on the basis of your birthday month (see footnotes) 4. Please include a plagiarism declaration with your submission, on the cover sheet

from the VULA site. 5. No extensions will be granted apart from medical grounds. Lateness penalties will

apply. 6. Show all working. If you make assumptions, state these. 7. All pages must be marked with your full name, student number and question

number (and option number where required). 8. Please staple sheets together, in question order 1 to 6. 9. Begin a separate sheet of paper for each new question. 10. Do not use a fancy folder or binding for your submission. 11. The concept maps MUST be drawn using CMap. Not hand drawn. 12. Some parts of the submission may require additional reading beyond the course

notes. Try the recommended textbooks at the end of the relevant sections, or a well-targeted internet search. Include references to justify your approach when necessary.

Important Notice To ensure that no two assignments are identical, for some questions students are assigned option numbers based on the month of your birthday. Please ensure you pay attention to this and include your option number at the TOP of the solution to each part.

Question 1 – 10 marks

An electric motor is sent to the repair shop because it is vibrating. The motor shaft is 1.0m long. A careful analysis reveals that the shaft can be idealised as being perfectly machined, except for three locations which can be described as out of balance masses attached as shown in Figure Q1. The unbalanced properties are provided in Table Q1, along with the angular velocity of the shaft.

Find the magnitude and angular position of the resultant out of balance force and the resultant couple with respect to the left hand bearing.

Figure Q1: Shaft idealisation Table A1: Unbalanced mass idealisation (select only your option)

Option number1

Mass 1 MR (kg

mm)

Mass 2 MR (kg

mm)

Mass 3 MR (kg

mm)

Mass 2 Angle measured cw from Motor1

mass,

Mass 3 Angle measured cw from Motor1

mass,

Rotational speed of

motor shaft (rpm)

1 10 12 12 65 170 700

2 10 13 12 60 170 700

3 10 12 13 65 167 700

4 10 13 13 65 170 700

5 9 13 13 60 170 650

6 11 13 13 65 167 700

7 11 13 14 65 170 650

8 11 12 14 60 170 700

9 10 14 13 53 167 700

10 11 12 12 65 170 650

11 12 14 14 55 170 700

12 12 14 14 61 171 700

Each student has been assigned an option, based on the month of your birthday1

1 Your option number is the same as the month of your birthday i.e. if your birthday is in January

choose option #1, if your birthday is in December choose option 12 and so on. Please include your option number at the top of the solution you submit.

Left hand

bearing

0.22m

Mass 1

Mass 2

Mass 3

0.5 m

0.61 m

Question 2 – 28 marks a. Draw typical displacement-time curves for vibrating systems subjected to an initial displacement x0, for (i) Undamped free vibration (ii) Under-damped free vibration (iiI) Critical damping (iv) Overdamped free vibration [4]

Give the value (or range) for each case. [2] b. What is resonance? How is it related to the system mass and stiffness? [2] c. Determine, and then plot (using Excel or Matlab) the variation of displacement x with time t for the oscillating mass shown in Figure Q2 subjected to external excitation force F1 for: i. The transient solution (choose a suitable time scale)

ii. The steady state solution (choose a suitable time scale) iii. The full solution (xc + xp) (choose a suitable time scale)

When: F1 = 20 sin 9t, k = 160 N/m, c = 15 Ns/m, mass = 4 kg, initial displacement = 0.05 m, and the mass is initially at rest. Comment on the effect of the transient vibrations on the response. Be sure to include all of your working out as part of your solution. Identify in your final answers the equations for x in each case as a function of the time t. See footnote2

[20]

2 Be careful with the phase angles, particular for the steady state solution: the phase angle can only

be between certain limits. See pages 623-4 from Meriam and Kraige 5th

Ed for more information.

Question 3 – 20 marks A car is travelling over a rough road. The suspension system on each wheel can be assumed to have a stiffness of 300 kN/m. The damping ratio was determined experimentally using the data in Table Q3. The car mass is approximately 1200kg. The following assumptions can be made to help in idealisation of the car:

(a) The road surface is sinusoidal with a peak to peak variation of 120 mm and a period of 10 m.

(b) The car vibrates only in the vertical direction. (c) The spring constant of the tyres is infinite. (d) The tyres do not leave the road surface.

Write a short report, indicating

The non-dimensional damping coefficient (zeta)

The amplitude of vertical vibration when the car is travelling at a speed of 60 km/h if it is assumed that damping can be neglected.

Speed at which the vibration will be greatest, including calculations to justify your answer, for no damping.

How accurate you think your assumptions are, and how your idealization might change if they were incorrect.

Table Q3: Information used to determine damping ratio

Time (sec) Cycle Number Displacement (mm)

0 1 25

4 2 5.0

8 3 1.0

Question 4 – 10 marks Draw at least one concept map3 (not more than two) to describe how gear transmit motion in automotive differentials, covering all the fundamental dynamics theory required to explain the kinematics, and including applications which relate to the real world. The concept map must be drawn using the free software CMap which can be downloaded from the web. Access to CMap is also available on the PCs in the Red Lab, Green Lab and new CAD Lab on Menzies level 2. Hand drawn concept maps will not be marked.

3 Concept maps are described in the introduction of the class handbook.

Question 5 – 12 marks An exhaust fan operating under the conditions shown in Table Q5 is causing unwanted vibrations in its surroundings.

Table Q4: Exhaust fan variables

Option4 # Harmonic force magnitude Operating speed Fan mass

N rpm kg

1 8000 1100 70

2 8500 1050 70

3 8000 1200 70

4 7500 1200 85

5 9000 1050 70

6 6500 1100 50

7 4000 1100 90

8 5600 850 110

9 6800 1200 70

10 8000 850 65

11 8500 1100 68

12 6800 1050 105

The isolator has a stiffness of 100 kN/m.

a) Calculate the transmitted force through the isolator. Also state this as a percentage of the excitation force. [8]

b) Report on how a damper could be used to isolate the fan and reduce the vibrations transmitted to the surroundings. [4]

4 Your option should be chosen on the month of your birthday. E.g. if your birthday is in January

choose option 1, December choose option 12 and so on.

Question 6 – 20 marks A schematic for a two gear automatic gearbox is show in Figure Q6. The tooth numbers for the various gears are given in Table Q6. Where planets are present, only one planet is shown for each epicyclic gear set. When gear 5 is fixed (and gear 4 is not fixed), the gearbox is in “first gear”. When gear 5 is free and gear 4 is fixed, the gearbox is in “second gear”. The power input to the gearbox is 6 kW. The gearbox is 92 % efficient when in “first gear” and 90 % efficient when in “second gear”.

a) Identify the functions (sun, annulus etc) for each gear and show that gear 2 must have 20 teeth.

b) Find the output speed of the gearbox when in “first gear” if the input speed is 2200 rpm cw.

c) Find the output speed of the gearbox when in “second gear” if the input speed is 2200 rpm cw.

d) Find the torque required to keep gear 5 fixed when in “first gear”.

Table Q6: Gear teeth numbers for the gearbox

Gear Number Number of teeth

1a 60

1b 30

2 20

3 40

4 100

5 110

Figure Q6: Gearbox schematic

Carrier arm for planet 2

Carrier arm for planet 3

OUTPUT INPUT 1a 1b

2

4 5

3