# LO1 Identify the relevance of mathematical methods to a variety of conceptualised engineering examples

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Assessment Brief

 Student Name/ID Number Unit Number and Title 2: Engineering Maths Academic Year 2021/22 Unit Tutor(s) Assignment Title Mathematical Concepts, Functions & Statistics Issue Date 23/11/21 Submission Date 23/02/22 IV Name & Date
 Submission Format You should present For Part 1, Part 2 and Part 3: Word processed report. For Part 4: A report featuring graphical data generated using computer software that could be understood by a non-technical audience.

Unit Learning Outcomes

LO1 Identify the relevance of mathematical methods to a variety of conceptualised engineering examples

LO2 Investigate applications of statistical techniques to interpret, organise and present data by using appropriate computer software packages

Assignment Brief and Guidance

Scenario:

You work as a production engineer for a company that manufactures engine and transformer parts. You have been investigating a number of issues in the production area and have been given access to a range of manufacturing data. Your manager has asked you to analyse the data in order to present this to non-technical colleagues within the organisation.

Activity:

Part 1:

1. Data has been gathered from a lifting system used to transport engine parts. The system consists of a drum and cable. The following data was obtained when the drum lowers the load (assume constant acceleration).
 Drum
 Cable

Drum diameter = 0.8m

Mass of load = 6kg

Initial velocity = 0 m/s

Time to descend = A secs

Distance travelled = 0.25m

You have been asked to use this information to determine:

a) The final linear velocity of the load

b) The linear acceleration of the load

c) The final angular velocity of the drum

d) The angular acceleration of the drum

e) The tension force in the cable

f) The torque applied to the drum

1. The velocity of propagation of a pressure wave (u) through a liquid can be expected to depend on the elasticity of the liquid represented by the bulk modulus K, and its mass density ρ. Derive using dimensional analysis, the equation that determines u.

Hint: u = f (C Ka ρb ) where C is a non-dimensional constant and bulk modulus is given by:

Pressure (P) is force per area

V is volume

Part 2:

a) You are investigating the electrical testing circuits and find that, in an inductive circuit, the relationship between instantaneous current i (amps) and the time t (secs) is given by:

i = 4.2(1-e-10t)

You have been asked to determine the time taken for the current to rise from 2350mA to B amps.

b) The drive system of a conveyor used to transport the engine parts is made up of an open belt which passes over two pulleys. The pulleys have diameters of 205mm and 325mm respectively and the distance between centres is T mm.

You have been asked to determine the length of the belt, using trigonometric techniques, assuming the belt is in tension.

c) The length l metres of an AAAC aluminium overhead line conductor at temperature t° C is given by

l = l0 eat, where l0 and a are constants.

Measurements are taken of the length at various temperatures:

at 80°C, length = 3.55m

at 250°C, length = Lm

Find the lengths of the bar at room temperature, 24.5°C and at 46.5°C.

Evaluate the practical implications of your results given that there are weather forecasts that temperature would rise up to around 49.5°C in the summer.

Part 3:

1). A drill used in the manufacture of the engine parts is designed to have eight speeds. The range of hole sizes anticipated is from 5mm to 30mm.

You have been asked to determine the seven spindle speeds (rev/min) given that the cutting speed is 15m/min. These spindle speeds are to be arranged in:

a) arithmetic progression

b) geometric progression

Note:

Where, N = spindle speed (rev/min), s = cutting speed (m/min) and D = hole diameter (mm).

2). The following are the equipment life (in years) and dielectric strength (in MΩ) of the winding of 24 apparently new transformers after pressure testing during factory acceptance tests (FAT).

 Dielectric Strength (MΩ) (y) Age (yrs) (x) Dielectric Strength (MΩ) (y) Age (yrs) (x) 121 129 142 127 135 129 137 133 141 145 136 130 45 53 62 20 60 54 26 20 31 22 60 68 129 137 147 125 144 131 125 122 127 124 127 121 20 43 64 26 53 33 58 46 58 70 26 46

(a) Find the correlation between age of equipment and dielectric strength using:

1.  Simple (Pearson’s) correlation coefficients, and
2. Spearman`s correlation coefficients, and comment on your results.

(b) Obtain the regression equation

(c) What is the predicted dielectric strength for a transformer aging 45 years?

(d) If the expected dielectric strength for a transformer to run for at least 40years is 132MΩ, is the quality control process adequate based on the data collected? Discuss your finding

LO1 Identify the relevance of mathematical methods to a variety of conceptualised engineering examples

Part 4:

The company considers itself to have quality processes in line with 6s principles

Measurement of inspection time, from a large sample of a specific engine part, gave the following distribution:

 Time (sec) 50-79 80-109 110-139 140-169 170-199 200-229 Number 4 8 10 13 4 3

You have been asked to determine the mean inspection time and the standard deviation (in seconds). You should also present a graphical illustration of the distribution using appropriate computer software.

From the data gathered and processed you have been asked to determine:

a) the maximum and minimum limits that inspection time might take (assuming the distribution to be normal and 6s compliant (all times within mean ±3s)

b) the probability that a randomly chosen inspection time will be greater than 155 seconds

c) the probability that a randomly chosen inspection time will be shorter than 82 seconds

One of the machines is causing quality problems as P% of the engine parts produced on this machine have been found to be defective. Find the probability of finding 0, 1, 2, 3, 4and 5 defective parts in a sample of 25 parts (assuming a binomial distribution). You should also present a graphical illustration of the probabilities using appropriate computer software.

Measurement of inspection time, from a large sample of outsourced components, gave the following distribution:

 Time (sec) 19 21 23 24 26 27 28 30 Number 2 4 3 4 5 4 4 5

Your manager thinks that the inspection time should be the same for all outsourced components. Using the data provided test this hypothesis and indicate whether there is a correlation or not.

Your manager has asked you to summarise, using appropriate software, the statistical data you have been investigating in a method that can be understood by non-technical colleagues. LO1 Identify the relevance of mathematical methods to a variety of conceptualised engineering examples

 Student A L B T P Initials 1 1.96 3.46 1.56 760 9.6 AH 2 1.97 3.47 1.57 770 9.7 SA 3 1.98 3.48 1.58 780 9.8 DB 4 1.99 3.49 1.59 790 9.9 DCo 5 2.00 3.50 1.60 800 10 DCr 6 2.01 3.51 1.61 810 10.1 KC 7 2.02 3.52 1.62 820 10.2 BD 8 2.03 3.53 1.63 830 10.3 AD 9 2.04 3.54 1.64 840 10.4 MD 10 2.05 3.55 1.65 850 10.5 JE 11 2.06 3.56 1.66 860 10.6 DF 12 2.07 3.57 1.67 870 10.7 GC 13 2.08 3.58 1.68 880 10.8 KH 14 2.09 3.59 1.69 890 10.9 LH 15 2.10 3.60 1.70 900 11.0 LT 16 2.11 3.61 1.71 910 101.1 LJ 17 2.12 3.62 1.72 920 11.2 RP 18 2.13 3.63 1.73 930 11.3 MR 19 2.14 3.64 1.74 940 11.4 RC 20 2.15 3.65 1.75 950 11.5 RW 21 2.16 3.66 1.76 960 11.6 KS 22 2.17 3.67 1.77 970 11.7 SF 23 2.18 3.68 1.78 980 11.8 TS 24 2.19 3.69 1.79 990 11.9 JT 25 2.20 3.70 1.80 1000 12.0 MW

 Learning Outcomes and Assessment Criteria Pass Merit Distinction LO1 Identify the relevance of mathematical methods to a variety of conceptualised engineering examples LO1 & 2   D1 Present statistical data in a method that can be understood by a non-technical audience P1 Apply dimensional analysis techniques to solve complex problems P2 Generate answers from contextualised arithmetic and geometric progressions P3 Determine solutions of equations using exponential, trigonometric and hyperbolic functions M1 Use dimensional analysis to derive equations LO2 Investigate applications of statistical techniques to interpret, organise and present data by using appropriate computer software packages P4 Summarise data by calculating mean and standard deviation, and simplify data into graphical form P5 Calculate probabilities within both binomially distributed and normally distributed random variables M2 Interpret the results of a statistical hypothesis test conducted from a given scenario

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